Differential Equations Volume 43, Number 6 June, 2007

c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 737–744. c B.A. B...

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 737–744. c B.A. Babazhanov, A.B. Khasanov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 723–730. Original Russian Text

ORDINARY DIFFERENTIAL EQUATIONS

Inverse Problem for a Quadratic Pencil of Sturm–Liouville Operators with Finite-Gap Periodic Potential on the Half-Line B. A. Babazhanov and A. B. Khasanov Urgench State University, Urgench, Uzbekistan Received February 10, 2006

DOI: 10.1134/S0012266107060018

Inverse problems for the Sturm–Liouville operator with ﬁnite-gap periodic and nonperiodic potentials were studied in [1–3] on the half-line and in [4–8] and other papers on the entire line. Inverse problems for a quadratic pencil of Sturm–Liouville operators on the entire line in the case of periodic and ﬁnite-gap periodic coeﬃcients were considered in [9–11]. In the present paper, we study the inverse problem for a quadratic pencil of Sturm–Liouville operators with ﬁnite-gap periodic potential on the half-line; more precisely, we derive a formula expressing the boundary condition via the spectral data and obtain the system of Dubrovin–Trubowitz diﬀerential equations and trace formulas. 1. In the space L2 (0, ∞), consider the pencil T (λ)y ≡ −y + q(x)y + 2λp(x)y − λ2 y = 0

(0 < x < ∞)

(1)

of Sturm–Liouville operators with the boundary condition y(0) cos α + y (0) sin α = 0,

α ∈ (0, π),

(2)

where p(x) and q(x) are real π-periodic functions. Throughout the following, we assume that the functions p(x) and q(x) satisfy the following conditions: (a) p(x) and q(x) are deﬁned on R1 and π-periodic and satisfy the inclusions q(x) ∈ L2 [0, π] and p(x) ∈ W21 [0, π]; (b) the inequality (L0 y, y) > 0, where L0 y ≡ −y + q(x)y, holds for all functions y(x) ∈ W22 [0, π], y(x) ≡ 0, such that [y(0) cos α + y (0) sin α] y (0) − [y(π) cos α + y (π) sin α] y (π) = 0. By W2n [0, π] we denote the Sobolev space of complex-valued functions deﬁned on the interval [0, π], having n − 1 absolutely continuous derivatives, and such that the nth derivative is square integrable on [0, π]. By c(x, λ), s(x, λ), θ(x, λ), and ϕ(x, λ), we denote the solutions of Eq. (1) satisfying the initial conditions c(0, λ) = 1, c (0, λ) = 0, s(0, λ) = 0, s (0, λ) = 1, θ(0, λ) = cos α, θ (0, λ) = sin α, and ϕ(0, λ) = − sin α, ϕ (0, λ) = cos α. The Weyl–Titchmarsh function for problem (1), (2) has the form [s (π, λ) − c(π, λ)] cos 2α − [s(π, λ) + c (π, λ)] sin 2α − Δ2 (λ) − 4 , (3) mα (λ) = 2s(π, λ) cos2 α + [s (π, λ) − c(π, λ)] sin 2α − 2c (π, λ) sin2 α √ √ where z = reiϕ/2 if z = reiϕ , 0 ≤ ϕ < 2π, or [ϕ (π, λ) − θ(π, λ)] cos α − [ϕ(π, λ) + θ (π, λ)] sin α − Δ2 (λ) − 4 . (4) mα (λ) = 2 (ϕ(π, λ) cos α + ϕ (π, λ) sin α) 737

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BABAZHANOV, KHASANOV

The function Δ(λ) = s (π, λ) + c(π, λ) is called the Lyapunov function, or the Hill discriminant, of problem (1), (2). Note that it is independent of α. It follows from the expression (4) that the continuous spectrum of problem (1), (2) has the form Eess = R

1

∞

+ λ− n , λn .

n=−∞ + The disjoint intervals (λ− n , λn ), n ∈ Z, are called the gaps of the pencil (1). Their endpoints + 2 , λ containing the point λ = 0 is always are zeros of the function Δ (λ) − 4. The gap λ− 0 0 − + nondegenerate, λ0 < λ0 . By ξ0− , ξ0+ , and ξn , n ∈ Z\{0}, we denote the roots of the equation

ϕ(π, λ) cos α + ϕ (π, λ) sin α = 0. They coincide with eigenvalues of the regular problem for Eq. (1) with the boundary conditions y(0) cos α + y (0) sin α = 0,

y(π) cos α + y (π) sin α = 0,

and the following inclusions hold: ξ0+ ∈ 0, λ+ ξ0− ∈ λ− 0 ,0 , 0 ,

+ ξn ∈ λ− n , λn ,

n ∈ Z\{0}.

Deﬁnition 1. The numbers ξ0− , ξ0+ , and ξn , n ∈ Z\{0}, and the signs

± ϕ π, ξ sin α 0 , − σ0± = sgn sin α ϕ π, ξ0± sin α ϕ (π, ξn ) − , n ∈ Z\{0}, σn = sgn sin α ϕ (π, ξn ) are called the spectral parameters of problem (1), (2). + Deﬁnition 2. The spectral parameters and the boundaries λ− n and λn , n ∈ Z, of the continuous spectrum are called the spectral data of problem (1), (2).

2. Equation (1) is ﬁnite-gap if the equation Δ2 (λ) − 4 = 0 has only ﬁnitely many simple roots. Changing the numbering, we denote them by − + + − + − + λ− 0 < λ0 < λ1 < λ1 < · · · < λn0 < λn0 < · · · < λN < λN ,

+ where λ− n0 , λn0 is the gap containing the point λ = 0. In the ﬁnite-gap case (see [9]), we have the representations 4 − Δ2 (λ) = 4d2 (λ)R(λ), c (π, λ) = −d(λ)S(λ), where R(λ) =

N

λ − λ− n

s(π, λ) = d(λ)P (λ), c(π, λ) − s (π, λ) = −2d(λ)Q(λ),

λ − λ+ n ,

N +1

N

λ − ξ˜n , n=0 n=n0

n=0

S(λ) =

P (λ) =

(5)

(λ − ηn )

+ ηn0 ∈ λ− , n0 , 0 , ηn0 +1 ∈ 0, λn0

n=0

DIFFERENTIAL EQUATIONS

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INVERSE PROBLEM FOR A QUADRATIC PENCIL OF STURM–LIOUVILLE . . .

739

Q(λ) is a polynomial of degree ≤ N − 2, and d(λ) is an entire function solely determined by the multiple roots of the equation Δ2 (λ) − 4 = 0. One can readily see that P (λ)S(λ) − Q2 (λ) = R(λ). By substituting the representation (5) into (3), we obtain Q(λ) cos 2α − 2−1 [P (λ) − S(λ)] sin 2α − i R(λ) . (6) mα (λ) = P (λ) cos2 α + Q(λ) sin 2α + S(λ) sin2 α Using the form (6) of the Weyl–Titchmarsh function, we introduce the functions A(λ) = P (λ) cos2 α + Q(λ) sin 2α + S(λ) sin2 α, C(λ) = Q(λ) cos 2α − 2−1 [P (λ) − S(λ)] sin 2α. Let B(λ) = P (λ) sin2 α − Q(λ) sin 2α + S(λ) cos2 α. Then one can readily see that A(λ)B(λ) − C 2 (λ) = R(λ), Q(λ) = C(λ) cos 2α + 2−1 [A(λ) − B(λ)] sin 2α.

(7) (8)

3. This section contains the main results of the present paper. Theorem 1. If problem (1), (2) has the gaps − + − + + λN , λN λ0 , λ0 , . . . , λ− n0 , λn0 , . . . , and the spectral parameters + + . . . , ξn0 −1 ∈ λ− ξ0 ∈ λ− 0 , λ0 , n0 −1 , λn0 −1 , + + . . . , ξN ∈ λ− ξn0 +1 ∈ λ− N , λN , n0 +1 , λn0 +1 ,

+ ξn−0 , ξn+0 ∈ λ− n0 , λn0 ,

σ0 = ±1, . . . , σn−0 = ±1, σn+0 = ±1, . . . , σN = ±1 + [where λ− n0 , λn0 is the gap containing the point λ = 0], then N σn−0 −R ξn−0 σn+0 −R ξn+0 σn −R (ξn ) − + , − ξn0 − ξn cot α = −ξn0 A˜ ξn−0 A˜ ξn+0 A˜ (ξn ) n=0

(9)

n=n0

where

N ˜ (λ − ξn ) . A(λ) = λ − ξn−0 λ − ξn+0 n=0 n=n0

Proof. Consider the case in which α = π/2. One can readily see that N (λ − ξn ) ξn−0 ∈ λ− A(λ) = sin2 α λ − ξn−0 λ − ξn+0 n0 , 0 ,

ξn+0 ∈ 0, λ+ , n0

n=0 n=n0

˜ , C(λ) = 2−1 sin 2α λN +2 + C(λ)

˜ where C(λ) is a polynomial of degree ≤ N + 1. For convenience of subsequent computations, we introduce the following new notation: β0 = ξ0 ,

...,

βn0 +1 = ξn+0 , ε0 = σ0 ,

βn0 +2 = ξn0 +1 , ...,

εn0 +1 = σn+0 , DIFFERENTIAL EQUATIONS

βn0 −1 = ξn0 −1 ,

Vol. 43

...,

εn0 −1 = σn0 −1 ,

εn0 +2 = σn0 +1 , No. 6

2007

...,

βn0 = ξn−0 , βN +1 = ξN , εn0 = σn−0 , εN +1 = σN .

740

BABAZHANOV, KHASANOV

Then we obtain A(λ) = sin2 α

N +1

+ βn0 ∈ λ− . n0 , 0 , βn0 +1 ∈ 0, λn0

(λ − βn )

n=0

Hence we derive the asymptotics C(λ) = cot α + O A(λ)

1 λ

(|λ| → ∞).

(10)

By using relation (10) and the Mittag-Leﬄer theorem, we obtain C (βn ) C(λ) = cot α + . (β ) (λ − β ) A(λ) A n n n=0 N +1

It follows from (7) that C (βn ) = εn the asymptotic formula

−R (βn ), n = 0, 1, . . . , N + 1. By virtue of (11), we have

d2 d1 C(λ) + 2 2 +O = cot α + 2 A(λ) λ sin α λ sin α where d1 =

N +1 n=0

C (βn ) , A˜ (βn )

(11)

N +1

C (βn ) d2 = , βn A˜ (βn ) n=0

1 λ3

˜ A(λ) =

(|λ| → ∞), N +1

(12)

(λ − βk ) .

k=0

Relation (12), together with the representation A(λ) = sin2 α λN +2 + a1 λN +1 + a2 λN + a3 λN −1 + a4 λN −2 + · · · , implies that C(λ) = λN +2 sin α cos α + λN +1 (d1 + a1 sin α cos α) + λN (d2 + a1 d1 + a2 sin α cos α) + O λN −1

(|λ| → ∞).

By matching the coeﬃcients of λ2N +3 and λ2N +2 in identity (7), we obtain (b1 − a1 ) sin2 α cos2 α − 2d1 sin α cos α = 0, b2 + a1 b1 − a2 − a21 sin2 α cos2 α − (4a1 d1 + 2d2 ) sin α cos α − d21 = 1.

(13) (14)

By multiplying identity (13) by (−a1 ) and by adding the resulting relation to (14), we obtain (b2 − a2 ) sin2 α cos2 α − (a1 d1 + d2 ) sin 2α − d21 = 1.

(15)

Now, by matching the coeﬃcients of λN +1 and λN in (8), we obtain d1 cos 2α + (a1 − b1 ) sin α cos3 α = 0, (d2 + a1 d1 ) cos 2α + (a2 − b2 ) sin α cos3 α = 0.

(16) (17)

By multiplying Eq. (13) by cot α and by adding the resulting relation to (16), we obtain d1 = 0. This relation permits one to represent (15) and (17) as (b2 − a2 ) sin2 α cos2 α − d2 sin 2α = 1, d2 cos 2α + (a2 − b2 ) sin α cos3 α = 0.

(18) (19)

By multiplying relations (18) and (19) by cos α and sin α, respectively, and by adding the resulting relations, we obtain cot α = −d2 . The deﬁnition of d2 implies (9). One can readily see that formula (9) is also valid for α = π/2. DIFFERENTIAL EQUATIONS

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INVERSE PROBLEM FOR A QUADRATIC PENCIL OF STURM–LIOUVILLE . . .

741

Theorem 2. Let problem (1), (2) have the continuous spectrum Eess

N − + =R λn , λn 1

(20)

n=0

and the spectral parameters + ξ0 ∈ λ− 0 , λ0 , σ0 = ±1,

...,

...,

+ ξn+0 , ξn−0 ∈ λ− n0 , λn0 ,

σn+0 = ±1,

σn−0 = ±1,

...,

...,

+ ξN ∈ λ− N , λN ,

σN = ±1

+ [where λ− n0 , λn0 is the gap containing the point λ = 0], then for all real values of the parameter t ∈ (−∞, ∞), the problem −y + q(x + t)y + 2λp(x + t)y − λ2 y = 0, y(0) cos α + y (0) sin α = 0,

0 < x < ∞, α ∈ (0, π),

(21) (22)

has the same continuous spectrum (20), and the spectral parameters ξn (t), σn (t), ξn±0 (t), and σn±0 (t) satisfy the system of Dubrovin–Trubowitz diﬀerential equations 2 − q(t) + cot2 α] σn (t) −R (ξn (t)) ˙ξn (t) = 2 [ξn (t) − 2ξn (t)p(t) , N ξn (t) − ξn−0 (t) ξn (t) − ξn+0 (t) k=0 (ξn (t) − ξk (t)) k=n,n0

n = 0, 1, . . . , n0 − 1, n0 + 1, . . . , N, 2 2 ± ± ± 2 ξn0 (t) − 2ξn0 (t)p(t) − q(t) + cot α σn0 (t) −R ξn±0 (t) , ξ˙n±0 (t) = N ξn±0 (t) − ξn∓0 (t) ξn±0 (t) − ξk (t) k=0

(23) (24)

k=n0

and the initial conditions ξn (0) = ξn ,

σn (0) = σn ,

ξn±0 (0) = ξn±0 ,

σn±0 (0) = σn±0 .

The sign of σn (t) [respectively, σn±0 (t)] changes to the opposite on each impact of the spectral parameter ξn (t) [respectively, ξn±0 (t)] with the boundary of its gap. Proof. For problem (21), (22), the functions A(λ, t), C(λ, t), and B(λ, t) have the form A(λ, t) = P (λ, t) cos2 α + Q(λ, t) sin 2α + S(λ, t) sin2 α, C(λ, t) = Q(λ, t) cos 2α − 2−1 [P (λ, t) − S(λ, t)] sin 2α, B(λ, t) = P (λ, t) sin2 α − Q(λ, t) sin 2α + S(λ, t) cos2 α.

(25)

A(λ, t)B(λ, t) − C 2 (λ, t) = R(λ).

(26)

Hence it follows that

By diﬀerentiating identity (25) with respect to t and by taking into account the relations [9] ˙ P˙ (λ, t) = −2Q(λ, t), S(λ, t) = 2 λ2 − 2λp(t) − q(t) Q(λ, t), ˙ Q(λ, t) = λ2 − 2λp(t) − q(t) P (λ, t) − S(λ, t), we obtain ˙ A(λ, t) = −2 cos2 α + λ2 − 2λp(t) − q(t) sin2 α C(λ, t) − sin 2αA(t, λ) 1 − λ2 − 2λp(t) − q(t) . DIFFERENTIAL EQUATIONS

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(27)

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BABAZHANOV, KHASANOV

By diﬀerentiating the identity N (λ − ξn (t)) A(λ, t) = sin2 α λ − ξn−0 (t) λ − ξn+0 (t) n=0 n=n0

with respect to t, we obtain ˙ A(λ, t)

N ± ± ± ∓ ˙ = − sin αξn0 (t) ξn0 (t) − ξn0 (t) ξn0 (t) − ξk (t) , 2

± λ=ξn (t) 0

˙ A(λ, t)

λ=ξn (t)

(28)

k=0 k=n0 N = − sin2 αξ˙n (t) ξn (t) − ξn−0 (t) ξn (t) − ξn+0 (t) (ξn (t) − ξk (t)) .

(29)

k=0 k=n,n0

By using (26), we obtain C ξn±0 (t) = σn±0 (t) −R ξn±0 (t) , C (ξn (t)) = σn (t) −R (ξn (t)),

(30) n = 0, 1, . . . , n0 − 1, n0 + 1, . . . , N.

(31)

Relations (27)–(31) imply (23) and (24). Theorem 3. If problem (1), (2) corresponds to the gaps − + − + + λN , λN λ0 , λ0 , . . . , λ− n0 , λn0 , . . . , and the spectral parameters + ξ0 ∈ λ− 0 , λ0 ,

+ ξn+0 , ξn−0 ∈ λ− n0 , λn0 ,

...,

...,

+ ξN ∈ λ− N , λN ,

σ0 = ±1, . . . , σn+0 = ±1, σn−0 = ±1, . . . , σN = ±1 + [where λ− n0 , λn0 is the gap containing the point λ = 0], then p(t) = −

N + − λ+ λk + λ− n0 + λn0 k + − − ξn0 (t) − ξn0 (t) − − ξk (t) , 2 2 k=0

(32)

k=n0

− 2 + λ 2 2 n0 − ξn+0 (t) − ξn−0 (t) q(t) + 2p2 (t) = 2 cot2 α − 2 2 2 N + λ− λ+ k k 2 − ξk (t) , − 2 k=0 2 λ+ n0

(33)

k=n0

where ξn+0 (t), ξn−0 (t), and ξk (t), k = 0, . . . , N, k = n0 , are the spectral parameters corresponding to the coeﬃcients p(x + t) and q(x + t). Proof. First, consider the case in which α = π/2. Let R(λ) = λ2N +2 + r1 λ2N +1 + r2 λ2N + · · · ,

P (λ) = λN + p1 λN −1 + p2 λN −2 + · · · ,

S(λ) = λN +2 + s1 λN +1 + s2 λN + s3 λN −1 + s4 λN −2 + · · · , Q(λ) = q0 λN −2 + q1 λN −3 + · · · DIFFERENTIAL EQUATIONS

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INVERSE PROBLEM FOR A QUADRATIC PENCIL OF STURM–LIOUVILLE . . .

743

Then A(λ) = sin2 αλN +2 + s1 sin2 αλN +1 + s2 sin2 α + cos2 α λN + s3 sin2 α + p1 cos2 α + 2q0 sin α cos α λN −1 + s4 sin2 α + p2 cos2 α + 2q1 sin α cos α λN −2 + · · · , B(λ) = cos2 αλN +2 + s1 cos2 αλN +1 + s2 cos2 α + sin2 α λN + s3 cos2 α + p1 sin2 α − 2q0 sin α cos α λN −1 + s4 cos2 α + p2 sin2 α − 2q1 sin α cos α λN −2 + · · · ,

(34)

(35)

+ s1 sin α cos αλ + (s2 − 1) sin α cos αλ C(λ) = sin α cos αλ + (s3 sin α cos α − p1 sin α cos α + q0 cos 2α) λN −1 + (s4 sin α cos α − p2 sin α cos α + q1 cos 2α) λN −2 + · · · N +2

N +1

N

(36)

By transforming the right-hand sides of the representations (34)–(36), we obtain A(λ) = sin2 α λN +2 + a1 λN +1 + a2 λN + a3 λN −1 + a4 λN −2 + · · · , B(λ) = cos2 α λN +2 + b1 λN +1 + b2 λN + b3 λN −1 + b4 λN −2 + · · · , C(λ) = sin α cos α λN +2 + c1 λN +1 + c2 λN + c3 λN −1 + c4 λN −2 + · · · .

(37) (38) (39)

Then for the coeﬃcients ak , bk , and ck , k = 1, 2, 3, 4, we have the relations a1 a2 a3 a4

= b1 = c1 , = c2 + sin−2 α, b2 = c2 + cos−2 α, = c3 + p1 sin−2 α + 2q0 sin−1 2α, b3 = c3 + p1 cos−2 α − 2q0 sin−1 2α, = c4 + p2 sin−2 α + 2q1 sin−1 2α, b4 = c4 + p2 cos−2 α − 2q1 sin−1 2α.

(40) (41) (42) (43)

We substitute (37)–(39) into (7) and match the coeﬃcients of λ2N +3 , λ2N +2 , λ2N +1 , and λ2N : (a2 + b2 − 2c2 ) + a1 b1 − c21 = 4 sin−2 2α, a1 + b1 − 2c1 = 0, (a3 + b3 − 2c3 ) + (a2 b1 + a1 b2 − 2c1 c2 ) = 4r1 sin−2 2α, (a4 + b4 − 2c4 ) + (a3 b1 + a1 b3 − 2c1 c3 ) + a1 b2 − c22 = 4r2 sin−2 2α.

(44) (45)

It follows from (40)–(45) that p1 = r1 − a1 ,

p2 + a1 p1 + a2 − cot2 α = r2 .

(46)

Let us now show that relations (46) remain valid for α = π/2. It follows from the identity P (λ)S(λ) − Q2 (λ) = R(λ) that p1 = r1 − s1 and p2 + s1 p1 + s2 = r2 . Since A(λ) = S(λ) in this case, we have s1 = a1 and s2 = a2 , where a1 and a2 are the coeﬃcients of λN +1 and λN , respectively, in the polynomial A(λ). Therefore, p1 = r1 − a1 and p2 + a1 p1 + a2 = r2 ; i.e., relation (46) is valid. From formulas (19) and (20) in [9], one can readily obtain the relations p(0) = p1 − r1 /2,

q(0) + 2p2 (0) = r12 /2 − r2 + 2p2 − p21 .

Hence it follows that p(0) = r1 /2 − a1 , DIFFERENTIAL EQUATIONS

q(0) + 2p2 (0) = 2 cot2 α − r12 /2 + r2 + a21 − 2a2 . Vol. 43

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BABAZHANOV, KHASANOV

By applying Vi`ete’s theorem to the polynomials R(λ) and A(λ), we obtain p(0) = −

− λ+ n0 + λn0 − ξn+0 − ξn−0 2

N + λk + λ− k − ξk , − 2 k=0,

(47)

k=n0

− 2 + λ 2 2 n0 − ξn+0 − ξn−0 q(0) + 2p2 (0) = 2 cot2 α − 2 2 2 N + λ− λ+ k k 2 − ξk . − 2 k=0, 2 λ+ n0

(48)

k=n0

If, instead of the functions p(x) and q(x) in (1), one considers p(x + t) and q(x + t), then relations (47) and (48) imply (32) and (33). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

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DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 745–756. c P.N. Nesterov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 731–742. Original Russian Text

ORDINARY DIFFERENTIAL EQUATIONS

Averaging Method in the Asymptotic Integration Problem for Systems with Oscillatory-Decreasing Coeﬃcients P. N. Nesterov Yaroslavl State University, Yaroslavl, Russia Received February 8, 2006

DOI: 10.1134/S001226610706002X

In the present paper, we consider the asymptotic integration problem for a class of linear systems with variable coeﬃcients. The so-called Levinson theorem is the most fundamental result in the theory of asymptotic integration of linear systems. The main diﬃculty encountered when applying this theorem is that one has to reduce the original system to an L-diagonal form. The Q-transformation technique [1] is one of the known methods that permit one to overcome this diﬃculty to some extent. A somewhat diﬀerent approach to the solution of this problem in the case of systems with oscillatory-decreasing coeﬃcients was suggested in [2]. The essence of this approach is to use the averaging method based on Shtokalo’s ideas. In the present paper, we put this method into a more general shape. As a sample application of the method, we construct the asymptotics of solutions of the second-order equation sin ϕ(t) d2 y y = 0, (1) + 1+a √ dt2 t where ϕ(t) = t + αtβ or ϕ(t) = t + α ln t. In Section 1, we discuss the averaging method for systems with oscillatory decreasing coeﬃcients, in Section 2, we consider some asymptotic results due to Levinson, needed in forthcoming considerations, and in Section 3, we construct the asymptotics of solutions of Eq. (1). 1. AVERAGING OF SYSTEMS WITH OSCILLATORY-DECREASING COEFFICIENTS The essence of the Shtokalo method [3, 4] is the possibility to construct a change of variables reducing the linear system k dx = A0 + εl Al (t) + O εk+1 x, (2) dt l=1 where 0 < ε 1, A0 is a constant matrix with real eigenvalues, and the entries of the matrix Al (t) are trigonometric polynomials, to the form k k+1 dy l = A0 + ε Al + O ε y, (3) dt l=1 where Al are constant matrices. By using this change of variables, one can reduce the stability analysis of solutions of system (2) to a similar problem for the averaged system (3). In [2], the Shtokalo method was generalized to some class of systems with oscillatory-decreasing coeﬃcients. In the present paper, we represent this method in the most general form. 745

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Consider the system dx = dt

A0 +

n

Ai1 i2 (t)vi1 (t)vi2 (t) + · · ·

1≤i1 ≤i2 ≤n

i=1

+

Ai (t)vi (t) +

Ai1 ...ik (t)vi1 (t) · · · vik (t) + R(t) x.

(4)

1≤i1 ≤···≤ik ≤n

Here x is an m-dimensional complex vector, A0 , Ai1 ...il (t), and R(t) are square matrices, and v1 (t), . . . , vn (t) are scalar functions. Let the following conditions be satisﬁed. (1◦ ) A0 is a constant matrix with real eigenvalues. (2◦ ) v1 (t) → 0, v2 (t) → 0, . . . , vn (t) → 0 as t → ∞. (3◦ ) v˙ 1 (t), v˙ 2 (t), . . . , v˙ n (t) ∈ L1 [t0 , ∞). (4◦ ) vi1 (t)vi2 (t) · · · vik+1 (t) ∈ L1 [t0 , ∞) for any sequence 1 ≤ i1 ≤ i2 ≤ · · · ≤ ik+1 ≤ n. (5◦ ) The entries of the matrices Ai1 ...il (t) are trigonometric polynomials; i.e., Ai1 ...il (t) =

ˆ S

(i ...il ) iλj t

Δj 1

e

,

j=1 (i ...i )

where λj are arbitrary real numbers and Δj 1 l are, in general, complex matrices. (6◦ ) R(t) ∈ L1 [t0 , ∞). (Here and in what follows, we write R(t) ∈ L1 [t0 , ∞), where R(t) is a square matrix, if R(t) ∈ L1 [t0 , ∞), where · is some matrix norm.) Theorem 1. The change of variables n Yi (t)vi (t) + x= I+

+

Yi1 i2 (t)vi1 (t)vi2 (t) + · · ·

1≤i1 ≤i2 ≤n

i=1

Yi1 ...ik (t)vi1 (t) · · · vik (t) y,

(5)

1≤i1 ≤···≤ik ≤n

where I is the identity matrix and the entries of the matrices Yi1 ...il (t) are trigonometric polynomials with zero mean value, reduces system (4) for suﬃciently large t to the form n dy = A0 + Ai vi (t) + Ai1 i2 vi1 (t)vi2 (t) + · · · dt i=1 1≤i1 ≤i2 ≤n

Ai1 ...ik vi1 (t) · · · vik (t) + R1 (t) y (6) + 1≤i1 ≤···≤ik ≤n

with constant matrices Ai1 ...il and with R1 (t) ∈ L1 [t0 , ∞). Proof. We substitute (5) into (4) and use (6). Omitting the limits on the sums for brevity, we obtain d Y˙ i vi (t) + · · · + Yi1 ...ik (t) (vi1 · · · vik ) + Y˙ i1 ...ik (t)vi1 (t) · · · vik (t) y Yi (t)v˙ i + dt

Yi1 ...ik (t)vi1 (t) · · · vik (t) + I+ Yi (t)vi (t) + · · · +

Ai vi (t) + · · · + Ai1 ...ik vi1 (t) · · · vik (t) + R1 (t) y × A0 +

Ai (t)vi (t) + · · · + Ai1 ...ik (t)vi1 (t) · · · vik (t) + R(t) = A0 +

Yi1 ...ik (t)vi1 (t) · · · vik (t) y. (7) × I+ Yi (t)vi (t) + · · · + DIFFERENTIAL EQUATIONS

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We collect terms of the class L1 [t0 , ∞) on both sides in (7). We have ⎤ ⎡ k d ⎣ Yi1 ...ij (t) vi1 · · · vij ⎦ dt j=1 1≤i1 ≤···≤ij ≤n ⎡ ⎞⎛ ⎞⎤ ⎛ 2k ⎢ ⎝ ⎥ +⎣ Yi1 ...ij (t)vi1 (t) · · · vij (t)⎠⎝ Ai1 ...ip vi1 (t) · · · vip (t)⎠⎦ j+p=k+1 1≤j, p≤k

1≤i1 ≤···≤ij ≤n

j+p=k+1 1≤j, p≤k

1≤i1 ≤···≤ij ≤n

1≤i1 ≤···≤ip ≤n

Yi1 ...ik (t)vi1 (t) · · · vik (t) R1 (t) + I+ Yi (t)vi (t) + · · · +

Yi1 ...ik (t)vi1 (t) · · · vik (t) = R(t) I + Yi (t)vi (t) + · · · + ⎡ ⎞⎛ ⎞⎤ ⎛ 2k ⎥ ⎢ ⎝ Ai1 ...ij (t)vi1 (t) · · · vij (t)⎠⎝ Yi1 ...ip (t)vi1 (t) · · · vip (t)⎠⎦. (8) +⎣

1≤i1 ≤···≤ip ≤n

Since, by virtue of condition (2◦ ) and the boundedness of the matrices Yi1 ...il (t) (which will be deﬁned below), the matrix

Yi1 ...ik (t)vi1 (t) · · · vik (t) I+ Yi (t)vi (t) + · · · + is invertible and the inverse is bounded for t ≥ t0 , it follows that the matrix R1 (t) can be expressed from (8) and obviously belongs to the class L1 [t0 , ∞). Now we match the terms containing vi1 (t) · · · vil (t), l ≤ k, on both sides in (7). By matching the free terms, we obtain the obvious identity IA0 = A0 I. In forthcoming considerations, we need some notation. We introduce the following symbolic function: κ = 1, 2, . . . , n, l = 1, 2, . . . , k, κ(l) = κ · · κ, · l

and κ(0) = 0. For example, 1(2) = 11, 2(4) = 2222, 5(0) = 0, and so on. Thus, we should match terms with vi1 (t) · · · vil (t). We represent the index sequence (i1 i2 · · · il ) ,

1 ≤ i1 ≤ i2 ≤ · · · ≤ il ≤ n,

in the form

κ1 < κ2 < · · · < κs , s ≥ 1, (κ1 (j1 ) κ2 (j2 ) · · · κs (js )) , where j1 + j2 + · · · + js = l. For example, the index sequence (112333) can be represented in the form (1(2)2(1)3(3)). Formulas below can contain index sequences involving zeros [for example, (1(1)03(1))]. All these zeros should be deleted. For instance, the index set in the last example should have the form (1(1)3(1)) = (13). We also assume that (00 . . . 0) = (0) and Y0 (t) = I. Finally, we introduce the functions 1 for x ≤ ji i (x) = 0 for x > ji for i = 1, . . . , s. Thus, by matching the coeﬃcients of vi1 (t) · · · vil (t) in (7), we obtain the following inhomogeneous diﬀerential equations for the matrices Yi1 ...il (t) and Ai1 ...il : Y˙ i1 i2 ...il + Yi1 i2 ...il A0 − A0 Yi1 i2 ...il =

l−1

r=0

l1 +···+ls =r 1 (l1 )...s (ls )=0

−

DIFFERENTIAL EQUATIONS

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Aκ1 (j1 −l1 )...κs (js −ls ) (t)Yκ1 (l1 )...κs (ls ) (t)

l−1

r=0

l1 +···+ls =r 1 (l1 )···s (ls )=0

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Yκ1 (l1 )...κs (ls ) (t)Aκ1 (j1 −l1 )...κs (js −ls ) .

(9)

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Here the integers l1 , . . . , ls range from 0 to r inclusively. We ﬁnd the matrix Ai1 ...il from the condition that the mean value of the right-hand side in (9) is zero; namely, ⎡ ⎤ Ai1 ...il

l−1 ⎢ ⎢ = M⎣ r=0

⎥ Aκ1 (j1 −l1 )...κs (js −ls ) (t)Yκ1 (l1 )...κs (ls ) (t)⎥ ⎦.

(10)

l1 +···+ls =r 1 (l1 )···s (ls )=0

We seek a solution Yi1 i2 ...il (t) of Eq. (9) in the form Yi1 i2 ...il (t) =

N

(i ...il ) iλj t

βj 1

e

λj = 0,

,

(11)

j=1 (i ...i )

where βj 1 l are constant matrices to be determined. We substitute the expression (11) into formula (9) and use the fact that the entries of the matrices Aκ1 (j1 −l1 )···κs (js −ls ) (t) and Yκ1 (l1 )···κs (ls ) (t) are trigonometric polynomials; then we match the coeﬃcients of like exponentials. We obtain the matrix equations (iλj I − A0 ) βj 1

(i ...il )

(i ...il )

+ βj 1

(i ...il )

A0 = Υj 1

,

j = 1, . . . , N,

which are solvable for λj = 0, since the matrix A0 has a real spectrum (e.g., see [5]). The proof of the theorem is complete. Remark 1. The condition that the matrix A0 has a real spectrum is not restrictive. Indeed, if the matrix A0 has complex eigenvalues and is represented in Jordan normal form, then in system (4), we make the change of variables x = exp{iR t}y, where R is the diagonal matrix formed by the imaginary parts of eigenvalues of A0 . This change of variables has coeﬃcients bounded with respect to t and reduces the matrix A0 to the matrix A0 − iR , all of whose eigenvalues are real. Remark 2. If all matrices Ai1 ...il (t) are periodic with the same period T > 0, then the assumption in Theorem 1 concerning the spectrum of A0 can be weakened. Namely, this matrix should not have eigenvalues related by the formula λr − λp =

2πq i , T

r, p = 1, . . . , m,

q = ±1, ±2, ±3, . . .

By using formula (10), one can readily show that the matrices of the ﬁrst and second approximations are deﬁned as follows: ⎛ ⎞ T 1 ⎝M[A(t)] := lim i = 1, . . . , n A(s)ds⎠ . Ai = M [Ai (t)] , T →+∞ T 0

Further,

1 ≤ i < j ≤ n, Aij = M [Aij (t) + Ai (t)Yj (t) + Aj (t)Yi (t)] , i = 1, . . . , n. Aii = M [Aii (t) + Ai (t)Yi (t)] ,

Now consider several special applications of Theorem 1. Suppose that the original system (4) contains only one function v1 (t); i.e., n = 1. In this case, Eqs. (9) are simpliﬁed, dY1(j) (t) − A0 Y1(j) (t) + Y1(j) (t)A0 = A1(j−l) (t)Y1(l) (t) − Y1(l) (t)A1(j−l) , dt l=0 l=0 j−1

j−1

DIFFERENTIAL EQUATIONS

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Accordingly, the expression (10) for the constant matrices acquires the form j−1 A1(j−l) (t)Y1(l) (t) , j = 1, . . . , k. A1(j) = M l=0

Now consider the case in which the original system has the form dx = [A0 + B(t)V (t) + R(t)] x, dt

(12)

where the m × m constant matrix A0 has only real eigenvalues, the entries of the m × pˆ matrix B(t) are trigonometric polynomials, R(t) is an m × m matrix of the class L1 [t0 , ∞), and V (t) is a pˆ × m matrix with the following properties. (1 ) V (t) → 0 as t → ∞. (2 ) V˙ (t) ∈ L1 [t0 , ∞). (3 ) V (t)2 ∈ L1 [t0 , ∞). It readily follows from Theorem 1 that system (12) can be reduced to the form dy = [A0 + BV (t) + R1 (t)] y, dt

(13)

where B = M(B(t)) and R1 (t) ∈ L1 [t0 , ∞). 2. LEVINSON THEOREM The asymptotic integration problem for a linear system with variable coeﬃcients is the problem of constructing the asymptotics of its solutions as t → +∞. As was mentioned at the beginning of the present paper, Levinson’s fundamental theorem is the main result dealing with the construction of asymptotics of linear systems (e.g., see [6, 7]). In its classical form, the theorem was stated for L-diagonal systems. Consider the system dx = (Λ(t) + R(t))x, dt

(14)

where Λ(t) = diag (λ1 (t), . . . , λm (t)) is a continuous1 diagonal matrix and R(t) ∈ L1 [t0 , ∞). Systems of the form (14) are referred to as L-diagonal systems. To construct the asymptotics of the fundamental matrix of system (14), we need the following conditions, known as dichotomy conditions: suppose that either the inequality t2 Re (λi (s) − λj (s)) ds ≤ K1 ,

t2 ≥ t1 ≥ t0 ,

(15)

Re (λi (s) − λj (s)) ds ≥ K2 ,

t2 ≥ t1 ≥ t0 ,

(16)

t1

or the inequality

t2 t1

is valid for each pair (i, j) of indices i = j, where K1 and K2 are some constants. Sometimes, instead of verifying conditions (15) and (16) it is much simpler to show that the quantity Re (λi (t) − λj (t)) preserves its sign for all i, j = 1, . . . , m for suﬃciently large t; i.e., Re (λi (t) − λj (t)) ≤ 0 (≥ 0),

t ≥ t3 .

(17)

Obviously, condition (17) is suﬃcient for the validity of the dichotomy conditions. Now we can state the main assertion. 1

It suﬃces to assume that the entries λi (t), i = 1, . . . , m, of the matrix Λ(t) are integrable on each ﬁnite interval [t0 , t1 ], t1 > t0 . DIFFERENTIAL EQUATIONS

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Theorem 2 (the Levinson theorem). Let the dichotomy conditions (15) and (16) be satisﬁed for system (14). Then the fundamental matrix Φ(t) of system (14) has the following asymptotics as t → ∞ : ⎫ ⎧ t ⎬ ⎨ Λ(s)ds , t > t∗ . (18) Φ(t) = (I + o(1)) exp ⎭ ⎩ t∗

Under some additional constraints on the behavior of quantities like (17), one can reﬁne the convergence rate of the term o(1) to zero in (18) (see [8]). The Levinson theorem is sometimes used for systems represented in a diﬀerent form. More precisely, consider the system dx = (A0 + V (t) + R(t)) x, (19) dt where A0 is a constant matrix with distinct eigenvalues, the matrix V (t) tends to the zero matrix as t → ∞, and the matrices V˙ (t) and R(t) belong to the class L1 [t0 , ∞). Let Λ(t) be the diagonal matrix whose diagonal entries are the eigenvalues λ1 (t), . . . , λm (t) of the matrix A0 + V (t). Theorem 3. If the eigenvalues λ1 (t), . . . , λm (t) of the matrix A0 + V (t) satisfy the dichotomy conditions (15) and (16) [or (17)], then the fundamental matrix Φ(t) of system (19) has the following asymptotics as t → +∞ : ⎫ ⎧ t ⎬ ⎨ Λ(s)ds , Φ(t) = [Π + o(1)] exp ⎭ ⎩ t∗

where Π is the constant matrix whose columns are the eigenvectors of the matrix A0 corresponding to the eigenvalues λ1 , . . . , λm [λ1 = limt→∞ λ1 (t), . . . , λm = limt→∞ λm (t)]. Thus, to construct the asymptotics of solutions of system (4), one should reduce system (6) to L-diagonal form and then use Theorem 2. But if all eigenvalues of the matrix A0 are distinct, then the asymptotics of solutions of system (6) is constructed in accordance with Theorem 3. Consider the case of a single function v1 (t), which often arises in applications. Thus, suppose that the original system (4) has been reduced to the form & dy % = A0 + A1 v1 (t) + A11 v12 (t) + · · · + A1(k) v1k (t) + R1 (t) y. dt

(20)

Let v1 (t), . . . , v1k (t) ∈ L1 [t0 , ∞) and R1 (t) ∈ L1 [t0 , ∞) in this expression. We also suppose that the function v1 (t) has a constant sign for t ≥ t0 . If the ﬁrst nonzero matrix A1(l) , 0 ≤ l ≤ k, in (20) has distinct eigenvalues, then, by using Theorem 3 [since the function v1 (t) is of constant sign, it follows that conditions like (17), suﬃcient for dichotomy, are satisﬁed], we ﬁnd that the fundamental matrix of the original system (4) has the following asymptotics as t → ∞ : ⎫ ⎧ t ⎬ ⎨ Λ(s)ds , X(t) = [Π + o(1)] exp ⎭ ⎩ t∗

where the columns of the matrix Π are the eigenvectors of the matrix A1(l) and Λ(s) is the diagonal 'k matrix whose diagonal entries are the eigenvalues of the matrix i=l A1(i) v1i (t). Note that the case in which v1 (t) = t−α and 0 < α ≤ 1 was considered in [2, 9]. 3. ADIABATIC OSCILLATOR In conclusion, we use the above-described method to analyze Eq. (1). This equation belongs to the class of equations referred to as adiabatic oscillators [10]. By [1, 2], if α = 0 and a = 0, then Eq. (1) has inﬁnitely growing solutions; this phenomenon is referred to as parametric resonance. DIFFERENTIAL EQUATIONS

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3.1. Let ϕ(t) = t + αtβ , a, α ∈ R, α = 0, and 0 < β < 1/2. Let us show that the additional power-law term αtβ in the argument of the sine makes all solutions of Eq. (1) bounded. First, we use the van der Pol change of variables y˙ = −x1 sin t + x2 cos t

y = x1 cos t + x2 sin t, to pass from system (1) to the system

x = (x1 , x2 ) ,

x˙ = A(t)g(t)x,

(21)

sin t + αtβ sin 2t 2 sin2 t √ , g(t) = a . t −2 cos2 t − sin 2t Since sin t + αtβ = sin t cos αtβ + cos t sin αtβ , it follows that system (21) can be reduced to the form (4), (22) x˙ = [A1 (t)v1 (t) + A2 (t)v2 (t)] x. where

1 A(t) = 2

Here A1 (t) = aA(t) sin t,

cos αtβ √ , v1 (t) = t

A2 (t) = aA(t) cos t,

sin αtβ √ v2 (t) = . t

All assumptions of Theorem 1 are valid for k = 2. Therefore, by using the change of variables & % x = I + Y1 (t)v1 (t) + Y2 (t)v2 (t) + Y11 (t)v12 (t) + Y12 (t)v1 (t)v2 (t) + Y22 (t)v22 (t) z, one can reduce system (22) to the form & % z˙ = A1 v1 (t) + A2 v2 (t) + A11 v12 (t) + A12 v1 (t)v2 (t) + A22 v22 (t) + R(t) z,

(23)

where Aij are constant matrices, and R(t) ∈ L1 [t0 , ∞). One can readily see that A1 = M (A1 (t)) = 0 and

A2 = M (A2 (t)) = 0.

Before computing the matrices of the second approximation, we note that cos2 αtβ 1 2 = 1 + cos 2αtβ , v1 (t) = 2t t β 2 sin αt 1 1 = 1 − cos 2αtβ , sin 2αtβ . v1 (t)v2 (t) = v22 (t) = t 2t 2t Thus, system (23) acquires the form β β 1 A11 + A22 + A12 sin 2αt + (A11 − A22 ) cos 2αt + R(t) z. z˙ = 2t By making the change of variables τ = tβ , we obtain the system 1 ¯ ) z¯, (A11 + A22 + A12 sin(2ατ ) + (A11 − A22 ) cos(2ατ )) + R(τ z¯ = 2βτ

(24)

¯ ) = R(t(τ ))t (τ ), and the prime stands for the derivative with respect to τ . where z¯(τ ) = z(t(τ )), R(τ & % ¯ in We use Theorem 1 once more and make the averaging change of variables z¯ = I + τ −1 Y¯1 (τ ) u system (24). We ﬁnally obtain the system % & ¯, (25) u ¯ = τ −1 Γ + R1 (τ ) u DIFFERENTIAL EQUATIONS

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where R1 (τ ) ∈ L1 [τ0 , ∞) and Γ = (A11 + A22 ) /(2β). To construct the asymptotics of the fundamental matrix of system (25), one can use the Levinson theorem. Therefore, one should compute the matrices A11 and A22 . It is unnecessary to ﬁnd the matrix A12 . We have

a2 0 5 a2 0 1 , A22 = M (A2 (t)Y2 (t)) = ; A11 = M (A1 (t)Y1 (t)) = − 24 1 0 24 5 0 therefore, a2 Γ= 12β

0 −1 1 0

.

Now one can readily construct the asymptotics of the fundamental matrix of system (21) :

sin ((a2 /12) ln t) cos ((a2 /12) ln t) + o(1), t → +∞. X(t) = sin ((a2 /12) ln t) − cos ((a2 /12) ln t) Therefore, the solutions of Eq. (1) have the asymptotics y(t) = C1 sin t − a2 /12 ln t + γ1 + o(1) as t → +∞, where C1 and γ1 are arbitrary real constants. In fact, by using the above-described method, one can show that the resulting asymptotic formulas for the solutions of Eq. (1) are valid for 1/2 ≤ β < 1 as well. 3.2. Let us now study the asymptotics of solutions of Eq. (1) for the case in which ϕ(t) = t + α ln t, a, α ∈ R, a = 0. The problem of constructing the asymptotics of solutions of this equation was posed for the ﬁrst time in [11] as an intermediate problem. The following results were obtained there: x1 (t) = t [− cos(t + 2α ln t − θ) + sin t] (1 + ε1 (t)) , x2 (t) = t− [− cos(t + 2α ln t − θ) + sin t] (1 + ε2 (t)) , √ 2 x (t) are linearly independent solutions of equation (1). Here = a 5/24, where x1 (t) and 2 √ θ = arctan 2/ 5 , 0 < θ < π/2, ε1 (t) = o(1), and ε2 (t) = o(1). In the present paper, we show that these results are invalid. It turns out that, in the plane of the parameters (a, α), the set −5a2 /24 ≤ α ≤ a2 /24,

a = 0,

(26)

is an instability domain (parametric resonance domain) of Eq. (1). Arguing as in the previous example, we pass from Eq. (1) to the system x˙ = [A1 (t)v1 (t) + A2 (t)v2 (t)] x,

(27)

where A1 (t) = aA(t) sin t,

A2 (t) = aA(t) cos t,

v1 (t) =

cos(α ln t) √ , t

v2 (t) =

sin(α ln t) √ t

and A(t) is a matrix of the same form as in the previous example. By using the change of variables & % x = I + Y1 (t)v1 (t) + Y2 (t)v2 (t) + Y11 (t)v12 (t) + Y12 (t)v1 (t)v2 (t) + Y22 (t)v22 (t) z, where Yi (t) and Yij (t), i, j = 1, 2, are matrices whose entries are trigonometric polynomials with zero mean, one can reduce system (27) for large t to the form & % (28) z˙ = A1 v1 (t) + A2 v2 (t) + A11 v12 (t) + A12 v1 (t)v2 (t) + A22 v22 (t) + R(t) z, DIFFERENTIAL EQUATIONS

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where Ai and Aij are constant matrices and R(t) = O t−3/2 belongs to the class L1 [t0 , ∞). Note that, just as above, A1 = M (A1 (t)) = 0 and A2 = M (A2 (t)) = 0. Unlike the preceding example, now we should ﬁnd all three matrices of the second approximation. We have

a2 0 5 a2 0 1 , A22 = M (A2 (t)Y2 (t)) = , A11 = M (A1 (t)Y1 (t)) = − 24 1 0 24 5 0 A12 = M (A1 (t)Y2 (t)) + M (A2 (t)Y1 (t)) = a2 /4 diag(−1, 1). Recall that Y˙ 1 = A1 (t) − A1 and Y˙ 2 = A2 (t) − A2 . Note again that 1 cos2 (α ln t) = (1 + cos(2α ln t)), t 2t 2 1 sin (α ln t) = (1 − cos(2α ln t)), v22 (t) = t 2t

v12 (t) =

v1 (t)v2 (t) =

1 sin(2α ln t). 2t

By virtue of the preceding, system (28) acquires the form 1 (A11 + A22 + A12 sin(2α ln t) + (A11 − A22 ) cos(2α ln t)) + R(t) z. z˙ = 2t By making the change of variables τ = ln t, we obtain the system % & ¯ ) z¯, (29) z¯ = A + sin(2ατ )B + cos(2ατ )C + R(τ ¯ ) = R(t(τ ))t (τ ) = O e−τ /2 , the prime stands for the derivative with where z¯(τ ) = z(t(τ )), R(τ respect to τ , and

1 a2 a2 −1 0 1 0 −2/3 , B = A12 = , A = (A11 + A22 ) = 2 8 2/3 2 8 0 0 1

a2 0 1 1 . C = (A11 − A22 ) = − 2 8 1 0 It is well known that there exists a Lyapunov transformation reducing system (29) to the form (30) w = [Γ + R1 (τ )] w −τ /2 . The construction of the asymptotics of with a constant matrix Γ and with R1 (τ ) = O e solutions of system (30) does not encounter any diﬃculties of essential nature. Unfortunately, in practical problems, it is rarely possible to construct the Lyapunov transformation in closed form. However, in our problem, this transformation can be constructed step-by-step. First, note that the matrix A has pure imaginary eigenvalues ±i (a2 /12). We ﬁrst perform the transformation z¯ = Gu, which reduces the matrix Ato a diagonal form. One can readily show that G can be chosen in the 1 1 . The transformed system acquires the form form of the matrix −i i

−τ /2 a2 2 2 0 1 0 i +O e diag i , −i − sin(2ατ ) − cos(2ατ ) u. u = 8 3 3 −i 0 1 0 Now we eliminate the constant matrix with the use of the change of variables ( ) ( 2 ) a a2 u1 . u = diag exp i τ , exp −i τ 12 12 DIFFERENTIAL EQUATIONS

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We obtain the system a2 u1 = 8

−ie−iγτ 0

0 ieiγτ

+ O e−τ /2 u1 ,

where γ = 2α + a2 /6. Here we have used Euler formulas. Now we make the change of variables u1 = S(τ )w, where

−ie−iγτ ie−iγτ , S(τ ) = 1 1 which leads to a system of the form (30),

%

−τ /2

w = Γ+O e

&

w,

Γ=

−iγ/2 a2 /8 + iγ/2 2 −a /8 + iγ/2 −iγ/2

.

(31)

First, let γ = 0, i.e., α = −a2 /12. In this case, the matrix Γ has a diagonal form. Therefore, by the Levinson theorem, the fundamental matrix W (τ ) has the following asymptotics as τ → +∞ : ( 2 ) ( 2 ) a a τ , exp − τ . W (τ ) = [I + o(1)] diag exp 8 8 Now, returning to Eq. (1), we obtain the following asymptotic representation of its solutions as t → +∞ : * 2 + 2 y(t) = C1 ta /8 sin(t + α ln t − π/4) + o ta /8 , where C1 is an arbitrary real constant. Now let γ = 0. The eigenvalues of the matrix Γ can be found from the characteristic equation λ2 − iγλ − a4 /64 = 0. Depending on the sign of the discriminant D = a4 /16 − γ 2 of this equation, one can single out the following cases. (a) |γ| < a2 /4; i.e., α ∈ (−5a2 /24, a2 /24). The eigenvalues of the matrix Γ have the form , def μ = (1/2) a4 /16 − γ 2 .

λ1,2 = iγ/2 ± μ,

Accordingly, we obtain the following asymptotics of the fundamental matrix W (τ ) as τ → +∞ :

) ( ) ( iγ iγ 1 1 . + o(1) diag exp μτ + τ , exp −μτ + τ W (τ ) = 2 2 i σ1 i σ2 Here σi = −(2/γ) (a2 /8 + (−1)i μ), i = 1, 2. In this case, the asymptotics of solutions of Eq. (1) has the form t → +∞, y(t) = C1 tμ sin(t + α ln t − ν) + o (tμ ) , where C1 is an arbitrary real constant and sin ν = -

1 − σ1 2

, 2

(1 − σ1 ) + (1 + σ1 )

cos ν = -

1 + σ1 2

.

(1 − σ1 ) + (1 + σ1 )

2

(b) |γ| > a2 /4; i.e., α ∈ (−∞, −5a2 /24) ∪ (a2 /24, +∞). For the eigenvalues of the matrix Γ, we obtain the expression , def μ∗ = (1/2) γ 2 − a4 /16. λ1,2 = iγ/2 ± iμ∗ , DIFFERENTIAL EQUATIONS

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The asymptotic representation of the fundamental matrix W (τ ) as τ → +∞ has the form

1 1 + o(1) W (τ ) = σ1 + iσ2 −σ1 + iσ2 * * * γ+ . γ + .+ τ , exp i −μ∗ + τ , × diag exp i μ∗ + 2 2 where σ1 = −2μ∗ /γ and σ2 = −a2 /(4γ). Returning to Eq. (1), we obtain the following asymptotics of its solutions as t → +∞ : y(t) = C1 [A sin (t + (α − μ∗ ) ln t − ν1 + γ1 ) + B sin (t + (α + μ∗ ) ln t − ν2 − γ1 )] + o(1), , , 2 (1 − σ1 ), B = 2 (1 + σ1 ), cos ν1 = where C1 and γ1 are arbitrary real constants, A = (σ1 − 1)/A, sin ν1 = σ2 /A, cos ν2 = σ2 /B, and sin ν2 = − (1 + σ1 )/B. It remains to consider the last case. (c) |γ| = a2 /4; i.e., α ∈ {{−5a2 /24} , {a2 /24}}. The matrix Γ has the double eigenvalue λ1 = λ2 = iγ/2. One can readily show that the geometric multiplicity of this eigenvalue is equal to unity. It follows that the matrix Γ is similar to a Jordan block. Thus, by making the change of variables w = T w1 in system (31), where

iγ/2 0 , T = 2 a /8 −1 we obtain the system

w1 =

iγ/2 1 iγ/2 0

+O

1

eτ /2

w1 .

Now, by performing the transformation w1 = exp{iγτ /2}P (τ )w2 , where

τ τ P (τ ) = , 0 1 we obtain the system

% & w2 = diag(−1, 0)τ −1 + O τ e−τ /2 w2 ,

which has an L-diagonal form. To construct the asymptotics of solutions of this system, one can use the Levinson theorem. We have τ → +∞. W2 (τ ) = [I + o(1)] diag τ −1 , 1 , By using the results in [8] (Theorem 1), one can show that here the quantity o(1) is actually O (τ −1 ). Hence it follows that the fundamental matrix W (τ ) of system (31) admits the asymptotic representation

( ) iγ O(1) iγτ /2 + O(1) exp τ , τ → +∞. W (τ ) = 2 O(1) a2 τ /8 + O(1) Thus, for the solutions of Eq. (1), we obtain the asymptotics y(t) = C1 ln t sin(t + α ln t − ν) + O(1) as t → +∞, where C1 is an arbitrary real constant and ( 0 for α = −5a2 /24 ν= π/2 for α = a2 /24. DIFFERENTIAL EQUATIONS

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Figure. The dashed domain represents the instability (parametric resonance) domain of Eq. (1) with ϕ(t) = t + α ln t.

The ﬁgure represents the parametric resonance domain for the equation sin(t + α ln t) d2 y √ y = 0, + 1+a dt2 t which is given by the expression (26). Therefore, the above-described method can be used to construct the asymptotics of solutions of a fairly wide class of linear systems that can be reduced to the form (4). It is unnecessary that the original system contains only decreasing oscillating terms; quite opposite, it can contain oscillatory-increasing terms (e.g., see [9]). ACKNOWLEDGMENTS The author is grateful to V.Sh. Burd for a number of useful remarks made during the preparation of the paper and to S.D. Glyzin, A.Yu. Kolesov, and V.V. Maiorov for attention to the research. REFERENCES 1. Harris, W.A.Jr. and Lutz, D.A., J. Math. Anal. Appl., 1977, vol. 57, no. 3, pp. 571–586. 2. Burd, V.Sh. and Karakulin, V.A., Mat. Zametki, 1998, vol. 64, no. 5, pp. 658–666. 3. Shtokalo, I.Z., Lineinye diﬀerentsial’nye uravneniya s peremennymi koeﬃtsientami (Linear Diﬀerential Equations with Variable Coeﬃcients), Kiev: Naukova Dumka, 1960. 4. Shtokalo, I.Z., Operatsionnoe ischislenie (obobshcheniya i prilozheniya) (Operational Calculus (Generalizations and Applications)), Kiev: Naukova Dumka, 1972. 5. Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Kiev: Fizmatlit, 2004. 6. Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Theory of Stability), Moscow: Nauka, 1967. 7. Coddington, E. and Levinson, N., Theory of Ordinary Diﬀerential Equations, New York, 1955. Translated under the title Teoriya obyknovennykh diﬀerentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958. 8. Bodine, S. and Lutz, D.A., J. Math. Anal. Appl., 2004, vol. 290, pp. 343–362. 9. Nesterov, P.N., Mat. Zametki, 2006, vol. 80, no. 2, pp. 240–250. 10. Harris, W.A.Jr. and Lutz, D.A., J. Math. Anal. Appl., 1975, vol. 51, no. 1, pp. 76–93. 11. Abdullayev, A.S., Studies in Applied Mathematics, 1997, vol. 99, no. 3, pp. 255–283.

DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 757–766. c F.F. Nikitin, S.V. Chistyakov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 743–752. Original Russian Text

ORDINARY DIFFERENTIAL EQUATIONS

Existence and Uniqueness Theorem for a Generalized Isaacs–Bellman Equation F. F. Nikitin and S. V. Chistyakov St. Petersburg State University, St. Petersburg, Russia Received December 21, 2005

DOI: 10.1134/S0012266107060031

1. INTRODUCTION The investigation of diﬀerential games leads to functional equations [1, p. 109; 2–7; 8, p. 15] that generalize the Isaacs–Bellman equation to the case in which the game value function (the Bellman function) is nonsmooth. In the class of games with terminal payoﬀ, for one such equation, we give a new proof of the theorem on the existence and uniqueness of a solution that satisﬁes a boundary condition resulting from the statement of the problem. Unlike the earlier proof [8, p. 52], it is not based on any known theorem on the existence of a solution of a diﬀerential game and does not use elements of any possible formalization of the game. Our theorem, together with other results pertaining to the method of programmed iterations [1, p. 145; 2–7; 8, p. 22], permits one to construct a closed-form theory of diﬀerential games covering a wide range of problems, including the existence of a solution of a diﬀerential game. 2. STATEMENT OF THE PROBLEM AND MAIN ASSUMPTIONS Consider a control process described by the system dx = f (t, x, u, v) dt

t ∈ R, x ∈ Rn , u ∈ P ∈ Comp Rm , v ∈ Q ∈ Comp Rl

(1)

with a given initial condition x (t0 ) = x0 .

(2)

We assume that the system is controlled by two players, who have complete information on the current position (t, x(t)). The ﬁrst player uses the control u and tries to minimize the terminal functional H(x(T )), H(·) ∈ C (Rn ) at a given time T > t0 , and the second player uses the control v and has the opposite objective. As to the right-hand side of system (1), we assume that the function f is continuous on R × Rn × P × Q, satisﬁes the local Lipschitz condition with respect to x with a constant independent of u and v, and satisﬁes the solution extendability condition f (t, x, u, v) ≤ λ(1 + x)

∀(t, x, u, v) ∈ R × Rn × P × Q

as well as the saddle-point condition max minl, f (t, x, u, v) = min maxl, f (t, x, u, v) v∈Q u∈P

u∈P v∈Q

∀(t, x) ∈ [t0 , T ] × Rn ,

∀l ∈ Rn

(3)

in the small game, where · , · is the inner product on the space Rn . 3. AUXILIARY ESTIMATE Lebesgue measurable programmed controls u(·) and v(·) ranging for almost all t in the sets P and Q, respectively, are said to be admissible. 757

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NIKITIN, CHISTYAKOV

Let A (t0 , x0 , T ) be the set of solutions of the Cauchy problem (1), (2) for all possible programmed controls u(·) and v(·), and let

D

= D (t0 , x0 , T ) = {(t, x) | t ∈ [t0 , T ] , x = x(t), x(·) ∈ A (t0 , x0 , T )}

be the segment of the integral funnel of system (1) on [t0 , T ] issuing from (t0 , x0 ). Lemma 1. There exists a continuous nondecreasing function c(·) : [0, T − t0 ] → R, c(0) = 0, such that the estimate f (t, x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x∗ , u, v) ≤ c(Δ)

(4)

is valid for arbitrary (t∗ , x∗ ) ∈ D , u ∈ P, v ∈ Q, Δ ∈ [0, T − t∗ ] , and t ∈ [t∗ , T ] , |t − t∗ | ≤ Δ, where x (t, t∗ , x∗ , u, v) is the solution of system (1) with the initial condition x (t∗ ) = x∗ for constant controls u and v. Proof. Let cD be the closure of the set D , let Prx (cD ) be the projection of cD onto the subspace of the phase coordinates {x}, and let L be the Lipschitz constant of the function f on the set [t0 , T ] × Prx (cD ). We take arbitrary (t∗ , x∗ ) ∈ D , u ∈ P , v ∈ Q, Δ ∈ [0, T − t∗ ], and t ∈ [t∗ , T ], |t − t∗ | ≤ Δ. Obviously, f (t, x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x∗ , u, v) ≤ f (t, x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x (t, t∗ , x∗ , u, v) , u, v) + f (t∗ , x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x∗ , u, v) ≤ f (t, x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x (t, t∗ , x∗ , u, v) , u, v) + L x (t, t∗ , x∗ , u, v) − x∗ .

(5)

But

t x (t, t∗ , x∗ , u, v) − x∗ = f (τ, x (τ, t∗ , x∗ , u, v) , u, v) dτ t∗

t

t ∗ +Δ

f (τ, x (τ, t∗ , x∗ , u, v) , u, v) dτ ≤

≤ t∗

f (τ, x (τ, t∗ , x∗ , u, v) , u, v) dτ ≤ KΔ,

(6)

t∗

where K = sup(t,x)∈D , u∈P, v∈Q f (t, x, u, v). In turn, f (t, x (t, t∗ , x∗ , u, v) , u, v) − f (t∗ , x (t, t∗ , x∗ , u, v) , u, v) ≤ c (Δ),

(7)

where c (Δ) =

f (t , x, u, v) − f (t , x, u, v) ,

(8)

G(Δ) = (t , t , x, u, v) ∈ [t0 , T ] × [t0 , T ] × Pr(cD ) × P × Q | |t − t | ≤ Δ .

(9)

max

(t ,t ,x,u,v)∈G(Δ)

x

Therefore, by setting c(Δ) = c (Δ)+LKΔ, from (5)–(7), we obtain the estimate (4); obviously, c(Δ) is a nondecreasing function, and c(0) = 0. To prove its continuity, we note that the multimapping G given by (9) is continuous. Therefore, under the above-stipulated assumptions on the function f , it follows from (8) (see [9, p. 125]) that the function c (Δ) is also continuous; therefore, so is c(Δ). The proof of the lemma is complete. We say that a pair (u∗ , v ∗ ) ∈ P × Q is an equilibrium in the small game with parameters (t∗ , x∗ , l) ∈ [t0 , T ] × Rn × Rn if l, f (t∗ , x∗ , u∗ , v) ≤ l, f (t∗ , x∗ , u∗ , v ∗ ) ≤ l, f (t∗ , x∗ , u, v ∗ )

∀u ∈ P,

∀v ∈ Q.

(10)

Under condition (3), there exists an equilibrium in the small game for an arbitrary choice of the parameters (t∗ , x∗ , l) ∈ [t0 , T ] × Rn × Rn [10, p. 14]. DIFFERENTIAL EQUATIONS

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Lemma 2. Let (t∗ , x− ) , (t∗ , x+ ) ∈ D , and let (u∗ , v ∗ ) ∈ P × Q be an equilibrium in the small game with parameters (t∗ , x− , l) , where l = x+ − x− . Then the estimate x (t∗ + δ, t∗ , x+ , u∗ , v(·)) − x (t∗ + δ, t∗ , x− , u(·), v ∗ ) ≤ (1 + βδ) x+ − x− + φ(δ)δ 2

2

is valid for arbitrary admissible controls u(·) and v(·), where the constant β and the function φ(δ) are independent of the positions (t∗ , x+ ) , (t∗ , x− ) and the controls u(·) and v(·); moreover, limδ→0+ φ(δ) = 0. Proof. Take arbitrary admissible controls u(·) and v(·) and set x+ (t) = x (t, t∗ , x+ , u∗ , v(·)) ,

x− (t) = x (t, t∗ , x− , u(·), v ∗ ) .

2

The function 2 (t) = x+ (t) − x− (t) is diﬀerentiable almost everywhere on the interval [t∗ , T ], and the relations d2 (t) = 2 x+ (t) − x− (t), f (t, x+ (t), u∗ , v(t)) − f (t, x− (t), u(t), v ∗ ) dt = 2 x+ − x− , f (t, x+ (t), u∗ , v(t)) − f (t, x− (t), u(t), v ∗ ) t [f (τ, x+ (τ ), u∗ , v(τ )) − f (τ, x− (τ ), u(τ ), v ∗ )] dτ, +2 t∗

f (t, x+ (t), u∗ , v(t)) − f (t, x− (t), u(t), v ∗ ) are valid at all points of diﬀerentiability. Since the right-hand side of system (1) is bounded on the compact set [t0 , T ] × Prx (cD ) × P × Q, it follows from the Cauchy–Schwarz inequality that the second term on the right-hand side in this chain of relations does not exceed κ 2 (t − t∗ ), where κ is a constant independent of the positions (t∗ , x+ ) and (t∗ , x− ) and the controls u(·) and v(·). Consequently, d2 (t) ≤ 2 x+ − x− , f (t, x+ (t), u∗ , v(t)) − f (t, x− (t), u(t), v ∗ ) + κ 2 (t − t∗ ) . dt We transform the right-hand side of this inequality and represent it in the form d2 (t) ≤ 2 x+ − x− , f (t∗ , x+ , u∗ , v(t)) − f (t∗ , x− , u(t), v ∗ ) dt + 2 x+ − x− , f (t, x+ (t), u∗ , v(t)) − f (t∗ , x+ , u∗ , v(t)) − 2 x+ − x− , f (t, x− (t), u(t), v ∗ ) − f (t∗ , x− , u(t), v ∗ ) + κ 2 (t − t∗ ) . Now, by using the Cauchy–Schwarz inequality and Lemma 1 to estimate the second and third terms on the right-hand side in this inequality, we obtain d2 (t) ≤ 2 x+ − x− , f (t∗ , x+ , u∗ , v(t)) − f (t∗ , x− , u(t), v ∗ ) + 4M c (t − t∗ ) + κ 2 (t − t∗ ) , dt

(11)

where M is a constant such that x − x ≤ M for arbitrary x , x ∈ Prx (cD ); in particular, x+ − x− ≤ M . We set φ∗ (t − t∗ ) = 4M c (t − t∗ ). Since the constant M and the function c(δ) are independent of the positions (t∗ , x+ ) and (t∗ , x− ) and the controls u(·) and v(·), it follows that so is the function φ∗ (δ). Like c(δ), the function φ∗ (δ) is obviously nondecreasing. By elementary transformations of the right-hand side of inequality (11), one can represent this inequality in the form d2 (t) ≤ 2 x+ − x− , f (t∗ , x+ , u∗ , v(t)) − f (t∗ , x− , u∗ , v(t)) dt + 2 x+ − x− , f (t∗ , x− , u∗ , v(t)) − f (t∗ , x− , u(t), v ∗ ) + φ∗ (t − t∗ ) + κ 2 (t − t∗ ) . DIFFERENTIAL EQUATIONS

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Since (u∗ , v ∗ ) is an equilibrium in the small game with parameters (t∗ , x− , l), l = x+ − x− , it follows from (10) that the second term on the right-hand side in the last inequality is nonpositive. Therefore, d2 (t) ≤ 2 x+ − x− , f (t∗ , x+ , u∗ , v(t)) − f (t∗ , x− , u∗ , v(t)) + φ∗ (t − t∗ ) + κ 2 (t − t∗ ) . dt Here we use the Cauchy–Schwarz inequality and the fact that the function f satisﬁes the Lipschitz condition with respect to x on the compact set [t0 , T ] × Prx (cD ). As a result, we obtain the inequality d2 (t) ≤ 2L2 (t∗ ) + φ∗ (t − t∗ ) + κ 2 (t − t∗ ) . dt By integrating this inequality over the interval [t∗ , t∗ + δ] and by using the fact that φ∗ (δ) is a nondecreasing function, we obtain 2 (t∗ + δ) ≤ (1 + 2Lδ)2 (t∗ ) + φ∗ (δ)δ + κ 2 δ2 /2. To prove the lemma, it remains to set φ(δ) = φ∗ (δ) + κ 2 δ/2,

β = 2L.

In view of the preceding, the function φ(δ) is independent of the positions (t∗ , x+ ) and (t∗ , x− ) and the controls u(·) and v(·), and the same is obviously true of the constant β. The proof of the lemma is complete. 4. VALUE OPERATORS AND A GENERALIZED ISAACS–BELLMAN EQUATION Let U C (D ) be the space of functions w(·) : D → R uniformly continuous on D . On this space, we deﬁne the operators Φc− ◦ w (t∗ , x∗ ) = max max inf w (t, x (t, t∗ , x∗ , u(·), v)) ,

(12)

Φc+ ◦ w (t∗ , x∗ ) = min min sup w (t, x (t, t∗ , x∗ , u, v(·))) .

(13)

t∈[t∗ ,T ] v∈Q u(·) t∈[t∗ ,T ] u∈P v(·)

[Here the inner inﬁmum and supremum are taken over all admissible programmed controls u(·) and v(·), respectively.] By [8, p. 17], Φc− ◦ w(·) ≥ w(·), Φc+ ◦ w(·) ≤ w(·)

(14) (15)

for any function w(·) ∈ U C (D ). Now the following assertion [8, p. 15] becomes obvious. Lemma 3. On the space

is equivalent to the equation

U C (D ), the system of equations Φc− ◦ w(·) = w(·), Φc+ ◦ w(·) = w(·)

(16) (17)

Φc− ◦ w(·) = Φc+ ◦ w(·).

(18)

As was established in [8, p. 55], in the class of continuously diﬀerentiable functions, system (16), (17) is equivalent to the Isaacs–Bellman equation [11, p. 90 of the Russian translation; 12, p. 137]

∂w ∂w (t, x) + max min (t, x), f (t, x, u, v) = 0. v∈Q u∈P ∂t ∂x Therefore, Eq. (18) can be called a generalized Isaacs–Bellman equation. DIFFERENTIAL EQUATIONS

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761

Consider the successive approximations w− (·) = Φc− ◦ w−

(·),

k = 1, 2, . . . ,

(19)

w+ (·) = Φc+ ◦ w+

(·),

k = 1, 2, . . . ,

(20)

(k)

(k−1)

(k)

(k−1)

to Eqs. (16) and (17), respectively, where for the initial approximations, we take the functions (0)

w− (t∗ , x∗ ) = max inf H (x (T, t∗ , x∗ , u(·), v)) , v∈Q u(·)

(0)

w+ (t∗ , x∗ ) = min sup H (x (T, t∗ , x∗ , u, v(·))) , u∈P v(·)

(t∗ , x∗ ) ∈ D ,

(21)

(t∗ , x∗ ) ∈ D .

(22)

Lemma 4. Each of the successive approximations (19) [respectively, (20)] with the initial approximation (21) [respectively, (22)], including the initial approximation itself, is a solution of Eq. (17) [respectively, (16)]; i.e., Φc− ◦ w+ (·) = w+ (·), Φc+ ◦

(k)

(k)

(23)

(k) w− (·)

(k) w− (·)

(24)

=

for each k = 0, 1, 2, . . . Proof. Let us establish, say, relation (24) for each k = 0, 1, 2, . . . To this end, we show by induction over k that (k) (k) (25) sup w− (t, x (t, t∗ , x∗ , u, v(·))) ≥ w− (t∗ , x∗ ) v(·)

for arbitrary (t∗ , x∗ ) ∈ D , t ∈ [t∗ , T ], and u ∈ P . If k = 0, then inf H (x (T, t, x (t, t∗ , x∗ , u, v(·)) , u (·), v )) sup w− (t, x (t, t∗ , x∗ , u, v(·))) = sup max (0)

v(·) v ∈Q u (·)

v(·)

inf H (x (T, t, x (t, t∗ , x∗ , u, v ) , u (·), v )) ≥ max v ∈Q u (·)

inf H (x (T, t, x (t, t∗ , x∗ , u (·), v ) , u (·), v )) ≥ max v ∈Q u (·)

inf H (x (T, t∗ , x∗ , u (·), v )) = w− (t∗ , x∗ ) = max (0)

v ∈Q u (·)

for arbitrary (t∗ , x∗ ) ∈ D , t ∈ [t∗ , T ], and u ∈ P . Now suppose that the above-stated assertion is valid for k − 1 but nevertheless there exist (t∗ , x∗ ) ∈ D , t ∈ [t∗ , T ], and u ∈ P such that (k)

(k)

sup w− (t, x (t, t∗ , x∗ , u, v(·))) < w− (t∗ , x∗ ) . v(·)

In view of the deﬁnition of the operator Φc− , the last inequality can be rewritten in the form (k−1)

inf w− sup max max v(·) t ∈[t,T ] v ∈Q u (·)

(t , x (t , t, x (t, t∗ , x∗ , u, v(·)) , u (·), v ))

(k−1) inf w− (t , x (t , t∗ , x∗ , u (·), v )) . < max max t ∈[t∗ ,T ] v ∈Q u (·)

(26)

By t∗ and v∗ we denote the values of the variables t and v at which the maxima on the right-hand sides in the last inequality are attained. Then it follows from (14) and (26) that (k−1)

sup w− v(·)

(k−1)

(t, x (t, t∗ , x∗ , u, v(·))) < inf w−

DIFFERENTIAL EQUATIONS

u (·)

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(t∗ , x (t∗ , t∗ , x∗ , u (·), v∗ )) .

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NIKITIN, CHISTYAKOV

All the more, (k−1)

sup w− v(·)

(k−1)

(t, x (t, t∗ , x∗ , u, v(·))) < w−

(t∗ , x (t∗ , t∗ , x∗ , u, v∗ )) .

Suppose that t > t∗ . Then the last inequality can be represented in the form (k−1) (k−1) (t, x (t, t∗ , x (t∗ , t∗ , x∗ , u, v(·)) , u, v(·))) < w− (t∗ , x (t∗ , t∗ , x∗ , u, v∗ )) . sup w− v(·)

In turn, this implies that (k−1) (k−1) (t, x (t, t∗ , x (t∗ , t∗ , x∗ , u, v∗ ) , u, v(·))) < w− (t∗ , x (t∗ , t∗ , x∗ , u, v∗ )) . sup w− v(·)

By setting x∗ = x (t∗ , t∗ , x∗ , u, v∗ ) here, we arrive at a contradiction with the induction assumption. Now let t ≤ t∗ . Then it follows from (26) that (k−1)

w− sup inf v(·) u (·)

(t∗ , x (t∗ , t, x (t, t∗ , x∗ , u, v(·)) , u (·), v∗ )) (k−1)

w− < inf u (·)

(t∗ , x (t∗ , t∗ , x∗ , u (·), v∗ )) .

(27)

Let us estimate the left-hand side of the last inequality: (k−1)

w− sup inf v(·) u (·)

(t∗ , x (t∗ , t, x (t, t∗ , x∗ , u, v(·)) , u (·), v∗ )) w− ≥ sup inf

(k−1)

v(·) u (·)

(k−1)

(t∗ , x (t∗ , t, x (t, t∗ , x∗ , u (·), v∗ ) , u (·), v∗ ))

(k−1)

(t∗ , x (t∗ , t∗ , x∗ , u (·), v∗ )) .

w− ≥ inf u (·)

w− = inf u (·)

(t∗ , x (t∗ , t, x (t, t∗ , x∗ , u (·), v(·)) , u (·), v∗ ))

Therefore, relation (27) implies the impossible inequality (k−1)

inf w−

u (·)

(t∗ , x (t∗ , t∗ , x∗ , u (·), v∗ )) < inf w−

(k−1)

u (·)

(t∗ , x (t∗ , t∗ , x∗ , u (·), v∗ )) .

We have thereby shown that inequality (25) is valid for arbitrary k = 0, 1, 2, . . . , (t∗ , x∗ ) ∈ t ∈ [t∗ , T ], and u ∈ P . Consequently, min min sup w− (t, x (t, t∗ , x∗ , u, v(·))) ≥ w− (t∗ , x∗ ) , (k)

(k)

t∈[t∗ ,T ] u∈P v(·)

D,

(t∗ , x∗ ) ∈ D ,

(k) (k) (·) ≥ w− (·). But, by (15), the opposite inequality is also for any k = 0, 1, 2, . . . , i.e., Φc+ ◦ w− valid. Therefore, relation (24) is valid for each k = 0, 1, 2, . . . In a similar way, one can show that relation (23) holds for all k = 0, 1, 2, . . . The proof of the lemma is complete.

5. EXISTENCE AND UNIQUENESS THEOREM Theorem 1. The generalized Isaacs–Bellman equation (18) with the boundary condition w(t, x)|t=T = H(x) has a unique solution in the space

(28)

U C (D ). DIFFERENTIAL EQUATIONS

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EXISTENCE AND UNIQUENESS THEOREM FOR A GENERALIZED . . .

763

Proof. By [6; 8, p. 20], the operators Φc− and Φc+ are continuous on the space U C (D ) equipped with the sup-norm; moreover, for an arbitrary initial approximation to a solution of Eq. (16) [respectively, Eq. (17)], chosen in that space, the corresponding successive approximations uniformly converge on the set D to a solution of this equation. Furthermore, if the initial approximation to a solution of Eq. (16) [respectively, (17)] satisﬁes condition (28), then this condition is also satisﬁed for the remaining approximations and hence for their uniform limit, that is, for the corresponding solution of Eq. (16) [respectively, (17)]. Consider the successive approximations (19) with the initial approximation (21). Let ∗ (·) = lim w− (·). w− (k)

k→+∞

∗ ∗ (·) = w− (·). Moreover, by passing to the limit as k → +∞ in (24), we obtain Then Φc− ◦ w− c ∗ ∗ ∗ Φ+ ◦ w− (·) = w− (·). Consequently, the function w− (·) is a common ﬁxed point of the operators Φc− c and Φ+ and hence, by Lemma 3, is a solution of Eq. (18). Next, since the initial approximation (21) ∗ (·) also satisﬁes this condition. We have thereby proved satisﬁes condition (28), it follows that w− the existence of a solution of Eq. (18) satisfying condition (28). Note in passing that the function (k) ∗ (·) = limk→+∞ w+ (·), that is, the limit of the successive approximations (20) with the initial w+ approximation (22), is one of these solutions. Now let us show that the solution of Eq. (18) satisfying the boundary condition (28) is unique. Consider the successive approximations (k) (k−1) (·) = Φc− ◦ g− (·), g−

(k) (k−1) g+ (·) = Φc+ ◦ g+ (·),

k = 1, 2, . . . ,

to solutions of Eqs. (16) and (17) with arbitrary initial approximations g− (·), g+ (·) ∈ U C (D ) (k) (k) (·) and g+ (·) = limk→+∞ g+ (·). satisfying the boundary condition (28). Let g− (·) = limk→+∞ g− As was mentioned above, the function g− (·) is a solution of Eq. (16), the function g+ (·) is a solution of Eq. (17), and both of them satisfy condition (28). Therefore, once we show that (0)

(0)

g+ (·) ≤ g− (·),

(29)

it will follow by virtue of the arbitrary choice of the initial approximations g− (·), g+ (·) ∈ U C (D ) that every solution of Eq. (17) satisfying condition (28) does not exceed every solution of Eq. (16) satisfying the same condition. Indeed, this follows from the fact that the initial approximation for a solution of Eq. (17) [respectively, (16)] can be chosen in the form of some solution of the same equation satisfying condition (28). This, together with Lemma 3, implies that if inequality (29) is valid, then w (·) ≤ w (·) and w (·) ≤ w (·) for two arbitrary solutions w (·) and w (·) of Eq. (18) satisfying condition (28). In other words, it follows that the solution of Eq. (18) satisfying condition (28) is unique. Thus, let us prove inequality (29). We take arbitrary (t∗ , x∗ ) ∈ D and ε > 0 and a positive integer N . Let δ = (T − t∗ ) /N , and let t∗ = τ0 < τ1 < · · · < τN = T be a partition of the interval [t∗ , T ] with increment δ. We choose an arbitrary pair (u1∗ , v∗1 ) ∈ P × Q of vectors. Then, on the one hand, there exists an admissible programmed control u1ε (·) such that 1 (N ) (N −1) 1 (t∗ , x∗ ) = 1max max inf g , x t , t , x , u (·), v t g− ∗ ∗ − ∗ ∗ t∗ ∈[t∗ ,T ] v ∈Q u (·) (N −1) τ1 , x τ1 , τ0 , x∗ , u1ε (·), v∗1 − ε/N ; (30) ≥ g− (0)

(0)

on the other hand, there exists an admissible programmed control vε1 (·) such that (N ) (N ) 1 (t∗ , x∗ ) = 1 min min sup g+ t∗ , x t1∗ , t∗ , x∗ , u , v (·) g+ t∗ ∈[t∗ ,T ] u ∈P v (·)

≤ g+

(N )

By setting

τ1 , x τ1 , τ0 , u1∗ , vε1 (·) + ε/N.

x1− = x τ1 , τ0 , x∗ , u1ε (·), v∗1 ,

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x1+ = x τ1 , τ0 , x∗ , u1∗ , vε1 (·) , 2007

(31)

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NIKITIN, CHISTYAKOV

we rewrite inequalities (30) and (31) in the concise form (N ) (N −1) τ1 , x1− − ε/N, g− (t∗ , x∗ ) ≥ g− (N ) (N −1) τ1 , x1+ + ε/N ; g+ (t∗ , x∗ ) ≤ g+

(32) (33)

in view of Lemma 2, we have the estimate 1 x − x1 2 ≤ φ(δ)δ + −

φ(δ) → 0 .

(34)

δ→0

Now let (u2∗ , v∗2 ) ∈ P × Q be an equilibrium in the small game with parameters τ1 , x1− , l , where l = x1+ − x1− . Then, by analogy with (32) and (33), we obtain (N −1) (N −2) τ1 , x1− ≥ g− τ2 , x2− − ε/N, g− (N −1) (N −2) τ1 , x1+ ≤ g+ τ2 , x2+ + ε/N, g+ where

x2− = x τ2 , τ1 , x1− , u2ε (·), v∗2 ,

(35) (36)

x2+ = x τ2 , τ1 , x1+ , u2∗ , vε2 (·)

for some admissible programmed controls u2ε (·) and vε2 (·). By Lemma 2 and the estimate (34), we have the inequalities 2 x+ − x2− 2 ≤ (1 + βδ) x1+ − x1− 2 + φ(δ)δ ≤ (1 + βδ + 1)φ(δ)δ, and consequently,

2 x+ − x2− 2 ≤ [(1 + βδ) + 1]φ(δ)δ.

(37)

Further, let (u3∗ , v∗3 ) ∈ P × Q be an equilibrium in the small game with parameters τ2 , x2− , l , where l = x2+ − x2− . Just as above, we obtain the inequalities (N −3) τ2 , x2− ≥ g− τ3 , x3− − ε/N, (N −2) (N −3) τ2 , x2+ ≤ g+ τ3 , x3+ + ε/N, g+ 3 x+ − x3− 2 ≤ (1 + βδ)2 + (1 + βδ) + 1 φ(δ)δ, (N −2)

(38)

g−

where

x3− = x τ3 , τ2 , x2− , u3ε (·), v∗3 ,

(39) (40)

x3+ = x τ3 , τ2 , x2+ , u3∗ , vε3 (·)

for some admissible programmed controls u3ε (·) and vε3 (·). By continuing similar considerations, at the last step of this process, we obtain the inequalities (1) (0) −1 τN −1 , xN ≥ g− τN , xN (41) g− − − − ε/N, (1) (0) −1 ≤ g+ τN , xN (42) g+ τN −1 , xN + + + ε/N, N 2 x − xN ≤ (1 + βδ)N −1 + · · · + (1 + βδ) + 1 φ(δ)δ. +

−

It obviously follows from the last inequality that N −1 N T − t∗ x − xN 2 ≤ N (1 + βδ)N −1 φ(δ)δ = 1 + β T − t∗ (T − t∗ ) . φ + − N N N −1

Since (1 + β (T − t∗ )/N )

→

N →+∞

exp (β (T − t∗ )) and φ ((T − t∗ )/N ) 2 N x+ − xN −

→

N →+∞

→

N →+∞

0, we have

0.

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From the chain of inequalities (32), (35), (38), and (41), we have (N ) (0) (t∗ , x∗ ) ≥ g− τN , xN g− − − ε,

(44)

and the chain of inequalities (33), (36), (39), and (42) implies that (N ) (0) (t∗ , x∗ ) ≤ g+ τN , xN g+ + + ε.

(45)

Now, by subtracting inequality (44) from (45) and by using the fact that τN = T and the functions (0) (0) g+ and g− satisfy the boundary condition (28), we obtain N (N ) (N ) (t∗ , x∗ ) − g− (t∗ , x∗ ) ≤ H xN g+ + − H x− + 2ε. N Since (a) the successive approximations {g− (·)} and {g+ (·)} are convergent; (b) xN + , x− ∈ Prx (D ) n for each N ; (c) a function H continuous on R is uniformly continuous on any bounded subset of the space Rn [in particular, on the set Prx (D )]; and (d) condition (43) is satisﬁed, it follows that one can pass to the limit as N → ∞ in the last inequality. Then we obtain (n)

(n)

g+ (t∗ , x∗ ) − g− (t∗ , x∗ ) ≤ 2ε, which, by virtue of the arbitrary choice of ε > 0 and (t∗ , x∗ ) ∈ proof of the theorem.

D , implies (29) and completes the

Remark 1. It follows from Theorem 1 and its proof that the unique solution w∗ (·) ∈ U C (D ) of Eq. (18) with the boundary condition (28) is, on the one hand, the uniform limit of the successive approximations (19) and (21) and, on the other hand, the uniform limit of the successive approximations (20) and (22). Therefore, each of these iterative processes can be used for the solution of the generalized Isaacs–Bellman equation with the boundary condition (28). 6. COROLLARIES OF THE MAIN THEOREM By Γ (t0 , x0 ) we denote the control problem (the diﬀerential game) described in Section 2, and by Γ (t∗ , x∗ ) we denote the similar problem that diﬀers from Γ (t0 , x0 ) only by the initial data. Each of the games Γ (t∗ , x∗ ), (t∗ , x∗ ) ∈ D , t∗ < T , is considered in the class of positional strategies [1]. Following [1], for any position (t∗ , x∗ ) ∈ D , t∗ < T , and any positional strategy V : D → Q of the ﬁrst player (respectively, U : D → P of the second player), we introduce the pencil of motions χ (t∗ , x∗ , V ) [respectively, χ (t∗ , x∗ , U )]. Recall that a number c (t∗ , x∗ ) ∈ R is called the value of the game Γ (t∗ , x∗ ) (t∗ < T ) in the class of positional strategies if, for each ε > 0, there exist positional strategies Vε and Uε such that H(x(T )) ≥ c (t∗ , x∗ ) − ε H(x(T )) ≤ c (t∗ , x∗ ) + ε

∀x(·) ∈ χ (t∗ , x∗ , Vε ) , ∀x(·) ∈ χ (t∗ , x∗ , Uε ) ;

(46) (47)

here the strategies Vε and Uε are called ε-optimal strategies of the ﬁrst and the second player, respectively, in the game Γ (t∗ , x∗ ). A function c : D → R, c (t∗ , x∗ )|t∗ =T = H (x∗ ), is called the value function of the game family Γ (t∗ , x∗ ), (t∗ , x∗ ) ∈ D , if c (t∗ , x∗ ) is the value of the game Γ (t∗ , x∗ ) at each position (t∗ , x∗ ) ∈ D , t∗ < T . A strategy Vε (respectively, Uε ) is called a universal ε-optimal strategy of the ﬁrst (respectively, second) player on the set D if, for each position (t∗ , x∗ ) ∈ D , it is the ε-optimal strategy of the same player in the corresponding game Γ (t∗ , x∗ ). We say that a game family Γ (t∗ , x∗ ), (t∗ , x∗ ) ∈ D , has a universal solution if, for each ε > 0, there exist universal ε-optimal strategies of both players on the set D . As was shown in [7; 8, p. 30], for each ε > 0 on the basis of the ﬁrst n− = n− (ε) approximations (19), (21), one can construct a positional strategy Vε such that, together with the function (n)

c (t∗ , x∗ ) = lim w− (t∗ , x∗ ) , n→+∞

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(t∗ , x∗ ) ∈ D ,

(48)

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NIKITIN, CHISTYAKOV

it satisﬁes inequality (46) at each position (t∗ , x∗ ) ∈ D , t∗ < T . Likewise, on the basis of the ﬁrst n+ = n+ (ε) approximations (20) and (22), one can construct a positional strategy Uε such that, together with the function (n)

c (t∗ , x∗ ) = lim w+ (t∗ , x∗ ) , n→+∞

(t∗ , x∗ ) ∈ D ,

(49)

it satisﬁes inequality (47) at each position (t∗ , x∗ ) ∈ D . Since relations (48) and (49) indeed deﬁne the same function (see Remark 1), we have the following assertion. Theorem 2. For each initial position (t0 , x0 ) ∈ (−∞, T ) × Rn , the game family Γ (t∗ , x∗ ) , (t∗ , x∗ ) ∈ D = D (t0 , x0 , T ) , has a universal solution; moreover, the unique solution of the generalized Isaacs–Bellman equation with the boundary condition (28) is the value function of this family. Remark 2. It was established in [12, pp. 131–137] that, under the additional assumption that |H(x)| ≤ μ(1+x) for all x ∈ Rn (μ = const), the value function is a generalized minimax solution of the Isaacs–Bellman equation. REFERENCES 1. Subbotin, A.I. and Chentsov, A.G., Optimizatsiya garantii v zadachakh upravleniya (Guaranteed Optimization in Control Problems), Moscow: Nauka, 1981. 2. Chentsov, A.G., Dokl. Akad. Nauk , 1975, vol. 224, no. 6, pp. 1272–1275. 3. Chentsov, A.G., Mat. Sb., 1976, vol. 99, no. 3, pp. 394–420. 4. Chistyakov, S.V. and Petrosyan, L.A., Vestnik Leningr. Univ. Ser. Mat., Mekh., Astron., 1977, vol. 1, pp. 77–82. 5. Chistyakov, S.V., Prikl. Mat. Mekh., 1977, vol. 41, no. 5, pp. 825–832. 6. Chistyakov, S.V., Prikl. Mat. Mekh., 1982, vol. 46, no. 5, pp. 874–877. 7. Chistyakov, S.V., Dokl. Akad. Nauk , 1991, vol. 319, no. 6, pp. 1333–1335. 8. Chistyakov, S.V., Operatory znacheniya antagonisticheskikh diﬀerentsial’nykh igr (Value Operators in Two-Person Zero-Sum Diﬀerential Games), St. Petersburg: St. Petersburg Univ., 1999. 9. Aubin, J.-P. and Ekeland, I., Applied Nonlinear Analysis, New York: Wiley, 1984. Translated under the title Prikladnoi nelineinyi analiz , Moscow: Mir, 1988. 10. Vorob’ev, N.N., Teoriya igr. Lektsii dlya ekonomistov-kibernetikov (Game Theory. Lectures for Economists and Cyberneticists), Leningrad, 1974. 11. Isaacs, R., Diﬀerential Games, New York: Wiley, 1965. Translated under the title Diﬀerentsial’nye igry, Moscow: Mir, 1967. 12. Subbotin, A.I., Obobshchennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka (Generalized Solutions of First-Order Partial Diﬀerential Equations), Moscow: Institute for Computer Studies, 2003.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 767–773. c S.B. Tkachev, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 753–759. Original Russian Text

ORDINARY DIFFERENTIAL EQUATIONS

Stabilization of Programmed Motions by the Virtual Output Method S. B. Tkachev Bauman Moscow State Technical University, Moscow, Russia Received November 20, 2006

DOI: 10.1134/S0012266107060043

INTRODUCTION For a smooth aﬃne system x ∈ Rn ,

x˙ = A(x) + B(x)u, T

u ∈ R1 , T

B(x) = (b1 (x), . . . , bn (x)) , A(x) = (a1 (x), . . . , an (x)) , ∞ n i = 1, . . . , n, ai (x), bi (x) ∈ C (R ) ,

(1)

with a scalar control, we consider the problem on the stabilization of a given programmed motion (x∗ (t), u∗ (t)), t ≥ 0, for a completely known state vector. For nonlinear systems, one known approach is based on the reduction of system (1) in Rn to a regular canonical [1, 2] or quasicanonical [3] form. To reduce the system to these forms, one should ﬁnd a function ϕ(x) whose derivatives along the trajectories of system (1) up to a given order do not contain the control and verify a number of additional conditions. It is convenient to treat the function ϕ as a virtual output of system (1) and use the terminology of the theory of a normal form of an aﬃne system with respect to a given output [4, p. 143]. An equilibrium stabilization method is known [4, p. 172] for stationary aﬃne systems with given output reducible to a normal form with asymptotically stable zero dynamics. If the zero dynamics is unstable, then for stationary systems, one can use the method suggested in [5, 6] for ﬁnding a virtual output, which is then used for the design of a normal form with asymptotically stable zero dynamics. The problem on the stabilization of a given change of the output of an aﬃne system was considered in [4, p. 180]. The results obtained for the problem on the stabilization of a programmed motion permit one to indicate conditions under which the programmed motion is stable in the variables of the normal form; however, the asymptotic stability problem remains open. The problem of the uniform asymptotic stabilization of a programmed motion can be reduced to the problem of uniform asymptotic stabilization for the zero equilibrium of a nonstationary aﬃne system, which is obtained from the original system by passage to deviations from the programmed motion. The normal form of a nonstationary aﬃne system was introduced in [7], and conditions providing the uniform asymptotic stabilization of the zero equilibrium in the variables of the normal form were presented there. The uniform asymptotic stability of the nonstationary zero dynamics is an important requirement. If the zero dynamics is unstable, then the problem of designing a stabilizing feedback requires a separate study. In the present paper, we consider stationary aﬃne systems reducible to a quasicanonical form of special structure and give a method for designing a nonstationary feedback providing the asymptotic stabilization of a given programmed motion. The method is based on the design of a virtual output of the nonstationary system in terms of the variables describing the deviation from the programmed trajectory such that the latter system can be reduced, with respect to this output, to a normal form with uniformly asymptotically stable zero dynamics. A related example is given. 767

768

TKACHEV

1. QUASICANONICAL FORM AND NORMAL FORM Suppose that system (1) can be reduced in Rn to the form z˙ = f (z, η) + g(z, η)u,

η˙ = q(z, η),

(2)

where z ∈ R1 , η ∈ Rn−1 , and (z, η) = Φ(x) is the corresponding smooth one-to-one correspondence between Rn and Φ (Rn ). This representation is a special case of a quasicanonical form [3] of system (1). If the coeﬃcient g(z, η) of the control is nonzero at each point of Φ (Rn ), then the quasicanonical form is said to be regular in Φ (Rn ). To write out conditions under which system (1) can be reduced to a regular special quasicanonical form (2), it is convenient to use the diﬀerential-geometric approach; to this end, we use the one-toone correspondence between aﬃne systems (1) on Rn and smooth vector ﬁelds A=

n i=1

ai (x)

∂ ∂xi

and

B=

n

bi (x)

i=1

∂ . ∂xi

Let ϕ(x) be a suﬃciently smooth function deﬁned on Rn and such that γ(x) = LB ϕ(x) = 0 in Rn , where LB ϕ(x) is the Lie derivative of ϕ(x) along the vector ﬁeld B. If γ(x) = 0, then the control occurs in the ﬁrst derivative of ϕ(x) along the trajectories of system (1), B(x) = 0, and [3] g(Φ(x)) = 0 in system (2) for z = ϕ(x). If in Φ (Rn ) there exist n − 1 functionally independent suﬃciently smooth ﬁrst integrals ηk (x) of the vector ﬁeld B such that (z, η) = Φ(x), z = ϕ(x), and ηk = ηk (x), k = 1, . . . , n − 1, is a nondegenerate change of variables in Rn , then in these variables, system (1) in Φ (Rn ) acquires the form (2). In this case, we say that system (1) can be reduced in Rn to a regular special quasicanonical form with reduction index = 1. Note that system (2) is a normal form [4, p. 139; 8, p. 106 of the Russian translation] of the stationary aﬃne system (1) with output y = ϕ(x); moreover, system (1) with the output ϕ(x) has the relative degree = 1 in Rn , and the deﬁnition of the relative degree implies that LB ϕ(x) = 0 in Rn . The problem on the reduction of the aﬃne system (1) with given output to a normal form is a special case of the problem on the reduction to a quasicanonical form, since the latter requires ﬁnding ﬁrst the function ϕ(x) and then the transformation (z, η) = Φ(x). 2. STABILIZATION OF A PROGRAMMED MOTION The function y = ϕ(x) will be treated as an output of system (1); this output will be referred to as a virtual output. A given programmed motion (x∗ (t), u∗ (t)), t ≥ 0, of system (1) uniquely determines a programmed motion (z ∗ (t), η ∗ (t), u∗ (t)), t ≥ 0, for system (2), where (z ∗ (0), η ∗ (0)) = Φ (x∗ (0)), and a given programmed change of the virtual output y = y ∗ (t) = z ∗ (t). Let us rewrite system (2) in deviations from the programmed motion and supplement the description of the system by the deviation y¯ = z−z ∗ (t) of the virtual output from its given change. Let T e = z − z ∗ (t), ψ = η − η ∗ (t), and δu = u − u∗ (t). By passing to the variables e, ψ = (ψ1 , . . . , ψn−1 ) , we obtain the nonstationary aﬃne system with output e˙ = f¯(e, ψ, t) + g¯(e, ψ, t)δu, where

ψ˙ = q¯(e, ψ, t),

y¯ = e,

(3)

f¯(e, ψ, t) = [f (z ∗ (t) + e, η ∗ (t) + ψ) − f (z ∗ (t), η ∗ (t))] + [g (z ∗ (t) + e, η ∗ (t) + ψ) − g (z ∗ (t), η ∗ (t))] u∗ (t), g¯(e, ψ, t) = g (z ∗ (t) + e, η ∗ (t) + ψ) , q¯(e, ψ, t) = q (z ∗ (t) + e, η ∗ (t) + ψ) − q (z ∗ (t), η ∗ (t)) , f¯(0, 0, t) = 0, q¯(0, 0, t) = 0.

(4)

The problem of uniform asymptotic stabilization of a given programmed motion (z ∗ (t), η ∗ (t), u∗ (t)) of system (2) is equivalent to the problem of uniform asymptotic stabilization of the equilibrium e = 0, ψ = 0 of system (3). DIFFERENTIAL EQUATIONS

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STABILIZATION OF PROGRAMMED MOTIONS BY THE VIRTUAL OUTPUT . . .

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The structure of system (3) is similar to that of system (2) with output y = z. Following [7], we say that the nonautonomous system (3) is represented in normal form, and the system of equations ψ˙ = q¯(0, ψ, t), (5) where q¯(0, 0, t) = 0, is referred to as the zero dynamics of the nonstationary system (3). If the equilibrium point ψ = 0 of the zero dynamics is (uniformly) asymptotically stable, then the aﬃne system (3) with output y = e is referred to as a (uniformly) minimal-phase system. Consider the problem on the uniform asymptotic stabilization of the zero equilibrium of system (3). To solve this problem, we use the nonlinear stabilization method [9, p. 398] and choose the control in the form g (e, ψ, t), (6) δu = − f¯(e, ψ, t) + c0 e /¯ where c0 > 0. The ﬁrst equation in system (3) closed by the control (6) acquires the form e˙ = −c0 e and is asymptotically stable. The following assertion, which is a corollary of the more general result in [7], provides conditions under which the zero equilibrium of system (3) with the control (6) is uniformly asymptotically stable. Theorem 1. If, for the nonstationary cascade system e˙ = −c0 e, ψ˙ = q¯(e, ψ, t),

(7) q¯(0, 0, t) = 0,

(8)

the function q¯(e, ψ, t) is continuously diﬀerentiable in O × R1+ , the Jacobi matrix ∂ q¯(e, ψ, t)/∂(e, ψ) is bounded in norm uniformly with respect to t in D × R1+ , where D and O are closed bounded and open neighborhoods, respectively, of the point (e, ψ) = (0, 0), D ⊂ O , system (5) of zero dynamics is uniformly asymptotically stable at the point ψ = 0, and the linear subsystem (7) is asymptotically stable at the point e = 0, then the dynamical system (7), (8) is uniformly asymptotically stable at the equilibrium (e, ψ) = (0, 0).

3. STABILIZATION PROBLEM AND DESIGN OF MINIMAL-PHASE SYSTEMS If the equilibrium ψ = 0 of equations (5) of zero dynamics is not uniformly asymptotically stable, then the problem on the stabilization of the programmed motion remains open, since a control of the form (6) cannot be stabilizing. A method for designing a state feedback stabilizing an equilibrium of a stationary nonminimalphase system was suggested in [5, 6] under the assumption that the complete state vector of the system is known. The method is, to ﬁnd a function, referred to as a virtual output, with respect to which the system has a given relative degree and is a minimal-phase system, which permits one to design a stabilizing feedback reducing the system to a cascade form with an asymptotically stable zero equilibrium. We generalize this method to the case of nonstationary systems. System (3) is the normal form of a nonstationary system with the output y¯ = e. We assume that system (3) is not a uniformly minimal-phase system; i.e., its zero dynamics is not uniformly asymptotically stable. We design a virtual output θ(e, ψ, t) such that system (3) with it has the relative degree = 1 and the uniformly asymptotically stable zero dynamics. If such an output exists, then, for the solution of the stabilization problem for the zero equilibrium of system (3), the system can be represented in normal form with respect to this virtual output. If the resulting system closed by a control of the form (6) satisﬁes the assumptions of Theorem 1, then its zero equilibrium is uniformly asymptotically stable. Existence conditions for a virtual output with the desired properties are provided by the following assertion. Theorem 2. System (3) has a virtual output with relative degree = 1 at the point (e, ψ) = (0, 0) and uniformly asymptotically stable zero dynamics if and only if the equilibrium point ψ = 0 of the nonlinear system ψ˙ = q¯(v, ψ, t) (9) DIFFERENTIAL EQUATIONS

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with the control v can be stabilized uniformly by a smooth feedback v = v(ψ, t), v(0, t) = 0, t ≥ 0. To each stabilizing feedback in system (9), there corresponds a virtual output θ = e − v(ψ, t) of system (3) with relative degree = 1 and with uniformly asymptotically stable zero dynamics. Necessity. Let some virtual output θ = θ(e, ψ, t), θ(0, 0, t) = 0, of the aﬃne system (3) have the relative degree = 1 at the point (e, ψ) = (0, 0) for t ≥ 0, and let the corresponding zero dynamics be uniformly asymptotically stable. Let us rewrite system (3) with this output in normal form. To this end, in system (3), we make a change of variables preserving time by setting e¯ = θ(e, ψ, t),

ψ¯ = ψ,

t = t.

(10)

Let us show that relations (10) is indeed a nondegenerate smooth time-preserving change of variables in a neighborhood of the point (e, ψ) = (0, 0). To this end, it suﬃces to show that ∂¯ e(0, 0, t)/∂e = 0. Since the relative degree of the aﬃne system (3) with output θ is equal to unity, it follows that the Lie derivative Lg¯ θ(e, ψ, t) is nonzero at the point (e, ψ) = (0, 0) for t ≥ 0, and at this point, it can ∂θ (0, 0, t)¯ g (0, 0, t). Consequently, be represented by the product ∂e ∂θ(0, 0, t) ∂¯ e(0, 0, t) = = 0. ∂e ∂e ¯ t , ψ = ψ, ¯ t = t permits one to rewrite system (3) The inverse change of variables e = θ −1 e¯, ψ, with output θ in the variables (10) in the form ¯ t , ψ, ¯ t , ¯ t + g˜ e¯, ψ, ¯ t δu, (11) e¯˙ = f˜ e¯, ψ, ψ˙¯ = q¯ θ −1 e¯, ψ, where

¯ t = θ (e, ψ, t)f¯(e, ψ, t) + θ (e, ψ, t)¯ q (e, ψ, t) + θt (e, ψ, t) e=θ−1 (¯e,ψ,t), f˜ e¯, ψ, e ψ ¯ ¯ t=t , ψ=ψ, ¯ t = (θ (e, ψ)¯ g (e, ψ, t))|e=θ−1 (¯e,ψ,t), g˜ e¯, ψ, ¯ ¯ t=t . e ψ=ψ,

System (11) with t ≥ 0 is a normal form of the nonstationary aﬃne system (3) with the virtual output θ. By setting e¯ ≡ 0 in the second equation, we obtain the system of equations of the zero dynamics, ¯ t , ψ, ¯ t , ψ˙¯ = q¯ θ −1 0, ψ, whose equilibrium point ψ¯ = 0 is uniformly asymptotically stable by virtue of the above-stipulated assumptions. The latter can be interpreted as follows: the control v = θ −1 (0, ψ, t) = v(ψ, t) uniformly stabilizes the equilibrium point ψ = 0 of system (9). Suﬃciency. Let the feedback v = v(ψ, t), v(0, t) = 0, uniformly stabilize the equilibrium point ψ = 0 of system (9). For the aﬃne system (3), we consider the virtual output that has the form y = θ(e, ψ, t) = e − v(ψ, t). If t ≥ 0, then the relative degree of this output at the point (e, ψ) = (0, 0) is equal to = 1, since Lg¯ θ(0, 0, t) = g¯(0, 0, t) = 0. The relations e¯ = θ(e, ψ, t) = e − v(ψ, t), ψ¯ = ψ, t=t (12) deﬁne a nondegenerate time-preserving change of variables in Rn . The feedback has the form ¯ t , ¯ e = e¯ + v ψ, ψ = ψ, t = t. (13) In the new variables, the aﬃne system (3) acquires the normal form corresponding to the virtual output θ(e, ψ, t) : ¯ t + g˜ e¯, ψ, ¯ t δu, ¯ t , ψ, ¯ t , e¯˙ = f˜ e¯, ψ, ψ¯˙ = q¯ e¯ + v ψ, (14) where

¯ t = f¯(e, ψ, t) − v (ψ, t)¯ q (e, ψ, t) + vt (ψ, t) e=¯e+v(ψ,t), f˜ e¯, ψ, ψ ¯ ¯ t=t . ψ=ψ, DIFFERENTIAL EQUATIONS

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By setting e¯ ≡ 0 in the second equation of system (14), we obtain the system of equations of zero dynamics, ¯ t , ψ, ¯ t , ψ˙¯ = q¯ v ψ, (15) which, after the change of variables ψ¯ = ψ, coincides with system (9) closed by the stabilizing feedback v = v(ψ, t). Consequently, the equilibrium point ψ¯ = 0 of system (15) of zero dynamics is uniformly asymptotically stable, which completes the proof of the theorem. Suppose that for system (3) there exists a virtual output θ = θ(e, ϕ, t) with a uniformly asymptotically stable zero dynamics. If the assumptions of Theorem 1 are valid for system (11), then the control ¯ t , ¯ t + c0 e¯ g˜ e¯, ψ, (16) δu = − f˜ e¯, ψ, where c0 > 0, uniformly stabilizes the equilibrium e¯, ψ¯ = (0, 0) of system (11). Under additional assumptions, one can show that this control uniformly asymptotically stabilizes the zero equilibrium of system (3). Conditions providing that the stability of the equilibrium is preserved under a time-preserving nonstationary change of variables are presented in [10, p. 157]. The following assertion provides conditions under which the uniform asymptotic stability of the equilibrium is preserved. Theorem 3. Let the equilibrium s¯ = 0 of the system s¯˙ = p (¯ s, t) , s ∈ Rn , be uniformly asymptotically stable, and let s¯ = F (s, t), F (0, t) = 0, be a suﬃciently smooth invertible time-preserving change of variables deﬁned in a neighborhood of the point s = 0 for t ≥ 0. If the mapping F (s, t) s, t) satisis continuous uniformly with respect to t in a neighborhood of the point s = 0 and F −1 (¯ ﬁes the local Lipschitz condition uniformly with respect to t in a neighborhood of the point s¯ = 0, then the equilibrium s = 0 of the system s˙ = (∂F/∂s)−1 (p(F (s, t), t) − ∂F/∂t) is uniformly asymptotically stable. Proof. Let us prove the Lyapunov stability of the equilibrium s = 0. Let some ε1 ≥ 0 be given. Consider the neighborhood s < ε1 of the point s = 0. Note that the Lipschitz property (uniform with respect to t) of the mapping F −1 implies that it is uniformly continuous with respect to t in a neighborhood of zero. Since the mapping F −1 is uniformly continuous with respect to t, it follows s < ε2 , then s < ε1 . By virtue of the stability of the that there exists ε2 > 0 such that if ¯ s (t0 ) < δ2 , then ¯ s(t) < ε2 for equilibrium s¯ = 0, for each ε2 , there exists δ2 > 0 such that if ¯ t ≥ t0 . Finally, since the mapping F is uniformly continuous with respect to t, we ﬁnd that there s < δ2 . exists δ1 > 0 such that if s < δ1 , then ¯ For ε1 > 0, we have thereby found δ1 > 0 such that if s (t0 ) < δ1 , then s(t) < ε1 for t ≥ t0 . The proof of stability is complete. Since the mapping F −1 satisﬁes the local Lipschitz condition uniformly with respect to t in a neighborhood of s = 0, we have s(t), t) ≤ L ¯ s(t) , s(t) = F −1 (¯ s(t) = 0, we have limt→∞ s(t) = 0, which where L is the Lipschitz constant. From limt→∞ ¯ implies the asymptotic stability of the equilibrium s = 0. The equilibrium s¯ = 0 is uniformly asymptotically stable; i.e., the choice of δ2 is independent of t0 . Then, by virtue of the uniform, with respect to t, continuity of the mapping F −1 , the choice of δ1 is also independent of t0 , which implies the uniform asymptotic stability of the equilibrium s = 0 and completes the proof of the theorem. Let us apply Theorem 3 to the above-considered stabilization problem for the zero equilibrium of system (3). The direct and inverse changes of variables have the form (12) and (13), respectively, and are nondegenerate in a neighborhood of the zero equilibrium. By virtue of the special form of the change of variables, the assumptions of Theorem 3 are satisﬁed if v(ψ, t) satisﬁes the local Lipschitz condition uniformly with respect to t in a neighborhood of the point ψ = 0. In this case, the zero equilibrium of system (3) closed by the feedback (16) is uniformly asymptotically stable. DIFFERENTIAL EQUATIONS

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4. EXAMPLE OF STABILIZATION OF A PROGRAMMED MOTION Consider the dynamical system of the Lotka–Volterra type with a control, z˙ = k1 z − a1 zη + u,

η˙ = −k2 η + a2 zη,

(17)

where k1 , k2 , a1 , a2 > 0. For the virtual output of the system; we take y = z. The relative degree of this output in Rn is constant and equal to unity, and the system is represented in normal form with respect to this output. For system (17), the problem on the stabilization of the programmed motion induced by a given change of the virtual output: (18) y ∗ (t) = z ∗ (t) = a + b sin(ωt), where a > b > 0. It corresponds to the programmed trajectory η ∗ (t) = d exp (a3 cos(ωt))

(19)

with respect to the variable η, where d = η(0) (cosh (a3 ) + sinh (a3 )) and a3 = (a2 b)/ω. If η(0) > 0, then η ∗ (t) > 0 for t ≥ 0. On the plane (z, η), the programmed trajectory forms a cycle. The programmed control u∗ (t) can be found by evaluating the derivative z˙ ∗ (t) from (18) and by substituting it, together with (18) and (19), into the ﬁrst equation in system (17). We do not represent the analytic expression of u∗ (t), since it is quite cumbersome. By using (4), we ﬁnd the system in deviations e˙ = f¯(e, ψ, t) = (k1 − a1 η ∗ (t) − a1 ψ) e − a1 z ∗ (t)ψ + δu, ψ˙ = q¯(e, ψ, t) = −k2 ψ + a2 (z ∗ (t) + e) ψ + a2 eη ∗ (t).

(20)

A straightforward veriﬁcation shows that since z ∗ (t) and η ∗ (t) are bounded for t ≥ 0, it follows that the norm of the Jacobi matrix ∂ q¯(e, ψ, t)/∂(e, ψ) is uniformly bounded in any closed bounded neighborhood of the zero equilibrium. The equation of zero dynamics of system (20) has the form ψ˙ = q¯(0, ψ, t) = − (k2 − a2 (a + b sin(ωt))) ψ, and is not uniformly asymptotically stable for k2 − a2 (a + b) < 0. To ﬁnd a virtual output whose relative degree is equal to unity and the corresponding zero dynamics is uniformly asymptotically stable, we use Theorem 2. Consider a system of the form (9) with the control v : (21) ψ˙ = −k2 ψ + a2 (z ∗ (t) + v) ψ + a2 vη ∗ (t). We design a feedback v = v(ψ, t) such that system (21) closed by it has the form ψ˙ = −c1 ψ − k2 ψ + a2 z ∗ (t)ψ. Since maxt≥0 {z ∗ (t)} ≤ a + b, it follows that the zero equilibrium of the last equation is uniformly asymptotically stable for c1 > max {0, a2 (a + b) − k2 }. Note that if η(0) > 0, then η1∗ ≥ η ∗ (t) ≥ η2∗ > 0, where η1∗ = d exp (a3 ) and η2∗ = d exp (−a3 ). Consider the neighborhood U0 = {ψ : ψ < η2∗ } of the point ψ = 0. The feedback v(ψ, t) = −

c1 ψ a2 (ψ + η ∗ (t))

is deﬁned in this neighborhood. The equilibrium η = 0 of system (21) closed by this feedback is locally uniformly asymptotically stable. In addition, the derivative v(ψ, t)ψ is continuous in U0 ×R1+ and, by virtue of the boundedness of η ∗ (t), is bounded uniformly with respect to t in any closed neighborhood U1 ⊂ U0 of the point ψ = 0. Consequently, the above-represented feedback satisﬁes the local Lipschitz condition uniformly with respect to t. DIFFERENTIAL EQUATIONS

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By Theorem 2, the virtual output θ(e, ψ, t) = e +

c1 ψ a2 (ψ + η ∗ (t))

has the relative degree 1, and the normal form of system (20) written out for this output has uniformly asymptotically stable zero dynamics. By evaluating the derivative of θ(e, ψ, t) along the trajectories of system (20), we obtain q (e, ψ, t)η ∗ (t) − ψ η˙ ∗ (t)) ˙ ψ, t) = f¯(e, ψ, t) + c1 (¯ θ(e, + δu. 2 a2 (ψ + η ∗ (t)) Then the feedback c1 (¯ q (e, ψ, t)η ∗ (t) − ψ η˙ ∗ (t)) , δu = −c2 θ(e, ψ, t) − f¯(e, ψ, t) − 2 a2 (ψ + η ∗ (t)) where c2 > 0, uniformly stabilizes the zero equilibrium of system (20) and hence the programmed motion (z ∗ (t), η ∗ (t), u∗ (t)) given by (18) and (19). CONCLUSION The method for solving the uniform stabilization problem for the zero equilibrium of a nonstationary system is a generalization of the stabilization method with the use of virtual outputs suggested in [5] for stationary systems. The same stabilizing control in the stationary case can be obtained by the method based on the use of a virtual control [8, p. 119 of the Russian translation] for the stabilization of an equilibrium of cascade systems. Note that the uniform stabilization of the programmed motion of system (1) obtained by a nonstationary feedback is guaranteed only in the variables of the normal form (2); however, this is often acceptable in applications. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 05-01-840) and the program OITVS of the Russian Academy of Sciences “Foundations of Information Technologies and Computational Systems.” REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Zhevnin, A.A. and Krishchenko, A.P., Dokl. Akad. Nauk , 1981, vol. 258, no. 4, pp. 805–809. Krishchenko, A.P., Izv. Akad. Nauk Ser. Tekh. Kibern., 1985, no. 6, pp. 103–112. Krishchenko, A.P., Tr. Bauman Mosk. Vyssh. Tekh. Uchil., 1988, no. 512, pp. 69–87. Isidori, A., Nonlinear Control Systems, London, 1995. Krishchenko, A.P., Panﬁlov, D.Yu., and Tkachev, S.B., Diﬀer. Uravn., 2002, vol. 38, no. 11, pp. 1483–1489. Krishchenko, A.P., Panﬁlov, D.Yu., and Tkachev, S.B., Diﬀer. Uravn., 2003, vol. 39, no. 11, pp. 1503–1510. Tkachev, S.B., Vestnik Moskov. Gos. Tekh. Univ. Ser. Estestv. Nauki, 2006, no. 4, pp. 43–60. Fradkov, A.L., Miroshnik, I.V., and Nikiforov, V.O., Nonlinear and Adaptive Control of Complex Systems, Mathematics and Its Applications, vol. 491, Dordrecht: Kluwer, 1999. Translated under the title Nelineinoe i Adaptivnoe Upravlenie Slozhnymi Sistemami, St. Petersburg, 2000. Krasnoshchechenko, V.I. and Krishchenko, A.P., Nelineinye sistemy: geometricheskie metody analiza i sinteza (Nonlinear Systems: Geometric Methods of Analysis and Synthesis), Moscow: Mosk. Gos. Tekh. Univ., 2005. Zhuravlev, V.F., Osnovy teoreticheskoi mekhaniki (Foundations of Theoretical Mechanics), Moscow: Fizmatlit, 2001.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 774–796. c A.A. Amosov, I.A. Goshev, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 760–779. Original Russian Text

PARTIAL DIFFERENTIAL EQUATIONS

Global Unique Solvability of the Longitudinal Vibration Equations of the Ishlinskii Viscoelastoplastic Material A. A. Amosov and I. A. Goshev Moscow Institute of Power Engineering (State University), Moscow, Russia Received March 13, 2006

DOI: 10.1134/S0012266107060055

1. STATEMENT OF THE PROBLEMS In the present paper, we prove the existence and uniqueness of global generalized solutions of initial-boundary value problems for the system of quasilinear operator-diﬀerential equations Dt e = Du in Q, Dt u = Dσ + g [xe ] in Q, ν[η] Du + σel [e] + F s0 , e, μ , σ= η Dt xe = u in Q

(1.1) (1.2) η = e+1

in Q,

(1.3) (1.4)

describing longitudinal vibrations of the Ishlinskii viscoelastoplastic material. The unknown vector function z(x, t) = (e(x, t), u(x, t), σ(x, t), xe (x, t)) is deﬁned on Q = QT = Ω × (0, T ), where Ω = (0, X). Here we have used the notation Dt = ∂/∂t, D = ∂/∂x; in addition, ν[η](x, t) = ν(η(x, t), x),

σel [e](x, t) = σel (e(x, t), x),

g [xe ] (x, t) = g (xe (x, t), x, t) ,

and F is the Prandtl–Ishlinskii hysteresis operator, whose deﬁnition and properties will be given in Section 3. Recall the physical meaning of the variables: x is the Lagrangian coordinate, t is time, is the undeformed material density, e is the strain, u is the velocity, xe is the Eulerian coordinate, σ is the stress, ν is the viscosity constant, σel [e] and F [s0 , e, μ] are the elastic and plastic stress components, and g is the external force density. Note that, by its physical meaning, the strain satisﬁes the inequality e > −1. A theory of plasticity in which the stress-strain relation is described by a hysteresis operator F was developed by Prandtl [1] and Ishlinskii [2]. The rigorous study of hysteresis operators has started relatively recently [3–7]. We supplement system (1.1)–(1.4) with the initial conditions e|t=0 = e0 (x),

u|t=0 = u0 (x),

xe |t=0 = x on

Ω

(1.5)

and one of the pairs of boundary conditions u|x=0 = u0 (t), σ|x=0 = σ0 (t), σ|x=0 = σ0 (t),

u|x=X = uX (t) on (0, T ), u|x=X = uX (t) on (0, T ), σ|x=X = σX (t) on (0, T ). 774

(1.61 ) (1.62 ) (1.63 )

GLOBAL UNIQUE SOLVABILITY OF THE LONGITUDINAL VIBRATION EQUATIONS

775

Thus, we simultaneously consider three initial-boundary value problems. We denote problem (1.1)–(1.5), (1.6m ) by Pm , m = 1, 2, 3. It is convenient to assume that the functions u0 , uX , σ0 , and σX are deﬁned for all m; moreover, the functions not occurring in the boundary conditions of a speciﬁc problem Pm are zero. A problem of type P2 not taking into account viscosity (i.e., for ν = 0) was studied in [6, 8]. The ﬁrst existence theorem for a global generalized solution of problem P1 in the case of a homogeneous Ishlinskii material and under some additional simplifying assumptions was established in [9]. Just as in [10–13], the proof of the existence of global generalized solutions is based on the use of a special semidiscrete method. The proof of the uniqueness is performed with the use of the technique and results in [14]. Section 2 contains some auxiliary results. In Section 3, we introduce the Prandtl–Ishlinskii operator and prove some of its properties. Section 4 deals with the statement of the main results of the present paper, namely, existence and uniqueness theorems for generalized solutions of problems Pm . The remaining part of the paper contains the proof of these theorems. In Section 5, we introduce problems Pmh , which are semidiscrete approximations to problems Pm . Global a priori estimates for their approximate solutions are derived in Section 6. They are used in Section 7 to justify the solvability of problems Pmh . In Section 8, after passing to the limit as h → 0, we prove the solvability of problems Pm . The uniqueness of the generalized solutions is proved in Section 9. 2. SOME NOTATION AND AUXILIARY RESULTS Throughout this section, B is a Banach space. Let w ∈ L1 (0, T ; B), v ∈ L1 (Ω; B), τ ∈ (0, T ), and δ ∈ (0, X). We set Δ(τ ) w(t) = w(t + τ ) − w(t) for t ∈ (0, T − τ ), Δ(τ ) w(t) = 0 for t ∈ (0, T − τ ), (Δδ v) (x) = v(x + δ) − v(x) for x ∈ (0, X − δ), and (Δδ v) (x) = 0 for x ∈ (0, X − δ). For f ∈ Lq (Ω; B), 1 ≤ q < ∞, we introduce the modulus of continuity ωq,δ (f, B) = sup Δα f Lq (Ω;B) 0<α<δ

and the averaging

1−x/X

fh (x) =

f (x + αh)dα. −x/X

¯ B . For f ∈ Lq (Ω), 1 ≤ q < ∞, we set ωq,δ (f ) = sup0<α<δ Δα f Obviously, fh ∈ C Ω; Lq (Ω) . Lemma 2.1. Let f ∈ Lq (Ω; B). Then one has the estimate fh − f Lq (Ω;B) ≤ 21/q ωq,h (f, B). Proof. Note that q

fh − f Lq (Ω;B)

(2.1)

q 1−x/X = (f (x + αh) − f (x))dα dx Ω −x/X B

1−x/X

f (x + αh) − f (x)qB dα dx

≤ Ω −x/X

1 X−αh f (x + αh) − f (x)qB dx dα ≤2 0

0 q

≤ 2 sup Δαh f Lq (Ω;B) = 2ωq,h (f, B)q . 0<α<1

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We need the Lebesgue spaces Lq,r (Q) of measurable functions on Q with ﬁnite norm anisotropic wLq,r (Q) = wLq (Ω) Lr (0,T ) , q, r ∈ [1, ∞]. Note that Lq,r (Q) = Lr (0, T ; Lq (Ω)) for q = ∞. We also use the spaces Lq (Ω; C[0, T ]) and C ([0, T ]; Lq (Ω)). One can readily see that Lq (Ω; C[0, T ]) ⊂ C ([0, T ]; Lq (Ω)) moreover,

for

f C([0,T ];Lq (Ω)) ≤ f Lq (Ω;C[0,T ])

1 ≤ q < ∞;

∀f ∈ Lq (Ω; C[0, T ]).

(2.2)

Note that C ([0, T ]; L∞ (Ω)) ⊂ L∞ (Ω; C[0, T ]); moreover, f L∞ (Ω;C[0,T ]) = f C([0,T ];L∞(Ω))

∀f ∈ C ([0, T ]; L∞ (Ω)) .

(2.3)

Theorem 2.1 (a precompactness criterion in Lq (Ω; C[0, T ])). A family F ⊂ Lq (Ω; C[0, T ]), 1 ≤ q < ∞, of functions is precompact if and only if it is bounded in Lq (Ω; C[0, T ]) and has the following properties of equicontinuity with respect to shifts : lim sup Δ(τ ) f C([0,T ];L1 (Ω)) = 0. (2.4) lim sup Δδ f Lq (Ω;C[0,T ]) = 0, τ →0 f ∈F

δ→0 f ∈F

Proof. Since the space Lq (Ω; C[0, T ]) is complete, it follows that F is precompact if and only if it has a ﬁnite ε-net [15, p. 109] for any ε > 0. Let F be bounded, and let condition (2.4) be satisﬁed. To a function f ∈ F , we assign its ¯ C[0, T ] = C Q ¯ . By virtue of the estimate (2.1), for each ε > 0, there exists averaging fh ∈ C Ω; an h such that the set Fh = {fh | f ∈ F } is an (ε/2)-net for F . In turn, if 1 < q < ∞, then the estimates (τ ) −1/q Δ fh ¯ ≤ h−1 Δ(τ ) f f Lq (Ω;C[0,T ]) , , (2.5) fh C(Q) ¯ ≤ h C(Q) C([0,T ];L1 (Ω)) −1/q Δδ f Lq (Ω;C[0,T ]) + 2(δ/X)1−1/q f Lq (Ω;C[0,T ]) (2.6) Δδ fh C(Q) ¯ ≤ h ¯ . Therefore, by the Arzel´ equicontinuous in C Q a–Ascoli imply that the family Fh is boundedand ¯ lemma, Fh is a precompact set in C Q and all the more in Lq (Ω; C[0, T ]). Therefore, for Fh , there exists a ﬁnite (ε/2)-net, which is an ε-net for F . Therefore, F is precompact. If q = 1, then inequality (2.6) should be replaced by the inequality 2

Δδ fh L2 (Ω;C[0,T ]) ≤ Δδ fh C(Q) ¯ Δδ fh L1 (Ω;C[0,T ])

≤ 2h−1 (X − h)−1 f L1 (Ω;C[0,T ]) X Δδ f L1 (Ω;C[0,T ]) + 2δf L1 (Ω;C[0,T ]) .

(2.7)

The above-performed considerations, together with the estimates (2.5) and (2.7), imply that the set Fh is precompact in L2 (Ω; C[0, T ]) and all the more in L1 (Ω; C[0, T ]). Therefore, F is also precompact for q = 1. Let F be a precompact family. Then F is bounded in Lq (Ω; C[0, T ]), and for each ε > 0 for F , there exists a ﬁnite ε-net F ε = {f1 , . . . , fN } ⊂ Lq (Ω; C[0, T ]). It follows that, for each f ∈ F , there exists an element fk ∈ F ε such that f − fk Lq (Ω;C[0,T ]) ≤ ε. Note also that, by virtue of (2.2), f − fk C([0,T ];L1 (Ω)) ≤ X 1−1/q ε. Therefore, lim sup Δδ f Lq (Ω;C[0,T ]) ≤ lim max Δδ fk Lq (Ω;C[0,T ]) + 2ε = 2ε, δ→0 1≤k≤N (τ ) lim sup Δ f C([0,T ];L1 (Ω)) ≤ lim max Δ(τ ) fk C([0,T ];L1 (Ω)) + 2X 1−1/q ε = 2X 1−1/q ε. δ→0 f ∈F

τ →0 f ∈F

τ →0 1≤k≤N

Since ε is arbitrary, it follows that properties (2.4) hold. The proof of the theorem is complete. DIFFERENTIAL EQUATIONS

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Let h = hn = X/n, n ≥ 2. We introduce the nodes xi = ih, 0 ≤ i ≤ n, and = (i − 1/2)h, xi−1/2 = (x , x ), 1 ≤ i ≤ n. We also set Ω = 0, x = x , Ω 1 ≤ i ≤ n, and set Ωi−1/2 i−1 i 0 1/2 i i−1/2 , xi+1/2 , 1 ≤ i < n, and Ωn = xn−1/2 , X . Let us introduce the following spaces of piecewise constant functions ranging in B and R : S h (Ω) = S h (Ω; R), S h (Ω; B) = {w | w(x) = wi ∈ B on Ωi , 0 ≤ i ≤ n} ,

h h h (Ω; B) = v | v(x) = vi−1/2 ∈ B on Ωi−1/2 , 1 ≤ i ≤ n , S1/2 (Ω) = S1/2 (Ω; R). S1/2 h : L1 (Ω; B) → We introduce the projection operators π h : L1 (Ω; B) → S h (Ω; B) and π1/2 that take each function f ∈ L1 (Ω; B) to the piecewise constant functions equal on to the corresponding mean values the sets Ωi (0 ≤ i ≤ n) and Ωi−1/2 (1 ≤ i ≤ n), xi respectively, h f (x) = h−1 xi−1 f (x ) dx for x ∈ Ωi−1/2 , 1 ≤ i ≤ n. of the function f . For example, π1/2

h S1/2 (Ω; B)

h f . Then the Lemma 2.2. Let f ∈ Lq (Ω; B), 1 ≤ q < ∞, and let f h = π h f or f h = π1/2 inequalities h f − f ≤ 21/q ωq,h (f, B), ωq,δ f h , B ≤ ωq,δ+h (f, B) (2.8) Lq (Ω;B)

hold, which imply that lim f h − f Lq (Ω;B) = 0,

h→0

lim sup ωq,δ f h , B = 0.

δ→0

(2.9)

h

Proof. The ﬁrst estimate in (2.8) and properties (2.9) are proved in [10]. Let us prove the h f . We set δ1 = [δ/h]h, δ2 = δ1 + h, and β = δ2 − δ. Note that second estimate in (2.8) for f h = π1/2 xi h −1 xi Δδ1 f (x ) dx for x ∈ (xi−1 , xi−1 + β) and Δδ f h (x) = h−1 xi−1 Δδ2 f (x ) dx for Δδ f (x) = h xi−1 x ∈ (xi−1 + β, xi ). Therefore, ⎡ q q ⎤ xi xi n ⎢ −1 −1 ⎥ Δδ f h q = Δ f (x ) dx + (h − β) h Δ f (x ) dx β h ⎣ δ δ 1 2 ⎦ Lq (Ω;B) i=1 xi−1

B

xi−1

B

h−β β q q q Δδ1 f Lq (Ω;B) + Δδ2 f Lq (Ω;B) ≤ sup Δα f Lq (Ω;B) h h 0<α<δ+h = ωq,δ+h (f, B)q .

≤

T We set (v, w)(0,T ) = 0 vw dt, (v, w)Ω = Ω vw dx, and (v, w)Q = Q vw dx dt. Note the identities h h h π1/2 ψ, w Ω = ψ, π1/2 w Ω ∀ψ, w ∈ L1 (Ω), π ψ, w Ω = ψ, π h w Ω , (2.10) h h π1/2 ψ, v Ω = (ψ, v)Ω π ψ, w Ω = (ψ, w)Ω , (2.11) h ∀ψ ∈ L1 (Ω), ∀w ∈ S h (Ω), ∀v ∈ S1/2 (Ω), which are used in forthcoming considerations without stipulation. 3. THE PRANDTL–ISHLINSKII OPERATOR ¯ + = [0, +∞). Let a parameter r ∈ R ¯ + , a number s0 ∈ [−r, r], and Let R+ = (0, +∞) and R 1 a function e ∈ W1 (0, T ) be given. Consider the following problem: ﬁnd a function s ∈ W11 (0, T ) satisfying the conditions s(t) ∈ [−r, r] ∀t ∈ [0, T ], (Dt s(t) − Dt e(t)) (φ − s(t)) ≥ 0 ∀φ ∈ [−r, r] for almost all s(0) = s0 . DIFFERENTIAL EQUATIONS

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t ∈ (0, T ),

(3.1) (3.2) (3.3)

778

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There exists a unique solution of this problem [3–6]. The operator Sr : [−r, r] × W11 (0, T ) → W11 (0, T ) that takes the data (s0 , e) to the solution s = Sr [s0 , e] of problem (3.1)–(3.3) is called the stop operator [4–6], or simply the stop [3]. The model of a perfectly plastic material in which the stress-strain relation is given by the formula σ = ESr [s0 , e] was introduced by Prandtl [1]. Here E is the modulus of elasticity, r is the yield strength, and s0 is the residual stress. This model has been comprehensively studied (e.g., see [3–6]). In particular, the following results were obtained. Lemma 3.1. One has the inequalities Dt Sr s0 , e (t) ≤ |Dt e(t)| for almost all t ∈ (0, T ), 1 0 2 Dt Sr s , e (t) ≤ Sr s0 , e (t)Dt e(t) for almost all t ∈ (0, T ). 2 ¯ + . The estimate Lemma 3.2. Let r1 , r2 ∈ R Sr1 s01 , e1 − Sr2 s02 , e2 C[0,t] ≤ max |r1 − r2 | , s01 − s02 + 2 e1 − e2 C[0,t]

(3.4) (3.5)

∀t ∈ [0, T ]

(3.6)

is valid for arbitrary numbers s01 ∈ [−r1 , r1 ] and s02 ∈ [−r2 , r2 ] and functions e1 , e2 ∈ W11 (0, T ). Corollary 3.1. The stop operator admits extension to an operator Sr : [−r, r] × C[0, T ] → C[0, T ] with preservation of the estimate (3.6). To describe the stress-strain relation in real plastic materials, Prandtl [1] and Ishlinskii [2] suggested a model in which the stress and the strain are related by the formula σ = E∞ e +

n

Ei Sri s0 (ri ) , e ;

i=1

to represent here 0 < r1 < · · · < rn , 0 ≤ E∞ , 0 < Ei , and |s0 (ri )| ≤ ri , 1 ≤ i ≤ n. It is convenient ∞ 0 S [s (r), e] dμ(r), the second term of this formula in the form of the Riemann–Stieltjes integral r 0 n where μ(r) = i=1 Ei χi (r), χi (r) = −1 for r < ri and χi (r) = 0 for r ≥ ri . + ¯ + with the norm ¯ be the Banach space of bounded continuous functions on R Let C R sC(R¯ + ) = sup |s(r)|. r≥0

We set

+ +

¯+ . ¯ = s∈C R ¯ | s(0) = 0, |s (r1 ) − s (r2 )| ≤ |r1 − r2 | ∀r1 , r2 ∈ R Lip1 R + ¯ ¯+ of nondecreasing right continuous functions μ deﬁned on R We also introduce the set M1 R + and satisfying the condition μ ∈ L1 (R ). Lemma 3.3. Let f ∈ L1 (1, ∞) be a nonnegative nonincreasing function on (1, ∞). Then the function f (r) = o (r −1 ) as r → ∞. ∞

Proof. Suppose the contrary. Then there exist a constant c > 0 and a sequence {rn }n=1 ⊂ (1, +∞) such that limn→∞ rn = ∞, 2rn < rn+1 , and f (rn ) ≥ crn−1 for all n ≥ 1. But then ∞ 1

r ∞ n+1 ∞ ∞ 1 1 rn = +∞. f (r)dr ≥ c dr = c 1− ≥c rn+1 rn+1 2 n=1 n=1 n=1 rn

The resulting contradiction completes the proof of the lemma. DIFFERENTIAL EQUATIONS

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+ ¯ . Then μ(r) = o (r −1 ) as r → +∞, and ∞ r dμ(r) = Corollary 3.2. Let μ ∈ M1 R 0 μL1 (R+ ) . + ¯ , e ∈ C[0, T ]. Then for all t ∈ [0, T ], there exists a generalized Lemma 3.4. Let s0 ∈ Lip1 R ∂ Sr s0 (r), e (t) ∈ L∞ R+ ; moreover, derivative ∂r ∂ Sr s0 (r), e (t) ≤ 1. (3.7) ∂r L∞ (R+ ) Proof. It follows from the estimate (3.6) that Sr [s0 (r), e] (t) satisﬁes the Lipschitz condition Sr1 s0 (r1 ) , e (t) − Sr2 s0 (r2 ) , e (t) ≤ |r1 − r2 | . Therefore, the assertion of the lemma follows from the Rademacher theorem [17, p. 65 of the Russian translation]. + + ¯ × C[0, T ] × M1 R ¯ → C[0, T ] given by the relation The operator F : Lip1 R

F

∞

s0 , e, μ (t) =

Sr

s0 (r), e (t)dμ(r)

for t ∈ [0, T ]

(3.8)

0

is referred to as the Prandtl–Ishlinskii operator . Since |Sr [s0 (r), e] (t)| ≤ r, we have 0 F s , e, μ (t) ≤ C[0,T ]

∞ r dμ(r) = μL1 (R+ ) .

(3.9)

0

Integration by parts with the use of the asymptotics μ(r) = o (r −1 ), r → +∞, gives the formula

F

∞

s , e, μ (t) = − 0

∂ Sr s0 (r), e (t)μ(r)dr. ∂r

(3.10)

0

+ + ∞ 2 ¯ × W 1 (0, T ) × M1 R ¯ and 0 [s0 (r)] dμ(r) < +∞. Lemma 3.5. Let (s0 , e, μ) ∈ Lip1 R 1 Then ∞ 1 0 2 Sr s , e (t) dμ(r) 2 0

1 ≤ 2

∞

0 2 s (r) dμ(r) +

0

t

F

s0 , e, μ (t ) Dt e (t ) dt

∀t ∈ [0, T ].

(3.11)

0

Proof. Inequality (3.11) can be derived from (3.5) by integration ﬁrst with respect to t and then with respect to r. Lemma 3.6. The estimate 0 F s1 , e1 , μ1 − F s02 , e2 , μ2 C[0,t]

0 0 ≤ |μ1 (0)| s1 − s2 C(R¯ + ) + 2 e1 − e2 C[0,t] + μ1 − μ2 L1 (R+ ) + + ¯ × C[0, T ] × M1 R ¯ . is valid for all (s01 , e1 , μ1 ) , (s02 , e2 , μ2 ) ∈ Lip1 R DIFFERENTIAL EQUATIONS

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∀t ∈ [0, T ]

(3.12)

780

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Proof. By using (3.10), we obtain

F

s01 , e1 , μ1

(t) − F

s02 , e2 , μ2

∞ (t) = 0

Sr

∞

−

s01 (r), e1 (t) − Sr s02 (r), e2 (t) dμ1 (r)

∂ Sr s02 (r), e2 (t) (μ1 (r) − μ2 (r)) dr. ∂r

0

Now inequalities (3.6) and (3.7) imply the estimate (3.12). Note that the estimate (3.12) was given in [3, pp. 244–246] without a complete proof. In the description of deformations of an inhomogeneous Ishlinskii material, the data s0 , e, and μ additionally depend on the space variable x ∈ Ω; i.e., s0 = s0 (x, r), e = e(x, t), and μ = μ(x, r). In this case, the Prandtl–Ishlinskii operator (3.8) acquires the form

F

0

∞

s , e, μ (x, t) =

Sr

s0 (x, r), e(x, ·) (t)dr μ(x, r).

0

We introduce the sets + + ¯ + = s0 ∈ L∞ Ω; C R ¯ ¯ | s0 (x, ·) ∈ Lip1 R L∞ Lip1 Ω × R and

x∈Ω

+ ¯ + = μ ∈ L∞ Ω; L1 R+ | μ(x, ·) ∈ M1 R ¯ L∞ M1 Ω × R for almost all

F

for almost all

x∈Ω

and μ(·, 0) ∈ L∞ (Ω) .

¯ + , e ∈ L1 (Ω; C[0, T ]), and μ ∈ L∞ M1 Ω × R ¯ + . Then Lemma 3.7. Let s0 ∈ L∞ Lip1 Ω × R [s0 , e, μ] ∈ L∞ (Ω; C[0, T ]); moreover, 0 F s , e, μ ≤ μL∞ (Ω;L1 (R+ )) . (3.13) L∞ (Ω;C[0,T ]) h h h s0 , eh = π1/2 e, and μh = π1/2 μ. One can readily see that Proof. We set s0,h = π1/2

+ ¯ , s0,h (x, ·) ∈ Lip1 R

eh (x, ·) ∈ C[0, T ],

¯ ; μh (x, ·) ∈ M1 R

+ h ¯ , e (x, ·) → e(x, ·) in addition (up to the choice of a subsequence), s0,h (x, ·) → s0 (x, ·) in C R h + in C[0, T ], and μ (x, ·) → μ(x, ·) in L1 (R ) as h → 0 for almost all x ∈ Ω. Since F s0,h , eh , μh for each t ∈ [0, T ] is a sequence of simple functions deﬁned on Ω, and since inequality (3.12) implies that 0,h h h F s , e , μ (x, ·) − F s0 , e, μ (x, ·) C[0,T ]

≤ |μ(x, 0)| s0,h (x, ·) − s0 (x, ·)C(R¯ + ) + 2 eh (x, ·) − e(x, ·)C[0,T ] + μh (x, ·) − μ(x, ·)L1 (R+ ) → 0 for almost all x ∈ Ω, it follows that the function F [s0 , e, μ] is measurable on Ω as a function ranging in C[0, T ]. The estimate (3.9) implies the property F [s0 , e, μ] ∈ L∞ (Ω; C[0, T ]) and inequality (3.13). DIFFERENTIAL EQUATIONS

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4. STATEMENT OF THE MAIN RESULTS Let us introduce the integration operators t (It w) (t) =

x

w (t ) dt ,

(Iv)(x) =

0

v (x ) dx .

0

Let V [0, T ] be the space of functions of bounded variation on [0, T ] with the norm wV [0,T ] = sup |w(t)| + var w. [0,T ]

[0,T ]

We also use the Banach spaces V2 (Q) and W (Q) and the Hilbert space S21,1 W (Q) of functions with the norms wV2 (Q) = wL2,∞ (Q) + DwL2 (Q) , wW (Q) = wL2 (Q) + Dt wL2 (Q) + DwL2,∞ (Q) ,

1/2 2 . wS21,1 W (Q) = w2W21 (Q) + Dt DwL2 (Q) ¯ . We introduce Note [18] that the spaces W (Q) and S21,1 W (Q) are compactly embedded in C Q the seminorm w 0,1/2 = sup τ −1/2 Δ(τ ) wL2 (Q) 0<τ
and the norm

wV 1,1/2 (Q) = wV2 (Q) + w 0,1/2 . 2

We introduce the function class N−1 (Q) = e ∈ C ([0, T ]; L∞ (Ω)) | ess inf e(x, t) > −1, Dt e ∈ L2 (Q) (x,t)∈Q

and the functions η (ν(ζ, x)/ζ)dζ,

Λ(η, x) =

L(η, x) =

1

η

e ν(ζ, x)/ζ dζ,

1

E(e, x) =

σel (ζ, x)dζ; 0

the last function has the meaning of potential elastic strain energy. Let us state conditions imposed on the data of problem Pm ; in these conditions, N > 1 is an arbitrarily large parameter. (C1 ) The function ν(η, x) is deﬁned on R+ × Ω. The property ν ∈ L∞ (Ω; C [a−1 , a]) is valid for all a > 1, and the inequalities ∀η ∈ a−1 , a , (4.1) 0 < c0 (a) ≤ ν(η, x) ≤ c¯0 (a) + ¯ ∀η ∈ R (4.2) Λ(η) ≤ Λ(η, x) ≤ Λ(η) ¯ hold for almost all x ∈ Ω, where Λ(η) → −∞ as η → 0+ and Λ(η) → +∞ as η → +∞. In addition, ∀η ∈ R+ for m = 1, (4.3) ν(η, x)η ≤ c1 L2 (η, x) + η + 1 2 + ∀η ∈ R for m = 2, 3 (4.4) η/ν(η, x) ≤ c2 L (η, x) + 1 for almost all x ∈ Ω. The function σel (e, x) is deﬁned on (−1, +∞) × Ω. For all a > 1, the property σel ∈ L∞ Ω; C −1 + a−1 , a DIFFERENTIAL EQUATIONS

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is valid, and the inequalities σel (e, x) ≥ 0 ∀e ∈ [0, +∞), σel (e, x) ≤ 0 ∀e ∈ (−1, 0], |σel (e1 , x) − σel (e2 , x)| ≤ c¯3 (a) |e1 − e2 | ∀e1 , e2 ∈ −1 + a−1 , a

(4.5) (4.6)

hold for all x ∈ Ω; moreover, if m = 1, then |σel (e, x)| (e + 1) ≤ c4 (E(e, x) + 1)

∀e ∈ (−1, +∞).

(4.7)

Condition (4.5) implies that E(e, x) ≥ 0. (C2 ) e0 ∈ L∞ (Ω), u0 ∈ L2 (Ω), and N −1 ≤ η 0 (x) ≡ e0 (x) + 1 ≤ N for almost all x ∈ Ω, 0 u L2 (Ω) ≤ N . If m = 1, then we additionally assume that N −1 ≤ η 0 L1 (Ω) + It (uX − u0 ) on (0, T ). (C3 ) ∈ L∞ (Ω) and N −1 ≤ (x) ≤ N for almost all x ∈ Ω. The function g (xe , x, t) is deﬁned on R × Q and has the property g ∈ L1 (Q; C[−a, a]) forall a > 1. Furthermore, |g (xe , x, t)| ≤ f¯(x, t) for almost all (x, t) ∈ R × Q, where f¯ ∈ L2,1 (Q) and f¯L2,1 (Q) ≤ N . (C4 ) u0 , uX ∈ V [0, T ], σ0 , σX ∈ L2 (0, T ), and u0 V [0,T ] + uX V [0,T ] ≤ N,

σ0 L2 (0,T ) + σX L2 (0,T ) ≤ N.

¯ + , μ ∈ L∞ M1 Ω × R ¯ + , and (C5 ) s0 ∈ L∞ Lip1 Ω × R ∞ 0 2 ≤ N, μL∞ (Ω;L1 (R+ )) ≤ N, s dr μ 0

μ ¯0 = μ(·, 0)L∞ (Ω) ≤ N.

L1 (Ω)

A generalized solution of problem

Pm

(1 ≤ m ≤ 3) is deﬁned as a vector function

z = (e, u, σ, xe ) ∈ N−1 (Q) × V2 (Q) × L2 (Q) × S21,1 W (Q) satisfying the following conditions. 1. Eqs. (1.1), (1.3), and (1.4) are satisﬁed in L2 (Q). 2. The integral identity − (u, Dt ϕ)Q + (σ, Dϕ)Q = u0 , ϕ|t=0 Ω + (g [xe ] , ϕ)Q + (σX , ϕ|x=X )(0,T ) − (σ0 , ϕ|x=0 )(0,T )

(4.8)

¯ such that ϕ|t=T = 0 and, in addition, ϕ|x=0,X = 0 for m = 1 and is valid for all ϕ ∈ C 1 Q ϕ|x=X = 0 for m = 2. 0 3. The initial conditions e|t=0 = e (x) and xe |t=0 = x are satisﬁed in the sense of the spaces ¯ , respectively. C([0, T ]; L∞ (Ω)) and C Q 4. The boundary conditions u|x=0= u0 for m = 1 and u|x=X = uX for m = 1, 2 are satisﬁed in ¯ L2 (0, T ) . the sense of the space C Ω; Remark 4.1. By [14, 16], identity (4.8) is valid if and only if there exists a generalized derivative DIt σ ∈ L2,∞ (Q) and DIt σ = u − u0 − It g [xe ] , (It σ)|x=0 = It σ0 for m = 2, 3, For the generalized solution of problem theorems.

Pm ,

(It σ)|x=X = It σX

for

(4.9) (4.10)

m = 3.

we have the following existence and uniqueness

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Theorem 4.1. Let conditions (C1 )–(C5 ) be satisﬁed. Then there exists a generalized solution of problem Pm (1 ≤ m ≤ 3) satisfying the estimates K0 (N )−1 ≤ e + 1 ≤ K0 (N ), uV 1,1/2 (Q) ≤ K(N ), Dt eL2 (Q) ≤ K(N ),

(4.11) (4.12)

2

It σW (Q) ≤ K(N ), xe S21,1 W (Q) ≤ K(N ), σL2 (Q) ≤ K(N ), 0 ∀q ∈ [2, ∞), ωq,δ (e, C[0, T ]) ≤ K(N ) ωq,δ e , Λ, σel , s0 , μ + δ1/2

(4.13) (4.14)

ωq,δ e0 , Λ, σel , s0 , μ = ωq,δ e0 + ωq,δ (Λ, B0 ) + ωq,δ (σel , B1 ) + ¯ + ωq,δ μ, L1 R+ , + ωq,δ s0 , C R −1 B0 = C a−1 0 , a0 , B1 = C a0 − 1, a0 − 1 , and a0 = K0 (N ). where

Here and throughout the following, K(N ), possibly, with indices, stand for positive nondecreasing functions of the parameter N (we omit N in the proofs): they can also depend on X, T , c0 , c¯0 , ¯ c1 , c2 , c¯3 , c4 , Λ, and Λ. Theorem 4.2. Let conditions (C1 )–(C5 ) be satisﬁed. In addition, suppose that the inequality ∀η1 , η2 ∈ a−1 , a (4.15) |ν (η1 , x) − ν (η2 , x)| ≤ c¯0 (a) |η1 − η2 | is valid for all a > 1 and for almost all x ∈ Ω, and |g (xe,1 , x, t) − g (xe,2 , x, t)| ≤ ba (x, t) |xe,1 − xe,2 |

∀xe,1 , xe,2 ∈ [−a, a]

(4.16)

for almost all (x, t) ∈ Q, where ba ∈ L2,1 (Q). Then the generalized solution of problem (1 ≤ m ≤ 3) is unique.

Pm

The remaining part of the present paper deals with the proof of Theorems 4.1 and 4.2. 5. SEMIDISCRETE METHOD FOR PROBLEMS Pm Let Ω = x1/2 , xn−1/2 , Ωh;2 = 0,xn−1/2 , Ωh;3 = Ω, and Qh;m = Ωh;m ×(0, T ). For w ∈ S h (Ω), ¯ that coincides with w at the nodes xi , 0 ≤ i ≤ n, and is by w ˆ we denote the function in C Ω h ¯ that coincides (Ω), by vˆ we denote the function in C Ω linear on Ωi−1/2 , 1 ≤ i ≤ n. For v ∈ S1/2 with v at the nodes xi−1/2 , 1 ≤ i ≤ n, and is linear on Ωi , 0 ≤ i ≤ n. We assume that the deﬁnition of the function vˆ is completed for x = 0, X in some way; moreover, vˆ(0) = v1/2 and vˆ(X) = vn−1/2 unless otherwise speciﬁed. For any completion of the deﬁnition, we have the integration by parts formula x=X h v |x=0 − (Dw, ˆ v)Ω ∀w ∈ S h (Ω), ∀v ∈ S1/2 (Ω). (w, Dˆ v )Ω = wˆ x h (Ω). We set (Ih v) (x) = 0 i v (x ) dx on Ωi , 0 ≤ i ≤ n. Let v ∈ S1/2 We construct a semidiscrete counterpart of the system of equations (1.1)–(1.4): h;1

uh in Q, Dt eh = Dˆ h σh + gh x ˆe in Qh;m , h Dt uh = Dˆ h ν h ηh h Dˆ uh + σel e + F h s0 , eh , μ , σh = h η h h Dt xe = u in Q.

(5.1) (5.2) η h = eh + 1

in Q, h

(5.4) x ˆhe = π h g x ˆhe .

h h h h ν, σel = π1/2 σel , F h [s0 , e, μ] = π1/2 F [s0 , e, μ], and g Here h = π h , ν h = π1/2 The unknown vector function h h (Ω) × S h (Ω) × S1/2 (Ω) × S h (Ω) z h = eh , uh , σ h , xhe ∈ C [0, T ]; S1/2

satisﬁes the condition Dt eh , Dt uh , Dt xhe ∈ L1 (Q), and eh > −1 in Q. DIFFERENTIAL EQUATIONS

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(5.3)

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AMOSOV, GOSHEV

We supplement system (5.1)–(5.4) with the initial conditions uh t=0 = u0,h , xhe t=0 = xh eh t=0 = e0,h ,

on

Ω

(5.5)

and one of the pairs of boundary conditions ) ) uh x=X = u(τ on (0, T ), uh x=0 = u(τ 0 , X (τ ) h h u x=X = uX on (0, T ), σ ˆ x=0 = σ0 , h σ ˆ h x=X = σX on (0, T ). σ ˆ x=0 = σ0 ,

(5.61 ) (5.62 ) (5.63 )

t+τ t+τ (τ ) (τ ) Here u0 (t) = τ −1 t u0 (t ) dt , uX (t) = τ −1 t uX (t ) dt . [If t > T , then we set u0 (t) = u0 (T ) (τ ) h e0 ; u0,h (x) = π h u0 (x) on Ωh;m , u0,h (x) = u0 (0+ ) and uX (t) = uX (T ).] In addition, e0,h = π1/2 on Ω0 for m = 1; u0,h (x) = uX (0+ ) on Ωn for m = 1, 2, and xh = xi on Ωi , 0 ≤ i ≤ n. The semidiscrete problem (5.1)–(5.5), (5.6m ) is referred to as problem Pmh (m = 1, 2, 3). Let us state an analog of Theorem 4.1 for it. Let τ ≤ N −2 for m = 1. (τ )

Theorem 5.1. Let conditions (C1 )–(C5 ) be satisﬁed. Then there exists a solution of problem (m = 1, 2, 3) satisfying the estimates

Pmh

K0 (N )−1 ≤ eh + 1 ≤ K0 (N ), h u Dt eh ≤ K(N ), ˆ V 1,1/2 (Q) ≤ K(N ), L2 (Q) h 2 h h It σ x σ ≤ K(N ), ˆ W (Q) ≤ K(N ), ˆe S 1,1 W (Q) ≤ K(N ), L2 (Q) h 0 2 ∀q ∈ [2, ∞), ωq,δ e , C[0, T ] ≤ K(N ) ωq,δ+h e , Λ, σel , s0 , μ + (δ + h)1/2

(5.7) (5.8) (5.9) (5.10)

ωq,δ+h e0 , Λ, σel , s0 , μ = ωq,δ+h e0 + ωq,δ+h (Λ, B0 ) + ωq,δ+h (σel , B1 ) + ¯ + ωq,δ+h μ, L1 R+ , + ωq,δ+h s0 , C R a0 = K0 (N ). B1 = C a−1 B0 = C a−1 0 , a0 , 0 − 1, a0 − 1 ,

where

The proof of this theorem is given in Sections 6 and 7. Remark 5.1. The estimates (5.8) and (5.10) (in view of the convergence h = hn → 0 as n → ∞) imply the following properties: lim sup Δδ eh Lq (Ω;C[0,T ]) = 0 ∀q ∈ [2, ∞). (5.11) lim sup Δ(τ ) eh C([0,T ];L2 (Ω)) = 0, τ →0

δ→0

h

h

The estimate (5.7), together with Theorem 2.1, provides the precompactness of the sequence eh in Lq (Ω; C[0, T ]) for all q ∈ [2, ∞). Remark 5.2. Equation (5.1) can be represented in the equivalent form uh . Dt η h = Dˆ

(5.12)

Remark 5.3. By taking into account (5.12), one can represent Eq. (5.3) either in the form h h (5.13) e + F h s0 , eh , μ σ h = Dt Λh η h + σel or in the form h σ ¯h, σ h = π1/2

where σ ¯ h = Dt Λ η h + σel eh + F s0 , eh , μ . DIFFERENTIAL EQUATIONS

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785

Remark 5.4. From (5.13), we have the important relation Λh η h = Λh η 0,h + It σ h − It σel eh − It F

h

0 h s ,e ,μ .

(5.15)

6. DERIVATION OF A PRIORI ESTIMATES FOR APPROXIMATE SOLUTIONS Consider an arbitrary solution of problem Lemma 6.1. For V h (t) =

Ω

Pmh .

η h (x, t)dx with m = 1, one has the two-sided estimate (2N )−1 ≤ V h (t) ≤ (X + 2T )N.

(6.1)

Proof. The desired assertion can be justiﬁed just as in [10, Lemma 3.1]. By integrating (τ ) (τ ) Eq. (5.12) over Qt , we obtain V h (t) = η 0 L1 (Ω) + It uX − It u0 , which, together with conditions (C2 ) and (C4 ), implies the estimate (6.1). It is at this point that the condition τ ≤ N −2 is used. Lemma 6.2. One has the energy estimate ∞ h 0 h 2 h 2 h 2 u + Dt L η L2 (Q) + E e L1,∞ (Q) + Sr s , e dr μ L2,∞ (Q) 0

≤ K(N ).

(6.2)

L1,∞ (Q)

(τ ) (τ ) (τ ) Proof. For m = 1, we set uhΓ = 1 − h u0 + h uX , h = Ih η h /V h . Note that uhΓ x=0 = u0 , (τ ) uhΓ x=X = uX , 0 ≤ h ≤ 1, and h ) (τ ) h h Dt uhΓ = 1 − h Dt u(τ 0 + Dt uX + dV u − uΓ ,

Dˆ uhΓ = dV η h ,

(τ ) (τ ) (τ ) where dV = uX − u0 /V h . We set uhΓ = uX for m = 2 and uhΓ = 0 for m = 3. Let us introduce the functions v h = uh − uhΓ and σΓ = (1 − x/X)σ0 + (x/X)σX . By taking the inner product of Eq. (5.2) by v h = uh − uhΓ in the sense of L2 Ωh;m and by taking into account the boundary conditions (5.6m ), we obtain h h h 1 h h Dt u , u Ω + σ ¯ , Dˆ uh Ω = Dt uh , uhΓ Ω − uh , Dt uhΓ Ω + Γhm + g x ˆe , v Ω , (6.3) 2 h (τ ) ¯ , Dˆ uhΓ Ω , Γh2 = −σ0 uh x=0 − uX , and Γh3 = σX uh x=X − σ0 uh x=0 . By using where Γh1 = σ Eqs. (5.12) and (5.14), we obtain

h 0 h ν ηh h h h h D η , D η + σ e + F , e , μ , D e e , D s t t el t t Ω Ω ηh Ω 2 = Dt L η h L2 (Ω) + Dt E eh L1 (Ω) + F s0 , eh , μ , Dt eh Ω .

uh Ω = σ ¯ h , Dˆ

By applying the operator It to Eq. (6.3) and by using inequality (3.11), we arrive at the energy inequality E h ≤ E h t=0 + uh , uhΓ Ω − u0,h, uhΓ t=0 Ω − uh , Dt uhΓ Qt + It Γhm + g xˆhe , vh Qt , (6.4) DIFFERENTIAL EQUATIONS

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AMOSOV, GOSHEV

where

Eh

E h t=0

∞ √ 1 1 2 2 2 h h h 0 h = u L2 (Ω) + Dt L η L2 (Qt ) + E e L1 (Ω) + dr μ , Sr s , e 2 2 0 L1 (Ω) ∞ 2 1 √ 1 0 2 = u0,h L2 (Ω) + E e0,h L1 (Ω) + d μ . s r 2 2 0

L1 (Ω)

By virtue of conditions (C2 )–(C5 ), from (6.4), we have

√ √ E h (t) ≤ K1 + N uh L2,∞ (Qt ) uhΓ L2,∞ (Q) + Dt uhΓ L2,1 (Q) + It Γhm + ¯ gL2,1 (Q) uh − uhΓ L2,∞ (Qt ) 1/2 h ≤ K2 Emax (t) + K3 + It Γhm , h (t) = E h C[0,t] . where Emax Let us estimate the term It Γhm . If m = 1, then, by taking into account (5.14), conditions (4.3) and (4.7), and inequality (3.13), we obtain h ¯ , dV η h Qt It Γh1 = σ

= dV ν [η h ] η h , Dt L η h + σel eh , dV η h Qt + F s0 , eh , μ , dV η h Qt Qt ! h h 1/2 Dt L η h ν η η ≤ dV + It σel eh eh + 1 C[0,T ]

L1 (Qt )

+ F s0 , eh , μ L∞ (Q) It V h ≤K

#

L2 (Qt )

L1 (Ω)

"

$ 1/2 h 2 h h L η +1 Dt L η L2 (Qt ) + It E e L1 (Ω) + 1 . L1 (Qt )

By using condition (4.4) for m = 2, we obtain

(τ ) = σ0 , Dt η h Qt = σ0 η h /ν [η h ], Dt L η h It Γh2 = − σ0 , uh x=0 − uX (0,t) Qt h 2 1 1 ≤ Dt L η L2 (Qt ) + It c2 σ02 L2 η h + 1L1 (Ω) . 2 2 Likewise, for m = 3, we obtain the inequality ˆh Qt + σΓ , Dˆ uh Qt It Γh3 = It σX uh x=X − σ0 uh x=0 = DσΓ , u

= (σX − σ0 )/X, uh Qt + σΓ η h /ν [η h ], Dt L η h Qt

2 1 ≤ (σX − σ0 )/XL1 (0,T ) uh L2,∞ (Qt ) + Dt L η h Qt 2 2 h 1 2 + It c2 σΓ L∞ (Ω) L η + 1 L1 (Ω) . 2 By using the relation 2 h L η (·, t)

L1 (Ω)

√ 2 0,h 2 Dt L η h 2 L η ≤ 1+ T + , L2 (Ω) L2 (Qt ) DIFFERENTIAL EQUATIONS

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GLOBAL UNIQUE SOLVABILITY OF THE LONGITUDINAL VIBRATION EQUATIONS

we derive the inequality h Emax ≤K

h Emax

1/2

+ It

787

h 2 + 1 Emax σ02 + σX +1 .

The Gronwall inequality permits one to obtain the estimate to (6.2). Lemma 6.3. One has the estimates DIt σ ˆ h L2,∞ (Q) ≤ K(N ),

h It σ

h Emax (T )

L∞ (Q)

≤ K, which is equivalent

≤ K(N ).

(6.5)

Qh;m .

(6.6)

Proof. It follows from (5.2) that h ˆ h = π h uh − u0,h − It g x ˆe DIt σ

in

This, together with the energy estimate (6.2), implies that √ √ √ DIt σ ˆ h L2,∞ (Q) ≤ N uh L2,∞ (Q) + u0,h L2 (Ω) + ¯ gL2,1 (Q) ≤ K1 . % h& h h be the mean = I Let us derive the second estimate in (6.5) for m = 1. Let I tσ t σ, η /V Ω h h h h h value of the function It σ with weight η /V (recall that η /V L1 (Ω) = 1). One can readily see & % that It σ h = σ h , η h /V h Qt + It σ h , Dt η h /V h Qt . By taking into account (5.14) and by using v h /V h , we obtain the relation Dt η h /V h = Dˆ

h h h ν [η h ] η h Dt L η h , 1/V h + σel eh η h , 1/V h Qt σ , η /V Qt = Qt + F s0 , eh , μ , η h /V h Qt , h v h /V h Qt = − DIt σ ˆ h , v h /V h Qt . It σ , Dt η h /V h Qt = It σ h , Dˆ Inequality (6.1), the energy estimate (6.2), and the ﬁrst estimate in (6.5) imply that ! 1/2

Dt L η h + σel eh eh + 1 L1,∞ (Q) It σL∞ (0,T ) ≤ K2 L2 η h L1 (Q) + 1 L2 (Q) " 0 h h h + F s , e , μ L∞ (Q) + DIt σ ˆ L2,∞ (Q) v L2,∞ (Q) ≤ K3 . % & ˆ h L1,∞ (Q) ≤ K4 , we have It σ h L∞ (Q) ≤ K. If m = 2, 3, Since It σ h − It σ h L∞ (Q) ≤ DIt σ ˆ + It σ0 , we obtain the desired estimate then from the relation It σ h = Ih DIt σ h h DIt σ It σ ≤ ˆ + σ0 L1 (0,T ) ≤ K. L∞ (Q) L1,∞ (Q) Lemma 6.4. The two-sided estimate (5.7) is valid. Proof. From (5.15), we obtain h

Λ

η

h

(x, t) = Λ η h (x, t∗ ) +

t

h

t∗

t −

F

h

s0 , eh , μ (x, t ) dt

t∗

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t

σ (x, t ) dt −

h

2007

h h σel e (x, t ) dt

t∗

(6.7)

788

AMOSOV, GOSHEV

for all 0 ≤ t∗ < t ≤ T . Let η h (x, t) < 1 for some (x, t) ∈ Q. Then there exists a t∗ ∈ [0,t) such that η h (x, t ) < 1 for all t ∈ (t∗ , t]; moreover, either η h (x, t∗ ) = 1 or t∗ = 0. Since It σ h L∞ (Q) ≤ K, h h e (x, t ) ≤ 0 for all η 0,h ≥ N −1 , F h s0 , eh , μ ≤ N , and, by virtue of condition (4.5), σel t ∈ [t∗ , t], it follows from (6.7) [in view of condition (4.2)] that ¯ η h (x, t) ≥ Λ η h (x, t∗ ) − 2K − N T ≥ Λ N −1 − K1 . Λ ¯ Since Λ(η) → −∞ as η → 0+ , we have η h (x, t) ≥ K0−1 . In a similar way, one can derive the estimate h η (x, t) ≤ K0 . Corollary 6.1. The estimates (5.8) and (5.9) are valid. Proof. By using the estimates (6.2), (4.1), (4.6), and (3.13), we obtain h u ˆ L2,∞ (Q) ≤ uh L2,∞ (Q) ≤ K, 2 h 2 2 Dˆ u L2 (Q) = Dt eh L2 (Q) ≤ K1 Dt L η h L2 (Q) ≤ K2 , h h h 0 h h h σ e F Dˆ σ ≤ K + + s , e , μ L2 (Q) ≤ K4 , u 3 el L (Q) L2 (Q) L (Q) h h 2 h 2 It σ ˆ W (Q) = It σ ˆ L2 (Q) + σ ˆ L2 (Q) + DIt σ ˆ h L2,∞ (Q) ≤ K5 . h 0,1/2

≤ K can be derived just as in [10, Lemma 3.5]. The estimate u ˆ Lemma 6.5. The estimate (4.14) is valid. h (x, t) = η h (x + δ, t) and eh+δ (x, t) = eh (x + δ, x). By applying the operator Δδ Proof. We set η+δ to relation (5.15), we obtain

h h 0,h 0,h − Λh η h = − Δδ Λh η+δ + Λh η+δ − Λh η 0,h + Δδ Λh η+δ + Δδ It σ h Λh η+δ h h h − It Δδ σel eh+δ − It σel eh+δ − σel eh − It Δδ F h s0 , eh , μ , where h (x, t) = Δδ Λh (ζ, x)ζ=ηh (x+δ,t) , Δδ Λh η+δ

h Δδ σel

(6.8)

h (ζ, x)ζ=eh (x+δ,t) . eh+δ (x, t) = Δδ σel

The estimate (5.7) and conditions (4.1) and (4.6) imply the inequalities h − Λh η h , K1−1 Δδ eh ≤ Λh η+δ h 0,h 0,h h ≤ K1 Δδ e0,h , Λ η η +δ − Λ h h h h σel e+δ − σel e ≤ K1 Δδ eh . Therefore, from (6.8), we obtain h Δδ eh + It Δδ F Δδ eh ≤ K Δδ e0,h + Δδ Λh + Δδ It σ h + Δδ σel + I t B0 B1

h

s ,e ,μ . 0

h

It follows from property (3.12) of the Prandtl–Ishlinskii operator and Lemma 2.2 that 0 + h ¯ Δδ F h s0 , eh , μ ≤ μ ¯ , C R e s + 2 Δ ω 0 q,δ+h δ Lq (Ω;C[0,t]) Lq (Ω;C[0,t]) + . + ωq,δ+h μ, L1 R DIFFERENTIAL EQUATIONS

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789

In addition, by taking into account the estimate (cf. [10, Lemma 3.6]) Δδ It σ h ≤ (δ + h)1/2 DIt σ ˆ h L2,∞ (Q) , L∞ (Ω;C[0,T ])

for the function y h (t) = Δδ eh Lq (Ω;C[0,t]) , we obtain

y h ≤ K2 It y h + ωq,δ+h e0 , Λ, σel , s0 , μ + (δ + h)1/2 . The use of the Gronwall inequality implies the estimate (4.14). 7. SOLVABILITY OF PROBLEM

Pmh

Let L be a ﬁnite-dimensional normed space, and let K ⊂ L . By C ([t0 , T ] ; K) we denote the metric space of continuous functions y : [t0 , T ] → K with the metric of the space C ([t0 , T ] ; L ). Let G be a domain in L , and let a nonlinear operator with values from f (t, y) ∈ L be deﬁned on the set M = {(t, y) | t ∈ (t0 , T ) , y ∈ C ([t0 , t] ; G)}. For s ∈ [t0 , T ], by ys we denote the restriction of the function y ∈ C ([t0 , T ] ; G) to [t0 , s]. Let us study the solvability of the Cauchy problem for the operator-diﬀerential equation t0 ≤ t ≤ T,

Dt y(t) = f (t, yt ) , y (t0 ) = y0 ,

(7.1) (7.2)

where y0 ∈ G. A solution of problem (7.1), (7.2) is deﬁned as a function y ∈ C ([t0 , T ] ; G) that has a generalized derivative Dt y ∈ L1 (t0 , T ; L ) and satisﬁes Eq. (7.1) in L1 (t0 , T ; L ) and the initial condition (7.2). Theorem 7.1. Let the operator f satisfy the following conditions. 1. The operator f (t, ·) : C ([t0 , t] ; G) → L is continuous for almost all t ∈ (t0 , T ]. 2. The function f (t, yt ) is measurable on (t0 , T ) for all y ∈ C ([t0 , T ] ; G). 3. For any compact set K ⊂ G, there exists a function FK ∈ L1 (t0 , T ) such that sup

f (t, y)L ≤ FK (t)

for almost all

t ∈ (t0 , T ) .

(7.3)

y∈C([t0 ,t];K )

Let the following “a priori estimate” be valid (for ﬁxed y0 ∈ G) for all T∗ ∈ (t0 , T ] and for any function y ∈ C ([t0 , T∗ ] ; G) that is a solution of problem (7.1), (7.2) on the interval [t0 , T∗ ] : y(t) ∈ K˜ ⊂ G for all t ∈ [t0 , T∗ ] , where K˜ is a ﬁxed time moment independent of T∗ . Then there exists a solution of problem (7.1), (7.2) deﬁned on the entire interval [t0 , T ]. In addition, suppose that the following local Lipschitz condition is satisﬁed. 4. For each t∗ ∈ [t0 , T ) , there exist an interval (t∗ , t∗ + δ) ⊂ (t∗ , T ) and a function a∗ ∈ L1 (t∗ , t∗ + δ) such that f (t, y) − f (t, z)L ≤ a∗ (t)y − zC([t∗ ,t];L ) for almost all t ∈ (t∗ , t∗ + δ) for all y, z ∈ C [t0 , t∗ + δ] ; K˜ coinciding on [t0 , t∗ ]. In this case, the solution of the Cauchy problem (7.1), (7.2) is unique. Proof. We set

K˜r =

y ∈ L | y, K˜ ≤ r

and choose r > 0 so as to ensure that K˜r ⊂ G. Note that, by virtue of the absolutely continuity of the Lebesgue integral, t+δ FK˜r (s)ds → 0 as δ → 0. ω(δ) = sup 0≤t
DIFFERENTIAL EQUATIONS

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t

No. 6

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AMOSOV, GOSHEV

We choose Δ = (T − t0 )/N so as to ensure that ω(Δ) < r and split the interval [t0 , T ] into subintervals [ti−1 , ti ] by the points ti = iΔ, 0 ≤ i ≤ N . [t0 , t1 ] by the points tnj = t0 + jh into subintervals n Step 1. We set h = Δ/n and split the interval n n tj , tj+1 , 0 ≤ j < n. We construct a function y ∈ C ([t0 , t1 ] ; G) such that y n (0) = y0 and y n (t) = y n tnj +

t

f s, ysn,j ds,

t ∈ tnj , tnj+1 ,

0 ≤ j < n.

(7.4)

tn j

Here y n,j (t) = y n (t) for t ∈ T0 , tnj and y n,j (t) = y n (tj ) for t ∈ tnj , T . t Formula (7.4), together with condition (7.3) and the inequality t01 FK˜r (s)ds ≤ ω(Δ) < r, implies that y n ∈ C [t0 ; t1 ] ; K˜r . In addition, |y n (t ) − y n (t )| ≤

t

FK˜r (s)ds ≤ ω (|t − t |)

∀t , t ∈ [t0 , t1 ] .

t ∞

The sequence {y n }n=1 is uniformly bounded and equicontinuous on [t0 , t1 ]. Consequently, there n ˜ exists a subsequence y → y in C [t0 , t1 ] ; Kr . It follows from (7.4) that t n

ψ n (s)ds

y (t) = y0 +

∀t ∈ [t0 , t1 ] ,

(7.5)

0

n,k(s)

and k(s) satisﬁes the condition s ∈ (tnk(s) , tnk(s)+1 ], 0 ≤ k(s) < n. where ψ n (s) = f s, ys By (7.4), we have n,k(s) ys − y C([t0 ,s];L ) ≤ y n − yC([t0 ,s];L ) + ω(h). Therefore, ψ n (s) → f (s, ys ) as → ∞. By using the dominated convergence theorem (FK˜r is used as a majorant) and by passing in (7.5) to the limit as n = n → ∞, we obtain t y(t) = y0 +

f (s, ys ) ds

∀t ∈ [t0 , t1 ] .

t0

At the ﬁrst step of the proof, we have thereby shown that there exists a function y that is a solution of the Cauchy problem (7.1), (7.2) on the interval [t0 , t1 ]. By virtue of the a priori estimate, y(t) ∈ K˜ for all t ∈ [t0 , t1 ]. Step 2. We split the interval [t1 , t2 ] by the points tnj = t1 + jh into the subintervals tnj , tnj+1 , 0 ≤ j < n. We construct the function y n ∈ C ([t0 , t2 ] ; G) such that y n (t) = y(t) for t ∈ [t0 , t1 ], and formula (7.4) is valid, where y n,j (t) = y n (t) for t ∈ t0 , tnj and y n,j (t) = y n (tj ) for t ∈ tnj , T . Note that y n (tn0 ) = y (t1 ) ∈ K˜ . By repeating considerations of the ﬁrst step, we ﬁnd that there exists a function y that is a solution of problem (7.1), (7.2) on the interval [t0 , t2 ]. By performing N steps of the proof, we prove the existence of a solution deﬁned on the entire interval [t0 , T ]. If condition 4 is satisﬁed, then the uniqueness of the solution can be proved by the use of the Gronwall inequality. Proposition 7.1. There exists a solution of problem

Pmh

(m = 1, 2, 3).

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GLOBAL UNIQUE SOLVABILITY OF THE LONGITUDINAL VIBRATION EQUATIONS

Proof. We rewrite problem is given by the formula

Pmh

791

in the form (7.1), (7.2), where y = eh , uh , xhe , the operator f

−1 h h h uh , h Dˆ σ + gh x ˆe , u , f (t, yt ) = Dˆ σ h being given by (5.3), and conditions (5.6m ) are additionally satisﬁed. Moreover, y 0 = e0,h , u0,h , xh . h (Ω) × S h (Ω) × S h (Ω) and G = {(e , u , xe ) ∈ L | e > −1}. The operator In this case, L = S1/2 f has properties 1–3. This follows from assumptions (C1 )–(C5 ). By Theorem 5.1, we have the a priori estimate y(t) ∈ K = (e , u , xe ) ∈ L | K0−1 ≤ e + 1 ≤ K0 , ∀t ∈ [0, T ]. u L2 (Ω) ≤ K, xe L∞ (Ω) ≤ K

By using Theorem 7.1, we justify the existence of a solution of the problem Theorem 5.1 is complete.

Pmh .

The proof of

8. PROOF OF THEOREM 4.1 Let us perform the passage to the limit in problem Pmh as h → 0, τ = τ (h) → 0. By Theorem 5.1, there exists a sequence of solutions z h of problem Pmh satisfying the estimates (5.7)–(5.9) uniformly with respect to h and the estimate (5.10). These estimates, together with Remark 5.1, guarantee the existence of a vector function z = (e, u, σ, xe ) ∈ N−1 (Q) × V2 (Q) × L2 (Q) × S21,1 W (Q) and a subsequence of solutions (denoted by the previous notation) such that eh → e in L2 (Ω; C[0, T ]), Dt eh → Dt e weakly in L2 (Q), η h → η = e+1 in L2 (Ω; C[0, T ]), uh → u ∗-weakly in L∞ (0, T ; L2 (Ω)), h ˆ h → DIt σ ∗-weakly Dˆ uh → Du weakly 2 (Q), DIt σ in L2 (Q),h σ → σ weakly in L in L∞ (0, T ; L2 (Ω)), 1,1 h ¯ ¯ . ˆ → It σ in C Q , and x ˆe → xe weakly in S2 W (Q) and strongly in C Q It σ Obviously, the estimates (4.11)–(4.14) are valid for the limit functions. By passing in Eqs. (5.1) and (5.4) and in the initial condition xhe t=0 = xh to the limit, we obtain Eqs. (1.1) and (1.4) and the initial condition xe |t=0 = x. We pick up an arbitrary function ϕ ∈ C0∞ (Q) and take the inner product of relation (6.6) by ϕ for suﬃciently small h in L2 (Q); then we obtain h h ˆ h , ϕ Q = uh − u0,h − It g x ˆe , π ϕ Q . DIt σ By passing to the limit, hence we obtain the relation (DIt σ, ϕ)Q = u − u0 − It g [xe ] , ϕ Q

∀ϕ ∈ C0∞ (Q),

¯ , it follows from (4.10) that ˆ h → It σ in C Q which implies (4.9). Since It σ h h ˆ x=0 = It σ0 for m = 2, 3, It σ ˆ x=X = It σX for It σ

m = 3.

By using Remark 4.1, we obtain identity (4.8). (τ ) ˆhx=X → u|x=X weakly in L2 (0, T ); moreover, u0 → u0 and Note that u ˆh x=0 → u|x=0 and u (τ ) uX → uX in L2 (0, T ). Therefore, the function u satisﬁes the boundary condition u|x=0 = u0 for m = 1 and the condition u|x=X = uX for m = 1, 2. It follows from the estimates (5.7) and (4.11) and assumptions (4.1) and (4.6) that h h h h h σel e − σel Λ η − Λh [η] ≤ K eh − e , [e] ≤ K eh − e . DIFFERENTIAL EQUATIONS

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AMOSOV, GOSHEV

Since η 0,h → η 0 in L2 (Ω) and eh → e in L2 (Ω; C[0, T ]), we have Λ η 0,h → Λ [η 0 ] in L2 (Ω), h eh → It σel [e] in L2 (Ω; C[0, T ]). In addition, the estimate Λh η h → Λ[η], and It σel h 0 h F s , e , μ − F s0 , e, μ L2 (Ω;C[0,T ]) ≤ 2¯ μ0 eh − e + F h s0 , e, μ − F s0 , e, μ L2 (Ω;C[0,T ])

L2 (Ω;C[0,T ])

h

implies that F s0 , eh , μ → F [s0 , e, μ] in L2 (Ω; C[0, T ]). By passing in (5.15) to the limit, we obtain the relation Λ[η] = Λ η 0 + It σ − It σel [e] − It F s0 , e, μ

in

Q.

(8.1)

By applying the operator Dt to it and by using the relation Dt Λ[η] = ν[η]η −1 Dt η = ν[η]η −1 Du, we ﬁnd that the limit functions satisfy Eq. (1.3). Since the right-hand side of formula (8.1) belongs to the space C ([0, T ]; L∞ (Ω)), it follows from the inequality K −1 e (·, t1 ) − e (·, t2 )L∞ (Ω) ≤ Λ[η] (·, t1 ) − Λ[η] (·, t2 )L∞ (Ω)

∀t1 , t2 ∈ [0, T ]

that e ∈ C ([0, T ]; L∞ (Ω)) and the initial condition e|t=0 = e0 is satisﬁed. The proof of Theorem 4.1 is complete. 9. PROOF OF THEOREM 4.2 The following proof is a modiﬁcation of the proof [14] of the continuous dependence on the data of solutions of a similar system of equations describing the one-dimensional motion of a viscous barotropic gas. First, we note some properties of an arbitrary generalized solution z of problem Pm . (It is not necessarily the same solution whose existence was proved in Theorem 4.1.) Lemma 9.1. If m = 1, then η(x, t)dx = η 0 L1 (Ω) + It (uX − u0 )

∀t ∈ [0, T ].

Ω

Proof. It suﬃces to integrate Eq. (1.1) represented in the form Dt η = Du over Qt and use the initial condition η|t=0 = η 0 and the boundary conditions u|x=0 = u0 and u|x=X = uX . Corollary 9.1. If m = 1, then Λ −1 (Λ[η](x, t), x)dx = η 0 L1 (Ω) + It (uX − u0 )

∀t ∈ [0, T ],

(9.1)

Ω

where Λ −1 (λ, x) is a function that is deﬁned on R × Ω and is the inverse function of Λ(η, x) for almost all ﬁxed x ∈ Ω. Remark 9.1. It follows from assumptions (C1 ) that the function Λ −1 exists, is measurable on R × Ω, and satisﬁes the relations Λ −1 (0, x) = 1 and Dλ Λ −1 (λ, x) = Λ −1 (λ, x)/ν Λ −1 (λ, x), x ; moreover, the estimate 0 < c(a) ≤ Dλ Λ −1 (λ, x) ≤ c¯(a)

∀λ ∈ [−a, a]

(9.2)

is valid for all a > 0 and for almost all x ∈ Ω. We set vΩ = X −1 Ω v dx for v ∈ L1 (Ω). Let us introduce the operators I 1 v = Iv − IvΩ , (1) (2) (3) I 2 v = Iv, and I 3 v = I (v − vΩ ). Let σΓ = σΩ , σΓ = σ0 , and σΓ = (1−x/X)σ0 +(x/X)σX . DIFFERENTIAL EQUATIONS

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Lemma 9.2. One has relation (8.1), and It σ = I m u − u0 − It g [xe ] + It σΓ m

in

Q.

(9.3)

Proof. By applying the operator It to (4.11), we obtain (8.1). To derive (9.3), it suﬃces to apply the operator I m to relation (4.9) and take into account (4.10). Corollary 9.2. One has m

Λ[η] = B m [z] + It σΓ ,

(9.4)

where B m [z] = Λ [η 0 ] − It σel [e] − It F [s0 , e, μ] + I m (u − u0 − It g [xe ]) (m = 1, 2, 3). Now we take two arbitrary generalized solutions z (k) = e(k) , u(k) , σ (k) , x(k) (k = 1, 2) of probe lem Pm . Suppose (without loss of generality) that (k) (k) ¯, ¯, ¯, ¯ −1 ≤ η (k) ≡ e(k) + 1 ≤ N u xe ≤N ≤N k = 1, 2, (9.5) N V2 (Q) C(Q) ¯ ≥ N. with a parameter N corresponding We introduce the diﬀerence operator Δψ = ψ (1) − ψ (2) , where ψ (k) is the function to the solution z (k) ; for example, Δe = e(1) − e(2) , Δσel [e] = σel e(1) − σel e(2) . By virtue of the estimate (9.5), assumption (C1 ), and the estimate (3.12), we have −1 ¯ |Δe| ≤ |ΔΛ[η]|, K1 N ¯ |Δe|, |Δσel [e]| ≤ K2 N 0 ΔF s , e, μ ≤ 2¯ μ0 ΔeC[0,t] .

(9.6)

Note also that ΔB m [z] = −It Δσel [e] − It ΔF

s0 , e, μ + I m (Δu − It Δg [xe ]) .

This, together with the estimate (9.6), implies that ⎛ t ⎞ ¯ ⎝ ΔeC[0,t ] dt + ΔuL1,∞ (Qt ) + Δg [xe ] ΔB m [z] ⎠. ≤ K3 N L1 (Qt ) C[0,t]

(9.7)

0

Note that Δxe = It Δu and, by virtue of assumption (4.16), we have t Δg [xe ]L1 (Qt ) ≤

¯b (t ) ΔuL (Q ) dt ≤ T ¯b ΔuL2,∞ (Qt ) , 2,1 t L1 (0,T )

0

⎡ Δg

2 [xe ]L2,1 (Qt )

≤⎣

t

⎤2

¯b (t ) ΔuL (Q ) dt ⎦ ≤ cX,T ¯b ∞,1 t L1 (0,T )

0

t

¯b (t ) Δu2 V2 (Qt ) dt ,

(9.8)

(9.9)

0

where ¯b(t) = bN¯ (·, t)L2 (Ω) , ¯b ∈ L1 (0, T ). Lemma 9.3. One has the estimate ¯ ΔuL2,∞ (Qt ) ΔeL∞ (Ω;C[0,t]) ≤ K N DIFFERENTIAL EQUATIONS

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∀t ∈ (0, T ].

(9.10)

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AMOSOV, GOSHEV

Proof. If m = 1, then it follows from (9.1) that M (x , t) ΔΛ[η] (x , t) dx = 0,

(9.11)

Ω

where

1

M (x , t) =

Dλ Λ−1 αΛ η (1) (x , t) + (1 − α)Λ η (2) (x , t) , x dα.

0

By virtue of the estimate (9.5) and assumption (C1 ), we have (1) ¯ . αΛ η + (1 − α)Λ η (2) ≤ K1 N Therefore, by virtue of inequality (9.2), the two-sided estimate −1 ¯ ¯ ≤ M (x, t) ≤ K2 N K2 N is valid. (1) Since the term It σΓ in (9.3) depends only on t, it follows from (8.1) that Λ[η](x, t) − B 1 (x, t) = Λ[η] (x , t) − B 1 (x , t)

for almost all

x, x ∈ Ω.

Therefore, relation (9.11) implies that ΔΛ[η](x, t) = ΔB

1

¯ (t)−1 [z](x, t) − M

M (x , t) ΔB 1 [z] (x , t) dx ,

Ω

¯ (t) = where M

Ω

M (x , t) dx . Hence, by taking into account (9.6), we obtain ¯ ΔB 1 [z] . ΔeC[0,t] ≤ K3 N L∞ (Ω;C[0,t])

By using the estimates (9.7) and (9.8), we derive the inequality ⎛ t ⎞ ¯ ⎝ ΔeL∞ (Ω;C[0,t ]) dt + ΔuL2,∞ (Qt ) ⎠ . ΔeL∞ (Ω;C[0,t]) ≤ K4 N

(9.12)

0

The use of the Gronwall lemma leads to the estimate (9.10) for m = 1. The cases in which m = 2, 3 are much simpler. From (9.4), we have ΔΛ[η] = ΔB m [z]. This, together with inequalities (9.6) and (9.7), implies inequality (9.12) and hence the estimate (9.10). Lemma 9.4. One has the estimate Δu2V2 (Qt )

¯ ≤K N

t

a ¯ (t ) Δe2L∞ (Ω;C[0,t ]) dt

∀t ∈ [0, T ],

(9.13)

0

2 where a ¯(t) = Du(2) (·, t)L2 (Ω) + 1. Proof. Note that the function Δu is a generalized [in V2 (Q)] solution of the linear parabolic problem for the equation Dt Δu = D(κDΔu + ψ) + Δg [xe ]

in Q

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795

(where κ = ν η (1) /η (1) and ψ = Δ(ν[η]/η)Du(2) + Δσel [e] + ΔF [s0 , e, μ]) with the homogeneous initial condition Δu|t=0 = 0 and boundary conditions Δu|x=0,X = 0 for m = 1, (κDΔu + ψ)|x=0 = Δu|x=X = 0 for m = 2, and (κDΔu + ψ)|x=0,X = 0 for m = 3. Therefore, we have the estimate

∀t ∈ (0, T ]. ΔuV2 (Qt ) ≤ K1 ψL2 (Qt ) + Δg [xe ]L2,1 (Qt ) By using inequalities (9.5) and (9.6) and assumption (4.15), we obtain ψ2L2 (Qt )

t

(2) 2 2 Du (·, t )2 ≤ K2 + 1 + μ ¯ 0 ΔeL∞ (Ω;C[0,t ]) dt L2 (Ω) 0

t ≤ K3

a ¯ (t ) Δe2L∞ (Ω;C[0,t ]) dt .

0

This, together with the estimate (9.9), implies the inequality ⎛ t ⎞ t ¯ (t ) Δe2L∞ (Ω;C[0,t ]) dt + ¯b (t ) Δu2V2 (Qt ) dt ⎠ , Δu2V2 (Qt ) ≤ K4 ⎝ a 0

0

whence one can derive the estimate (9.13) with the use of the Gronwall inequality. The proof of the lemma is complete. It follows from (9.10) and (9.13) that Δu2V2 (Qt )

¯ ≤K N

t

a ¯ (t ) Δu2V2 (Qt ) dt

∀t ∈ [0, T ].

0

This, together with the Gronwall inequality, implies that Δu2V2 (QT ) = 0, i.e., Δu = 0. By returning to inequality (9.10), we obtain Δe = 0. Therefore, Δσ = 0 and Δxe = 0. The proof of Theorem 4.2 is complete. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 04-01-00539). REFERENCES 1. Prandtl, L., Z. Angew. Math. Mech., 1928, vol. 8, pp. 85–106. 2. Ishlinskii, A.Yu., Izv. Akad. Nauk Ser. OTN , 1944, no. 9, pp. 580–590. 3. Krasnosel’skii, M.A. and Pokrovskii, A.V., Sistemy s gisterezisom (Systems with Hysteresis), Moscow: Nauka, 1983. 4. Visintin, A., Diﬀerential Models of Hysteresis, Berlin: Heidelberg, 1994. 5. Brokate, M. and Sprekels, J., Hysteresis and Phase Transitions, Appl. Math. Sci., vol. 121, New York, 1996. 6. Krej˘c´ı, P., Hysteresis. Convexity and Dissipation in Hyperbolic Equations, GAKUTO Int. Ser. Math. Sci. Appl., vol. 8, Tokyo, 1996. 7. Dr´ abek, P., Krej˘c´ı, P., and Taka´c, P., Nonlinear Diﬀerential Equations, CRC Press, 1999. 8. Francˇ u, J. and Krej˘c`ı, P., Contin. Mech. Thermodyn., 1999, vol. 11, pp. 371–391. 9. Goshev, I.A., Vestnik Moskov. Energ. Inst., 2005, no. 6, pp. 82–100. 10. Amosov, A.A. and Zlotnik, A.A., Diﬀer. Uravn., 1994, vol. 30, no. 4, pp. 596–608. 11. Amosov, A.A. and Zlotnik, A.A., Izv. Vyssh. Uchebn. Zaved. Mat., 1997, no. 4, pp. 3–19. DIFFERENTIAL EQUATIONS

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12. 13. 14. 15.

Amosov, A.A., Diﬀer. Uravn., 2000, vol. 36, no. 4, pp. 486–499. Amosov, A.A., Tr. Mat. Inst. Steklova, 2002, vol. 236, pp. 11–19. Amosov, A.A. and Zlotnik, A.A., Mat. Zametki, 1994, vol. 55, no. 6, pp. 13–31. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1976. 16. Amosov, A.A. and Zlotnik, A.A., Diﬀer. Uravn., 1997, vol. 33, no. 1, pp. 83–95. 17. Evans, L.C. and Gariepi, R.F., Measure Theory and Fine Properties of Functions, Boca Raton: CRC Press, 1992. Translated under the title Teoriya mery i tonkie svoistva funktsii, Novosibirsk: Nauchnaya Kniga, 2002. 18. Amosov, A.A. and Zlotnik, A.A., Zh. Vychisl. Mat. Mat. Fiz., 1996, vol. 36, no. 2, pp. 87–110.

DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 797–805. c Dang Han Hoi, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 780–787. Original Russian Text

PARTIAL DIFFERENTIAL EQUATIONS

On the Structure of the Period Set of Periodic Solutions to Some Linear Evolution Systems of Diﬀerential Equations Dang Han Hoi Novgorod State University, Novgorod, Russia Received August 25, 2005

DOI: 10.1134/S0012266107060067

1. It was discovered in [1, 2] that, in the investigation of the problem on periodic solutions of some partial diﬀerential equations, the set of periods for which the periodic solution is unique has a rather complicated structure. In the present paper, we study this problem for a special class of linear equations in the space of smooth diﬀerential forms on the circle. Let CS∞ be the space of smooth diﬀerential forms on the circle S = Π = R/(2Z), and let HSk be the Sobolev space of diﬀerential forms on S whose coeﬃcients have square integrable generalized derivatives of order ≤ k [3]. By A we denote the operator i(d + δ) (the so-called natural diﬀerential operator on S), where d is the exterior diﬀerential and δ = d∗ is its formal adjoint with respect to the inner product in CS∞ induced by the Riemannian structure on S. As is known [3, 4], d + δ is a ﬁrst-order elliptic diﬀerential operator. Consider the problem on periodic solutions for the equation 1 ∂ + aA − λ u(x, t) = νG(u − f ) (1) (L − λ)u ≡ i ∂t with the t-periodicity condition u|t=0 = u|t=b , where

u(x, t) = 1 dx

(2)

u0 (x, t) u1 (x, t)

≡

u0 (x, t) u1 (x, t)

is a complex form on the circle with coeﬃcients depending on t ∈ [0, b], a = 0, λ, and ν are given complex numbers, f (x, t) is a given form, and u0 (y, t) dy Gu(x, t) = g(x, y) u1 (y, t) Π

is an integral operator with smooth kernel 2

g(x, y) ≡ (gls )l,s=1 on the space L2 ([0, b], HS0 ). Consider the most interesting and typical case in which the operator (L − λ)−1 is deﬁned but unbounded. In particular, the parameters a and λ should be real. In this case, the operator (L − λ)−1 ◦ G can be bounded and even compact, which permits one to use Fredholm theory for the analysis of the solvability of problem (1), (2). 797

798

DANG HAN HOI

The change of variables t = bτ reduces the problem in question to a problem with ﬁxed period but for a new equation, in which the coeﬃcient of the derivative with respect to τ is equal to 1/b : ∂ 1 + aA − λ u(x, bτ ) = νG(u(x, bτ ) − f (x, bτ )) i b∂τ with the τ -periodicity condition u|τ =0 = u|τ =1 . 2. Thus, problem (1), (2) can be reduced to the problem on periodic solutions for the equation ∂ 1 + aA − λ u(x, t) = νG(u − f ) (3) (L − λ)u ≡ i b ∂t with the ﬁxed t-periodicity condition u|t=0 = u|t=1 . Here we assume that a = 0 and λ are given real numbers. We assume that the operation ∂ 1 ∂ 1 + aA = + a(d + δ) i b ∂t ib ∂t

(4)

is deﬁned on the space of forms u(x, t) ∈ C ∞ ([0, 1], CS∞ ) satisfying condition (4). 1 ∂ + a(d + δ) in H = L2 ([0, 1], HS0 ). Thus, an By L we denote the closure of the operation ib ∂t ∂ 1 + aA if there exists element u ∈ H belongs to the domain D (L) of the operator L = i b ∂t a sequence {uj } ⊂ C ∞ ([0, 1], CS∞ ), uj |t=0 = uj |t=1 , such that lim uj = u and lim Luj = Lu in H as j → ∞. Lemma 1. The forms ekmγ = 2−1 exp{iπ(γkx + 2mt)}(1 + γi dx), k, m ∈ Z, γ = ±1, are eigenforms of the operator L with the corresponding eigenvalues λkm = π(2m/b + ak)

(5)

on the space H = L2 ([0, 1], HS0 ). These forms form an orthonormal basis in this space. The domain of the operator L is

2 2 D (L) = u = ukmγ ekmγ |ukmγ | < ∞ . |λkm ukmγ | < ∞, The spectrum σ(L) of the operator L is the closure of the set {λkm }. Lemma 2. Let M0 = maxΠ2 g(x, y) . Then G ≤ 2M0 . Here 2 2 2 2 (x, y) + g12 (x, y) + g21 (x, y) + g22 (x, y)

g(x, y) = g11

is the Euclidean norm of the matrix g. Proof. We have

2 2 2 2 2 2 2 2

Gu(x, t) =

u H = gu dy ≤ g dy u dy, 0 0 0 0 0 ⎞ ⎛ 2 2 1 1 2 2 ⎝ g 2 dy u 2 dy ⎠ dx dt

Gu 2 dx dt ≤

Gu(x, t) 2H = 21

u20 + u21 dx dt,

0 0

0 0

22

g 2 dx dy

≤ 0 0

0

0

21

u 2 dy dt ≤ 4M02 u 2H ,

G ≤ 2M0 .

0 0

The proof of the lemma is complete. DIFFERENTIAL EQUATIONS

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799

2 By Gx2 we denote the integral operator with kernel gx2 (x, y) = gjl x2 j,l=1 . We set { g , gx2 }. M = max 2 Π

Lemma 3. Let v = Gu =

vkmγ ekmγ ; then 2 ≤ 4M 2 u 2/(|k| + 1)4 . vkmγ

(6)

2 ≤ |αkmγ | /(|k| + 1)4 , where αkmγ = Gx2 u, ekmγ H . Moreover, if k = 0, then vkmγ 2

Proof. We have 21 2 vkmγ = Gu, ekmγ H = 21 =

⎛ ⎝

0 0

gu dy, ekmγ 0 0

2

21 dx dt =

0

0 0

⎞

⎞ ⎛ 2 ⎝ gu, ekmγ dy ⎠ dx dt 0

gu, ekmγ dx⎠ dy dt.

0

If k = 0, then ⎞ ⎛ 2 1 ⎝ gx2 u, ekmγ dx⎠ dy dt = (−iπkγ)2 0 0 0 ⎞ ⎛ 21 2 1 ⎝ gx2 u, ekmγ dy ⎠ dx dt = (−iπkγ)2 21

vkmγ

0 0

1 = (−iπkγ)2

0

21 2 0 0

gx2 u dy, ekmγ

dx dt =

1 Gx2 u, ekmγ H , (−iπkγ)2

0

2 where Gx2 u = 0 gx2 u dy. We set αkmγ = Gx2 u, ekmγ H . Then, by virtue of the Parseval relation and Lemma 2, we have the inequality

2 α2kmγ ≤ α2kmγ = Gx2 u ≤ 4M 2 u 2 . k=0

Hence it follows that

2

1 |αkmγ | 4M 2 u 2 ≤ . |vkmγ | ≤ 4 4 α2kmγ ≤ π k (|k| + 1)4 (|k| + 1)4 2

2

Further, for k = 0, we have |v0mγ | = Gu, e0mγ H . By using the Parseval relation and Lemma 2, we obtain the estimate

2 2 |vkmγ | = Gu 2 ≤ 4M 2 u 2 ; |v0mγ | ≤ 2

therefore, |vkmγ | ≤ 4M 2 u 2 /(|k| + 1)4 for all k, m ∈ Z and γ = ±1. The proof of the lemma is complete. Since a and λ are real numbers, it follows from Lemma 1 that the spectrum σ(L) of the operator L lies on the real axis. The following cases are possible. (1) ab is a rational number. In this case, we have σ(L) = {λkm }, and this is a discrete set. Therefore, if λ = λkm for all k, m ∈ Z, then there exists an inverse operator (L − λ)−1 , and its norm DIFFERENTIAL EQUATIONS

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DANG HAN HOI

can be estimated via the distance from λ to the spectrum. Note that the set of rational numbers has zero measure, and this case is exceptional. (2) ab is an irrational number (this is a typical case). In this case, the set of numbers λkm is everywhere dense on the real axis, and σ(L) = R. Now we also assume that λ = λkm for all k, m ∈ Z; then the inverse operator (L − λ)−1 is deﬁned but unbounded.

More precisely, there exists a subsequence of eigenvalues λkm(k) such that λkm(k) − λ → 0. In the expression for the inverse operator (L − λ)−1 , there appear small denominators (L − λ)−1 v(x, t) =

vkmγ ekmγ , λkm − λ

(7)

where vkmγ are the Fourier coeﬃcients of the series

v(x, t) =

vkmγ ekmγ .

k,m∈Z, γ=±1

For positive numbers σ and C, by Aσ (C) we denote the set of positive numbers b such that the inequality C ≡ C˜ (8) |λkm − λ| ≥ (|k| + 1)1+σ is valid for all integers m and k. It follows from the deﬁnition that the sets Aσ (C) grow as C decreases and σ increases. Therefore, in the following, to prove that such a set or its part is nonempty, we have to require that C is suﬃciently small and σ is suﬃciently large. By Aσ we denote the union of the sets Aσ (C) with respect to C > 0. If inequality (8) is valid for some b for all m and k, then it holds for m = 0, which implies a necessary condition for the set Aσ (C) to be nonempty: C ≤ (|k| + 1)1+σ |akπ − λ|

∀k ∈ Z.

(9)

We set d = min(|k| + 1)1+σ |akπ − λ| > 0. Theorem 1. The sets Aσ (C) and Aσ are Borel sets. Furthermore, Aσ is a set of full measure; i.e., its complement on the half-line R+ has the zero measure. ∞ Proof. Obviously, the sets Aσ (C) are closed. Further, Aσ = n=1 Aσ (1/n) is a Borel set, since it is a union of closed sets. Let us show that Aσ has full measure on R+ . Suppose tat b, l > 0 and C ≤ d/2 and consider the complement (0, l)\Aσ (C). This set consists of positive numbers b for which there exist m and k such that ˜ (10) |λkm − λ| < C. By solving this inequality for b, we ﬁnd that, for ﬁxed m and k, these numbers form an interval of the form Ik,m = (mαk , mβk ), m = 1, 2, . . . , where αk =

2π , |akπ − λ| + C˜

βk =

2π . |akπ − λ| − C˜

The length of this interval is mδk , where δk =

4π C˜ . |akπ − λ|2 − C˜ 2

By virtue of the assumption that C ≤ d/2, we have

˜ 3|akπ − λ|2 . δk ≤ 16π C/ DIFFERENTIAL EQUATIONS

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801

For a ﬁxed k and for diﬀerent m, there exist ﬁnitely many above-mentioned intervals Ikm meeting the given interval (0, l). Such intervals exist for m = 1, 2, . . . such that mαk < l, i.e., the double inequality 0 < m < l(2π)−1 |akπ − λ| + C˜ is valid. Since C˜ ≤ |akπ − λ|/2, we have the simpler constraints for the above-mentioned values of m : 0<m<

l l 3 |akπ − λ| < |akπ − λ|. 2π 2 π

(12)

The measure of the union of the above-mentioned intervals (for a ﬁxed k) can be estimated from above by the number δk S˜k , where S˜k = S˜k (l) is the sum of positive integers m satisfying inequality (12). By the formula for the sum of an arithmetic progression, we have l S˜k ≤ 2 |akπ − λ|{l|akπ − λ| + π}. 2π

(13)

By considering the union of the above-mentioned intervals with respect to k and with respect to m and by using (11), we obtain μ ((0, l)\Aσ (C)) ≤

δk S˜k ≤ CS(l),

S(l) =

k∈Z

k∈Z

8l{l|akπ − λ| + π} . 3π(|k| + 1)1+σ |akπ − λ|

Note that the quantity (l|akπ−λ|+π)/(π|akπ−λ|) is bounded above by some constant D; therefore, S(l) ≤

1 8

lD < ∞. 1+σ 3 (|k| + 1) k∈Z

We have μ ((0, l)\Aσ ) ≤ μ ((0, l)\Aσ (C)) ≤ CS(l) for all C > 0. It follows that μ ((0, l)\Aσ ) = 0 for all l > 0. Therefore, μ ((0, ∞)\Aσ ) = 0, and Aσ is a set of full measure. The proof of the theorem is complete. Theorem 2. Let g(x, y) be a matrix deﬁned on Π2 and such that the entries of the matrices g, and gx2 are continuous on Π2 , 0 < σ < 1, b ∈ Aσ (1/n). Then the inverse operator (L − λ)−1 is deﬁned, and the product (L − λ)−1 ◦ G is a compact operator. gx ,

Proof. Since b ∈ Aσ (1/n), we have λkm = λ for all k, m ∈ Z, and the inverse operator (L − λ)−1 is deﬁned; its expression has the form (7). Note that lim|k|→∞ (|k| + 1)−2+2σ = 0. Therefore, for a given positive number ε > 0, there exists an integer k0 > 0 such that (|k| + 1)−2+2σ < ε2 (2M n)−2 for all k, |k| > k0 . We set v = Gu. (L − λ)−1 v(x, t) = Qk0 1 v + Qk0 × 2v, Here Qk0 1 v =

|k|≤k0

vkmγ ekmγ , λkm − λ

Qk0 × 2v =

|k|>k0

vkmγ ekmγ . λkm − λ

Consider the operator Qk0 1 . We have 2

Qk0 1 v =

|vkmγ |

|k|≤k0

|λkm − λ|

2 2.

Note that if |k| ≤ k0 , then the limit relation lim|m|→∞ |2mπ/b + ak − λ|−2 = 0 is valid. Therefore, the quantity under the sign of the limit is bounded above by the constant C (k0 ). Then we have the inequality

2 2 |vkmγ | C (k0 ) ≤ C (k0 ) v 2 .

Qk0 1 v ≤ |k|≤k0

It follows that Qk0 1 is a bounded operator. DIFFERENTIAL EQUATIONS

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802

DANG HAN HOI

Consider the operator Qk0 2 ◦ G. By Lemma 3 and condition (8) with C = 1/n, we have 2

2

Qk0 2 v = Qk0 2 ◦ Gu =

|vkmγ |

2 2

≤

|λkm − λ| |k|>k0 ε 2

2 ≤ n2 |αkmγ | ≤ ε2 u 2 . 2M n

|k|>k0

=

|k|>k0

n2 α2kmγ (|k| + 1)2−2σ

α2kmγ n2 (|k| + 1)2+2σ (|k| + 1)4

|k|>k0

It follows that Qk0 2 ◦ G ≤ ε. Since G is a compact operator and Qk0 1 are bounded operators, it follows that Qk0 1 ◦ G is a compact operator. Further, we have (L − λ)−1 ◦ G − Qk0 1 ◦ G = Qk0 2 ◦ G < ε. Thus (L − λ)−1 ◦ G is the limit of a sequence of compact operators and hence a compact operator itself. The proof of the theorem is complete. We set K = Kb = (L − λ)−1 ◦ G. Theorem 3. Let b ∈ Aσ (1/n); then problem (1), (2) has a unique periodic solution with period b for all ν ∈ C except for an at most countable discrete set of values. Proof. Equation (1) can be reduced to an equation of the form

(L − λ)−1 ◦ G − 1/ν u = (L − λ)−1 ◦ G(f ). We set (L − λ)−1 ◦ G − 1/ν = K − 1/ν. Since K = (L − λ)−1 ◦ G is a compact operator, it follows that its spectrum σ(K) is at most countable, and zero is the only possible limit point of this set. Therefore, the set S = {ν = 0 | 1/ν ∈ σ(K)} is at most countable and discrete, and the operator (K − 1/ν)−1 is deﬁned [i.e., Eq. (1) has a unique solution] for all ν = 0, ν ∈ S. The proof of the theorem is complete. Let us study the solvability of problem (1), (2) for a given value of the parameter ν. To this end, one should analyze the structure of the plane set E ⊂ C × R+ of pairs (ν, b) such that ν = 0 and 1/ν ∈ σ (Kb ), where Kb = (L − λ)−1 ◦ G. Theorem 4. The set E is a measurable set of full measure in C × R+ . To prove Theorem 4, we need a number of auxiliary assertions. ˜ b < ε Lemma 4. For each positive integer ε, there exists a positive integer k0 such that Kb − K for arbitrary b ∈ Aσ (1/n), 0 < σ < 1, where −1

Kb u = (Lb − λ) ˜ bu = K

|k|≤k0

−1

◦ Gu = (Lb − λ)

v=

|k|>k0

vkmγ ekmγ , λkm (b) − λ

vkmγ ekmγ . λkm (b) − λ

Proof. Note that, for each ε > 0, there exists a positive integer k0 such that (|k| + 1)−2+2σ ≤ (ε/(2nM ))2

for

|k| ≥ k0 ,

0 < σ < 1.

DIFFERENTIAL EQUATIONS

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ON THE STRUCTURE OF THE PERIOD SET OF PERIODIC SOLUTIONS

We have

803

˜ b u = Kk0 b u = Kb − K

vkmγ ekmγ , λkm (b) − λ |k|>k0 2 2

vkmγ

n2 α2kmγ 2 ˜ − K =

K u

= ≤ u K b b k0 b λkm (b) − λ (|k| + 1)2−2σ |k|>k0 |k|>k0 ε 2

ε 2 2 ≤ n2 |αkmγ | ≤ n2 × 4M 2 u 2 = ε2 u 2 . 2nM 2nM |k|>k0

˜ b = Kk0 b < ε, which completes the proof. Thus, Kb − K Lemma 5. The operator function Kb is continuous with respect to b ∈ Aσ (1/n). Proof. Let b, b + Δb ∈ Aσ (1/n) and ε > 0 be given. By Lemma 4, there exists a positive integer k0 such that ˜ b ˜ b+Δb Kb+Δb − K = Kk0 b < ε, = Kk0 (b+Δb) < ε. Kb − K ˜ b+Δb + Kk0 (b+Δb) − K ˜ b + Kk0 b , Kb+Δb − Kb = K

We have which implies that

˜ ˜

Kb+Δb − Kb ≤ K b+Δb − Kb + Kk0 (b+Δb) + Kk0 b . ˜ b . We have ˜ b+Δb and K Consider the operators K

1 1 ˜b u = ˜ b+Δb − K − vkmγ ekmγ , K λkm (b + Δb) − λ λkm (b) − λ 2 ˜ ˜ b+Δb u = Kb u − K

|k|≤k0

2

|Δb|2 |vkmγ | 4m2 π 2 2 2. |b(b + Δb)|2 |λkm (b + Δb) − λ| |λkm (b) − λ| |k|≤k0

(14)

For b + Δb ∈ Aσ (1/n), we have |vkmγ |

2

|λkm (b + Δb) − λ|

2

2

≤ n2

|αkmγ | . (|k| + 1)2−2σ 2

By virtue of the limit relation limm→∞ 4m2 π 2 /|λkm (b) − λ| = b2 and the condition |k| ≤ k0 , the quantity 2 4m2 π 2 / |λkm (b) − λ| = 4m2 π 2 /|2mπ/b + ak − λ|2 is bounded by the constant C (k0 ) depending on k0 . Therefore, by virtue of the representation (14), 2 ˜ ˜ b+Δb u ≤ Kb u − K

|αkmγ |2 n2 |Δb|2 C (k0 ) |b(b + Δb)|2 (|k| + 1)2−2σ |k|≤k0

≤

|Δb| 2 |αkmγ | n2 C (k0 ) |b(b + Δb)|2

≤

|Δb| n2 C (k0 ) × 4M 2 u 2 . |b(b + Δb)|2

2

|k|≤k0

DIFFERENTIAL EQUATIONS

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804

DANG HAN HOI

It follows that

2 ˜ ˜ b ≤ |Δb|2 |b(b + Δb)|−2 × 4M 2 n2 C (k0 ) . Kb+Δb − K

We take Δb such that

|Δb|2 |b(b + Δb)|−2 × 4M 2 n2 C (k0 ) < ε. Then Kb+Δb − Kb < 3ε. It follows that the operator function Kb is continuous with respect to b on Aσ (1/n). The proof of the lemma is complete. Lemma 6. The spectrum σ(K) of a compact operator K continuously depends on K in the space Comp(H ) of compact operators on H in the sense that, for each positive ε, there exists a δ > 0 such that the inclusions σ(B) ⊂ σ(K) + Vε (0),

σ(K) ⊂ σ(B) + Vε (0)

(15)

are valid for all compact (and even bounded) operators B satisfying the inequality B − K < δ. Here Vε (0) = {λ ∈ C | |λ| < ε} is the ε-neighborhood of zero in C. Proof. Let K be a compact operator, and let ε > 0. It follows from the structure of the spectrum σ(K) of K that there exists an ε1 < ε/2 and neighborhoods Vε1 (0) and Vε1 (λ), where λ ranges over the ﬁnite set S = {λ1 , . . . , λk } of points of the spectrum with |λ| > ε1 , such that the boundaries of these neighborhoods do not contain points of the spectrum. Let V = λ∈S∪{0} Vε1 (λ). Then V is a neighborhood of the spectrum σ(K), and V ⊂ σ(K) + Vε (0). By the well-known properties of spectra (e.g., see [5, Th. 10.20]), there exists a δ > 0 such that σ(B) ⊂ V for all bounded operators B with B − K < δ. In addition (e.g., see [5, p. 293 of the Russian translation, Exercise 20]), the quantity δ > 0 can be chosen so as to ensure that σ(B) ∩ Vε1 (λ) = ∅ for all λ ∈ S ∪ {0}. Then the desired inclusions σ(K) ⊂ σ(B) + V2ε1 (0) ⊂ σ(B) + Vε (0),

σ(B) ⊂ V ⊂ σ(K) + Vε (0)

are valid for all bounded operators B such that B − K < δ. The proof of the lemma is complete. Lemma 6 readily implies the following assertion. Assertion. The function (λ, K) = dist(λ, σ(K)) is continuous on C × Comp(H ). Proof. Let λ ∈ C, K ∈ Comp(H ), and ε > 0. By Lemma 6, there exists a positive number δ such that the inclusion (15) is valid for any operator H lying in the δ-neighborhood H − K < δ, which readily implies the estimate | (λ, K) − (λ, H)| < ε. Then

| (μ, K) − (λ, H)| ≤ | (μ, K) − (λ, K)| + | (λ, K) − (λ, H)| < |μ − λ| + ε < 2ε

for all μ ∈ C, |μ − λ| < ε, and H, H − K < δ; since ε > 0 is arbitrary, it follows that the function

(λ, K) is continuous function. The proof of the assertion is complete. The above-proved assertion, together with Lemma 5, implies the following corollary. Corollary 1. The function (λ, b) = dist (λ, σ (Kb )) is continuous with respect to (λ, b) ∈ C × Aσ (1/n). Proof of Theorem 4. By Corollary 1, (1/ν, b) is a continuous function of the variables (ν, b) ∈ (C\{0}) × Aσ (1/n); therefore, the set Bn = {(ν, b) | (1/ν, b) = 0, b ∈ Aσ (1/n)} is measurable. Therefore, B = n Bn is measurable. Obviously, B ⊂ E and E = B ∪ B0 , where B0 = E\B. Clearly, B0 lies in the set C × (R+ \Aσ ) of zero measure (recall that, by Theorem 1, the set Aσ has full Lebesgue measure on R+ ); and since the Lebesgue measure is complete, it follows that B0 is a measurable set. Thus, E is a measurable set, since it is a union of measurable sets. Further, by Theorem 3, the cross-section E b = {ν ∈ C | (ν, b) ∈ E} has full measure for b ∈ Aσ , since its complement {1/ν | ν ∈ σ (Kb )} is at most countable. Therefore, the set E has a complete plane Lebesgue measure. The proof of the theorem is complete. This theorem implies the following assertion. DIFFERENTIAL EQUATIONS

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805

Corollary 2. For almost all ν ∈ C, problem (1), (2) has a unique periodic solution for almost all values of the period b ∈ R+ . Proof. Since the set E is measurable, it follows that, for almost all ν ∈ C, the cross-section Eν = {b ∈ R+ | (ν, b) ∈ E} = {b ∈ R+ | 1/ν ∈ σ (Kb )} has full measure, therefore, problem (1), (2) has the unique b-periodic solution for these b. The proof of the corollary is complete. ACKNOWLEDGMENTS The author is grateful to E.Yu. Panov for attention to the research. REFERENCES 1. Dang Han Hoi, Tez. dokl. mezhdunar. konf. “Diﬀerentsial’nye uravneniya i smezhnye voprosy.” Moskva, 16–22 maya 2004 g. (Abstr. Int. Conf. “Diﬀerential Equations and Related Topics,” May 16–22, 2004, Moscow, Russia), Moscow, 2004, p. 48. 2. Dang Han Hoi, Vestnik Novgor. Gos. Univ. Tekhn. Nauki, 2004, no. 28, pp. 77–79. 3. Palais, R. et al., Seminar on the Atiyah–Singer Index Theorem, Princeton: Princeton University Press, 1965. Translated under the title Seminar po teoreme At’i-Zingera ob indekse, Moscow: Mir, 1970. 4. Pham Ngoc Thao, Diﬀer. Uravn., 1969, vol. 5, no. 1, pp. 186–198. 5. Rudin, W., Functional Analysis, New York: McGraw-Hill, 1973. Translated under the title Funktsional’nyi analiz , Moscow: Mir, 1975.

DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 806–812. c M.T. Dzhenaliev, M.I. Ramazanov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 788–794. Original Russian Text

PARTIAL DIFFERENTIAL EQUATIONS

On a Boundary Value Problem for a Spectrally Loaded Heat Operator: II M. T. Dzhenaliev and M. I. Ramazanov Institute of Mathematics, Ministry of Education and Science, Almaty, Kazakhstan Received February 17, 2005

DOI: 10.1134/S0012266107060079

The present paper is a continuation of [1]. Here we give a solution of adjoint Volterra integral equations by the method of regularization by a solution of characteristic equations (Section 5). The solution of the original boundary value problems is given (Section 6). Finally, Sections 7 and 8 deal with spectral problems. In conclusion, we present a number of problems whose analysis can be performed by a similar scheme. We continue the numbering of formulas and sections in [1]. 5. ON THE SOLVABILITY OF THE INTEGRAL EQUATIONS (17) AND (22) BY THE REGULARIZATION METHOD [2] We introduce the notation

K˜

(t, τ ) = K2 (t, τ ) − K (t, τ )

(66)

and, by using (26), rewrite the integral equation (17) in the form t Kλ μ ≡ (I − λK)μ = λ

K˜

(t, τ )μ(τ )dτ + f1 (t),

t ∈ R+ .

(67)

0

We solve the last equation as a characteristic equation temporarily assuming that the right-hand side is a known function. Then, by (63), we obtain t

K˜

μ(t) = f1 (t) + λ ⎡

t (t, τ )μ(τ )dτ + λ ⎤

0

× ⎣f1 (τ ) + λ

τ −2 rλ− t−1 − τ −1

0

τ

K˜ (τ, η)μ(η)dη⎦ dτ +

N2

ck exp −izk t−1 ,

t ∈ R+ .

(68)

k=−N1

0

We reduce Eq. (68) to the form t

ˆ μ ≡ μ(t) − λ Kˆ (t, τ )μ(τ )dτ ˆ λ μ ≡ I − λK K 0

= fˆ(t) +

N2

ck exp −izk t−1 ,

k=−N1

806

t ∈ R+ ,

(69)

ON A BOUNDARY VALUE PROBLEM FOR A SPECTRALLY LOADED HEAT . . . : II

807

where t

Kˆ

η −2 rλ− t−1 − η −1 K˜ (η, τ )dη ≡ K˜ (t, τ ) + λK˜˜ (t, τ ),

(t, τ ) = K˜ (t, τ ) + λ τ

t fˆ(t) = f1 (t) + λ

τ −2 rλ− t−1 − τ −1 f1 (τ )dτ.

0

By using the estimates for (30) and (52), we ﬁnd that the kernel Kˆ (t, τ ) has a weak singularity; i.e., t3/4 ˆ , 0 < τ < t < ∞. (70) K (t, τ ) ≤ C 3/4 √ τ t−τ Since Kˆ (t, τ ) admits the representation K˜ (t, τ ) + λK˜˜ (t, τ ), we ﬁnd that the estimate (70) is a consequence of (30), (52), and the following relations. First, we prove the inequality

K˜˜ (t, τ ) ≤ M1

t τ

√ √ η δ0 (t − η) 1 tη

√ exp − dη η 2 τ (η − τ ) t − η tη

√ η t3/2 η 3/2 δ0 tη 1

dη exp − + M2 η 2 τ (η − τ ) (t − η)3/2 t−η τ t t I1 (t, τ ) + M2 I2 (t, τ ). = M1 τ τ t

Further, we represent each of the integrals I1 (t, τ ) and I2 (t, τ ) as the sums I1 (t, τ ) = I11 (t, τ ) + I12 (t, τ );

I2 (t, τ ) = I21 (t, τ ) + I22 (t, τ )

of two integrals; the integration in I11 and I21 is performed from τ to (t + τ )/2, and the integration in I12 and I22 , from (t + τ )/2 to t. For each of the last integrals, we have (t+τ )/2

I11 (t, τ ) ≤ τ

√ √ t dη 2

≤ √ δ0 (t − η) η(η − τ ) δ0 (t − τ )

√ 1 2 2 t1/4

≤ τ 1/4 δ0 (t − τ ) √ τ /t

2 √ I12 (t, τ ) ≤ (t + τ ) t − τ

√

2t I21 (t, τ ) ≤ δ0 (t − τ ) √

M t ≤√ t−τ

τ

(t+τ )/2

τ

DIFFERENTIAL EQUATIONS

τ

t dη (t − η)η(η − τ )

√ 2 2π dy t1/4 √ √ = ,

δ0 τ 1/4 t − τ (1 − y) y − τ /t

(t+τ )/2

t

δ0 (t − η) exp − t2

(t+τ )/2

(t+τ )/2

δ0 (t − η) 1 √ exp − dη t−η tη

t

−4t √ ≤√ δ0 (t + τ ) t − τ

√

(t+τ )/2

√ δ0 (t − η) 2 π , ≤ d t δ0 (t − τ )

√ δ0 tη dη δ0 tη √

exp − t−η t−η η(t − η)(η − τ )

dη ; η(t − η)(η − τ )

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DZHENALIEV, RAMAZANOV

by analogy with the estimate for I11 (t, τ ), we ﬁnally obtain 2πM t1/4 √ , τ 1/4 t − τ √ t 2 I22 (t, τ ) ≤ √ t−τ

I21 (t, τ ) ≤

δ0 t(t + τ ) t dη exp − (t − η)3/2 2(t − η)

(t+τ )/2

√ 2 =√ t−τ

t

δ0 tτ δ0 t2 t − dη exp − (t − η)3/2 2(t − η) 2(t − η)

(t+τ )/2

√ √ t 2 π 1 δ0 t2 2 t ≤√ . dη ≤ √ √ exp − (t − η)3/2 2(t − η) t−τ δ0 t − τ 0

These inequalities imply the desired estimate (70). Thus, by virtue of the estimate (70), for a given right-hand side, Eq. (69) has a unique solution whose existence can be justiﬁed by the successive approximation method. It follows from (17) and (69) that the homogeneous equation t K2λ μ ≡ (I − λK2 ) μ ≡ μ(t) − λ

K2 (t, τ )μ(τ )dτ

= 0,

t ∈ R+ ,

(71)

0

is equivalent to the inhomogeneous equation t ˆ λ μ ≡ μ(t) − λ K

Kˆ

(t, τ )μ(τ )dτ =

N2

ck exp −izk t−1 ,

t ∈ R+ .

(72)

k=−N1

0

Instead of (72), we consider the family of integral equations ˆ λ μ = exp −izk t−1 , k = −N1 , . . . , 0, . . . , N2 , K

t ∈ R+ .

(73)

Further, since each of Eqs. (73) has a unique nontrivial solution μλk (t), k = −N1 , . . . , 0, . . . , N2 (corresponding to its right-hand side), it follows that, for each parameter value λ ∈ C\D0 , these functions μλk (t), k = −N1 , . . . , 0, . . . , N2 , are the corresponding eigenfunctions of the homogeneous equation (71) [and, therefore, the homogeneous equation for (17)]. Lemmas 1 and 2 imply the following assertion. Lemma 4. The values λ ∈ D0 in (45) are regular numbers of the operator K2 given by (17). Lemma 5. The set C\D0 consists of the characteristic numbers of the operator K2 given by (17). Moreover, if λ ∈ Dm ∪ Γm−1 \{(−1)m emπ } , m = 1, 2, . . . , then dim Ker (K2λ ) = κ(λ) = m, and the corresponding eigenfunctions are given by the solutions of Eq. (73), −1 ˆλ exp −izk t−1 , k = 1, . . . , m = κ(λ) = N1 + N2 + 1. μλk (t) = K Remark 5. A general solution of the inhomogeneous integral equation (69) as well as (17) is m=N 1 +N2 +1 −1 ˆ λ fˆ(t) + ck μλk (t), μλ (t) = K

t ∈ R+ ,

(74)

k=1

where ck , k = 1, . . . , m, are arbitrary constants. DIFFERENTIAL EQUATIONS

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809

Let is proceed to the analysis of the integral equation (22), which is the adjoint equation for (17). We show that the homogeneous integral equation corresponding to (22) has only the trivial solution −1 −1 for any λ ∈ C. To this end, we make the change −1 2of independent variables t = t1 , τ = τ1 in the homogeneous equation and set ν1 (t1 ) = ν t1 /t1 ; then we obtain 3/2 3/2 t1 1 τ1 √ t1 2 π (t1 − τ1 ) τ1 0 t1 × exp − ν1 (τ1 ) dτ1 = 0, 4 (t1 − τ1 ) τ1

t1 ν1 (t1 ) − λ

t1 ∈ R + .

(75)

Now the desired assertion follows from the boundedness of the kernel of the integral operator (75). Thus, in view of Lemmas 3–5, we have the following assertion. Lemma 6. 1. Each value λ ∈ C is a regular number of the operator K∗2 given by (22). 2. The inhomogeneous integral equation (22) is uniquely solvable for any right-hand side g1 (t) if λ ∈ D0 (45). 3. If λ ∈ Dm ∪ Γm−1 \ {(−1)m emπ } , m = 1, 2, . . . , then for the unique solvability of the inhomogeneous integral equation (22), it is necessary and suﬃcient that the function g1 (t) satisfy the orthogonality conditions ∞ μλk (t)g1 (t)dt = 0,

k = 1, . . . , m = κ(λ) = N1 + N2 + 1.

(76)

0

Remark 6. By Lemma 6, the solution of the inhomogeneous integral equation (22) is given by −1

νλ (t) = [K∗2λ ]

g1 (t),

t ∈ R+ .

(77)

Remark 7. The above-represented results readily imply that μλ (t) ∈ M (R+ ) ,

νλ (t) ∈ L1 (R+ ) .

(78)

This is consistent with conditions (6) and (7). 6. ANALYSIS OF THE BOUNDARY VALUE PROBLEMS (1) AND (2) By using (11), we write out the solution of problem (1) in the form t u(x, t) = −λ

τ

−1/2

t∞

K0 (x, t − τ )μλ (τ )dτ +

0

τ −1/2 G(x, ξ, t − τ )τ 1/2 f (ξ, τ )dξ dτ,

(79)

0 0

where the functions μλ (t) [see (74)] and t1/2 f (x, t) are bounded and continuous on R+ and Q, respectively. Since the functions K0 (x, t − τ ) and G(x, ξ, t − τ ) are nonnegative, from (79), we have the estimate t (80) |u(x, t)| ≤ (C1 |λ| + C2 ) τ −1/2 K0 (x, t − τ )dτ ≤ C3λ x + t1/2 , 0

where C3λ = C1 |λ| + C2 . The derivatives of the solution u(x, t) given by (79) satisfy the relation √ √ t [ut (x, t) − uxx (x, t)] = −λμ(t) + tf (x, t) ∈ M (Q) [which follows from Eq. (1)]. DIFFERENTIAL EQUATIONS

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(81)

810

DZHENALIEV, RAMAZANOV

Thus, the function (79) satisﬁes Eq. (1) in the sense of (81). One can directly verify that the solution u(x, t) given by (79) satisﬁes the initial and boundary condition in (1). Therefore, by (80) and (81), the function (79) completely satisﬁes the boundary value problem (1) and belongs to the class (6). Further, by (20), we rewrite the solution of problem (2) in the form ∞ v(x, t) = −λ

∞∞ τ

1/2

Gξξ (x, ξ, τ − t)|ξ=τ νλ (τ )dτ +

t

G(x, ξ, τ − t)g(ξ, τ )dξ dτ,

(82)

t 0

where νλ (t) ∈ L1 (R+ ) is the function given by (77). For the function v(x, t) to belong to the class (7), it is suﬃcient that 1

∞ τ 1/2 Gξξ (x, ξ, τ − t)|ξ=τ ν(τ )dτ ∈ L1 (Q),

t1/2 t

(83)

∞∞

1

G(x, ξ, τ − t)g(ξ, τ )dξ dτ ∈ L1 (Q).

t1/2

(84)

t 0

The inclusion (84) is indeed valid by virtue of condition (5), and the inclusion (83) is equivalent to the inequality ∞∞

1

∞ τ 1/2 Gξξ (x, ξ, τ − t)|ξ=τ ν(τ )dτ dt dx ≤ νL1 (R+ ) ≤ ∞.

t1/2 0 0

t

Obviously, the derivatives vt (x, t) and vxx (x, t) of the function v(x, t) satisfy the inclusion

√ x + t (vt + vxx ) ∈ L1 (Q). Remark 8. From Eq. (2), we additionally obtain

√ t 1 + t ν(t) ∈ L1 (R+ ) . Let us state the obtained results on the solvability of the boundary value problems (1) and (2) in the form of the following assertions. Theorem 1. If λ ∈ D0 , then, for each function f satisfying condition (5), the boundary value problem (1) has a unique solution u ∈ U . If λ ∈ {C\D0 } ∩ {Dm ∪ Γm−1 \ {(−1)m emπ }} , then for each function f satisfying condition (5), the boundary value problem (1) has the general solution u ∈ U that is the sum of the solution uhom (x, t) =

m

ck uλk (x, t)

(85)

k=1

of the homogeneous equation, where t uλk (x, t) = −λ

τ −1/2 K0 (x, t − τ )μλk (τ )dτ,

k = 1, . . . , m,

(86)

0

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ON A BOUNDARY VALUE PROBLEM FOR A SPECTRALLY LOADED HEAT . . . : II

811

−1 ˆλ μλk = K [exp (−izk t−1 )] , k = 1, . . . , m, and ck , k = 1, . . . , m, are arbitrary constants, and the particular solution t upart (x, t) = −λ

−1 ˆλ τ −1/2 K0 (x, t − τ ) K fˆ(τ )dτ

0

t∞ +

τ −1/2 G(x, ξ, t − τ )τ 1/2 f (ξ, τ )dξ dτ.

(87)

0 0

Theorem 2. If λ ∈ D0 , then, for each function g satisfying condition (5), the boundary value problem (2) has a unique solution v ∈ V . If λ ∈ {C\D0 } ∩ {Dm ∪ Γm−1 \{(−1)m emπ }} , then for the unique solvability of the boundary value problem (2) in the class V , it is necessary and suﬃcient that the function g satisfying condition (5) satisﬁes the orthogonality conditions ∞ uλk (x, t)g(x, t)dx dt = 0,

k = 1, . . . , m = κ(λ) = N1 + N2 + 1.

(88)

0

7. ON THE SPECTRUM OF THE OPERATORS L1 GIVEN BY (3) AND L∗1 GIVEN BY (4) The following assertion is a straightforward consequence of Lemmas 4–6. Theorem 3. The open set D0 given by (45) is the resolvent set of the operator L1 given by (3), and its complement C\D0 is the spectrum of L1 . Moreover, if λ ∈ Dm ∪ Γm−1 \ {(−1)m emπ } ,

m = 1, 2, . . . ,

then dim Ker (L1 ) = κ(λ) = m, and the corresponding eigenfunctions of L1 are given by the formulas t uλk (x, t) = −λ

τ −1/2 K0 (x, t − τ )μλ (τ )dτ,

k = 1, . . . , m = κ(λ) = N1 + N2 + 1,

(89)

0

where

−1 ˆλ exp −izk t−1 , μλk (t) = K

k = 1, . . . , m = κ(λ) = N1 + N2 + 1.

Theorem 4. The set of values λ ∈ C is the resolvent set of the operator L∗1 given by (4).

8. ON THE SPECTRUM OF THE OPERATORS Lλ GIVEN BY (1) AND L∗λ GIVEN BY (2) Now consider the spectral problems for the operators Lλ given by (1) and L∗λ given by (2), i.e., the problems of ﬁnding pairs {α, uα (x, t)} and {α, vα (x, t)} such that Lλ u = αu, L∗λ v = αv, DIFFERENTIAL EQUATIONS

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(90) (91)

812

DZHENALIEV, RAMAZANOV

where

Uα Vα

= u ⎧ ⎨ = v ⎩

√ −αt √ −αt e−αt √ u, te (ut − uxx ) ∈ M (Q), te uxx (x, t) ∈ M (R+ ) , x=t x+ t ⎫ ∞ ⎬

αt αt √ e e √ v, eαt x + t (vt + vxx ) ∈ L1 (Q), √ v(ξ, t)dξ ∈ L1 (R+ ) , ⎭ t t

(92) (93)

0

D (Lλ,α ) = {u | u ∈ Uα , u(x, 0) = 0, u(0, t) = 0} , D L∗λ,α = {v | v ∈ Vα , v(x, ∞) = 0, v(0, t) = 0, v(∞, t) = 0,

vx (∞, t) = 0} .

The change of variables u(x, t) = u1 (x, t)eαt , α ∈ C, reduces problem (90) to the aboveconsidered spectral problem for the operator L1 given by (3). The following assertion is a straightforward consequence of Theorem 3. Theorem 5. If a value λ belongs to the resolvent set D0 of the operator L1 given by (3), then each value α ∈ C belongs to the resolvent set of the operator Lλ given by (90); i.e., the spectrum of the operator Lλ is empty in this case. But if λ belongs to the spectrum C\D0 of the operator L1 , then each value α ∈ C belongs to the spectrum of the operator Lλ given by (90); i.e., in this case, the resolvent set of the operator Lλ is empty. Just as in problem (90), by using the change of variables v(x, t) = v1 (x, t)e−αt , one can derive the following assertion for the spectral problem (91) from Theorem 4. Theorem 6. The set of values α ∈ C is the resolvent set of the operator L∗λ given by (91). Obviously, the classes of solutions of the corresponding inhomogeneous boundary value problems for (90) and (91) are the function classes (92) and (93). CONCLUSION Along with the above-considered heat operator, it would be of interest to study operators of the form (with appropriate conditions) ut − uxx + λ ut (x, t)|x=t , ut − uxx + λu(x, t)|x=t , ut − uxx + λux (x, t)|x=t , ut − uxx + λuxx (x, t)|x|=t , ut − uxx + λuxx (x, t)x=¯x ,

{x, t} ∈ R2+ ; {x, t} ∈ R2+ ; {x, t} ∈ R2+ ;

u(x, 0) = u(0, t) = 0, u(x, 0) = u(0, t) = 0,

u(x, 0) = u(0, t) = 0, {x, t} ∈ R × R+ ; u(x, 0) = 0, 2 {x, t} ∈ R+ ; u(x, 0) = u(0, t) = 0,

where x ¯ ∈ R+ is a given point. Situations similar to the above-considered ones appear for these operators as well. However, the partition of the complex plane of the parameter λ into the resolvent and spectral sets is performed in a diﬀerent way for the latter operators. ACKNOWLEDGMENTS The authors are grateful to A.M. Nakhushev, T.Sh. Kal’menov, and M.O. Otelbaev for useful discussion of the results. REFERENCES 1. Dzhenaliev, M.T. and Ramazanov, M.I., Diﬀer. Uravn., 2007, vol. 43, no. 4, pp. 498–508. 2. Gakhov, F.D., Kraevye zadachi (Boundary Value Problems), Moscow: Nauka, 1963.

DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 813–832. c F.E. Lomovtsev, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 795–812. Original Russian Text

PARTIAL DIFFERENTIAL EQUATIONS

Cauchy Problems for Quasi-Hyperbolic Factorized Even-Order Diﬀerential Equations with Smooth Operator Coeﬃcients Having Variable Domains F. E. Lomovtsev Belarus State University, Minsk, Belarus Received April 22, 2004

DOI: 10.1134/S0012266107060080

Cauchy problems for quasi-hyperbolic factorized operator-diﬀerential equations of higher even orders with constant domains were considered in [1]. The Cauchy problem for hyperbolic operatordiﬀerential equations with variable domains was investigated in [2, 3] for second-order equations. The present paper deals with the proof of the well-posed solvability in the strong sense of Cauchy problems for some quasi-hyperbolic factorized operator-diﬀerential equations of higher even orders with unbounded operator coeﬃcients whose domains vary; mixed problems for some hyperbolic partial diﬀerential equations with coeﬃcients in the boundary conditions smoothly depending on time can be reduced to such problems. For the proof, we use modiﬁcations and generalizations of the functional method of energy inequalities in [1]. Unlike [1], in the present paper, the derivation of a priori estimates with the use of abstract smoothing operators is generalized to the case of variable domains of variable unbounded operator coeﬃcients; the proof of the solvability by induction, decomposition of operators into operator factors, and the use of the Lemmas 5 and 6 below is a new technique; and a formula for their strong solutions is derived for the ﬁrst time [see formula (25) below]. This formula generalizes a similar formula for smooth (classical) solutions and shows that, by analogy with smooth solutions, strong solutions of these Cauchy problems can be found in a recursive way on the basis of operator factors. In addition, unlike [1], in the present paper, we do not repeat any steps of the proofs in [4]. In conclusion, we consider an example of new well-posed mixed problems for hyperbolic factorized partial diﬀerential equations of even order with coeﬃcients in the boundary conditions depending on t and smooth with respect to t. 1. STATEMENT OF THE CAUCHY PROBLEMS Let H be a Hilbert space with inner product (· , ·) and norm | · |. On a bounded interval ]0, T [, we consider the diﬀerential equations Lm (t)u ≡ d2 /dt2 + Am (t) · · · d2 /dt2 + A1 (t) u = f, t ∈ ]0, T [, (1) with the initial conditions lj u ≡ dj u/dtj t=0 = ϕj ∈ H,

j = 0, . . . , 2m − 1,

m = 1, 2, . . . ,

(2)

where u and f are functions of the variable t ranging in H and Ak (t), t ∈ [0, T ], are positive self-adjoint operators in H with domains D (Ak (t)), k = 1, . . . , m, depending on t. We assume that all operators Ak (t) satisfy the following conditions. I. For each t ∈ [0, T ], the operators Ak (t) are therestrictions [to D (Ak (t))] of some linear unbounded operators A˜k (t) in H with domains D A˜k independent of t such that D (Ak (t)) ⊂ D A˜k and A˜k (t)u = Ak (t)u for all u ∈ D (Ak (t)), t ∈ [0, T ], k = 1, . . . , m. 813

814

LOMOVTSEV

II. For each t ∈ [0, T ], the inverse operators A−1 k (t) ∈ B ([0, T ], L(H)), k = 1, . . . , m, have strong (t)/dt ∈ B ([0, T ], L (H)) with respect to t [5, p. 22] in H, and these derivatives derivatives dA−1 k satisfy the inequalities (1) −1 ∀g ∈ H, k = 1, . . . , m, (3) Ak (t)g, g − dA−1 k (t)/dt g, g ≤ ck where B ([0, T ], E) is the set of all functions of t ∈ [0, T ] bounded in the norm of a Banach space (1) E and the constants ck ≥ 0 are independent of g and t. III. For all t ∈ [0, T ], the operators dA−1 k (t)/dt, k = 1, . . . , m, have strong derivatives 2 d2 A−1 k (t)/dt ∈ B ([0, T ], L(H))

which satisfy the inequalities 2 −1 d A (t)/dt2 g, v ≤ c(2) |g| A−1/2 (t)v k k k

in

H,

∀g, v ∈ H,

k = 1, . . . , m,

(4)

(2) (t) are the inverses of the square roots A1/2 ≥ 0 are constants where A−1/2 k k (t) of Ak (t) and ck independent of g, v, and t. IV. For each t ∈ [0, T ] and for all operators Ak (t), k = 1, . . . , m, the norms

|As (t)u| ∼ |Ak (t)u| ∼ |As (t)u − Ak (t)u| t ∈ [0, T ], 1 ≤ s = k ≤ m, ∀u ∈ D (Ak (t)) , m are equivalent, and the domains D (Am k (t)) of their powers Ak (t), t ∈ [0, T ], are dense in H. Following [1] and using induction over i, one can show that the norms i A (t)u ∼ Ai (t)u ∀u ∈ D (Am (t)) , t ∈ [0, T ], i = 1, . . . , m, 1 ≤ s, k ≤ m, s k k m are equivalent. Hence it follows |Am (t) · · · A1 (t)u| ∼ |Am 1 (t)u| for all u ∈ D (A1 (t)), t ∈ [0, T ]. that α/2m α/2m (t) of positive fractional orders A (t) of the self-adjoint By equipping the domains D A α/2 m operators A(t) = A1 (t) in H with the norms |v|α,t = |A1 (t)v| for each t ∈ [0, T ], we obtain Hilbert spaces W α (t), t ∈ [0, T ], α ≤ 2m, W 0 (t) = H. Obviously, we have the continuous dense embeddings W β (t) ⊂ W α (t), t ∈ [0, T ], provided that β > α. It follows from conditions I and IV that

|As (t)u − Ak (t)u|α,t ≥ cs,k |u|α+2,t

∀u ∈ W α+2 (t),

α ≤ 2m − 2,

1 ≤ s < k ≤ m,

(5)

for all t ∈ [0, T ], where cs,k > 0 are constants independent of u and t. These inequalities can ﬁrst be proved for even integer α just as above and then generalized to the remaining values of α with the use of the Heinz inequality [5, pp. 177–179]. V 2i , i = 0, . . . , m, independent of t and such that V 0 = H, V. There2 exist Banach spaces D A˜k ⊂ V , the spaces V 2j are continuously embedded in the spaces V 2i for j > i, the spaces W 2i (t) are continuously embedded in the spaces V 2i , t ∈ [0, T ], i = 0, . . . , m, and in H, there exist strong t-derivatives [5, p. 218] di A˜k (t)/dti ∈ B [0, T ], L V 2[j/2]+2 , V 2[j/2] , j = 0, . . . , 2m − 2 − i, i = 0, . . . , 2m − 2, k = 1, . . . , m, where [·] is the integer part of a number. VI. If t ∈ [0, T ], then all operators Ak (t), k = 1, . . . , m, satisfy the inequalities |As (t)Ak (t)u − Ak (t)As (t)u|α,t ≤ c˜s,k |u|α+3,t ∀u ∈ W α+4 (t),

α ≤ 2m − 4,

(6)

1 ≤ s = k ≤ m,

where c˜s,k ≥ 0 are constants independent of u and t. DIFFERENTIAL EQUATIONS

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CAUCHY PROBLEMS FOR QUASI-HYPERBOLIC FACTORIZED EVEN-ORDER . . .

815

Condition I is a new (compared with [6, pp. 150–158; 7]) expression of the same speciﬁc feature of some diﬀerential operators Ak (t): they usually consist of diﬀerential expressions A˜k (t) and boundary conditions, each of which can have speciﬁc independence. In applications, the role of such operators A˜k (t) can be played by some elliptic diﬀerential operators without boundary conditions, and the role of their restrictions Ak (t) can be played by the same elliptic diﬀerential operators but with some t-dependent boundary conditions. Additional conditions imposed on the operators Ak (t) will be stipulated in the statements of lemmas and theorems. 2. AUXILIARY ASSERTION Throughout the following, in the derivation of a priori estimates for strong solutions and in the proof of the solvability of Cauchy problems, we need the interpolation inequalities (8) (see below) in the positive Hilbert scale of the spaces {W α (t)}, t ∈ [0, T ], α = 0, . . . , 2m, generated by self-adjoint operators with variable domains. Lemma 1. Suppose that linear positive self-adjoint operators A1 (t), t ∈ [0, T ], in a Hilbert space H with t-dependent domains D (A1 (t)) have inverses A−1 1 (t) ∈ B ([0, T ], L(H)) for which the strong (t)/dt ∈ B ([0, T ], L (H)) exists in H for all t ∈ [0, T ]. If the ﬁrst strong derivative derivative dA−1 1 (t) of the operators A(t) = Am of the inverse operators A−1 (t) = A−m 1 1 (t) satisﬁes the relation (7) dA−1 (t)/dt ∈ B [0, T ], L H, W 2m−1 (t) in H for all t ∈ [0, T ], then the inequalities i 2 τ i+1 2 τ d u d u di u 2 ≤ c1 i+1 dt + c1 1 + 2Mmi /(2m) dt dti dti dt mi ,t t=τ mi+1 +1,t mi +1,t 0 0 i 2 d u mi = 2m − 2 − i, i = 0, . . . , 2m − 2, + i , dt mi ,t

(8)

t=0

are valid for all u ∈ E m (the spaces E m will be deﬁned in Section 3) and for all τ ∈ [0, T ], where the constants c1 and Mγ independent of u and t will be speciﬁed below. −1 [I − A−1 Proof. For each t ∈ [0, T ], the operators Aε (t) = A(t)A−1 ε (t) = ε ε (t)], ε > 0, where −1 = (I + εA(t)) , are bounded, self-adjoint, and positive in H. One can directly show that, for all t ∈ [0, T ], they have the strong derivative dAε (t)/dt = −Aε (t) dA−1 (t)/dt Aε (t) ∈ B ([0, T ], L(H))

A−1 ε (t)

in H satisfying the inequalities −β Aε (t) (dAε (t)/dt) Aε−1 (t) = Aε1−β (t) dA−1 (t)/dt L(H) L(H) = A−(1−β) (t)A1−β (t) dA−1 (t)/dt L(H) ≤ M , ε where M = sup0 0, 0 < ≤ 1, with = 1 − β. In H1 = H, the operators A = B = Aεβ−1 (t) and T = Aε−β (t) (dAε (t)/dt) Aε−1 (t) satisfy Remark 7.1 in [5, p. 179] for all t ∈ [0, T ] and, in particular, the inequality ∀x ∈ H, |BT x| = dA−1 (t)/dt A1−β (t)Aε−(1−β) (t)Aεβ−1 (t)x ≤ M |A x| since

(dA−1 (t)/dt) A1−β (t)

DIFFERENTIAL EQUATIONS

L(H)

Vol. 43

= A1−β (t) dA−1 (t)/dt L(H) , No. 6

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t ∈ [0, T ],

816

LOMOVTSEV

where the bar stands for the closure of operators by continuity in H [5, p. 228]. By applying the Heinz inequality (7.6) in [5, pp. 177–178] to them, we obtain the inequality −β−α A (t) (dAε (t)/dt) Aε−1+α (t)x ≤ M |x| ∀x ∈ H, α = 0, . . . , 1 − β, (9) ε for all t ∈ [0, T ]. By diﬀerentiating the integral representation of positive fractional powers of the operators Aεγ (t) [5, p. 140] and by using the representation of the resolvents Rε (−s) = (1 + εs)−1 (I + εA(t))R(−s/(1 + εs)) via the resolvent R(−r) = (A(t) + r)−1 , the inclusions (7), and the estimates β A (t)R(−r) ≤ Nβ /(1 + r)1−β , r > 0, 0 ≤ β < 1, L (H) where Nβ are constants known from operator calculus, we obtain the following integral representation of the derivative of these fractional powers for all t ∈ [0, T ] : dAεγ (t) sin πγ x= dt π

+∞ dAε (t) sγ Rε (−s) Rε (−s)x ds dt

∀x ∈ W 2m (t),

0 < γ < 1 − β,

(10)

0 −1

where Rε (−s) = (Aε (t) + s) and ε > 0. Indeed, by taking into account these representation and estimates, for all t ∈ [0, T ], we obtain the estimates +∞ dAεγ (t) 1 sγ x ≤ π dt (1 + εs)2

1−β dA−1 (t) β −s −s A (t) R ds|A(t)x| A (t)R 1 + εs dt 1 + εs

0

1 ≤ Nβ M N0 π

+∞

sγ ds |A(t)x| (1 + εs)β (1 + s + εs)2−β

0

≤

1 Nβ M N0 π

+∞

sγ ds |A(t)x| < +∞ ∀x ∈ W 2m (t) (1 + s)2−β

0

uniformly bounded for all ε > 0 if 0 < γ < 1 − β and β = 1/(2m). If Q = Aε−β (t)Rε (−s) (dAε (t)/dt) Aε−γ (t)Rε (−s) and x, y ∈ W 2m (t), then |(Qx, y)| = Aε−β−(1−γ)/2 (t) (dAε (t)/dt) Aε−1+(1−γ)/2 (t)Aε(1−γ)/2 (t)Rε (−s)x,

Aε(1−γ)/2 (t)Rε (−s)y ≤ Aε−β−(1−γ)/2 (t) (dAε (t)/dt) Aε−1+(1−γ)/2 (t)L(H) × Aε(1−γ)/2 (t)Rε (−s)x Aε(1−γ)/2 (t)Rε (−s)y . By using the integral representation (10) and the estimates (9) for γ ≥ 2β − 1, we obtain the inequalities γ Aε−β (t) dAε (t) Aε−γ (t)x, y dt +∞ sin πγ M sγ Aε(1−γ)/2 (t)Rε (−s)xAε(1−γ)/2 (t)Rε (−s)y ds ≤ π 0

≤

1 M π

⎞1/2 ⎛ +∞ ⎞1/2 ⎛ +∞ 2 2 ⎝ sγ Aε(1−γ)/2 (t)Rε (−s)x ds⎠ ⎝ sγ Aε(1−γ)/2 (t)Rε (−s)y ds⎠ . 0

0

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817

For each ε > 0, for the positive deﬁnite operators (Aε (t)h, h) ≥ cε |h| ,

h ∈ H,

2

cε =

−1 −1 sup A (t) L(H) + ε ,

0
there exists a unique resolution of identity Eλ (ε) such that +∞ +∞ +∞ 2 (1−γ)/2 γ s Aε (t)Rε (−s)x ds = sγ λ1−γ 0

0

+∞

+∞ 1−γ

λ

= cε

+∞ =

cε

sγ ds d (Eλ (ε)x, x) = (λ + s)2

0

+∞ +∞ cε

σγ dσ (1 + σ)2

0

1 d (Eλ (ε)x, x) ds (λ + s)2

+∞

+∞

d (Eλ (ε)x, x) = cε

(s/λ)γ d(s/λ)d (Eλ (ε)x, x) (1 + s/λ)2

0

σγ dσ|x|2 (1 + σ)2

∀t ∈ [0, T ].

0

It follows that −β Aε (t) (dAεγ (t)/dt) Aε−γ (t)x ≤ Mγ |x|

∀x ∈ H,

γ = β, . . . , 1 − β,

(11)

+∞ for all t ∈ [0, T ], where Mγ = π −1 M 0 σ γ (1 + σ)−2 dσ < +∞, if the elements x ∈ H in (11) are approximated by some sequences xn ∈ W 2m (t) for each t ∈ [0, T ]. By using the Schwartz and Cauchy–Schwarz inequalities, the estimates (11) and the δ-inequality 2ab ≤ δa2 + δ−1 b2 for all δ > 0 on the right-hand side of the obvious identities τ m /(2m) di u 2 dAεmi /(2m) (t) di u di u −1/(2m) (mi +1)/(2m) Aε i (t) = 2 Re A (t) , A (t) dt ε ε dti dt dti dti t=τ

0

τ

di+1 u di u Aε (t) i+1 , Aεmi /(2m) (t) i + 2 Re dt dt 0 m /(2m) di u 2 i (t) i ∀u ∈ D (Lm ) , + Aε dt mi /(2m)

dt

t=0

we obtain m /(2m) di u 2 A i (t) i ε dt

≤ c1 + ε1/(2m)

t=τ

τ m /(2m) di+1 u 2 A i (t) i+1 dt ε dt 0

1/(2m)

+ c1 + ε

1 + 2Mmi /(2m)

m /(2m) di u 2 (t) i + Aε i dt

τ (m +1)/(2m) di u 2 i Aε (t) i dt dt 0

,

i = 0, . . . , 2m − 2,

t=0

for all τ ∈ ]0, T ], where the constant c1 = sup0
−1 1/2 A1 (t) is also independent of ε. By letting L(H)

ε in the last inequality tend to zero and by using the property |Aεα/(2m) (t)v − A1 (t)v| → 0 for all v ∈ W α (t), t ∈ [0, T ], α ≤ 2m, as ε → 0, we obtain the estimate (8) for arbitrary functions u in the sets D (Lm ) deﬁned in Section 3. Then the estimate (8) for all u ∈ D (Lm ) can be generalized to all u ∈ E m by passage to the limit. α/2

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3. DEFINITION OF STRONG SOLUTIONS OF CAUCHY PROBLEMS By H we denote the Hilbert spaces L2 (]0, T [, W α (t)) with Hermitian norms · α , α ≤ 2m, H = H . The space H α is the set of all measurable functions u : [0, T ] t → u(t) ∈ H such that u(t) ∈ D Aα/2m (t) , t ∈ [0, T ], and h(t) = Aα/2m (t)u(t) ∈ H = L2 (]0, T [, H). Let the Hilbert spaces H p,q be the sets of all functions u ∈ H with ﬁnite Hermitian norms p 1/2 2 di u/dti ;

u p,q = α

0

q−i

i=0

let the Banach spaces

E p,q

be the sets of all functions u ∈ B([0, T ], H) with ﬁnite norms |||u|||p,q =

p i d u(t)/dti 2 sup q−i,t

0
1/2 ;

i=0

and let the Banach space B ([0, T ], H) be the set of all bounded functions of t ∈ [0, T ] ranging in H, equipped with the uniform convergence norm · B = sup0 0, which are not necessarily valid for any operator A1 (t) with variable domain D (A1 (t)). In the case of variable domains D (A1 (t)), suﬃcient conditions for the well-posedness of this deﬁnition and the existence of all derivatives in the below-introduced spaces H 2m,2m and E 2m−1,2m−1 will be given in Lemma 1. The continuous embeddings [1] H p,q ⊂ E p−1,q−1 are usually valid for constant domains of the operators A1 (t). These embeddings are not necessarily valid for variable domains of the operators A1 (t). Let H 2m,2m and E 2m−1,2m−1 be the closures of the above-deﬁned sets D (Lm ), whose deﬁnition contains the requirement of the corresponding smoothness of the operators A˜k (t) with respect to t, in the norms u 2m,2m and |||u|||2m−1,2m−1 , respectively. The following assertion describes the case of variable domains D (A1 (t)) for which the continuous embeddings H 2m,2m ⊂ E 2m−1,2m−1 are valid. Assertion 1. If the assumptions of Lemma 1 are valid, then i+1 2 i d u d u(t) 2 di u 2 −1 ≤ c1 i+1 + c1 1 + 2M(mi +1)/(2m) + T c1 i , dti dt dt 2m−i mi +1,t m1 +1 i = 0, . . . , 2m − 1, for all u ∈ H

2m,2m

and all t ∈ [0, T ].

Proof. The proof of the assertion is similar to that of Lemma 1 with the only diﬀerence: in the integration with respect to t from 0 to τ (see the end of the proof of Lemma 1), one should take the variable lower integration limit s < τ instead of the lower integration limit t = 0 and then perform the estimate with the use of additional integration with respect to s from 0 to T . However, condition (7) is quite restrictive for the diﬀerential operators A1 (t) for which the dependence of the domains on t is caused by the dependence of the coeﬃcients in the boundary conditions on t. Examples of operators A1 (t) with variable domains D (A1 (t)) that satisfy or do not satisfy this condition will be given in Section 6. Therefore, in the following, we simply assume where necessary (see Remark 1) that D (Lm ) ⊂ E 2m−1,2m−1 , i.e., (m +1)/2 (t) di u/dti ∈ B([0, T ], H), (12) 0 ≤ i ≤ 2m − 1, ∀u ∈ D (Lm ) . A1 i DIFFERENTIAL EQUATIONS

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As spaces of strong solutions of the Cauchy problems (1), (2), we take the Banach spaces E m that are the closures of the sets ˜ (Lm ) : ds u/dts ∈ H 2m−2[(s+1)/2] , s = 0, . . . , 2m − 1 , D (Lm ) = u ∈ D where

˜ (Lm ) = D

u∈H :

ds u ∈ L2 ]0, T [, V 2m−2[(s+1)/2] , s = 0, . . . , 2m; s dt

p d2m−2 u dαj A˜kj (t) d2m−2p−2−|α(p)| u , ∈ H 2 , |α(p)| ≤ 2m − 2p − 2, dt2m−2 j=1 dtαj dt2m−2p−2−|α(p)|

1 ≤ p ≤ m − 1, 1 ≤ k1 , . . . , kp ≤ m, ki = kj , [·] is the integer part of a number, α(p) = (α1 , . . . , αp ) ∈ Zp+ , and |α(p)| = α1 + · · · + αp , in the norms 1/2 2m−1 di u(t) 2 . |||u|||m = sup dti 0
The Cauchy problems (1), (2) correspond to the linear unbounded operators Lm ≡ {Lm (t), l0 , . . . , l2m−1 } : E m ⊃ D (Lm ) → F m with dense domains D (Lm ), m = 1, 2, . . . In forthcoming considerations, we use the following suﬃcient conditions for their closability. Lemma 2. If Conditions I, II [without inequality (3)], IV, and V and condition (12) are satisﬁed, then the operators Lm , m = 1, 2, . . . , admit closure. Proof. First, let us show that the set H01,1 = {v ∈ H 1,1 : v(0) = v(T ) = 0} is dense in H . T Suppose the contrary: there exists some function 0 = w ∈ H such that 0 (v, w)dt = 0 for −1 −1 , ε > 0, all v ∈ H01,1 . In this integral, we set v = A−1 1,ε (t)h, where A1,ε (t) = (I + εA1 (t)) h, dh/dt ∈ H , and h(0) = h(T ) = 0; then we obtain the relation T

A−1 1,ε (t)h, w dt = 0,

0

where A−1 1,ε (t) are operators satisfying the properties 1 and 2 at the beginning of the proof of Theorem 1. By using the known property (15), we pass to the limit in this relation as ε → 0, use the passage to the limit to generalize the resulting relation to all h ∈ H , set h = w, and obtain

w 20 = 0, i.e., w = 0. Let us now verify the validity of the closability criterion for the linear operators Lm ; this criterion says that if un ∈ D (Lm ), un → 0 in E m , and Lm un = {Lm (t)un , l0 un , . . . , l2m−1 un } → F = DIFFERENTIAL EQUATIONS

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{f, ϕ0 , . . . , ϕ2m−1 } in F m as n → ∞, then F = 0. Since lj : E m → W 2m−1−j (0), j = 0, . . . , 2m − 1, are bounded operators, it follows that ϕj = 0, j = 0, . . . , 2m−1, and consequently, after integration by parts with respect to t, we have T

T

T (Lm (t)un , v) dt = − lim

(f, v)dt = lim

n→∞

0

n→∞

0

dv d Mm−1 (t) · · · M1 (t)un , dt dt

dt

0

T + lim

n→∞

1/2 A1/2 m (t)Mm−1 (t) · · · M1 (t)un , Am (t)v dt = 0,

0

d2 Mk (t) = 2 + A˜k (t) dt for all v ∈ H01,1 . It follows that f = 0, since H01,1 is dense in H . The proof of the lemma is complete. m m ¯ ¯ Then we construct the closures Lm : E ⊃ D Lm → F of the operators Lm , mm = 1, 2, . . . ¯ m are deﬁned to contain all functions u ∈ E for each of ¯ m of the operators L The domains D L which there exists a sequence un ∈ D (Lm ) and an element F ∈ F m such that |||un − u|||m → 0 ¯ mu = and Lm un − F m → 0 as n → ∞, m = 1, 2, . . . In this connection, we assume that L limn→∞ Lm un = F , m = 1, 2, . . . ¯ m [respectively, u ∈ D (Lm )] of the operator equations Deﬁnition 1. Solutions u ∈ D L ¯ m u = F , F ∈ F m , m = 1, 2, . . . [respectively, Lm u = F , F ∈ R (Lm ) = Lm (D (Lm )), L m = 1, 2, . . .] are referred to as strong (respectively, smooth) solutions of the Cauchy problems (1), (2). 4. UNIQUENESS THEOREM FOR CAUCHY PROBLEMS First, we derive a priori estimates for smooth solutions of the Cauchy problems (1), (2). Theorem 1. If Conditions I, II, IV–VI and condition (7) are satisﬁed for m > 1, then there exist constants c0 (m) > 0 independent of u such that 2

|||u|||2m ≤ c0 (m) Lm u m

∀u ∈ D (Lm ) ,

m = 1, 2, . . .

(13)

Proof. In view of Conditions I and V, we set

Mk (t) = d2 /dt2 + A˜k (t), 1 ≤ s ≤ k ≤ n ≤ m, and

Lk(n,s) (t) = Mn (t) × · · · × Mk+1 (t) × Mk−1 (t) × · · · × Ms (t),

Lk(k,k) (t) = I

and write

Lm (t) = Mk (t)Lk(m,1) (t) + Pk,m (t), where, by virtue of (6), 2

|Pk,m (t)u| ≤ c˜k

2m−3 i i=0

2 d u dti

∀u ∈ D (Lm )

(14)

mi +1,t

for all t ∈ [0, T ] with constants c˜k ≥ 0 independent of u and t. The smoothing operators A−1 k,ε (t) = −1 (I + εAk (t)) , ε > 0, k = 1, . . . , m, have the following properties [2]. 1. We have −1 A (t)v − v → 0 ∀v ∈ H (15) k,ε

for all t ∈ [0, T ] as ε → 0. DIFFERENTIAL EQUATIONS

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CAUCHY PROBLEMS FOR QUASI-HYPERBOLIC FACTORIZED EVEN-ORDER . . . −1 2. The operators A−1 k,ε (t) have the strong derivatives dAk,ε (t)/dt ∈ all t ∈ [0, T ]. By integrating by parts, we obtain the identities

B ([0, T ], L(H))

821

in H for

(m,1) (m,1) (t)u, A−1 (t) L (t)u Ak (t)Lk k,ε k

t=τ

τ = 2 Re

(m,1) (t)u, A−1 k,ε (t) k

Ak (t)L 0

d (m,1) L (t)u dt dt k

τ d Ak (t)A−1 k,ε (t) Lk(m,1) (t)u, Lk(m,1) (t)u dt + dt 0

(m,1) (m,1) (t)u, A−1 (t) L (t)u m = 1, 2, . . . , + Ak (t)Lk , k,ε k t=0

˜ (Lm ). By using the formulas [2] for all u ∈ D −1 −1 −1 d Ak (t)A−1 k,ε (t) /dt = −Ak (t)Ak,ε (t) dAk (t)/dt Ak (t)Ak,ε (t) and inequalities (3) in the second integral on the right-hand side, we obtain the inequalities

(m,1)

Ak (t)Lk

(t)u, A−1 k,ε (t)Lk

(m,1)

τ ≤ 2 Re

(t)u

t=τ

(m,1) (t)u, A−1 k,ε (t) k

Ak (t)L

d (m,1) L (t)u dt dt k

0

+

(1) ck

τ

A−1 k,ε (t)Lk

(m,1)

(t)u, A−1 k,ε (t)Ak (t)Lk

(m,1)

(m,1) (m,1) (t)u, A−1 (t) L (t)u + Ak (t)Lk k,ε k

(t)u dt

0

. t=0

In these relations, we use property (15) and pass to the limit as ε → 0; this yields

Ak (t)Lk(m,1) (t)u, Lk(m,1) (t)u

t=τ

τ τ

d (m,1) (m,1) (1) ≤ 2 Re (t)u, Lk (t)u dt + ck Lk(m,1) (t)u, Ak (t)Lk(m,1)(t)u dt Ak (t)Lk dt 0 0

(m,1) (m,1) (t)u, Lk (t)u . (16) + Ak (t)Lk t=0

By integrating by parts, we obtain the identities 2 d (m,1) L (t)u dt k

d (m,1) d2 (m,1) = 2 Re L (t)u, Lk (t)u dt dt2 k dt 0 2 d (m,1) ˜ (Lm ) . (t)u ∀u ∈ D + Lk dt τ

t=τ

t=0

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By adding these identities to inequalities (16), we obtain 2 2 d (m,1) 1/2 (m,1) L (t)u + Ak (t)Lk (t)u dt k τ ≤ 2 Re

t=τ

d Lm (t)u, Lk(m,1) (t)u dt + dt

τ Φk (u, u)dt

2 2 d (m,1) 1/2 (m,1) (t)u + Ak (t)Lk (t)u + Lk dt 0

0

˜ (Lm ) , ∀u ∈ D

(17)

t=0

where Φk (u, u) =

(1) ck

(m,1) (t)u, Ak (t) k

L

(m,1) (t)u k

L

− 2 Re

d (m,1) Pk,m (t)u, Lk (t)u . dt

By virtue of Condition IV, the left-hand sides of inequalities (17) do not exceed the quantity 2 2 d (m,1) (m,1) (t)u + Lk (t)u , (18) c2 Lk dt 1,t t=τ

where c2 > 0 is a constant independent of u, t, and k. By using inequalities (5), one can justify the following assertion. Lemma 3. If the assumptions of Theorem 1 are valid, then there exist constants c3 > 0 and c4 ≥ 0 independent of u and t such that 2 m 2m−1 2m−3 2 di u 2 di u 2 d (m,1) (m,1) L (t)u + Lk (t)u − c4 (19) ≥ c3 dti dti dt k 1,t +1,t m mi ,t i i=0 i=0 k=1 for all u ∈ D (Lm ) and t ∈ [0, T ]. Proof. We prove the desired assertion by induction on m. If m = 1, then Lemma 3 is valid. Suppose that it holds for m − 1 distinct factors Mk (t); in particular, inequalities of the form (19) are valid for two sums, ⎛ ⎞ 2 m−j 2 dL (m−j,2−j) (t)v (m−j,2−j) ⎝ k Sm−j,2−j (v) = (t)v ⎠ , j = 0, 1. + Lk dt 1,t k=2−j

By denoting the sum on the left-hand side in (19) by

Sm,1 (u), we ﬁnd that it satisﬁes the relation

Sm,1 (u) = (1/3) [Sm,2 (M1 (t)u) + Sm−1,1 (Mm (t)u)] + Pm (t)u; when estimating it from below, the last four nonnegative terms in the expression ⎞ 2 m−1 ⎛ (m−1,1) 2 1 (t)u (m−1,1) ⎝ dMm (t)Lk Pm (t)u = 2 (t)u ⎠ + Mm (t)Lk 3 dt 1,t k=1 dLm(m,1) (t)u 2 + 2 Lm(m,1) (t)u2 − Sm−1,1 (Mm (t)u) + 2 1,t dt 2 dL (m,1) (t)u 2 (m,1) + 1 (t)u + L1 dt 1,t DIFFERENTIAL EQUATIONS

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can be omitted. If in the remaining terms of this expression Pm (t)u, we use an estimate of the form (1 + δ)|x|2 − |y|2 ≥ − (1 + δ−1 ) |x − y|2 , x, y ∈ H, δ > 0, with δ = 1, then, by virtue of inequalities (6), we obtain the estimate

Pm (t)u ≥

2m−3 i

−˜ c(1) 4

i=0

2 d u dti

∀u ∈ D (Lm )

mi ,t

≥ 0 independent of u and t. Therefore, by the induction for all t ∈ [0, T ] with a constant c˜(1) 4 assumption, we have i 2m−3 2m−3 di M1 (t)u 2 di u 2 d Mm (t)u 2 (1) (2) Sm,1 (u) ≥ c˜3 + , − c˜4 dti dti dti mi −1,t mi −1,t mi ,t i=0 i=0 where c˜3 > 0 and c˜4 ≥ 0 are constants independent of u and t. Since, by virtue of Condition V, the diﬀerentiation of the operators A˜1 (t) and A˜m (t) with respect to t, the evaluation of their restrictions and the restrictions of their derivatives with respect to t ˜ (Lm ) to D (Lm ), and elementary estimates lead to the inequality with D (1)

(2)

i 2 M1 (t) d u dti

mi −1,t

2 di u + Mm (t) i dt

mi −1,t

i i d M1 (t)u 2 d Mm (t)u 2 ≤ 2 + 2 i dti dt mi −1,t mi −1,t i 2 2m−3 d u (3) (3) + c˜4 , c˜4 ≥ 0, dti mi ,t i=0

it follows from the identity |x + y|2 = |x|2 + |y|2 + 2 Re (x, y) and the Schwarz inequality that the left-hand side of the last inequality is not less than the quantity i+2 2

di u 2 d u 2 2 + |A1 (t)w|mi −1,t + |Am (t)w|mi −1,t i 2 i+2 dt dt mi −1,t mi −1,t i+2 i d u d u − 2 |A1 (t)w + Am (t)w|mi −1,t i+2 dti dt mi −1,t

,

mi +1,t

di u w= i dt

i d u , dti mi +1,t

which, in turn, by virtue of the δ-inequality, is not less than the quantity i+2 2 2 2 d u |A1 (t)w|mi −1,t + |Am (t)w|mi −1,t + (2 − δ) i+2 dt δ mi −1,t 2 di u 2 |A1 (t)w + Am (t)w|mi −1,t . × δ− 2 2 |A1 (t)w|mi −1,t + |Am (t)w|mi −1,t dti mi +1,t Since, by virtue of the parallelogram identity and inequalities (5), 2

δ0 = sup w

|A1 (t)w + Am (t)w|mi −1,t 2

2

|A1 (t)w|mi −1,t + |Am (t)w|mi −1,t

2

= 2 − inf w

|A1 (t)w − Am (t)w|mi −1,t 2

2

|A1 (t)w|mi −1,t + |Am (t)w|mi −1,t

< 2,

it follows that δ can be chosen so as to ensure that δ0 < δ < 2. This implies that there exist constants c3 > 0 and c4 ≥ 0 independent of u and t such that inequalities (19) are valid for m distinct factors Mk (t), k = 1, . . . , m. The proof of Lemma 3 is complete. By summing inequalities (17) in view of the estimates (18) and by using the estimates (19) on the left-hand sides of the resulting inequalities and the estimates (14), the Schwarz inequality, the DIFFERENTIAL EQUATIONS

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Cauchy–Schwarz inequality, the δ-inequality, and elementary estimates on the right-hand sides, we ﬁnd constants c5 , c6 , c7 > 0 independent of u and t such that τ 2m−1 τ 2m−1 di u 2 di u 2 2 ≤ c5 dt + c6 |Lm (t)u| dt c2 c3 dti dti mi +1,t t=τ mi +1,t i=0 i=0 0 0 2m−1 2m−3 di u 2 2 |lj u|mj +1,0 + c2 c4 ∀u ∈ D (Lm ) . (20) + c7 dti mi ,t i=0 i=0 t=τ

In the last sum in inequality (20), we use the interpolation inequalities (8) and ﬁnd a constant c8 > 0 independent of u and t such that τ 2m−1 τ 2m−1 di u 2 di u 2 2 ≤ c8 dt + c6 |Lm (t)u| dt c2 c3 dti i dt mi +1,t mi +1,t i=0 i=0 t=τ

0

0

+ c7 + c21 c2 c4

2m−1

2

|lj u|mj +1,0

∀u ∈ D (Lm ) .

j=0

Then, in these inequalities, we use the following Gronwall lemma [4, p. 23 of the Russian translation]. Lemma 4. If v and g are nonnegative functions on [0, T ], v is integrable, and g is nondecreasing, τ then the inequality v(τ ) ≤ c 0 v(t)dt + g(τ ) implies the inequality v(τ ) ≤ ecτ g(τ ), τ ∈ [0, T ]. Then we obtain the inequalities ⎛ ⎞ τ 2m−1 2m−1 di u 2 2 2 ≤ ec9 τ ⎝c6 |Lm (t)u| dt + c7 + c21 c2 c4 |lj u|mj +1,0 ⎠ , c2 c3 dti mi +1,t i=0 j=0 t=τ

(21)

0

where c9 = c8 /(c2 c3 ). By evaluating the least upper bound with respect to τ in inequality (21), we obtain (13) for all u ∈ D (Lm ) with constants c0 (m) = exp (c9 T ) max {c6 , c7 + c21 c2 c4 }/(c2 c3 ). The proof of Theorem 1 is complete. Remark 1. In general, if the energy inequalities (13) can be derived without using the inclusion (12), then condition (12) is satisﬁed. If, in the derivation of the energy inequalities (13), instead of the integration with respect to t from 0 to τ , we use the integration with respect to t from s < τ to τ and then the additional integration with respect to s from 0 to T , then |||u|||22m−1,2m−1 ≤ c˜0 (m)||u||22m,2m ,

c˜0 (m) > 0,

m = 1, 2, . . . ,

for all functions u ∈ D (Lm ). Now let us derive a priori estimates for strong solutions of the Cauchy problems (1), (2). The following assertion is a straightforward consequence of Theorem 1. Corollary 1. If the assumptions of Theorem 1 are valid, then the energy inequalities ¯ m u 2 |||u|||2m ≤ c0 (m) L m ¯ m , m = 1, 2, . . . are valid for all u ∈ D L Proof. By taking into account Remark 1, we ﬁnd that condition (12) and Lemma 2 on the closeness of the operators Lm , m = 1, 2, . . . , are valid. Therefore, by using passage to the limit, one can generalize the energy inequalities (13) from smooth solutions u ∈ D (Lm ) to all strong solutions ¯ m of the Cauchy problems (1), (2). u∈D L DIFFERENTIAL EQUATIONS

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5. EXISTENCE THEOREM FOR CAUCHY PROBLEMS From Theorem 1 and Corollary 1, we ﬁnd that if a strong solution of the Cauchy problems (1), (2) exists, then it is unique and continuously depends on the data f and ϕj , j = 0, . . . , 2m − 1. The following assertion justiﬁes the strong solvability of the Cauchy problems (1), (2) for all F ∈ F m . Theorem 2. Let Conditions I–VI be satisﬁed. If condition (7) is valid for m > 1, then for any f ∈ H and ϕj ∈ W mj +1 (0), j = 0, . . . , 2m − 1, there exists a strong solution u ∈ E m of the Cauchy problems (1), (2). Proof. We perform the proof by induction on m. If m = 1, then, by virtue of inequalities (3) ¯ 1 u = F is solvable [2, 3] for any F ∈ F 1 . By induction, we suppose and (4), the equation L ¯ m−1 u = F are solvable for any F ∈ F m−1 and for any set and order of that the equations L ¯ m u = F for m − 1 distinct factors Mk (t) in Lm−1 (t) and prove the solvability of the equations L m all F ∈ F , m = 2, . . . (m,1) (m,1) ≡ {Lk (t), l0 , . . . , l2m−3 } : E m−1 ⊃ D (Lm−1 ) → F m−1 in Consider the linear operators Lk (m,1) m 1,m : E ⊃ D (Lm ) → E , k = 1, . . . , m, where E 1,m = E 1 × W 2m−2 (0) × · · · × other spaces, Lk 1 W (0) are the Banach spaces with norms |||u|||1,m =

|||u|||21 +

2m−3

1/2 2

|lj u|mj ,0

.

j=0

(m,1) of the last operators is given by the restriction of the closure of the ﬁrst operators The closure Lk (m,1) (m,1) (m,1) (m,1) (m,1) (m,1) Lk to E m ; i.e., Lk = Lk , since Lk ⊂ Lk and the Lk : E m ⊃ D (Lm ) → E 1,m , m E k = 1, . . . , m, are continuous operators. In addition, consider the linear operators −1/2 −1/2 Mk = Mk (t), A1 (0), . . . , A1 (0), l0 , l1 : E 1,m ⊃ D (L1 ) × W 2m−2 (0) × · · · × W 1 (0) → F m , ¯ −1 : F m → E 1,m , k = 1, . . . , m. By virtue ¯ k , by [2, 3[, have bounded inverses M whose closures M k ¯ m u = F for F ∈ F m are simultaneously of the estimates (14), the solutions of the equations L (m,1)

solutions of the equations Mk Lk u = Fk for Fk = F + Mk (t)Lk(m,1) (t) − Lm (t) u, 0, . . . , 0 ∈ F m ,

k = 1, . . . , m.

Lemma 5. Let X, Y, and Z be Banach spaces. If S : X → Y is a linear operator closable by continuity to a bounded operator S¯ and P : Y → Z is a linear operator that admits closure P¯ , then ¯ the product P · S : X → Z admits closure P · S, and P · S ⊂ P¯ · S. Proof. By the closability criterion for linear operators in Banach spaces, to prove that the product P · S is closable in X × Z, we show that if un ∈ D(P · S), un → 0 in X, and (P · S)un = P (Sun ) → g in Z as n → ∞, then g = 0. It follows from the assumptions of this criterion that vn = Sun ∈ D(P ), vn → 0 in Y since S¯ is a bounded operator, and P vn → g in Z as n → ∞. Consequently, g = 0, since P admits closure in Y × Z. ¯ Let P · S u = g; i.e., there exist It remains to prove the algebraic embedding P · S ⊂ P¯ · S. un ∈ D(P · S) such that un → u in X and (P · S)un → g in Z as n → ∞. Then, obviously, ¯ in Y since S¯ is a bounded operator, and P vn → g in Z as n → ∞. vn = Sun ∈ D(P ), vn → Su ¯ ∈ D P¯ and P¯ · S¯ u = g. The proof of Lemma 5 is complete. Hence it follows that Su (m,1)

By applying Lemma 5 to the operators S = Lk

(m,1)

and P = Mk in the spaces X = E m , Y = E 1,m , ¯ k L(m,1) , k = 1, . . . , m. Hence we ﬁnd that ⊂M

and Z = F m , we obtain the embeddings Mk Lk k (m,1) ¯ m can be represented in the form M ¯ k L(m,1) u = Fk , the equations Mk Lk u = Fk for all u ∈ D L k DIFFERENTIAL EQUATIONS

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¯ −1 Fk ∈ E 1,m for all Fk ∈ F m , u=M k −1 −1 (m,1) ¯ −1 Fk ∈ E m−1 , where L(m,1) M and, by the induction assumption, they have solutions u = L (m,1)

k = 1, . . . , m. By [2, 3], the last equations have solutions Lk

k

k

k

are the inverses of the operators Lk , k = 1, . . . , m. It remains to justify the inclusion u ∈ E m . Let us show that the smoothness of this solution u ∈ E m−1 can be increased by unity owing to ¯ −1 Fk ∈ F m−1 , k = 1, . . . , m. ¯ −1 Fk ∈ E 1,m instead of M the smoothness of the right-hand side, M k k Lemma 3, together with Lemmas 1 and 4, implies the inequalities (m,1)

c10 |||u|||2m

m (m,1) 2 ≤ Lk u k=1

1,m

,

∀u ∈ D (Lm ) ,

c10 > 0,

which, by passage to the limit, can be generalized to all functions u ∈ E m in the domains of (m,1) (m,1) are homeomorphic mappings the closure Lk . These inequalities imply that the operators Lk (m,1) equipped with the norm of the space E 1,m . To complete of the space E m onto their ranges R Lm (m,1) = E 1,m . In turn, to this end, it suﬃces to show that the the proof, it remains to show that R Lm equations L(m,1) v = Φk , k = 1, . . . , m, with an arbitrary right-hand side Φk in a set dense in E 1,m k have solutions in v ∈ E m . By virtue of the estimate (14) with m − 1 instead of m, the solutions v ∈ E m−1 of the equations Lk v = Φk ∈ F m−1 are simultaneously solutions of the equations (k) (m,k) (k−1,1) ˜ L(k) Lk−1 and m−2 Mk−1 v = Φk−1 , where Lm−2 = Lk (m,1)

˜ k−1 = Φk + Φ

(k−1,1) (m,1) Lk(m,k) (t)Lk−1 (t)Mk−1 (t) − Lk (t)

v, 0, . . . , 0 ∈ F m−1

(1) (m,2) ˜ m , where L(1) for k = 2, . . . , m and the equation Lm−2 Mm v = Φ and m−2 = Lm

˜ m = Φ1 + Φ

Lm(m,2) (t)Mm (t) − L1(m,1) (t)

v, 0, . . . , 0 ∈ F m−1

for k = 1. Below we increase the smoothness of solutions of the operators L(m,1) owing to increasing k ¯ k , k = 1, . . . , m. the smoothness of solutions of the operators M Let the Hilbert space H˜ 2,2 be deﬁned as the set D (L1 ) equipped with the Hermitian norm

1/2 2 . |u|2,2 = d2 u/dt2 0 + ||du/dt||20 + ||u||22 ˜0 = Φ ˜ m . If the function v ∈ E m−1 To shorten the notation, we set M0 (t) = Mm (t), M0 = Mm , and Φ (k) ˜ k−1 for Φ ˜ k−1 ∈ F 2,m = H˜ 2,2 × W 2m−1 (0) × · · · × is a solution of the equations Lm−2 Mk−1 v = Φ (k) ˜ k−1 ∈ F m , where Mk ≡ {Mk (t), I, . . . , I, l0 , l1 } : F 2,m → F m , W 2 (0), then Mk Lm−2 Mk−1 v = Mk Φ k = 1, . . . , m, are bounded linear operators. Lemma 6. Let X, Y, and Z be Banach spaces. If P : X → Y is a linear operator that admits a closure P¯ , S : Y → Z is a linear bounded operator, and their product S · P : X → Z admits a closure S · P , then S · P¯ ⊂ S · P . Proof. Let S · P¯ u = g. Then u ∈ D P¯ ; i.e., there exist un ∈ D(P ) such that un → u in X and P un → P¯ u in Y as n → ∞. Since the operator S is bounded, we have S (P un ) → S P¯ u in Z as n → ∞. Hence it follows that un ∈ D(S · P ), un → u in X, and S (P un ) = (S · P )un → g in Z as n → ∞; i.e., u ∈ D S · P and S · P u = g. The proof of the lemma is complete. DIFFERENTIAL EQUATIONS

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827

(k)

By applying Lemma 6 to the operators P = Lm−2 Mk−1 and S = Mk in the spaces X = E m , Y = F 2,m , and Z = F m , we obtain the embeddings

(k) (k) k = 1, . . . , m. (22) Mk Lm−2 Mk−1 ⊂ Mk Lm−2 Mk−1 , By applying Lemma 5 to the operators S = Mk−1 ≡ {Mk−1 (t), l0 , l1 } : E m ⊃ D (Lm ) → E m−1 × (k) (k) −1/2 −1/2 W 2m−2 (0) × W 2m−3 (0) and P = Mk Lm−2 ≡ {Mk (t)Lm−2 (t), A1 (0), A1 (0), l0 , . . . , l2m−3 } : (k) E m−1 × W 2m−2 (0) × W 2m−3 (0) ⊃ D (Lm−1 ) × W 2m−2 (0) × W 2m−3 (0) → F m , where Lm−2 (t) are the (k) ﬁrst operator coordinates of the vector operators Lm−2 , we obtain the embeddings

(k) (k) Mk Lm−2 Mk−1 ⊂ Mk Lm−2 Mk−1 ,

k = 1, . . . , m.

(23)

(k) ˜ k−1 , k = 1, . . . , m, which, by the From (22) and (23), we have the equations Mk Lm−2 Mk−1 v = Mk Φ induction assumption, have the solutions Mn (t)v ∈ E m−1 , n = 1, . . . , m, for n = k − 1.

Lemma 7. If the assumptions of Theorem 1 are valid, then there exist constants c11 > 0 and c12 ≥ 0 independent of v and t such that m 2m−3 di Mk (t)v 2 dti k=1 i=0

≥ c11

mi −1,t

2m−1 i i=0

2 d v dti

− c12

2m−3 i

mi +1,t

i=0

2 d v dti

,

m = 2, 3, . . . ,

(24)

mi ,t

for all v ∈ D (Lm ) and t ∈ [0, T ]. Proof. Lemma 7 can be proved by induction over m by analogy with the proof of Lemma 3. By using Lemmas 1 and 4, from (24), we obtain c13 |||v|||2m

≤

m

|||M

2 k (t)v|||m−1

+

2m−3

|lj v|mj ,0 ,

c13 > 0.

j=0

k=1

By passing to the limit, we generalize these inequalities from the solutions v ∈ D (Lm ) to solutions ˜ k−1 such that Mk (t)v ∈ E m−1 , k = 1, . . . , m, of the desired equations with right-hand sides Mk Φ m m−1 . Therefore, the earlier-found solution u ∈ E m−1 of the original and obtain v ∈ E , since v ∈ E equation with arbitrary right-hand side F ∈ F m indeed belongs to the space E m , m = 2, . . . By induction over m, hence we obtain ¯ −1 · · · M ¯ −1 F ∈ E m ¯ −1 F = M u=L m 1 m

∀F ∈ F m ,

m = 1, 2, . . .

(25)

Remark 2. In the same way, the assertions of Theorem 1 and Corollary 1 [ possibly, with larger values of the constants c0 (m)] and the assertion of Theorem 2 (with the use of continuation with respect to a parameter) can be generalized to the equations with lower terms

Lm (t)u +

2m−1

Bk (t)

k=0

dk u = f, dtk

t ∈ ]0, T [,

m = 1, 2, . . . ,

(26)

if Bk (t) ∈ B ([0, T ], L (W mk +1 (t), H)), k = 0, . . . , 2m − 1. The lower terms of Eqs. (26) should be treated not only as additional terms subjected to the leading terms of these equations but also as the factorization remainder of arbitrary quasi-hyperbolic even-order diﬀerential-operator equations, i.e., as terms preserved under the reduction of quasi-hyperbolic even-order diﬀerentialoperator equations to their factorized (divergent) form (1). Remark 3. The analysis of the proof of Theorem 1, Corollary 1, and Theorem 2, shows that if all operators A˜k (t) = A˜k are independent of t and commute with each other, then the interpolation DIFFERENTIAL EQUATIONS

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inequalities (8) become unnecessary, and consequently, in these assertions, the suﬃcient condition (7) for m > 1 is unnecessary even if the domains D (Ak (t)) of the restrictions Ak (t) depend on t. Note that, as a rule, this condition is satisﬁed even if all A˜k (t) smoothly depend on t and do not commute with each other, but the domains D (Ak ) of the operators Ak (t) are independent of t [1], i.e., if Ak (t) = A˜k (t). In addition, such variable operators Ak (t) with constant domains D (Ak ) were subjected in [1] to the more restrictive condition dA−1 (t)/dt ∈ B ([0, T ], L (H, W 2m (0))) for all m ≥ 1. 6. EXAMPLE OF MIXED PROBLEMS In the bounded domain G = ]0, T [ × ]0, l[ of the variables t and x, we consider the following mixed problems: the diﬀerential equations 2 2 (t, x) ∈ G, (27) ∂ /∂t − a2m ∂ 2 /∂x2 · · · ∂ 2 /∂t2 − a21 ∂ 2 /∂x2 u(t, x) = f (t, x), where ak > 0 are distinct constants and a1 = 1, with the boundary conditions ∂ 2i+1 u(t, 0)/∂x2i+1 − β(t)∂ 2i u(t, 0)/∂x2i = 0, 2i ˜ u(t, l)/∂x2i = 0, ∂ 2i+1 u(t, l)/∂x2i+1 + β(t)∂

t ∈ [0, T ],

i = 0, . . . , m − 1,

(28)

˜ are nonnegative functions that do not simultaneously vanish for any t ∈ [0, T ] where β(t) and β(t) and are twice continuously diﬀerentiable with respect to t, with the initial conditions ∂ j u(0, x)/∂tj = ϕj (x),

x ∈ ]0, l[ ,

j = 0, . . . , 2m − 1,

m = 1, 2, . . .

(29)

Let us show that, by Remark 3, the diﬀerential operators Ak (t) obtained as the restriction of the diﬀerential expressions A˜k u(t, x) = −a2k ∂ 2 u(t, x)/∂x2 , t ∈ [0, T ], to the domains D (Ak (t)) = {u ∈ L2 (0, l) : u(x) ∈ (26) for m = 1; A˜k (t)u(x) ∈ L2 (0, l)}, t ∈ [0, T ], satisfy the suﬃcient assumptions of Theorems 1 and 2 in the Hilbert space H = L2 (0, l). The operators A1 (t), t ∈ [0, T ], are self-adjoint in L2 (0, l), since they are obviously symmetric in L2 (0, l) and have bounded inverses A−1 1 (t)g

x =−

l (x − s)g(s)ds + (A1 (t) + B1 (t)x)

0

l (l − s)g(s)ds

g(s)ds + (C1 (t) + D1 (t)x) 0

0

on L2 (0, l), where A1 (t) = 1/ β + β˜ + lβ β˜ , B1 (t) = β A1 (t), C1 (t) = β˜A1 (t), and D1 (t) = β β˜A1 (t). Obviously, they are positive in L2 (0, l). Their boundedness in L2 (0, l) follows from the inequalities −1 A1 (t)g2 ≤ c13 g 20,Ω ∀g ∈ L2 (0, l), t ∈ [0, T ], 0,Ω where · 0,Ω is the norm in L2 (Ω), Ω = ]0, l[, and c13 = l2 max l2 + 3(1 + lβ)2 A12 (t) + l2 (1 + lβ)2 C12 (t) . 0≤t≤T

The operators A−1 1 (t), t ∈ [0, T ], have the strong derivative

dA−1 1 (t) g = A˙ 1 (t) + B˙ 1 (t)x dt

l

l

g(s)ds + C˙1 (t) + D˙ 1 (t)x 0

(l − s)g(s)ds

0

in L2 (0, l), where the dots above functions stand for the ﬁrst derivatives with respect to t. This 2 2 strong derivative is a bounded operator in L2 (0, l), since dA−1 1 (t)/dt g 0,Ω ≤ c14 g 0,Ω for all t ∈ [0, T ], all g ∈ L2 (0, l), t ∈ [0, T [, where c14 = 4l2 max A˙ 12 (t) + l2 /3 B˙ 12 (t) + l2 /3 C˙12 (t) + l4 /9 D˙ 12 (t) , 0≤t≤T

DIFFERENTIAL EQUATIONS

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and satisﬁes inequality (3), since −

dA−1 1 (t)/dt

A1 (t)u, A1 (t)u

0,Ω

2 1/2 ≤ c15 A1 (t)u

0,Ω

for all u ∈ D (A1 (t)), t ∈ [0, T ], where (· , ·)0,Ω is the inner product in L2 (0, l) and ! √ √

2 A˙ 1 (t) + l(1 + l) B˙ 1 (t) + l C˙1 (t) + l2 D˙ 1 (t) 1 + β˜ , c15 = max 0≤t≤T √ √

˙ ˙ ˙ 2 ˙ 2 A1 (t) + l B1 (t) + l(1 + l) C1 (t) + l D1 (t) (1 + β), √ " √ l B˙ 1 (t) + l C˙1 (t) + 2l D˙ 1 (t) . Here we have used the relation 2 1/2 A1 (t)u

0,Ω

∂u(t, l) 2 2 ˜ ∂x + β(t)|u(t, l)| 2 ∂u(t, 0) 2 1 + β(t)|u(t, 0)|2 + ∂u , + ∂x 1 + β(t) ∂x 0,Ω

1 = ˜ 1 + β(t)

t ∈ [0, T ].

(30)

The operators dA−1 1 (t)/dt have the strong derivative

d2 A−1 1 (t) ¨1 (t) + B ¨ 1 (t)x g = A dt2

l

l

g(s)ds +

C¨1 (t) + D¨ 1 (t)x

0

(l − s)g(s)ds

0

in L2 (0, l) for all t ∈ [0, T ], which is bounded in L2 (0, l), since 2 −1 d A1 (t)/dt2 g2

0,Ω

≤ c16 g 20,Ω

∀g ∈ L2 (0, l)

for all t ∈ [0, T ], where the constant c16 is obtained from the constant c14 by the replacement of functions with a single dot by the same functions with double dots standing for their second 2 derivatives with respect to t. The operators d2 A−1 1 (t)/dt satisfy inequalities (4), since 1/2 2 −1 2 ∀g ∈ L2 (0, l), ∀u ∈ D (A1 (t)) d A1 (t)/dt g, A1 (t)u 0,Ω ≤ c17 ||g||0,Ω A1 (t)u 0,Ω

for all t ∈ [0, T ], where c17 =

√

! # √ ¨ √ ¨ 2 ¨ ˜ ¨ l sup 3 A1 (t) + l C1 (t) + 3l B1 (t) + l D1 (t) 1 + β, 0
" √ √ ¨ ¨ $ 3/2 ¨ ¨ 3 A (t) + l C (t) 1 + β, 3l B (t) + l D (t) 1 1 1 . 1

Obviously, the operators Ak (t), t ∈ [0, T ], also satisfy Conditions IV–VI. Moreover, the Banach spaces V 2k are just the Sobolev spaces W22k (0, l) with their ordinary norms · 2k,Ω , k = 0, . . . , m, 2k (0, l) of the V 2 = D(A˜1 ), V 0 = L2 (0, l). The Hilbert spaces W 2k (t) are the closed subspaces W2,Δ(t) 2k Sobolev spaces W2 (0, l), namely, the sets & % u ∈ W22k (0, l) : u ∈ (28), t ∈ [0, T ], i = 0, . . . , k − 1 DIFFERENTIAL EQUATIONS

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equipped with the Hermitian norms · 2k,t,Ω inherited from W22k (0, l), k = 1, . . . , m. The Hilbert 2k+1 (0, l), which are the closures of the sets spaces W 2k+1 (t), t ∈ [0, T ], are the spaces W2,Δ(t) %

u ∈ W22k+2 (0, l) : u ∈ (28), t ∈ [0, T ], i = 0, . . . , k

&

in the Hermitian norms

1/2

u(t, x) 2k+1,t,Ω = A1 (t)∂ 2k u(t, x)/∂x2k

k = 0, . . . , m − 1.

, 0,Ω

The Hilbert spaces W α (t), t ∈ [0, T ], for noninteger α ∈ ]0, 2m[ are deﬁned in a similar way. For the mixed problems (27)–(29), for the Banach spaces E m (G) of their strong solutions, we take the closures of the intersections of the closed subspaces of the Sobolev–Slobodetskii spaces D (Lm ) = {u ∈ D˜ (Lm ) : u ∈ (28)}, where 2m ' ∂ 2m−2−2[(s+1)/2]+s u D˜ (Lm) = u ∈ W2i,2m−2[(i+1)/2] (G) : vs ≡ s 2m−2−2[(s+1)/2] , ∂t ∂x i=0 ∂vs (t, l) ˜ ∂vs (t, 0) − β(t)vs (t, 0) = + β(t)vs (t, l) = 0, t ∈ [0, T ], s = 0, . . . , 2m − 2 , ∂x ∂x in the norms

|||u(t, x)|||m =

sup

1/2

2m−1

0
i=0

∂ i u(t, x) 2 ∂ti

.

mi +1,t,Ω

For the mixed problems (27)–(29), for the spaces of the right-hand sides f (t, x) and the initial 2m−1 (0, l) × · · · × L2 (0, l) of functions data ϕj (x), we take the Hilbert spaces F m (G) = L2 (G) × W2,Δ(0) F (t, x) = {f (t, x), ϕ0 (x), . . . , ϕ2m−1 (x)} with Hermitian norms F (t, x) m =

⎧ T ⎨ ⎩

f (t, x) 20,Ω dt +

2m−1

2

ϕj mj +1,0,Ω

j=0

0

⎫1/2 ⎬ ⎭

,

s (0, l) are the closures of the sets of all functions u(x) where, just as above, the Hilbert spaces W2,Δ(0) 2[(s+1)/2]

in the Sobolev spaces W2 the Hermitian norms

(0, l) satisfying condition (28) for t = 0 and i = 0, . . . , [(s + 1)/2] in

(s−2[s/2])/2 (0)∂ 2[s/2] u(x)/∂x2[s/2]

u(x) s,0,Ω = A1

,

0,Ω

s = 1, . . . , 2m − 1.

By taking into account Remark 3 saying that condition (7) with m > 1 becomes unnecessary in the case of constant coeﬃcients ak , from Theorem 1, Corollary 1, and Theorem 2, one can obtain the following theorem on the existence and uniqueness of strong solutions of problems (27)–(29) and their continuous dependence on the right-hand sides of the equations. Theorem 3. If the coeﬃcients β and β˜ satisfy the above-mentioned conditions for the functions mj +1 (0, l), j = 0, . . . , 2m − 1, then the mixed problems (27)–(29) f (t, x) ∈ L2 (G) and ϕj (x) ∈ W2,Δ(0) have a unique strong solution u(t, x) ∈ C (2m−1) ([0, T ], L2 (0, l)) ∩ E m (G) such that |||u(t, x)|||2m ≤ c0 (m) F (t, x) 2m , F (t, x) = {f (t, x), ϕ0 (x), . . . , ϕ2m−1 (x)} ,

(31)

m = 1, 2, . . .

Note that Theorem 3 was announced in [8]. DIFFERENTIAL EQUATIONS

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Remark 4. In these mixed problems, the requirement that the functions β and β˜ do not simultaneously vanish for any t ∈ [0, T ] is not important and is caused only by the fact that the inverse operators A−1 1 (t) are bounded in L2 (0, l) to simplify the veriﬁcation of the assumptions −1 of Theorems 1 and 2 and Corollary 1. If, in the proof of Theorem 3, we choose (A1 (t) + δ1 I) , ˜ δ1 > 0, instead of A−1 1 (t), then in this case, the functions β and β can simultaneously vanish, and the assertion of Theorem 3 remains valid in this case. 7. DISCUSSION OF THE CAUCHY PROBLEMS In the Hilbert space H = L2 (Rn ), n ≥ 1, condition (7) with m > 1 in Theorems 1 and 2 is valid for the diﬀerential operators p(t)

A(t) = (I − Δx )

,

p(t) > n/2,

p(t) ∈ C (1) [0, T ],

, . . . , xn ) ∈ Rn with t-dependent domains where Δx isthe Laplace operator with respect to x = (x1 p(t) D(A(t)) = u(x) ∈ L2 (Rn ) : (I − Δx ) u(x) ∈ L2 (Rn ) , t ∈ [0, T ]. Here fractional-order partial derivatives and fractional-order powers of the given operators A(t) in the deﬁnition of the spaces α α/(2m) −1 2 p(t)α/(2m) (t)u = F F [u] , α > 0, for all (1 + |ξ| ) W (t) are given by the expressions A α/(2m) −p(t)α/(2m) (t) and A−α/(2m) (t)g = F −1 (1 + |ξ|2 ) ∗ g, α > 0, for all g ∈ L2 (Rn ), u∈D A where F and F −1 are the direct and inverse Fourier–Plancherel integral transforms and ∗ stands for the convolution of functions. By using the properties of the Fourier–Plancherel transforms, we ﬁnd that A1−1/(2m) (t) (dA−1 (t)/dt) ∈ B ([0, T ], L (L2 (Rn ))) for m > 1, since 2 1−1/(2m) −1 2 −p(t)/(2m) 2 −1 A (t) dA (t)/dt g = (p (t)) F ln 1 + |ξ|2 F [g] 1 + |ξ|2 2 2 (p (t)) 2 −p(t)/(2m)+ ≤ F [g] 1 + |ξ| 2 n (e) (2π) (p (t)) (p (t)) 2 ≤

F [g]

=

g 2 (e)2 (2π)n (e)2 2

2

∀g ∈ L2 (Rn )

for all t provided that the parameter satisfying the estimate ln z ≤ (1/e)z for all z ≥ 1 is 0 < ≤ min[0,T ] p(t)/(2m). Unfortunately, condition (7) with m > 1, which provides the interpolation inequalities (8), is rarely valid for the elliptic diﬀerential operators A1 (t) with t-dependent coeﬃcients in boundary conditions. We show that it fails for the operators A1 (t) in the mixed problems (27)–(29) with m = 2. For them, this condition with m = 2 is equivalent to the condition −2 A3/2 1 (t) dA1 (t)/dt ∈ B ([0, T ], L (L2 (0, l))) , which is not necessarily valid. On the right-hand sides of the relation A3/2 1 (t)

dA−2 dA−1 dA−1 3/2 1 (t) 1 (t) 1 (t) −1 = A1/2 + A A1 (t), (t) (t) 1 1 dt dt dt

(32)

˜ the terms are not necessarily bounded operators if at least one of the coeﬃcients β(t) and β(t) −1 depends on t, since in this case the derivative dA1 (t)/dt can “lose” boundary conditions neces1/2 ˜ in (28) with m = 1 vanish in sary for the square root A1 (t). If the coeﬃcients β(t) and β(t) some open neighborhood V0 of the point t0 , then it is well known that the boundary conditions ∂v(x)/∂x|x=0 = 0 and ∂v(x)/∂x|x=l = 0 for t ∈ V0 are not necessarily satisﬁed by all functions 1/2 1/2 v(x) ∈ D A1 (t) in the domains of the operators A1 (t), since in this case their graph norm is equivalent to the norm of the Sobolev space W21 (0, l) [see (28) and (30)]. However, if there exists a t0 ∈ [0, T ] such that β (t0 ) = 0 and β˜ (t0 ) = 0, then, by virtue of their continuity, there exists a DIFFERENTIAL EQUATIONS

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832

LOMOVTSEV

˜ = 0 for all t ∈ V0 . Then, from the formula neighborhood V0 of this point t0 such that β(t), β(t) 1/2 for the graph norm of the operators A1 (t), the ﬁrst boundary condition (28) for m = 1, and 1/2 formula (30), we ﬁnd that, for each t ∈ V0 , the functions v(x) ∈ D A1 (t) are continuous with respect to x, and the derivatives ∂v(x)/∂x belong to the space L2 (0, l) and have trace for x = 0 in the generalized sense as the limit ∂v(x)/∂x|x=0 = lim ∂vn (x)/∂x|x=0 = β(t) lim vn (x)|x=0 = β(t)v(0) n→∞

n→∞

of some functions vn (x) ∈ D (A1 (t)) = {w ∈ W22 (0, l) : w ∈ (28) for m = 1} in R by virtue of 1/2 the deﬁnition of the square root A1 (t). The above-mentioned assertions are valid for the trace 1/2 in this sense, foreach t ∈ V0 , ∂v(x)/∂x|x=l of all functions v(x) ∈ D A1 (t) , t ∈ V0 . Therefore, ˜ = 0 are the boundary conditions [∂v(x)/∂x − β(t)v(x)]|x=0 = 0 and ∂v(x)/∂x + β(t)v(x) x=l 1/2 preserved for all functions v(x) ∈ D A1 (t) , while they can fail for the derivative dA−1 1 (t)/dt in ˜ the case of the coeﬃcients β(t) and (or) β(t) depending on t ∈ V0 . Indeed, if, for example, β(t) = t, ˜ = 1, and l = 1, then the functions β(t) x−2 dA−1 1 (t) g= v(x) = dt (2t + 1)2

1 (2 − s)g(s)ds

∀g ∈ L2 (0, 1)

0

do not satisfy the t-dependent boundary condition .

/ 1 1 ∂v(x) − tv(x) = (2 − s)g(s)ds = 0 ∂x 2t + 1 x=0

∀t > 0,

0

˜ for example, for g(x) = 1. Therefore, for such β(t), β(t), and l, the ﬁrst and, all the more, second products of operators on the right-hand side in (32) cannot be bounded in L2 (0, l); in addition, this disadvantage cannot cancel by summation. that, for the above-mentioned values of β(t) = t, Note −1 ˜ β(t) = 1, and l = 1, the functions v(x) = dA1 (t)/dt g satisfy the t-independent second boundary condition [∂v(x)/∂x + v(x)]|x=1 = 0, t ∈ [0, T ], for all g ∈ L2 (0, 1). How to eliminate or, at least, weaken condition (7) for m > 1 in the Cauchy problems (1), (2) for the case in which the operators Ak (t) depend on t, have t-dependent domains, and do not commute with each other? REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

Radyno, Ya.V. and Yurchuk, N.I., Diﬀer. Uravn., 1976, vol. 12, no. 2, pp. 331–342. Lomovtsev, F.E., Diﬀer. Uravn., 1992, vol. 28, no. 5, pp. 873–886. Lomovtsev, F.E., Dokl. Nats. Akad. Nauk Belarusi, 2001, vol. 45, no. 1, pp. 34–37. G˚ arding, L., Cauchy’s Problem for Hyperbolic Equations, Chicago, 1957. Translated under the title Zadacha Koshi dlya giperbolicheskikh uravnenii, Moscow: Inostrannaya literatura, 1961. Krein, S.G., Lineinye diﬀerentsial’nye uravneniya v banakhovom prostranstve (Linear Diﬀerential Equations in a Banach Space), Moscow: Nauka, 1967. Lions, J.-L., Equations diﬀ´erentielles op´erationnelles et probl`emes aux limites, Berlin, 1961. Lomovtsev, F.E., Diﬀer. Uravn., 1997, vol. 33, no. 10, pp. 1394–1403. Lomovtsev, F.E., Tez. dokl. mezhdunar. konf. “Diﬀerentsial’nye uravneniya i nelineinye kolebaniya” (Chernovtsy, 27–29 avg. 2001 g.) (Abstr. Int. Cong. “Diﬀ. Eqs. and Nonlin. Oscill.,” Chernovtsy, August, 27–29, 2001), Kiev, 2001, p. 98.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 833–839. c A.I. Noarov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 813–819. Original Russian Text

PARTIAL DIFFERENTIAL EQUATIONS

Generalized Solvability of the Stationary Fokker–Plank Equation A. I. Noarov Institute for Computational Mathematics, Russian Academy of Sciences, Moscow, Russia Received October 3, 2005

DOI: 10.1134/S0012266107060092

The present paper continues the study [1] of the solvability of the stationary Fokker–Plank equation Δu − div(uf ) = 0. (1) Equation (1) is considered on the entire space Rn , and the vector ﬁeld f is subjected to the condition of smallness in some norm and rapid decay at inﬁnity. We prove the existence of a solution of Eq. (1) other than identical zero in the class of functions of tempered growth. Here, just as in [1], the simplest special case is the Laplace equation Δu = 0, that is, Eq. (1) that corresponds to the zero vector ﬁeld and has the solution equal to identical unity. A simpliﬁcation (weakening) of the condition imposed on the vector ﬁeld f is the main diﬀerence of the present paper from [1]. This condition does not contain derivatives of f , and the smoothness of f is not assumed. The main result of the present paper, Theorem 3, is that the solvability of Eq. (1) takes place in this case but is weaker (generalized). Theorem 4, the assertion on the classical solvability of Eq. (1), is a corollary of Theorem 3 under the condition of smoothness of f . The proof of the solvability of Eq. (1) is performed on the basis of the orthogonal decomposition of the space of vector ﬁelds in conjunction with the contraction mapping method. 1. THE WEYL EXPANSION 2

n

We introduce the space L (R ) of vector ﬁelds with inner product (g, h) =

n

gi (x)hi (x)dx1 . . . dxn .

i=1 Rn

The components of these vector ﬁelds belong tothe function space L2 (Rn ) with inner product (g, h) = Rn g(x)h(x)dx1 . . . dxn and norm g = (g, g). The spaces of vector ﬁelds and functions, as well as the corresponding inner products and norms, are denoted by the same symbols; which space is meant is always clear from the context. We introduce the following subspaces: G (Rn ) is the set of gradients of all locally square integrable functions that have ﬁrst derivatives in L2 (Rn ), and J (Rn ) is the closure in L2 (Rn ) of the set of all divergence free vector ﬁelds in W21 (Rn ). In this section, we perform the orthogonal decomposition of L2 (Rn ) into the subspace G (Rn ) of gradient vector ﬁelds and the subspace J (Rn ) of solenoidal vector ﬁelds. Decomposition theorems go back to Weyl’s work; similar results were obtained in [2]. Nevertheless, the proof of Theorem 1 given in [2, p. 38] did not rigorously solve the problem on the convergence of a sequence of functions in L2 in the case of convergence of their gradients in L2 . The most severe diﬃculties are encountered for the case in which the values of the functions at some ﬁxed point are assumed to be zero [2, p. 39]. In forthcoming considerations, the functions satisfy the integral condition (2), and convergence problems are solved with the use of Lemma 1 in [1]. 833

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Theorem 1. Each vector ﬁeld F in the space L2 (Rn ) of vector ﬁelds can be represented in the form F = h + ∇U, where U is a locally square integrable function such that ∇U ∈ G (Rn ) and h ∈ J (Rn ). Moreover, ∇U ≤ F, h ∈ J (Rn ) is determined uniquely, and U is determined to within a constant term. Proof. The space W21 (Rn ) is everywhere dense in L2 (Rn ); therefore, there exists a sequence ∞ {Fk }k=1 of vector ﬁelds in W21 (Rn ) that converges in L2 (Rn ) to the vector ﬁeld F occurring in the assumptions of the theorem. By Theorem 2 in [1], each Fk ∈ W21 (Rn ) can be represented in the form Fk = ∇Uk + hk , where Uk is a locally square integrable function determined to within a constant term; moreover, ∇Uk ∈ W21 (Rn ), hk ∈ W21 (Rn ), div hk = 0, and ∇Uk ≤ Fk . ∞ By applying Theorem 2 in [1] to a diﬀerence of elements of the sequence {Fk }k=1 , we obtain ∞ 2 the inequality ∇Uk − ∇Ul ≤ Fk − Fl . The sequence {Fk }k=1 converges in L (Rn ) and hence ∞ is a Cauchy sequence; therefore, by virtue of the last inequality, {∇Uk }k=1 is a Cauchy sequence in L2 (Rn ); and since L2 (Rn ) is dense, we ﬁnd that it converges to some vector ﬁeld g ∈ L2 (Rn ). We set h = F − g and note that the sequence {Fk − ∇Uk } converges in L2 (Rn ) to h = F − g, ∞ ∞ since the sequences {Fk }k=1 and {∇Uk }k=1 converge in L2 (Rn ) to F and g, respectively. Moreover, Fk − ∇Uk ∈ W21 (Rn ) and div (Fk − ∇Uk ) = div hk = 0; consequently, h ∈ J (Rn ). By passing to the limit in the inequality ∇Uk ≤ Fk , we obtain the estimate g ≤ F. The derivation of the decomposition F = h + g is complete. Let us now prove the existence of a locally square integrable function U (x) such that ∇U (x) = g(x). We assume that the functions Uk satisfy the relation Uk (y)y1 . . . dyn = 0 (2) Q1/2

(the integration is performed over the cube Qσ = {x : |xi | < σ, i = 1, 2, . . . , n}, σ = 1/2); the validity of this condition can always be achieved by adding appropriate constants to Uk . The se∞ quence {∇Uk }k=1 is a Cauchy sequence in L2 (Rn ); therefore, by (2) and Lemma 1 in [1] applied ∞ ∞ to a diﬀerence of terms of the sequence {∇Uk }k=1 , we ﬁnd that {Uk (x)}k=1 is a Cauchy sequence ∞ in L2 (Qσ ) for each σ > 1/2. Moreover, the Cauchy sequence {∇Uk }k=1 in L2 (Rn ) is also a Cauchy ∞ sequence in L2 (Qσ ) for any σ > 1/2. Therefore, {Uk (x)}k=1 is a Cauchy sequence in W21 (Qσ ) for ∞ each σ > 1/2. Since the spaces W21 (Qσ ) are complete, it follows that the sequence {Uk (x)}k=1 con1 verges in W2 (Qσ ) for each cube Qσ (σ > 1/2) to some function U (x) that is deﬁned on Rn and is locally square integrable, and all of its ﬁrst derivatives belong to L2 (Rn ). The convergence of ∇Uk to ∇U in L2 (Qσ ) for all σ > 1/2 and the convergence of ∇Uk to g in L2 (Rn ) imply that ∇U = g. It remains to prove the uniqueness of the decomposition. To this end, it suﬃces to use the relations 0 = h + ∇U , ∇U ∈ L2 (Rn ), h ∈ J (Rn ) (U is a locally square integrable function) to ∞ show that 0 = h = ∇U . Since h ∈ J (Rn ), it follows that there exists a sequence {hk }k=1 of vector 2 n 1 n ﬁelds converging in L (R ) to h and such that hk ∈ W2 (R ), div hk = 0. Let Z ∈ C0∞ (Rn ); by performing integration by parts, we obtain (hk , ∇Z) = − (Z, div hk ) = 0. By passing to the limit, we obtain (h, ∇Z) = 0, which implies that (∇U, ∇Z) = 0 for all Z ∈ C0∞ (Rn ). By performing integration by parts once more, we get Rn U (x)ΔZ(x)dx1 . . . dxn = 0 for all Z ∈ C0∞ (Rn ). This, together with the corollary of the Friedrichs theorem (or the Weyl lemma, see [3, p. 248 of the Russian translation; 4, p. 113 of the Russian translation]), implies that U ∈ C ∞ and ΔU = 0; moreover, ∇U ∈ L2 (Rn ). We ﬁx a positive integer i between 1 and n and set V = ∂U/∂xi . Then V ∈ L2 (Rn ), V ∈ C ∞ , and ΔV = 0. Let us show that V ≡ 0. To this end, we take an arbitrary point y ∈ Rn and the ball Bσ with radius σ and center y and use the mean-value property for a harmonic function and the Cauchy–Schwarz inequality: ⎛ ⎞1/2 V 1 ⎝ 1 μ (Bσ ) V 2 (x)dx1 . . . dxn ⎠ ≤ V (x)dx1 . . . dxn ≤ . |V (y)| = 1/2 μ (Bσ ) μ (B ) σ μ (Bσ ) Bσ

Bσ

[Here μ( ) is the Lebesgue measure.] By letting σ to inﬁnity, we obtain V (y) = 0, which implies that V ≡ 0 and ∇U ≡ 0. The proof of the theorem is complete. DIFFERENTIAL EQUATIONS

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Theorem 1 permits one to deﬁne a linear continuous operator mapping the space L2 (Rn ) of vector ﬁelds into itself and taking each vector ﬁeld F ∈ L2 (Rn ) to the gradient vector ﬁeld g ∈ L2 (Rn ) such that F − g ∈ J (Rn ). One can show that such an operator performs an orthogonal [in L2 (Rn )] decomposition of the space of vector ﬁelds [2], but in the present paper, just as in [1], orthogonality is not used. ´ 2. SOME POINCARE-STEKLOV TYPE INEQUALITIES

∞ ∞ Lemma 1. Let {ak }k=1 be a sequence of positive numbers such that the series k=1 ak is conver ∞ gent. Let W (x) be the function deﬁned on Rn by the condition W (x) = k=l ak /kn+2 if x ∈ Ql \Ql−1 for some positive integer l. Then the integral I1 ≡ Rn W (x)u2 (x)dx1 . . . dx n is convergent integral for any locally square integrable function u such that ∇u ∈ L2 (Rn ) and Q1/2 u(y)dy1 . . . dyn = 0; moreover, ∞

n+2 ak |∇u|2 dx1 . . . dxn . (3) I1 ≤ n × 2 k=1

Rn

Proof. By Lemma 1 in [1], the inequality 2 n+2 n+2 u dx1 . . . dxn ≤ n × 2 k |∇u|2 dx1 . . . dxn Qk

Qk

is valid for each positive integer k. By multiplying it by ak /kn+2 and by summing the resulting inequality with respect to k, we obtain ∞ ∞ ak 2 n+2 u dx1 . . . dxn ≤ n × 2 ak |∇u|2 dx1 . . . dxn . n+2 k k=1 k=1 Qk

Qk

The right-hand side of this inequality does not exceed ∞

n+2 ak |∇u|2 dx1 . . . dxn ; n×2 k=1

Rn

therefore, both sides are ﬁnite, and ∞ ∞ ak 2 n+2 u dx . . . dx ≤ n × 2 ak |∇u|2 dx1 . . . dxn . 1 n n+2 k k=1 k=1 Rn

Qk

It remains to show that the left-hand side of the last inequality is equal to I1 . To this end, we split the integrals over the cubes Qk into integrals over smaller sets, rearrange the terms, and use the deﬁnition of the function W (x) : ⎛ ⎞ ∞ ∞ k ak ak ⎜ ⎟ u2 dx1 . . . dxn = u2 dx1 . . . dxn + u2 dx1 . . . dxn ⎠ n+2 n+2 ⎝ k k k=1 k=1 l=2 Qk

Ql \Ql−1

Q1

∞ ak u2 (x)dx1 . . . dxn = n+2 k k=1 Q1

∞ ∞ ak + u2 (x)dx1 . . . dxn = I1 . n+2 k l=2 k=l Ql \Ql−1

The rearrangement of terms is justiﬁed by the absolute convergence. The proof of the lemma is complete. DIFFERENTIAL EQUATIONS

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Lemma 2. Let n ∈ N and α > 0. Then there exists a constant C > 0 such that the integral I2 ≡ (1 + |x|)−n−α−2 u2 (x)dx1 . . . dxn Rn

is convergent for any locally square-integrable function u satisfying the conditions ∇u ∈ L2 (Rn ) and u(y)dy1 . . . dyn = 0, Q1/2

and I2 ≤ C

|∇u|2 dx1 . . . dxn .

(4)

Rn

The constant C = C(n, α) can depend on α and the number n of space variables but is independent of u. ∞ Proof. We use Lemma 1. We set ak = 1/k1+α . Then the series k=1 ak is convergent, and all assumptions of Lemma 1 are satisﬁed; therefore, we obtain the inequalities ∞

1 n+2 |∇u|2 dx1 . . . dxn , (5) I1 ≤ n × 2 1+α k k=1 Rn

where W (x) =

∞ k=l

1/kn+3+α if x ∈ Ql \Ql−1 . It remains to note that W (x) =

∞ k=l

1 kn+3+α

+∞ l−n−2−α ; ≥ y −n−3−α dy = n+2+α l

(1 + |x|)−n−2−α ; therefore, n+2+α I1 ≥ (n + 2 + α)−1 I2 . By comparing the last inequality with inequality (5), we obtain the desired assertion. The proof of the lemma is complete. in addition, l − 1 ≤ |x|, since x ∈ Ql−1 . Then l ≤ |x| + 1 and W (x) ≥

3. CONTRACTION MAPPING OF THE SPACE OF VECTOR FIELDS Let f be a given vector ﬁeld, and let r be a given real number. We deﬁne a mapping A of the space L2 (Rn ) of vector ﬁelds into itself, that takes each vector ﬁeld f1 to a vector ﬁeld f2 by the following algorithm. 1. We expand the vector ﬁeld f1 into the components f1 = j + g, j ∈ J (Rn ), g ∈ G (Rn ). 2. We ﬁnd a function U (x) such that Q1/2 U (x)dx1 . . . dxn = r and ∇U = g. 3. We set f2 = U f . In this section, we show that the mapping A is well deﬁned and is a contraction mapping under some conditions imposed on f . In the following, we use the contraction mapping method to prove the solvability of Eq. (1). Lemma 3. Let r ∈ R, n ∈ N, let f : Rn → Rn be a measurable vector ﬁeld, and let the condition supx∈Rn (1 + |x|)n/2+α+1 |f (x)| < +∞ be satisﬁed for some α > 0. Then the operator A : L2 (Rn ) → L2 (Rn ) given by the above-represented algorithm is well deﬁned. Proof. Let f1 ∈ L2 (Rn ). The expansion f1 = j + g, j ∈ J (Rn ), g ∈ G (Rn ), performed at the ﬁrst step of the algorithm exists and is unique by virtue of Theorem 1. Consider some locally square integrable function V (x) such that ∇V = g. We set U (x) = V (x) − Q1/2 V (y)dy1 . . . dyn + r. Then DIFFERENTIAL EQUATIONS

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∇U = g and Q1/2 U (x)dx1 . . . dxn = r. The existence of the function U obtained on the second step of the algorithm is thereby justiﬁed; by Theorem 1, this function is uniquely determined. It remains to show that U f ∈ L2 (Rn ). To this end, we note that (U (x) − r)dx1 . . . dxn = 0, ∇(U − r) = g, 2α > 0; Q1/2

therefore, by Lemma 2, the integral (1 + |x|)−n−2α−2 (U (x) − r)2 dx1 . . . dxn Rn

is convergent; i.e., (1 + |x|)−n/2−α−1 (U (x) − r) ∈ L2 (Rn ). Since the function (1 + |x|)n/2+α+1 |f (x)| is bounded, we have (U (x) − r)f (x) = (1 + |x|)−n/2−α−1 (U (x) − r)(1 + |x|)n/2+α+1 f (x) ∈ L2 (Rn ) . Now the inclusion U f ∈ L2 (Rn ) follows from the condition f ∈ L2 (Rn ), which is valid by virtue of the boundedness of the function (1 + |x|)n/2+α+1 |f (x)| and the convergence of the integral (1 + |x|)−n−2α−2 dx1 . . . dxn . Rn

The proof of the lemma is complete. Theorem 2. Let r ∈ R and n ∈ N. Then for each α > 0, there exists an m = m(n, α) > 0 such that for each measurable vector ﬁeld f : Rn → Rn satisfying the condition sup (1 + |x|)n/2+α+1 |f (x)| < m,

(6)

x∈Rn

the operator A : L2 (Rn ) → L2 (Rn ) given by the above algorithm is a contraction operator ; i.e., there exists an L ∈ [0; 1) such that Af1 − Af2 ≤ L f1 − f2 for all f1 , f2 ∈ L2 (Rn ). The constant L = L(n, α) can depend on n and α but is independent of f1 and f2 . Proof. Let f1 , f2 ∈ L2 (Rn ). By using the deﬁnition of the operator A and Theorem 1 (the uniqueness of the expansion), we ﬁnd that the passage from f1 − f2 to Af1 − Af2 can be performed in accordance with the following procedure. 1. We expand the vector ﬁeld f1 − f2 into two components, f1 − f2 = j + g, j ∈ J (Rn ), g ∈ G (Rn ). 2. We ﬁnd a function u(x) such that Q1/2 u(x)dx1 . . . dxn = 0 and ∇u = g. 3. Then Af1 − Af2 = uf . Therefore, the proof of Theorem 2 consists in relating the norms uf and f1 − f2 . By Theorem 1, at the ﬁrst step, the L2 (Rn )-norm does not grow: g ≤ f1 − f2 .

(7)

The function u(x) obtained at the second step (it diﬀers from the function used in the proof of Lemma 3) does not necessarily belong to L2 (Rn ), but all of its ﬁrst derivatives belong to L2 (Rn ), since they are the components of a vector ﬁeld g ∈ L2 (Rn ), and inequality (7) is valid. It follows from (7) that ∇u ≤ f1 − f2 ; therefore, the relation u(x)dx1 . . . dxn = 0 Q1/2

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and Lemma 2 imply that 2 (1 + |x|)−n−2α−2 u2 (x)dx1 . . . dxn ≤ C∇u2 ≤ C f1 − f2 . Rn

By using the last inequality and (6), for uf , we can estimate the L2 (Rn )-norm as 2 Af1 − Af2 = uf 2 = u2 (x)|f (x)|2 dx1 . . . dxn =

Rn

2 (1 + |x|)−n−2α−2 u2 (x) (1 + |x|)n/2+α+1 |f (x)| dx1 . . . dxn

Rn 2

≤ C (f1 − f2 m) , where C is determined by 2α and n. By choosing m > 0 from the condition m < C −1/2 , we obtain a contraction operator A. The proof of the theorem is complete. 4. GENERALIZED AND CLASSICAL SOLVABILITY OF THE STATIONARY FOKKER–PLANK EQUATION Theorem 3. Let r ∈ R and n ∈ N. Then for each α > 0, there exists an m = m(n, α) > 0 such that, for each measurable vector ﬁeld f : Rn → Rn satisfying condition (6), there exists a unique 2 n 2 n n locally square integrable function u such that ∇u ∈ L (R ) , uf ∈ L (R ) , ∇u − uf ∈ J (R ) , and u(y)dy1 . . . dyn = r. Q1/2 Proof. We use the contraction mapping method. Let A : L2 (Rn ) → L2 (Rn ) be the operator constructed in Section 3. We choose a number m > 0 such that A is a contraction operator under condition (6). (This is possible by Theorem 2.) We take some vector ﬁeld f0 ∈ L2 (Rn ), for example, ∞ f0 ≡ 0. Let us construct a sequence {fk }k=1 with the use of the recursion formula fk = Afk−1 . Since ∞ A is a contraction mapping, it follows that {fk }k=1 is a Cauchy sequence in L2 (Rn ). The space ∞ L2 (Rn ) is complete; therefore, {fk }k=1 converges in L2 (Rn ) to some vector ﬁeld a ∈ L2 (Rn ). ∞ The contraction mapping A is continuous; therefore, the sequence {Afk−1 }k=1 converges in L2 (Rn ) 2 n to Aa. The left-hand side of the relation fk = Afk−1 converges in L (R ) to a; consequently, a = Aa. Thus, a is a ﬁxed point of the mapping A; in particular, a lies in the range of A. Then (see the deﬁnition of the operator A) there exists a locally square integrable function u(x) such that Q1/2 u(y)dy1 . . . dyn = r and a = uf ; moreover, ∇u ∈ L2 (Rn ) and a − ∇u ∈ J (Rn ). But a = uf ; therefore, ∇u − uf ∈ J (Rn ). Uniqueness follows from the fact that a contraction mapping cannot have more than one ﬁxed point. The proof of the theorem is complete. Theorem 3 implicitly introduces the notion of a generalized solution of Eq. (1) and justiﬁes the unique generalized solvability of this equation. The uniqueness of a generalized solution is treated in the sense that two arbitrary solutions coincide almost everywhere in the sense of the Lebesgue measure. The uniqueness theorem can be stated separately and can be strengthened by eliminating the condition uf ∈ L2 (Rn ). This study was motivated by the following simple considerations on the solvability of Eq. (1). Let f be a gradient vector ﬁeld; i.e., ∇V = f for some suﬃciently smooth function V . We set u(x) = exp(V (x)); then ∇u = uf , and relation (1) is valid. But the relation ∇u = uf is also valid for the case in which f is only continuous, V is continuously diﬀerentiable, and the expression Δu is not necessarily meaningful. The idea arises to deﬁne a solution of Eq. (1) also in this case as well as in the case of nongradient functions f . To this end, we generalize the notion of divergence and introduce the space J (Rn ), which leads to Theorem 3. In conclusion, we note that Theorem 3 can be used for the derivation of the following assertion on the classical solvability of Eq. (1). DIFFERENTIAL EQUATIONS

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Theorem 4. For each α > 0, there exists an m = m(n, α) > 0 such that, for each inﬁnitely diﬀerentiable vector ﬁeld f : Rn → Rn satisfying condition (6), there exists a function u ∈ C ∞ (Rn ) satisfying the inclusions ∇u ∈ L2 (Rn ) and uf ∈ L2 (Rn ) and the relations u(y)dy1 . . . dyn = 1, Δu − div(uf ) = 0. Q1/2

Proof. This theorem diﬀers from Theorem 3 by the additional condition f ∈ C ∞ (Rn ). We set r = 1. Let u(x) be the function found in Theorem 3; it suﬃces to show that u ∈ C ∞ (Rn ) and Δu − div(uf ) = 0. To this end, we note that ∇u − uf ∈ J (Rn ); therefore, (∇u − uf , ∇Z) = 0 for each Z(x) ∈ C0∞ (Rn ). (The derivation of a similar relation was given in the proof of Theorem 1.) By performing integration by parts, we obtain

n ∂Z = 0. (8) (∇u − uf , ∇Z) = −(u, ΔZ) − (uf , ∇Z) = − u, ΔZ + fi ∂xi i=1 n The operator that takes each function Z(x) ∈ C0∞ (Rn ) to the function ΔZ + i=1 fi ∂Z/∂xi is the formal adjoint of the operator that takes Z(x) ∈ C0∞ (Rn ) to ΔZ − div(Zf ). Therefore, by relation (8) and the corollary of the Friedrichs theorem [3, p. 248 of the Russian translation], u ∈ C ∞ (Rn ) and Δu − div(uf ) = 0. The proof of the theorem is complete. REFERENCES 1. Noarov, A.I., Diﬀer. Uravn., 2006, vol. 42, no. 4, pp. 521–530. 2. Ladyzhenskaya, O.A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti (Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid), Moscow: Fizmatlit, 1961. 3. Yosida, K., Functional Analysis, Berlin, 1965. Translated under the title Funktsional’nyi analiz , Moscow: Mir, 1967. 4. McKean, H.P., Stochastic Integrals, Providence: Chelsea Publishing, 1969. Translated under the title Stokhasticheskie integraly, Moscow: Mir, 1972.

DIFFERENTIAL EQUATIONS

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 840–853. c Z.M. Gogniashvili, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 820–832. Original Russian Text

INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS

On Some Integral Equations on a Surface with a Conical Point Z. M. Gogniashvili Tbilisi State University, Tbilisi, Georgia Received May 25, 2004

DOI: 10.1134/S0012266107060109

INTRODUCTION The aim of the present paper is to generalize the method of integral equations, which is a classical method for studying boundary value problems and proving existence theorems, to surfaces with conical points. Let S k be a curvilinear cone with maximum opening angle less than π and vertex the origin of the given by the equation x3 = φ (x1 , x2 ), where coordinate system Ox1 x2 x3 . We assume that the cone is φ (x1 , x2 ) is a homogeneous function of ﬁrst order. If x21 + x22 > a, a > 0, then ∂φ (x1 , x2 ) /∂xi belongs to C 0,λ , λ ≤ 1, i = 1, 2. Obviously, ∂φ/∂xi is a homogeneous function of zero order. We denote the domain bounded by the surface S k by Dk . Let D be the domain bounded by a surface S that is smooth of class L1 (α) (a Lyapunov surface with exponent λ) everywhere outside an arbitrary neighborhood of the point O and coincides with S k in C(0, A), A > 0, where C(0, A) is the ball with radius A and center O. Deﬁnition. A function K(x, y) is referred to as a function of the class B if the following conditions are satisﬁed. (1◦ ) K(x, y) = ψ(x, y)/|x − y|2 , x, y ∈ S\{0}, |ψ| < M , where ψ is a homogeneous function of zero order on S k ∩ S. (2◦ ) If x, y ∈ S, |x| > a, |y| > a, 0 < a < A/2, then |ψ(x, y)| < M |x − y|λ . (3◦ ) If |x|, |y|, |x |, |x |, |y |, and |y | are greater than a, then |ψ (x , y) − ψ (x , y)| < M |x − x | ,

|ψ (x, y ) − ψ (x, y )| < M |y − y | .

λ

λ

One can readily see that the kernels of a classical double layer potential and of the normal derivative of a simple layer potential are kernels of the class B on S. We also show that the kernels of integral equations obtained after the regularization of singular integral equations of the main boundary value problems of three-dimensional elasticity [1–4] are kernels of the class B on S. Consider an integral equation with kernel of the class B : x ∈ S\{0}. (1) ϕ(x) − σ K(x, y)ϕ(y)dSy = f (x), S

older functions with exponent γ; the We use the following function classes: the class C (0,γ) of H¨ (0,δ) α older class Cα,δ such class Cα such that ϕ(x) ∈ Cα if ϕ(x) = ψ(x)/|x| , |ψ| < C; the weighted H¨ (0,δ) that ϕ ∈ Cα,δ if ϕ ∈ Cα and |x|δ ψ(x) = |x|α+δ ϕ(x) ∈ C (0,δ) . Boundary value problems in the case of a surface with a conical point were considered by numerous authors (see [5–9] and the bibliography therein). Our aim is to generalize the method of integral equations to the case of a surface with a conical point. 840

ON SOME INTEGRAL EQUATIONS ON A SURFACE WITH A CONICAL POINT

841

1. AUXILIARY SURFACES We approach the surface S by some smooth surfaces Sh , 0 < h < A. For example, let the surface Sh coincide with S for φ (x1 , x2 ) ≥ h and be given by the relation x3h = φ2 (x1 , x2 )/(2h) + h/2 for φ (x1 , x2 ) < h; |x3h − x3 | < h and x3h → x3 uniformly for arbitrary points x (x1 , x2 , x3 ) and xh (x1 , x2 , x3h ) as h → 0. We introduce the following notation: Sh \ (Sh ∩ S) = Sh and Sh ∩ S k = Sh . We consider the surfaces Sh and Sh1 , h1 < h. We say that x ∈ Sh and z ∈ Sh1 are corresponding points and write x ∼ z if x1 /z1 = x2 /z2 = h/h1 . Then we have x3 /z3 = h/h1 , and the points x and z lie on a same ray issuing from the point O. Indeed, let x ∈ Sh and z ∈ Sh 1 ; then φ2 (h1 h−1 x1 , h1 h−1 x2 ) h1 h1 φ2 (x1 , x2 ) h h1 φ2 (z1 , z2 ) h1 = = + = x3 , + + z3 = 2h1 2 2h1 2 h 2h 2 h as desired. For each point x ∈ D k , there exists an h such that x ∈ Sh . If x (x1 , x2 , x3 ) ∈ Sh , then z (tx1 , tx2 , tx3 ) ∈ Sth ; furthermore, Sh is a Lyapunov surface with exponent λ. We set |¯ x| = max (|x | , |x |) ,

|x| = min (|x | , |x |) ,

y (x , x ) = min (|x − y| , |x − y|) . λ

x| , where the constant c is Lemma 1. If x , x ∈ Sh , then |n (x ) − n (x )| < c |x − x | /|¯ independent of h. Proof. (a) Let x , x ∈ S k ∩ S. We consider points z , z ∈ S k on the same rays (as the points x and x , respectively) issuing from the point O and such that a < |z| < A and |x |/z = |x |/|z |. Then |x − x |/(z − z ) = |x|/|z| and

c |x − x |

λ

|n (x ) − n (x )| = |n (z ) − n (z )| < c |z − z | < λ

λ

|x|

.

(b) Let x , x ∈ Sh . Consider the surface Sh1 , h1 > a. Let z , z ∈ Sh 1 , z ∼ x , and z ∼ x ; then we have |z − z |/|x − x | = |z|/|x| , |n (x ) − n (x )| = |n (z ) − n (z )| < c |z − z | < c |x − x | /|x| , λ

λ

λ

|¯ x| |n (x ) − n (x )| < |x| |n (x ) − n (x )| + |x − x | |n (x ) − n (x )| < c |x − x | . λ

λ

λ

λ

The proof of the lemma is complete. For each h, 0 < h < A, we deﬁne the kernel Kh (x, y) so as to ensure that it coincides with K(x, y) for x, y ∈ Sh \Sh , i.e., on the common part of Sh and S, and satisﬁes the following requirements, similar to conditions (1◦ )–(3◦ ). (1 ) Kh (x, y) = ψ(x, y)/|x − y|2 , |ψ| < M , x, y ∈ Sh , where ψ(x, y) is a homogeneous function of zero order. (2 ) If x, y ∈ Sh , h > 0, then |ψ(x, y)| < M |x − y|λ . (3 ) The inequalities ψ (x , y) − ψ (x , y) < M |x − x | , λ

ψ (x, y ) − ψ (x, y ) < M |y − y |

λ

are valid on Sh , h > a, where M is independent of h. For example, let K be the kernel of a double layer potential. We set Kh (x, y) = cos(x − y, n(y))/|x − y|2 ,

x, y ∈ Sh .

The kernel thus deﬁned satisﬁes conditions (1 )–(3 ). ˜, y˜ ∈ Sh1 , x ∼ x ˜, and y ∼ y˜, then It follows from the condition (1 ) that if x, y ∈ Sh , x ψ(x, y) = ψ (˜ x, y˜). DIFFERENTIAL EQUATIONS

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Theorem 1. The kernel Kh (x, y) can be represented in the form Kh (x, y) = ψ1 (x, y)/ |x|α |y|β |x − y|2−α−β , |ψ1 | < c, ∀α ≥ 0, β ≥ 0, α + β ≤ λ, x, y ∈ Sh , where c is independent of h. ˜ and y˜ on the same generatrices of Proof. Let |x| < |y|. In case (a), x ∈ S k ∩ Sh , we choose x the cone as the points x and y, respectively, a < |x| < A, and |˜ x|/|x| = |˜ y|/|y|. Then |˜ x − y˜|/(x − y) = x ˜/|x|,

λ

|ψ(x, y)| = |ψ (˜ x, y˜)| < c |˜ x − y˜| < c|x − y|λ /|x|λ .

˜, y˜ ∈ Sh1 , x ∼ x ˜, and y ∼ y˜; In case (b), x ∈ Sh , we consider the surface Sh1 , a < h1 < A. Let x λ then |˜ x − y˜|/|x − y| = |˜ x|/|x|, a/2 < |˜ x| < A, and |ψ(x, y)| = |ψ (˜ x, y˜)| < c |˜ x − y˜| < c|x − y|λ /|x|λ . In cases (a) and (b), we have |x|λ |ψ(x, y)| < c|x − y|λ . Then |y|λ |ψ(x, y)| < c|x − y|λ , i.e., |ψ(x, y)| < c|x − y|λ /(max(|x|, |y|))λ , c is independent of h. For α + β ≤ λ, we have Kh (x, y) = ∀α ≥ 0,

ψ(x, y) c|x − y|α+β c < < 2 2 α+β 2−α−β |x − y| |x − y| (max(|x|, |y|)) |x − y| |x|α |y|β β ≥ 0, α + β ≤ λ.

We set ψ1 (x, y) = Kh (x, y)|x − y|2−α−β |x|α |y|β , |ψ1 | < c, which implies that |Kh (x, y)| =

ψ1 (x, y) , |x|α |y|β |x − y|2−α−β

|ψ1 | < c,

∀α ≥ 0,

β ≥ 0,

α + β ≤ λ,

where c is independent of h. The proof of the theorem is complete. All lemmas and theorems proved for the kernel K(x, y) on the surface S with a conical point are valid for the kernel Kh on Sh as well. They can be proved on the basis of Lemma 1 and Theorem 1 by analogy with the case of the kernel K. 2. AUXILIARY EQUATION Along with Eq. (1), we consider the auxiliary equation x ∈ Sh , ϕh (x) − σ Kh (x, y)ϕh (y)dSy = fh (x),

fh (x1 , x2 , x3h ) = f (x1 , x2 , x3 ) .

(2)

Sh

After p − 1 iterations of Eqs. (1) and (2), we obtain ϕ(x) − σ ¯ Kp (x, y)ϕ(y)dSy = F (x),

x ∈ S\{0},

(3)

S

where σ ¯ = σ p , F (x) = f + σKf + σ 2 K2 f + · · · + σ p Kp f , and Kp (x, y) = ψp−1 (x, y)/ |x|α |y|β , α > 0, β > 0, α + β = 2, and the equation with continuous kernel ¯ Khp (x, y)ϕh (y) = Fh (x), (4) ϕh (x) − σ Sh

where Fh = fh (x) + σKh fh + σ 2 Kh2 fh + · · · + σ (p−1) Kh(p−1) fh and Khp (x, y) = ψp (x, y)/ |x|α |y|β , α > 0, β > 0, α + β = 2. DIFFERENTIAL EQUATIONS

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Theorem 2. The relation x (x1 , x2 , x3 ) , y (y1 , y2 , y3 ) ∈ S\{0},

x, y¯) = ψp (x, y), lim ψhp (¯

h→0

x ¯ (x1 , x2 , x3h ) , y¯ (x1 , x2 , x3h ) ∈ Sh , is valid uniformly on the set {x, y ∈ S | |x| ≥ a, |y| ≥ a}, for any a > 0 and p. Let us ﬁrst prove the following auxiliary assertion. Lemma 2. Let |y| > hν , 0 < ν < 1, and |x| > h. Then there exists a number γ, 0 < γ ≤ ν, such that |ψhp (x, y) − ψp (x, y)| < chγ . Proof. Obviously, y ∈ Sh ∩ S and x ∈ Sh ∩ S. We set S ∩ {x ∈ S | x3 < h} = S¯h and S\S¯h = Sh \Sh . If either t ∈ Sh or t ∈ S¯h , then 1 |t − y| ≥ |y| − |t| ≥ hν − h = hν 1 − h1−ν > hν 2 for suﬃciently small h. The lemma is valid for p = 1, since ψ1 (x, y) = ψ1h (x, y), x, y ∈ Sh \Sh . Consider the diﬀerence ψ2h (x, y) − ψ2 (x, y). We have 2−α−β α β |x| |y| Kh (x, t)Kh (t, y)dSt ψ2h (x, y) = |x − y| Sh

= |x − y|2−α−β |x|α |y|β

Kh (x, t)Kh (t, y)dSt Sh

+ |x − y|

2−α−β

|x| |y| α

β

Sh \Sh

ψ2 (x, y) = |x − y|2−α−β |x|α |y|β

K(x, t)K(t, y)dSt ¯h S

+ |x − y|

2−α−β

K(x, t)K(t, y)dSt = ψh2 + ψh2 ,

|x| |y| α

β

K(x, t)K(t, y)dSt = ψ2 + ψ2 .

¯h S\S (x, y) and Obviously, if x, y ∈ S\S¯h , then ψ2 (x, y) = ψh2 − ψ2 | < |ψh2 | + |ψ2 | , |ψh2 (x, y) − ψ2 (x, y)| = |ψh2 |x − t|2−α−β + |t − y|2−α−β (x, y)| < c dSt |ψh2 |t|β |x − t|2−α−β |t|α/2 |t − y|2−β−α/2 Sh

<

ch

c hν(2−α/2−β)

r dr r β+α/2

+

hνα/2

0

(1−ν)(2−α/2−β)

< ch

+

c

c a+h

hνα/2

Sh

dSt − x|2−α−β

|t|β+α/2 |t

r dr r 2−α/2

< ch(1−ν)(2−α/2−β) + chα(1−ν)/2

a α(1−ν)/2

< ch

γ

< ch ,

0 < γ < α/2,

x ∈ Sh ,

|y| > hν .

In a similar way, we obtain the estimate |ψ2 | < chγ , γ > 0, x ∈ S, |y| > hν , i.e., |y| > hν , x ∈ Sh \Sh = S\S¯h . |ψh2 (x, y) − ψ2 (x, y)| < chγ , DIFFERENTIAL EQUATIONS

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GOGNIASHVILI

Let us now estimate |ψh3 (x, y) − ψ3 (x, y)|. We have 2−α−β α β |x| |y| Kh (x, t)Kh2 (t, y)dSt ψh3 (x, y) = |x − y| Sh

= |x − y|2−α−β |x|α |y|β

Kh (x, t) Sh

ψh2 (t, y)dSt = ψh3 (x, y) + ψh3 (x, y), |t|α |y|β |t − y|2−α−β ⎡

where

(x, t) = |x − y|2−α−β |x|α |y|β ⎣ ψh3

ψh2 (t, y)dSt α β |t| |y| |t − y|2−α−β ⎤

K(x, t) Sh

+

K(x, t) Sh

(x, y) = ψ3h

ψh2 (t, y)dSt ⎦, α |t| |y|β |t − y|2−α−β

K(x, t) Sh \Sh

ψh2 (x, y)dSt . α β |t| |y| |t − y|2−α−β

(x, y) = ψ3 (x, y), Just as above, we obtain the relations ψ3 (x, y) = ψ3 (x, y) + ψ3 (x, y) and ψh3 x, y ∈ S\S¯h . Then (x, y) − ψ3 (x, y)| < |ψh3 (x, y)| + |ψ3 (x, y)| , |ψh3 (x, y) − ψ3 (x, y)| = |ψ3h dSt (x, y)| < chγ |x − y|2−α−β |x|α |y|β K(x, t) α β |ψh3 |t| |y| |t − y|2−α−β Sh |x − t|2−α−β + |t − y|2−α−β +c dSt |x − t|2−α−β |t|α/2+β |t − y|2−α/2−β Sh

< ch |x − y| γ

2−α−β

|x| |y| α

β

K(x, t) Sh

+

ch

c hν(2−α/2−β)

r dr r β+α/2

+

0

0 < γ < α/2,

x ∈ Sh ,

c

dSt − y|2−α−β

|t|α |y|β |t

a+ch

hνα/2

r dr r 2−α/2

< chγ ,

a

|y| > hν .

In a similar way, one can obtain the estimates |ψ3 (x, y)| < chγ , x ∈ S, |y| > hν , and |ψh3 (x, y) − ψ3 (x, y)| < chγ , |ψhp (x, y) − ψp (x, y)| < chγ ,

..., γ > 0,

|y| > hν ,

|x| > h.

In the same way, for |x| > hν , 0 < ν < 1, |y| > h, we obtain |ψhp (x, y) − ψp (x, y)| < chγ ,

γ > 0.

Now the validity of the theorem is obvious. Indeed, we choose an arbitrarily small number ε. Let |x| ≥ a, |y| ≥ a, a > 0. The assertion of the theorem is justiﬁed if there exists a number h0 such that |ψhp (x, y) − ψp (x, y)| < ε, x, y ∈ {x, y ∈ S | |x| ≥ a, |y| ≥ a}, for h < h0 . We set h0 = min a1/ν , ε1/γ . If h < h0 , then |x| > a > hν , |y| > a > hν , and, by Lemma 2, |ψhp (x, y) − ψp (x, y)| < chγ < ε; that is, limh→0 ψhp (x, y) = ψp (x, y) uniformly on the set {x, y ∈ S | |x| ≥ a, |y| ≥ a} for all α > 0. It follows that limh→0 ψhp (x, y) = ψp (x, y), x, y ∈ S\{0}, which completes the proof. DIFFERENTIAL EQUATIONS

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845

3. THE FREDHOLM DENOMINATOR OF THE KERNEL Kph Consider Eq. (4) with continuous kernel Khp . For the kernel Khp , we construct the Fredholm σ ) and Dh (x, y, σ ¯ ). series Dh (¯ Lemma 3. There exists a function α(h), limh→0 α(h) = 0, α(h) > chε , where ε > 0 is an σ ) is uniformly convergent with respect to h, arbitrarily small number, such that the series α(h)Dh (¯ σ ¯ ∈ R, h ≤ A/2. Proof. We have [10, pp. 32–34] σ) = 1 + Dh (¯

∞

(−1)n

n=1

where

σ ¯n n!

···

Sh

Kph = Kph

t 1 t 2 . . . tn t 1 t 2 . . . tn

Khp dt1 . . . dtn , Sh

K (t , t ) . . . K ph 1 1 ph(t1 ,tn ) = , ... ... ... K (t , t ) . . . K ph n 1 ph(tn ,tn )

Kph = ψ(x, y)/ |x|α |y|β , α + β = 2, α > 0, β > 0. In the determinant Khp , one can assume that [10, p. 74] Khp (ti , ti ) = 0. We represent the kernel Khp in the form Khp (x, y) = Khp (x, y) (e(x, y) + e¯(x, y)), where 1 if |y| ≤ |x| 0 if |y| ≤ |x| e¯(x, y) = e(x, y) = 0 if |y| > |x|, 1 if |y| > |x|. 1 for i = k We introduce the following notation: e (ti , tk ) = eik , e¯ (ti , tk ) = e¯ik , |ti | = i , δik = 0 for i = k, ψ (ti , tk ) 1 e (ti , tk ) + e (ti , tk )) Khp (ti , tk ) δik = δik α β (¯ = 2 2 · · · 2 δik ψ (ti , tk ) (eik + e¯ik ) 1 2 n i k 1 ∗ i, k = 1, . . . , n, δik ψ (ti , tk ) eik , = 2 1 · · · 2n where the sum contains 2n terms and each term is an nth-order determinant in which the ∗ row δik (ψ (ti , t1 ) ei1 , . . . , ψ (ti , tn ) e∗in ) coincides with either δik (ψ (ti , t1 ) ei1 , . . . , ψ (ti , tn ) ein ) or δik (ψ (ti , t1 ) e¯i1 , . . . , ψ (ti , tn ) e¯in ). By using the Hadamard inequality (since |ψ| < M ), we ﬁnd that each term of the abovementioned sum is less than 1/2 ∗2 1/2 M n ∗2 ∗2 1/2 ∗2 · · · e∗2 , e12 + · · · + e∗2 e21 + e∗2 1n 23 + · · · + e2n n1 + · · · + en(n−1) 2 2 1 · · · n 1/2 2 1/2 < nei = n1/2 ei , δik e2i1 + e2i2 + · · · + e2in 1/2 2 1/2 < n¯ ei = n1/2 e¯i , δik e¯2i1 + · · · + e¯2in

where ei =

e¯i =

1 on {k , i ∈ (h, A) | k < i } 0 on {k , i ∈ (h, A) | k > i }, 1 on {k , i ∈ (h, A) | k > i } 0 on {k , i ∈ (h, A) | k < i },

2 M n n1/2 (˜ e1 , e˜2 , . . . , e˜n )m ; Khp < 2 2 1 2 · · · 2n m=1 n

DIFFERENTIAL EQUATIONS

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k ∈ {1, . . . , n}\{i},

846

GOGNIASHVILI

here, in each of 2n products e˜1 , e˜2 , . . . , e˜n , some (or all) factors ei are equal to ei , and the remaining ones are equal to ei , σ ¯n σ ¯ n dhn < n! n!

M n nn/2 Khp dt1 . . . dtn < |¯ σ | n!

···

n

Sh

Sh

Sh

2 dSh (˜ e1 · · · e˜n )m . 21 · · · 2n m=1 n

For example, let us estimate the term e1 e2 · · · en−1 dSh c dSh ··· e1 e2 · · · en−1 e˜n < δ ··· 2 2 1 · · · n h 21 22 · · · 2n−1 2−δ n Sh

Sh

Sn

c < δ h

A

d1 1

h

<

1

d2 ··· 2

h

c hδ δ2

A

1

d2 ··· 2

h

Sh

c dn < δ 1−δ h δ n

h

d1 1

h

n−1

n−3

h

A

1

d1 1

h

d2 ··· 2

n−2

h

h

A

c dn−2 < · · · < δ n−1 1−δ h δ n−2

dn−1 1−δ n−1

c d1 < δ n, 1−δ h δ 1

h

where δ > 0 is an arbitrarily small number. Consider the term dSh e¯1 e¯2 · · · e¯n−1 δn ··· e ¯ e ¯ · · · e ¯ e ˜ < · · · 1 2 n−1 n 21 · · · 2n 21 22 · · · 2+δ n Sh

Sh

Sn

A δ

< cA

d1 1

c δn−1

A h

A

d2 ··· 2

1

h

<

A

Sh

dn−1 n−1

n−2

A n−1

cAδ dn < 1+δ δ n

A h

A

d1 1

A

d2 2

1

n−2

dn−1 < ··· 1+δ n−1

c d1 < δ n. 1+δ h δ 1

In the remaining terms, there are also factors of the form ek as well as e¯k , k = 1, 2, . . . , n. For example, let us estimate the term e4 e5 · · · en−1 cAδ e¯1 e¯2 e¯3 e4 e5 · · · en e¯1 e¯2 e¯3 ··· dSh < δ ··· dSh 2+δ 2 2 2 2 2 2 2 1 2 · · · n h 1 2 3 4 5 · · · 2n−1 2−δ n Sh

Sh

Sh

cAδ < δ h

A

d1 1

cAδ hδ δn−4

d2 2

1

h

<

A

A h

d1 1

A

d3 3

2

A 1

d2 2

A 3

A 2

d3 3

Sh

d4 2+δ 4 A 3

A h

d5 5

5

d6 ··· 6

h

n−1

h

dn 1−δ n

cAδ cAδ d4 < = . hδ δn−4 hδ δn−4 h2δ 1+δ 4

The remaining integrals can be estimated in a similar way. Therefore, n n n n n/2 c |¯ σ d ¯ hn < σ| M × 2 n . n! δn n!h2δ n

σ | dhn (−1)n /n! is convergent uniBy the d’Alembert criterion, the series with general term n2δ |¯ n 2δ n n n n n/2 σ | dhn (−1) /n! < σ ¯ M × 2 n /(δn n!). Here δ can be chosen formly with respect to h, since n |¯ DIFFERENTIAL EQUATIONS

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∞ n arbitrarily close to zero. Therefore, the series n=1 hε |¯ σ | dhn (−1)n /n! is convergent uniformly with respect to h, where ε > 0 is an arbitrarily small number. Then, obviously, there exists a function α(h) inﬁnitely small as h → 0, α(h) > hε , where ε > 0 is an arbitrarily small number, and such that the series ∞ σ ¯ n dhn (−1)n σ) = α(h) α(h)Dh (¯ n! n=1 is convergent uniformly with respect to h and σ ¯ , which completes the proof. ¯ ). The following assertion can be proved by analogy with Now consider the series Dhp (x, y, σ Lemma 3. an arbitrarily Lemma 4. There exists a function A(h), limh→0 A(h) = 0, A(h) ≥ h, where ∞ ε is α β α β n ¯ ) = |x| |y| A(h) n=1 σ ¯ dhn (x, y, σ ¯ )/n! small number, such that the series |x| |y| A(h)Dhp (x, y, σ is convergent uniformly with respect to x, y, and h. ¯ ). Then We denote the sum of this series by A(h)ψhp (x, y, σ ¯) = Dhp (x, y, σ

ψh (x, y, σ ¯) , |x|α |y|β

|A(h)ψh | < c,

α + β = 2.

If K is the kernel of a double layer potential, then ¯ ) = |x|γ ψhp (x, y, σ ¯ )/y 2+γ , Dhp (x, y, σ

|A(h)ψh | < C.

σ ) and Dhp (x, y, σ ¯ ) satisfy the equations The functions Dh (¯ ¯ ) = Khp (x, y)Dh (¯ σ) + σ ¯ Khp (x, t)Dhp (t, y, σ ¯ ) dSt , Dhp (x, y, σ Sh

¯ ) = Khp (x, y)Dh (¯ σ) + σ ¯ Dhp (x, y, σ

(5) Dhp (x, t, σ ¯ ) Khp (t, y)dSt .

Sh

¯ ) is bounded for x, y ∈ Sh . Then the expression By Lemma 4, the expression |x|α |y|β Dhp (x, y, σ ¯ ) dSt |x|α |y|β Khp (x, t)A(h)Dph (t, y, σ Sh

is bounded as well. Indeed, α β ¯ ) dSt |x| |y| Khp (x, t)A(h)Dhp (t, y, σ Sh

c|x|α |y|β ≤ |x|α+ε |y|β

|x| 0

c|x|α |y|β + |t|2−α−ε |t|α |y|β |x|α−ε dSt

A |x|

dSt 2−α+ε |t| |t|α

< c.

σ ) is a bounded expression, i.e., Ah Dh (¯ σ ) is a It follows from Eq. (5) that |x|α |y|β Khp (x, y)Ah Dh (¯ σ ) can be an inﬁnitely large function of order ≤ A(h) as h → 0, bounded function. Therefore, Dh (¯ and a(h) ≥ A(h). ¯ ) such By [10, vol. IV, p. 74], for the original equation (2), there exists a function Ehp (x, y, σ that ¯ ) = Kh (x, y)Dh (σ p ) + σ Kh (x, t)Ehp (t, y, σ ¯ ) dSt , Ehp (x, y, σ Sh

p

¯ ) = Kh (x, y)Dh (σ ) + σ Ehp (x, y, σ

Ehp (x, t, σ ¯ ) Kh (x, y)dSt ; Sh

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here

p

p

p

¯ ) = Hh (x, y, σ)Dh (σ ) + σ Dhp (x, y, σ ) + σ Ehp (x, y, σ

p

Hh (x, t, σ)Dhp (t, y, σ)dSt Sh

= Hh (x, y, σ)Dh (σ p ) + H1h (x, y, σ), Hh (x, y, σ) = Kh (x, y) + σK2h (x, y) + · · · + σ p−2 Kh(p−1) (x, y), p p p Hh (x, t, σ)Dhp (t, y, σ p ) dSt . H1h (x, y, σ) = σ Dhp (x, y, σ ) + σ Sh

It follows from properties of the compositions of the kernels Kh that |Hh (x, y, σ p )| < c/ |x|ν |y|δ for all ν > 0, δ > 0, and ν + δ ≤ λ. One can readily see that c Hh (x, t, σ)A(h)Dhp (t, y, σ p ) dSt < , α + β = 2, |x|α |y|β Sh

and consequently, |A(h)H1h (x, y, σ)| < c/ |x|α |y|β and A(h) > hε . Indeed, let τ1 = Sh ∩ c(0, |x|/2), τ2 = Sh ∩ c(0, |x|/2), and τ3 = (Sh \τ1 )\τ2 ; then dSt 1 Hh (x, t, σ)A(h)Dhp (t, y, σ p ) dSt < |x|ν |y|β 2−ν−ε |x − t| |t|ε |t|2−β τi

Sh

c < |x|ν |y|β |x|2−ν−ε c < , α |x| |y|β

|x|

d 2−β+ε

0

c + ν β 2−β+ε |x| |y| |x|

|x|

d 2−ν−ε

0

c + ν β |x| |y|

A

d 4−β+ν

|x|

α + β = 2.

4. ALTERNATIVE THEOREM We ﬁx σ ¯ and set σ) , a(h) = 1/Dh (¯

A(h) = 1/ sup

¯ ) |x|α |b|β = 1/ sup ψhp (x, y, σ ¯) . Dhp (x, y, σ

x,y∈Sh

x,y∈Sh

I. Let a(h) and A(h) be inﬁnitesimal functions of the same order as h → 0. We have a(h)Dhp =

Dhp (x, y, σ ψ¯hp (x, y, σ ¯) ¯) ¯ hp (x, y, σ =R ¯) = , α β Dh (¯ σ) |x| |y|

α + β = 2,

ψ¯ < c.

(7)

By multiplying both sides of Eq. (5) by a(h), we obtain ¯ hp (t, y, σ ¯ ¯ ) = Khp (x, y) + σ ¯ Khp (x, t)R ¯ ) dSt , Rhp (x, y, σ Sh

(8) ¯ hp (x, t, σ ¯ ) Khp (t, y)dSt . R

¯ hp (x, y, σ ¯ ) = Khp (x, y) + σ ¯ R Sh

¯ hp (x , y, σ ¯ hp (x , y, σ ¯) − R ¯ ). Let |x | < |x |; then we have the Let us estimate the diﬀerence R following. DIFFERENTIAL EQUATIONS

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(a) If |x | < |x − x |, then |x | < 2 |x − x | and ¯ hp (x , y, σ R ¯ hp (x , y, σ ¯) − R ¯ ) <

c |x | c |x − x | + < < . α α α+γ α+γ |x | |y|β |x | |y|β |x | |y|β |x | |y|β c

γ

c

γ

(b) Let |x − x | < |x |. By (8), if γ < λ/2; then ¯ hp (t, y)dSt (Khp (x , t) − Khp (x , t)) R Sh γ dSt c |x − x | < α+γ+ε 2−ε−α β |t| |t|α |x | |y| Sh ∩c(0,|x |)

c |x − x | + α+γ−ε |x | |y|β

|x|

c |x − x | < . α+γ |t|2−γ+ε |t|α |x | |y|β γ

dSt

Sh \c(0,|x |)

Therefore, from (8), we obtain the estimate c |x − x |γ ¯ hp (x , y, σ R ¯ hp (x , y, σ ¯) − R ¯ ) < , α+γ |x | |y|β

γ<

λ , 2

α + β = 2.

¯ hp (x, y , σ ¯ hp (x, y , σ ¯) − R ¯ ). Then one can A similar result can be obtained for the diﬀerence R α+γ β+γ ¯ γ γ ¯ |y| Rhp (x, y, σ ¯ ) = ψhp (x, y, σ ¯ ) |x| |y| is the set of uniformly bounded readily see that |x| equicontinuous functions. Therefore, one can single out a sequence hk → 0, k → ∞, such that there exists a limit ¯ (x, y, σ ¯ h p (x, y, σ ¯ ) |x|α+γ |y|β+γ = lim |x|γ |y|γ ψhk p (x, y, σ ¯) = R ¯ ) |x|α+γ |y|β+γ . lim R k

k→∞

k→∞

In addition, for an arbitrary η > 0, there exists a number k0 such that ¯ (x, y, σ ¯ h p (x, y, σ R ¯) − R ¯ ) < cη/ |x|α+γ |y|β+γ , k ¯) Ehp (x, y, σ p−1 ¯ p ¯h p (t, y, σ = Hhk (x, y, σ ¯ ) + σ Rhk p (x, y, σ ¯) + σ Hhk (x, t, σ)R ¯ ) dSt k Dh (¯ σ)

(9)

Shk

for k > k0 . By Theorem 2, lim Hhk (x, y, σ) = H(x, y, σ) = K(x, y) + σK2 (x, y) + · · · + σ p−2 Kp−1 (x, y),

k→∞

and if |y| > h and |x| > a > hν , then |Hhp (x, y, σ) − H(x, y, σ)| < chγ ,

γ > 0.

Let us show that

lim

k→∞ Shk

¯h (t, y, σ)dSt = Hhk (x, t, σ)R k

DIFFERENTIAL EQUATIONS

¯ y, σ)dSt dSt . H(x, t, σ)R(t, S

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GOGNIASHVILI

We suppose that |x| ≥ a and |y| ≥ a, a > 0, and estimate the diﬀerence ¯h p (t, y, σ)dSt − H(x, t, σ)R(t, ¯ y, σ)dSt Hhk (x, t)R k Shk S ¯ h p dSt H (Rhk p − R) dSt + (Hhk − H) R < k Sh Sh k k 4 ¯ h p dSt + (Hhk − H) R HR dS Ii . = + t k S\Sh i=1 Sh k

k

By (9), we have |I1 | < cη

|x −

Sh

⎛

dSt 2−λ t| |t|α+γ+ε ⎞

k

⎜ < cη ⎝

∩c(0,|x|/2) Sh

dSt α+γ+ε |t|

k

< cη,

|x| ≥ a,

+ Shk ∩c(x,|x|/2)

dSt + |x − t|2−λ

A a

dSt ⎟ ⎠ 2+α−λ |t|

|y| ≥ a.

The term I2 can be estimated in a similar way with the use of inequality (10). Next, we have dSt dSt dSt < + +c dSt |I3 | < |x − t|2−λ |t|α+ε |t|α+ε |x − t|2−λ Sh

∩c(0,|x|/2) Sh

= o(h) < cη,

|x| ≥ a,

Sh

∩c(x,|x|/2) Sh

k

k

k

k

|y| ≥ a.

In a similar way, one can obtain an estimate for the integral I4 . Then there exists a limit Ehk p (x, y, σ) ¯ (x, y, σ ¯ (t, y, σ = Hh (x, y, σ) + R ¯ ) + σ p H(x, t, σ)R ¯ ) dSt lim k→∞ Dh (¯ σ) S

= R(x, y, σ),

|x| ≥ a,

|ϕ| ≥ a,

and the function R(x, y, σ) satisﬁes the equations K(x, t)R(t, y, σ)dSt ,

R(x, y, σ) = K(x, y) + σ S

(11) R(x, t, σ)K(t, y)dSt .

R(x, y, σ) = K(x, y) + σ S

Since the number a > 0 is arbitrary, it follows that relations (11) are valid for arbitrary x, y ∈ S\{0}. Therefore, in case I, Eq. (1) has the resolvent R(x, y, σ), which satisﬁes Eq. (11). II. Let A(h) be an inﬁnitesimal function of higher order than a(h). Then the expression σ ) = φh (x, y, σ) is unbounded as h → 0. a(h)|x|α |y|β H1h (x, y, σ) = H1h (x, y, σ)|x|α |y|β /Dh (¯ We set σ ) = 1/β(h), sup φh (x, y, σ) = sup H1h (x, y, σ)|x|α |y|β /Dh (¯ x,y∈Sh

x,y∈Sh

DIFFERENTIAL EQUATIONS

lim β(h) = 0.

h→0

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Suppose that this least upper bound is attained at (xh , yh ). We extract a sequence {hk } such that the sequence {xhk , yhk } has a limit. Let limk→∞ (xhk , yhk ) = (x0 , y0 ). (a) First, suppose that x0 = 0 and y0 = 0. Then |φhk (x, y, σ)βhk | ≤ 1,

φhk (xhk , yhk , σ) βhk = 1.

Just as in case I, from this sequence, one can extract a convergent sequence. We introduce the following notation: limk→∞ φhk (x, y, σ)β (hk ) = B(x, y, σ), B (x0 , y0 , σ) = 1. By multiplying Eq. (6) by α (hk ) β (hk ), we obtain φhk (x, y, σ)β (hk ) |x|α |y|β φhk (x, y, σ)β (hk ) = Khk (x, y)β (hk ) + σ Khk (x, t) Hhk (x, y, σ)β (hk ) + dSt . |x|α |y|β

Hhk (x, y, σ)β (hk ) +

Shk

In this relation, we pass to the limit as hk → 0 : B (t, y, σ ¯) B (x, y, σ ¯) = σ ¯ K(x, t) dSt . α β α β |x| |y| |t| |y| S

α

¯ )| < c. By assumption, B (x0 , y0 , σ ¯ )/|x0 | = 0 and This, with y = y0 = 0, implies that |B (x, y0 , σ ¯ )/|x|α is a nonzero eigenfunction of Eq. (1). Therefore, in the case under consideration, B (x, y0 , σ Eq. (1) has an eigenfunction of the class Cα . ¯ ) is not bounded as (b) Now let either (i) x0 = 0 or (ii) y0 = 0; therefore, the function φh (x, y, σ h → 0 if either x → 0 or y → 0. Then the following assertions are valid. ¯ ) = φh (x, y, σ ¯ )/δ1h (|x|) or φh (x, y, σ ¯ ) = φh (x, y, σ ¯ )/δ2h (|y|), where δih , (i) Either φh (x, y, σ i i = 1, 2, are inﬁnitely small functions together with the argument, a |φh | < c. For example, ¯ ). In this case, the relation δ1h (|x|) can be given by the relation δ1h (|x|) = 1/ supx∈Sh φh (x, y, σ δ1h (|x|) > |x|ε should be valid, where ε > 0 is an arbitrarily small number. Indeed, suppose that δ1h (|x|) is an inﬁnitesimal function of higher order than |x|ε ; then we ﬁnd that hε

hε φh (x, y, σ H1h (x, y, σ ¯) α β ¯) |x| |y| = hε φh (x, y, σ ¯) = Dh (¯ σ) δ1h (|x|)

is not bounded for |x| = h, which contradicts Lemma 4. Consequently, the quantity ¯) |x|ε |y|ε φh (x, y, σ is bounded as h → 0 and either |x| → 0 or |y| → 0, φh (x, y, σ φh (x, y, σ)|x|ε |y|ε φ¯h (x, y, σ ¯) ¯) ¯) H1h (x, y, σ = = = , α β α+ε β+ε α+ε β+ε Dh (¯ σ) |x| |y| |x| |y| |x| |y| where φ¯n is a function bounded as h → 0 and either |x| → 0 or |y| → 0. (b ) In addition, let φ¯h < c uniformly with respect to h, x, and y ∈ Sh . Then, following case I, ¯ ) and we prove the existence of a sequence hk such that φ¯hk has the limit φ¯2 (x, y, σ φ2 (x, y, σ H1hk (x, y, σ ¯) ¯) φ¯hk (x, y, σ) = lim = = R2 (x, y, σ ¯) . α+ε β+ε α+ε β+ε hk →0 hk →0 |x| Dhk (¯ σ) |y| |x| |y| lim

By passing in (6) to the limit as hk → 0, we obtain the resolvent equations (11). The relation Kphk (x, t)Rhk (t, y, σ)dSt = Kp (x, t)R(t, y, σ)dSt lim hk →0 Shk

DIFFERENTIAL EQUATIONS

S

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can be proved just as in case I. The fact that the exponents in the denominator R are increased by ε does not lead to any essential diﬃculty. (b ) Let φ¯hk be an unbounded function as hk → 0 with (x0 , y0 ), x0 = 0, and y0 = 0 (φ¯hk is bounded as hk → 0 and either |x| → 0 or |y| → 0). Then, just as in case (a), we show that Eq. (1) has a nonzero eigenfunction of the class Cα+ε . ¯ ) |x|α |y|β be bounded as h → 0. Then Dh (¯ σ ) is a bounded III. Let the expression H1h (x, y, σ σ ) = D (¯ σ ) and function as well. We extract a sequence hk such that there exist limits lim Dhk (¯ ¯ ) |x|α |y|β = H1 (x, y, σ)|x|α |y|β . If D (¯ σ ) = 0, then H1 (x, y, σ)/D (¯ σ ) = R2 (x, y, σ), lim H1hk (x, y, σ the function R = R1 + R2 satisﬁes Eq. (11), and Eq. (1) has the unique solution. If σ0 is a root of the equation D (¯ σ ) = 0, then Eq. (1) has an eigenfunction of the class Cα . By combining all considered cases, we obtain an alternative-like theorem. Theorem 3. Either Eq. (1) has an eigenfunction of the class Cα+ε , where ε > 0 is an arbitrarily small number, or there exists a resolvent of the form R = R1 + R2 ,

∀ν > 0, δ > 0, ν + δ ≤ λ, R1 = φ1 (x, y, σ)/ |x|ν |y|δ |x − y|2−ν−δ α+ε β+ε ¯ )/ |x| |y| , α > 0, β > 0, α + β = 2, |ψ1 | < c, R2 = φ2 (x, y, σ

|ψ2 | < c,

which satisﬁes Eq. (11); in the latter case, the solution of Eq. (1) is unique and is given by the formula ϕ(x) = f (x) + σ S R(x, y, σ)f (y)dSy . If K is a kernel of a double layer potential, then R2 =

|x|α+ε ψ(x, y, σ) ψ2 (x, y, σ) = , α+ε α+ε β+ε |y| |x| |y| |y|2+ε

|ψ2 | < c,

R1 =

ψ(x, y) , − y|2−δ

δ ≤ λ.

|y|δ |x

Theorem 4. If γ < λ/2, α + β = 2, 0 < ν < λ, then γ

|R (x , y, σ) − R (x , y, σ)| ≤

γ

c |x − x | c |x − x | + α+γ+ε . 2+γ−λ y (x , x ) |y|λ−ν |x|ν |x| |y|β+ε

Let K be the kernel of a double layer potential. Theorem 5. Either Eq. (1) has an eigenfunction of the class C (0,γ) , γ < λ/2, or there exists a resolvent R(x, y) satisfying Eq. (11) and the inequality c |x − x | c |x − x | + , |R (x , y, σ) − R (x , y, σ)| < 2+γ−λ |y|2+γ+ε |y|λ y (x , x ) γ

γ

γ<

λ , 2

ε > 0,

which implies that if f ∈ C (0,γ) , then ϕ ∈ C (0,ν−ε) , where ν = min(γ, λ/2). By using the above-obtained results, one can study the Dirichlet and Neumann boundary value (0,δ) problems for the Laplace equation in the classes C (0,γ) and Cα,δ and the main boundary value problems of three-dimensional theory of elasticity on a surface with a conical point of the aboveconsidered form. One can prove existence and uniqueness theorems. REFERENCES 1. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., and Burchuladze, T.V., Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Amsterdam, 1979. 2. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., and Burchuladze, T.V., Trekhmernye zadachi matematicheskoi teorii uprugosti (Three-Dimensional Problems in the Mathematical Theory of Elasticity), Tbilisi: Izdat. Tbilis. Univ., 1968. DIFFERENTIAL EQUATIONS

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3. Kupradze, V.D., Potential Methods in the Theory of Elasticity, Yerusalem, 1965. 4. Kupradze, V.D., Metody potentsiala v teorii uprugosti (Potential-Theoretic Methods in the Theory of Elasticity), Moscow: Fizmatlit, 1963. 5. Eskin, G.I., Uspekhi Mat. Nauk , 1963, vol. 18, no. 3, pp. 241–242. 6. Kondrat’ev, V.A., Tr. Mosk. Mat. Obs., 1967, vol. 16, pp. 210–292. 7. Maz’ya, V.G. and Plamenevskii, B.A., Asimptotika reshenii ellipticheskikh kraevykh zadach pri singulyarnykh vozmushcheniyakh granitsy (Asymptotic Behavior of Solutions of Elliptic Boundary Value Problems Under Singular Perturbations of the Boundary), Tbilisi: Izdat. Tbilis. Univ., 1981. 8. Burago, Yu.D. and Maz’ya, V.G., Nekotorye voprosy teorii potentsiala i teorii funktsii dlya oblastei s neregulyarnymi granitsami (Some Problems of Potential Theory and Function Theory for Irregular Regions), Leningrad: Stekl. Math. Inst., 1967. 9. Plamenevskii, B.A., Algebry psevdodiﬀerentsial’nykh operatorov (Algebras of Pseudodiﬀerential Operators), Moscow: Nauka, 1968. 10. Smirnov, V.I., Kurs vysshei matematiki (A Course of Higher Mathematics), Moscow: Fizmatgiz, 1953, vols. 4, 5.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 854–861. c T.A. Dzhangveladze, Z.V. Kiguradze, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 833–840. Original Russian Text

INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS

On the Stabilization of Solutions of an Initial-Boundary Value Problem for a Nonlinear Integro-Diﬀerential Equation T. A. Dzhangveladze and Z. V. Kiguradze Tbilisi State University, Tbilisi, Georgia Institute of Applied Mathematics, Tbilisi, Georgia Received March 31, 2005

DOI: 10.1134/S0012266107060110

1. INTRODUCTION Mathematical modeling, analysis, and numerical solution of diﬀusion problems for magnetic ﬁelds in media are a topical problem in applied mathematics. Nonlinear diﬀerential and integrodiﬀerential equations and systems of such equations arise in the modeling of such processes as well as other numerous problems. Suppose that a magnetic ﬁeld propagates in a medium whose electric conductivity substantially depends on the temperature. In the quasistationary approximation, the corresponding Maxwell system has the form [1, p. 238] ∂H = − rot (νm rot H) , ∂t ∂θ = νm (rot H)2 , cν ∂t

(1.1) (1.2)

where H = (H1 , H2 , H3 ) is the magnetic ﬁeld intensity vector, θ is temperature, and cν and νm characterize the heat capacity and the electric conductivity of the medium. Equations (1.1) describe the propagation of the magnetic ﬁeld in the medium, and Eq. (1.2) describes the change in temperature due Joule heating. If cν = cν (θ) and νm = νm (θ), then, after the integration of Eq. (1.2) with respect to time and the substitution into (1.1), we obtain the system [2] ⎡ ⎛ t ⎞ ⎤ ∂H = − rot ⎣a ⎝ | rot H|2 dτ ⎠ rot H ⎦ , (1.3) ∂t 0

where a = a(S) is a function deﬁned for S ∈ [0, ∞). The above-represented integro-diﬀerential model is complicated and can be studied eﬀectively only in special cases. Note that, for metal conductors, the nonlinearity has the form a(S) = exp(S). Consider a plane magnetic ﬁeld H = (0, 0, U ), where U = U (x, t) is a scalar function of time and one space variable. Then rot H = (0, −∂U/∂x, 0), and system (1.3) acquires the form ∂ ∂U ∂U = a(S) , (1.4) ∂t ∂x ∂x where

t S(x, t) = 0

854

∂U ∂x

2 dτ.

(1.5)

ON THE STABILIZATION OF SOLUTIONS OF AN INITIAL-BOUNDARY . . .

855

The study of equations of the form (1.3) and (1.4) was initiated in [2], where, in particular, existence theorems for a generalized solution of the ﬁrst boundary value problem for Eq. (1.4) for a(S) = 1 + S and the uniqueness in more general cases were proved. Equation (1.4) with and uniqueness theorem a(S) = (1 + S)p , 0 < p ≤ 1, was studied in [3], where the existence ˚ 1 (0, 1) ∩ W 2 (0, 1) was proved for the ﬁrst boundary value problem in the space L2p+2 0, T ; W 2p+2 2 with the use of the compactness method [4; 5, pp. 118–132 of the Russian translation]; moreover, ∂U/∂t ∈ L2 (QT ), where QT = (0, 1) × (0, T ) and T is a given positive number. The analysis of spatially multidimensional versions began in [6]. Equations of the type (1.3) and (1.4) were investigated in [7, 8] by the scheme of conditionally weakly closed operators. Note that equations of the type (1.4) were investigated in [9] as well. The paper [10] and other papers also deal with the analysis of the existence and uniqueness of solutions to problems considered in the present paper and similar problems. Our aim is to investigate the asymptotic behavior of the solution of the ﬁrst boundary value problem for Eq. (1.4) as t → ∞. Some issues concerning solution stabilization as t → ∞ were considered in [10–14] for initial-boundary value problems for equations of type (1.4). 2. STATEMENT OF THE PROBLEM AND ASYMPTOTIC BEHAVIOR OF THE SOLUTION IN THE NORM OF THE SPACE W21 Consider the nonlinear integro-diﬀerential equation (1.4), where a = a(S) is a given function of the argument S given by (1.5). In the cylinder (0, 1)×(0, ∞), we pose the following initial-boundary value problem for Eq. (1.4): U (0, t) = U (1, t) = 0, t ≥ 0, x ∈ [0, 1]. U (x, 0) = U0 (x),

(2.1) (2.2)

We proceed to the analysis of the asymptotic behavior of the solution of problem (1.4), (2.1), (2.2) as t → ∞. To prove the main theorem of this section, we obtain necessary a priori estimates. The following assertion can readily be proved. Lemma 2.1. If a(S) ≥ a0 = const > 0 and U0 ∈ L2 (0, 1), then the solution of problem (1.4), (2.1), (2.2) satisﬁes the estimate U ≤ C exp (−a0 t) . Here and throughout the following, C stands for positive constants independent of t and U , and · is the norm of the space L2 (0, 1). Therefore, the stabilization of the solution of problem (1.4), (2.1), (2.2) takes place in the norm of the space L2 (0, 1) as t → ∞; moreover, the convergence is exponential. Note that the stabilization of the solution takes place in the norm of the space W21 (0, 1) as well. Let us proceed to the proof of this assertion. Let the functions a = a(S) and U0 = U0 (x) satisfy the additional conditions a (S) ≥ 0, a (S) ≤ 0, U0 ∈ W22 (0, 1), and U0 (0) = U0 (1) = 0. We multiply Eq. (1.4) by U and integrate the resulting relation over the domain (0, 1) × (0, t). By using the formula of integrating by parts and by taking into account the boundary conditions (2.1) and the constraint a(S) ≥ a0 , we obtain t ∂U 2 2 U + 2a0 ∂x dτ ≤ U0 . 2

0

Hence we obtain the a priori estimates U ≤ C,

t ∂U 2 ∂x dτ ≤ C. 0

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Let us estimate the following integral: 2 2 t t 1 t d ∂U ∂U d dx dτ ≤ 2 dτ = dτ ∂x ∂x dτ 0

0

0

0

t1 t ∂U ∂ 2 U ≤2 ∂x ∂x ∂τ dx dτ ≤ 0 0

1 2 ∂U ∂ U ∂x ∂x ∂τ dx dτ 0

t ∂U 2 dτ + ∂x

0

2 2 ∂ U ∂x ∂τ dτ.

0

By taking into account (2.3), we obtain t d dτ

t 2 2 ∂U 2 ∂ U ∂x dτ ≤ C + ∂x ∂τ dτ.

0

(2.4)

0

Let us now estimate the integral occurring on the right-hand side in inequality (2.4). To this end, we diﬀerentiate Eq. (1.4) with respect to t and take the inner product of the resulting relation by ∂U/∂t. By using the formula of integrating by parts, by taking into account the boundary condition (2.1), and by performing simple manipulations, we obtain 2

4

2 2 1 1 ∂U ∂U 1 ∂ ∂ U 1 d + a (S) dx + a(S) dx = 0. 2 dt ∂t 4 ∂t ∂x ∂x ∂t 0

0

Let us integrate the resulting identity over the interval [0, t] : 2 ∂U 2 ∂U 1 1 − 2 ∂t 2 ∂t

+

t=0

1 − 4

t1 a(S)

∂2U ∂x ∂τ

0 0

1

a (S)

∂U ∂x

0

4

2

dx

1 dx dτ + 4

1

a (S)

∂U ∂x

4 dx

0

1 − 4

t=0

t1

a (S)

∂U ∂x

6 dx dτ = 0.

0 0

By taking into account the initial condition (2.2) and the constraints imposed on the coeﬃcient a = a(S), we obtain t 2 2 ∂ U (2.5) ∂x ∂τ dτ ≤ C. 0

Finally, from (2.3)–(2.5), we obtain the estimates t t ∂U 2 d ∂x dτ ≤ C, dτ 0

∂U 2 ∂x dτ ≤ C,

0

which are valid for arbitrary t. Therefore, ∞ ∂U 2 ∂x dτ ≤ C,

∞ d dτ

0

0

∂U 2 ∂x dτ ≤ C.

The last a priori estimates imply the relation [15; 16, pp. 101–103] ∂U (·, t) ∂x → 0 as t → ∞. DIFFERENTIAL EQUATIONS

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Consequently, we have shown that the solution is stabilized in the norm of the space W21 (0, 1) as t → ∞. We have thereby justiﬁed the following assertion. Theorem 2.1. If a(S) ≥ a0 = const > 0, a (S) ≥ 0, a (S) ≤ 0, U0 ∈ W22 (0, 1), U0 (0) = U0 (1) = 0, then the solution of problem (1.4), (2.1), (2.2) satisﬁes the relation U (·, t)W21 (0,1) → 0

t → ∞.

as

(2.6)

Relation (2.6) justiﬁes the stabilization of a nonstationary solution of problem (1.4), (2.1), (2.2) but does not provide any information on the convergence rate. 3. ESTIMATE FOR THE ORDER OF THE ASYMPTOTIC BEHAVIOR OF THE SOLUTION IN THE NORM OF THE SPACE W21 Let us proceed to estimating the order of the asymptotic behavior of the solution of problem (1.4), (2.1), (2.2). In particular, let us show that the convergence occurring in relation (2.6) is also exponential. Note that, in this section, we derive an asymptotics with indication of order for ∂U/∂t as well. Theorem 3.1. If the assumptions of Theorem 2.1 are valid, then the solution of problem (1.4), (2.1), (2.2) can be estimated as ∂U ∂U + ≤ C exp − a0 t . ∂x ∂t 2 Proof. We multiply Eq. (1.4) by ∂U/∂t and integrate the resulting relation over the interval [0, 1]. By using the formula of integrating by parts and by taking the boundary conditions (2.1), we obtain

2 1 ∂U 2 1 ∂ ∂U + a(S) dx = 0. (3.1) ∂t 2 ∂t ∂x 0

Let us diﬀerentiate relation (1.4) with respect to t : ∂2U ∂ ∂a(S) ∂U ∂2U + a(S) = 0; − ∂t2 ∂x ∂t ∂x ∂t ∂x

(3.2)

we multiply (3.2) by U and integrate the resulting relation over the interval [0, 1] : 1

∂2U U dx + ∂t2

0

1

∂a(S) ∂t

∂U ∂x

2

1 dx + 2

0

1

∂ a(S) ∂t

∂U ∂x

2 dx = 0.

0

It follows from (3.1) and (3.3) that 1

2

2 1 1 ∂U 2 ∂ ∂U ∂2U ∂a(S) ∂U U dx + + a(S) dx + dx = 0, ∂t2 ∂t ∂t ∂x ∂t ∂x

0

0

0

which, in view of the relation 1

∂U 2 1 d2 ∂2U 2 U dx + ∂t = 2 dt2 U , ∂t2

0

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acquires the form d 1 d2 U 2 + 2 2 dt dt

1 a(S)

∂U ∂x

2 dx = 0.

(3.4)

0

Let us multiply Eq. (3.2) by ∂U/∂t and integrate the resulting relation over the interval [0, 1]. By using the formula of integrating by parts, the boundary conditions (2.1), and the Poincar´e– Friedrichs inequality, we obtain 2 2

4 1 d ∂U + 2a0 ∂U + 1 a (S) ∂ ∂U dx ≤ 0. ∂t dt ∂t 2 ∂t ∂x

(3.5)

0

By multiplying Eq. (1.4) by U and by integrating the resulting relation over the interval [0, 1], we obtain 2

1 ∂U 1 d 2 U + a(S) dx = 0, (3.6) 2 dt ∂x 0

which, together with the Poincar´e–Friedrichs inequality and the condition a(S) ≥ a0 , implies that 1 d U 2 + a0 U 2 ≤ 0. 2 dt

(3.7)

We multiply relations (3.4), (3.6), and (3.7) by γ, α, and β, respectively, where α, β, and γ are positive constants. By summing the resulting relations and inequality (3.5), we obtain α+β d U 2 + α 2 dt

1

2

∂U γ d2 a(S) dx + βa0 U 2 + U 2 ∂x 2 dt2

0

d +γ dt

1

2 2

4

1 ∂U 2 1 ∂U d ∂ ∂U ∂U a(S) dx + + 2a0 + a (S) dx ≤ 0, ∂x dt ∂t ∂t 2 ∂t ∂x

0

0

which, after simple manipulations, acquires the form ⎤ ⎡ 2 2

1 1 2βa0 ∂U α ∂U d α+β d U 2 + U 2 + γ ⎣ a(S) dx + a(S) dx⎦ 2 dt α+β dt ∂x γ ∂x 0

0

2

4 1 ∂U 2 γ d2 ∂U 1 ∂ ∂U d 2 + 2a0 + U + a (S) dx ≤ 0. + dt ∂t ∂t 2 dt2 2 ∂t ∂x

(3.8)

0

Suppose that 2βa0 /(α + β) = α/γ = a0 ; then α = β = (α + β)/2 = a0 γ, and inequality (3.8) can be represented in the form ⎡ ⎤ 2

1 ∂U 2 d ∂U d d ⎣ 2 exp (a0 t) U + γ exp (a0 t) a(S) exp (a0 t) dx⎦ + a0 γ ∂t dt dt ∂x dt 0 ⎡ ⎤

4 1 ∂U 2 γ d2 ∂U 1 ∂ 2 dx⎦ ≤ 0. + exp (a0 t) ⎣a0 ∂t + 2 dt2 U + 2 a (S) ∂t ∂x 0

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By integrating the resulting inequality over the closed interval [0, t] and by taking into account the initial condition (2.2), we obtain the inequality

1 a0 γ exp (a0 t) U + γ exp (a0 t) 2

a(S)

∂U ∂x

2

∂U 2 dx + exp (a0 t) ∂t

0

t ∂U 2 γ d2 exp (a0 τ ) dτ + exp (a τ ) U 2 dτ 0 ∂τ 2 dτ 2

t + a0 0

0

t

1 + 2

1 exp (a0 τ )

0

∂ a (S) ∂τ

∂U ∂x

4 dx dτ ≤ C.

(3.9)

0

Note that t

1 exp (a0 τ )

0

∂ a (S) ∂τ

∂U ∂x

4 dx dτ

0

1 = exp (a0 t)

a (S)

∂U ∂x

4

1 dx −

0

1 exp (a0 τ ) 0

t

1 exp (a0 τ )

0

a (S)

0

≥ −a0

a (S)

∂U ∂x

4

0

t − a0

4

∂U ∂x

t dx dτ −

∂U ∂x

t=0

1

exp (a0 τ ) 0

a (S)

dx

a (S)

∂U ∂x

6 dx dτ

0

4 dx dτ − C,

(3.10)

0

and we have the identity t exp (a0 τ )

d2 d U 2 dτ = exp (a0 t) U 2 2 dτ dt

0

t − a0 0

d d 2 2 exp (a0 τ ) U dτ − U dτ dt

t=0

1 = 2 exp (a0 t)

U

∂U dx − a0 exp (a0 t) U 2 ∂t

0

t +

a20

d 2 exp (a0 τ ) U dτ + a0 U0 − U . dt t=0 2

0

2

By using the ε-inequality in the last relation, we obtain t

∂U 2 d2 1 2 2 exp (a0 τ ) 2 U dτ ≥ − (ε + a0 ) exp (a0 t) U − exp (a0 t) ∂t − C. dτ ε

0

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We multiply relation (3.3) by μ exp (a0 t), where μ is a positive constant, and integrate the resulting relation over the interval [0, t] : t

1 exp (a0 τ )

μ 0

∂2U U dx dτ + μ ∂τ 2

0

μ + 2

t

1

exp (a0 τ )

a (S)

0

0

t

1 exp (a0 τ )

0

∂ a(S) ∂t

∂U ∂x

4 dx dτ 2

∂U ∂x

dx dτ = 0.

0

Hence, by performing integration by parts, we obtain 1 μ exp (a0 t)

∂U U dx − a0 μ ∂t

0

1 exp (a0 τ )

0

t 0

∂U U dx dτ − μ ∂τ

t

0

1 exp (a0 τ )

+μ

t

a (S)

∂U ∂x

4

∂U 2 dτ exp (a0 τ ) ∂τ

0

μ dx dτ + 2

0

t

1 exp (a0 τ )

0

∂ a(S) ∂τ

∂U ∂x

2 dx dτ = C.

0

By using the ε-inequality with parameter ε and by taking into account (3.1) and (3.6), we obtain 2 2

t 1 μ exp (a0 t) ∂U ∂U dx dτ −ε μ exp (a0 t) U − ∂t + a0 μ exp (a0 τ ) a(S) ∂x 4ε

2

0

t −μ 0

t −μ

0

4

t 1 ∂U 2 ∂U exp (a0 τ ) dτ + μ exp (a0 τ ) a (S) dx dτ ∂τ ∂x 0

0

∂U 2 dτ ≤ C. exp (a0 τ ) ∂τ

(3.12)

0

From Lemma 2.1 with η > 0, we obtain the inequality η exp (a0 t) U 2 ≤ C.

(3.13)

By combining inequalities (3.9)–(3.13), we obtain ∂U 2 γ (ε + a0 ) 2 − με + η U + γa0 exp (a0 t) a0 γ − 2 ∂x 2 t ∂U 2 μ γ ∂U dτ − + 1− + (a0 − 2μ) exp (a0 τ ) 2ε 4ε ∂t ∂τ 0

a0 + μ− 2

t

1 exp (a0 τ )

0

a (S) 0

∂U ∂x

4

t dx dτ +

a20 μ

∂U 2 dτ ≤ C. exp (a0 τ ) ∂x

0

To complete the proof of Theorem 3.1, it remains to show that the coeﬃcients occurring on the left-hand side in the last inequality can be chosen in an appropriate way. For example, this can be achieved by an appropriate choice of the free parameters as follows: μ = a0 /2, ε = ε = a0 , γ < 7a0 /4, and η ≥ a20 /2. The proof of Theorem 3.1 is complete. DIFFERENTIAL EQUATIONS

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REFERENCES 1. Landau, L.D. and Lifshits, E.M., Elektrodinamika sploshnykh sred (Electrodynamics of Continuous Media), Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1957. 2. Gordeziani, D.G., Dzhangveladze, T.A., and Korshiya, T.K., Diﬀer. Uravn., 1983, vol. 19, no. 7, pp. 1197–1207. 3. Dzhangveladze, T.A., Dokl. Akad. Nauk , 1983, vol. 269, no. 4, pp. 839–842. 4. Vishik, M.I., Mat. Sb., 1962, vol. 59 (101), pp. 289–325. 5. Lions, J.-L., Quelques m´ethodes de r´esolution des probl`emes aux limites nonlin´eaires, Paris: Dunod, 1969. Translated under the title Nekotorye metody resheniya nelineinykh kraevykh zadach, Moscow: Mir, 1972. 6. Dzhangveladze, T.A., Diﬀer. Uravn., 1985, vol. 21, no. 1, pp. 41–46. 7. Laptev, G.I., Mat. Sb., 1988, vol. 136, no. 4, pp. 530–540. 8. Laptev, G.I., Quasilinear Evolutionary Partial Diﬀerential Equations with Operator Coeﬃcients, Doctoral (Fiz.–Mat.) Dissertation, Moscow, 1990. 9. Long, N.T. and Dinh, A.P.N., Math. Mech. Appl. Sci., 1993, vol. 16, pp. 281–295. 10. Jangveladze (Dzhangveladze), T.A., Rep. Sem. I. Vekua Inst. Appl. Math., 1997, vol. 23, pp. 51–87. 11. Jangveladze (Dzhangveladze), T.A. and Kiguradze, Z.V., Rep. Enl. Sess. Sem. I. Vekua Inst. Appl. Math., 1995, vol. 10, no. 1, pp. 36–38. 12. Jangveladze (Dzhangveladze), T.A. and Kiguradze, Z.V., Georgian Math. J., 2002, vol. 9, no. 1, pp. 57–70. 13. Jangveladze (Dzhangveladze), T.A. and Kiguradze, Z.V., Appl. Math. Inform. Mech., 2003, vol. 8, no. 2, pp. 1–19. 14. Kiguradze, Z.V., Rep. Enl. Sess. Sem. I. Vekua Inst. Appl. Math., 2004, vol. 19, no. 1, pp. 52–55. 15. Kazhikhov, A.V., Diﬀer. Uravn., 1979, vol. 15, no. 4, pp. 662–667. 16. Antontsev, S.N., Kazhikhov, A.V., and Monakhov, V.N., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei (Boundary Value Problems of the Mechanics of Inhomogeneous Fluids), Novosibirsk: Nauka, 1983.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 862–872. c I.K. Lifanov, A.S. Nenashev, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 841–851. Original Russian Text

INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS

Generalized Functions on Hilbert Spaces, Singular Integral Equations, and Problems of Aerodynamics and Electrodynamics I. K. Lifanov and A. S. Nenashev Air-Force Engineering Academy, Moscow, Russia Received November 27, 2006

DOI: 10.1134/S0012266107060122

INTRODUCTION External ﬂow suction devices are used in aerodynamics to increase the lift of an airfoil in an ideal incompressible ﬂow [1]. When solving this problem, the airfoil is modeled by a vortical layer [2], and the external ﬂow suction device is modeled by a sink [3]. The desired velocity ﬁeld around the proﬁle should have the following property. In a neighborhood of the sink, the velocity ﬁeld should have a sink-type singularity on the external side of the airfoil (where the sink lies), and it should be smooth on the opposite side. Naturally, if the airfoil satisﬁes the no-ﬂow condition on the sink side, the sink point itself is excluded from this condition. The airfoil ﬂow problem is thereby reduced to the solution of a singular integral equation on the proﬁle contour with right-hand side undeﬁned at the sink point. It follows from the analysis of tangential components of the velocity ﬁeld in a neighborhood of the sink on the airfoil side where the velocity ﬁeld is smooth that the solution of the resulting singular integral equation should be sought in the class of functions that have a singularity of the type of 1/x at the sink point [1]. In the numerical solution of this singular integral equation by the discrete vortex method, grid points are chosen so as to ensure that the sink point is one of them, and when replacing the singular integral equation by a system of linear algebraic equations, the equation corresponding to the sink point is omitted [1]. The omitted equation is replaced by another equation, which is derived from some physical considerations. This proves to be inconvenient, since, when using several external ﬂow suction devices, one always has to decide how to ﬁll in the missing equations. Hence it was suggested in [4] to satisfy the no-ﬂow condition not on the side of the airfoil, where the sink lies, but on the opposite side, where the velocity ﬁeld is smooth. This approach results in the appearance of a delta function supported at the sink point on the right-hand side of the corresponding singular integral equation. Now the singular integral equation should be treated as a pseudodiﬀerential equation in the class of distributions. A version of such interpretation for a singular integral equation with a Hilbert kernel in the class of periodic distributions was presented in [5]. In this case, periodic distributions were treated as a subset of distributions on the entire real line. But this is inconvenient for singular integral equations on an interval in the class of distributions. However, it became simpler to use the discrete vortex method for the numerical solution of singular integral equations for the case in which the righthand side contains a delta function. The method acquired the same classical form [1] as for singular integral equations in the class of absolutely integrable functions, the delta function being replaced by the corresponding step function [6]. Note also that the solution of the problem on the computation of the input resistance of a thin wire antenna [7] powered by a current source leads to the analysis of a hypersingular integral equation on an interval, where the right-hand side contains a function with a singularity of the type of 1/x inside the solution domain. The solution of the corresponding characteristic equation was performed with the use of a spectral relation for a hypersingular integral operator on an interval, which provided a solution with a jump discontinuity at the point of the singularity of the right-hand side. In general, this solution should be considered in the class of distributions. This extension of the classical interpretation of singular and hypersingular integral equations and the corresponding 862

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operators resulted in studies in the theory of distributions. These studies are presented in this paper. 1. A VERSION OF GENERALIZED FUNCTIONS ON HILBERT SPACES Consider an inﬁnite-dimensional real Hilbert space H. (For a complex space, the forthcoming considerations remain valid with the corresponding modiﬁcations.) We denote the inner product of vectors f, g ∈ H by (f, g). Let a vector system {ψ1 , ψ2 , . . . , ψn , . . .} be an orthonormal basis in the space H, i.e., satisfy the conditions (ψi , ψj ) = δi,j ,

(1.1)

where δi,j is the Kronecker delta. Then each vector f in H can be represented in the form f=

∞

ak ψk ,

(1.2)

k=1

∞ where the coeﬃcients ak satisfy the relation k=1 a2k < ∞. Moreover, the norm of the vector f is given by the formula ∞ 1/2 a2k . (1.3) f = (f, f ) = k=1

Now, in the space H, we deﬁne a test set S of vectors. Deﬁnition 1. A vector f in the space H belongs to the test set S if its expansion in the basis has ﬁnitely many nonzero coeﬃcients, i.e., if for the expansion (1.2), there exists a positive integer K such that k > K. (1.4) ak = 0, Note that the set S = S(H) is linear and everywhere dense in the space H but is not closed in H. Indeed, each vector f in H with inﬁnitely many nonzero coeﬃcients in the representation (1.2) is the limit of the sequence of vectors n ak ψk (1.5) fn = k=1

in the metric of the space H. Therefore, in the set S, we introduce a diﬀerent notion of convergence of vectors. Deﬁnition 2. We say that a sequence of vectors f (1) , f (2) , . . . , f (n) , . . . in the set S converges to a vector f in the ﬁnite-dimensional sense if there exists a common index K such that (n)

ak = 0, and the condition

k > K,

n = 1, 2, . . . ,

(1.6)

(n) lim ak − ak = 0,

k = 1, 2, . . . ,

(1.7)

n→∞

is satisﬁed for each n = 1, 2, . . . We brieﬂy write this in the form f (n) → f (S).

(1.8)

In other words, a sequence of vectors f (1) , f (2) , . . . , f (n) , . . . in S converges to a vector f in H if all of them lie in a same ﬁnite-dimensional subspace of H and relation (1.7) is satisﬁed. Therefore, the vector f lies in the same ﬁnite-dimensional subspace and hence belongs to the set S. It follows that the ﬁnite-dimensional convergence in S provides that S is closed. Now the set S with the above-introduced ﬁnite-dimensional convergence is referred to as the space S of test vectors. One can also say that the ﬁnite-dimensional coordinate convergence is introduced in S. DIFFERENTIAL EQUATIONS

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We also introduce the notion of a distribution. Deﬁnition 3. A distribution is an arbitrary continuous linear functional F on the space S of test vectors. The value of the functional F on a test vector f is denoted by F (f ) or (F, f ). The continuity of a functional is understood as follows. A functional F is said to be continuous on S if the ﬁnite-dimensional convergence of a sequence f (1) , f (2) , . . . , f (n) , . . . of vectors in the space S to a vector f in the same space implies that (1.9) lim F, f (n) = (F, f ). f (n) →f

The set of all distributions is denoted by S = S (H). The set S is linear if the linear combination λF + μG of distributions F and G is deﬁned as the functional acting by the rule f ∈ S.

(λF + μG, f ) = λ(F, f ) + μ(G, f ),

(1.10)

Following [8], one can show that the functional λF + μG is linear and continuous on S, i.e., belongs to the set S . Note that the coordinate convergence in S implies the following assertion. Theorem 1. An arbitrary linear functional on S is continuous on S in the sense of Deﬁnition 3. Proof. Indeed, let a sequence f (1) , f (2) , . . . , f (n) , . . . of vectors in the space S converge in the (n) ﬁnite-dimensional sense to a vector f in S. It follows that, for the coeﬃcients {ak } in the expansion (n) of the vectors f (n) , there exists a K such that ak = 0, k > K, n = 1, 2, . . . Therefore, if F is a linear functional on S, then lim

f (n) →f

F, f

(n)

= lim

n→∞

=

F,

K

(F, ψk ) a(n) k

k=1 K

=

K

ak ψk

(F, ψk ) lim

n→∞

k=1

a(n) k

=

K

(F, ψk ) ak

k=1

= (F, f ).

k=1

This completes the proof of Theorem 1. ∞ Note that each series G = k=1 bk ψk (and hence each element of the space H) deﬁnes a linear functional on S by the following rule. Let f=

K

fk ψk ∈ S;

k=1

then, by deﬁnition, we set G(f ) = (G, f ) =

K

bk fk .

(1.11)

k=1

The functional G is linear, since G(λf + μg) =

K

bk (λfk + μgk ) = λ

k=1

K

bk fk + μ

k=1

K

bk gk = λG(f ) + μG(g).

k=1

It follows from Theorem 1 that G is a continuous functional on S and hence a distribution on H. Now we note that the value of a distribution F on an element f in S is given by the relation F (f ) = (F, f ) =

K

fk (F, ψk ) .

(1.12)

k=1

Let us now justify the following assertion. DIFFERENTIAL EQUATIONS

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Theorem 2. Let F be some distribution on H. Then the series ∞ (F, ψk ) ψk G=

865

(1.13)

k=1

is a distribution in S and coincides with F . Proof. Let f be an arbitrary element in S. Then, by deﬁnition, we have [see (1.11) and (1.12)] G(f ) =

K

(F, ψk ) fk = F (f ).

k=1

∞ Thus, we ﬁnd that the set of arbitrary series of the form k=1 bk ψk and the set of distributions on H coincide for this construction. Now we deﬁne convergence in S as the weak convergence of a sequence of functionals. Deﬁnition 4. A sequence of distributions F1 , F2 , . . . , Fn , . . . in S converges to a distribution F in S if (Fn , f ) → (F, f ) as n → ∞ for each vector f in S. In this case, we write Fn → F , n → ∞, in S . The linear set S with the introduced convergence is referred to as the space S of distributions. Now one can prove the following assertion. Theorem 3. Let F1 , F2 , . . . , Fn , . . . be a sequence in S such that, for each vector f in S, the numerical sequence (Fn , f ) is convergent as n → ∞. Then the functional F deﬁned on S by the relation f ∈ S, (1.14) (F, f ) = lim (Fn , f ) , n→∞

is also linear and continuous on S; i.e., F ∈ S . Proof. By virtue of Theorem 1, it suﬃces to show that F is a linear functional. Indeed, we have (F, αf + βg) = lim (Fn , αf + βg) = lim (α (Fn , f ) + β (Fn , g)) n→∞

n→∞

= α lim (Fn , f ) + β lim (Fn , g) = α(F, f ) + β(F, g), n→∞

n→∞

where α and β are arbitrary numbers and f and g are elements of the space S. It follows from Theorem 3 that the space S of distributions is complete with respect to the introduced convergence. 2. SINGULAR INTEGRAL EQUATIONS ON THE CLASS OF PERIODIC FUNCTIONS In applications [1, 9–11], problems are often reduced to the solution of linear singular integral equations of the ﬁrst kind in the class of 2π-periodic functions, for which the corresponding characteristic equations have the form 1 L (g(ϕ), ϕ0 ) = π

2π

ϕ0 − ϕ g(ϕ)dϕ = f (ϕ0 ) , ln sin 2

ϕ0 ∈ [0, 2π],

(2.1)

0

1 S (g(ϕ), ϕ0 ) = 2π

2π

ϕ0 − ϕ g(ϕ)dϕ = f (ϕ0 ) , 2

ϕ0 ∈ [0, 2π],

(2.2)

g(ϕ)dϕ = f (ϕ0 ) , sin ((ϕ0 − ϕ) /2)

ϕ0 ∈ [0, 2π],

(2.3)

cot 0

H (g(ϕ), ϕ0 ) =

1 4π

2π

2

0

where f (ϕ) is a 2π-periodic function. DIFFERENTIAL EQUATIONS

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By investigating the spectral properties of the operators L (g(ϕ), ϕ0 ), S (g(ϕ), ϕ0 ), and H (g(ϕ), ϕ0 ) with respect to the functions cos nϕ and sin nϕ, n = 0, 1, . . . [12], we obtain the relations

2π ϕ0 − ϕ inϕ 1 −(1/n) sgn(n)einϕ0 for n = ±1, ±2, . . . e dϕ = ln sin (2.4) −2 ln 2 for n = 0, π 2 0

1 2π

2π

ϕ0 − ϕ inϕ e dϕ = −i sgn(n)einϕ0 , 2

n = 0, ±1, ±2, . . . ,

(2.5)

einϕ dϕ = −n sgn(n)einϕ0 , sin2 ((ϕ0 − ϕ) /2)

n = 0, ±1, ±2, . . .

(2.6)

cot 0

1 4π

2π 0

From relations (2.4)–(2.6), we obtain the following. If we choose an arbitrary series of exponentials and formally apply one of the operators L, S, and H to it, then we obtain a similar series whose coeﬃcients diﬀer from the coeﬃcients of the original series by a factor of the order of nλ , nλ = max{1, |n|}. Thus, it is a natural idea to use the spaces H λ similar to Sobolev spaces of distributions [4, 5]. To this end, we consider the set of complex-valued 2π-periodic functions on [0, 2π] with square-integrable absolute value. In this setof functions, one can introduce the 2π inner product (f, g) = 0 f (ϕ)g(ϕ) dϕ and the norm f 2 = (f, f ). We obtain the Hilbert space L2 of complex-valued 2π-periodic functions on [0, 2π] with square integrable absolute value. In this space, an orthonormal basis is given by the function system ψn = (2π)−1/2 einϕ , n = 0, ±1, ±2, . . . Now let f (ϕ) be a 2π-periodic distribution on L2 ; then f (ϕ) =

f˜(n)ψn ,

n∈Z

1 f˜(n) = f (ϕ), ψ¯n = √ 2π

2π

f (ϕ)e−inϕ dϕ,

(2.7)

0

where f (ϕ), ψ¯n is treated as the value of a distribution (a linear functional) f (ϕ) on the function ψ¯n = (2π)−1/2 e−inϕ . By H λ we denote the set of distributions f (ϕ) such that 2 1/2 n2λ f˜(n) < ∞; (2.8) f λ = n∈Z

i.e., the function f ∗ (ϕ) = n∈Z nλ f˜(n)ψn (ϕ) belongs to the space L2 . Now, for functions in H λ , we introduce the inner product by the relation n2λ f˜(n)˜ g (n). (2.9) (f, g) = n∈Z

This inner product makes H λ a Hilbert space; moreover, H 0 = L2 is the space of functions with square-integrable absolute value on [0, 2π]. Now for each distribution g(ϕ) = n∈Z g˜(n)ψn , we assume that 1 L (g(ϕ), ϕ0 ) = π

2ϕ

ϕ0 − ϕ g(ϕ)dϕ ln sin 2

0

=−

1 sgn(n)˜ g(n)ψn (ϕ0 ) − 2 ln 2˜ g (0)ψ0 (ϕ0 ) , n n∈Z

Z0 = ±1, ±2, . . . ,

(2.10)

0

1 S (g(ϕ), ϕ0 ) = 2π

2ϕ cot 0

ϕ0 − ϕ g(ϕ)dϕ = − i sgn(n)˜ g (n)ψn (ϕ0 ) , 2 n∈Z DIFFERENTIAL EQUATIONS

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GENERALIZED FUNCTIONS ON HILBERT SPACES, SINGULAR INTEGRAL . . .

1 H (g(ϕ), ϕ0 ) = 4π

2ϕ 0

1 g(ϕ)dϕ = − n sgn(n)˜ g (n)ψn (ϕ0 ) . sin2 ((ϕ0 − ϕ)/2) n∈Z

867

(2.12)

From (2.10)–(2.12), one can make the following conclusions. Equation (2.1) has a unique solution for any distribution f (ϕ), Eqs. (2.2) and (2.3) have solutions neglecting a constant, and their solvability condition is given by the relation

2π f (ϕ)dϕ = 0,

(2.13)

0

where the value of the integral of f (ϕ) over [0, 2π] is understood as the quantity For these equations, we obtain the inversion formulas 1 g(ϕ) = 4π

2π 0

1 f (ϕ0 ) dϕ0 − 2 2 ln 2 sin ((ϕ − ϕ0 )/2)

for Eq. (2.1), 1 g(ϕ) = − 2π

2π cot

√

2π f˜(0).

2π f (ϕ)dϕ

(2.14)

0

ϕ − ϕ0 C f (ϕ0 ) dϕ0 + 2 2π

(2.15)

0

for Eq. (2.2), and 1 g(ϕ) = π

2π

ϕ − ϕ0 C f (ϕ0 ) dϕ0 + ln sin 2 2π

(2.16)

0

for Eq. (2.3); in addition, C in (2.15) and (2.16) satisﬁes the relation

2π g(ϕ)dϕ = C,

(2.17)

0

since condition (2.13) should be satisﬁed for the solvability of these equations. However, to construct numerical methods for Eqs. (2.1)–(2.3), it is necessary to estimate the closeness of solutions of these equations via the closeness of their right-hand sides. To this end, it is useful to note that the operator L is a mapping of the space H λ into the space H λ+1 , the operator S is a mapping of the space H λ into H λ , and the operator H is a mapping of the space H λ into H λ−1 ; moreover, the operator L is a one-to-one mapping, and the kernels of the operators S and H consist of a set of constants. The considerations carried out in this section permit one to solve the following problem in a simple way. The problem on a circulation-free ideal incompressible ﬂow past a circular cylinder with an external ﬂow suction device can be reduced to the solution of the equation [4] 1 2π

2π cot

ϕ0 − ϕ 1 g(ϕ)dϕ = δ (ϕ0 − q) − , 2 2π

ϕ0 , q ∈ [0, 2π],

(2.18)

0

where the function δ(ϕ − q) is deﬁned as the 2π-periodic delta function given by the relations 2π δ(ϕ − q) = 0 for ϕ = q and δ(ϕ − q) = +∞ for ϕ = q, ϕ, q ∈ [0, 2π], and 0 δ(ϕ − q)dϕ = 1; i.e., 2π δ(ϕ − q)ψn (ϕ)dϕ = ψn (q) for any function ψn (ϕ), n = 0, ±1, ±2, . . . Hence it follows that if a 0 distribution w(ϕ) is continuous in some neighborhood of the point q, then

2π δ(ϕ − q)w(ϕ)dϕ = w(q). 0

DIFFERENTIAL EQUATIONS

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LIFANOV, NENASHEV

Now from (2.15), we ﬁnd that the solution of Eq. (2.18) is given by the function g(ϕ) = −

ϕ−q C 1 cot + . 2π 2 2π

(2.19)

Therefore, we obtain [4] 1 − 2π

2π cot

ϕ−q ϕ0 − ϕ 1 1 cot dϕ = δ (ϕ0 − q) − , 2 2π 2 2π

ϕ0 , q ∈ [0, 2π].

(2.20)

0

The same result can be obtained [4] if the distributions δ(ϕ − q) and (2π)−1 cot((ϕ − q)/2) are represented by Fourier series in the functions ψn = (2π)−1/2 einϕ and are substituted into Eq. (2.18). It follows from the representations by Fourier series that both above-mentioned functions belong to any space H λ for λ < −1/2. Note also the following. Relations (2.4) and (2.5) imply that Eq. (2.2) can be integrated term by term; therefore, 1 − π

2π

1 ϕ−q ϕ0 ϕ − ϕ 0 cot dϕ = − + F (δ (ϕ0 − q)) + Cq , ln sin 2 2π 2 2π

ϕ0 , q ∈ [0, 2π],

(2.21)

0

where F (δ(ϕ − q)) = 0, 0 ≤ ϕ < q; F (δ(ϕ − q)) = 1/2, ϕ = q; F (δ(ϕ − q)) = 1, q < ϕ ≤ 2π and 0 < q < 2π; F (δ(ϕ)) = 0 and ϕ = 0 for q = 0; F (δ(ϕ)) = 1/2, 0 < ϕ < 2π; and F (δ(ϕ)) = 1, ϕ = 2π. Since the constant term in the Fourier series of the function occurring on the right-hand side in (2.21) should vanish, we have Cq = (2π)−1 (q − π). Note that the function F (δ(ϕ − q)) is an antiderivative of the function δ(ϕ − q) and vanishes for ϕ = 0. The solution of the problem on the computation of the input resistance of a thin wire antenna powered by a power source leads [7] to the investigation of a hypersingular integral equation on an interval, whose right-hand side is a function with a singularity of the type of 1/x inside the solution domain. An analog of this equation in the periodic case is given by the equation 1 4π

2ϕ

ϕ0 − q 1 g(ϕ)dϕ = cot , 2 sin ((ϕ0 − ϕ) /2) 2

0

ϕ0 , q ∈ [0, 2π].

(2.22)

By virtue of the inversion formula (2.16) for the considered equation and formula (2.21), the solution of Eq. (2.22) is given by the function g(ϕ) =

C ϕ − F (δ(ϕ − q)) + Cq + , 2π 2π

ϕ ∈ [0, 2π],

(2.23)

which satisﬁes relation (2.17). For the approximate method of the solution of Eq. (2.20) by the discrete vortex method, it is important that the function δ(ϕ − q) is the limit of a sequence of 2π-periodic functions δh (ϕ − q) = 1/h, ϕ ∈ [q − h/2, q + h/2]; δh (ϕ − q) = 0, ϕ ∈ [q − h/2, q + h/2], q ∈ [0, 2π], in the sense of Deﬁnition 4; i.e., limh→0 δh (ϕ − q) = δ(ϕ − q). 3. SINGULAR INTEGRAL EQUATIONS ON THE INTERVAL [−1, 1] Now we consider singular integral equations of the ﬁrst kind on the interval [−1, 1], to which numerous applied problems can be reduced [1, 9–11]. In this case, the characteristic equations can be represented in the form 1 L1 (g(x), x0 ) = π

1 √ −1

1 ln |x0 − x| g(x) dx = f (x0 ) , 1 − x2 DIFFERENTIAL EQUATIONS

x0 ∈ (−1, 1), Vol. 43

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GENERALIZED FUNCTIONS ON HILBERT SPACES, SINGULAR INTEGRAL . . .

1 S1 (g(x), x0 ) = π 1 S2 (g(x), x0 ) = π 1 S3 (g(x), x0 ) = π 1 S4 (g(x), x0 ) = π 1 H2 (g(x), x0 ) = π

1 √

1 g(x)dx = f (x0 ) , 2 1 − x x0 − x

x0 ∈ (−1, 1),

(3.2)

√

g(x)dx = f (x0 ) , x0 − x

x0 ∈ (−1, 1),

(3.3)

1 − x g(x)dx = f (x0 ) , 1 + x x0 − x

x0 ∈ (−1, 1),

(3.4)

1 + x g(x)dx = f (x0 ) , 1 − x x0 − x

x0 ∈ (−1, 1),

(3.5)

−1

1

869

1 − x2

−1

1

−1

1

−1

1 √

1 − x2

−1

g(x)dx 2

(x0 − x)

x0 ∈ (−1, 1),

= f (x0 ) ,

(3.6)

√ (1 − x)/(1 + x), and 4 (x) = −1 where 1 (x) = 1/ 1 − x2 , 2 (x) = −1 1 (x), 3 (x) = 3 (x). One can show that the operators L1 (g(x), x0 ), Sk (g(x), x0 ), k = 1, 2, 3, 4, and H2 (g(x), x0 ) satisfy the spectral relations [12, 13] 1 π

1 √ −1

1 1 ln |x0 − x| Tn (x)dx = − Tn (x0 ) , 2 n 1−x

1 π

1 √ −1

1 π 1 π

1 √ −1

1 −1

1 π 1 π 1 π where

1 ln |x0 − x| dx = − ln 2, 1 − x2

√

1

1

−1

1 −1

√

x0 ∈ (−1, 1),

1 Tn (x)dx = −Un−1 (x0 ) , 1 − x2 x0 − x

1 − x2

−1

x0 ∈ (−1, 1),

Un−1 (x)dx = Tn (x0 ) , x0 − x

1 − x Pn (x)dx = −Qn (x0 ) , 1 + x x0 − x

1 + x Qn (x)dx = Pn (x0 ) , 1 − x x0 − x

1 − x2

Un (x)dx 2

(x0 − x)

n = 1, 2, . . . ,

n = 0,

x0 ∈ (−1, 1),

x0 ∈ (−1, 1),

x0 ∈ (−1, 1),

x0 ∈ (−1, 1),

= −(n + 1)Un (x0 ) ,

(3.8)

n = 0, 1, 2, . . . ,

n = 1, 2, . . . ,

n = 0, 1, 2, . . . ,

n = 0, 1, 2, . . . ,

x0 ∈ (−1, 1),

(3.7)

(3.9)

(3.10)

(3.11)

(3.12)

n = 0, 1, 2, . . . , (3.13)

Un (x) = sin((n + 1) arccos x)/ sin(arccos x), Tn (x) = cos(n arccos x), Qn (x) = Un (x) − Un−1 (x). Pn (x) = [Tn+1 (x) − Tn (x)]/(1 − x),

For forthcoming considerations, we ﬁrst recall the notion of the space L2, of functions on the interval [−1, 1] of the axis OX whose absolute values are square integrable on this interval with DIFFERENTIAL EQUATIONS

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1 weight = (x), that is, functions f (x), x ∈ [−1, 1], such that −1 (x)|f (x)|2 dx, where = (x) > 0 almost everywhere on [−1, 1]. In a natural way, we introduce the inner product of functions in this 1 space as (f (x), g(x)) = −1 (x)f (x)g(x) dx. This makes L2, a Hilbert space. Therefore, in it there exists an orthonormal system of functions ψn, , n = 0, 1, 2, . . . , that is a basis in it; i.e., 1 (x)ψn, (x)ψm, (x) dx = δn,m , where δn,m is the Kronecker delta. Any function f (x) ∈ L2, can −1 be represented by a Fourier series in the system of these functions, which is convergent in the norm of L2, :

f (x) =

∞

1 f˜(n) =

f˜(n)ψn, (x),

n=0

(3.14)

−1

1 f 2L2, =

(x)f (x)ψn, (x) dx,

(x)|f (x)|2 dx =

∞ ˜ 2 f (n) < +∞. n=0

−1

• system is formed by the functions T (x) = In the space L2, , a complete orthonormal n 2/πTn (x), • 1/πT0 (x), for = 1 (x); the functions U (x) = 2/πUn (x), n = 1, 2, . . . , and T0• (x) = n • n = 0, 1, 2, . . . , for = 2 (x); the functions Pn (x) = 1/πPn (x) for = 3 (x); and the functions Q•n (x) = 1/πQn (x) for = 3 (x). Now let f (x) be a distribution on L2, in the sense of Section 1; i.e., f (x) ∈ S = S (L2, ); then

f (x) =

∞

f¯(n)ψn, (x),

1 ¯ f (n) = f (x), ψn, (x) = (x)f (x)ψn, (x) dx,

n=0

(3.15)

−1

where f (x), ψn, (x) is treated as the value of the distribution (linear functional) f (x) on the function ψn, (x). By Hλ we denote the set of distributions f (x) on L2, such that f λ, =

2λ

n

˜ 2 f (n)

1/2 < ∞;

(3.16)

n∈Z

i.e., the function f ∗ (x) = n∈Z nλ f˜(n)ψn, (x) belongs to the set L2, . Now for functions in Hλ , we introduce the inner product by the relation (f, g) =

n2λ f˜(n)˜ g (n).

(3.17)

n∈Z

The inner product thus deﬁned makes Hλ a Hilbert space; moreover, H0 = L2, is the space of function with square integrable absolute value on [−1, 1]. Now for each distribution g(x) =

∞

g˜(n)ψ2, (x),

(x) = k (x),

k = 1, 2, 3, 4,

n=0

we assume that the values L1 (g(x), x0 ), Sk (g(x), x0 ), k = 1, 2, 3, 4, and H2 (g(x), x0 ) are obtained by interchanging the integral and sum in these operators. Now relations (3.7)–(3.13) imply the following. The operator L1 (g(x), x0 ) is a one-to-one mapping of the space of distributions S (L2,1 ) ; the operator S1 (g(x), x0 ) is a mapping of into itself and the space Hλ1 into the space Hλ+1 1 S (L2,1 ) onto S (L2,2 ) (the kernel of the mapping is a set of constants) and is a mapping of Hλ1 DIFFERENTIAL EQUATIONS

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GENERALIZED FUNCTIONS ON HILBERT SPACES, SINGULAR INTEGRAL . . .

871

onto Hλ2 with the same property; the operator S2 (g(x), x0 ) is a mapping of S (L2,2 ) into S (L2,1 ), the functions f (x) in its range satisfy the relation

1 −1

f (x) √ dx = 0, 1 − x2

(3.18)

and it is a mapping of Hλ2 into Hλ1 with the same property; the operator S3 (g(x), x0 ) is a one-to-one mapping of S (L2,3 ) into S (L2,4 ) and of Hλ3 into Hλ4 ; the operator S4 (g(x), x0 ) is a one-to-one mapping of S (L2,4 ) into S (L2,3 ) and of Hλ4 into Hλ3 ; and ﬁnally, the operator . Now we H2 (g(x), x0 ) is a one-to-one mapping of S (L2,2 ) into S (L2,2 ) and of Hλ2 into Hλ−1 2 make the following important remark. The well-known inversion formulas [7, 12, 14] are valid for Eqs. (3.1)–(3.6) on basis elements of the corresponding spaces. Therefore, by virtue of the deﬁnition of values of the operators L1 (g(x), x0 ), Sk (g(x), x0 ), k = 1, 2, 3, 4, and H2 (g(x), x0 ), the following inversion formulas are valid on distributions in the corresponding spaces for Eqs.(3.1)–(3.6): ⎛ 1 ⎞

1 1 1 − xx0 f (x)dx ⎠ 1 √ , x ∈ (−1, 1), g(x) = ⎝ 2 f (x0 ) dx0 − 2 π ln 2 1 − x2 1 − x0 (x − x0 ) −1

−1

for Eq. (3.1), 1 g(x) = − π

1 −1

1 − x20 f (x0 ) dx0 C + , x − x0 π

for Eq. (3.2), 1 g(x) = − π for Eq. (3.3), 1 g(x) = − π

1 −1

1 −1

1 √

x ∈ (−1, 1), −1

f (x0 ) dx0 1 − x20 x − x0

,

g(x) dx = C 1 − x2

(3.19)

x ∈ (−1, 1),

1 + x0 f (x0 ) dx0 , 1 − x0

x ∈ (−1, 1),

1 − x0 f (x0 ) dx0 , 1 + x0

x ∈ (−1, 1),

under condition (3.18) for Eq. (3.4), 1 g(x) = − π

1 −1

for Eq. (3.5), and 1 g(x) = √ π 1 − x2

1 −1

x − x0 ln √ f (x0 ) dx0 , 1 − xx0 + 1 − x2 1 − x20

x ∈ (−1, 1),

for Eq. (3.6). As an example of the use of inversion formulas in the class of distributions, consider Eq. (3.2). On its right-hand side, we choose the function f (x0 ) = δ (x0 − q), q ∈ (−1, 1); then, by virtue of (3.19), we ﬁnd that the solution of this equation is given by the function (C = 0) √ 1 1 − q2 . g(x) = − π x−q DIFFERENTIAL EQUATIONS

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LIFANOV, NENASHEV

Now, by substituting the last function into Eq. (3.2), we obtain 1 π

1 −1

1 √ 2 1 − x (x0 − x)

√ 1 1 − q2 − dx = δ (x0 − q) , π x−q

x0 ∈ (−1, 1).

ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 05-01-00951). REFERENCES 1. Belotserkovskii, S.M. and Lifanov, I.K., Chislennye metody v singulyarnykh integral’nykh uravneniyakh (Numerical Methods in Singular Integral Equations), Moscow: Nauka, 1985. 2. Belotserkovskii, S.M. and Nisht, M.I., Otryvnoe i bezotryvnoe obtekanie tonkikh kryl’ev ideal’noi zhidkost’yu (Separable and Nonseparable Flow of an Ideal Fluid Around Thin Wings), Moscow: Nauka, 1978. 3. Koening, D.G. and Falarski, M.D., NACA TMX 62029 , 1972. 4. Vainikko, G.M. and Lifanov, I.K., Diﬀer. Uravn., 2000, vol. 36, no. 9, pp. 1184–1195. 5. Sarannen, J. and Vainikko, G., Periodic Integral and Pseudodiﬀerential Equations with Numerical Approximation, Berlin, 2001. 6. Vainikko, G.M., Lebedeva, N.V., and Lifanov, I.K., Mat. Sb., 2002, vol. 193, no. 10, pp. 3–16. 7. Lifanov, I.K. and Nenashev, A.S., Diﬀer. Uravn., 2005, vol. 41, no. 1, pp. 121–137. 8. Vladimirov, V.S., Uravneniya matematicheskoi ﬁziki (Equations of Mathematical Physics), Moscow: Nauka, 1976. 9. Belotserkovskii, S.M. and Lifanov, I.K., Method of Discrete Vortices, Boca Raton, 1993. 10. Zakharov, E.S. and Pimenov, Yu.V., Chislennyi analiz difraktsii radiovoln (Numerical Analysis of Diﬀraction of Radio Waves), Moscow: Radio i Svyaz’, 1982. 11. Panasyuk, V.V., Savruk, M.P., and Datsyshin, A.P., Raspredelenie napryazhenii okolo treshchin v plastinakh i obolochkakh (Distribution of Stresses Near Cracks in Plates and Shells), Kiev: Naukova Dumka, 1976. 12. Lifanov, I.K., Metod singulyarnykh integral’nykh uravnenii i chislennyi eksperiment (The Method of Singular Integral Equations and Numerical Experiment), Moscow: Yanus, 1995. 13. Dovgii, S.A. and Lifanov, I.K., Metody resheniya integral’nykh uravnenii. Teoriya i prilozheniya (Methods for Solving Integral Equations. Theory and Applications), Kiev: Naukova Dumka, 2002. 14. Lifanov, I.K., Diﬀer. Uravn., 2006, vol. 42, no. 4, pp. 556–559.

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c Pleiades Publishing, Ltd., 2007. ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 6, pp. 873–883. c the Editorial Board, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 6, pp. 852–860. Original Russian Text

CHRONICLE

On the Seminar on Qualitative Theory of Diﬀerential Equations at Moscow State University Moscow State University, Moscow, Russia DOI: 10.1134/S0012266107060134

This issue contains the abstracts of reports made in the spring semester, 2007. [For the previous publication on the seminar, see Diﬀerentsial’nye Uravneniya, 2006, vol. 42, no. 11 (Diﬀerential Equations, 2006, vol. 42, no. 11).] Astashova, I.V. (Moscow, Russia). On the Oscillation Property of Solutions of Quasilinear Differential Equations (March 2, 2007). We consider the diﬀerential equation y (n) +

n−1

ai (x)y (i) + p(x)|y|k−1 y = 0,

(1)

i=0

where p(x) and ai (x) are continuous functions, n ≥ 1, and k > 1. Suﬃcient conditions for the existence of a nonoscillating solution of Eq. (1) and an oscillation criterion for all solutions of an even-order equation (1) are given. Theorem 1. Let the functions p(x) and aj (x), j = 0, 1, . . . , n − 1, occurring in Eq. (1) satisfy the conditions ∞ xn−1 |p(x)|dx < ∞; (2) ∞

x

xn−j−1 |aj (x)| dx < ∞.

(3)

x

Then Eq. (1) has a nonoscillating solution that tends to a nonzero limit as x → ∞. Theorem 2. In (1), let n be an even number, let p(x) > 0, and let the functions aj (x), j = 0, 1, . . . , n − 1, satisfy condition (3). Then the following two assertions are equivalent : ∞ (1) x xn−1 p(x)dx < ∞; (2) Eq. (1) has a nonoscillating solution. Remark 1. The above-represented results generalize the well-known Atkinson oscillation criterion [1] proved for Eq. (1) with ai (x) = 0 and n = 2 and generalized in [2] to the case in which ai (x) = 0 and n ≥ 2. Remark 2. The proof of the above-represented results can be performed on the basis of methods [3, 4] used in the derivation of uniform estimates for solutions of Eq. (1) which have a common domain. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 06-01-00715) and the Program “State Support for Leading Scientiﬁc Schools of the Russian Federation” (project no. NSh-2538.2006.1). 873

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REFERENCES 1. Atkinson, F.V., Paciﬁc J. Math., 1955, vol. 5, no. 1, pp. 643–647. 2. Kiguradze, I.T. and Chanturiya, T.A., Asimptoticheskie svoistva reshenii neavtonomnykh obyknovennykh diﬀerentsial’nykh uravnenii (Asymptotic Properties of Solutions of Nonautonomous Ordinary Diﬀerential Equations), Moscow: Nauka, 1990. 3. Astashova, I.V., Tr. Semin. im. I.G. Petrovskogo, 2006, vol. 25, pp. 21–34. 4. Astashova, I.V., Dokl. Akad. Nauk , 2006, vol. 409, no. 5, pp. 586–590.

Barabanov, E.A. (Minsk, Belarus). A Generalization of the Bylov Reduction Theorem (March 2, 2007). By Mn we denote the set of systems of linear ordinary diﬀerential equations with piecewise continuous and uniformly bounded coeﬃcients on the half-line. The well-known Bylov criterion [1] provides a necessary and suﬃcient condition for the reducibility of a system in Mn to a blocktriangular form by a Lyapunov transformation. The following theorem extends this criterion to generalized Lyapunov transformations. Theorem. A system x˙ = A(t)x in Mn can be reduced by some generalized Lyapunov transformation x = L(t)y satisfying the additional condition supt≥0 L(t) < +∞ to a system y˙ = B(t)y in Mn , whose coeﬃcient matrix is block-triangular (B(·) = diag [B1 (·), . . . , Bq (·)] , where Bi (·) is an upper-triangular ni × ni matrix, i = 1, . . . , q) if and q only if the set XA of its solutions can be expanded as a linear space into a direct sum XA = k=1 Xk of subspaces such that dim Xi = ni k and the angles ϕk (t) = ∠{Xk+1 (t), i=1 Xi (t)} satisfy the relation limt→+∞ t−1 ln ϕk (t) = 0 for all k = 1, . . . , q − 1. A clear geometric proof of the Basov criterion for the properness of a system in Mn is one of the applications of the above-represented theorem. Indeed, the Vinograd criterion [3] of the system properness implies that there exists a basis {x1 , . . . , xn } of its solutions satisfying the assumptions (q = n) of the above-formulated theorem. Therefore, a proper system can be reduced by a generalized transformation to the diagonal form y˙ = diag [a1 + o1 (t), . . . , an + o1 (t)] y, where ai = limt→+∞ t−1 ln xi (t), and oi (t) → 0 as t → +∞, i = 1, . . . , n. But, obviously, the last system can be reduced by a generalized transformation to the system z˙ = diag [a1 , . . . , an ] z. REFERENCES 1. Bylov, B.F., Diﬀer. Uravn., 1965, vol. 1, no. 12, pp. 1597–1600; 1987, vol. 23, no. 12, pp. 2027–3031. 2. Basov, V.P., Vestnik Leningr. Univ., 1952, no. 5, pp. 3–8. 3. Vinograd, R.E., Uspekhi Mat. Nauk , 1954, vol. 9, no. 2, pp. 129–136.

Bykov, V.V. (Moscow, Russia). On the Lebesgue Measurability of Extraordinary Auxiliary Logarithmic Exponents of a Linear System (March 16, 2007). Throughout the following, K is treated as either R everywhere or C everywhere. Theorem. Let x˙ = Aμ (t)x be a family of linear systems, where M is a closed set in K, and let the mapping R+ × M → End Kn given by the formula (t, μ) → Aμ (t) be continuous and bounded for each μ ∈ M . Then the functions ϕk : M → R given by the formula μ → alk (Aμ ) , k = 1, . . . , n, where alk is the kth extraordinary auxiliary logarithmic exponent [1], are Lebesgue measurable, and the restrictions of these functions to some set of the type Gδ dense in M are continuous. This theorem provides a partial solution of a problem in [1]. REFERENCES 1. Millionshchikov, V.M., Diﬀer. Uravn., 1995, vol. 31, no. 12, p. 2090.

Sergeev, I.N. (Moscow, Russia). On the Diﬀerent Dependence of the Leading Frequencies of Zeros, Signs, and Roots of a Linear Equation on a Parameter (March 16, 2007). We equip the set En of all equations y (n) + a1 (t)y (n−1) + · · · + an−1 (t)y˙ + an (t)y = 0,

t ∈ R+ ,

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each of which is speciﬁed by the set a ≡ (a1 , . . . , an ) of continuous bounded coeﬃcients, with the uniform topology on R+ . The higher and lower frequencies of zeros of an equation a ∈ En are deﬁned by the formulas ω10 (a) ≡

inf

lim

y∈S∗ (a) t→∞

π 0 ν (y, t), t

ωn0 (a) ≡ sup lim

y∈S∗ (a) t→∞

π 0 ν (y, t), t

where S∗i (a) stands for the set of all nonzero solutions of the equation a and ν 0 (y, t) stands for the number of zeros of the solution y on the interval (0, t). By using similar formulas with the replacement of the number ν 0 of zeros by the number ν − of sign alternations or the number ν + of roots, that is, all zeros with regard to their multiplicities (of the same solution on the same interval), one can deﬁne the higher and lower frequencies ω − of the signs and the higher and lower frequencies ω + of roots (of the same equation with the same indices). In the space E n , we deﬁne an almost rectilinear arc with initial point a0 ∈ E n and terminal point a1 ∈ E n as any continuous (with respect to the parameter μ ∈ [0, 1]) family of equations σ(μ) ∈ C R+ , σ(0) = 0, σ(1) = 1. aμ ≡ a0 + σ(μ) a1 − a0 ∈ E n , Each function ω : E n → R generates a function Ω(μ) ≡ ω (aμ ) of the parameter μ ∈ [0, 1] along the arc aμ . Now let n = 3. By using methods in [1], one can prove the following assertions. Theorem 1. In the space E 3 , there exist two almost rectilinear arcs with a common initial point and a common terminal point such that the functions induced by the higher and lower frequencies of zeros, signs, and roots are step functions along one arc and are everywhere discontinuous along the other. Theorem 2. In the space E 3 , there exist two almost rectilinear arcs with a common initial point such that, along one arc, the functions induced by the frequencies ω10 , ω1+ , and ω3+ are continuous, and the functions induced by the frequencies ω1− , ω30 , and ω3− are discontinuous, and the opposite takes place along the other arc.

REFERENCES 1. Sergeev, I.N., Tr. Semin. im. I.G. Petrovskogo, 2006, vol. 25, pp. 249–294.

Braun, S.A. (Moscow, Russia). On Estimates for the Minimal Eigenvalue of the Sturm–Liouville Problem with a Mixed Boundary Condition (March 23, 2007). Consider the Sturm–Liouville problem y (x) + λq(x)y(x) = 0, y(0) = 0, y (1) + ky(1) = 0,

(1) (2)

where q(x) > 0 is an integrable function on [0, 1] such that 1 q(x)β dx = 1,

β = 0.

(3)

0

The function y(x) is referred to as a solution of Eq. (1) satisfying condition (2) if y is absolutely continuous and satisﬁes condition (2) and Eq. (1) is valid almost everywhere. An eigenvalue of problem (1), (2) is deﬁned as λ such that there exists a nontrivial solution of Eq. (1) satisfying condition (2). This solution is referred to as an eigenfunction. It is known (e.g., see [1, pp. 133–139]) that the set of eigenvalues is an inﬁnite discrete set, and the eigenfunctions corresponding to distinct values of λ are linearly independent and orthogonal DIFFERENTIAL EQUATIONS

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with weight q(x). Moreover, there exists a minimum eigenvalue of problem (1), (2). It can be found by the variational method as 1 λ1 = inf L[q, y],

L[q, y] =

y + ky (1) dx 2

1

2

q(x)y 2 dx,

y(x)∈H

0

0

where H = y : y(x) ∈ H 1 [0, 1], y(0) = 0. One can show that λ1 > 0 if and only if 1 + k > 0. In the present work, we investigate the dependence of the eigenvalue λ1 of problem (1), (2) on k for various β. We set mβ = inf λ1 , q∈Aβ

Mβ = sup λ1 , q∈Aβ

where Aβ is the set of functions satisfying condition (3). Estimates for mβ and Mβ were obtained in [1, pp. 153–176] for Eq. (1) with the zero Dirichlet boundary conditions and in [2] for the problem with boundary conditions of the form y (0) − k12 y(0) = 0,

y (1) + k12 y(1) = 0.

Obviously, mβ = Mβ = 0 in problem (1), (2) with k = −1. Theorem 1. Let 1 + k > 0. If β > 1, then 0 < mβ < +∞ and Mβ = +∞; moreover, there exist functions u(x) ∈ H = 0 and q(x) ∈ Aβ such that infy(x)∈H L[q, y] = L[q, u] = mβ . If β = 1, then m1 = 1 + k > 0 and M1 = +∞. If 1/2 < β < 1, then mβ = 0 and Mβ = +∞. If β ≤ 1/2, then mβ = 0 and Mβ < +∞; moreover, there exist functions u(x) ∈ H and q(x) ∈ Aβ such that inf L[q, y] = L[q, u] = Mβ .

y(x)∈H

Theorem 2. Let k < −1. If β > 1, then mβ = −∞ and Mβ < 0; moreover, there exist functions u(x) ∈ H and q(x) ∈ Aβ such that infy(x)∈H L[q, y] = L[q, u] = Mβ . If β = 1, then m1 = −∞ and M1 = 1 + k. If 0 < β < 1, then mβ = −∞ and Mβ = 0. If β < 0, then Mβ = 0 and mβ > −∞; moreover, there exist functions u(x) ∈ H and q(x) ∈ Aβ such that infy(x)∈H L[q, y] = L[q, u] = mβ .

ACKNOWLEDGMENTS The work was performed in the framework of the program “State Support for Leading Scientiﬁc School of the Russian Federation” (project no. NSh-2538.2006.1). REFERENCES 1. Kondratiev, V. and Egorov, Yu., On Spectral Theory of Elliptic Operators. Operator Theory, vol. 89, Basel, 1996. 2. Muryshkina, O.V., Diﬀer. Uravn., 2001, vol. 37, no. 6, p. 854.

Rudakov, I.A. (Bryansk, Russia). Periodic Solutions of a Nonlinear Wave Equation with Boundary Conditions of the Third Kind (March 23, 2007). Consider the problem 0 < x < π, t ∈ R, p(x)utt − (p(x)ux )x = g(x, t, u) + f (x, t), u(π, t) + h2 ux (π, t) = 0, t ∈ R, u(0, t) − h1 ux (0, t) = 0; u(x, t + T ) = u(x, t), 0 < x < π, t ∈ R. DIFFERENTIAL EQUATIONS

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Here h1 ≥ 0, h2 ≥ 0, h1 + h2 > 0, T = 2πb/a, a, b ∈ N, (a, b) = 1. In the study of the spectrum of the linear part of the equation, we consider the auxiliary problem on the asymptotic estimate for eigenvalues of the Sturm–Liouville problem − (pϕx )x = pλ2 ϕ, ϕ(0) − h1 ϕ (0) = ϕ(π) + h2 ϕ (π) = 0.

(4) (5)

Let {ϕn (x)} and {λn }, n ∈ N, be sequences of eigenfunctions and eigenvalues of problem (4), (5) such that λn ↑ +∞ [a positive function p(x) satisﬁes the corresponding conditions]. By the Steklov theorem, one can assume that the function system {ϕn (x)} is orthonormal in L2 (0, π) with the inner product p(x)ϕ(x)ψ(x)dx,

(ϕ, ψ) =

ϕ, ψ ∈ L2 (0, π).

[0,π]

The existence of constants b0 , b1 ∈ (0, +∞) such that b0 < (λn − n + 1) n < b1 for all n is proved for boundary conditions of the third kind at both endpoints (h1 > 0, h2 > 0). If a boundary condition of the third kind is posed at one endpoint and the Dirichlet homogeneous condition (h1 h2 = 0) is posed at the other endpoint, then we have the estimate b0 < (λn − n + 1/2) n < b1 for all n. Theorem. Let h1 h2 = 0, and let the function g(x, t, u) satisfy the following conditions : g(x, t, u) is continuous on [0, π] × R2 and T -periodic with respect to t; g(x, t, u) g(x, t, u) ≤ lim ≤ β(x, t) u→∞ p(x)u u→∞ p(x)u

α(x, t) ≤ lim

uniformly with respect to (x, t) ∈ Ω = [0, π] × [0, T ], where α(x, t), β(x, t) ∈ L∞ (Ω), α ≤ α(x, t) ≤ β(x, t) ≤ β, almost everywhere in Ω and [α, β] ∩ σ = ∅, where σ is the spectrum of the linear part of Eq. (1). If b is an odd number, then problem (1)–(3) has a generalized solution u ∈ C(Ω) for any T -periodic function f ∈ L2 (Ω) of t. Burlutskaya, M.Sh. (Voronezh, Moscow, Russia). On the Expansion in Eigenfunctions of the Sturm–Liouville Operator on a Graph-Pencil (March 30, 2007). On a graph-pencil Γ of three edges, we consider the Sturm–Liouville operator L with the Dirichlet boundary conditions at the boundary vertices of Γ and with the continuity and transmission conditions at the node [1]. By virtue of the vector approach [1], y = Rλ f , where Rλ is the resolvent of the operator L, is a solution of the vector boundary value problem d−2 j yj (x) + qj (x)yj (x) = λyj (x) + fj (x),

y1 (0) = y2 (0) = y3 (0),

yj (1) = 0

(j = 1, 2, 3),

x ∈ [0, 1], 3

j = 1, 2, 3,

(1)

αk yk (0) = −α0 y1 (0).

(2)

k=1

(Here yj , fj , and qj are the components of y, f , and q; qj (x) ∈ L[0, 1], dj > 0, and

3 k=1

αk dk = 0.)

Theorem 1 (on the equiconvergence). One has lim Sr (f, x) − Σr (f, x)C[δ,1−δ] = 0

r→∞

T

for any vector function f (x) = (f1 (x), f2 (x), f3 (x)) (T stands for the transposition), fi (x) ∈ L[0, 1], and for any δ ∈ (0, 1/2), where Sr (f, x) is the partial sum of the Fourier series of the function f (x) in eigenfunctions and associated functions of the operator L including terms for which |λk | < r, T Σr (f, x) = σ√rd1 (f1 , x) , σ√rd2 (f2 , x) , σ√rd3 (f3 , x) ,

+∞ and σrj (fj , x) is the partial sum of the Fourier series of the function fj in the system e2kπix k=−∞ for k such that |2πk| < rj . DIFFERENTIAL EQUATIONS

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Theorem 2 (an analog of the Jordan–Dirichlet theorem). If the components of the vector function f (x) are continuous, have bounded variation, and satisfy condition (2), then f (x) can be expanded in a uniformly convergent series in eigenfunctions of the operator L on the closable interval [0, 1]. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (projects nos. 04-01-00049, 06-01-00003, and 04-01-00697). REFERENCES 1. Pokornyi, Yu.V., Penkin, O.M., Pryadiev, V.L., Borovskikh, A.V., Lazaev, K.P., and Shabrov, S.A., Diﬀerentsial’nye uravneniya na geometricheskikh grafakh (Diﬀerential Equations on Geometric Graphs), Moscow: Fizmatlit, 2005.

Prokhorova, R.A. (Minsk, Belarus). On the Instability of Diﬀerential Systems in the Linear Lp -Dichotomous Approximation (March 30, 2007). Consider the linear systems x˙ = A(t)x (1) with piecewise continuous coeﬃcient matrix A(·) : [0, +∞) → Hom (Rn , Rn ) and fundamental matrix X(t). Deﬁnition [1, p. 131; 2]. We say that system (1) has the property of Lp -dichotomy on R+ with the parameter p > 0 if there exists a pair of complementary projections P1 and P2 such that t Cp ≡ sup

X(t)P1 X −1 (τ ) p dτ < +∞,

t≥0 0 +∞

X(t)P2 X −1 (τ ) p dτ < +∞.

Dp ≡ sup t≥0 t

Along with the linear system (1), we consider the perturbed system y˙ = A(t)y + f (t, y)

(2)

with a vector function f : [0, +∞)×U(f ) → Rn , U(f ) = {y ∈ Rn : y < (f )}, (f ) > 0, piecewise continuous with respect to t, continuous with respect to y, and satisfying the m-perturbation condition f (t, y) ≤ Cf ym ,

Cf = const,

m > 1,

y ∈ U(f ) ,

t ≥ 0.

(3)

The problem of stability analysis in the linear approximation in the extreme cases in which P1 = E and P1 = 0 was solved in [3, 4]; moreover, if p ≥ 1, then the zero solution of system (2) with an m-perturbation (3) is asymptotically stable in the ﬁrst case and is unstable in the other. In the general case, we have the following assertion. Theorem. If system (1) has Lp -dichotomy on R+ with parameter p ≥ 1 and with nonzero second projection P2 , then the zero solution of system (2) is unstable under an arbitrary m-perturbation f of order m > 1.

1. 2. 3. 4.

REFERENCES Coppel, W.A., Stability and Asymptotic Behavior of Diﬀerential Equations, Boston, 1965. Conti, R., Funkcialaj Ekvacioj , 1966, vol. 9, no. 1, pp. 23–26. Izobov, N.A. and Prokhorova, R.A., Diﬀer. Uravn., 2004, vol. 40, no. 12, pp. 1608–1614. Izobov, N.A. and Prokhorova, R.A., Diﬀer. Uravn., 2005, vol. 41, no. 1, pp. 61–72. DIFFERENTIAL EQUATIONS

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Shabrov, S.A. (Voronezh, Russia). On the Nonlinear Spectral Problem with Derivatives with Respect to a Measure (April 6, 2007). We investigate the nonlinear spectral problem with derivatives with respect to a measure,

− (pu )σ + uQσ = λf (x, u),

u(0) = u(1) = 0,

(1)

where σ(x) is an increasing function inducing a measure on the interval [0, 1] (we assume that it is continuous at the endpoints), the function p(x) belongs to BV [0, 1] (that is, the space of functions of bounded variation on [0, 1]); moreover, inf[0,1] p(x) > 0, and Q(x) is nondecreasing and σ-absolutely continuous. We construct solutions of problem (1) in the space E of absolutely continuous functions u(x) whose derivatives u (x) are σ-absolutely continuous. The outer derivative with respect to σ occurring in (1) is treated in the Radon–Nikodym sense. Theorem. Let the function f (x, u) satisfy the Carath´eodory condition. Let f (x, 0) = 0, and suppose that the function does not increase with respect to u for u ≥ 0 and for each x. Finally, let f (x, u)/u be a strictly decreasing function of u for u > 0. Then the set Λ of values λ ≥ 0 for which problem (1) has at least one nontrivial nonnegative solution has the following properties : (1) Λ is nonempty and coincides with some interval (λ0 , λ∞ ) ⊂ (0, +∞); (2) each λ ∈ Λ corresponds to exactly one nontrivial nonnegative solution uλ (x) of the boundary value problem (1); moreover, uλ → 0 as λ → λ0 and uλ → ∞ as λ → λ∞ . (Here u = max |u(x)| is the norm in the space of continuous functions on the interval [0, 1]); (3) The function uλ (x) is monotone with respect to λ; i.e., (λ1 − λ2 ) (uλ1 (x) − uλ2 (x)) ≥ 0 for all x ∈ [0, 1]. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 04-01-00049). Kulikov, D.A. (Yaroslavl, Russia). Some Attractors of a Finite-Dimensional Analog of the Ginzburg–Landau Equation (April 13, 2007). In the space C2 , we consider the diﬀerential equation ξ˙ = ξ − (1 + ic)ξ|ξ|2 + d exp(−iα)Dξ,

(1)

where ξ = ξ(t) = col (ξ1 (t), ξ2 (t)), ξj (t) ∈ C (j = 1, 2), d and c are real constants, and α ∈ [0, π/2]. Let

2 ξ1 |ξ1 | −1 1 2 . D= , ξ|ξ| = 2 ξ2 |ξ2 | 1 −1 Equation (1) is one of simplest ﬁnite-dimensional (ﬁnite-diﬀerence) analogs [1] of the Ginzburg– Landau equation ut = u − (d0 + ic0 ) u|u|2 + (a0 + ib0 ) uxx , if this equation is considered with the no-ﬂow boundary conditions ux (t, 0) = ux (t, l) = 0 or the periodic boundary conditions u(t, 0) = u(t, l),

ux (t, 0) = ux (t, l).

Here x ∈ [0, l], t ≥ 0, u(t, x) is a complex-valued function, and a0 , b0 , c0 , d0 ∈ R; moreover, a0 ≥ 0 and d0 > 0. For Eq. (1), we analyze (see also [2]) the problem on the existence, stability, and local bifurcations of asymmetric self-similar cycles, that is, solutions of the form ξ(t) = y exp(iσt) such that |y1 | = |y2 | (y ∈ C2 , σ ∈ R). DIFFERENTIAL EQUATIONS

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Consider the quadratic equation η 2 + 4p − q 2 1 + h2 η + 4 p2 − q 2 = 0, where

(3)

q = d1 (1 − d cos α)/d, p = 1 + cos2 α − c sin α cos α, h = (c cos α + sin α)/d1 . d1 = c sin α − cos α,

A root of Eq. (3) is said to be “proper” if it satisﬁes the inequalities η > 0 and (η + 2p)q > 0. Theorem 1. Let d1 > 0. Then each “proper” root of the bilinear equation (3) corresponds to two asymmetric cycles of Eq. (1). The parameters of these cycles are given by the formulas √ y2 = 2 exp(iψ),

1 = R/λ,

2 = Rλ, y1 = 1 , R = dq η + 4/(d1 (η + 2p)) , ψ = arctan h 1 − λ2 / 1 + λ2 , η + 4 cos ψ − (1 + d1 /d) , σ = −c − d/ sin α √ where λ is a root of the equation λ + 1/λ = η + 4 and η is a “proper” root of Eq. (3). The converse is also valid: each asymmetric cycle corresponds to a “proper” root of Eq. (3). Theorem 2. For each symmetric cycle, there exists a positive constant d∗ = d∗ (α, c) such that this cycle is orbitally asymptotically stable if d > d∗ and is unstable if d < d∗ . We ﬁx one of these cycles and set d = d∗ − ε. Theorem 3. There exist two domains B1 , B2 ∈ R2 such that the following assertions are valid. (1) If ε > 0 and (α, c) ∈ B1 , then a two-dimensional asymptotically stable invariant torus bifurcates from the chosen cycle. (2) If ε < 0 and (α, c) ∈ B2 , then a two-dimensional unstable invariant torus bifurcates from the chosen cycle.

REFERENCES 1. Mishchenko, E.F., Sadovnichii, V.A., Kolesov, A.Yu., and Rozov, N.Kh., Avtomodel’nye protsessy v nelineinykh sredakh s diﬀuziei (Autowave Processes in Nonlinear Media with Diﬀusion), Moscow: Fizmatlit, 2005. 2. Kulikov, D.A., Mezhdunar. konf. molodykh uchenykh: Tez. dokl. (Abstr. Int. Conf. of Young Scientists), N. Novgorod, 2006, pp. 91–92.

Izobov, N.A. (Minsk, Belarus). Diﬀerential Systems with a Linear Lp -Dichotomous Coppel–Conti Approximation (April 20, 2007). Consider the diﬀerential system y˙ = A(t)y + f (t, y),

y ∈ Rn ,

t ≥ 0,

(1)

with a matrix A(t) piecewise continuous with respect to t ≥ 0 and with a vector function f (t, y) satisfying the condition f (t, y) ≤ Lym ,

m > 1,

(t, y) ∈ [0, +∞) × U ,

(2)

in some neighborhood U ≡ {y ∈ Rn : y < } ⊂ Rn , ∈ (0, 1], of the origin y = 0 and continuous with respect to y in this neighborhood; this function is referred to as an m-perturbation. Note that DIFFERENTIAL EQUATIONS

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nontrivial solutions of system (1) are not unique [1, p. 260], although the zero solution is always unique. We assume that the linear approximation system x ∈ Rn ,

x˙ = A(t)x,

t ≥ 0,

(3)

for system (1) with the fundamental solution matrix X(t), X(0) = E, belongs to the Coppel–Conti set Lp D, p > 0, of linear Lp -dichotomous systems. (For its deﬁnition, see [2].) Theorem. Let system (3) belong to the set Lp D, p ≥ 1, with mutually complementary projections P1 and P2 , P1 +P2 = E, of ranks k, 1 ≤ k < n, and n−k, respectively and with constants Cp (A) > 0 and Dp (A) > 0. Then for any δ ∈ (0, 1), there exists an r-neighborhood Ur ⊂ U of the origin with radius r > 0 found from the condition L [C1 (A) + D1 (A)] [2C(A)]m r m−1 < δ,

r(c) ≡ 2C(A)r <

with C(A) ≥ X(t)P1 ≥ 1 for t ≥ 0 such that the following conditions are satisﬁed. (1◦ ) For any vector y0 ∈ Rn with projection P1 y0 ∈ Ur(c) \{0}, there exists a solution y : [0, +∞) → Ur(c) \{0} of system (1), which satisﬁes the estimates y(0) − P1 y0 ≤ δ P1 y0 ,

y(t) ≤ (1 + δ)C(A) P1 y0 ,

t≥0

−1

and has the characteristic exponent λ[y] < − [eC1 (A)] < 0. (2◦ ) Two arbitrary distinct values P1 y1 , P1 y2 ∈ Ur(c) correspond to at least two distinct (not coinciding on the entire half-line [0, +∞)) solutions y1 (t) and y2 (t) of system (1) with property (1◦ ). REFERENCES 1. Izobov, N.A., Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to Theory of Lyapunov Exponents), Minsk: FPMI, 2006. 2. Prokhorova, R.A., Diﬀer. Uravn., 2007, vol. 43, no. 6, p. 856.

Masterkov, Yu.V. and Rodina, L.I. (Izhevsk, Russia). Lyapunov Functions of Control Systems with Random Parameters (April 20, 2007). We investigate the problem on the construction of a nonanticipating control for a linear nonstationary system with random parameters. A control u(t, x) is said to be nonanticipating if its construction at time t = τ requires information on the system behavior only for t ≤ τ . We consider the control system (t, ω, x, u) ∈ R × Ω × Rn × Rm , (1) x˙ = A f t ω x + B f t ω u, where u ∈ U , U is a convex compact set in Rm , 0 ∈ intr U , and (Ω, F, μ) is a probability space. . t t The random function ξ (f ω) = (A (f ω) , B (f t ω)) induced by the ﬂow f t takes constant values s ϕk for t ∈ [τk , τk+1 ) and is stationary in the narrow sense. Here ϕk ∈ Ψ, Ψ = {ψj }j=1 is a ﬁnite . set of matrix pairs ψj = (Aj , Bj ). System (1) is identiﬁed with the function ξ : Ω → Ψ. For each ﬁxed ω, the function ξ (f t ω) deﬁnes a linear deterministic system. For admissible controls of the system ξ, we take all possible bounded Lebesgue measurable functions uω : R × Rn × Rn → U ∈ Rm . Solutions of the system ξ corresponding to uω (t, x, x0 ) are treated in the Filippov sense. Deﬁnition. Let D[t0 ,t1 ] (ω) be the controllability set of the system ξ on the interval [t0 , t1 ], and let D[t0 ,t1 ] (ω) be the set of nonanticipating controlled states of the system ξ on [t0 , t1 ]. The system ξ is said to be locally controllable with probability μ0 on the interval [t0 , t1 ] if

μ ω : 0 ∈ int D[t0 ,t1 ] (ω) = μ0 and nonanticipating locally controllable with probability μ0 on [t0 , t1 ] if

μ ω : 0 ∈ int D[t0 ,t1 ] (ω) = μ0 . DIFFERENTIAL EQUATIONS

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By ξi we denote the system x˙ = Ai x + Bi u, (x, u) ∈ Rn × Rm ; L (ξi ) is the controllability space of the system ξi , dim L (ξi ) ≤ n, i = 1, . . . , s. Let the function V ∈ C (Rn , R) have the following properties. (a) V (x) > 0 for x = 0, and V (0) = 0. (b) There exists a v1 > 0 such that for any 0 < v0 < v1 the surface V (x) = v0 is unique and has an n-dimensional Lebesgue measure equal to zero. (c) V (x) is piecewise continuously diﬀerentiable in some neighborhood Oεn . Theorem. Let a sequence W = (ψi1 , . . . , ψik ) , where ψij ∈ Ψ, a function V ∈ C (Rn , R) , and subspaces M , . . . , Mk−1 satisfy the following conditions. (α)M +1 , = 1, . . . , k − 2, Mk−1 ⊂ L (ξik ) , and M1 + L (ξi1 ) = Rn . (a) M ⊂ L ξi+1 + Xi−1 +1 (b) minu∈U ∂V (x)/∂x, Ai x + Bi u ≤ 0 for all i = 1, . . . , s and for almost all x ∈ Oεn0 . (c) There exist controls u (x) ∈ U such that Ai x + Bi u (x) ⊂ M and ∂V (x) , Ai x + Bi u (x) ≤ 0 ∂x for almost all x ∈ M ∩ Oεn0 , = 1, . . . , k − 1. (d) V (xj + j ) ≥ V (xj ) for all xj ∈ Mj ∩ Oεn0 , j ∈ L ξij , j = 1, . . . , k − 1. Then the system ξ is nonanticipating locally controllable on the interval [0, T ] with probability μ(T ) such that if T ≥ [r(N + 3) − 2]β, r, N ≥ 1, then r

1 − (qi1 − pqik ) . μ(T ) ≥ pQ 1 − qi1 + pqik Here p = pi1 ,i2 · · · pik −1,ik , pij is the probability of the transition from the state ψi to ψj , qj is the probability of the ﬁrst transition from ψj to ψi1 in at most N steps, and Q is the probability of the ﬁrst transition to ψi1 in at most N steps. ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (project no. 06-01-00258). Panasenko, E.A. and Tonkov, E.L. (Tambov, Russia; Izhevsk, Russia). Lyapunov Functions and Positively Invariant Sets of Diﬀerential Inclusions (April 27, 2007). For a ﬁxed topological dynamical system (Σ, f t ) and a given function F : Σ × Rn → comp (Rn ), we consider (for each σ ∈ Σ) the diﬀerential inclusion t ∈ R, (1) x˙ ∈ F f t σ, x , and the “convexized” diﬀerential inclusion

x˙ ∈ co F f t σ, x .

(2)

Next, for each σ ∈ Σ, we assume that the function t → f t σ satisﬁes the local Lipschitz condition, the function (t, x) → F (f t σ, x) satisﬁes the Carath´eodory condition [1], and each solution of the inclusion (1) is deﬁned for all t ≥ 0. To each point ω = (σ, X) ∈ Ω = Σ × comp (Rn ) and any t, we assign the cross-section S(t, ω) of the integral funnel of the inclusion (2) and introduce the dynamical system (Ω, gt ), where gt ω = (f t σ, S(t, ω)). Further, for a given continuous function σ → M (σ) ∈ comp (Rn ), we construct the set . M = {ω = (σ, X) ∈ Ω : X ⊂ M (σ)} . and the r-neighborhood Mr = {ω = (σ, X) ∈ Ω : X ⊂ M r (σ)} of the set M, where M r (σ) is the r-neighborhood of the set M (σ). DIFFERENTIAL EQUATIONS

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Deﬁnition. A set M is said to be (a) positively invariant with respect to the inclusion (1) if gt M ⊂ M for all t ≥ 0; (b) stable positively invariant with respect to the inclusion (1) if M is positively invariant and, for each ε > 0, there exists a δ > 0 such that gt Mδ ⊂ Mε for all t ≥ 0; (c) invariant with respect to the inclusion (1) if gt M = M for all t ∈ R; (d) minimal if M is invariant and does not contain a proper invariant subset. Singletons {x} in the space comp (Rn ) are identiﬁed with points of the space Rn and are denoted by small letters without curly braces. We set Nr = {ω = (σ, x) ∈ Mr : ω ∈ M}. A continuous function V : Mr → R is referred to as a Lyapunov function (with respect to Mr ) if V (ω) = 0 for all ω ∈ ∂ M and V (ω) > 0 for ω ∈ Nr . Further, a Lyapunov function V is said to be positive deﬁnite (with respect to M) if for any ε ∈ (0, r), there exists a δ > 0 such that V (ω) ≥ δ for all ω ∈ ∂ Mε . For r > 0 and for a locally Lipschitz function V : Mr → R, the limit V f δτ f ϑ σ , y + δh − V f ϑ σ, y . 0 lim sup V (ω; q) = δ (ϑ,y,δ)→(0,x,+0) is referred to as the generalized derivative of the function V at the point ω = (σ, x) in the direction of the vector q = (τ, h) ∈ R × Rn (the Clarke derivative [2, p. 32]). Further, if q = (1, h), then . VF0 (ω) = maxh∈F (ω) V 0 (ω; q) is referred to as the derivative of V along the trajectories of the inclusion (1). Theorem 1. If there exists a Lyapunov function V : Mr → R such that VF0 (ω) ≤ 0 for all ω ∈ Nr , then the set M is positively invariant. Theorem 2. If a positive-deﬁnite Lyapunov function V : Mr → R satisﬁes the condition ≤ 0 for all ω ∈ Nr , then the set M is stable positively invariant.

VF0 (ω)

Theorem 3. If M is a positively invariant set and there exists a Lyapunov function W : Mr → R such that minh∈F (ω) W 0 (ω; (1, h)) ≥ 0 for all ω ∈ Nr , then M is invariant. Theorem 4. If Σ is a compact space and M is a positively invariant set, then in the set M, there exists a compact minimum subset.

ACKNOWLEDGMENTS The work was ﬁnancially supported by the Russian Foundation for Basic Research (projects nos. 04-01-00324 and 06-01-00258). REFERENCES 1. Blagodatskikh, V.I. and Filippov, A.F., Tr. Mat. Inst. Steklova, 1985, vol. 169, pp. 194–252. 2. Clarke, F.H., Optimization and Nonsmooth Analysis, Montreal, 1983. Translated under the title Optimizatsiya i negladkii analiz , Moscow: Nauka, 1988.

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