Approximate solutions of operator equations

sup

^ " ^ K

geH,9^0

n=l,2,-.-.

(3.39)

\\9\\H

Proof: First, in Equation (3.37') letting v = un — u gives [y>, A{un -u)]v

= [Jig, un-u]v

= {g, un - u)H .

Second, it follows from Equations (3.32') and (3.35') that

[A(un-u\iP]v=0,

V^€Vn.

By combining the above two equations, we have (un - u , g)H = [A(un

-u),
V ^ G Vn .

Thus, substituting this result in the following identity \\un-u\\H=

SUp

9eH,9^o

TTM— | ( W n - W , ^ ) H | ,

\\9\\H

and then applying (3.30), which was established under Condition {Hi), we arrive at \\v>n - u\\H < M\un

- u\v

sup

——

inf

geH,g^0

\\g\\H

tf€Vn

|y> - ^ |

T h e resulting error b o u n d (3.39) then follows. We next discuss the p e r t u r b e d operator equation of (3.27).

[]

Operator Equations (2)

85

Suppose, moreover, that the embedding of V into H is compact. Con­ sider the operator equation Au + Bu = f,

f eH,

(3.40)

where operator A satisfies Conditions (Hi) and (H2) stated above, B : V(B) —> H is a linear operator, with V(B) D T>(A), and satisfies the follow­ ing additional condition: (Hz) There are constants M\ > 0 and 0 < 6 < 1 such that \(Bx,y)H\<M1\x\v\y\1v-6\\y\\H,

Vi,^P(B);

(3.41)

or (#3) B = Bi+B2, such that

and there are constants Mi, M2 > 0 and 0 < £1, £2 < 1

| ( B i x l y > w | < M 1 | x | v | » | J r - ^ | | i , | | § > Vx,y£V(B), | (B 2 x,y) H | <

M2\y\v\x\1v-S*\\x\\lg

(3.42')

, V x,» € V(B). (3.42")

We note, here, that a special case of Condition (#3) is that B :V —» H is a bounded linear operator. According to the Riesz Representation Theorem, there is a linear oper­ ator S : V(B) -> V such that {Bx,y)H

= [Bx,y]v>

xeV(B),

y

eV.

If, furthermore, Condition (H$) is satisfied, then there are linear operators Bi, S 2 : P(JB) -* V such that
x e V(B),

y E ^ , A; = 1,2.

Similar to Proposition 3.5, we can extend operator B (or Si, or S2) to the entire space V. Proposition 3.7. Suppose that Condition (H3) is satisfied. Then operator B can be uniquely extended to a completely continuous linear operator B : V(B) -> V, with V(B) = V, such that

|[Bx,tf] v |<M 1 |x|Jtf|J r -*||tf||i r , If Condition (H'3) holds, similar result holds.

Vs,y€V.

(3.43)

Chapter 3

86

Proof: Similar to Proposition 3.5, operator B can be uniquely extended to the entire space V, with the Inequality (3.43). It suffices to verify the complete continuity of the operator. According to Inequality (3.43), we have |£*y|v=

sup

|[x,yy]

I^Afilyl^Hyllir,

VyeV.

(3.44)

For any bounded set G C V, it follows from the compactness of the embed­ ding of V into H that G contains a convergent sequence {yj} inH. Accord­ ing to (3.44), the sequence {B*yj} is also convergent in V, implying that B*G C V is a relatively compact set. Hence, operator Z3* is compact, and hence completely continuous, so is B. If Condition (H^) holds, then it can be shown in the same way that similar result holds. \\ Instead of operator Equation (3.40), we now consider Au + Bu = Kf,

feV,

(3.45)

or equivalently, [Au + Bu,v]v

= [Kf,v]v,

V veV.

(3.45')

We note that its corresponding equations of projection approximations are

nn(*4 + B)un = nnKf,

/ e v,

(3.46)

or [Aun + Bun, vn] v = [Rf, vn]v ,

V vn € Vn , n = 1,2, • • • .

(3.46')

A direct application of Theorem 3.2 yields the following result. Theorem 3.7. Suppose that Conditions {Hi), (H2), and (H3) (or {H^}) are all satisfied, and that M{A + B) = {0} (or —1 is not an eigenvalue of the operator A~lB). Let Tn = {Vn,n n } be an orthogonal projection approximation scheme. Then for any f € H, Equation (3.45) is uniquely and strongly approximate-solvable under the scheme Tn.

Operator Equations (2)

87

3.2.4 K-positive definite operator equation in a Hilbert space setting As usual, let H be a Hilbert space with inner product (-,-)„ and induced norm || • ||#, and let T : H —» H be a linear operator with V(T) = H. Definition 3.2. Operator T is said to be K-positive definite, if there exists a closable linear operator K : H -> H, with V(K) D £>(T) and KV(T) = H such that (Tx,Kx)H > c i | | x | & , (Tx,Kx)H > c2\\Kx\\2H,

VxeD(T), V x G D(T),

where ci and c2 are positive constants. Operator T is said to be if (Tx,Ky)H = (Kx,Ty)H , Vx ) 2 /G 2>(T).

(3.47') (3.47") K-symmetric, (3.48)

Now, we extend operator T following the following procedure: 1. V(T) completion. Under the conditions that T is If-positive definite and if-symmetric, (Tx,Ky)H can be used as a new inner product in V(T), denoted by [x,y]HQ :=(Tx,Ky)H,

Vx,yeV(T),

with the induced norm | • | H o , where HQ = V{T)

(under inner product [•, -)HQ) .

Then by taking a limit on Inequality (3.47') we can show that \\x\\H
Vxetfo-

(3.49)

Since H = V(T) under the metric defined by (•, -) H , we also have V(T) CH0CH.

(3.50)

2. Extension of operator K. Under the condition that K : V(K) —» H is a closable linear operator, there is a minimal closed extension K. It then follows from (3.47;/) that \\Kx\\H
VXGP(T),

88

Chapter 3 namely, K\V(T) is a densely defined bounded linear operator from V{T) C HQ to H, and so can be uniquely and continuously extended to be a bounded linear operator, denoted KQ, from the entire set Ho to H, with \\Kox\\H < c^'2\x\Ha

,

V x e HQ .

(3.51)

Obviously, this extension satisfies K0CK,

(3.52)

in the sense that V(K0) C V(K) and K0x = Kx for all x e V{K0). 3. Extension of operator T. Now, we assert that operator T can be uniquely extended to be a closed linear operator from HQ to H, denoted T 0 , with T>(T0) C H0, H{To) = H, and having a bounded inverse T 0 _1 that is Ko-positive definite and Ko-symmetric, namely, (To*, Kox)H = \x\2Ho > Cl\\x\\2H , 2

(T0x,K0x)H

> C2\\K0x\\ H,

(T 0 z, K0y)H

= (Kox,T0y)H

V x e V{To), V*€P(To),

,

(3.53') (3.53")

V x,y € P ( T 0 ) .

(3.54)

In the following, we further explain this extension procedure and verify the aforementioned properties of the extended operator. First, for any y e H, it follows from (3.51) that (K0x, y)H is a bounded linear functional on Ho'. \{y,Kox)H\<\\y\\H\\Kox\\H


V x € H0 .

Then, by the Riesz Representation Theorem, there exists a bounded linear operator G : H —► H0 such that (y,K0x)H

= [Gyyx]Ho,

Vxeffo,

y€H.

(3.55)

It can be shown that Af{G) = {0}, so that G~l exists. Indeed, if y e H and Gy = 0, then it follows from (3.55) that (y,K0x)H=0,

VXGP(T).

Since KT>(T) = # , this implies that <
V.Gff.

Operator Equations (2)

89

Next, let T 0 = 6 T 1 . Then X>(T0) C H0 and K(T0) = H. Since G is a bounded linear operator defined on H, it is closed, and so is T 0 . We need to show that T C To. For this purpose, take any xo G V(T) and denote yo = TXQ. It follows from (3.55) and the definition of [•, -)H that [Gyo, AH0 = (yo, K0x)H

= (Tx 0 , Kx)H = [x0, x]Ho ,

V x € X>(T).

Since P(T) = #0 under [•, •] , we know that the above equation holds also for all x e Ho. Thus, we obtain Gyo = xo, i.e., yo = Toxo, implying that TcT0. Finally, we show that To is Ko-positive definite and Jfo-symmetric. In­ deed, it follows from (3.55) that
= [GTox, x]Ho =

\X\2HQ

,

Vx e

V(T0),

which is (3.53'). It then follows from this equation and (3.51) that (T0x,K0x)H

= \x\lo > C2\\K0x\\2H ,

VXGD(TO),

which is (3.53"). Moreover, it follows from (3.55) that (T0x, K0y)H

= [GT0x, y]Ho = [x, y]Ho ,

V x, y e T>(T0),

implying that To is Ko-symmetric. We need moreover to show that the above operator-extension is unique. Let T\ be another extension that satisfies the same properties. Then for any xo € T>(To) there exists a sequence {x^} C 2>(T) such that Xj —> xo within Ho as j —> oo. Since Ko is continuous, for any y e P ( T ) , we also have (T1xj1K0y)H

= (Txj,Koy)H = (K0Xj,Ty)H -> {K0x0, Ty)H = (T 0 x 0 , K0y)H

(j -* oo).

Observe, furthermore, that KV(T) = H. Hence, we have Tix^ w—* Toxo in H as j —► oo. Since Tf 1 is a continuous linear operator, it is weak continuous, so that Xj w-+ T1_1Toxo as j —> oo. Thus, we have xo = Tf 1 Toxo and so TiX0 = T 0 x 0 , implying that Ti = T 0 . Such an extension, To, is usually called the generalized Friedrichs solvable extension of T. It follows from the basic properties of the extended operator To that for any / € H the operator equation Tou = / always has a unique solution u = Gf G Ho, where operator G was defined in (3.55) above. Based on the above development of the general theory, we are now in a position to discuss the solvability and approximate solutions of a class of unbounded operator equations defined by if-positive definite operators.

Chapter 3

90 Consider the operator equation Au = f,

feH,

(3.56)

and its perturbed equation Au + Bu = f,

f€H,

(3.57)

where A and B are both densely defined linear operators from H to itself, with V(A) = V(T) C V{B). We have the following assumption on operator A: there exist constants r/i > 0 and r/2 > 0 such that | {Ax, Kx)H | > m (Tx, Kx)H, V x € V(A), (3.58') | {Ax,Ky)H\ < rniTx^KxYJ^Ty^Ky)]!2, V x,y € X>(A). (3.58") Proposition 3.8. Let T be a K-positive definite and K-symmetric linear operator and To be its generalized Friedrichs solvable extension. Assume that the linear operator A : T)(A) -+ H satisfies Conditions (3.58') and (3.58"). Then operator W = T$1A, considered as a linear operator from V(A) C Ho to HQ, has a minimal closed extension W, with T>(W) = TZ(W) — Ho, and \WX\HQ



| [Wx, x]Ho | > 771 \x\

2 Ho

Vxetfo,

(3.59')

,

(3.59")

V . 6 4

where [Wx, y]Ho is the unique continuous extension of (Ax, Ky)H on Ho x Ho. Proof: First, it follows from (3.58") that \Wx\Ho =

sup

| [To"1 Ar, y]


\=

H

yeH0,\y\Ho
«

sup

| (Ax, K0y)H \

yeH0,\y\Ho
VxeV(A),

namely, W is a bounded linear operator from a dense subspace V(W) = V(A) of Ho to Ho. Hence, W can be extended to be a bounded linear operator defined on the entire Ho, and this extension is norm-preserving. The extended operator, denoted W, satisfies Condition (3.59'), with V(W) = H0. On the other hand, it follows from (3.58') that

\[Wx,x]Ho\>m\x\lo,

VxeP(i).

Then Inequality (3.59") follows from this inequality and a limiting process. Since W : HQ —► HQ is a bounded and definite linear operator, it follows from

Operator Equations (2)

91

Theorem 3.3 t h a t TZ(W) = Ho- T h e rest of t h e assertation can be easily verified. [] Instead of Equation (3.56), we now consider operator equation Wu = g,

ueH0,

(3.60)

where g = T Q - 1 / e HO. Obviously, any solution of Equation (3.56) is also a solution of Equation (3.60). yet the converse may not be true in general. We call the solution of Equation (3.60) a generalized solution of Equation (3.56). To find approximate solutions of Equation (3.60), we put it into the framework where the Hilbert space H is separable, so that it is possible to construct an orthogonal projective approximation scheme Tn = {V^,IT n }, where {Vn} is a sequence of finite-dimensional subspaces of HQ, satisfying KcK+i,

n=l,2,..-,

with

\J™=1Vn = H0,

and {II n } are orthogonal projection operators from Ho to Vn, n = 1,2, • • • Let Wn = UnW\v . Then the projection approximate equations are given by Wnun = Ung, u n € Vn , (3.61) or equivalently, \Wuny

Vn]

^ = [
V Vn € Vn .

(3.610

If, furthermore, we choose Vn C 2?(T), n = 1,2, • • •, then we rewrite (3.61') as
Chapter 3

92

Galerkin method; where if K ^ I then it is the generalized Ritz method. Also, when A = K = T, HQ has the inner product (Ax,Ax)H, and the scheme provides a least-squares method. Since operators T and K have such flexible choices, the above approxi­ mation scheme covers a very wide range of applications. For instance, it can be used for odd-order differential equations. Besides, using this scheme ap­ propriately can provide better convergence result than the Galerkin method. To support the validity of this scheme theoretically, we show the follow­ ing result, which follows immediately from Theorem 3.3: Theorem 3.8. Let T be a K-positive definite and K-symmetric linear operator, and A : T>(A) —► H satisfy Conditions (3.58') and (3.58"), where T>(A) = V(T). If To is the generalized Friedrichs solvable extension ofT and W is the minimal closed extension ofW = T^A, then Equation (3.60) is uniquely and strongly approximate-solvable under the orthogonal protective approximation scheme Tn = {Vn,Hn}, and its approximate solutions {un} satisfy \un-u\H

<—

inf

\u-vL

.

(3.62)

To provide ff-estimates for the approximate solutions {un}, we can con­ sider the adjoint problem of Equation (3.60'): find a tp G HQ such that \Wv,
h]Ho,

V v € Ho,

(3.63)

where heV(K). Observe that W, and hence W*, is a definite bounded linear operator. Therefore, Equation (3.63) is uniquely and strongly approximatesolvable under the scheme r n , with

k n - ¥ > U < ^ jrf I?-*!„„,

(3.64)

where tp is the unique solution of Equation (3.63), and {(pn} are approximate solutions of the equation under scheme Tn. Proposition 3.9. Suppose that all the conditions stated in Theorem 3.8 are satisfied. Then the approximate solution un discussed above has the following H-estimate: \\un - u\\H < r}2\un - u\]H

o

\\\H

(3.65)

Operator Equations (2) Proof: Since KV(K)

93

= H, we have

\\un-u\\H=

SUp K^n-^feOgl h'eH,h'^0 11^1 Iff sup

—ir^fcM

heV(K),h^o

Letting i; = un — u in (3.63) gives \W(un-u),(p]

^3-66^

•

ll^^llff

= (un-u,Kh)H

.

Then, it follows from (3.60') and (3.61') that \W(un-u),Yln
=0.

Combining this equation and the last one yields (un - w, Kh)H = \W(un -u),ip-

Un(p]

.

This, together with (3.66) and (3.59"), yields the expected result (3.65). Q Finally, in this section, we discuss the perturbed operator Equation (3.57), in which we have the following assumptions on operator B: There exists a constant rjs > 0 such that | (Bx, Ky)H|

< V3(Tx, Kx)]>2{Ty, Ky)lJ2 ,

V x,y € V{A).

(3.67)

We first observe that the operator S = T^B is bounded on V(A) in the Ho-norm: \SX\HQ <7/3|*|„o,

Vx€P(i).

Hence, S can be continuously extended, with the norm-preserved, to a bounded operator S on the entire space HQ. Consequently, (Bx, Ky)H can be uniquely and continuously extended to the linear functional [Sx,y]H on Ho x Ho. Instead of Equation (3.57), we now consider the perturbed operator equation Wu + Su = g, ueH0, (3.68) where g = T 0 - 1 / € Ho. Its approximate equation is Wnun + Snun = Ilng, where Wn = IinW\v \Wun, vn]

and Sn =HnS\y + [Sun, v n ]

-"0

uneVn,

, which is equivalent to

= [(?, v«] -"0

(3.69)

V v B € VB ■

(3.69')

V vn € K ,

(3.69")

" 0

If, furthermore, Vn C £*(-A), then it becomes (Aun, Kvn)„ + {Bun, Kvn)H n=l,2,....

= (/, Kvn)H

,

Chapter 3

94 The next result then follows from Theorem 3.2.

Theorem 3.9. Let T be a K-positive definite and K-symmetric linear operator, with a generalized Friedrichs solvable-extension To. Assume that A : V{A) -► H satisfies Conditions (3.58') and (3.58"), with V{A) = V{T). Also, assume that B : T>{B) —► H is a linear operator satisfying Condi­ tion (3.67), where V(B) D T>(A). Let W and S be the closed extension^ of operators W = T$lA and S = TCi~1B, respectively and assume that S is completely continuous, with Af(W + 5) = {0}. Then the perturbed opera­ tor Equation (3.68) is uniquely and strongly approximate-solvable under the orthogonal projective approximation scheme Fn = {Vn, Hn}. 3.3 Stability of Approximation Schemes In this section, we furthermore discuss the numerical stability of the projec­ tive approximation scheme that was introduced for solving the bounded and unbounded operator equations above. To be more general, we consider two reflexive and separable Banach spaces X and F , together with the linear operator A : X —► Y. Let Tn = {X n , P n ; Yn,Qn} be the projective approximation scheme defined in Section 3.1.1 that satisfies the properties (i)—(iv) stated therein. Recall that the projection approximate solutions of the operator equation Au = f,

feY,

(3.70)

satisfy Anun = Qnf,

feY,

(3.71)

where An = QnA\x . The numerical stability of the approximate solution problem is referred to how closed are the approximate solutions of Equation (3.71) and that of its perturbed equation (An + En)un

= Qnf + hn,

(3.72)

where En : Xn —> Yn is the linear perturbed operator of An, and hn 6 Yn is the perturbation of Qnf, n= 1,2, • • •. More precisely, we have the following concept. Definition 3.3. Suppose that for any / € Y there is an integer N > 0 such that the operator Equation (3.71) is uniquely solvable for all n > N. The projective approximation scheme Tn is said to be stable, if there are nonnegative constants a and (3 and a positive constant 6, independent of / and n, such that for any hn € Yn and any En, with ||25 n || < 6 for all

Operator Equations (2)

95

n> JV, the perturbed operator Equation (3.72) always has a unique solution un e Xn, satisfying I K - un\\x

< a \\En\\ • \\un\\x + 0\\hn\\x ,

V n > N.

(3.73)

We have seen in the last section, from the discussion on approximate solvability, that a typical approach to solving a unbounded and densely de­ fined linear operator equation is to reformulate it as a bounded linear oper­ ator equation defined on a certain new space, where the technique used was to extend certain operators associated with certain new subspaces. For this reason, we only discuss a bounded linear operator A in the following. Theorem 3.10. Let X and Y be reflexive and separable Banach spaces, with a bounded linear operator A : X —► Y. Let Fn be a projective approx­ imation scheme satisfying properties (i)-(iv) stated in Section 3.1.1. Then this approximation scheme is stable if and only if (a) the operator Equation (3.70) is uniquely and strongly approximatesolvable under Tn; or (b) there exist an integer N > 0 and a constant v > 0 such that \\Anx\\Y>is\\x\\x,

VxGln,

n>N.

(3.74)

Proof: The equivalence of parts (a) and (b) follows from Theorem 3.1. Hence, we only prove part (a) using (b). Sufficiency. Suppose that Condition (3.74) is satisfied. Then for any / G Y and n > N, Equation (3.71) has a unique solution un € Xn. Let 0 < 6 < v. Then when \\En\\ < 6 we have \\(An + En)x\\Y>(v-6)\\x\\x,

Vxeln,

n>N.

Hence, for any hn € Yn, where n> N, Equation (3.72) has a unique solution un € Xn, and

IKAn + E n ) - 1 ! ! ^ ^ ,

Vn>N.

Consequently, it follows from (3.71) and (3.72) that un-un

= (An -f En)-1^

-

Enun),

and

||fin-«„IU<-^( 11^.11-IKIU + IKIIy). V — 0

This implies that the scheme Fn is stable.

96

Chapter 3

Necessity. Suppose that the projective approximation scheme Tn is sta­ ble. Let En = 0. Then for any / € Y and hn € Yn, when n> N Equations (3.71) and (3.72) have a unique solution un,un € Xn, respectively. Let vn = un — un. Then we have Anvn = hn ,

n>

N.

It then follows from this equation and (3.73) that \\Vn\\X
Since An : Xn —► Yn is one-to-one, Condition (3.74) must be satisfied. 3.4 Numerical Solutions of Boundary Value Problems In this section, we discuss numerical solutions for boundary value problems, for both ordinary differential equations and partial differential equations. This will demonstrate applications of the projective approximation solution method developed in this chapter to linear operator equations. 3.4.1 Ordinary differential equations In this subsection, we show two examples of numerical solutions for bound­ ary value problems of ordinary differential equations, so as to illustrate a di­ rect application of the projective approximation solution method developed above. Example 3.1.

Consider the boundary value problem -u" + a1u' + a2u = f, ii(0)=ti,(l)=0,

0 < t < 1,

where ak(t) e L ^ O , 1), k = 1,2, and / <E L 2 (0,1). Let H = L 2 (0,1) and

V={v

| veH^l),

v(0) = 0 } .

Define two operators A:V{A)-*H, B:V{B) -^H, where

Au = -u"", Bu = axu' + a2u,

V{A) = C 2 [0,1] n {t, | v(0) = v'{l) = 0} c V, V{B) = C%

1] n {v I v(0) = v ; (l) = 0} C V .

(3.75) (3.76)

[]

Operator Equations (2)

97

Obviously, V{A) C V{B), and the embedding of V into H is compact. Introduce some inner products and norms as follows:

<**.-/>*"*'■ and

(

jj

H = <*,<> 1 1/2 2

IMb = |X>l [

•

Recall the Hardy inequality [18], which states that if y(t) is absolutely continuous on [0,1], with y(0) = 0 (or y(l) = 0), then

/W**£j[W*Hence, in space V the norm | • |i and || • ||i are equivalentt We now rewrite the boundary value problem (3.75) and (3.76) as the following equivalent operator equation: Au + Au + Bu Bu == f,f,

(3.77)

/f €€LL2 (2 (0,1). 0,1).

Using the above-defined inner products and norms, the problem under consideration has a weaker form (u,v)1+(a u' +a2u,v) <«,«)1 + 1
VtigV. V»€V.

(3.78)

Here, we should observe that (u,v}1 is a continuous extension of {Au,v)0 via integral by parts, and that this extension is unique o n K x V . Hence, Equation (3.78) is actually the same as Equation (3.45'). To solve the problem, we choose a sequence {V^} of subspaces from V such that Vn C Vn+l and u ~ tVn = V. Then the corresponding projecttve approximation equation is (u (<*!< + n,vn)00 ( « nnnv , «nn)1) 1 + + <<*!< + a2u a2Mn,fn)

= n )0 , , = (j,v «,

V vW GVV„ n„ G n ..

(3.79)

It is clear that we have |{Ax,y) |J|<|x|i|w|i, < x , y ) J | < |1x\y\| i1|,w | i , 0\ 00|| = | (Ar,x) (Ac,x) (Ax,x)00|\|

= | (x.x), |x|? , (x,x)t | -= \x\\

V x , yVx,y€V(A), €P(i4), V xx G PP (( 4 ) , V

||{Bx,y) (Bx,y)00\ \ = |(a | {alx' \ < M|s|i\y\o, ix' + a22x,y)00\<M|s|i\y\o,

V Vx,y B), xi ,, yy € P (V{B),

Chapter 3

98

namely, operators A and B satisfy the Conditions (Hi), (H2) and (#3) stated in (3.28), (3.29) and (3.41), respectively. Thus, it follows from Theorem 3.7 that if the corresponding homogeneous boundary value problem (with / = 0) of problem (3.75)-(3.76) does not have a nontrivial weak solution, then for any / € £2(0,1), the original problem (3.75)—(3.76) is uniquely strongly approximate-solvable under the projection approximation scheme T n = {V n ,II n }, namely, the projective approximate Equation (3.79) has a unique solution un € Vn, and {un} converges in norm | • |i to u € V, where u is the generalized (weak) solution of the original problem. If, however, u 6 V has a one-degree higher smoothness, then this limit u is indeed the classical solution of the problem. Example 3.2.

Consider the boundary value problem - u,n + aiu" + a2u' + a3u = f, u(0) = u'(0) = u'(l) = 0,

0 < t < 1,

(3.80) (3.81)

where ak(t) € too(0,1), k = 1,2,3, and / € L 2 (0,1). We use the framework established in Subsection 3.2.4. Let H = L 2 (0,1) and define operators A, T ( = A), B, and K as follows: A:V(A)^H, B : V(B) -+ H, K:V(K)->H,

Au=-v!", Bu = am" + a2u' + a 3 w, Ku = u',

where 2>(A) = {v I t; G # 3 ( 0 , 1 ) , v(0) = v'(0) = t/(l) = 0} , £>(£) = {v I t; € # 2 ( 0 , 1 ) , v(0) = v'(0) = v'(l) = 0} , P ( K ) = {v I v € ff1^, 1), v (0) = 0} . It is clear that KV(T) = H under the Z/2-norm, K is a closed linear operator in £T, and T is both K-positive definite and If-symmetric, namely, (Tx, Ky)0 = (-*'", y \ = (x", y")0, ^ |s|? > ^ |x|g,

V x, y € 2>(T), V x, y € P ( T ) .

Now, taking the closure of V(T) with respect to the norm \X\Q, = (Tx, we obtain a Hilbert space H0 = {v I v € H 2 (0,1), «(0) = «'(0) = v'(l) = 0} .

Kx}^2,

nnerntnr Emmtinr,* Operator Equations

f9.) (2)

99 y»

In this case, the bilinear form (x,y> {x,y)22 = = /" f Jo

x"y"dt

is the continuous extension of the inner product (Ax,Ky) (Ax, Ky) 0= 0 = f Jo

-x'"y'dt

from V{A) V{A) ) x V(A) V{A) to Ho H0 x H0, and is corresponding to tl the inner product [Wx,y]Ho \Wx,y) that we discussed in Subsection 3.2.4, i.e., Ho {Wx,y} \Wx,y]Ho Ho = (x,y)2,

Vx,yeH0.

The Che bilinear form associated with operator B needs not be extended extended, that is, we have [Sx,y} = (Sx, (Bx,Ky) = ({a + aa3x,y') [Sx, Ky) = a lX j x"" + a2x' + x, y')o Ho = a 0,) , , px,y\y] + H Ho — {&x,n.y)0 0 = \<*ix +a2x 3x,y 0

H0 .•• V v x,y x,y € € #o ilo

Thus, the problem under consideration, (3.80)-(3.81), can be written A h p ffollowing n l l n w i n o - r>r>f>rftt.r>r Pnnnt.innasS tthe onerator eauation: operator equation:

Au + Bu = f,

/ €6 Ll 22 ( 0 , 1l ) .

(3.82)

ote that this problem eeneralized form as finding a u € g ffi, Note jblem has the generalized H. H00 such that _ [[Wu,v] W « , «Ho ] a+[Su,v] , t ; ] H o ==< / , i f(f,Kv) t ; > 0 ,0, o + [ 5 t t H(> or, more more precisely, precisely, cisely, or, v") + (oxti" (a +a a2u' + («", »")„ (ai«" + a3u, (f,v') lU" + (u", v")00 + u, v') v')00 = = ({f,v') / , v '00), 0 , 2u' + a3

V V vv € €H H00 ..

(3.83) (3.83)

To solve solve the the problem, problem, we choose aa sequence {Vnn}} of of subspaces subspaces from from H H00 To we choose sequence {V V = H such that Vn C Vn+i and U~=1 F ff Then the corresponding projective n 0 =1 approximation equation is to find un € Vn such that + a3W„, a3un, v' V »vn € V„ « , < ) 0 + { « ! < + a*2u' 2 K . n)0 ,

(3.84)

Since ,4 = T, it can be easily verified that operator A satisfies the Conditions (3.58') and (3.58") stated in Theorem 3.9. On the other hand, it follows from the assumption ak{t) € 1^(0,1), k = 1,2,3, that there exists a constant c> 0 such that I| [Sx, y]HoHo\ | = {alX" + a2x' + a33x,y') x, y') c\x\22\y\i, [Sx,y] = \| {atx" ||/|i, 0\ 0 \ < c|z|

V x,y € H0 ,

100

Chapter 3

so that | 5 * l y2
V jy, e Jf ?l o -

l1 L ogether with H (0,1) Hence, ff^O.1) Hence, together together with the the fact fact that that the the embedding embedding of of H H00 into into H H {U,1) is is compact, , this implies that the operator S* is completely continuous, continuous compact, this implies that the operator S* is completely continuous, and and so so is is 5 5 .. To from Theorem that n if corresponding hoTo this this end, end, it it follows follows Theorem 3.9 if the the corresponding hofouows from Theorem 3.9 -i.9 that m™n>r.a™ic KA,inHorv ,ral„0 ™ ™ k l o m fwitVi f— (\\ nf r.rr.hloTn 9A\ Am*, tint mogeneous boundary value problem (with / = 0) of problem (3.83) does mogeneous boundary value problem (with / = 0) of problem (% (3.83) does not not have nontrivial solution, then the original problem (3.80)-(3.81) is uniquely have nontrivial solution, then the original problem (3.80)-(3.81) is uniquely strongly strongly approximate-solvable approximate-solvable under under the the projective projective approximation approximation scheme scheme T . In other words, the approximate Equation (3.84) n T n . In other words, the approximate Equation (3.84) has has a a unique unique solution solution €Vn, and the sequence {un} of approximate solutions converges tto un €V„, o the unique solution u € H0 of of Equation Equation (3.83), (3.83), which which is is the the generalized generalized solution solution of of problem (3.80)-(3.81). If, furthermore, u has one-degree higher smoothness, then this solution, w, is also the classical solution of the problem. Indeed, in this case, it follows from (3.83) that for u € # 3tf(30(0,1), ,1), I

III

.

II

I

l\

/ #•

/\

{-v.'" aiu" -ta22u' u' ++-t =(/,v') (/,«')„ , (-u'" am" ++a aa3ce u, v')00 = 3u,v') 0, \—u + -|-+<x\u OC2U 3u, v ; 0 = \j ,v ; 0 ,

\ /

TT

Hn00o. . VVV vv€v€€ H

Since KV{T) KV(T) = = H, the above equation holds also for any v v6H,so 6 H, so that -u'" -v!" u' = - « ' " + ai a u" iu" «"+ + aao222u'+ «' + + aa33uu = = f, ai almost everywhere on (0,1). 3.4.2 Partial differential equations equations 3.4.2 Partial differential examDles nrohlems We of boundary We show show in m this this subsection subsection two two examples examples of boundary value value problems problems for for linlinear elliptic partial differential equations, and discuss how the general ear elliptic partial differential equations, and discuss how the general theory theory and developed in and 3.2 be applied and methods methods developed in Sections Sections 3.1 3.1 and 3.2 can can be applied to tonumerical numerical solutions of partial differential equations. solutions of partial differential equations. Example 3.3. Consider the following boundary value problem of a secondorder linear elliptic equation:

Au Au=-^2

x)tt=/ = -- J2 ^—( alaijix)-^-) ^ x ) ^ ) + c ( +c{x)u ( x )f(x) >

«\ = 0, «|dQ 8 n = °>

y

x €xn€

> ft,(3(3.85 -85) (3-86) (3-86)

where ft Q C Rn is a bounded region, n > > 2, with boundary dQ,. dSl. In this equation, all the coefficients aay(x) ( = a^x)) and their partial derrvattves tj(x) are bounded and continuous real functions, c(x) is a nonnegative, bounded, and measurable real function, and f(x) f{x) G L2(£l). L2(^).

Operator Equations (2)

101

Suppose, furthermore, that Equation (3.85) is uniformly elliptic on 17, namely, there exists a constant v > 0 such that for any constant vector n

n

a

J2 a (*)&&■ ^ * S &2 >

xen

( 3 - 87 )

•

Under the above condition, operator .A satisfies the following three prop­ erties: (Au,v)0 = (u,Av)0 = Yl / <*v>^—^—dx + / ^ ^ x , ^ ! in <*** <**j in V u, v € 2>(i4) = C 0 (n) n C 2 (Q),

(3.88)

where Co(Q) = {v\v € C(Q), suppv CC 0, v|a^ = 0}, and | {Au,v)0\ < M H i | v | i , (Au, u)0 > v\u\\,

V I*,v e V(A), V U G D(A),

(3.89) (3.90)

in which M > 0 is a constant. Now, let i7 = L2(n) and F = HQ(Q). Instead of studying the original problem (3.85)—(3.86), we may consider the following variational problem (a weak form): find & u eV such that 7^

/ aa~———dx + / cuvdx = / fvdx,

/TWn

J

dxidxj

JQ

Vv€V.

(3.91)

JQ

Solutions of this variational problem are called the generalized (weak) solu­ tions of the original problem (3.85)—(3.86). To solve the problem, we choose a sequence {Vm} of subspaces from V such that Vm C Vm-\-i and U™=1Vm = V. Since V = HQ(Q) is a separable Hilbert space, such a sequence of subspaces exists. Then the corresponding projective approximation equation is to find um e Vm such that /

/ oiii~n

^—dx-\- I CUmVm dx =

fv m

dx,

V Vm € V . (3.92) m

We apply the framework established in Subsection 3.2.3. Properties (3.89) and (3.90) imply that operator A satisfies the Conditions (H\) and (H2) stated in (3.28) and (3.29), respectively. Hence, it follows from Theorem 3.6 that for any / G 1/2(0), Equation (3.92) has a unique solution um € Vm, and the solution sequence {um} converges to a limit u in HQ(Q). In addition,

Chapter 3

102

this limit u € HQ(SI) is the unique solution of Equation (3.91), which is the weak solution of problem (3.85)-(3.86). Moreover, the approximation errors of the solution satisfy M \Um-U\l<

. _

,

,

ml V VrnGVn

\U-Vm\l.

We finally remark that if we induce an operator equation directly from the weak form of the original problem, then we can obtain an operator equa­ tion defined on the entire space V, so that the framework developed in The­ orem 3.3 can be applied. As a matter of fact, for any u € HQ(Q), the bilinear form

,

v^

x

du dv

f

f

cuv dx

is a bounded linear functional of v € HQ(Q). Thus, it follows from the Riesz Representation Theorem that there exists a unique Tu € HQ(Q) such that a(u,v) = (Tu,v), ,

Vv e

fl^(n).

(3.93)

Here, operator T is defined on the entire space HQ(Q) and is bounded and positive definite: V^Fo1^),

|Tt*|i <M\u\u (Tu,u)1

>v\u\l,

Vu€H£{n).

Then, by the Poincare inequality [18, 32] (see Appendix A.2.5), for any / € L 2 (fi), 0 is a bounded linear functional of v € H^(il). Similarly, the Riesz Representation Theorem implies that there is a unique TZf e H&(Q) such that l / » o = W^v),

,

V v € HfoSl) •

(3-94)

Consequently, the variational problem (3.91) can be represented by Tu = Kf,

feL2(Q).

To this end, Theorem 3.3 yields the claim.

(3.95)

Operator Equations (2)

103

Example 3.4. Consider the following Dirichlet problem of a system of general higher-order strong-elliptic equations: {Lu), := £ k=1

£

(-l)ND«(a«£(x)Z?V) = /,(*),

|a|,|/?|<m

x € l 7 , j = l,...,JV, dR

ti = 0,

/

(3.96)

i = 0,l,--,ro-l,

(3.97)

where a^f is a function defined on 17, and d/dn stands for the outer normal derivative along the boundary 517. Obviously, any system of 2mth-order equations N

(Lu)j = J2

dlk(x)Dauk

E

k=l

= fj(x),

xeQ,

j = l,..-,AT,

|a|<2m

with smooth enough coefficients (e.g., a?fc € C'QI(17)), can be written in the form of Equation (3.96). Suppose that 17 is a bounded region and Equation (3.96) is uniformly strong-elliptic on 17, namely, its coefficients a?]f (x) is a bounded, measurable, and complex-valued function of x € 17, and there exists a constant v > 0 such that for any real vector £ = (£i, • • • ,£ n ) and any complex vector rj =

Re

{E

E

e^ernnu} > v ierM2

almost everywhere on 17. In applications, many important elliptic equations and systems of equa­ tions are strong-elliptic, such as multi-harmonic equations, the bending equa­ tion of a thin plate, and the Blasov equation of an oblate cell [11, 27, 36], etc. Hence, the technique discussed above is very useful. To close this section, we mention that the classical Dirichlet problem is stated as follows: find a vector function u(x) = (wi(x), • • •, UN(X)), with uj € C2m(17) H C m_1 (17), such that u{x) satisfies Equations (3.96)-(3.97). Correspondingly, its generalized form is to find & u e H^(Q) such that

E j7k=l

E |a|,|0|<m

(«$&**>

0*^)0

= (Mo,

Vf;€fl2T(n).

(3.98)

104

Chapter 3

Here, we note that the vector function u G ilj'(^), namely, each of its components .mponents belongs to flj*(n), Now, to study the solvabUity suppose solvability of the generalized problem, we suppo aa*f(x)€C(n), CC((fni )) ,, #« (f ( *x)) €€eCfJS), n . a ,

«££(*)€L00(),, N

T

,~^

|a| = |/3\ == m, m, i

i

i ~i

-

i

H , |/?|<m, |a|,

I .

I ^i

^ ^

|a| + + |/3\ < 2m - 1,

and nd apply ly the Carding inequality: there exist a constant cx > 0 (depei (depending only nly on (i) the modulus of continuity of the highest-order term aaQff (x) \where \a\ |/?| = \f3\ == m, and and (ii) (ii) the the (uniform) (uniform) strong-elliptic strong-elliptic constant constant v> v> 0)0) and = m, ?andaa (\ = ILXI —— I KJ I —~* »!€/« CbllVl I 11 I V/X1V/ V U.H1J.WX 111 1 O U l v / l l C V l l l l / U l v ViV/llOUClllV IS ^^ ' - ' 7 t W l V l Vb constant mstant c2 (depending on the maximum modulus of the coefficients), coefficients), such that lat Re{ Cl fl^(n). 2\\u\\l, WjW WjW Re{{Lu,u} |H|^ calMlS, fl^(n). W When1 C2 C2 < < 0, 0, operator operator L L is is definite, definite, so so that that Theorem Theorem 3.6 3.6 can can be be applied applied A , m the. u n i n i i A aand n r l ostrong f r r t n a approximate » n n m v i m ^ o ssolvability r . W a h i l i h i r nf h e protective r.rrti«vtivo to deduce the unique of tthe approximation scheme. In the general case, we let A = L + c2I

and

B = -c2I,

and so L = A + B. B. ItIt can can be be easily easily seen seen that that K iA 4utut,,t«ti> | |>>CCCl |llH| ||t2C, , (Au,v)v)00\\0\ < cc33\\u\\ II (Au, 3 N mmU\\v\\ MmmU ,, (Au,v) _. _.\ I ^ _ 11..11 11 11 (B«,t;>J < C4ll«llol||;||n. II {Bu,v) \ < < C4|H|o||v||o, C4||u||o|||;||o, (Bu,v) 00\

\/u€H^(Q), Vu€H?(Sl), Vu€H?(n), flj*(n), V H^(Q), V u,v u,v G G H^(Q), ^ TTfntrw m m V V u,v u,vM € F/€0Ttf„ ( n)) (m V € O ,, ,

w

where C3 and C4 are constants. Therefore, according to Theorem 3.7, if the corresponding homogeneous boundary value problem (with / = 0) of Equations (3.96)-(3.97) does not have a nontrivial solution, then for any / € G L2{Q) tne original problem (3.96)-(3.97) has a unique generalized solution u € jffJ*(Q), which can be approximated by a sequence of projective solutions.

Operator Equations (2)

105 Exercises

3.1. Prove the following Babuska Theorem: Let U and V be real Hilbert spaces, with inner products [•, -]v and [•, -] v and induced norms | • \u and | • |v, respectively. Let a(«, •) be a bilinear form on U x V, satisfying the following two conditions: (i) boundedness: there exists a constant M > 0 such that \a(u, v)\ < M\u\u\v\v

,

V ueU,

v eV;

(ii) weak coerciveness: there exists a constant a > 0 such that inf lultf

sup \a(u, v)\ > a

= 1

\v\v = l

and sup \a(u, v) | > 0,

V v € V\{0} .

Then for any / € V*, the dual space of V, there is a unique u € U such that a(u,v) = / ( v ) ,

VVGF,

and

Ha <-ll/llv.. a 3.2. Let U and V be real Hilbert spaces, and {Un} and {Vn} be sequences of their finite-dimensional subspaces, respectively, satisfying Un C Un+i, iioo

IT

_

Vn C Vn+i,

FT

MOO

y

n = 1,2, • • • ,

_ Y

Suppose that a ( , •) is a bilinear form on U x V, satisfying the following conditions: (i) there exists a constant M > 0 such that \a{u,v)\ < M\u\u\v\v

,

VueU,

veV]

(ii) there exists a constant a > 0 such that inf

sup \a{un,vn)\

>a.

Recall that the generalized Galerkin method for solving the equation a(u,v) = f(y),

VveV,

(a)

Chapter 3

106 is to find a un € Un such that a(t*n, vn) = f(vn),

V vn e V n -

(6)

Prove that both Equations (a) and (6) have unique solutions u and un, with the error estimates \u - un\u < ( 1 + — ) Jnf \u - u\u , y a J ueun for all n = 1,2, • •. 3.3. Let V be a separable real Hilbert space, and suppose that a(-, •) and /(•) are bilinear and linear forms defined respectively o n F x F and V. Assume that the equation a(w, v) = f(v),

Vv € V

has a unique solution w 6 ^ . We construct its corresponding discrete problem as follows: find a Uh 6 Vh such that Q>h{uh,vh) = fh(vh),

Vvfc€Vfc,

where V^ is a finite-dimensional space, and ah(-, •) and /&(•) are bilinear and linear forms as approximations of a ( , •) and /(•), respectively, sat­ isfying the following condition: there exist constants M > 0 and a > 0, independent of V^, such that a>h(yh,vh) > a | K | | ^ ,

Vvfc€Vh,

M v , ti;)| < M||v|| f c |H| f c ,

Vv€V+Vh,weVh.

Here, || • ||fc is a norm defined on space Vh, called the discrete norm, which is also defined on space V. Show that there exists a constant C, independent of V&, such that \\u-uh\\h
sup ! / » ( » * ) - ^ ( » . " * ) i y whevh \\wh\\h )

3.4. Let V be a Banach space, and K be a nonempty, closed and convex subset of V. Suppose that a(-, •) : V x V —► # is a continuous, positive definite, and symmetric bilinear form, and / : V —► i? is a continuous linear form. Define a functional J :V ^ Rby J(v)=

2 ^ ' ^

- / H '

Operator Equations (2)

107

and consider the following minimization problem: find a u e K such that J(u)= inf J(v). J(v). (c) Prove that (i) problem (c) has a unique solution; (ii) u e K is a. solution of problem (c) if and only if u satisfies the variational inequality a(u,v-u)>f(v-u), a(u,v -u)>f(v-u),

V neif; VveK;

(d)

(iii) when K is a closed subspace then u is a solution of problem (c) if and only if u satisfies the variational equation a(u,v) = f(v), f(v),

MvzK. VveK.

(e)

3.5. Let H be a separable Hilbert space, with inner product (•, •) and norm || • ||. Let also V be a subspace of H such that V = H under || • ||. Suppose that with the inner product [•, •] and norm | • |, V is itself a Hilbert space. Assume that the embedding of V into H is continuous. Moreover, let a(-, •) : V x V -► R be a continuous, positive definite, and symmetric bilinear form, and / : V -► R be a continuous linear form. For a nonempty, closed and convex set K C V, consider the following variational inequality: find a u e K such that a(u,v-u) a{u,v-u)

>f(v-u), >f(v-u),

V v VveK, € K,

and its corresponding discrete problem: find a un e Kn such that a(u aK,n,vnv-u -un)>n-un),f(vn -un), n n)>f(v

Vvn<EK , n, Vvnn€K

where Kn is a nonempty, closed and convex subset of Vn and Vn is a finite-dimensional subspace of V, n = 1,2, •••. Introduce a mapping A : V -f V*, the dual space of V, by Vv€V; AveV*, Av € V* , \/veV; VweV. (Av) {w) = a{v,w), a{v,w), (Av){w) Vw€V. Show that under the Condition Au-feH, independent of Vn and Kn, such that | « - « „ | < c ( inf |«-«„|
there exists a constant C,

2 \\u[|«-t*.| -vnf \\Au-f\\.\\u-v \\\ + +P«-/||.||*.-« Bn||l 1/2

+ ||^-/||W|| | | ^ - / | | W | |WWnn-,||}1/2. -,||}

108

Chapter 3 (Note Sfote that many one-sided constrained and time-invariant mechanical and tid physical problems can introduce this kind of variational inequalities. This his problem is by nature nonlinear: since K is not a subspace, and so introduces troduces nonlinearity to the mapping / e V* -» u € K.) In the rest of the exercises, we assume that £2 C IT is an open region igion with a Lipschitz-continuous boundary dQ, and that ^ 2u d2u ^d

A A u ::: = AU U =E^2

£^J =3=1 E ^ 233 j=l j=l 3=1

3

is the standard Laplace operator. ator. The reader is €expected to show the equivalence between the original inal problem and its corresponding weak w form, and to discuss the unique strong approximate solvability of the protective approximation scheme developed in this chapter. Here, the equivalence between the original problem and its corresponding weak form means that the classical solution of the original problem is also its weak solution (where the latter is called a generalized solution of the original problem) and conversely if a generalized solution is smooth enough then it must also be a classical solution. 3.6. Consider the following nonhomogeneous Dirichlet problem:

jf - Au Au + qu == fo, f/o f00,,, in in Q ft, ft, fi,, u == uo on dQ, uo, wo on dQ, |1 -u ixo,,, on dQ,, dfl, where 9qa e€ L Loffl) (fl) I with SI. fn G e€ 1L2(fl), (ft), L22(fl) {Q) with 9qa >> 00 almost almost everywhere everywhere on on fi, Q, /„ (Q), (fi), and ft, /0 un0 €G iH^t). Tts weak form is: find a u?1 == uM0n++ uW1 7,, ^e VV = H&(Sl) tfl/W such si,,* wo /tfifftV w H^tl). ui G H^(Q) H$(Q) ^^( Q f l ) . Its u u x e that o(ulyuv) = f(v), f(v), V vVveV, e GV, V, o(ui,v) a(u v) =f(v), /(v), a(ui,v) = \fveV, where where

•

n

^

o(«i,t>) = / ( « ) , a(u.v) = I I >^ f(v)=

3.7. the following following :.7.. Consider Consider the f(v)— (

{(

^

VueV, \-auv\dx

/ fovdx -a(u0,v). ./n nonhomogeneous Neumarj proble Neumann problem: nonhomogeneous Neumann / fovdx - a(u0,v). 4- qu au = fo fc\,, Au + /o in Q,, fftft,, — Aw JQ.

8 ier the following8^nonhomogeneous problem: -£=9, = g, on 5Neumann 0, onnft, £=9, ( — Aw + <ju = /o , in f2,

I du

Operator Equations (2)

109

where q e L2(ft) with o ^), q > 0 almost everywhere on ft, /o € W L2(0)> # € L2(dft), and du/dv is the outer normal derivative of u along the boundary 5 0 (assuming that it is smooth). Its weak form is: find a ueV=H1(Q) such that a{u,v) = / ( v ) , a(u,v)

V,, Vv GV

where /* / ^ - ^ du dv

\

/(v) = f(v)=

ff00vdx+ vdx+ gvdv. gvdv. Jn JdQ 3.8. Consider the the following boundary value problem of a linear system of elastic equations: lations: (— (T——/xAu /xAu— —(A (A+H-fi) /x)/x)grad graddiv divuu==f,f,i n inin ^ft, ft, , u = 0, on 8Qi = 0, 0, on f t iily,, Iu u= oonncddft M u l y ] aao-iiiu)^ Qi, 'E ij( X ^ () u'j) ^ = = ^9i, '

oonndddft fdft t*i = , 2 , 3 on =l1,2,3 1,2,3 on ftt22.2.

This Phis system of equations describes the equilibrium state of an object ob ft1 under small variations of displacement and strain, where ft C C R3-, dft )ft = dftx dQ,i U dft2, mess(dfti) dfti meas(dfti) > 0, v = (^1,^2,^3) is the outer norinormal nal vector along dft, u = (ui,u2,u3) {ui,u2,u3) € [ i / 1 ^ ) ] 3 is the displacem (ui,U2,u3) displacement, 1 _ (f/1,/2,/3) r /xi , /x , / 3 f )\ € <- [L f r (r&)]3 / r ^ \ l 3 : „ 4-v,^ c A ^^:*-„ ~ „ 4-V,^ ^ u ; 4- body, u~J„ f = 2 22(r2)l3 is the force density on the object b< 3 g = (gi,g2,g3) € [L2(8ft2)] is the force density on the object surface, and moreover, , ,

1 fdm

duA

3 (( 3

X x -.

a^-(u) = A ( X>fcfc(u) ^2 £fc(u) (u)W j bij ++222/x£ij / x ^ ( (uu) ), , M^(^)> ^X>fc e f c *fc fc( u )W W + 2/iey(u),

< , i << 3, 11l1< << t»»,i , j 33, ,

are the ie strain and stress tensors, respectively, in which whic] b^ 6^ is the th Kroconsta neckerr delta, and A > 0 and \i > 0 are the Lame constants. Let ' = f v = (vuv2,v3)

e [#W| [HHil)]3 I Vvj=0ondn j=0 onujdQ,,

=

l,2,s},

ff o or r U v 6 and ||v|| = E j = i \^J\\H IMIjfi(n)) e V' V\ The form of I 1 V 1 all AiiV V i i x the IMIfP(n)) a - md — ii'ii — ( y£^j=i {Q)J " " T ^ ' • The ^ weak "weak ^ " - i V iform " vof "the — problem is: find a u e V such that

a(u,v) / ( v ^) , "V**? ; = - M ¥

f

Vv€F,

v ▼ vz r ,

110

Chapter 3 where a (UuV), v ) = / f

a( ' = / n £ '7^y ((un) eMy ( vv) t)f a* (A div(u)div(v) div(u)div(v) + 22M/ iEE e % « ( (uu) )eeyy((vv )) W = // (A W / ( vv)) == / / ]Svdx+ Fvv
r

jdx

3 j dv.

Hint: Use the Korn inequality

x>

/ 3 3

(

\ 1/2 1/2

//

3 3

1/2 \ 1/2

3 3

E i< )lH0(n)+EK^(O)) \ J -»-

£ vtf(v)|^«

i\\2wl

/

»*>j

V

J •*-

J.

I

where C is a constant, to show the following F-elliptic property of the bilinear form a(«, •): there exists a constant a > 0 such that a(v,v)>a||v||2,

VvGV.

3.9. Consider the following Dirichlet problem of a biharmonic equation:

{

(AA2 utx = f/ 0o, , du < u= <9u — = 0, V dv av

in i n fft, t, on9fi, on9n,

where /«, 6 L2(ft) and du/3i/ is the outer normal derivative of u along the boundary dQ (assuming that it is smooth). The weak form of the problem is: find a u € V = flg(n) such that a(w,v) = /(v),

Vv€F,

where a(u,v)=

f[

JQ

AuAvdx,

/(») = f fovdx. This kind of problems can often be found in fluid mechanics.

Operator Equations (2)

111

3.10. Consider the following plate bending problem with a fixed boundary. The original problem is the same as the above biharmonic Dirichlet problem, but its weak form is: find &u eV = HQ(Q) such that a(u,v) = / ( v ) ,

V v € V,

where Q € R2, and a(w, v) = /

| AuAv + (1 - a) d2u

d2?;

#^<9^

" dx\dx2 dx\dx2 =

/

|
■£[•' d u d v 2

2

Kdx\dx\

/('

dx\dx\

3*ti0V\"

dx

dx\dx\)\

-a)

d2u d2v dx\dx\

d2v d2u WJ dx\dx
where a is the Poisson coefficient, 0 < a < 1/2.

CHAPTER

4

Topological Degrees and Fixed Point Equations

The theory of topological degrees, or simply degrees, of operators is an impor­ tant mathematical tool in studying nonlinear problems and solving nonlinear operator equations. Fixed point theorems and the local topological degrees are closely re­ lated. Generally speaking, topological analysis only gives qualitative but not quantitative results about a problem. However, the topological degree the­ ory has an advantage over the classical fixed point theory in that it gives significant information about the number of distinct solutions, continuous families of solutions, and the stabilities of solutions, for a fixed point opera­ tor equation. The concept of a local topological degree in a finite-dimensional Euclidean space, namely, the degree with respect to a neighborhood of an isolated solution of the operator equation f(x) = y, can be traced back to Kroneker's investigation in 1869. Brouwer extended this local degree con­ cept to a degree in a global sense in 1912, and then used it as a basis to develop many important results on general fixed point theories and operator equations. Traditionally, topological degrees were defined for "mappings," and hence also called mapping degrees, which not only refer to operators but often are defined for set-valued maps. Since set-valued maps are not studied in this book, the name operator degrees instead of mapping degrees will be used throughout the rest of the book. In the following, the notion of topological degrees (the Brouwer degrees) of continuous operators in Euclidean spaces is first introduced, followed by

113

Chapter 4

114

the topological degrees (the Leray-Schauder degrees) of compact fields in Banach spaces, and then by the generalized topological degrees of the socalled A-proper operators. In this chapter, the focus is the study of the solvability problem of the operator equation

[ J - / ] ( * ) = <>,

xeX,

where X is generally a Banach space, I the identity operator, and / an oper­ ator to be specified. This is indeed a standard fixed point equation, usually written in the form x = / ( x ) , x e X. Several important fixed point theorems will be derived from the study of this general operator equation. Moreover, by the end of the chapter, several projection approximation schemes and the protective approximate solvability problem will be discussed for several different fixed point equations. 4.1 Topological Degrees of Continuous Operators in Euclidean Spaces In this section, we first introduce the concept of topological degrees for regular operators. Later, we will extend this concept to some operators that have more general properties. Throughout this section, we let Rn be the r^dimensional Euclidean space equipped with the norm || • ||. 4.1.1 Topological degrees of regular operators and their integral representations We first give some definitions. Definition 4.1. Let Q C Rn be a bounded open set. If an operator n / :fi, —» R is continuous on ft and continuously differentiate on fl, namely, / € C(Q) fl C ,1 (H), then / is said to be a regular operator from Q, to Rn. Definition 4.2. Let Q C Rn be a bounded open set and / € C^fi). If for any solution x e ft of the equation / ( x ) = o,

aeRn,

we have det [/'(#)] ^ 0, then this a e Rn is called a regular value of / on Q. A point x eft satisfying det [f'(x)] = 0 is called a critical point of / and the image f(x) of a critical point is called the corresponding critical value of / on Q,. Our purpose is to establish a measure, called degree, which depends topologically on / , Q and a, and provides some information about the number

Topological Degrees

115

of solutions of the equation f(x) = a on ft, such that this degree is robust with respect to both / and a in the sense that it remains unchanged for small variations of / or a. Figure 2.1 shows a simple case of an equation f(x) = a in the onedimensional Euclidean space. Obviously, the solution x which is located at the boundary is not robust since a small variation of / or a will disqualify this boundary point as a solution. For this reason, we exclude all a € /(dft) where, as usual, dft denotes the boundary of the set ft. On the other hand, observe that a small variation of / or a can change the number of solutions of the equation dramatically. For example, in Figure 2.1 if either the constant a moves up or the function curve / moves down a little, then the number of solutions of the equation will be completely different. Hence, we cannot simply use the total number of solutions as the degree for the measure since it is not a robust quantity. However, if for each solution xv of the equation we associate with it a signum function of det[/'(x„)], denoted sign det [/'(av)], and use the algebraic sum, Yl,v ^S 11 det [/'(^i/)]* t o define the degree, then this degree will be robust to small variations of both / and a. Example 4.1 given below will clearly illustrate this viewpoint. To this end, we first give a formal definition for this degree measure. k

Figure 2.1. Relation between the number of solutions and / or a. Definition 4.3. Let fl C ff1 be a bounded open set, / : ft —* Rn be a regular operator, and a € iT 1 \/(5Q) be a regular value of / on ft. Then, deg (f,ft,a)

:= ] T sign det [/'(x„)]

is called the topological degree, or operator degree, of / with respect to the point a and the set ft, where xv is a solution of the equation f(x) = a in ft. If the equation has no solutions in ft for a particular value a, then we define deg(/,n,a) = 0. Note that under our assumptions described above, the number of solu­ tions of the equation must be finite. Otherwise, it follows from the funda­ mental limit theory that in the bounded region ft there will be a subsequence

Chapter 4

116

of solutions converging to a limit point, x, which also satisfies the equation: f(x) = a. However, since a ^ /(9fl), we have x € ft] and since a is a regular value and x is a solution of the equation, we have det [/'(£)] ^ 0. Thus, it follows from the well-known Implicit Function Theorem that x is an isolated solution, contradicting the fact that x is a limit point of the solutions. Example 4.1. Let ft = (a,/?) and the regular operators fj'.ft—* R, j =1,2, be given by Figures 2.2 (a) and (b), respectively. It can be seen from the two figures that a*[/i(oO,/i(0)]u{Cl},

deg ( / i , n , a ) =

aeC/^a),/!^))^},

and deg (/ 2 , ft, o) =

0, -1,

a*[/2(/?),/2(a)]u{Cl}, ae(/2(/3),/2(a))\{c2}.

It is easy to see that the topological degrees of / i and / 2 are not changed when f or a has a small variation.

Figure 2.2. Graphical meaning of two topological degrees. We next derive the integral representation of a topological degree, which will be useful in discussing basic properties of the degree and its generaliza­ tion. We first have the following. Lemma 4.1. Let Qn = ( - 1 , l ) n and (x) € C^(Qn) be a continuously differentiable real-valued function defined on Qn with support contained in Qn, where the support of <j> is defined as usual by supp — closure of {x e Qn\

4>{x) ¥" ®} •

Thus, if (j> satisfies JQ 4>{x)dx = 0, then it can be decomposed as

^( X ) = m^'^)' J= l

x

€Qn,

(4.1)

117

Degrees Topological Degrees 1 where ^ €G Q C&{Q and (Qn) n) and

j|y

j{x .-,xjjj-ut,x -UUt,xj+l ,...,xnn)dt jj{xlrlr lr--,x j+1 j+1,---,x

== 0, 0,

V (xi,---,Xj-i,X ( x 1 , . . . , X ij+_i,---,X )+1,---,Xn) 1 , X Jn-

€ Q eQn-ln-

Here,:, note that if we let PXj Xj

i/>j(x)= l/>j(x)=

/

<j>j{x ■ t, ixj+u ■nnJXn) xn) dt, ^^- i((xxi i^U, -.•■ .-•j.X ,•,xj ■ jx+*i+i, i,^- --•^-•X , x•, )■c f■, t , dt, j(*U *-i _ij ,^-iXj-u ij, ,Xt ,*, i+

/

then Equation lation (4.1) can be rewritten as

«„-£Mfe>, namely, namely, Proof: Proof:

.7=1

•<

3d asasthe <£ can be expressed thedivere-ence divergenceofof(1/,,. (V>i,• • ••.•,i/>_ ^ nV) . can be expressed as the divergence of (V>i, • • •, 4>n)«(t) 6€ Ctf(-1,1) ^ ( - l , 1) be such that / ^^«u{i)dt (*)* = = 1. Let also Let u{t) Let u{t) 6 Ctf(-1,1) be such that / ^ «(t)«ft = 1. Let also /•l

V3ji{x) ) W {X)=U(X w = u(x1l)---u(x i)---U(X J{x)=u{x!)---U(x j) J )

(f

^.l

dt &! dt Xxl J[

/.l

■■■ ■■■Jf •••

f

cj>{t1,---,tj,Xj+ cf>(tl,---,tj,xj+
Bvwritinff ji == ll.--.n-l. , -- -, ,nn -- ll . By writing cf,= (<j +' (w -w2) + --+ (•Wn - ~_W +n~r_!) «;„_!, 4> {4> ++1{w K 2w == (4> wt) Kt ^-2- ;w2) w2) •^_2 ++ «;„_!)+«;„_!, )-ww1-i)-; wi) n _ni--)l y W n _2 1^1 T *++ *■ '■ -r■ V■ n - 2(w t ^ n+- i «;„_!, » hat je verified that it can be <j>1=<j>-w =-w 4>i= wi wi 1

and

4> -wj,j,—-Wj, j <j> ^- i=w =j=w =w^-i Wj-t Wj;, j-j1-i-w

jj= j ==2, • 2,---,n-l, • •,2,--,n-l, n — 1,

and <j> «;„_!isisthe theeYnented expectedresult result. cj)n n= =«;„_! the expected result.

0

Lemma 4.2. Let / : fi -> i? n be a regular operator and a € ir\f(dil), so that p := min x e an ||/(x) - o|| > 0. Let, also, Vp be the cube centered at the origin with the major diagonal equal to p/(2<Jn). Then for any real-valued function 4>{y) e C&(VP), we have [I 4>{f{x) - a) det[/'(*)] dx = K f[ <j>(y) <j>(y) dy, (
(4.2) (4.2) (4.2)

118

Chapter 4

where A KT isis aa constant constant independent independent of of4>. <£. We remark that in this lemma, a € K*\f(dto) i ? " \ / ( d « ) is not required to be a reg regular value of / on Q, ft, and the constant K is independent of ^ € C%{V This ular C%(VPP). Thi observation will be important for an the concept of topological observation topologics observation will be important for an extension extension of of the concept of topological degree Definition 4.4). degree later (given in degree later (given in Definition 4.4). We cf> outside tttht hh tt cucu cucu V then it folfows from from We also also remark remark that that if if <j> cf> = == = == outside outside cucu VVppp,,, then then itit folfows folfows from the definition of // on the definition of of pp and and the the uniform uniform continuity continuity of on the the closed closed bounded bounded set set Q that |||/(*) | / ( * )-"- a«|l «|l| l>> |f§»», ||/(«)

V x e€e ttti

with with

< ££,, ddiisstt((x , d5^ if)li)<<

provided that e > 0 is small enough. Consequently, e e&(VP) implies that tht 4>(f(x) aa ) Gb(fl), 4>{f{x) - aa Gb(n), ss that the integral oo nhe left-hand side dide ef Equatioi Equation (4.2) exists. Proof: We first show that under the constraint f/ (y)dy (y)dy == 0, 0, the ;he left-hand side of Equation (4.2) is zero. If this can be proved, then for my 4>i C&(V 1,2,this thisresult resultcan canbebeapplied appliedtoto ==Cl c2<£itoto any & 6 6C&(V p),i i==1,2, Cl P), 2 2- -c2<£i obtain obtain the lemma, where a = fv <j>i{y)dy, i = = 1,22 ie remark given too Lemma Lemms 4.1 above we first notice tlthat To do so, from the nn ssed as sasthe (j> P cac cacbebeexpressed expressed as t ht ht tdivergence divergence divergencei, of>(^ [e&[Vp)] namely, nameb p)]^ ^ namely, (of>(^ V ' l •,1-1•-?-•?••ip ,,V>n) ^••ipn)n) €«€ [e&[V n

4>{y) = X

™'~U

»>» '

It then follows that

)]^^.(/(x)-q)de

V y=^^(/W-a)det[ *(/(*) tt[[//' (<x(x)x) ] = A ' det[f/,{(x)]] "Ml^lzA det[/'(x)] * ( / ( * ) -- a)a)d)deedet[/'(x)] =V££ ?r 3 m T± det[/'(x)l

" = = £ = V J2

x d^ X j)(f(x)-a) A , , dMf( ^ (X)/~a) i)-«) " a ) Ajkjk{),, (),, d^

„x €_ onfl,, M .
,4jkjfc(x) (x) is the algebraic complement of [/'(x)] with respect to where A dfj(x)/dxk, namely, (-lp+ f c det[/((x)j in which the j t h row and the fcth column of the determinant are deleted.

Topological Degrees

119

If / € C 2 (fi), then according to the differential formula of a determinant (see Exercise 4.1) we have

| : ^ = ° -

*«<>■

(">

so that by the Gaussian formula in Calculus and Eqs. (4.3) and (4.4), we have / (/(*) -a) det[f'(x)]dx

Jn

= -J2

f ^

j=1Jn

E

^

£J dxk

^ = 0.

Even if / ^ C2(Q), we still have this result by applying the standard smooth­ ing technique as follows: First, extend / continuously to outside of fi, and let {/&}, with fk(x) e C°°(Rn), be a sequence of approximants to / that "smooth" / in such a way that as k —> oo, /&(#) —► / dfk(x) — OXj

uniformly on Q,,

df(x) ► —

OXj

Then pick any <j> e

uniformly on any proper open subset QQ with Qo C ft. CQ(VP).

For large enough k, we have

/ 4>{y)dy= j 4>{y)dy, JvPk Jvp where pk := m i n ^ n \fk(x) - a\ > 0, satisfying l i m ^ o o ^ = p. Thus, by the result shown above, if Jv (j>(y)dy = 0 then / 4>(fk{x) - a) det[f'k(x)]dx Since {fk(x)}

= 0,

k = 1,2, • • • .

(4.5)

converges uniformly on Q, we have

<j>(fk{x) - a) = 0,

V xefi

with

dist(z, dft) < e,

for large enough fc, provided that e > 0 is small enough. Hence, letting k —» oo in (4.5), where the limit and integral can be interchanged, yields the result. D

120

Chapter 4

The following theorem gives an integral representation of the topological degree. Theorem 4.1. Let f : ft —> Rn be a regular operator, and a € Rn\f(dQ) be a reguiar vaiue of f on ft. Also, let Vp be deGned as in Lemma 4.2. Then for any <j> € CQ(VP) satisfying fv 4>{y)dy = 1, we have d e g ( / , f t , a ) = [ <j>(f(x)-a)det[f'(x)]dx

= K,

(4.6)

where the constant K is independent of' <j>. Proof: We first consider the case where the equation f(x) = a has no solutions in ft. In this case, deg(/, ft, a) = 0. We now show that the integral in Equation (4.6) is also zero. To do so, let <£o € CQ(VPO) be such that Jv o € CQ(VP) and Jv (f>o(y)dy = 1, so that by Lemma 4.2 we have K=

f
(4.7)

Since \\f(x) - a\\ > po for all x € ft, we have 0(f{x) - a) — 0, and hence, the right-hand side of Equation (4.7) with this ^>0 is zero. We next consider the case where equation f(x) = a has all its solutions XW — <,XN in ft. Since a is a regular value, we have det [/'(av)] ^ 0 for all v = 1, • • •, N. It then follows from the Implicit Function Theorem that there is a small 77 > 0 such that the operator / is one-to-one, and so invertible, from Uu = {x I x € ft, \x - xu\ < 77} to a neighborhood Wu of a, and moreover, that det [/'(#)] has no sign changes on each UV1 v = 1, • • •, TV". Let p'=

_min xen\uy=1uu

||/(x)-a||>0,

and po = min { p', dist (a, dWu), v = 1, • • •, N } . It is clear that VPo C V, and a + VPo C W„, z/ = 1, • • •, JV. Let 0O € Co (V^J be such that fv <j>o{y)dy = 1. Then it follows from Lemma 4.2 that it is sufficient to show /
= deg (/,ft, a).

Topological Degrees

121

Indeed, we have [
<j>o(f(x)-a)det[f(x)]dx

N

-a) det [/'(*)]<&.

+ V / MfW

In this equality, the first term has been shown to be zero above, and the second term, according to the transform of volume-integrals, is N

f y^ /

^°(2/) sign d e t [f'(xv)]dy

u=zlJWu-a

N

=

N

r

11, / ^°^)

sign det

x

d

[f'( v)) y = ] C s i g n

det

[A*")] •

Note that according to Definition 4.3, the right-hand side of the above is deg(/,fi,a). D We now return to Definition 4.3. As remarked above, Lemma 4.2 does not require a be a regular value. Hence, we can extend the definition of topological degrees as follows. Definition 4.4. Let / : Q —> K*1 be a regular operator, and suppose that n a G R \f(dQ,). Let also p \— m i n ^ a ^ ||/(a0 — a\\ > 0. According to Lemma 4.2, for any <j> e CQ(VP) satisfying fv (y)dy = 1, integral / <j>(f(x) - a) det[f(x)]dx

= K

for a constant K independent of <j>. This constant K is defined to be the topological degree, deg(/, £i, a), with respect to the point a and the set Q. We note that this definition extends the concept of topological degree from a regular value a € Rn\f(dft) to include also irregular values, so that the degree is also well defined for those critical values of / . We also note that in this extended definition, namely, in deg (/, fl, a) = f (f(x) - a) det [f'(x)] dx,

Chapter 4

122

the function can also be taken from C0(VP), provided that fv 4>(y)dy = 1. In this case, the degree calculated by the above integral is still single-valued. As a matter of fact, we can apply the aforementioned "smoothing" technique again, to approximate cf> by a smooth e with a small enough e > 0 such that € € CQ(VP), and then let e —► 0 in the equation / <j>£ (f(x) - a) det [/'(*)] dx = deg(/, Q, a) / £(y)dy, to obtain / cj>(f(x) - a) det [/'(*)] dx = deg(/, fl, a ) .

4.1.2 Basic properties of topological degrees We have extended the definition of the topological degree to cover critical values of / in Q. In the following discussion of basic properties of the topo­ logical degree, however, we only focus on regular values a € Rn\f(dQ), since the set of critical values of / is ignorable, as can be seen from the following result. Theorem 4.2. (Sard Theorem) Let ft c Rn be an open set and f e C 1 (fi). Then the family of critical values of f onQ, has zero Lebesgue measure. Proof: Since the open set Q can be decomposed as a union of countably many closed cubes in BJ1, and the union of countably many zero-measured sets has zero measure, it suffices to show the result for the case where Q = [0, l ] n and

feC1^). First, we divide Q, as mn closed cubes with edge-length 1/m, denoted ^ i , ' , ^ m n - Consider only those cubes Qk that contain a critical point, denoted xo, of / . Under the linear transform y = / O o ) + f\x0)

(x - xQ),

the entire set Q,k will be mapped into the interior of a ball, with radius My/n/m and centered at /(xo), of an (n- l)-dimensional hyperplane in i? n , where M := max x€ Q ||/'(#)||. Observe that f(x) = / ( x 0 ) + f'(x0)

(x - x 0 ) + ry(x, x 0 ) ,

x € ttk,

Topological Degrees

123

where r](x,x0) = (x-x0) IMs,*o)|| r{m)=

/ [f'(xQ + t{x-x0)) Jo

-

f'(x0)]dt,


\\*'-*"\\
Hence, /(fife) is contained in an n-dimensional cylinder of volume less than 2v n _i \

M^— + r{m)^— m m )

^— m

r(m),

where vn-\ is the volume of the (n — l)-dimensional unit ball. Consequently, the outer measure of the set of critical points of / is not greater than 2 V n _ 1 (M + r ( r n ) ) n " 1 ( ^ ) n r ( m ) . Since / ' is uniformly continuous on ft, we have lim™-^ r(m) = 0, and so the measure of the set of critical points is zero. [] Next, we give three basic properties of the topological degree, leaving other general properties to the next section where the concept of topological degrees for general continuous operators will be provided. Theorem 4.3. The topological degree has the following basic properties: (i) (Unity property) For the identity operator I(x) = x on ft, (l,

a€ft,

(ii) (Integer-valued property) If f € C(fi) n C x (fi), then deg(fyft,a) is an integer-valued continuous function of a in i ? n \ / ( 5 0 ) . Hence, by the Mean-Value Theorem of Calculus, deg(f, ft, a) assumes a constant value on any simply connected region of i? n \/(51]). (iii) (Stability) If ft(x) := f{x\t) € C(U x [0,1]), with £ft{x) € C(Q x [0,1]), i = 1, • • •, n, and a € ^ftidil), for all t e [0,1]* then &eg(fufl,a) for a constant K independent ofte

= K [0,1].

Chapter 4

124

Proof: Part (i). It follows immediately from the definition of topological degree. Part (ii). For any OQ e Rn\f{dQ), let p0 = m i n ^ a o ||/(z) - ao|| and pa = m i n ^ a o ||/(a;) - a\\. It is clear that if \\a - oo|| < po/2 then pa > po/2. Hence, for any fixed e CQ(VPO/2) such that fv 4>{y)dy = 1, we have d e g ( / , n , o ) = / 4>(f{x) - a) det [/'(*)]dz, for all a satisfying \\a — ao|| < po/2. It then follows that lim deg(/,ft,a) = / (f{x) - ao) det[f'{x)]dx a-+a0

= deg(/,ft,ao),

Jn

namely, deg(/, Q, a) is continuous at ao- Recall from Theorem 4.2 that the set of regular values of / is dense in Rn\f(dQ,). Since the topological degree for a regular value is an integer, so is deg(/, Q, oo). This implies the integer-valued property and the continuity of deg(/, ft, a). Part (iii). Let

^= ^ x£a\t

H-f*(x) ~ aH

and

P ^ n5S?i 0
Pt =

"SK ll-kfc) ~ aH ' xean o
Then 0 < p < pt for all t e [0,1]. For a fixed cf> e C&(VP) satisfying JVp4>{y)dy = 1, we have deg(/ t , O, a) = / ^ ( / t ( x ) - a) det [/t'(x)](ft(x) - a) in the integrand is nonzero only on an proper open subset fic^ independent of t, where Cl C Q>. It then follows from the condition -£r:ft(x) € C(Q x [0,1]), i = 1, • • •, n, that the right-hand side of the above equation is continuous in t. Moreover, according to Part (ii), deg(/*, Q, a) assumes only integer-values, which implies that deg(/ t , Q, a) — K, a constant independent of (/>. [] 4.1.3 Topological degrees of continuous operators The topological degree of a continuous operator can be defined via that of a regular operator, where the key is the degree stability. Definition 4.5. Let / : Q, —► Rn be a continuous operator, a e and / : Q, —► iT1 be a regular operator satisfying

I!/(*)-/(*)||
xeda.

Rn\f(dQ),

Topological Degrees

125

The Brouwer topological degree of / with respect to the point a and the set ft is defined by deg(/, ft, a) = deg(/, ft, a). It is important to point out that this definition of Brouwer degree for / is independent of the choice of / . Indeed, for any fi € C(Q)DC1(Q), i = 1,2, satisfying \\fi(x) - f(x)\\ < \\f(x) - a||,

x € dft,

i = 1,2,

we can introduce an operator £ = (l-t)/l(*)+*/2(*),

t€[0,l].

For x € 9ft and £ G [0,1], we have | | £ ( s ) - / ( x ) | | < (1 - 1 ) ||/i(z) - / ( x ) | | + 1 ||/ 2 (x) - / ( x ) | | <||/(x)-a||, which implies that a £ /t(<9ft) for all t 6 [0,1]. Hence, it follows from Part (iii) of Theorem 4.3 that deg(/i, ft, a) = deg(/ 2 , ft, a). We now show some useful properties of the Brouwer degree of a contin­ uous operator [6, 14, 21]. Theorem 4.4. The topological degree of a continuous operator f : ft —» iT1 has the following properties: (i) (Kronecker principle) If deg(f, ft, a) ^ 0, then equation f(x) = a has a solution in ft. (ii) (Domain decomposition) IfQ. — fti Uft 2 , fti Hft 2 = 0, and a ^ /(<9fti)U /(dft 2 ), then deg(/, ft, a) = deg(/, fti? a) + deg(/, ft2, a ) . (iii) (Stability) If ft{x) := f(x;t) then

€ C(ft x [0,1]) and a g /t(<9ft), £ € [0,1], deg(/ t ,ft,a) = K

for a constant K independent ofte [0,1]. (iv) (Shift-invariant property) If a £ f{dft) then deg(/,ft,a) = d e g ( / - a , f t , 0 ) .

126

Chapter 4

(v) (Removability) If F is a closed subset ofQ, such that a & / ( F ) , then

deg(/,ft,a)=deg(/,ft\F,a). (vi) (Connectivity) Let a be in a simply connected region ftc of Rn\f(dQ)J then deg(f, ft, a) assumes the same value over the entire region ftc. (vii) (Boundary value dependence) deg(/, ft, a) depends only on the values of f on the boundary dft. We recall that two operators fi : ft —» Rn, i = 1,2, are_said to be continuously homotopic if there exists a continuous operator F : ft x [0,1] —+ Rn such that F(x,0)=/i(x)

and

F(x, 1) = / 2 ( x ) ,

VxeH.

It follows from the stability (iii) that any two continuously homotopic opera­ tors have the same Brouwer degree, provided that a ^ F(dQ x [0,1]), that is, the Brouwer degree is invariant in the continuous homotopy, and this prop­ erty is called the continuous homotopy-invariant property of the Brouwer degree. Proof: Part (i). Suppose the conclusion is not true, and let f(x) =^ a for all x € ft. Then, we define p = min | | / ( x ) - a | | > 0 , and let / € C(ft) n C x (ft) be such that max | | / ( * ) _ / ( * ) | | < £ . Select the regular value a of f such that \\a - a\\ < p/4. Then we have ||/(*) - a|| > ||/(x) - a|| - \\f(x) - f(x)\\ - \\a - a\\

> ^ >o,

Vxeft.

When a is sufficiently close to a, it follows from Definitions 4.5 and 4.3, as well as Theorem 4.3, that deg(/, ft, a) = deg(/, ft, a) = deg(/, ft, 2) = 0, contradicting our assumption. Hence, equation f(x) = a has a solution in ft.

Topological Degrees

127

Part (ii). Let fe

C(ft) 0 C ^ Q ) be an approximant of / such that

ll/fr)-/(*)ll<£,

VxeH,

where x6aQiUaQ2

Choose the regular value a of f such that \\a - a\\ < p/4. x € dCli U 80,2, we have

Then for all

ll/(*) - 211 > \\f(x) - a|| - ||/(x) - f(x)\\ - \\a - o|| > P- > 0 . Thus, when a is sufficiently close to a, we have deg(/,ft,a) = d e g ( / , f t , a ) ,

(4.8)

deg(/,^,a) = deg(/,a,a),

t = l,2.

(4.9)

Let the solutions of the equation f(x) = a in SI1USI2 be {x^}. Then it follows from Definition 4.3 that deg(/,ft,2) = ^ s i g n det[/'(x„)] = J^

sign det [/'(*,,)] + ] T sign det [/'(*,,)]

= deg(/, nua)+

deg(/, fi2,8).

The result follows from this and from Equations (4.8) and (4.9). Part (iii). Extend ft(x) continuously to Rn x R1, and then apply the standard smoothing technique to find an approximant ft(x) of /t(x), which is smooth in both x and t, such that \\ft(x)-Mx)\\
V (*,*)€ fix [0,1],

where ?=

, x™}? r n i J I ^ ^ " " (x,t)edQx[0,l]

0

"

> 0

'

It then follows from Definition 4.5 and Theorem 4.3 that deg(/ t , n, a) = deg(/ t , Jl, a) = X , where If is a constant independent of t e [0,1].

128

CAapter Chapter 4

Parts (iv) and (v) follow immediately from the definition, and Part (vi) follows from Definition 4.5 and Theorem 4.3 (ii). Part (vii). Let /< € C(Q), » = 1,2, such that h(x) h(x)== h(x), /2(x),

V VxeM. xeSO.

Define an operator Ft(x) = (1-t)fi(x) (1 - t)fi(x)

+ tf2(x), tf2(x),

tt € [0,1].

Then, since a £ f(dii), we see that a # Ft(dCl) = f^dSl) follows from the stability of the Brouwer degree that

= /2(da).

It then

deg(/!, Q, n, a) = deg(F 0 , Q, a) = deg(F1;u Q, a) = deg(/ 2 ,ft, n , a). a).

0

The following property of the topological degree for a composite continuous operator is important. Theorem 4.5. (Composition Theorem) Let f € C(ft), g e C ( / ( 0 ) ) , and a £ (go /) (9ft), where o denotes the operator composition. Then deg(go/,Q,a) deg(go/,Sl,a)

=

Y, deg(#,£>i, a) deg(/,n,2><). deg(/, O,V t ), ^ deg( fl,2><,a) x>iC/(n)

(4.10)

where Vt is a simply connected region in / P \ \ / ( a n ) , i = 1,2, • ••, , n d deef f « P ) = W ^7 ' 3 ) ' degc/.n.po-l^ ' J; [ 0, '"^* u \0,

vv «aeePVi , C c/ (/(n), fi), A not not tompletely tompletely yontained yontained di di n(fl). n(Q). A

We remark that i?"\/(dfi) contains at most countably many simply connected regions, which completely cover the solution set 5 of equation 9(y) = 0 within /(ft). Since 5 is a closed bounded set, a finitely many X>< can cover the entire 5, and so the right-hand side of Equation (4.10) is indeed afinitesum. Proof: First, let us suppose that both / and g are regular operators, and a is a regular value of go / . In this case, by the chain rule we have d(gof)(x) d*

\dg(y)] \dg(y)] df(x) a/(g) i dy Jy= / ( x ) 5x dx ■

Topological Degrees

129

It is then clear that the regular value a of g o f is not a critical value of #, and that if x is a solution of equation (g o / ) (x) = a then y = f(x) must be a regular value of / . Hence, by Definition 4.3, we have deg(g o /, ft, a) =

]P *e(0o/)-i(a)

=

Y^

df(x) sign det sign det #0(3/)' dx &/ y=f(x)

sign det #0(2/)'

y]

J

=

Y^

sign det

_1

2/€0 (a)

Since Rn\f(dQ)

dy

sign det

xef-^y)

df(x) dx

deg(/,ft,y).

= Vl&i and deg(/, ft,y) = 0 for y £ /(ft), we have

<*eg(# o / , ft, a) = ^

]T

sign det

\dgiy)' dy

deg(/,ft,A).

Moreover, since a is a regular value of g\v , deg(# o / , ft, a) =

] T deg(s, £>*, a) deg(/, ft, P»), x>
as claimed. Next, suppose that / and g are as assumed in the theorem statement. Let Gk = {aeRn\f(dn)

I deg(/,ft,a) = fc}, * = 0, ± 1 , ± 2 , • • • .

If we can show that deg(ffo/10,a) = ^fc-deg(5,G'fc,a),

(4.11)

then the result follows from the domain decomposition property (Theorem 4.4, Part (ii)). To do so, let two regular operators / and g be close enough to / and #, respectively, such that

H/(*)-/(x)||
xeU,

(4.12)

Mv) - 9(V)\\ < dist(a, (g o f) (3J1) U (g o / ) (an)) , y € f(ty , (4.13) ||(?o/)(x)-(go/)(x)||
xeQ,

(4.14)

130

Chapter 4

where S is the solution set of equation g(y) = a within /(ft). It follows from (4.14) and Definition 4.5 that deg(
(4.15)

Let Gk = {a € Rn\f(dSl)

| deg(/,n,o) = fe}, fc = 0,±1,±2, • • • .

Then it can be easily verified that dist(o, (
(dft)) < dist (a,
This, together with Equation (4.13) and Definition 4.5, imply that deg(#, Gfc, a) = deg(£, Gk, a) .

(4.16)

Now, observe that for any y G S, it follows from Equation (4.12) and Defini­ tion 4.5 that deg(/,ft,y) =deg(/,ft,s/), so that GkC\S = GkC\S. Therefore, by Theorem 4.4, Parts (i) and (ii), we know that deg(#, Gk, a) = deg(#, Gk, a).

(4.17)

By Theorems 4.2 and 4.3, Part (ii), on the other hand, we may assume that a is a regular value ofgof. Hence, from what we have shown at the beginning of this proof, we obtain deg(£o/,ft,a)=]TVdeg(£,Gfc,a).

(4.18)

k

The final result now follows from Equations (4.15)-(4.18).

0

The following result shows that some higher-dimensional topological de­ grees can be reduced to lower-dimensional ones. Theorem 4.6. (Reduction Theorem) Let Rm C Rn (m < n), ft C Rn be a bounded open set, f : ft -> R1 be a continuous operator, and (I - f) (ft) C R71. Then for any a € Rm\f(dn), we have deg m (/, ftm, a) = deg n (/, ft, a ) , where ftm = R71 n ft and deg m stands for the topological degree in an mdimensional space.

Topological Degrees

131

Proofc First, it follows from (J - / ) (ft) C RT that / ( x ) 6 R™ for all x € ftm. Hence, / | H is a continuous operator from ftm to i? m , and so deg m (/, n m , a ) is well defined. Choose coordinates of i? n such that its first m coordinates are located within i? m . Since (I - / ) (ft) c # m , y = / ( # ) has the following expression: f fj(xU" - ? ^ n ) , [ /jO^i, •••,:&„) = ^ ,

l<j<m, m + 1 < j < n.

Let / be a regular operator that approximates / such that ||/(x)-/(x)||<min

||/(x)-o||.

Since / has the above expression, / can be chosen in such a way that fj (x) = Xj ,

m + 1 < j < n.

Observe that / 1^ is also a regular operator from ftm to R™. By Definition 4.5, we have deg n (/,ft,a) = d e g n ( / , f t , a ) , deg m (/, ftm, a) = d e g m ( / , ftm, a ) . Without loss of generality, let us suppose that a is a regular value of / . Notice that a G Rm implies that the last n — m components of a are all zero. This implies that if we add n — m zero components to the solutions in ftm of equation f(x) — a, then we obtain all the solutions of the equation in ft. Moreover, note that the Jacobian det[/'(x)] is the same for x 6 Rm or x e Rn. Thus, we indeed have deg m (/, ftm, a) = deg n (/, ft, a ) .

Q

We close this section by introducing the concept of index for isolated points of a continuous operator. Definition 4.6. Let / : ft —► Rn be a continuous operator, a € Rn, and xo € ft be an isolated point of f~l{a). The index of the isolated point xo of / with respect to a is defined to be ind(/, x 0 , a) = deg(/, G, a), where the region G C ft contains XQ and G 0 / - 1 ( a ) = {XQ}.

Chapter 4

132

We note that for any region G satisfying the above condition, Part (v) of Theorem 4.4 implies that deg(/, G, a) is a constant. Theorem 4.7. (Index Formula) Let f € C(ft), a g f(dQ), and equation f(x) = a has finitely many solutions {xi, • • •, x/v} within ft. Then N

deg(/, ft, a) = ] T ind(/, x„, a ) .

Proof: For each av, let Gv C ft be a sufficient small region containing xv such that Gp C\ G„ = $ if JJ, ^ v. It then follows from Parts (ii) and (v) of Theorem 4.4 that N

deg(/, ft, a) = ^

N

deg(/, Gu, a) = ] P ind(/, av, a ) .

[]

Finally, we remark that since a finite-dimensional Banach space is isomorphic to a Euclidean space of the same dimension, the topological de­ gree of a continuous operator from an n-dimensional Banach space to an n-dimensional Banach space can be defined by the Brouwer degree of a con­ tinuous operator on Rn; all the aforementioned properties are still preserved for the finite-dimensional Banach space setting. 4.2 Topological Degrees of Compact Fields In this section, we first develop a general theory of approximating a compact field by a sequence of finite-dimensional continuous operators. Then we es­ tablish the concept of the topological degree and discuss its basic properties for a compact field in an infinite-dimensional Banach space. For simplicity of notation, throughout the rest of this chapter we let 11 • 11 be either a Banach space norm, oftentimes associated with the Banach space X, or a Euclidean norm associated with Rn, unless otherwise indicated. Next, we recall from Chapter 2, Section 2.1, that an operator T from a subset ft of a Banach space X to a Banach space Y is compact if it is continuous and for any bounded subset S C ft, T(S) is a compact set in Y. It should be pointed out, however, for a nonlinear case, a compact operator may not be completely continuous, and conversely a completely continuous operator may not be compact (see Exercise 4.3). The latter holds only in a reflexive Banach space (see Proposition 2.1). We introduce the following concept.

Topological Degrees

133

Definition 4.7. Let X be a Banach space. Anoperator F is called a compact field if it has the form F = I - /, / , where / : Q -+ X X ii s aompact tperator on a bounded open set Q. fi. We next discuss the topological degree of a compact field in a Banach space. For this purpose, we need some preliminaries. First, recall that a subset S in a Banach space X is said to be compact if its every open covering contains a finite subcovering, and is said to be sequentially compact if its every sequence contains a convergent subsequence with the limit in the set 5S.. By t.hf. wf»ll-knnwn Borel-Lebesgue T W P U .phPsmiA Theorem Thorn-Pm f^l cnruvnt.fi of romnact. the well-known [35], the t.wn two concepts compact and sequentially compact sets are equivalent in a metric space: they are distinctive tive only in a general topological space. Moreover, a subset S C X is said to be relatively compact if S is compact, and is said to be totally bounded finitely (orprecompact) precompact) if canbe coveredby finitelymany manyballs ballsof radius uounaea>4(or (or precompact) n ifit ititcan can bebecovered covered bybymutely many baiis ofofradius radius ££ for any s ny e > 0. It is also well known that a subset in a complete metric for any £ > 0. It is also well known that a subset in a complete metric space space is iss relatively relatively compact compact if if and and only only if if it it is is totally totally bounded bounded (the (the Hausdorff Hausdorff Theorem). an). Theorem). Proposition 4.1. Let K C X be a relatively reiativeiy compact subset subset. closed covex hull of K is also relatively compact.

Then the

Proof: It suffices to show that the total boundedness of K implies the total boundedness of K, the closed convex hull of K. Since K is totally bounded, there is a finite collection of points {ai, • • •• ,N } of K such that each x € K is contained in a ball Bj{ah e) of radius e > 0 centered at aj for some j = 1, • • •• N, namely, K C \jf=1 B^a^e). Lee W be the convex set spanned by {au • • •, aNv} ,hat is, W={\ M ^ / 1iai A .... + \Avav | A , > 0 , jj = = ll,---,AT. , - -- -, ,AATT,,V W={\ a1w ^ ++. --~ \j>0, ££ff"== 11 AA, , = l}. NaN| Obviously, W is a relatively compact set, so that there exist h,•-••,&' bu • • •• ,&. €€X* Uf=i B(bj,e). such that W C UjLi |jf=i I j j i B(b We now show that K is covered by the union of balls |jfLi B(bjyjt3e) 3e), and so is totally bounded. Let x € K. It follows from the definition of K that there are #i • • • xi 6 K and A Aiii , • •• \e ^ >, with £*. = 1 A^ = 1, such J that Hl'dLl

ll*-£$-iVill<*. \\*-Y,U*4<e-

For each Xj, j = 1, • • •, I, £, there ii sn akjk. eG {{a {{u 1 ? • •• aN} such that \\xj-ak i\\ IN-°*>||

<e <£

134

Chapter 4

Thus, it follows from the triangular inequality that

||*-£kv*,||<2 ||-E^v fe ,||<2 £. Observe, on the other hand, that e1

N' N' x a

^2 j *j

ewe

3=1 J=I

U B^ . * ) •

3=1 J=l

Hence, there is a bjo e {bu • • •, bN,} such that

||£kA^,~^o||<e, ||£S«iV*~*i.||<«, so that |\\x-b | x - 6j0\\<3e, ,-0||<3e, as claimed.

[]

Proposition 4.2. Let K be a relatively compact set in a Banach space X. Then for any e > 0, there exists a continuous operator T£ : K -> X, such that its range U{TS) is contained in a finite-dimensional subspace Rm ofX, and | | T «a (( xx )) -- x | | < e , VxeK, m T T.(A-)ciJ nx, £(Jf)crnf,

where K is the convex closure of K in X. The operator Te denned in this proposition is usually called the Schauder operator. Proof: Since K is relatively compact, there are a i , • • •• aN e K such that K C u f = 1 S ( a j ; e). Letfl™be the linear subspace spanned by {a l7 • • •• ,N} and let Km be the convex set spanned by {ai, • • • , a N } . Then, obviously, Km C Rm n K. Let AT N

AT i, N

T = J^ > x - 0" i > ) 0 ij // 5 Te(x) £ 3 „(x Mx -~ a,), Oi), £{x) = M (( x where W

'

\e-||x||,

||x||<e.

x €€/ X f, , x

135

Topological Degrees

Since A e), for any x € K, we have £ " A*(* - aj) 0> which K- C uf ^yB{a <*i) > > 0> =1 £(a.,-, h implies that Ts£ is is continuous on K. Moreover, we have x x ) | | / ^ ^ M ( x - aa ) || || T = H ^ x( a-Ja- oj j)) ((aja ~ J - )\\ i T £e (( ax :))--xx| || | = H ^^^jLLi M / £ £ = !1* * ( * - i)

<Er
finitA.HimAncir.r.al ™ntin„™« continuous It fouows from me above proposition inat finite-dimensional nmie-dimensional continuous operators can can be be constructed constructed to uniformly approximate compact operator operator operators to uniformly approximate a a compact with arbitrarily high accuracy. with arbitrarily high accuracy. Proposition 4.3. Let X be a Banach space and SI C X be aa bounded bounded set. set. Let also f/ :: SI f! -> -» IX be be a compact operator. Then for any e > 00 ,here is always a finite-dimensional finite-dimensional subspace rR™ C C and a compact operator f£:Si►R :Sl►Rm such that

||\f| /£e(x)-f(x)|\<£, (x)-/(ar)||<£,

V xx €e Xl .

Proof: It follows immediately Prop mmediately from Proposition 4.2 with fe = Te£of.

\\ fl

Next, let_X. be be aa Banach Banach space space and and SI MC CX A be be aa bounded bounded open open set. set. Let also // :: SI -► bb e aompact tperator, F = I f / h e r e -+ X - ^ I b b e aompact tperator, F = I - f / h e r e II is is tthhtt and a€ X\F(dSl). We We then the concept concept of identity operator, r, and and a e X\F(dtt). then introduce introduce the of the the for the compact field F , denoted deg(F, SI, a), with respect topological degree t ee for the compact field F , denoted deg(F, ft, a), with respect to the point a and nd the the set set SI. ft. First, it is easy to to F maps maps a a closed set to to a easy to see see that that F closed set a closed closed set. set. Since Since oa <£ £ F(dSl), F(3fl), we wehave have have p= p=mm min | | F (\\F{x)-a\\>0. x)-a||>0. xedn xedu It follows from Proposition oposition 4.3 4.3 that finite-dimension finite-dimensional subIt then then m follows from Proposition m a t there therem is is a a nnite-dimensional sur> m and a compact operator fm : SI -► ►m such that space R space R and a compact operator fm : ft -* ► such that \\fm(x)-f(x)\\
V*€ft. V * €€ f0t .

w * examine P V Q m i n . n}™*W the tho n r n n f of n f P r n n n « i t i ^ n 4.2 A 9 tViPn we WP n a n To J-\J this I f l l l O end, ^11V-1, if 11 we WC/ CA.CU11111V/ closely ^-IVyoOiy 1/llV^ proof p i W W l V^l Proposition X I V J / U O I U I U I I ~X.^i then V/AAVyXi »» V/ can v^vnii see that this R™ Rm can be the one that contains within it the point o. a. Thus, we m R™ n SI may let Sl Qm Q and then consider the the finite-dimensional finite-dimensional continuous m= R operator F mm = -*• i? Rmm.. Since for = I/ - /mm : Q ftm -> for any x € dSl 5ft m C dSl 5ft we have I||F I P m(x) / ^ -- oa|| i l > \\7P(> 0 > ||F(x) a|| -- ||/ ||F (x) -- / (/f(xrx^))l||l|| > 00 ,, m (x) - a | | > | | F ( x ) - a | | - ||/ m

the m-dimensional topological degree deg m (F rmo , nftmm,, a) is well defined. There­ m we can show that deg wm (F m ft,, and //mm,, then fore, if we m , ft m, a) is independent of R

Chapter 4

136

we can use it to define the topological degree of an operator in a Banach space. For this purpose, we first have the following. Proposition 4.4. Let X be a Banach space and ft C X be a bounded open set. Let also f : ft —► X be a compact operator and F = I — f. Moreover, for two finite-dimensional subspaces Rn and i? m of X, where n and m can be different, let a e (Rn n i r ^ V ^ d f t ) , and fn : ft -> Rn and fm : ft -> R™ be continuous operators satisfying ||/„(*)-/(aO||
and

||/ m (x) - f(x)\\ < p ,

VxGO,

where p= min | | F ( x ) - a | | . Furthermore, denote Fn = I — fn, Fm = I - fm, ftn = Rn n ft and ftm = RmnQ. Then deg n (F n , ftn, a) = deg m (F m , ft Proof: Let Re = R" + R"1 and ft* = Re 0 ft. Since (J - Fn) (ft) = fn(Tt) C Rn, it follows from Theorem 4.6 that deg n (F n , ftn, a) = deg*(F n , ft*, a ) . Similarly, deg m (F m , ftm, a) = deg*(F m , ft*, a ) . Hence, we only need to show that deg^(F n , ft*, a) = deg*(F m , ft*, a ) . To do so, let Ft(x) = tFn(x) + (1 - t)Fm{x),

x € ft*, t € [0,1].

For x e 9ft* C 9ft, we have \\Ft(x) - a|| > ||F(x) - a|| - t||/ n (x) - / ( x ) | | -{l-t)\\fm(x)-f{x)\\>0, so that by the stability of topological degrees we have deg € (F n , ft*, a) = deg £ (F m , ft*, a ) . The above proposition reveals the following definition.

[]

Topological Degrees

137

Definition 4.8. Let ft be a bounded open set in a Banach space X, f : ft —► X be a compact operator, F = I — / , and a € X\F(dffc). Suppose that ae R71, where # n is a finite-dimensional subspace of X, and / n : ftn = ft fl i? n —» i? n be a continuous operator satisfying ||/n(x)-/(x)||<min

||F(x)-a||.

Let moreover Fn = I — / n . Then the topological degree, called the LeraySchauder degree, of the compact field F with respect to the point a and the set ft is defined to be deg(F, ft, a) = deg n (F n , ftn, a ) . The topological degree of a compact field preserves many properties of the topological degree of a continuous operator defined in a finite-dimensional space [6, 14, 21]. Theorem 4.8. The topological degree of a compact Geld in a Banach space has the following properties: (i) (Leray-Schauder principle) If deg(F, ft, a) ^ 0 then equation F(x) = a has a solution in ft. (ii) (Domain decomposition) If ft = ft± Uft2, ^1 ^ ^ 2 = 0, and a g F(dfti)l) F(dtt2), then deg(F, ft, a) = deg(F, il1,o)+

deg(F, Q2, a ) .

(iii) (Stability) Suppose that ft(x) is compact with respect to its two vari­ ables onQx [0,1], Ft = I - ft, and a € X\Ft(dft) for all t e [0,1]. Then deg(Ft,n,a)=*T, where K is a constant independent ofte

[0,1].

Proof: Part (i). Suppose not. Then a ^ F(ft), where F(ft) is a closed set, so that p = min \\F(x)-a\\ >0. Let Rn be a finite-dimensional subspace of X with a e Rn and / n : Q —► iT1 be a continuous operator such that

||/ n (x)-/(x)||
Vxefi.

138

Chapter 4

It follows from definition that deg n (F n , t ) n , a) = deg(F, ft, a) ^ 0, where Fn = I - fn and ftn = 0, n R*1. By the Kronecker principle (Theorem 4.4, Part (i)), there exists an XQ € ttn such that Fn(xo) = a. On the other hand, however, we have \\Fn(x0) - a|| > ||F(x 0 ) - a|| - | | / n ( z 0 ) - / ( * 0 ) | | > 0, a contradiction. Hence, conclusion (i) holds. Part (ii). It is clear that min | | F ( x ) - a | | > x€o\l

min

||F(x) - a\\ := p0 > 0.

xEoiliUd\l2

Choose a finite-dimensional subspace R71 with a G Rn and a continuous operator fn:Q^Rn such that ||/»(s)-/(*)||
*€$!.

Let Qn = RnDQl and ^ n j = R^nQj, j = 1,2. Then it follows from definition that deg(F,ft,a) = d e g n ( F n , n n , o ) , deg(F, % a) = deg n (F n , ftnJ, a ) , j = 1,2 . The conclusion follows from the domain decomposition property of the topo­ logical degree for continuous operators on finite-dimensional subspaces (The­ orem 4.4, Part (ii)). Part (iii). According to Proposition 4.3, there is a finite-dimensional subspace Rn with a € R™ and a continuous operator / n t : ft x [0,1] —► R71 such that ll/n,t(z)-/t||
min

||Ft(z)-a||,

x€ft,

t € [0,1].

x£dQ,t£[Qyl\

Hence, we have deg(F t , ft, a) == deg n (F n , t , ftn, a ) , where F n>t = J — / n > i . By the stability of the topological degree in a finitedimensional subspace (Theorem 4.4, Part (iii)), we know that deg(F t ,ft,a) is a constant independent of t € [0,1], \\ Other important properties of the topological degree in a finite- dimen­ sional space, such as the removability, connectivity, boundary value depen­ dence, and composition property, also hold for the topological degree of a

Topological Degrees

139

compact field in a Banach space. We have to point out, however, that Theo­ rems 4.8 and 4.4 impose different conditions on the stability of the topological degrees, respectively. In a finite-dimensional space the homotopy is continu­ ously invariant, yet in a Banach space the homotopy is compactly invariant. Here, that F\ = I — / i and F2 = I — fi are compactly homotopic in a Ba­ nach space means there exists an operator f{x,t) : ft x [0,1] —► X, which is compact with respect to its two variables, such that / ( x , 0) = fi(x)

and

f(x, 1) = / 2 ( x ) ,

x € ft .

4.3 Generalized Topological Degrees of A-Proper Operators In the last section, we used a sequence of finite-dimensional continuous oper­ ators to uniformly approximate a compact operator, establishing the concept of the topological degree for a compact field. In this section, we introduce, via the familiar projective approximation method, the concept of generalized topological degree for a more general class of operators. Let X and Y be Banach spaces and Fn = {Xn, P n ; Yn, Qn} be a projec­ tive approximation scheme, satisfying Xn C X n + i , Qny -+y

Yn C y n + i ,

(n -> 00),

dim Xn = dim Yn,

n = 1, 2, • • ,

VyeY.

We first have the following concept. Definition 4.9. open set, with Q n approximate-proper T n , if fn = Qnf\x satisfying

Let X be a Banach space and ft C X be a bounded = Xn n ft. An operator / : ft —> Y is said to be an operator, or simply an A-proper operator, with respect to : ftn —>Yn is continuous and for any sequence xnic € ftnk fnk{Xnk)~>yeY

(k-*00),

there exist an x € X and a subsequence {xnk.} x

nkj ^>x

ti

-+ °°)

with

such that f(X) = V •

Definition 4.10. Let / : ft —» Y be an ^4-proper operator with respect to r n , and let a € Y\f(dft). The generalized topological degree of / with respect to the point a and the set ft is defined to be Deg(/, ft, a) = {z e Z* |

there exists {n^} C {n} such that degi*(/n fc ,fln fc ,Qn fc (a)) ^ ^

(fc-»oo)},

140

Chapter 4

where Z is the set of integers, {n} is the sequence of natural numbers, Z* = {—ex)} U Z U {oc}, and degB is the Brouwer degree. Here, it must be pointed out that we will certainly have Qn(a) £ fn{d®>n) for large enough n, so that d e g B ( / n , Q n , Qn{a)) is well defined and Deg(/, H, a) is a nonempty subset of Z*. Otherwise, there would be a subsequence xnk € dQ> such that fnk{xnk) = Qnk{o) —> a> as k —> oo. Since / is Aproper, there is a subsequence x nfc . —► x G <9fJ with f(x) = a, contradicting the fact that a £ /(#$)). The following theorem shows that an ^-proper operator is a generaliza­ tion of a compact field, and its generalized topological degree is a generaliza­ tion of the Leray-Schauder degree [34]. Theorem 4.9. Let Q be a bounded open set of the Banach space X, f : Q —► X a compact operator, and F = I — f :Sl —> X the compact Geld. Also, let T n = {XnyQn;Xn,Qn} be a projective approximation scheme, where Xn C X n + i C X, Qn : X —» Xni and Qn converges pointwise to the identity operator I. Then F is an A-proper operator with respect to Tn, and Deg(F,ft,a) = (deg L 5 (F,ft,a)} ,

(4.19)

where degLS is the Leray-Schauder degree. Proof: The operator Fn := QnF\x : Q,n —► Xn is obviously continuous. Let {xnk } be a sequence with xnk € fJnfc = Q Pi Xntc, such that F

nk{xnk)

= Xnk ~ Qnkf{Xnk)

~> V € X

{k - > Oo) .

Then it follows from the compactness of / that there is a subsequence [xnk } such that f{xnkj) -► 2/1 € X (j -» oo). It then follows from the basic properties of the projective approximation scheme Tn that X

nkj

= Fnkj (xnkj)

+ Qnk. [f(Xnh. ) ~ Vl] + <9nfc.?/l

-+ y + yi

(j -+ oo),

so that F x

( n f c . ) = xnkj

- f{xnkj)

-+y + yi-yi=y

{j -+ oo).

The continuity of F then implies that F(y + yi) — yy and so F is an A-proper operator.

Topological Degrees

141

We next verify Equation (4.19). First, we show that for large enough n, \\Qnf(x)-f(x)\\
mm \\F(x)-a\\,

Vxeft.

(4.20)

xEoit

Otherwise, there would be a subsequence {x nj} C ft such that \\Qnif(xn.) - f(xnj)\\ >p, j = 1,2, • • • . (4.21) Since / is compact, without loss of generality we may assume that f{x ) —► yi as j —» oo. Consequently, nj

Qnjixnj)

~

f(xnj)

= Qnj [f(xni) - yi] + [Qnjyi - yi] + [yi - f(xn.)]

-> 0 (j -* oo)

contradicting (4.21). Hence, we have (4.20). Next, we let Xn = X n 0 {a} and ftn = ft fl X n . Then, it follows from the definition of the Leray-Schauder degree that for large enough n, de

SLs(Fi

A , <0 =

de

S i ? ( J - Q n / , ftn, a ) .

Furthermore, according to the shift-invariant property of the Brouwer degree, we have deg L 5 (F, ft, a) = degB(l

- Qnf + Qn{a) - a, ftn, Q n (a)) .

(4.22)

Thus, we can assert that for large enough n, (I - Qnf) (x)+t(Qn(a)

- o) # o,

V x € dft n ,

* € [0,1].

Otherwise, there would be a subsequence {xn.} C dft n C 3ft and a sequence {t n ,.}C [0,1] such that (I - Qn.f)

{xn.)+tnj

(Q

n

» -a)=a

Without loss of generality, let f{xn.) x n i = a + Qnj (f(xn.) -* a + y2

j = 1,2, • • • .

(4.23)

—» y oo. It then follows that

- y2) + Q ni !/2 - t n i (Q n i (a) - <*)

(j -» oo).

Since F is continuous, however, we have F(#o) = a with xo = a + 2/2 € 3ft, contradicting the fact that a & F(dft). Hence, (4.23) holds.

142

Chapter 4

Consequently, by the continuous homotopy-invariant property and the reduction theorem (Theorem 4.6) of the Brouwer degree, for large enough n, we have deg L 5 (F, O, a) = degB (I - Q n / , ftn, Qn(a)) = degB(J-Qn/,nn,Q»(a)) = degB(Fn,Qn,Qn(a)). Thus, (4.19) follows directly from Definition 4.10.

0

Theorem 4.10. The generalized topological degree of A-proper operators has the following properties: (i) (Solvability) If Deg(f,ft,a) ^ {0}, then the operator equation f(x) = a has a solution in ft. (ii) (Domain subdecomposition) If ft = ft\ U ft2 with ft\ n Q2 = 0, and a<£ /{dft^U f(dft2), then Deg(/, n, a) C Deg(/, Qua) + Deg(/, n 2 , a ) ,

(4.24)

where if either degree on the right-hand side is a singleton then the above inclusion becomes an equality. Here, the sum of two integer sets E\,E2 C Z* is defined to be E1 + E2 = { z I z = zi + z2,

*i € £ 1 , 2 2 € E2 } ,

and we use the convention that (+00) 4- (—00) can be any integer in Z*. (iii) (Continuous homotopy-invariant property) Suppose that ft :ftx [0,1] —» y is continuous and this continuity is uniform with respect to x € ft. Suppose also that ft:fi—>Yis A-proper, and a £ ft(dft) for all t € [0,1]. Then Deg(ft,ft,a) is invariant with respect tot € [0,1]. Proof: Part (i). Suppose that Deg(/,Q,a) ^ {0}. Then there exists a sequence {nk} such that degB(/nfc,finfc,Qnfc(a))^0,

*=1,2-...

It follows from the Kronecker principle (Theorem 4.4, Part (i)) that there are xnk 6 ftnk C ft such that fnk(Xnk)

= Qnh(o)

,

k = 1, 2 , • • • .

Since / is Aproper, there exist a subsequence {xnk } and xQ e ft such that #nfci -* x 0

(j —► oo)

with

f(x0) = a.

Topological Degrees

143

But a £ f(dft). Hence, we have XQ 6 ft. Part (ii). Let ftn = ft n Xn and ftnJ enough n, we must have

= ftj n Xn, j = 1,2. For large

Q»(a) £ Qnf(dnnA u an n i 2 ).

(4.25)

Otherwise, there would be a sequence {xn.} C #ft n ,i U dftn^ such that QnJ(xnk) = Qnk(a), fc = l , 2 , . - - .

C #fii U df&2

Since / is an A-proper operator, {xnk } has a convergent subsequence {xnk } such that x nfc . —► xo € <9fh U df&2 and /(#o) = a- This contradicts the fact that a $ f(dfti U 80,2). Hence, (4.25) holds. Consequently, it follows from the domain decomposition property of the Brouwer degree that for large enough n, 2

degB(fnyQn,Qn(a))

= J2de^B(fn^nij,Qn(a))

.

(4.26)

j=i

Pick any z € Deg(/, ft, a). Then there is a sequence {n^} such that deg s (f nk , ftnk,Qnh(a>))-+z

(k-+oo).

There is a subsequence {nfk} of {n&} such that de

gB{fn'h,ton'h,i,Qn'h{a))

-+ zi € Deg(/,fii,a)

(k -> oo),

and there is a subsequence {n'fc'} of {n'fc} such that de

g 5 ( / n - , ^ n - , 2 , < 9 n - ( « ) ) ~> *2 € D e g ( / , f t 2 , a )

(fc -► 0 0 ) .

Therefore, it follows from Equation (4.26) that z = zx + z2 e Deg(/, fii, a) + Deg(/, ft2, a ) , which gives (4.23), where the equality conclusion follows also from Equation (4.26). Part (iii). We first point out that for large enough n, Qn{a) t Qnft(dQn),

t € [0,1].

(4.27)

Otherwise, there would be a sequence {n fc }, with nfe -* oo(A: -» oo) and a sequence {tk} G [0,1] such that xnk € dfJnfc C dft and that On f c /« f c (x n h )=0„ f c (a) l

* = 1,2,.-..

(4.28)

Chapter 4

144

Without loss of generality, let us assume that tk —► to G [0,_1] as k —► oo. Since the continuity of ft in £ is uniform with respect to x € Q, we have ||/t»(*»J-/t.(*»j||->0

(fc-»oo).

This, together with (4.28), imply that Qnkft0{xnk)

->a

(fc-*00).

Note also that / t o is an A-proper operator, there exists a subsequence of {xnk } such that x

nkj -> xo€dQ

(j -> oo)

with

{xnk.}

ftofro) = a,

contradicting the fact that a ^ ft(dQ). Hence, relation (4.27) holds. More­ over, the continuous homotopy-invariant property of the Brouwer degree en­ sures that deg B (Q n /t,r& n ,(5 n (a)) is invariant with respect to t € [0,1], so that the result follows from definition. \\ Finally, we point out that ^4-proper operators also have other properties, such as the shift-invariant property, removability, connectivity, and boundary value dependence (see Theorem 4.4), which can be similarly derived. 4.4 Fixed Point Theorems In this section, we show that the topological degrees introduced in the pre­ vious sections are very useful tools for solving various operator equations. Using these tools, we study the solvability of the general operator equation of the form ( I - / ) ( * ) = 0, xeX, where X is a Banach space, for some operators / under different conditions. A solution of this equation, XQ 6 X satisfying XQ — / ( X Q ) , is called a fixed point of the operator / . The above equation is also called a fixed point equation. We will derive some important fixed point principles in finite and infinite-dimensional Banach spaces. Roughly speaking, the basic idea in this pursuit is to reformulate a fixed point of the operator / as an equivalent problem of computing the operator degree of / , where the latter will be further reduced to a simple operator case by using the stability or homotopy-invariant property of the topological degree.

Topological Degrees

145

4.4.1 Brouwer fixed point and open-set invariant theorems We first introduce the Brouwer fixed point theorem for finite-dimensional operators in this subsection. Let K C Rn be a bounded, closed, and convex set and G be the largest open set contained in K. G is usually called the kernel of K. Lemma 4.3. Suppose that there is no subspace of dimension lower than n that contains the entire K in Rn. Then K = G. Proof: Obviously, G C K. Suppose that K c G i s not true. Then there is an isolated point x0 e K\G, with no other points of G in its small neighbor­ hood. As a consequence, G must be empty. Otherwise, there would be an x e G such that (1 —t)xo+tx € G for any t € (0,1] since K is convex, but this contradicts the fact that xo $ G. Thus, G is empty, namely, the only possi­ bility for K C G being false is that G — 0. In this case, any group of n + 1 points in K must fall into the same (n — 1 )-dimensional hyperplane of Rn. Otherwise, there would be (n + 1) points of K such that the /^dimensional simplex with these points as its vertices would be entirely contained inside the bounded, closed and convex set K. This however contradicts the fact that G = 0. Hence, the closed covex hull of K has a dimension lower than n, which contradicts the assumption of the lemma. \\ Theorem 4.11. (Brouwer's Fixed Point Theorem) Let K C Rn be a bounded, closed, and convex set, and let f : K —► K be a continuous operator. Then f has at least one fixed point in K. Proof: Without loss of generality, suppose that K is not contained in any hyperplane of dimension lower than n; otherwise this hyperplane can be used to replace R?1 in the following discussion. According to Lemma 4.3, we have K = G for the kernel G of K. Consider the operator g.xeK -* g(x) =x- f(x) € Rn . If 0 € g{dG) then / has a fixed point on dG. So suppose 0 ^ g(dG). By the Kronecker principle of the Brouwer degree (Theorem 4.4), it suffices to show that deg(
x0) + tg[x).

146

Chapter 4

Then gt{x) = x-

[x0 + t ( / ( x ) - x 0 ) ] ^ 0 ,

VxedG,

t € [0,1].

Hence, the homotopy-invariant property of the Brouwer degree implies that deg(0, G, 0) = deg(x - x 0 , G, 0) = 1, namely, / has at least one fixed point in K.

\\

To introduce the next theorem we first recall that an operator / , from a set A of a metric space onto a set B of another (probably different) metric space, is called a topological operator from A onto B if it is one-to-one and both / and / _ 1 are continuous. Theorem 4.12. (Open-Set Invariant Principle) Let Q C Rn be a bounded open set and f : ft —> /(ft) C Rn be a topological operator. Then /(ft) is an open set in Rn and for any xo € ft, deg(/,ft,/(x0))=±l.

Proof: Let g be the inverse operator of / . Then for XQ € ft, by the compo­ sition theorem (Theorem 4.5) we have 1 = deg(# o / , ft, xo) =

J2

de

^ > Vi> x o) * de g(/> Q> Vi) >

z>iC/(n)

where V{ is a simply connected region in Rn\f(dtt). must be an i such that

It follows that there

deg(#, Vu xo) • deg(/, ft, 2>«) ± 0 . Thus, according to the Kronecker principle and the inverse relation between / and #, we must have x 0 € g(T>i) and V{ C /(ft), so that / ( x 0 ) € D» C /(ft), with deg(/, ft, A ) = deg(/, ft, /(xo)) = ± 1 . [] Corollary 4.1. A topological operator from a set to another set in Rn maps inner points to inner points and boundary points to boundary points.

Topological Degrees

147

4.4.2 Schauder and Krasnosel'skii fixed point theorems The Brouwer fixed point theorem introduced in the last subsection can be extended to those infinite-dimensional operators that can be uniformly ap­ proximated by finite-dimensional operators. This is the following result. Theorem 4.13. (Schauder Fixed Point Theorem) Let K be a closed convex set in a Banach space X and f : K —► K be a continuous operator such that f(K) is a relatively compact set. Then f has at least one fixed point in K. Proof: Since f(K) is a relatively compact set, Proposition 4.1 implies that the closed convex hull of f{K), denoted K, is also a relatively compact set. Since f(K) C K and K is a closed convex set, we must have K C K. Hence / | - is a continuous operator from K to K. Thus, it follows from Proposition 4.2 that there are a sequence of finite-dimensional subspaces R™* in X and a continuous operator Tj : K —► K C\ iT 71 ', such that IIT.OrO-xll^, 3 Tj(K) C K n BT1',

xeK, j = l,2,... .

Let fj = Tjf:K-+KC) Ft171* C X be an approximant of / . Then clearly fj \Rmy is a continuous operator from Rmj C)K to itself. Hence, it follows from the Brouwer fixed point theorem that there exists an xj e K C\ RmJ C K such that x* = fj(xi) = Tj(f(xi)), j = l,2,.... Since f(xj) e K and K is relatively compact, the sequence {f(xj)} has a convergent subsequence. Without loss of generality, let this sequence be convergent: f(xj)-*x0eK (i->oo). Then, it can be verified that this x 0 is a fixed point of / . Indeed, we have

11^-/(^)11 = \\TAnxj))-f(xj)\\<)->°

tf-oo),

so that xi —► xo as j —► oo. It then follows from the continuity of / that \\xo-Hxo)\\<\\xj-xo\\

+

\\xj-f(xi)\\

+ ||/(s'')-/(*o)||->0

(J-o°),

Chapter 4

148 namely, x0 = f{x0).

Q

We remark that contraction mapping theorems in Banach spaces [5,25] are different types of fixed point theorems. Krasnosel'skii combined the con­ traction principle with the Schauder fixed point theorem, and hence gave a very useful result which shows that a contraction mapping perturbed by a relatively compact operator still has a fixed point. Theorem 4.14. (Krasnosel'skii Fixed Point Theorem) Let K be a closed convex set in a Banach space X, and let g and h both be continuous operators from K to itself, satisfying (i) g(x) + h{y) eKforallx,y€ K; (ii) g is a contraction mapping, namely, there is a a € [0,1) such that \\g(xi) -0(a;2)|| < ot\\xx - x 2 | | for aUxux2 € K; and (iii) h(K) is a relatively compact set. Then the operator f := g+h has at least one fixed point in K. Proof: Pick any z € h(K). Consider the equation ^ = 9(x) + z • It follows from Condition (i) that the operator g(-) + z : K —* K. Since K is a closed set in a complete metric space X, K has a complete metric defined in it. Hence, Condition (ii) and the Banach contraction mapping principle together imply that there exists a unique x = r(z) G K satisfying T(Z) = g(r(z)) + z.

(4.29)

Moreover, r is a continuous operator in h(K). Consider the equation r o h(y) = y,

yeK,

where the composite operator r o h : K -» K is continuous and, clearly, roh(K) is a relatively compact set in K. Hence, it follows from the Schauder fixed point theorem that there exists a yo € K such that r o h(yo) = yo. Let­ ting z = h{y0) in (4.29) yields y0 = g(y0) + h(y0). []

4.4.3 Leray-Schauder fixed point theorem In this subsection, we apply the topological degree theory in Banach spaces to derive a more general fixed point theorem.

Topological

Degrees

149

Theorem 4.15. (Leray-Schauder Fixed Point Theorem) Let X be a Banach space, and the operator equation x = /(t,x),

xeX,

i€[0,l],

(4.30)

satisfies the following conditions: (i) / : [ 0 , l ] x I - > I is compact; (ii) all solutions of the equation, if exist, are uniformly bounded with respect tot € [0,1], namely, \\x\\ < M < oo; and (iii) there is a ball, B, of radius r > M centered at 0, such that deg(Fo, B, 0) ^ 0, where F t (-) : = / ( • ) - / ( * , - ) • Then for each t e [0,1], Equation (4.30) has a solution in B. Proof: Condition (ii) implies t h a t 0 g Ft(dB). Hence, it follows from the homotopy-invariant property of the Leray-Schauder degree and Condition (iii) t h a t d e g ( F t , B , 0) = d e g ( F 0 , B , 0) ^ 0 , t e [0,1]. According t o P a r t (i) of Theorem 4.8, for each t € [0,1], Equation (4.30) has a solution in B. []

Corollary 4.1.

Condition

(iii) in Theorem

/ ( 0 , x ) = xo,

4.15 can be replaced

by

V x € X,

for a given and fixed XQ € X. Corollary 4.2. Let f : X —> X be a compact operator. Suppose that there is a constant M such that for all t € [0,1], the solutions (if exist) of the equation x-t/(x)=0 (4.31) are uniformly bounded: \\x\\ < M. Then for each t e [0,1], Equation (4.31) lias a solution. Finally, we point out that the key in applying the Leray-Schauder fixed point theorem is to find an estimate of the solution of the fixed point equation a priori. 4.4.4 Boundary conditions and fixed point theorems We next investigate the existence of a fixed point for an operator with certain boundary conditions. According to the boundary value dependence property

150

Chapter 4

of a topological degree, we know that the degree depends only on the bound­ ary values of the operator, and so boundary conditions indeed are key to the fixed point problems. The main result in this section is the following. Theorem 4.16. Let Q, be a bounded open set in a Banach space X, with 0 € Q, and suppose that f : Q —► X is a compact operator. Assume, moreover, that either one of the following four conditions holds: (i) The Leray-Schauder boundary value condition: f(x) ^ ex,

V x € <9ft,

c > 1.

(ii) The Altman boundary value condition:

||/(x)-x||2>||/(x)||2-||x||2,

VxedQ;

or, particularly,

||/(x)||<||x||,

Vzedfi.

(iii) The Krasnosel'skii boundary value condition: X is an inner product space, and

(/(*),*) <||x||2,

Vxedn.

(iv) The Rothe boundary value condition: Q is a convex set, and

/(x)eft,

v xedn.

Then, f has at least one fixed point in ft. We remark that in Case (iv) above, by a translation we can remove the condition Q € ft. Proof: Case (i). If / has a fixed point on dft, then the conclusion holds. Suppose that x ^ /(x) for all x e dQ. Let Ft(x) = (1 - t)x + t(x - f{x)) =x-

tf{x).

We can show that 0 £ Ft(dQ) for all t e [0,1]. If not, then there would be an xo G dft and a to € [0,1] such that Fto{xo)=xo-tof(xo)=0. However, since x 0 ^ / ( x 0 ) , we know t0 ^ 1, and since 0 e ft, we also know t0 ^ 0. Thus, we can write x0 = tQ1f(x0), where t^1 > 1. This contradicts the given boundary value condition. Hence, 0 £ Ft(dQ). Consequently, d e g ( J - / , f t , 0 ) = deg(/,ft,0) = l ,

Topological Degrees

151

implying that / has a fixed point in f&. Condition (ii) or (iii) can be used to derive Condition (i), and so the conclusion holds. To show this implication, let's suppose that this is not true. Then there exists an xo € dft and CQ > 1 such that f(xo) = CQXQ. It follows that ||/(*o)-a;o||2=(<*-l)2||zo||2 <{<%-l)\\x<>\\2 = \\f{x0)\?

-\\xo\\\

contradicting to the given boundary value Condition (ii); or (/(a?o),* 0 ) = c o | k o | | 2 > ||x 0 || 2 , contradicting to the given boundary value Condition (iii). _ Case (vi). Since Q is convex, for x € dQ and c > 1, we have ex $ Q. But f(dft) C fi. Hence, f(x) ^ ex, giving Condition (i) and, hence, the conclusion. [] 4.5 Approximate Solutions of Nonlinear Fixed Point Equations In this section, we discuss the projective approximate solvability problem for different fixed point equations. 4.5.1 Projective approximate solvability of fixed point equations We first point out that if we slightly strengthen the existence condition on the fixed points given in Theorem 4.16, then we can immediately obtain the projective approximate-solability for the fixed point equation u = Ku,

ueX,

(4.32)

where K : X —> X is a compact operator on the Banach space X. Let {Xn} be a sequence of finite-dimensional subspaces of X, satisfying Xn C Xn+i for all n > 1, and let Pn : X —► Xn be bounded linear pro­ jections, converging pointwise to the identity operator I on X. Denote, as usual, a projective approximation scheme by Tn — {XnjPn} and consider the projective approximation equations un = K n u n , where Kn := PnK

un € Xn ,

(4.33)

\Xn.

Theorem 4.17. Suppose that there is an open ball B(0,r) of radius r > 0 centered at 0 such that one of the following four conditions holds:

152

Chapter 4

(i) Kx^cx for allxe dB(0, r) and c > 1; (ii) \\Kx-x\\2 > | \Kx\ | 2 - 1 |x| | 2 for all x e dB(0,r), or particularly, \\Kx\\ < \\x\\ for allxe dB(0,r); (iii) X is an inner product space, with (Kx, x) < \\x\\2 for all x e <9B(0, r); (iv) Kx e P ( 0 , r ) for allxe <9£(0,r). Then Equation (4.32) is strongly approximate-solvable, namely for large enough n, Equation (4.33) has a solution un e Xn, with \\un\\ < r, such that {un} has strongly convergent subsequences, each of which converges to a solution of Equation (4.32). Proof: Similar to the proof of Theorem 4.16, either Condition (ii), (iii), or (iv) implies Condition (i). Hence, we only show the strong approximate solvability of Equation (4.32) by the scheme Tn under Condition (i). We first point out that there must be an integer N > 0 such that for n> JV, Knx^cx, Vx €<9B(0,r)OXn, c > l . (4.34) If not, then there would be sequences Xj e dB(0, r) n Xnj and Cj > 1, where rij —► oo as j —» oo, such that PUjKxj

= CjXj ,

j > 1.

(4.35)

Since K is compact and | \XJ \ \ = r for all j > 1, we can assume that KXJ —* ?/ as j —► oo. Since {P n } converges pointwise to I, we have PnjKxj^y

(j-oo).

(4.36)

Therefore, c, := HP^.ifxjU/r is bounded, so that we can assume cj —► c > 1 as j —► oo. This, together with (4.35) and (4.36), imply that Xj —► x G 5P(0, r)

(j —► oo).

Let j -^ oo in (4.35). Then by the continuity of K, we have Kx = cx,

xe <9B(0, r ) ,

c > 1,

contradicting the Condition (i). Hence, (4.34) must hold. Next, we observe that Kn : Xn —► X n are compact, so that Theorem 4.16, Part (i), implies that for n> N, Equation (4.33) has a solution un e X n , with ||w n || < r. Thus, it follows from the compactness of K and the pointwise conver­ gence of {Pn} that the bounded sequence {un} has a strongly convergent

Topological Degrees

1153 53

subsequence un. -> u € 95(0,r) n Xn, which yields PnjKuni j -► ooo Hence, ,etting j -> oo in tht following equations: PnjKuni =un,,= «„,.,

— ►fu, ,a

j = 1,2, • • • •

gives if« = w, namely, w is a solution of Equation (4.32).

Q

Now, if we apply Theorem 4.17 to the following operator equation: (XI-K)u

fex, feX,

= f, f,

A^^Oo,, A

(4.37)

then we obtain the following corollaries. Corollary 4.3. In Equation (4.37), if K : X -► X is a compact operator and there is a constant r > 0 such that Kx + f = fi,x, fix,

xe&B(0,r) x€&B(0,r)

implies /x < A. Then Equation (4.37) is strongly approximate-solvable respect to the scheme Tn. Proof:

with

Let

Kx=\(Kx + /)./). (4.38) Rx=\(Kx v A '' Av Then it can be easily verified that this operator satisfies Condition (i) of Theorem 4.17, and so the conclusion follows. 0 Corollary 4.4. satisfying

In Equation (4.37), if K : X -> X is a compact operator

||tf*||- - 0n

1 R [

(||x||->oo), (IWI-oo),

then for any A jk 0 and / € X, Equation (4.37) is strongly solvable with respect to the scheme Tn.

approximate-

Proof: It suffices to verify that the operator K defined in (4.38) satisfies Condition (ii) of Theorem 4.17. fl Corollary 4.5. satisfying

In Equation (4.37), if K : X -► X is a compact operator |||lif|||:= infill := '"

'"

inf

i\ sup

0 r

U3>
I

154

Chapter 4

then for any A satisfying |A| > |||K||| and for any / e l , Equation (4.37) is strongly approximate-solvable with respect to the scheme Tn. Proof: Let e0 be such that 0 < e0 < |A| - \\\K|||. definition of \\\K\\\ there is an r > 0 such that

Then according to the

sup Mf<\\\K\\\ + e0. IW|>r IFII Hence, for large enough ||x||, we have ||£x||

1 /||iTx||

||/||\

IWI - IAIV IWI \\\K\\\+e0 |A|

iNi; +

1 11/H |A| | N | ^

'

namely, Condition (ii) of Theorem 4.17 holds, so that the conclusion fol­ lows. [1 Corollary 4.6. In Equation (4.37), if X is a Hilbert space and K : X —» X is a compact operator satisfying {K(x),x)

< (#(0),x),

Vxel,

then for any A > 0 and / € X, Equation (4.37) is strongly solvable with respect to the scheme Tn.

approximate-

Proof: When ||x|| is large enough, we have (Kx,x)=(\-1(Kx

+ f),x)

< A " 1 ( X ( 0 ) , x ) + A- 1 (/,x)

^A^di^ojii + ii/iDiwi^iixii 2 , which is Condition (iii) of Theorem 4.17.

\\

4.5.2 Projective solutions of nonlinear integral equations Let fe(t, s, v) and f(t) be continuous functions defined on [0,1] x [0,1] x R1 and [0,1], respectively. Consider the following nonlinear integral equation: u \u(t)= = / fc(t,s, fc(t,s,u(s))dsH-/(t), Jo

A^O.

(4.39)

Topological Degrees

155

Let (Ku)(t):=

f k(t,s,u{s))ds. (4.40) Jo Then Equation (4.39) is indeed a special case of the operator Equation (4.37). To verify this, it suffices to verify that the operator K : C[0,1] —» C[0,1] defined in (4.40) is compact. For any v,v 6 C[0,1], we have \\Kv — Kv\\ = sup / [k(t,s,v(s)) o
SUp |fc(t, 5, v(s)) 0
—

k(t,s,v(s))]ds\ I

— k(t, S, v(s))\

.

This, together with the uniform continuity of k(t, s, v) on [0,1] x [0,1] x [a, b] for —oo < a < 6 < oo, imply that K is a continuous operator. For any v 6 C[0,1], we have \\Kv\\ < sup / |fc(£, s,v(s))\ds o<«
<

sup |A:(t, s, v(s))\, o
I /ifvCti) - Jfifv(t2) I < / Jo < sup

\k{tus,v(s))-k{t2,s,v(s))\ds \k(tus,v(s))-k{t2,s,v(s))\.

0<s
Hence, the compactness of K follows from these two inequalities, the conti­ nuity of &, and the well-known Ascoli-Arzela Theorem [13, 39, 40]. We next discuss the projective approximation scheme Tn for solving the nonlinear integral Equation (4.39). Using the same notation, we can write the projective approximation equations of (4.39) as follows: \un - PnKun

= Pnf,

un € Xn .

(4.41)

We now choose the finite-dimensional subspace Xn and the projection oper­ ator Pn in the following way: Let {<£i, • • •, cj>n} be B-splines defined on the partition 0 = ti < t2 < ■ • • < tn = 1, as discussed in Example 1.1, Chapter 1, and let Xn = span {0i, • • •, n} C C[0,1]. Define Pn : C[0,1] -+ Xn by n

(pnv) (*) = ^ vWW),

Vve c[o, l].

Chapter 4

156 Suppose that the partition is chosen such that hfenn:= :=

max

(t (*i+1 _««)-<) i+1-ti)-*0

(n-» (n-> oo).

*oo), ( »n - >

V » €e C [ 0 ,,1l ] ,

V

l < »i << n - l

Then, it is easy to verify that |\\p | PnVn_v\\^Q V_v||^0

where || • || is the norm of continuous functions denned on C[0,1]. Under this framework, we can rewrite Equations (4.41) as n

n

A^tin^JM*) = £ fa,---,K

+

3j=l= 1

3= 1

Since

n

(Kun) (t^fait)

'£fnti)i(t). 3=1

are linearly independent, the above equation reduces to

\un{tj) {t3)

= (Kun) (tj) + f{tj), f(tj),

j = 1, • • •, • , nn,,

or (4.42) f k{t s,un{s))ds + j = l l, ,. .- .- ,,nn.. k(tjth8,tln(8))ds + f{tA, f(t4), Jo This is a system of n nonlinear algebraic equations, with unknowns {w„(ti), • • -,«»(*»)}, and is actually the approximation equations obtained from the collocation method discussed in Chapter 1. Solving this system of equations usually requires some effective numerical integration methods. In order to obtain the approximate solvability for Equation (4.39), we assume that the kernel function k(t, s, v) denned on [0,1] x [0,1] x R1 satisfies \u Xunn{t (tj)= 3)=

\k{t,s,v)\ 0, p > 0 and 6 < 1. Under this condition, we have, for all v € C[0,1], the following:

W-MIIJT^HI 1

<_i j[a^1|v( jr |,(| s)r -iRih^ s) ^s] (fS] +/?IHI 1 0 0 - j|R| R | + / ? I H I ^ 1^ "

(IHI oo) (IMI^°°)^ -

Therefore, it follows from Corollary 4.4 that for any A ^ 0 and / € C[0,1], Equation (4.39) is strongly approximate-solable, namely, for large enough n,

Topological Degrees

157

Equation (4.41) has a solution un € Xn, and {un} has strongly convergent subsequences, each of which converges strongly to the solution of Equation (4.39). Furthermore, if the solution of (4.39) is unique, then {un} itself converges. We finally remark that different conditions on the kernel function k(t, s, v) can also be imposed, so that the operator K satisfies the other conditions stated in Corollaries 4.3, 4.5 and 4.6, respectively. Then the same strongly approximate solvability conclusion about Equation (4.39) can also be drawn, which however will not be further discussed in this chapter.

158

Chapter 4 Exercises

4.1. Verify Equation (4.4) by using the determinant differential formula from linear algebra. 4.2. Let ft = {(x,y) : x2+y2 < l } . Find the topological degree, deg(/,ft,0), of the following operator / : ft -* R2 with respect to the point 0 and the set ft: (a) f(x,y) = (y,y-x3); (b) / ( * , y ) = ( e * - l , s / 2 ) ; (c) f(x,y) = (y-\x\,x2+y2). 4.3. (a) Let X be an infinite-dimensional Banach space. Show that the func­ tional f(x) = \\x\\x is bounded and compact, but is not completely continuous. (b) Let / : C[0,1] —► I/2[0,1] be an operator defined by f(x) = x. Show that / is completely continuous but is not compact. 4.4. For the topological degree of a compact field in a Banach space, state and verify its (i) Removability, (ii) Connectivity, (iii) Boundary value dependence, and (iv) Composition property. 4.5. Let u=[ui • • • un]T and f{t,u) = [/i(£, u) • • • / n ( ^ ^ ) ] T - Suppose that f{t,u) is continuous on a given rectangular region 71 : \t —1§\ < a and Iw —tzo | < 6, where a and b are positive constants, UQ € Rn, and t0 6 R1. Prove that the initial value problem

(V = /(t,u), | U(t0)

=U0y

has a continuously differentiable solution u = u{t), which is well defined at least in the region \t - IQ\ < h, where h = min{a, b/M} with M = max( t , u ) G tt|/(t,t*)|. Hint: Use the Schauder Fixed Point Theorem to show that its equivalent integral equation u(t) =u0+

f(s, u(s)) ds Jto

has a solution in the region ft=

{u(t)eC[t0-h,t0

C C[to-h,t0

+ h].

+ h] I \u(t)-u0\

<6,

\t-t0\
Topological Degrees

159

4.6. Let u, t, and / be the same as Problem 4.5 above. Suppose that f(t, u) e C1^1 x Rn). Consider the following periodic boundary value problem:

(V = /(*,«), \ u(0) = u(r), where r > 0 is to be determined. Let ft C Rn be a bounded region such that (i) for each c 6 ft, the initial value problem

(u' = f(t,u), \u{0) = c, has a unique solution u = u(t, c), defined at least on 0 < £ < TO for some ro < oc, and for each c € dft, u(t, c)^c

(0 < t < r 0 ) ;

(ii) for every w e M satisfying /(O, w) 7^ 0, deg(/(0, u), fl, 0) ^ 0. _ Under the above conditions, we define a parametric operator Tr : ft —» i? n as follows: for 0 < r/r0 < 1, ii(Y, c) — c

—

TTc = -L2-J

VceO;

T

and for T/TQ = 0, dr

T=0

Prove that for each r G [0, ro], this operator TT has at least a zero in the region ft, so that for any r G (0, ro], the original periodic boundary value problem is solvable. 4.7. Consider the following boundary value problem: x" = / ( t , x , x ' ) , 0
(a) (6)

where / € C([0,1] x R1 x i? 1 ) and a 0 , a i , $>, /?i are all constants. Sup­ pose that (i) in the space of polynomials of first degree, only zero satisfies Con­ dition (b); and

Chapter 4

160

(ii) the continuously differentiable solution, if exists on [0,1], of the following parametric boundary value problem f x " = A/(i,x,x'), 0
(c) W

is uniformly bounded with respect to the parameter A € [0,1] in the norm of the space C1 [0,1]. Under the above conditions, prove the following results: (1) The boundary value problem (a) and (b) is equivalent to the integral equation x(t)=

(d)

f G(t,s)f(s,x(s),x'(s))ds,

7o

in which

G(t, ,) = (*_ ,)+ -

°° + f°* a 0 p i -t- aim

+ POPi

[ai +A(l - *)] ,

where 0 < s,t < 1 and (t - s)+ = t - s if t > s and = 0 otherwise. Also, here, the equivalence means that if x(t) e C*2[0,1] is a solution of problem (a) and (b) then x(t) satisfies the integral Equation (d); and if x(t) € C^O,1] is a solution of the integral Equation (d) then x(t) € C 2 [0,1] and satisfies the problem (a) and (b). (2) Use the Leray-Schauder Fixed Point Theorem to prove that the boundary value problem (a) and (b) has a solution in C 2 [0,1]. 4.8. Let X be a Banach space and K : X -» X be an operator satisfying (i) K is compact; and (ii) there exists a bounded linear operator K^ : X -* X, with WK^W < 1, such that \\Kx-Ko6*\\_ ||x||moc ||x|| "UProve that the equation x = Kx is strongly approximate- solvable with respect to the scheme Tn = {Xn,Pn}, where Xn C Xn+1 C X for all n = 1,2, • • •, and Pn : X — Xn is a bounded linear projection operator converging pointwise to the identity operator / as n -> oo. 4.9. Let / ( t , x ) € C([0,1] x R1) and fc(t,») € C([0,1] x [0,1]). Define two operators F and AT, from L2(0,1) to L 2 (0,1), by (Fx)(*) = / ( « , x ( t ) ) ,

Vx(t)€L2(0,l),

(Kl/) (t) = / fc(t, s)i/(s) ds, JO

V y(t) e L 2 (0,1).

Topological Degrees

161

(1) Prove that K o F is a compact operator from £2(0,1) to £2(0,1). (2) Suppose that /(£, x) and k(t, s) also satisfy the following conditions: |/(t, a?) - x| < M ,

L

V t € [0,1],

x e R1,

( |fc(t,s)|2
where M > 0 is a constant. Prove that for any g(t) G ^2(0,1), the intergal equation x{t)=

f k(t,s)f{s,x(s))ds Jo

+ g(t)

has a solution in L 2 (0,1), and is strongly approximate-solvable with respect to the projective approximation scheme Tn = { X n , P n } , where both Xn and Pn are the same as that defined in Problem 4.8 above.

CHAPTER

5

Nonlinear Monotone Operator Equations and Their Approximate Solutions

In this chapter, a class of important nonlinear operator equations — the nonlinear monotone operator equations — and their perturbations will be investigated. The study of monotone operators theory has more than thirty-year his­ tory, and has reached a fairly high, profound and complete level [41]. To date, monotone operator theory has found many important applications in various areas such as nonlinear differential and integral equations, nonlinear control systems, and fluid dynamics. In this chapter, a theoretical framework will be established for monotone operator equations in the real space setting unless otherwise indicated, with emphasis on solvability and approximate solvabil­ ity problems. Moreover, their perturbed equations will be studied, along with some applications of the established framework for numerical solutions of boundary value problems of some typical nonlinear differential equations. Some fundamental concepts are first reviewed in Section 5.1, including continuity, derivatives and differentials of nonlinear operators. Then the basic theory of nonlinear monotone operators is reviewed in Section 5.2. Section 5.3 discusses the approximate solvability of monotone operator equations. The so-called If-monotone operator theory is then introduced in Section 5.4. Numerical solutions of boundary value problems of some typical nonlinear differential equations are finally discussed by examples in Section 5.5.

163

Chapter 5

164 5.1 Continuity, Derivative and Differential of Operators

To prepare for further theoretical development throughout the rest of the book, we need some more fundamental concepts and results in the Banach space setting, which are of central importance in dealing with many nonlinear analytic problems. 5.1.1 Continuity of operators It has been seen, from the studies of various operator equations in the last few chapters, that in order to guarantee the solvability of an operator equation and the convergence of approximate solutions to the unknown true solution, a certain continuity of the operators involved in the equation is extremely im­ portant and necessary. For linear operators, the problem of continuity is quite simple, since the concepts of boundedness, continuity and weak-continuity etc. are all equivalent. However, the situation is much more complicated for nonlinear operators. To analyze the basic properties of a nonlinear operator, different concepts of continuity and convergence are needed. Definition 5.1. Let X and Y be Banach spaces, with a subset G C X. An operator F : G —► Y is said to be (i) hemicontinuous on G, if for any x € G and h € X such that x + th G G, we have F(x -f th) w-+ Fx as t —► 0; and (ii) demicontinuous on G, if for any x € G and {XJ} C G such that Xj —» x, we have FXJ w-^> Fx as j —► co. Recall also from Definitions 3.1 and 2.1 that the operator F : G —> Y is said to be weakly continuous on oo; and is said to be completely continuous if for any x € G and {XJ} C G such that Xj w—► x, we have FXJ — ► Fx as j —► oo. The following implications are immediate. Proposition 5.1. Complete continuity = » continuity ==> demicontinuity ==> hemicontinuity. On the other hand, complete continuity ==> weai con­ tinuity = » demicontinuity. Definition 5.2. An operator F : G —► Y" is said to be locally bounded on G, if for each XQ € G there exists a ball U{XQ,8) centered at xo with radius 6 > 0, such that sup ||Fx||y < oo. xeu(x0,s)nG

Monotone

Operator

Equations

165

Obviously, t h e local boundedness of F on G is equivalent to t h a t for any Xo G G and {XJ} C G such t h a t Xj —> xo (j —> oo), t h e sequence { . F X J } is bounded in Y. Hence, t h e following result holds. Proposition 5.2.

Demicontinuity

= > local

boundedness.

It is well known t h a t in a finite-dimensional space t h e strong and weak convergence are equivalent, and so t h e above classification of continuity is not necessary. For linear operators, we have t h e following result. Proposition 5.3. A linear operator is hemicontinuous. Also, for a linear operator, continuity 4=> weak continuity <£=> demicontinuity. Proof: T h e first part follows from definition. As to t h e second p a r t , we recall from Proposition 3.1, Chapter 3, t h a t a continuous linear operator is necessary weakly continuous, and by Proposition 5.1, weak continuity im­ plies demicontinuity. On t h e other hand, t h e demicontinuity implies t h e local boundedness; and for linear operators this means t h e continuity. [] For a nonlinear operator defined on an infinite-dimensional space, dif­ ferent concepts of continuity and boundedness have different meaning and properties. Hence, it is very important to understand t h e continuity and boundedness of an operator in t h e study of nonlinear operator equations and t h e convergence of a sequence of approximate solutions to t h e t r u e solution. 5 . 1 . 2 . D e r i v a t i v e a n d differential of o p e r a t o r s Similar to Calculus, the concepts of derivative and differential can be defined for operators, and t h e n be used for local approximation of nonlinear operators by linear operators. To introduce these concepts, in this section we always let X and Y be Banach spaces, with an open subset G C X, and let F : G —> Y be a suitably defined nonlinear operator. Definition 5.3.

For an XQ € G, if for every h G X t h e limit hm t-»o

F(x0 +

th)-Fx0 t

exists, then t h e operator F is said to be weakly differentiable, or Gateaux differentiable, at XQ, a n d t h e limit is called t h e weak differential, or Gateaux differential, of F at XQ with respect to its increment h, which will be denoted by DF(XQ, h). Furthermore, if t h e weak differential is linear in h, t h e n there

166

Chapter 5

exists a linear operator DF(x0) independent of h, such that m F{xo±th)-Fxo hiimF(x 0+th)-Fx0= t->o

that need not be bounded but is

:X-*Y,

yv f c

^

(5.1)

tt

and is called the derivative of F at x0. Moreover, F is said to be weakly differentiable, or Gateaux differentiable, on G C X if it is so at every point xo € G. Obviously, it follows from (5.1) that F(x0 + th) = Fx0 + DF(x Z)F(xo) a(x00, , t, h), 0) (th) +1 ■ a{x

(5.2)

where a(x0, t,h) eY satisfies the property that for any h € X, | |a(x 0 , t, h) | | y -» 0 as t ->■ 0. Also, the above expression can be rewritten as = Fx0 + DF(i DF(x00)) (x {x - xo) Fx = ((

,, ■■

iiII

* — — #0 *o

x

+ ||x - xo||x • a ( x 0 , ||x - xollx, ,, _x

X

,,

\\

J ■

//

(5.3)

X

If, in addition, the derivative of F at x 0 , DF(x0), is bounded and if ||a(x 0 ,t,/i)||K -» 0 uniformly with respect to h on the sphere ||ft||x = 1 as t -+ 0, then we have the following concept. Definition 5.4. For an x0 € G, if there is a bounded linear operator A{x0) :X-^Y such that

Hfcfi% Hfcfi%

[|F(xo +

fe)-Fxo-^(x0)fe||y

pu

~ '

(

}}

then the operator F is said to be strongly differentiable, or Frechet differen­ tiable, at x 0 , and the linear operator A{x0) is called the Frechet derivative of F at x 0 , which will also be denoted by DF(x0), i.e., A(x0) = DF(x0). Moreover, the term DF{x0)h is called the strong differential, or Frechet dif­ ferential, of F at x0 with respect to the increment h. Moreover, F is said to be strongly differentiable, or Frechet differentiable, on G C X if it is so at every point x 0 € G. The Frechet derivative, if exists, is unique. Indeed, Formula (5.4) implies that for any x in the ball U(x0,6)r\G, centered at x0 with some radius 6 > 0, we first have \\Fx - Fx0 - i4(x P(\\x A(xo) II* -~ xM\x), 0 ) (x - x 0 )||y < ||x - x0|U • P( 0\\x),

(5.5) (5-5)

Monotone Operator Equations

167

where (3{r) - ^ O a s r - 4 0-h Furthermore, if there is another bounded linear operator A(xo) that satisfies also \\Fx - Fx0 - A{x0) (x - x0)\\y < \\x - x0\\x

• 7( 11^ -

x0\\x),

where 7(r) - > 0 a s r - 4 0-h, then 11 A{x0) - A{x0) 11 < /?(r) + 7 ( r ) ,

0 < r < 6,

so that ||A(xo) - ^.(^o)ll —> 0 as r —► 0. Hence, A(x 0 ) = A(x0). It is clear that if an operator has the Frechet differential then it also has the Gateaux differential, and they are identical. On the other hand, if an operator F has a bounded derivative DF(xo) and the limit (5.1) exists uniformly on the sphere \\h\\x = 1, then it also has a Prechet derivative, and they are identical. If an operator F has a derivative, DF(x), on G C X, where x G G, then DF is a (linear or nonlinear) operator mapping from G to C(X, Y), i.e., DF : G —► £(X, F ) , where as usual we use C(XyY) to denote the family of linear operators from X to Y. Moreover, if DF(x) is bounded for every x G G, then DF : G —> CB(X, Y), where similarly we use CB{X, Y) to denote the family of bounded linear operators from X to Y. It is well known that equipped with the usual operator norm, £ # ( X , Y) is itself a Banach space. Next, suppose that the operator F : G —» Y has a derivative DF(x), x € G C X and D F : (7 —+ £j3(X,y). We can furthermore consider the derivative of DF. If indeed DF has a derivative, D(DF) (x0), at the point xo G G, then it is called the second derivative of F at xo, which will be denoted by D2F(x0). Note that D2F(x0) is a linear operator from X to CB{X,Y). Hence, for any (increment) k G X, we have the operator D2F(xo)k G CB{X, Y). Furthermore, for any h G X, by applying the linear operator D2F(xo)k to it, we have the value D2F(xo)kh G F , which will be denote by D2F(xo){k,h). Very often, D2F(x0)(k,h) is symmetrical with respect to k and fe, namely, D2F(x0)(k,h)=D2F(x0)(h,k), and, in this case, we can consider the second derivative as a bilinear operator mapping from X x X to Y. Higher-order derivatives can be defined in the same way, which we will not pursue any further, however. We next give some basic properties of the derivative of an operator. Theorem 5.1. Let X and Y be Banach spaces, with an open subset G C X. Let F : G —» Y be a (linear or nonlinear) operator.

Chapter 5

168

(i) If F is strongly (weakly, resp.) differentiable at xo € G, then F is continuous (hemicontinuous, resp.) at xo. (ii) If F is linear from X to Y, then for every x e X, DF(x) exists, with DF(x) = F. IfF is a constant operator, i.e., Fx = yo eY for allx e X, then DF{x) = 0. (iii) (Linearity) If two operators Fj : G —► Y, j = 1,2, have derivatives at x0 e G, DFj(x'o), then for any constants a and b, the operator aFi + bF2 also has a derivative at x§, with D(aF1 + bF2) (xo) = aDFxixo) +

bDF2(x0).

Moreover, if Fj, j = 1,2, are strongly (weakly, resp.) differentiable at xo, then so is aF\ + bF2. (iv) (Chain rule) Let X, Y and Z be Banach spaces and Gi C X and G2 C Y be open subsets. Let Fi : (?i —► G2 and F2 : G2 —► Z be operators. If F\ has a derivative DF\(xo) at XQ G G I and F2 is strongly differentiable at yo = Fi(xo), then the composite operator F2 o F\ has a derivative at xo, denoted D(F2 o F\) (x0), and D{F2 o Fi) {x0) = DF2{y0) o DFx{xo),

(5.6)

namely, D(F2 o Ft) (xQ)h = DF^yo^DF^ti).

(5.6')

Proof: Parts (i)-(iii) follow directly from definitions. To verify Part (iv), observe that F2 is strongly differentiable at yo, and so F2(y0 + Ay) - F2y0 = DF2(y0)Ay

+

a(Ay),

where

|KAy)||z<||Ay||y-/?(||Ay||y), in which (3{r) -* 0 as r —* 0+. Consequently, ^ [ ( ^ 2 0 Fx) (xo + th) - (F 2 o F1)x0] = DF2(y0) ■ 1 [Fi(x 0 + th) - FlXo] + 1 a ( F ! ( x 0 + *fc) - F i x 0 ) ,

Vh€X.

Thus, since DF2(yo) is a bounded linear operator from Y to Z and since Fj has a derivative at xo, it can be seen that as t —> 0, the first term on the

169 169

Monotone Operator Equations right-hand side above has a limit DF2(y0)(DF1(x0)h) therein gives * / i ) - F i xxo0 ) - a ( F i ( x 00 -+M

11 X

and the second term

11 z

< | | i [F^xo [Fdxo + th) - F Fixo] +th) th)-F - FtxoWr) y • P{\\F x{xo + 1xo]\\\\ Y•P(\\F 1{xo 1x0\\r) -> WDFUXOWIY ||DFi(xo)W|y ■ • lim lim /?((r -* / ? ( ( r== 00,, r—0+ r_>0+

which implies the relation shown in (5.6).

Q

Next, we point out that although the mean-value formula in Calculus does not hold for nonlinear operators in general, but we still have the following result in a weaker form. Theorem 5.2. Let X and Y be Banach spaces and G C X be an open and convex subset. LetF:G^Y have a derivative on G. Then (i) If the derivative DF(x) is bounded for each xeG, then for any xux2e G there exist 0U92€ (0,1) such that FX.WY < \\DF(x 0i(x(x2 2 -- Xi))|| \\Fx2 - FXiHy \\DF(xix + 01 Xl))\\ •• ||x 2 - * i l U ,,

(5.7) (5-7)

and | | F x 22 -Fx \\Fx - F x 11-DF(x - D F ( xl1)(x ) ( x2-x 2 - 1x)\\ 1 )Y| | y < |||| D < F ( X 1 + 0 2 (2x(x - -xi))-I?F(xi)||•||x2-x x i ) ) - Z > F ( x i ) | | • | | * 2 - X!\\x J DF(xi+^ 1 |U. 2 2 (ii) IfY = X*, then for any xux2 (0,1) such that

.

(5.8) (5.8)

€ G and h € X there exists a0 = 6(h) €

(h, Fx2 - Fxj) Fxx) = (h, DF(Xl + 0(x 9(x2 - xj)) xi)) (x2 -- xj), xj),,,

(5.9) (5.9)

where (h, £) is the value of I e X* at h € X. Proof: Part (i). A direct application of Inequality (5.7) to the operator Fx - DF(xi) (x - xi) gives immediately y5.8)8 .ence, it tuffices s t oerify Inequality (5.7). If Fx! = Fx2, then (5.7) holds. Suppose, therefore, Fxi ^ Fx2. By the Hahn-Banach Theorem [13, 40] (see Appendix A.6.1), there exists a n f e T such that € ( F x 2 - F x 1 ) = \\Fx | | F x22-Fxi\\Y• -Fx1||y• (5.10) ||l|| i{Fx2-Fxi) ||l|| = 1,

170

Chapter 5

Let ip(t) = £(F(xi 4- t(x2 - *i))),0 < t < 1. It can be verified that y>(i) is everywhere differentiate on (0,1), with tp'{t) = £{DF(x1 + t(x2 - x 2 )) (x 2 - x{)) . It then follows from this derivative, Equation (5.10), and the Lagrange meanvalue formula, that \\Fx2-Fx1\\Y

=
+ 01(x2 - xt)) (x2 - a?i))

< \\DF(X! + 91{x2 - xi))|| • ||*2 - * i | | x . Hence, Inequality (5.7) holds. Part (ii). Let
sup

\\DF(x)\\ • ||x 2 - x i | | x ,

(5.11)

ac = x 1 + t ( x 2 - a ; 1 ) 0
and, if F has a bounded second derivative at each x G G,

\\Fx2-Fx1-DF(x1)(x2-x1)\\Y <\

sup

||£2F(x)|H|x2-xi|&.

(5.12)

0
Here, (5.11) follows directly from Theorem 5.2, Part (i). To verify (5.12), we consider F ( x i + t(x2 — xi)) as a function taking values in space Y, and write Fx2-Fx!=/

— F(xi+t(ar2-xi))dfc

= /

D F ( x i + t(x 2 - Xi)) (X2 - Xi) eft

= /?F(xi)(x2-xi) + / [£>F(xi + *(x2 - xi)) - DF(xi)] (x 2 - xi) dt. Jo

Monotone Operator Equations

171

Then, by applying Inequality (5.11), we have ||Fx 2 - Fxt - DF(Xi)

(x 2 - Xi)||y

< / \\DF(xi + t(x2 - xi)) - DF(xi)\\ ■ \\x2 Jo <

\\D2F(x)\\-\\x2-x1\\2x

sup

[

x=xi+t(x2-x1) 0
< i

tdt

JO

||D2F(x)||.||*2-zi|&.

sup

Z

xi\\xdt

i=x1+t(i2-a:i) 0
Corollary 5.1. Let X and Y he Banach spaces and G C X be an open convex set. Suppose that operator F : G —► Y has a bounded derivative at each x e G. Then (i) If DF is bounded in G, then F is continuous in G; and (ii) If DF is continuous in G then F is strongly differentiable in G. Furthermore, we have the following extension of the above results. Theorem 5.3. Let X and Y be Banach spaces and G C X be an open convex set. Suppose that operator F : G -+Y has Frechet derivatives up to the nth order in G. Then for each xo € G and h G X, satisfying XQ + h € G, there exists a 0 € (0,1) such that F(x0 + h) - Fx0 - DF(x0)h 1

(n-1)! where DkF(x0)hk

■~Dn-1F{xQ)hn^1

= DkF(x0)

<-t

\\DnF(xo

+

0h)hn\\Yy

(fc, • • •, fe) with k variables in h.

Proof: Let ^n— 1 TPfry.

—D^F^U\un—l

yo = F(xQ + h) - Fx0 - DF(x0)h

(n-1)! If yo = 0 then the conclusion follows. So suppose that yo ^ 0. Then by the Hahn-Banach Theorem, there exists an £ € Y* such that ||*|| = 1 Let tp(t) = e(F(x0+th)),

and

% o ) = ||yo||y.

with 0 < t < 1. Then


+ th)hk),

k = 1, • ■ •, n .

172

Chapter 5

By the Taylor expansion formula in Calculus, there exists a 0 € (0,1) such that n—1 ..

*,(!):

1

^ (fe) (0) +

°(»)-



fe=0 fc=0

Hence, we have £(yo) = -e(DnvF(xo n!

+

0h))hn, "

so that n n Hitolly F(x0 + Oh)h I M I r = % o ) < ^i \\D HU-FOro 0ft)ft n\\||yy •.

0

Two remarks are in order. First, if F : G -* Y has continuous nthorder Frechet derivatives, then we have the following Taylor formula with a residual: 1 F{x )h +■■■+ F(x0 + ft) =Fx0 + DF(x0)h+■■+ * ( n - 1)! n 1

Dn-11^*,,)**"-1 F(x0)hn-1 U"-

/ ( l - t ) n - 1 £{l-t) +th)hn n F-( D x 0 F(x + 0t/i)/i J0 ([n-l)\ n - 1 ) ! Jo

-+T ,^ - T*T T,

n

n

Indeed, by using the following Taylor formula (with a residual, for the function y>(t) = £(F(* 0 +*ft))):

(fc) 1 (B) vM = £ET^ T^lv j k\ (°) + (n - 1)! y0A -*)-V (*)A,

f^k\

(n-l)\J0

and the fact that t € Y* is arbitrary, the conclusion follows immediately. Second, under the above conditions, we have the following estimate: \\F(xo FXQ F(ar0 + ft) - Fx 0 - DF(x0 )h

-r

DnF(x0)hn\\

n! ! II ™ l lly ly {1(1 * (^Ty.Io (^i "-'^[^* ^ H [^*+*> M ** i ** + *> - -J ^™M n n n n < -1, sup \\D | | l ^F(x F (0a H+ , + t f t ) - Dth)-D F ( a ;F(xo)\\-\\h\\ o ) | | - | | f t | &x. n\ o
We next show that when Y = R1, namely, when F is a functional on X, we not only have the Lagrange mean value formula but also have a finite Taylor expansion formula. We first review some useful concepts. Definition 5.5. Let X and Y be Banach spaces, and G C X be an open set on which there defines a functional F. Then the weak differential DF(x0, ft)

Monotone Operator Equations

173

of F at xo G G is also called the variation of F at xo with respect to the increment h £ X, and is denoted by dhF(x0). Also, the derivative DF(XQ) of F at xo G G is also called the gradient of F at xo, and is denoted by gradF(xo). Finally, if every gradient gradF(x) is bounded at x € G, then gradF is an operator mapping from G to the dual space X* of X, and so is called the gradient operator. Theorem 5.4. (The Lagrange Mean Value Formula for Functional) Let X and Y be Banach spaces, and G C X be an open and convex set on which there defines a functional F . If for each x G G, the variation 5/ l F(x) exists, then for any x i , x 2 G G, there is a 0 G (0,1) such that F x 2 - F x i = a ( x 2 _ X l ) F ( x ! + 0(x 2 - x i ) ) . Proof: Let (^(t) = F ( x 1 + t ( x 2 - x 1 ) ) . Then we have
0

Theorem 5.5. (The Finite Taylor Expansion Formula for Functional) Let X and Y be Banach spaces, and G C X be an open and convex set on which there defines a functional F.IfF has nth Frechet derivatives, then for each xo G G and h G X, satisfying xo -f h G G, there is a 0 G (0,1) such that F ( x 0 + h) = F x 0 + DF(x0)h + • • • + , * . ( n - 1J!

Dn-1F{xQ)hn~1

n + ^-D F(x 0 + ^ ) ^ n , n!

where DkF(x0)hk

= DkF(x0)

(fe, • • •, ft) with k variables in ft.

Proof: Let
th)hk.

174

Chapter 5

Consequently, applying the function Taylor formula yields

^1)

=

Erf^ (fc) W

+

^^(n)W'

o<0
so that

F(x0 + h) = Y,l

DkF(x0)hk + - DnF(x0 + 6h)hn .

Q

k=0

5.2 Monotone Operators from a Banach Space to Its Dual Space In this section, we study a special class of operators, the monotone operators, in a Banach space setting. Throughout this section, we let X be a Banach space and X* be its dual space. For any £ e X*, its value at x € X* is denoted as usual by (x, £). 5.2.1 Monotone operators We first introduce the concept of monotone operators. Definition 5.6. Let G C X be an open set and F : G —► X* be an operator. F is called a monotone operator on G if for any xi,X2 € G, (x2 - xuFx2

- Fxt) > 0 .

(5.13)

A monotone operator F is said to be strictly monotone, if the equality in (5.13) holds only when x\ = x%To familiarize the concept of monotone operators, we first show some examples. Example 5.1. Let f(x) be a real-valued non-decreasing function of x G R1. Then / is a monotone operator (monotone function), which is perhaps the simplest example. Example 5.2.

Let F be a linear and positive definite operator satisfying (x,Fx)

> c\\x\\x,

Vx € X ,

for some constant c > 0. Then F is a (linear) monotone operator. As an example, if X = V is a Hilbert space in which there defines a ^-elliptic bilinear form a ( , •), namely, a(v,v)>c\\v\\l,

VveV,

then the induced linear operator F : u € V —► a{u, •) 6 V* is positive definite. This operator was discussed in Chapter 3.

Monotone Operator Equations

175

Example 5.3. Consider the following problem from the study stu of plasticity elasticity twist phenomenon:

!\-IUT,& -£(«?>&)--, K)-*..> -eir"*^ -^rln)Ty < ) ;=»<*■»>• - * \u(x,y)

= 0,

^»'€t

{x,y)edQ,

where SI C R2 is a bounded region, # G L2(ft), <> / is a given function determined by the properties of the material under stress, and

MSD'*®'The corresponding weak form of the problem is to find a u 6 H^{Q) such that a(u v\=

fI f 6(Tu) MTu)
= m,drdu. = jf[ ff[ gvdxdy,

dx7 dx7 Xy

vVv Vn€H£(U). ^«=^Hf^iO)I .

Here, H$(Sl) is the first-order Sobolev space defined in Q with zero bound boundary a n . Note that the left-hand side of the equation actu; actually value conditions on 8Q. defines a nonlinear operator F : H^(Q) -» [H^{Q)]] via t

. _ v

r

(v, Fu) = a{u,v),

+ . _ . -i *

Wv€H^{il),

and the right-hand side is an element Ug 6 [H&(0)]*, ».c, {v, Hg)=

j

I gv dxdy,

Vv €

H£(0).

Hence, the weak form of the problem is equivalent to solving the following operator equation: Fu = = TZg. Ka. Now, we point out that under a certain condition (for instance, when {t) > Co > 0), the operator F defined above is a monotone operator. Indeed, under this condition, for

Chapter 5

176 any Ul,u2 € H^(n) we have ( t X -2 -ui, t i i ,Fu F «2 2-- FFu « i )x) («2

= ^//>TU2)+
[/ao^_-«i) y i

+ -tfTiiO] (T«w22" - T«0 + I\ Ji Ij ^(Tui) hTw2)" ^(Twi)](T Tui)dxdy dxdy

-//J(^)M(^)> >0.

Example 5.4. Consider the gradient operator of the functional defined in a Hilbert space V. For x ^ 0, we have

}{x)-\\x\\v

f(x th)-f{x) ||x _ <X,h) ,. /(x + + th)-/(x) ,. ||g + + tMlv-Hx||^ tfeHy-||xH|r {x,h)vv Vfce V , 1 *™ * " *t-™ ||x||v) £3 t ™, t(||x *(||x + +tfc|| *Mlv l N k~) "||x||v l R i 7' ' v ++

with g r aadd| | x| x| ||v| =v - 7 * r V ,, IFIIV

| | g r a d ( | | x | | vv ) | L |v = l ,

and (| |vx=||x|| | | v .v. Next, define an operator by fgrad||x||v, \ 0 ,, U

0, x ^# O x X= = 0. 0.

Then for any x l 9 x 2 € V, we have <X x 22--xa ir ,i F, Fxx2 2--FFxx11 > v = ||x2||v + Hxillv - (x <X2,Fxi>v (xi, Fx22)vv 2 ,Fx 1 )v - <xi, > I||x2|k M I v + Hxillv 11*1 Hv -- ||Fxi||v||x ||Fxi||v||x22||v ||v -- ||Fx ||Fx22||v||xi||v ||v||xi||v == 0, 0, showing that F is a monotone operator. We next give some equivalent conditions for monotone operators.

'

Monotone Operator Equations

177

Theorem 5.6. Let G C X be an open convex set, and F : G -+ X* be an operator. Then F is a monotone operator on G if and only if one of the following conditions is satisfied: (i) For any i 6 ( ? and h € X satisfying x -f h G G, the function (fx,h{t) = {h, F(x -f- th)) is nondecreasing as t increases from 0 to 1. (ii) The operator F is locally monotone in the sense that for each x € G there exists a ball of radius 6 > 0 centered at x, denoted U(x,6), such that F is monotone on the set G n C/(x, 6). (iii) Under the condition that F has a derivative on G, (h,DF(x)h)

>0,

VxeG,

VheX.

Proof: Part (i). The necessity follows from the following equality: (t 2 - h) [(pXlh{h) -
-

Fxi)

= (x2 - xu F ( x i + (x 2 - xi)) =

(

fxi7x2—xi (1 j

— (

Fxi)

Pxi,X2— x\ (y) ■

Part (ii). The necessity is obvious. As to the sufficiency, we let #i, x<2 e <2, and then show (x2 — xi, F x 2 — F x i ) > 0. To do so, we only need to show the function ¥?W = (x2 - Xu F(X! + t{x2 ~ Xi))) is nondecreasing as t increases from 0 to 1. By the locally finite covering property [35], we only need to show that for any to € [0,1], the function (fito 4- r ) is nondecreasing in a neighborhood of r = 0. Indeed, by the sufficient condition, there exists a 8 > 0 such that the operator F is monotone in the set G n (/(xo, 6), where xo = x\ + to(x2 — xi). Hence, the function ^(s) = (v - x 0 , F ( x 0 + s(v - x 0 ))) is nondecreasing as s increases from —1 to 1, where 6

V = X0 +

T:

jT- (X 2 ~ X i ) .

||X2 - X i l l x

Chapter I5

178 Since

« - > = ( T i a < « - « ) - F ( - + i i ^ ' * ■ - * ■ >)) = =

8 ( 8s \ — 7 7 ^ ¥» (
f-l^ar that that, t.h* fnnct.ion Vcp(t cn(t, nondecreasintrin theneighborhood it isi clear at the <^(< (i he function {t0 0n+ +4-rr) r))r^lis isisnondecreasing nondecreasing ininthe tt the neneighborhood \<S/\\x l\\ . \T\<S/\\X \\ . 2 XX -x ^ i l lllx2 XI xPart (iii). an let x € G ) . Necessity. Necessity. Suppose Suppose that that F F is is monotone on on GG, and id fth € X. Thenn for small enough th and andheX. t > > 0, we have i x + tfc eG tteGand and enourfi 0. t f t e i an

xx + F(F(x FF x ^ Q„ ^/ , F Fix ^tlt ) -Fx ++ th)-Fx\

(h^ (h, t {h,

?)>o. ^- ')>o.

Letting etting £ -► -+ -* 00+ (++ gives givesthe the result. result. As Asto tothe the sufficiency, sufficiency, we werecall recall from fro from Theorem Theorem 5.22 (ii) that there is a 09 € (0,1) such that 0h)f), (h, F{x h) - Fx) = (h, F(x + h){h, DF(x + 6h)h), which hich yields the result immediately.

[]

Next, we establish an important result that the gradient o] operator of a convex >nvex lvex functional is a monotone operator. We first recall the cconcept of a convex >nvex lvex functional. Definition efinition finition 5.7. Let G C X be a convex set and J(x) be ia functi. functional denned efined ined on G. J(x) is called a convex functional if for any xxi,X2 x1l; x2 defined € G and v a, b>>>00satisfying satisfying aa+ wehave havethe the inequality inequality a,b>0 ++bb==1,1,we aJ(xi) J(aXl + 6x2). aJ( + bJ{x2) bJ(x2) > J{a Xl) LetGcX Proposition 5.4. Let G C X be an open convex set and J(x) be a functional functionai defined on G.IfJis weakly differentiabJe on G, then either of the following denned conditions is necessary and sufficient for J to be a convex functional: (i) Foranyxux2€G, J(xi) - J(x2) - (xi (x! --x2x,2, gradJ(x 2 )) > 0. (ii) gradJ(x) is monotone on G. » J is convex. Proof: We show that J is convex =* (i) = »* (ii) = ==►

Monotone fonofone Operator Equations

179

J is convex = > (i). Since J is convex, for any xi,X2 G G and t € (0,1], wee have J ( s 2 + t(xx t(gi - x))) ~ J(x2) J(x2) ^ tJ(xx) U(Xl) + (l - t)J(g2) - J(x 2a ) t ~ t = J(xi) - J ( x 2 ) . Letting ->■> 0+ ijerang tz — u-h gives gives (xi (xi -- xx 22 ,, gradJ(x gradJ(x 22 )) )) < < J(xi) J ( x i ) -- JJ (( xx 22 )) .. (i) (i) = => > (ii). (ii). It It follows follows from from (i) (i) that that for for any any xxuuxx222€ €€ G, G, we we have have (xi ( H - x 2 , gradJ(x 2 )) < J(xi) J(X!) - J ( x 2 ) , and

(x ( x2-xi,g™dJ(xi))<J(x , gr ar da Jd( ^x xi2)-J(xi). )O) <^ J (fxo2J) - J (f xo i)).. 22- -xxi ,i g

Adding Adding these these two two inequalities inequalities together together yields yields -(x )-gradJ(xi))<0, -(xo 2d - ( x2-xi,gradJ(x , g.r aeiadJ(xo) J ( x 22 ) - g- r sarda.d7J(f x ,i ^) ) < 00. , 22 --x ixi which implies (ii) > JJ (ii) = = (ii) => > J we define we define

that gardJ(x) is monotone on G. is gradJ(x) is monotone on G, any xi, X2 € G is convex. convex. Since Since G, for for is convex. Since gradJ(x) gradJ(x) is is monotone monotone on on G, for an any xi, X2 € G

p(t) = J(txi + (1 - t)x t)x2)2 ) - tJ(xi) t J(xi) - (1 - t)J(x t) J ( 2x),2 ) ,

t G [0,1].

ihow that (t) is fort tGG[0,1]. [0,1]. Suppose Supp differentiable •ble on [0,1], we have cp'(t) gardJ(toi 0 0 (0)>0 (/?(») >0

and and

= O. O. yp'(0) ^0) == O.

V J'(0)

Thus, Thus, by the monotonicity of gradJ(x), for t G (0,1] we have .itj.\ J/J.\ Jia\ '(0) ip'(t) =='(*) ^'(0) V'(9)
= 0)x 9)x = (( xx :: -- xx 22 ,, gradJ(te gradJ(te 11 + + (( (( -- t)x^) t)x^) -- gradJ(9x gradJ(9x11 + + (( (( -- 9): 9)x222)) )) t)x 0)x t)x > (0xi Xi + > _^ L__ (txi (txi + + (1 (1 -- t)x t)x222 -- (^ (0xi + (1 (1 -- 0)x 0)x222))) ,, gradJ(txi gradJ(txi + + (1 (1 -- t)x t)x222)) r — cr v - g rraaddJJ(^0f xex i + (( 1 - ^0^)Xx 2a)^) > ^0.

Chapter 5

180

This implies that tp(t) is nondecreasing on [0,1], which contradicts the fact that (p(0) > 0 and 0,

VxGG,

then FXQ = yo. Proof: It suffices to show that (h, F x o - y o ) = 0 ,

V

heX.

For this purpose, we observe that by the given condition for any h € X we have (th, F(xo + t/i) — yo) > 0, for small enough \t\. Hence, it follows that (th, Fx0 - yo) = (th, Fx0 - F(x0 + th)) + (th, F(x0 + th) - y0) > (th, F x 0 - F(x0 4- t h ) ) . Then, dividing both sides by t ^ 0 gives (h, Fxo-yo)

> -(fc, F ( x 0 - f th) - F x 0 ) ,

t >0,

(h, F x o - y o ) < -(fe, F ( x o - h t h ) - F x o ) ,

t <0,

so that, by letting t —► 0 and using the hemicontinuity of F , (h, F x o - y o ) = 0 ,

\fheX.

0

We recall from Subsection 5.1.1 that the demicontinuity implies both the hemicontinuity and the local boundedness. We now show that for monotone operators, both the hemicontinuity and the local boundedness together imply the demicontinuity in a reflexive Banach space setting.

Monotone Operator Equations

181

Proposition 5.5. Let X be a reflexive Banach space and G C X be an open subset. Then a monotone operator F : G —* X * is demicontinuous if and only if F is both hemicontinuous and locally bounded on G. Proof: According to Propositions 5.1 and 5.2, we only need to verify the sufficiency. For any xo € G and {XJ} C G satisfying Xj —> xo (j -+ oo), the local boundedness of F implies that {FXJ} is bounded in X*. Since X is reflexive, there exists a weak*-convergent subsequence {Fxnj} with a weak*-limit y € X*. In the following inequality (x - xn., Fx - Fxnj)

> 0,

WxeG,

j = 1,2, • • • ,

letting j —► oo gives (x — XQ , Fx — yo)>0,

V x € G.

By Theorem 5.7, we have yo = FXQ] that is, any weak* convergent subse­ quence of {FXJ} converges to the same limit FXQ. Hence, lim (h,Fxj)

= (h,Fx0),

VheX.

0

j—*oo

We next point out that for an operator defined on an open set, monotonicity implies local boundedness, so that the above proposition implies the equivalence of the hemi- and demi-continuities. To show this, we first need the following lemma. Lemma 5.1. Let X be a Banach space, with {XJ} C X and {yj} C X*, such that XJ-^XQ

and

\\yj\\x* -» oo

(j -> oo).

Then, for any r > 0, there exist an x e X, satisfying \\x\\x < r, and a subsequence {ynj} in X*, such that (xnj - xo - x, yn.) —► - o o

(j -> oo).

(5.14)

Proof: Suppose it is not true. Then there exists an ro > 0 such that for every x € S = {x\x G I , \\%\\X < ^o} there is a constant c(x) that satisfies (XJ -XQ-X,

yj) > c(x),

j = 1, 2, • • • .

(5.15)

Chapter 5

182 For each integer k > 1, set Sk = { x | x € S, (XJ - x0 - x, yj) > -k,

j = 1,2, • • • } .

It is clear that Sk is a closed set and, by (5.15), S = U^=1Sk- Since S is itself complete in metric, it follows from the Baire property of a Banach space [40] (see Appendix A.3.2) that there exists an Sko which contains a closed ball U(x,8) of radius 8 > 0 centered at some x € X, with \\x\\x < ro and 0 < 6 < r0 - \\x\\x, s u c n t h a t Inequality (5.15) implies (2(XJ - x0) + x - x , yj) > -k0

+

c(-x),

VxeU(x,6),

j = l,2,- •,

that is, (2{XJ - x 0 ) + x , %■) > -fc 0 + c ( - x ) ,

V \\x\\x <6,

j = 1,2, ••• .

From this inequality and the fact Xj —> x 0 (j —► oc), it follows that for large enough j , (*,%) > -k0 + c(-x),

\\x\\x < g •

Replacing x by - x , we also have ( x , % ) < A:0 - c ( - x ) ,

||x||x < - .

Thus, by combining these two inequalities, we obtain I (x,Vj) | < | - k0 + c ( - x ) I,

||x||x < - ,

j »

1,

which implies that

11%-llx* <

2| - /c0 + c ( - x ) |

jr-^

,

3» 1 •

This contradicts the assumption that ||s/j||x* —► oo as j —> oc. Proposition 5.6. Let X be a Banach space and G C X be an open set. If F : G —» X* is a monotone operator, then F is locally bounded inside G. Proof: Suppose not. Then there exist an x 0 € G and a sequence {x^} C G such that Xj->

x()

and

\\FXJ\\X*->

oo

(j — ► oo).

[]

Monotone Operator Equations

183

It then follows from Lemma 5.1 that there exist an x € X and a subsequence {xn.} such that x0 + x € G and (xn. - x 0 - x, Fxnj)

—> - o o

(j —► oo).

Since F is monotone, we have (xn. - x 0 - x, Fxnj)

> (xn. - x 0 - x, F(x 0 + x)) .

Letting j —^ oo in the above yields a contradiction: (x, F ( x 0 -fx)) > + o o . Hence, F must be locally bounded on X.

[]

Finally, by combining Propositions 5.5 and 5.6, we arrive at the following conclusion. Proposition 5.7. Let X be a reflexive Banach space and G C X be an open set. Let F : G —> X* be a monotone operator on G. Then F is demicontinuous on G if and only if F is hemicontinuous on G. 5.2.3 Strongly monotone operators Again, we let X be a Banach space and G C X be an open set. Recall from (5.13) that a monotone operator F : G —► X* is strictly monotone on X if (x 2 - xuFx2

- Fxi) > 0,

Vx1,x2€G,

x1 ^ x2 .

Obviously, a strictly monotone operator on X must be one-to-one. However, the operator equation

Fu = f,

feX\

need not have a solution u € G for any / € X*, even if F is a strictly monotone operator. In order to guarantee that for every / € X* the above operator equation has a unique solution, and to ensure the strong solvability of projective approximation schemes, we have to impose stronger monotonicity condition on the operator F (see, for example, [25, 38, 41]). Definition 5.8. An operator F : G —► X* is called a strongly monotone operator on G if for any x\,x a ( | | x 2 - x i | | x ) ,

Chapter 5

184

where a(t) : [0,oo) -> [0,oo) is a strictly monotonically increasing function satisfying o(0) = 0. For a strongly monotone operator defined on the entire space X, we have the following equivalence conditions [42]. Theorem 5.8. For an operator F : X -+ X*, the following conditions are equivalent: (i) F : X -> X* is a strongly monotone operator. (ii) There exists a strictly increasing function r(«) : [0, oo) -» [0, oo) satisfying r(0) = 0 and r{t) - * < » ( « - ♦ oo), such that for all xux2e X, (x2 - xi, X!, Fx2 -- Fxi) FXl) > ||X2 ||x2 - xxxlU i | U • r(||x 2 - *xii|l| U x ))•. (iii) There exists a function v(t) : (0, oo) -+ (0, oo) such that with xx±x2,

forallxux2eX

( x 22 - x 11 , F x 22 - F x 11 ) > ^ ( | | x 22 - x i1 l| U ) . Proof: The implications (ii) =*■ (i) =*> (iii) are obvious. Hence, it suffices to verify the implication (iii) = » (ii). For this purpose, let r(t)=

inf

-- (x2 - Xlxi,, Fx22 - FFx Xlx),),

te(0,oo).

It follows from Condition (iii) that r(t) > 0 for every t > 0. We will show that r(h V tuut2t2 € (0, oo). (5.16) r(t1+t+1 (0,oo). 2) > 2)>r(t 1) r(h) + r(t2), Indeed, for any xux2eX

with ||x 2 -x1\\x=h+12,

we have

—!—(x2-x1,Fx2-Fx1)

'H^k^-^-F''-F{-^kn+rrh^)) +

hXirh^-^F(^h^irh-)-F^

> r ( *t i ) ++r (r(< * 2 2)),, which is (5.16). On the other hand, (5.16) shows that r(t) is strictly increas­ ing. Finally, repeatedly applying (5.16) yields r(«)>r([t])>[«]r(l) r(t)>r([t})> [*]r(l) — ooo, o,

(t -► oo), (t

where [t] is the largest integer that is less than or equal to the real number * € (0,oo). By defining r(0) = 0 and using the definition of r{t), we obtain («)• []

Monotone Operator Equations

185

5.3 Approximate Solvability of Monotone Operator Equations We are now ready to discuss the monotone operator equations: their unique solvability and protective approximate solutions. 5.3.1 Monotone operator equations In this subsection, we let X be a reflexive and separable Banach space and A : X —► X* be an operator. Consider the following equation: Au = f,

f€X*.

(5.17)

We first establish an important result on the existence of a solution to this operator equation in the finite-dimensional case. Proposition 5.8. Let G c f f 1 b e a nonempty bounded open set with 0 € G, and F : G —» FT1 be a continuous operator. If

(x,Fx)>0,

VxedG,

then the homogeneous operator equation Fu = 0 has at least one solution on G. Proof: Suppose the opposite, namely, Fx ^ 0,

V x e G.

Let Ftx = (1 - t)x + tFx, 0 < t < 1. Then Fox = x^0,

FiX = F x ^ 0 ,

WxedG,

and

(x,Ftx)

=

(l-t)\\x\fx+t(x,Fx)

>(l-*)||x||x,

V*€(0,1),

Vx€#G.

Consequently, we have Ftx^Q,

Vt€[0,l],

xedG.

Thus, it follows from the continuous homotopic invariance of topological de­ grees that deg(F, G, 0) = deg(F 0 , G, 0) = 1.

186

Chapter 5

According to the Kronecker Principle (Theorem 4.4, Part (i)), we know that Fu = 0 has a solution in G, which contradicts the assumption. \\ Now, observe that X is separable, and so we can choose a sequence of finite-dimensional subspaces of X, { X n } , with Xn C X n +i and X = U^=1Xn. Consider the discretized problem: Find a un € Xn such that (v1Aun)

= (vJ),

V^eXn,

n=l,2,.-..

(5.18)

As usual, let Pn : X —► Xn be a bounded linear projection, and P* : X* —► X* be its dual operator. Then, we have a projective approximation scheme r n = {X n , Pn}, and so obtain a sequence of approximate equations of (5.17) as follows: Anun=P^f, (5.19) where the operator An = ^ n ^ l x • Clearly, no matter how Pn is defined, Equation (5.19) is always equivalent to Equation (5.18). Hence, we will only discuss Equation (5.18) in the following, where as usual we will denote the approximation scheme by Tn = {Xn}. We have not yet specified the operator A in Equations (5.17) and (5.18). We now introduce a new concept of coercive operators. Definition 5.9. An operator A : X —► X* is said to be coercive with respect to / € X*, if there is a constant r > 0 such that (x,Ax-/)>0,

VXGX

with

||x||x=r.

(5.20)

Then, we have the following important result. Theorem 5.9. Let X be a separable reflexive Banach space and A : X —► X* be a hemicontinuous monotone operator. Suppose that A is coercive with re­ spect to an f e X*, and {Xn} is a sequence of finite-dimensional subspaces in X , satisfying Xn C X n + i and U ^ 1 X n = X. Then Equation (5.17) is weakly approximate-solvable with respect to the approximation scheme Tn — {Xn}, in the sense that for each n= 1,2, • • •, the approximate Equation (5.18) has a solution un e Xn, the sequence {un} has a weakly convergent subsequence, and each of such subsequence converges to the solution of Equation (5.17). Proof: Without loss of generality, let / = 0, dimX n = n, and Xn = span{wi, • • •, Wn}- We first show that there exists a « n G X n , with ||w n ||x < r, such that (wj,Aun)=0, j = l,...,n. (5.21)

Monotone Operator Equations

187

For this purpose, let U n = / ^

'3^3 >

5= 1

and =
C = [ci,---,Cn]

,

I '


X^^'^l

ll i = 1

llx

f-

)

Then consider the operator : G —► i? n defined by

$(c)=[$ 1 (c),... ; $ n (c)] T : where A

(^c3w3jy

W,

*<(c) =

i = 1, • • • , n .

It follows from the given conditions and Proposition 5.7 that 3>: G —* i?71 is a continuous operator, satisfying

(c, ^ ( C ) ) = ( £ C ^ ^ ( E C ^ ) ) ^°>

VcedG.

It then follows from Proposition 5.8 that there exists a c G G such that $(c) = 0 , which is (5.21). Observe, moreover, that {un} is a bounded sequence in a reflexive Banach space, it has a subsequence {unj} such that unj

w

->u0eX,

(j->oo).

Thus, for any v € Xk, we have (v - it n , , Av) = (v - unj , At; - Aun.) + (v - wn, , A u n i ) . When n, > A:, v - w ni € X nj ., so that (v - unj, Aun.) = 0. Hence, (v - unj , Av) > 0,

V v e Xk ,

n3>k.

Letting j —► oo gives (v — wo , Av) > 0,

V t; G Xk .

Chapter 5

188

Observe that \J™=1Xn = X, and that the hemi- and demi-continuities are equivalent for a monotone operator. Hence, we actually have (v —

UQ

, Av) > 0,

V v G X.

Thus, it follows from Theorem 5.7 that Auo = 0.

[]

We remark that if A is strictly monotone, then Equation (5.17) is weakly and uniquely approximate-solvable with respect to T n , namely, the approx­ imate Equation (5.18) has a unique solution un, with un w-> uo as n —» oo, where UQ is the unique solution of Equation (5.17). Corollary 5.2. If the coerciveness condition stated in Theorem 5.9 is re­ placed by the following general coerciveness condition: lim ^ l = o o , IMIx-oo \\x\\x

(5.22)

then for any f € X*, the conclusion of Theorem 5.9 holds. Moreover, we have the following result. Theorem 5.10. Let X be a reflexive and separable Banach space, and A : X —* X* be a hemicontinuous and strongly monotone operator. Suppose that {Xn} satisfies Xn C Xn+1 and U^Xn = X. Then for any f e X*, Equation (5.17) is uniquely and strongly approximate-solvable with respect to the scheme Tn — { X n } , that is, Equation (5.18) has a unique solution un, satisfying un —► uo as n —» oo, where u$ is the unique solution of Equation (5.17). Proof: The equivalent Condition (ii) of Theorem 5.8 on the strong mono­ tonicity can be applied to show that the strong monotonicity condition of this theorem implies the general coerciveness condition (5.22). Hence, for any / € X*, Equation (5.17) is weakly and uniquely approximate-solvable with respect to the scheme Tn. What is left is to show that un —► u§ as n —► oo. To do so, we first observe that since {Xn} satisfies Xn C X n + i and is ultimately dense in X, which implies that there exists a Vj € Xj for each j = 1,2, • •, such that *>j -► uo ,

(j ~>

oo).

It then follows from the strong monotonicity of A and Equation (5.18) that when n > j , <x(\\un - Vj\\x) < (un - Vj , Aun - AVJ) = (un -Vj,

f - AVJ) .

Monotone Operator Equations

189

Thus, since un ™—» uo (n —■> oo) and Vj -* no (j -+ oo), it follows from the demicontinuity of operator A that lim j—too

lim

(un —Vj,f

— AvA = 0,

n-+oo

so that lim

lim a( \\un - vA\x) = 0.

j—>-oo n—>-oo

Consequently, for any e > 0, there is a jo > 0 such that lnh~ a(\\un

- vjo\\x)

< a(e/2),

\\vjo-uQ\\

< e/2.

n—>-oo

It then follows that there is an N(> jo) such that oc(\\un -vjo\\x)

< <*(e/2),

n>N.

By the strict increasing property of a(t), we have

IK-^olU <^/2,

n> N.

Hence, when n > N, we have I K - u o | | x < \\v>n - vjo\\x + \\vjo -u0\\x

<e,

which proves un —»> UQ as n —► oo, as expected.

[]

We remark that if Theorem 5.10 applies to linear operators, since a linear operator is always hemicontinuous, we do not need any continuity condition imposed on the operator. Hence, the following conclusion can be drawn. Corollary 5.3. Under the conditions of Theorem 5.10 on the spaces X and {X n }, if A : X —► X* is a positive definite linear operator, then for any f £ X* Equation (5.17) is uniquely and strongly approximate-solvable with respect to the scheme Tn. We should remark that this result improves some results that we ob­ tained in Chapter 3. More importantly, here we no longer require the uniform boundedness condition for the sequence of projection operators that are used in a general projective approximation scheme: | | P n | | < c for all n = 1,2, • •. As a matter of fact, the Lax-Milgram Theorem (Corollary 3.1) can be considered as a special case of Theorem 5.10 when the operator is linear, and its conditions can be formally weakened. More precisely, we have the following [7].

Chapter 5

190

Corollary 5.4. (Modified Lax-Milgram Theorem) Let V bea separable Hilbert space and o(-, •) bea bilinear functional defined on V x V. Suppose that for every u eV, a(u, v) is a continuous functional ofveV and that a(-, •) is positive definite: a(v,v) > a | M & ,

VveV,

for some constant a > 0. Then for any f € V*, equation a(ti,v) = ( t ; , / ) ,

V v € V\

is uniquely solvable. Proof: It amounts to note that in this case the equation under consideration is equivalent to Au = / , where A is the operator induced by a(-, •), namely, (y, Au) = a{u, v),

V v € V.

Hence, the unique and strong approximate-solvability implies the unique solv­ ability. [] It should be pointed out that if we only consider the existence of a solu­ tion but not the approximate solvability of the equation, then the separability of the space is not needed. This can be verified by introducing the so-called finite intersection property and using the compactness theory of a topological space [5]. Note that Theorem 5.10 only points out that the hemi- or demi-continuity is a sufficient condition for the unique and strong approximate-solvability of a strongly monotone operator equation. We next point out that this condition is also necessary. Proposition 5.9. Let X be a reflexive Banach space and A : X —► X* be a strongly monotone operator. If Equation (5.17) is uniquely solvable for any f € X*, then A must be demicontinuous. Proof: Axn

w

Let {xn}

C X such that xn —> x0 as n —► oo. We show that

—+ AXQ as n —> oo.

Since A is monotone, it is locally bounded. Hence, {Axn} is bounded, so that there is a weakly convergent subsequence Axn. w-+ y e X* as j —► oo. According to the assumption of unique solvability of Equation (5.17), there exists an x e X such that Ax = y. Since A is strongly monotone, we have a

( Ikn,- - x\\x)

< (xn. - x, Axn. -

Ax).

Monotone Operator Equations

191

Letting j —> oo yields lim a ( | | x n - -x\\x)

=0.

Furthermore, since a(t) is strictly increasing and a(0) = 0, we have xn. —► x, so x = xo and Axnj. ™—► Axo as j —► oo. This has also verified that every weakly convergent subsequence of {Ax n } converges weakly to AXQ, namely, {Ar n } weakly converges to Axo, which implies that A is demicontinuous. [] We next derive some error bounds, under some conditions on the oper­ ator A, for the approximate solutions discussed above. Theorem 5.11. Suppose that under the conditions of Theorem 5.10, u and un are solutions of Equations (5.17) and (5.18), respectively. (i) If operator A is bounded, then there exists a constant c > 0 such that r(\\un-u\\x)-\\un-u\\x
inf vexn

{\\u-v\\x},

(5.23)

where r(t) is the function defined in Theorem 5.8, Part (ii), satisfying the strong monotonicity equivalent condition stated therein. If r(t) = r0t for some constant ro > 0, then \\un-u\\x<^/^

{\\u-v\\%2}.

inf UtA

(5.24)

n

(ii) If operator A is Lipschitz continuous, namely, for any given M > 0 there exists a constant CM > 0 such that \\Ax1 - Ax2\\x* < CM • Iki - x2\\x ,

V x l 7 x 2 e X , ||*i||x, \\x2\\x<M, then there is a constant c > 0 such that r(|K-«|U)
inf

{\\u-v\\x}.

(5.25)

Ifr(t) = rot for some constant ro > 0, then

IK-«IU<-

inf {||«-t;||x}.

ro vexn

(5-26)

Chapter 5

192

Proof: Part (i). Since A is strongly monotone, for any v e Xn: we have r(\\un -u\\x)

-|K

-u\\x

< (un - u, Aun - Au) = (y — u, Aun — Au) <\\Aun-Au\\x.\\u-v\\x.

(5.27)

Since {un} is a bounded sequence a n d A is a bounded operator, respectively, there exists a constant c > 0 such t h a t \\Aun — Au\\x* < c. Hence, t h e error estimate (5.23) holds. P a r t (ii). T h e given conditions a n d t h e result (5.27) established above together yields the error estimate (5.25) immediately. []

5.3.2 The perturbation problem In this subsection, we consider the perturbed operator equation Au + Bu = f,

(5.28)

where A : X —► X* is a monotone operator and B : X —► X* is assumed to be completely continuous. Let {Xn} satisfy Xn C X n + i and be ultimately dense in X. Denote by Fn = {Xn} the corresponding approximate scheme as usual, and then consider the approximate equations (v.Atin + Bun) = ( « , / ) ,

VveXn,

u = 1,2, • • • .

(5.29)

We have the following result. Theorem 5.12. Let X be a reflexive and separable Banach space and A : X —* X* be a hemicontinuous monotone operator and B : X —> X* be a completely continuous operator. If for the given f € X* operator A-\- B is coercive, then Equation (5.28) is weakly approximate-solvable with respect to the scheme Tn. Proof: Without loss of generality, let / = 0. Since A + B is demicontinuous and coercive, similar to the proof of Theorem 5.9 we can show that the approximate Equations (5.29) have solutions un € Xn, with ||w n ||x < r f° r some constant r > 0 for all n = 1,2, • • •. It then follows that there exist a subsequence {unj} and a UQ G X such that uUj

w

-> u0,

(j -* oo).

Monotone Operator Equations

193

Thus, for any v € Xk, when nj > k we have (v -unj,

Av +

Bunj)

= (v - unj, Av - Aunj) > (v-

unj, Aun. + Bunj)

+ (v- un., Aunj +

Bun.)

= 0.

Since B is completely continuous, letting j —► oo leads to (v - w0 , Av + Buo) > 0,

Vt/elfc.

Since {X n } is ultimately dense in X and A is demicontinuous, the above in­ equality holds for all v € X. Hence, according to Theorem 5.7, AUQ + BUO =

o.

D

In the above, we have seen that when a monotone operator has a com­ pletely continuous perturbation, if the operator after perturbation is still co­ ercive, then the approximate solvability of the equation remains unchanged before and after the perturbation. In the following, we examine the case where the operator is strongly monotone. Theorem 5.13. Let X be a reflexive and separable Banach space, A : X —» X* be a hemicontinuous and strongly monotone operator, and B : X —► X* be a completely continuous operator. Let {Xn} satisfy Xn C Xn+\ and be ultimately dense in X, with the approximation scheme denoted by Tn = {Xn}. Suppose that for the given f € X* the operator A-\- B is coercive. Then Equation (5.28) is strongly approximate-solvable with respect to the scheme Fn. Proof: Based on Theorem 5.12, we will furthermore prove that unj —► u$ as j —> oo. To do so, observe that {Xn} satisfies Xn C X n + i and is ultimately dense in X, and so there are uk € Xk, k = 1,2, • • •, such that w^ -> u 0 as k —» oo. For any k > 1, when nj > k we have, by the strong monotonicity of A, the following: a

(

ll^n,- - &k\\x)

< {unj

- Uk , Aun.

= (unj -uk,

-

Auk)

f - Auk - Bunj) .

It then follows from the complete continuity of B and the demicontinuity of A that lim lim (un - uk, f - Auk - Bun) = 0, so that lim k-+oo

lim a ( \\un - uk\\x) j—K>O

= 0.

Chapter 5

194

Similar to the proof of Theorem 5.10, we can finally verify that unj —► u 0 as j —> oo. [] 5.3.3 Some remarks on the complex Banach space s e t t i n g When X is a complex Banach space, we can slightly modify the definitions of monotonicity and coerciveness. Definition 5.10. Let X be a complex Banach space and G C X be an open set. An operator F : G —> X* is called a monotone operator on G if for any Xi,X2 E G,

Re (x2 - xi, Fx2 — Fx\) > 0 , where Re(-) is the real part of a complex function (or number). A monotone operator F is said to be strictly monotone, if the above equality holds only when #i = X2. Moreover, F is called a strongly monotone operator on G if for any x\,x a( ||x 2 - x i | | x ) , where a(t) : [0, oo) —► [0, oo) is a strictly monotonically increasing function satisfying a(0) = 0. Definition 5.11. Let X be a complex Banach space. An operator A : X —> X* is said to be coercive with respect to / G X*, if there is a constant r > 0 such that |(x,

A E - / ) | ^ 0 ,

VXGX

with

||x||x=r.

Moreover, A is said to be general coercive, if hm IMIx-oo

,, ,, ||x||x

= oo.

Under these modifications, all results given in Theorems 5.9-5.13 as well as their corollaries still hold true. To verify this, it amounts to modifying their proofs accordingly, by taking into account the following remarks. 1. To prove the corresponding Theorem 5.7, it suffices to replace the lefthand side of the given condition (inequality) by its real part, and then to use it to replace t in the proof, where i = \/-T> since in doing so it follows that the imaginary part of (ft, F x 0 - t/o) equals zero. 2. To prove Proposition 5.7, it suffices to observe that Lemma 5.1 still holds when restricted to the real parts.

Monotone Operator Equations

195

3. To establish the existence of approximate solutions, the following Browder Theorem [3] can be used to replace Proposition 5.8. Theorem 5.14. (The Browder Theorem) Let E be a finite-dimensional Banach space with dimE > 1, and let A : E —> E* be a continuous operator satisfying (x,Ar)^0,

with

VXGE,

||x||£7 = r ,

for some constant r > 0. Then there exists a u G E, with that Au = 0.

\\U\\E

< r, such

5.4 Solvability and Approximate Solutions of K-Monotone Operator Equations In this section, we consider the Hilbert space setting, and study the solvability and approximate solutions of K-monotone operator equations. 5.4.1 K-monotone operator equations Let if be a Hilbert space, with inner product (•, -)H and the induced norm l l l l / / . Let V be another Hilbert space with inner product [•, -] v and induced norm | • \y. For a nonlinear operator A : V(A) C V —> if, where V{A) is a dense set in V, consider the operator equation Au = f,

feH,

(5.30)

and study its approximate solvability problem. We will introduce a bounded linear operator K : V -» H and use it to establish a framework of K-monotone operator equations [10, 12]. We first have the following concept. Definition 5.12. An operator A : V(A) C V —> H is said to be K-monotone if there exist a bounded linear operator K : V —> if, satisfying K(V) = H with respect to the norm || • ||H> such that (Ax2 - Axi, K{x2 -

X1))H

> 0,

V xi, x2 G V(A),

Moreover, operator A is said to be strongly K-monotone if (Ax2 - Axi, K(x2

-XI))H

> <x(\x2-xi\v),

V x i , x 2 G V{A),

for a strictly increasing function a(t) : [0, oo) —► [0, oo) satisfying a(0) = 0. Finally, operator A is said to be K-coercive if hm I*I V ^°° xe-D(A)

| (Ax, ——~

Kx)H\ — = oo . \x\v

Chapter 5

196 Now, let a(u, v) = {Au, a{u, (Au, Kv)H, , f(v) = (f,Kv)H,

uu€G V(A), V{A), vzV.

v€V, v€V,

Clearly, for each u € V{A) and / € H, both a(u, •) and /(•) are continuous linear functional on V. Hence, it follows from the Riesz Representation Theorem that there exist a unique Au € V and a unique Kf €V such that (Au,Kv) {Au,Kv)HH=[Au,v] v,

,

(f,Kv)HB=[Hf,v] =[llf,v]vv,, {f,Kv)

Vv€V, WveV.

Then the following result can be established. Proposition 5.10. Suppose that the operator A introduced above satisfies the following condition: (Hi) For every w € V and every Cauchy sequence {vn} in V{A) V{A) C V, (Avn-Avm,Kw)H-+>

(n,m->oo).

Then the operator A has a demicontinuous extension, A:V-^V. A : V -> V. Proof: We extend the operator A to the entire space V as follows: If v € V{A) then let Av = Av. If v € V\V(A), since V{A) is dense in V, there is a sequence {vn} C P(i4) such that vn -* / 6 V ss - -♦ o.. Thus, ac­ cording to Condition {Hi), we know {«4»„} C V is a weak Cauchy sequence. Consequently, there is a v G V such that v = (weak) limn-oo Avn. It is clear that v only depends on » but is independent of the choice of {vn}. Hence, we can let Av = v. The above construction shows that A : V -> V ii s demicontinuous operator. 0 Based on this proposition, we can continuously extend the functional a(-, •) to a two-variable functional denned on V x V by a(u,v)=

[Au,v]yv,

u,veV. u,veV.

This continuous extension is obviously unique. Next, we introduce the concept of generalized solution for Equation (5.30) as follows: A u G V is said to be a generalized solution of Equation (5.30) if a{u,v) f(v), VveV, a(u,v) = f(v), \lveV, (5.31)

Monotone Operator Equations

197

namely, [Au,v]v

= [nf,v]v,

VveV,

(5.31')

or equivalently, Au = Kf,

feH.

(5.31")

To find approximate solutions for the above equations, we choose a se­ quence of finite-dimensional subspaces Vn C V such that Vn C F n + i

and

U~ =1 Vn = V ,

and use the approximate scheme Tn = {Vn}. Under this framework, the approximation problem is to find un G Vn such that a(un,v) = f{v),

VveVn,

(5.32)

namely, [.Atin , v]y = [Kf, v]v yveVn.

(5.320

If we choose Vn C T)(A), then the approximate equation is (Aun ,Kv)H=(f,Kv)H,

VveVn,

(5.32")

where n = 1,2, • • •. We have the following result. Theorem 5.15. Suppose that operator A satisfies Condition (Hi). (i) If operator A is K-monotone and K-coercive, then for any f G H Equa­ tion (5.30) is (in a generalized sense) weakly approximate-solvable under the scheme Tn, namely, the approximate Equation (5.32) has a solution u>n £ Vn f°r eacn n — 1>2,---, and the sequence {un} has a weakly convergent subsequence, and the limit of any such weakly convergent subsequence is the generalized solution of Equation (5.30). (ii) If operator A is strongly K-monotone, then for any f G H Equation (5.30) is (in a generalized sense) uniquely and strongly approximatesolvable under the scheme Fn, namely, the approximate Equation (5.32) has a unique solution un G Vn for each n — 1, 2, • • •, and the sequence {un} is strongly convergent to the unique generalized solution of Equa­ tion (5.30).

Proof: According to Proposition 5.10, the extension A of the operator A is a demicontinuous operator from V to V. If A is if-monotone and_/f-coercive, then by going through a limiting process it can be verified that A : V —» V is

Chapter 5

198

monotone and coercive, since V(A) is dense in V and A is demicontinuous. Thus, we have [x2 - X\ , AX2 ~ Axi] and hm |x| v ->oo

> 0,

V Xi, X2 € V ,

_ [ar,Ac] v . . = oo. p|v

Here, the Riesz Representation Theorem enables us to identify V* with V. To this end, Theorem 5.9 and its corollary together imply that Equation (5.30) is (in a generalized sense) weakly approximate-solvable. If A is strongly Kmonotone, then similarly by going through a limiting process we can verify that A : V —> V is strongly monotone: [x2 - x

u

Ax2 - Axi]v

> a(\x2 - x i | v ) ,

Vxi,rc2 € V , x± ^ rc2 ,

where a ( t ) = l i m 3 ^ t a ( g ) is a strictly increasing function from (0, oo) t o (0, oo). It then follows from Theorem 5.10 t h a t Equation (5.30) is (in a gen­ eralized sense) uniquely a n d strongly approximate-solvable. [] In t h e case t h a t Equation (5.30) is (in a generalized sense) uniquely a n d strongly approximate-solvable, we can estimate t h e error bound of t h e ap­ proximate solution un as compared t o t h e generalized solution u of Equation (5.30). For this purpose, we need a n additional condition: (H2) For each M > 0, there exists a constant CM > 0 such t h a t | (/ - Ax\, K{x2 -x\))H\

< cM \u - x i | v | x 2 -xi\v

Vxux2eVnr)T)(A),

| x i | v , |x 2 | v

, M

< -

Proposition 5.11. Suppose that A : T>(A) C V —► H is in the following form strongly K-monotone: (Ax2 - Axx, K{x2 -

Xl))H

> a0\x2 - xx\2v ,

V xu x2 e V(A),

(5.33)

for some constant ao > 0. Suppose also that Conditions (Hi) and (H2) are satisfied. Then we have the following error bound: \un-u\v

< f l + — } inf

|ti-t;L.

(5.34)

Monotone Operator Equations

199

Proof: Since V(A) is dense in V and A is demicontinuous, by Conditions (5.33) and (H2) we have [x2 - xi,

Ax2 - Ax\\

> &o\x2 - xi\2v ,

Vxi,x2€V,

(5.35)

and \[x2 -xi,

nf -Axi]v\

< cM\u-xi\v\x2

Vxi,x2GVn,

\xx\v,

-xi\v

,

\x2\v <M.

(5.36)

Consequently, we have a o M v < [un,

Aun-A0]v

= (Xun, / ~ A 0 ) H <||/-^0||-||^||-|un|v. Hence, by letting M 0 = -^ \\f - A0\\ • ||K||, we arrive at \un\v < M0 and |it| v < M 0 . Furthermore, if we set M = 2MQ1 then iox v ^Vn with |v| v > M, we have \u - v\v > \v\v - \u\v > M0> |u| v . Hence, inf

\u-v\v=

inf

\u-v\v.

(5.37)

|V|V<M

Thus, for v eVn with |v| v < M, it follows from (5.35) and (5.36) that \un - v\2 < — [un-v, ao

Aun

-Av]

v

=

[Un ~ V , Hf - Av ] v a0 < — \u-v\v\un -v\v ,

so that \un - u\v < \un - v\v -f \v - u\v <(l

+ ^)-\u-v\v,

VveVn,

\v\v

<M.

This, together with (5.37), imply (5.34).

[]

It should be noted that in many applications, we have the following stronger but more easily verified condition: (Hi2) For any M > 0, there exists a constant CM > 0 such that | (Ax2 - A c i , Ky)H\ \/Xl,x2eV(A),


V y e V,

- Xi\v\y\v

,

| x i | v 1 | x 2 | v , \y\v

<M.

200

Chapter 5

Proposition 5.12. Suppose that operator A has the strong monotonicity as shown in (5.33), and assume that Condition (H12) is satisfied. Then Equation (5.30) is (in a generalized sense) uniquely and strongly approximate-solvable under the scheme Tn, with the error hound given by (5.34). Proof: It suffices to show that Condition (#12) implies both Conditions (Hi) and (H2). First, it is obvious that (#12) implies (Hi). Second, (#12) implies that I [Ax2 -Aci,y]v\

< CM \%2 - xi\v\y\v

V x i , x 2 , 2 / € V\

\xi\v,

i

\x2\v, \y\v < M.

In the above, replace x2 by w, and let y — x2 — xi. Then for large enough M > 0, we obtain I (f -Axi,

K(x2

= I [x2 - x i , An< CM

\U-Xi\v\x2

-xi))H\ Axi]v\ -Xi\v

,

V xux2eV(A),

\xi\v, \x2\v < M,

which gives (H2).

\\

5.4.2 The perturbation problem We now discuss the perturbation problem of the if-monotone operator equa­ tion: Au + Bu = f, / € H, (5.38) where A : V(A) C V -* H is K-monotone and B : P ( B ) c V -+ H is a nonlinear perturbation operator, with V(A) C T>(B). Let 6(i*, v) = (J5w, Kv)H, ue V(B), v eV. Then by the Riesz Representation Theorem, for each u e T>(B) there exists a unique Bu eV such that (Bu,Kv)H

= [Bu,v]v,

ueV(B),

veV.

We next extend the operator B to the entire space V. For this purpose, we impose the following condition: (Hs) : For any weak Cauchy sequence {vn} C T>(B) C V, we have {Bvn - Bvm , Kw )H —► 0

(n,m -+ 00)

Monotone Operator Equations

201

uniformly with respect to w € {v \ v e V , \v\v = 1}. Then we have the following result. Proposition 5.13. Suppose that Condition (H%) is satisGed. Then operator B has a completely continuous extension, B, from V to V. Proof: For each v G V(B) = V, there is a sequence {vn} C V(B) such that vn w~* v i n F a s n - ^ o o . It then follows from Condition (H$) that I Bvn - Bvm I =

sup I [Bvn - Bvm , w] \ Wv=i = sup I (Bvn — Bvm , Kv)HI —> 0

(n, ra —± 00).

This implies that operator B maps a weak Cauchy sequence {vn} C V(B) to a strong Cauchy sequence {Bvn} C V. Therefore, there exists a unique v eV such that £ = (strong) lim n _,oo^?; n . Obviously, this v depends only on v but is independent of the choice of {vn}. Hence, we can let Bv = v. It is clear from the construction that B is the extension of B, which maps a weakly convergent sequence of V to a strongly convergent sequence, and so is completely continuous. \\ Thus, we have uniquely and continuously extended 6(w, v) to a twovariable functional b(u,v) — [Bu,v]v defined on V x V. Under the Conditions (Hi) and (H2), the perturbation problem (5.38) has a generalized solution given by a u € V satisfying a(u, v) + 6(u, v) = f(v),

Vv e V ,

(5.39)

VveV,

(5.39')

namely, [Au + Bu,v]v=

[Hf,v]v,

or equivalently, Au + Bu = Kf,

feH.

(5.39")

Clearly, its approximation problem is to find un 6 Vn such that a(unyv) + 6(t*n,t;) = f(v),

V.€^n,

(5.40)

which, in the case that Vn C T>(A), becomes (Aun + Bun , ^ ) ^ = (/, Kv)H ,

V t; € Vn ,

(5.40')

Chapter 5 5

202

wheren = 1,2,--• Based on the above discussions, combining Theorems 5.12 and 5.13 yields immediately the following result. Theorem 5.16. Suppose that Conditions (Hi) and (H%) are satisfied. Then (i) If A is K-monoton,, and if A + B is K-coercive, then for any f € H the perturbation problem (5.38) is (in a generalized sense) weakly approximate-solvable. (ii) If A is strongly K-monoton,, and ifA + B is K-coercive, then for any f G H the perturbation problem (5.38) is (in a generalized sense) strongly approximate-solvable. Finally in this section, we remark that when K = I, the identity operator, the above framework reduces to a general problem of finding approximate solutions for a densely defined monotone operator equation and its perturbed equation. In this case, the continuity assumption on operator K is the continuous embedding requirement for K : V -> H. 5.5 Application Examples: Numerical Solutions of Boundary Value Problems In this section, we give some examples to show how monotone and Kmonotone operator equations, as well as their perturbations, can be applied to approximate numerical solutions of nonlinear differential equations. Example 5.5. (The homogeneous Dirichlet problem) Let Q be a region in B? with boundary dtl, and let / € Lq(Q) with p~l + q-1 = I where p > 2. Consider the following boundary value problem:

= /(*). Iu = = 0, [u 0,

x € Q,

x ex€dQ, on,

where V is the gradient operator and || • || is the Euclidean norm. We are looking for a weak solution of the problem. Notice that the corresponding variational problem is to find a u € W^P(Q.) such that 2 \\Vu\r^udVvdx fvdx, I[ \\Vu\r VudVvdx = [[ fvdx,

Jn Jn

Jn Jn

p

V v£W>), Vv€WZ*(tt),

°

;

(5.41)

V

;

where W0' (Q) is the homogeneous Sobolev space, a subspace of the standard Sobolev space W1*^), defined by W1*^)

= {u6 LP(Q) (fi) || v has generalized derivatives (£l), dv/dii€Lp(Q),

i = l,2}

Monotone Operator Equations

203

and W01,p(fi) = C£°(ft), where the closure is with respect to the norm of

W1'P(Q).

In particular, if p = 2, we actually have Wli2(Q) = H1^) and Wo' 2 (^) = 1,p HQ(Q). As usual, the norm of space W (ft) is defined by

Mo,P,n + Mi, P ,nJ

>

where lvlo,p,n

(/j-H-*) 1 ". 2

'

MI,P,H i=i

P

I

\

I/P

ctei

in which | • |i,p>n is indeed an equivalent norm of the W 0 l j , (n) -norm. In the following, to simplify our computation we will use the following norm:

"-a

\

||Vv|| p dx) n /

i/p

Vv^01,p(fl),

,

in which ||Vv|| is the Euclidean norm of Vv e WQ'P(Q). It is easy to verifythat this || • || is a W^P(Q) -norm and is equivalent to the norm | • |i, p ,n defined above. For any fixed u G W 0 , p (fi), by the Holder inequality we have

/

Jn

\\Vu\\p-2Vu-Vvdx ^WuW^Wvl

Vve

W^P(Q).

This implies that for any u € W0'P(Q), the integral on the left-hand side of (5.41) is a bounded linear functional on space W 0 1,p (n), which defines an operator A : W01,p(fi) -> [W 0 1,P (n)]* via v , ; 4 u ) = / ||Vu|| p - 2 Vu-V7;dx,

V v e ^ f l ) ,

where, as before, (v, Au) stands for the functional value of Au evaluated at v. Observe that the right-hand side of (5.41) is also a bounded linear functional on space W 0 ' p (fi), and can be rewritten as

/ Jn

fvdx = {v,n,

l.p/ V D G ^ f t ) ,

204

Chapter 5

where / * e [Wr01,P(^)]*- T n u s > Equation (5.41) can be reformulated as the following operator equation: An = r • (5-410 Using the identity ab - cd = - (a - c) (b + d) + - (a + c) (6 - d), for any wi, ^2, v €

WQ'^Q)

we have

(v, Aw2 --Awi) (||V^2ir~2Vw2-Vt;-||Vwi|^-2V^1.Vv)dx

= /

= 11 (iiv^ir^-iiv^ir^v^+^o-v^dx + 5 / (llVual^ + IIVuilp-^V^a-iiiJ-Vvdx.

(5.42)

We next use this identity to show that operator A is strongly monotone (see Part (a) below) and Lipschitz continuous (see Part (b) below). Part (a). Operator A is strongly monotone. We show that there exists a constant ao > 0 such that (u2 - uuAu2

- Am) > a0\\u2 - u ^ ,

V uuu2

€ Wtf'^fi).

(5.43)

To do so, let v = u2 - u\ in Equation (5.42). Thus, the first integral on the right-hand side becomes

\ I {\\^u2\f-2-\\Vu1\r2)V(u2 = \ f (

+

u1)-V(u2-u1)dx

llVualp- 2 - H V t i i i r 2 ) ( IIV^II 2 - | | V ^ | | 2 )dx > 0,

so that (u2 — u\, Au2 — Au{) > | /

( l | V w 2 | r 2 + H V w i i r 2 ) ||V(« 2 -

2 Ul)\\ dx.

Then, we introduce an auxiliary function Mfr*

ii^r 2 +iMr 2

(e

.

r

(5.44)

Monotone Operator Equations

205

77) |£,»? | £, r\ e R2 , £ # where G = {(£, r?) ^ 77}. We next show that inf

(€,V)€G

4>^,r])>0. <£(£,»?) > 0 .

(5.45)

To do so, we first note that hat since 4.(0,77) = = !1,, 4.(0,77)

V rTy7^^#O0 ,

it suffices to verify that lat inf

U,n)ea (f,n)eo

4>(£,77)>0. 4>(£,T?) >0.

(5.46)

We also note that for any ^(t^, try)v) = <£(£,T?). 4.(^,77). Hence, by the the y t > 0, we have <j>{t^t rotational invariance of an m inner product about the original in a Euclidean space, instead of showing (5.46), it suffices to verify that inf

4.(1,77) > 0 , 4.(1,7?) 4-(l,r?)>0,

=( (1,0). I= 1,0).

(5.47)

To do so, we first first point int out that lim 4.(|,77) 4.(1,77) = iJ 22 IMI—00 ^ ' " \ ll IMI—00

2 llfp f p = 2

,. f 11 Ll- s llim i m ^4>{ln) (|,t7)= (\ * = »7->{ »j-»€ [00

i fl pf =P22 , /^ ' if p *> Zl' 22..

'' ii ff pp >> 22 ,

and

These two limits, together with the fact that 4>(l 4>(|, 77) > > 0 for 77 ^ f, imply the Inequality (5.47). Inequalities (5.44) and (5.45) give the expected (5.43), with

^-SttSU*^Part (b). Operator A is Lipschitz continuous. We show that there exists a constant M > 0 such that

2 2 I1(v(v, , AuAu ) I <1 M|| - U2l |-| Wl ( n| K( iHr 2t i +i i|r| W + 2 |ir n2 )r| H) | ,IMI , Am) < MU 3 \\u 22 -AUl

W

+ IWJIMI,

Vuuu«2,v€W^(Q). ,«6^(n). U l l

This implies that A is Lipschitz continuous.

2

(5.48) (5.48)

Chapter 5

206

To show (5.48), we observe that when p = 2, this inequality holds. Hence, suppose p > 2. Thus, applying the Holder inequality to the second integral of (5.42), we obtain I / (||Vti2ir2 + l|Vtii|r2)V(t*2-«i)-Vt;cfe I Jn (p-2)/p

< { £ (iivn 2 ir 2 + nv W l ir 2 ) p/(p - 2) <**}

|T*2 — 1*l|| " | | f |

<(iiU2ir2+iKir2)iK-Wlii.ibii.

(5.49)

Next, in order to estimate the first integral of (5.42), we introduce the following auxiliary function

HCv)

iie-^iKiieii^ + iMip- 2 )'

and then show that sup

(5.50)

rl>(£,ri)
where G is the same as defined above. Similarly, it suffices to show that sup V(&»7)<«>,

£ = (1,0).

(5.51)

It is clear that lim ^>(f, rj) = 1 IMI->oo

and

lim ^(£, v) = V - 2 •

v->£

Then, (5.51) follows from these two limits and the continuity of ip(i,rj). Now, applying the Holder inequality to (5.50) yields I / ( \\VU2\\P-2 I Jn <

sup (£,V)€G ;,»?)€G

<

SUp

- HVtiiir- 2 ) V(t*i + u2)

-Vvdx\ I

^(£,77) / ( | | V W l | r 2 + | | V W 2 | r 2 ) | | V ( W 2 - m ) | | . | | V i ; | | ^ JQ

^,T/)(||til|p-2 + ||tl2|r2)||fl2-t*l||-|H|.

Finally, combining (5.49), (5.52) and (5.42) gives (5.48), with M = - ( l +

sup

iC(&*?)).

(5.52)

Monotone Operator Equations

207

In the above, we have shown that operator A is strongly monotone and demicontinuous. Hence, under a suitable approximation scheme Tn = {Xn}, with Xn C X n + i C W^p(ft) and U™=1Xn = W^P(Q), by Theorem 5.10 we know that the approximate equation {v,Aun) = ( v , / * ) ,

Vv

eXn,

has a unique solution un 6 X n , n = 1,2, • • •. Moreover, the solution sequence {un} strongly converges in WQ'P(Q), with the limit u e WQ'P(Q) being the unique solution of Equation (5.41). Inequality (5.48) implies that operator A is Lipschitz continuous. Ac­ cording to Theorem 5.11, we have the following error bound: IK-ti||
inf

Wu-vW^^-V.

(5.53)

v€Xn

If ft is a polygonal region, then it is possible to give a more precise es­ timate. In this case, partition ft by a sequence of triangular subregions Ah such that ft = UxeAhK- Let hx be the minimal diameter of the disk that contains triangle K, and let h = msxKeAh ^K- Then let pK > 0 be the max­ imal diameter of the disk contained in triangle K, and let p = max^e Ah pK • Suppose that h —► 0 when the above partition becomes dense in 17, and that there exists a constant VQ > 0 such that hK <

»

"OPK

VK

eAh.

For each of such partitions, A&, choose a finite-dimensional subspace of W£'p(tl) as follows: Xh = {wh | wh € C(ft), wh\K is a polynomial of degree one Wfc(&)=o,

v&e^ndft},

where fth is the family of all angular points of triangles K in A&. Then, define an interpolation operator 11^ : w —► n^w € Xn by (nfcti;) (6) = w(b),

V 6 € tlh .

According to the interpolation theory in a Sobolev space, there exists a con­ stant c such that \w ~ n H i , P , n ^ ch\™kP,n ,

V w e w^p(n)

n

tf^(ft).

Since the norms || • || and | • \i,Pln are equivalent, the above inequality can be rewritten as | \w - Uhw\ | < ch\w\2^n

,

V w € W^p(n) n W ^ f l ) ,

Chapter 5

208

or

inf

||w-i»||
Vffl£<''(fl)n^(l!).

v€Xn

This, together with (5.53), give the following error bound for the approximate solution un: Hue W^'p(n) n W2
IK-«ll
-E^(H^ir 2 £)+Kx,.) = /(x), w = 0,

xefl,

x € dfi.

The weak form of this problem is to find u € W0'P(Q) such that / ||Vix|| p " 2 Vu-Vv Jn = f fv dx,

dx+ / b(sc,iA)v cfc Jn

Vv G^ ( H ) .

(5.54)

Next, we suppose that for a fixed x G fl, b(x,£) is continuous with respect to £ € it, and for a fixed £ € i?, 6(x, £) is measurable with respect to x G f t . Suppose also that there exists a constant c > 0 such that

!&(*,$)i^(I + K P - 1 ) ,

VXGQ,

Then, it is clear that for each fixed u € WQ'P(£1), the integral is a bounded linear functional of v G W0'P(Q), namely, \jb(x,u)v

dx I < c ( l + K ^ n ) | | f ||,

(5.55)

veei?.

J f i b(x,u)vdx

V v € Wo 1,P («).

Thus, it defines an operator B : W01,p(ft) - * [W 0 1,p (n)]* by / b(x, u)v dx = (v, Bu), in

V v € W 0 1,p (ft),

so that Equation (5.54) can be rewritten as Au + Bu = r ,

/ * € [W01,p(fi)] *.

(5.54')

Monotone Operator Equations

209

Now, we show that the operator B introduced above is completely con­ tinuous. If not, then there would be a sequence {UJ} C W 0 ' p (ft) such that Uj ™—> u e W0 ,P(Q) as j —> oo and \\Buj -Bu\\>e0,

for all j = 1,2, • • • ,

(5.56)

for some €Q > 0. By the compactness of the embedding W0'P(CL) into L p (fi), we know that {UJ} would contain a subsequence {unj} converging strongly to u e LP(Q,). Without loss of generality, let us suppose that {unj} converges to u(x) almost everywhere in ft. Thus, it would follow from the assumption on 6(x, £) and the Lebesgue (Dominated) Convergence Theorem [35] that \\Bunj-Bu\\=

sup

/

.6W 0 l l P (n) \Jnl

6(a;,Mni) -b(x,u)\v

< c < I \ 6(x, unj (x)) - 6(x, u(x)) \ dx>

-* 0

dx\ J

I

(j —> oo),

which contradicts Inequality (5.56). We finally remark that if 6(x, £) is so defined that the operator A + B satisfies the general coercive condition, then by Theorem 5.13 we know that Equation (5.54) is strongly approximate-solvable with respect to the approximation scheme Tn defined above. Example 5.6. Let ft be a bounded region in R2 with boundary dd. Consider the boundary value problem J - Au 4- \u\6u = f(x), x e ft, 1 u(x) = 0, x e dQ,, where 8 > 0 is a constant, and / £ Z^ft). The weak form of this classical problem is to find a u £ HQ (Q) such that I Vu-Vv

dx+

JQ

j \u\6uv dx= JQ

To this end, since f^ fvdx can be rewritten as

V v e H%(Si).

(5.57)

is a bounded linear functional of v G HQ(Q),

[ fvdx = (v,f*), Jn

[ fv dx, Jn

V.G^),

it

Chapter 5

210

>oint out that for any u e H&(il), i # ( n ) , the leftwhere / * e6 [ \Hi(il J # ( n ) ] * . We next point ffoW hand side of Equation (5.57) is alsoo a bounded linear functional of v € H&(0). Equat i(fi) into is continuous, Indeed, since the eembedding of H&(Cl) intoLa(« L2(<5+1+1)(n) )(£2)is continuous,we wehave have fi), and |u|*u € La(n), L2(Q), anc c |IH 2,n = H o i + i ) , Q- ^*-rii,2,n N i i n -• I«*I^ l"o ,lo,2,n ~ rio,2(*+i),o

,r we we obtain the H Holder inequality inequality we obtain obtain Then, usingSthe Holder >dx+ jf \u\ I\ f VuVvdx+ \u\ssuv uv dx\ dx\ Jn \Jo \Jn Jn Jn II ^ Hl i < , 2 ,2n l>v l U i , 2n2,n f l + lNM>,2,nMo,2,n H«No,2,nHo,2,n lluMoAnMo,2,n

<(K^c\u\^)K, 2,a.2,aC|
f \ufuv W
VV«v€e f H^(Cl), fo1^),

where the operator operate A : H^(ft) —* [H&(0)]*. [i/o1 ( f l ) ] \ Thus, Thus, Equation Equation (5.57) (5.57) can can be be rewritten as Au (5.57') i4« = f/ *. . Now, we examine exan the monotonicity 0 nicity of the operator A introduced above. Since \t\H is a mo: monotonically increasing •easing function of* € (-00,00), (-oo, oo), we have fi

tf

(( || ** il || 5 ** il --N N ' ** 2 2 )) (( *' li -- * *<2 22)))>^>0°0,,,

V *t*1!1;,;it22 €e (-00,00). V (( -- 00 00 ,, 00 00 )) ..

Consequently, ently, ({Ul Ul-u22,AUl Ul-Au22) 2 2 s ss - |t*2|*« \u2\ u2)(u -u2)dx u1-\u2\ = dx + fIf ( |KW|l |VV {\ui\ («i u2)dx == /I [V(«i [V(t --u2u)]2)]dx+ 2)(-Ul 1-u2)dx 2) u Jn Jo Jn Jn > « i -- «2li,2,n» « > |l«i V t,1i,u2eH^{Q), i.^eflo1^),

which implies thai ^A is a stronglyy monotone operator. lplies that Next, t, we exa: examine the continuity luity of the operator A. For this purpose, we first point out that there exists ts a constant c > 0 such that

II\Jo// ('|«i|*«i - \u2\ u2) V dx\ Ss

u2) V dx\

S < cc( (|u2| | , li 2,n + |«! < | t t l - ««22|lflf| i2,>>0 Mo oAAnn,, t*2|i,2,n 2o),)°n )|«i V - -t-«2|i,2,n|t;|o,2,n, tW 2 2|i,2,n k 2 , n MMo,2,n, w .. .. .. ^ r r l / v ^ \ ff(j(fi). V ui,u«2,«>€ € Ho{il). 2,v

Monotone Operator Equations

211

To verify this, we observe that if 6 = 0 then the Holder inequality yields immediately the above result. So suppose S 6 > 0. It is easily seen that there are constants a and f30 satisfying a,(3,a8>2

and

| + - + i = 1.

According to the Sobolev Embedding Theorem [1] (see Appendix B.4.4), the embeddings H^{Q)^L H£(Q) <-►aS(Q) LaS{Q)

and

fl^(n) H&(0) «-> <-> 1^(0) L^O)

are both continuous, so that u € / # ( n ) =4> « e L a « ( f i ) n ^ ( Q ) , \u\o,as,n, |«|o,/»,n < c|u|ii |«|o,o«,n, |w|o,a«,n, c|«|i, c|tt| l l2 ,n. s4 Furthermore, observe f^(|t|'«) + 6)\t\ 6)\t\\4e, so that an application of aserve that £(|t| t(\t\ t) *) = (1 +«)|*| the Holder inequality gives

{\ul\6u1-\u2tu2)vdx\

\[ =

/ ( l1++« S)) |«2 |« u2 + + 0(ui |'(«i — -« u«2)» [(1+6) 2 2)) |*(«1 22 ) v« dx\
Iin

M + ul «i i - « 2 | ) aa*S«dfxa } < ( (! 1 +5* ) { / (l«2l

- + ) { X ( + | ~u^

}

x6

I

<11)) ((00 <
|w|«i-ti2|o i tt2| 1j9An,nHo,2,n |v|o 2 n

~ °' ' -

Mo,2,n < (1 + «)(|u «)(|«2|o,a«,n « 2 |20|o,a«,n) ,a«,n) |«i - «2|o,/»,n |»|o,2,n 2 |o,««,n + |«i - M < c( |« 2 | li 2,n + K oAn , |«i - « 2 ||il il2, n )*|«! ) V - ««2li,2 Mo,2,o 2 |i,2,n ln M so that \(v,

Au AUl1-Au2)\

< | tt iiii--««22| |i i, ,2 2l nl nMMi ,u2 , n

+ c( |«2| li 2,n + |«i - «2|i,2,n) { |«i - «2| « 2 | llll2,n |«|i,2,n, Mi,2,n,

w ,u2 e mm,

11 andforallw1,w2€F («), andforallMi,M2€H Ul 0 0 (S2),

4 \\Am« i - Au2\\ < [l + c( |tx2| \u2\1> 2,a + |«i |t*i - ti2| |«i - ««2|i P « 2 |i,2,n) ] l«i 2 | >i 2,n li 2 i n)'] 1?2,n An •

212

Copier Chapter 5

This implies that the operator A is Lipschitz continuous on H&(Q). To this end, under the approximation scheme r „ = {Xn} that satisfies the required conditions, Theorems 5.10 and 5.11 together imply that the approximate equation I/ Vun •Vvdx \sunv dxdx= = f [fvdx, • Vz> dx+ +I / \un|ti„|V,« fvdx, JQ

JQ JCI

VveX \/veX n, n,

JQ. JQ

always has a unique solution un € Xn, n = 1,2, • • •• and the solution sequence {un} strongly converges within the space H^(Sl), whose limit u € H£(Q) is the unique solution of Equation (5.57). Moreover, there is a constant c> 0 such that K - « l i 2 n < c - inf | t t - « | i 2 n ' '' vex vexn n ' ' Example 5.7. We now consider a second-order boundary value problem of a quasi-linear partial differential equation. Let SI C E71 be a bounded region with boundary dSl. Consider the following first kind of homogeneous boundary value problem of an elliptic equation:

- ^ f ^ ^ V ^ +^ ^ V ^ ^ f x ) ,

xx€Q, eQ,

(5.58)

»»=i = i a XaXii

«u(x)= ( x ) = 00,,

(5.59)

x€d£2, x € dSl,

where x = (xi, • ■ •• xnn) V« = (du/dxu • • •• du/dxn), and d / e2(Sl). We have the following assumptions. (i) ^ ( x . t i . p i , • • • ,pn) are eontinuous so SI x Rnn+, with hi(x,0,0) ) €2(Sl) for alH = 1, • • •, n, and there are constants cx and c2 such that

ci

, and

\daAI idot —
kl-

I dai I \dai\ h~H— p —

ij l,---,n. h3 = = 1, •••,«• (ii) There is a constant CQ > 0 such that o

n n

da*

t j ' == l i,3 1

^

E

nn

6& > coE^2. i=l

for all &, i = 1, • • •• n, uniformly on f2 x # " + ^ . (hi) b(x,u,Pu• ■ • ,Pn) is sifferentiable eo nI x Rn+\ , i t h h(x,x,00 ) L2(Sl), and there are constants c3, c4, and c = C(co, 02,04) such that —

< c4

and

0 < cc < ^ < C3, < |— C3 ,

o-

Monotone Operator Equations Equations

213 213

for all j = 1, - , n . (iv) There is a constant c 5 , with 0 < c5 < CQ, such that ^(c2 ^(c2 + c4) c 4 ) 2 -4c(«,-c - 4 c ( c o - c55)<0. )<0. On the other hand, we note that the generalized Dirichlet form of problem (5.58)-(5.59) is to find a u G H$(Q) such that /

/ a^(x, zz,

Jn J n Vfr[ L ^

dx

= fvdx, =f [fvdx,

V.e^O). VveHfcn).

Jn

i #*<

J J (5.60) (5.60)

Under Conditions (i) and (iii), it is easily seen that for each u G H&{tt), the integral on the left-hand side of Equation (5.60) is a bounded linear functional on fl^(n), denoted Au, namely, c r J?-.

(v,Au)= I Jn ./n

fin

i

li^a^w^u)^ h b(x,w,Vu)u\dx, Voj(i)w,Vtt)f-h6(i,w,Vtt)u Lb, Vv€H&(Cl). V«€^(n). Yjr[ oxi J L^ &*i

Similarly, the right-hand side of Equation (5.60) is also a bounded linear functional on H&(il), denoted /*, namely, ((*,/*)= *,/*)= f / f-vdx, fvdx, Jn Jn

( n ) ,, vVv, ^^ ^H£(Q)

Hence, by using these notations Equation (5.60) can be rewritten as

Au = r . To establish the approximate-solvability of Equation (5.60) with respect to an appropriate approximation scheme, we next analyze the continuity and monotonicity of the operator A defined above. First, it follows from Conditions (i) and (iii) that for any uuu2,^ € \(v,AUl-Au2)\ fr fr ,A ,A|0(tii-u |a(«i-«22)|)| \9v\ 1^1

+ c3\Ul-u2\.\v\

+ i=1 i = l

|\

fv, - ,

, 1. \dv\ ^1

ccf]\d{Ul'U2)\.\v\)dx,

CXj OXi

\|

J)

214

Chapter 5

and so there exists a constant M > 0 such that for all u\, 112, v G

HQ(£1),

| (v, Au± - Au2) | < M\ui - U2|i,2,nMi,2,n , which implies that A is Lipschitz continuous. Indeed, we have \\Au! -Au2\\

< M|t*i -t*2|i,2,n,

Vwi,w 2 €

ffo(fl)-

Second, for any wi,^2,vG ifo(fi), (v, Aui - ^4w2)

=/„{ ^

[ai(x,Wi, VlXi) ~ Oi(x,W2, V w 2 ) ] 7T~

• t=l

*

[6(x, wi, Vwi) — b(x, ti2, Vw2)]v >
a(t*i - w2) #"

f

~— ( * , & , » 7 i )

S2 k , • _ ! < % n o

dv

+ ^(a?,6,^2)(wi -W2>

E ^db (*.&,•») d(u\

— U2) 1 ,

dxi

*>*»,

where ik = u2(x) + 0fc[t*i(x) - u2(x)] , r/fc = Vu2{x) + 6>fc[V^i(x) - Vti 2 (x)] O<0fc
k =1,2.

By Conditions (i)-(iii), we have (1*1 -tt 2 ,^4wi - Au 2 ) >

"9(^i - i t 2 )

dxi n (C2 + C 4 ) ^ | W 1 - t * 2

9(^! - u2) -h c(tq — U2)2 > dx. dxi

Monotone Operator Equations

215

It then follows from Condition (iv) that (co - c5)

d(ux

-u2)

dxi

~ (C2 + C 4 ) | ^ i -

U2\

d{u\ — u2) dxi

n so that ( l i l - U2 , Au\

- Au2)

> C5|ui - 1A2|l,2 Q >

which implies that operator A : HQ(Q) —► [ H Q ( Q ) ] * is strongly monotone. Finally, if we choose an appropriate approximation scheme Tn = {X n }, then by Theorems 5.10 and 5.11 the approximate equation /

^

a%(x, un, Vun)—

= I fv dx,

Vv e

+ b(x,un,S7un)v

dx

Xn,

JQ

always has a unique solution un € Xn, and the solution sequence {un} con­ verges strongly to a limit u € HQ (Q), which is the unique solution of Equation (5.60). Moreover, for the approximate solutions, we have the following error bound: \Un - u\iy2tn <

inf C5

|w-vi,2,n.

v£Xn

Example 5.8. We again consider Example 5.7, but with nonlinear boundary value conditions: - ] T —
x e ft,

(5.61)

t=i

Y^ Oi(x, w, Vw) cos(7, Xi) + a(x)w(x) = 0(x),

x € <9ft ,

(5.62)

where <£ € I/2(Q), cr is a continuous nonnegative function on d£l, cos(7,x^) denotes the ith component of the unit outer normal vector of 7 on dQ, in which dfl is assumed to be piecewise smooth. In addition to the Conditions (i)-(iv) stated in Example 5.7, we strengthen Condition (iv) as (iv') or add Condition (v), as follows: (iv') n(c 2 + c 4 ) 2 - 2c(co - c5) < 0.

Chapter 5

216 (v)

There is a piece dfti C dft with measdfti > 0, such that a > a0 > 0 on dft i for some constant (To > 0. We note that the weak form of problem (5.61)-(5.62) is to find a u G if 1 (ft) such that / y^ai(:r,u, V u ) h 6(x, u, Vu)v \ dx + / cruvds 9x Jn L^zt * J -/an = [ fvdx+ Jn

VveH1^).

[ 4>vds, JdQ

Observe that the embedding H1^) is a constant c > 0 such that

(5.63)

<^-> Z,2(dft) is continuous, namely, there

\ 1/2

an

v2ds)

/

< c|H|i|2fn,

VuGtf^ft).

(5.64)

Similar to the discussions given in Example 5.7, for any u G if 1 (ft) the lefthand side of Equation (5.63) is a bounded linear functional on Hl{Q) and is denoted by Au: r r n f) (v, Au) = / 5 ^ fli(^i u, Vu) + / auvds, Van

h b(x, u, Vw)i; dx

V v G iJ x (ft).

Also, for every / G L 2 (ft) and