Topics in Almost Automorphy

5

the

space of all classes of equivalence (with respect to the equality a.e on I) of measurable functions f : I —> X such that \\f\\ € LP{I)If we define a norm on

LP(I'J

X) by

\\f\\p = (f\\mrdt)-p,

if i
and WfWp- \\f\\oo = esssnp\\f(t)l

if p = 00,

I

then U{I\X)

is a Banach space.

We shall denote by Lf^(/;X) the space of all (equivalence classes of) measurable functions f : I —^ X such that the restriction of / to every bomided subinterval of / is in I ^ ( / ; X),

1.2 Sobolev Spaces Let i? C R'^ be an open boimded subset. Definition l ^ l l , A function g € L\^{Q) is said to be the weak derivative of a function f € Ll^J^Q) (or a derivative in the sense of distributions of order a), if f g .(l>dx = {-lp

Ja

f

Jo

f,D''(t>dx,

forall

(t>eC^{Q).

In this case, we write g = Z?^/. Recall that D^f denotes the Of-derivative defined by: dMf n« f .= Z—1 where a = (ai, 0^2,..., an), a^ (1 < i < n) is a nonnegative integer, and \a\ = ai + a2 + ... + otn-

6

1 Introduction and Preliminaries

Definition 1.12* Let k be a non-negative integer and let 1

by for all \a\ < k}.

In W^'^{fi), we define a norm by iN)k,p

\\f\\l,,:=

iN)k,oo

f

E

PV(x)rdx,

p
||/|U,oo:=niax||D-/|U, \oi\
and for p = 2, we define an inner product

We have the following Theorem 1.13. The space W^'^{fi) is a Banach space. Ifp < oo, it is separable. Also we have Definition 1.14. By Wjf'^(i?) we denote the closure ofC^{Q)

in

14^*^'P(i7). (CQ^{Q) denotes the space of functions of class C^ with compact support in Q. Definition 1.15. We define H^{Q) := W^^^{n) and H^{Q) :=

Theorem 1.16. H^{Q) and HQ{0) are Hilbert spaces when endowed with the inner product {N)k,2' Theorem 1.17. C^(i7)nif^(r2) is dense inH^{Q), where C^(r2) is the space of functions defined on Q of class C^.

1.3 Semigroups of Linear Operators

7

1.3 Semigroups of Linear Operators Definition 1.18. Let A : X -¥ X be a linear operator with domain D{A) cX,ina

Banach space (X, |H|). The family T = {T{t))teR+

of bounded linear operators on X is said to be a Co-semigroup if i) For all x E X, the mapping T(t)x :R'^ -^ X is continuous; ii) T(t + s) = T{t)T(s) for all <, 5 € M"^ (semigroup property); Hi) T(0) = I, the identity operator. The operator A is called the infinitesimal generator (or generator in short) of the Co-semigroup T if Ax = lim

T{t)x - X

where D{A) ^ixeXj (^

lim ^^^^"^ t-^Q-^

"^ exists \ .

t

J

It is observed that A commutes with T{t) on D{A). We define a Co-group in a similar way, replacing R"^ by M. For a bounded linear operator A, we have °° t'^A'^

T{t) := e'^ = Y. n=0

We also have the exponential growth: Proposition 1.19. (see [90] page 232). If T ^ (T(t))t€R+ is a Co-semigroup then there exist K > 0 and a; < oc such that \\T{t)\\ < Kc""',

forallteR^.

If u; < oo, we say that T is exponentially stable.

8

1 Introduction and Preliminaries

Proposition 1.20. IfT = {T{t))teisi^ is a Co-semigroup, then: a) the function t —> ||T(f))||; M"^ -^ R"*" is measurable and bounded on any compact interval o/R"^. b) the domain D{A) of its generator is dense in X. c) the generator A is a closed linear operator.

1.4 Fractional Powers of Operators Let (X, ||.||) be a (complex) Banach space and let C : D{C) C X »-> X be a densely defined closed imboimded linear operator acting in X. Assume that —C is the infinitesimal generator of an analytic semigroup {R{t)) and that 0 € p(C7), where p{C) is the resolvent of the operator C Then one can define, for 0 < a < 1, the fractional powers of C^. It is well-known that C : D ( C ) C X H^ X is a densely defined closed linear operator. Further, its domain D{C^) is endowed with the norm defined as ||x|U = IIC^xll,

ioTxeD{C^).

Since C is closed, then it can be easily shown that X^

=

{D{C^)^ \\.\\a) is also a Banach space. Recall that if — C is the infinitesimal generator of an analytic semigroup {R{t)) and that 0 € p(C), for a > 0, the fractional powers C" of C are implicitly defined as

1 r^ r{oi) Jo where r{a) is the classical Gamma function.

1.5 Evolution Equations

9

In the case where 0 < a < 1, since 0 € p(C7), then the operator C""^ is boimded, that is, there exists K >Q such that ||C~''|| < K. Theorem 1.21. Under the above assumptions on the operator C, we have

(ii) Uma_>o C^ = I (strong operator topology). Proof. See [83] for instance. We also recall the following. Lemma 1.22. Let —C be the infinitesimal generator of an analytic semigroup R{t). Assume that 0 G p(C). Then for a > 0, we have the following: 1. for every u G i5(C^), R{t)C'^u = C''R{t)u. Moreover

CRit)

is bounded, with an estimate of the form

2. If Q < a
and u € D{C^), we have an estimate of the form \\Rit)U'-u\\
More details on fractional powers of operators can be found in the literature, especially in [83].

1.5 Evolution Equations Unlike the finite dimensional case, the infinite dimensional theory of evolution equations has several notions of solutions. We will

10

1 Introduction and Preliminaries

present two of them in this section : the so-called classical solutions and mild solutions. We first consider in a Banach space (X, ||.||) the Abstract Cauchy Problem (ACP) x\t) = Ax{t) (t > 0) x{0) =xoe

D{A)

(1.1)

where A : D(A) C X —>- X is a linear operator densely defined in X, We have: Definition 1.23. The problem (1.1) is said to be well-posed if p{A) ^ 0 and for each XQ G D{A), there exists a unique (classical) solution X : [0,oo) »-> D{A) in C^([0,oo),X). It is well-known that "well-posedness" of (1.1) involves more than existence and uniqueness of solutions, and continuous dependence on the initial data. Indeed we have: Theorem 1,24. The problem (LI) above is well-posed iff A generates a Co-semigroup of linear operators T = T(t))t^^^^ on X. In this case, the (classical) solution of (LI) is given by the formula x(f) = T(t)xo, t e R + . For the proof, refer for instance to [45], pages 83-84. We now consider the linear non-homogeneous equation x\t)=Ax{t)-^f{t),

teM+

(1.2)

with initial data x{Q) = xo € X

(1.3)

We assimae that A is the infinitesimal genarator of a C(0" semigroup T = (T{t))teR^, and / € L/^(R+,X). Then we have:

1.5 Evolution Equations

11

Definition 1«25. A function x :W' \-^ X is said to he a mild solution of the Abstract Cauchy Problem (1,2)'(1.3) ifx € C(M+,X) and satisfies the equation x{t) = T{t)x{a) + f T{t-

s)f{s)ds,

(1.4)

Ja

for any a G R, and any t> a. This formula is also called the Variation of Constants Formula. We also have: Definition 1-26. Iffe

C{R-^,X),

then a function x : R+ i-> X

is said to be a classical solution of the Problem (1.2)-(1.3) if: (i) X e C(R+, X) and x{0) =

XQ

(a) X is strongly differentiable in X at each t € R"*" (Hi) X satisfies the equation (1.2) in X everywhere on (0, oo) (iv)x\t)

e L/^(R+,X) and satisfies x{t) = x{a) + / x\s)ds,

for all a G R, all t> a.

Ja

It is clear that classical solutions are mild solutions too. The converse is not always true. We have the following (see [45, p. 84]): Theorem 1.27* Suppose A generates a Co-semigroup on X and Xo G D{A). Assume either (i) f G C{R-^,X)

takes values in D{A) and Af G (7(R+, X), or

(n)feC\R^,X). Then the Abstract Cauchy Problem (1.2)-(1.3) has a unique (classical) solution with x : [0,oo) ^ D{A) such that x{0) = XQ*

12

1 Introduction and Preliminaries

Theorem 1.28. Assume that A generates a Co-semigroup on X

andfeLUR-^,X), Then a mild solution x : [0, oo) ^ X of the (ACP) is a classical solution if and only if both of the following conditions are satisfied: (i)x{t) € D{A) almost everywhere on [0,oo)^ and (ii)Ax e LUR'-^X),

i.e. X G L]^{

R-'.D(A)).

See for instance [87, p. 149] for the proof.

1.6 Almost Automorphic Functions Definition 1.29. Let {X, ||.||) be a (real or complex) Banach space and f : R -^ X a (strongly) continuous function. We say that f is almost automorphic if for every sequence of real numbers (s!^), there exists a subsequence (sn) such that: lim lim f{t + 5n - Sm) = fit) for each t

eR.

This limit means that: g{t):= \\m f{t + Sn) o

is well defined for each t £R and lim g{t - Sn) = f{t) n~^oo

for each t € R.

1.6 Almost Automorphic Functions

13

Remark 1.30. We observe that the function g{t) m Definition 1.29 is measurable but not necessarily continuous. If the convergence above is uniform on R, the function / is said to be almost periodic (in Bochner's sense). Also if the convergence above holds in the weak topology of X, that is g{t) ~w-

lim f{t + Sn)

is well defined for each t € E and It; - hm 5(< - Sn) = f{t) n->oo

for each t e M, we say that / is weakly almost automorphic. Clearly almost automorphy implies weak-almost automorphy. The reader can find more informations on weak almost automorphy m [80]. We end these remarks by the following important result (see Theorem 2.1.10 in [80]): If /n : M >-> X, n = 1,2..., is a sequence of almost automorphic functions such that limn-^oo /n(*) =/(*)> uniformly in i G R, then / is also almost automorphic. Theorem 1 . 3 1 . / / / , / i , / 2 : K -> X are almost automorphic functions , then the following are true: i) / i + /2 ^ almost automorphic . ii) cf is almost automorphic for every scalar c. Hi) fa(t) = f(t + a) is almost automorphic for each fixed a € R. iv)sup^^jg^ \\f{t)\\ < 00; that is f is a bounded function.

14

1 Introduction and Preliminaries

v) The range Rf = {/(<) : t E R} of f is relatively compact in X. Proof: See [80] . D Proposition 1.32. We have (i) sup^^K \\9{t)\\ = supt^R \\f{t)\\ (a) Rg Q Rf, where g is the function that appears in Definition 1.29. Let us first recall the following. Lemma 1.33. (see [90]) (i) Weakly hounded sets are hounded (in norm) in any Banach space {X, \\.\\)j and (a) if W'limn^oo ^n = ^y then \\x\\ < liminf„,^oo H^^nll; o,nd the sequence {\\Xn\\) is bounded. Now we prove Proposition 1.32. Proof, (i) Since g{t) = limn->oo / ( * + Sn), we may use the above Lemma to get, for each t G M, the inequality

11^(011
D

We also establish the following composition result:

1.6 Almost Automorphic Functions

15

Theorem 1*34. Let X^Y be two Banach spaces with norms H-Hx and

\\.\\Y

respectively, and let f : R -^ X be an almost automor-

phic function. If (f> : X —^ Y is a continuous function, then the composite function (f>{f{t)) :R-^Y

is almost automorphic.

Proof: Since / is almost automorphic, /(R) is compact in X by Theorem 1.31 v). We deduce that <^ restricted to /(M) is uniformly continuous. So given e > 0, there exists ^ > 0 such that \\(l){xi)-(t){x2)\\Y <e,

for any 0:1,0:2 e / ( M )

with \\xi — X2\\x < <^Let {s*^ be an arbitrary sequence of real numbers. There exists a subsequence (^n) C (5^) such that g{t) := lim f{t + Sn) n—>oo

is well-defined for each i € M and lim g{t - Sn) = fit) n~>oo

for each t

eR.

Note that <j) is well-defined on p(R), since g{R) C f{R) (Proposition 1.32 ii)). Therefore there exists N eN such that if n > iV, we have

Wfit + sr.) - git)\\x < S for each t € R.

mf{t+sn))-4>{9m\Y<e for each t € M; which shows that

16

1 Introduction and Preliminaries

lim (f{t + Sn)) = (f>(9(t)) for each t €

Analogously, we can prove that lim (g{t-sn)) = (f>{m) n—><X)

for each < € R. So that t —^ 0(/(O) is almost automorphic. The proof is now complete. D Proposition 1.35. Let <;6 : R -> C and / : R -> X 6e almost automorphic where X is a complex Banach space. Then (/>f :R --^ X defined by {<})f){t) = ^(t)f{t)

is also almost automorphic.

Proof. Both functions 0 and / are bounded since they are gJmost automorphic {Theorem 1.31 iv)). So we let Ki = sup^^j^ 1^(01Let (in) be an arbitrary sequence of real nmnbers. Then there exists a subsequence {sn) of {t^) such that: v{t) := lim (t>{t + Sn) n->oo

is well-defined for each t € R, and lim v{t — Sn) = (t>{t) for each

teR.

Also g{t) := lim f(t + Sn) n->oo

is well-defined for each i € R, and

1.6 Almost Automorphic Functions

17

lim g(t - Sn) = f(t) n-foo

for each

teR.

Now write 'i>(t + Sn)f(t + S„) - V(t)g(t) = {t + Sn)f(t + S„) - 4>(t + 5„)p(<) + it +

Sn)9{t)-v{t)g(t).

So that im + Sn)f{t + S„) - V{t)git)\\ < KiWfit + Sn) - ^(t)|| + K2\Ht +

Sn)-v{t)\,

where K2 = supt^jj ||^(t)|| = sup^eR | | / ( 0 | | < co (Proposition 1.32 i) and Theorem 1.31 iv)). Clearly when n -> 00, we obtain lim (f>(t + Sn)fit + Sn) = v(t)g(t) n—Kx)

for each t e R. It is also easy to check that lim vit - Sn)g{t - Sn) = 4>{t)fif) for each t e M. The proof is now complete. D The following result is important in view of its applications to the theory of evolution equations (see for instance Chapter J2, [80]). Theorem 1*36. Let T = (T(i))tGR be a one parameter group of strongly continuous linear operators such that sup^^j^ ||T(t)|| =

18

1 Introduction and Preliminaries

M < oo. Let f :R -^ X be an almost automorphic function

and

S = f{Q), where Q denotes the set of rational numbers, with the property that the function T{t)x : R ^ X is almost automorphic for each x G 5 . Then T{t)f{t)

: R ~> X Z5 almost automorphic.

Proof: Let B = {f{t)

: t e R} be the range of / . Then S is

a countable subset of B. It is also dense in S , the closure of B, Indeed it is known that if ^ is a continuous function g : Xi -^ X2 where Xi and X2 are two topological spaces and A C Xi, then g{A) C 9(A). Since / is continuous and Q = R, then we have B = /(R) = /(Q) C /(Q) = S That is S = B since 5 is a subset of B] which proves oin: claim. Let S = (xn); then T(t)xn is almost automorphic for each n = 1,2,

Consider an arbitrary sequence of real numbers {s'^. Using

the well known Cantor diagonal procedure we can show that there exists a subsequence {s^) of (5^) such that lim T{sn)x exists for every x E S. Pick XQ arbitrary in B, For any n, m, k we have \\T{Sn)Xo - T{s^)Xo\\ < \\T{Sn)Xo - T{Sn)Xk\\ +

\\T{Sn)Xk-T{Sm)Xk\\

+

\\T{Sm)Xk-T{Sm)xo\\

1.6 Almost Automorphic Functions

19

<||T(5„)||||a:o-a:,|| + \\TiSn)Xk -

T{Sra)Xk\\

+ ||rM||||x,-a:o|| < 2M||Xo - XkW + ||T(5n)Xfc - T(s^)Xfc||.

Now, since X]^ € 5, (T(5i)xfc)ieN is a Cauchy sequence because it is convergent as seen above. Thus lim

\\T{Sr,)Xk-T{Sm)Xk\\=Q

and consequently: lim \\T{sn)x^ - T{s^)x^\\ < 2M||xo - Xi,\\. Therefore, using the fact that S is dense in B , we can say that lim T'(5n)xo exists for every XQ € B. Now we observe that lim^-^oo ^(^n)^ = V defines a mapping F from the linear subspace spanned by B into X, namely Fx = y if

lim T(sn)x = y.

The map F has the following properties: i) F is linear, ii) ||Fx|| = ||y|| < limn->oo ||T(5n)x|| < M||x||

for every x in the

subspace spanned by S, iii) F is one-to-one, iv) If (xn) is a given sequence in B such that strong-limn->oo Xn = x exists, then strong-limn-^oo T{sn)xn = Fx, and strong-limn--)^oo Fxn = Fx.

20

1 Introduction and Preliminaries

Let Rp = {Fx : x E B} be the range of F, Then we observe that F{S) = RF and Rp is compact in X. Let us show that lim T{—Sn)y exists for every y e Rp* n->oo

It suffices to prove that hm T(sn)ym

exists for every y^ € F(S),

n->oo

where ym = Fxm, m = 1,2,.... Since T(t)xm is almost automorphic for each m = 1,2,... we have lim T{t + Sn)xm = lim T{t)T{sn)xm n—foo

n->oo

= T(t) lim T{Sn)Xm

= T(t)ym pointwise on R. Also we have lim T{t - Sn)ym =

n->oo

T{t)Xm

= T{t) lim

T{-s^)ym.

n—yoo

Now, for t = 0, we get lim T(—5n)t/m exists for m = 1,2,... and T(0)a:m = ^m- Hence, we have lim T{—Sn)y exists for every

y € /?F-

n—^oo

This defines a linear function G on the hnear subspace spanned by Rp where

1.6 Almost Automorphic Functions

21

Gy = lim T{-Sn)yIt is easy to verify that G has the same properties as did F and we observe that GFx = X for every x £ B. Let (5^) be an arbitrary sequence of real numbers. Then we can extract a subsequence {sn) such that lun fit + Sn) = g{t) n—>oo

and Um g{t - Sn) = f{t) n-4oo

pointwise on M, and Um T{'-Sn)x = y exists for each x £ B. Now let us observe that for every t € M and n = 1,2,... we have f{t + Sn),

9it)eB.

Let t be arbitrary in R. Then for every n = 1,2,..., we get

T{t + Sn)f{t + Sn) = T{t)T{Sn)fit + 5^) so that lim T{t + Sn)f{t + Sn) = T{t)Fg{t) and lim T{t - Sn)Fg{t ~ Sn) = T{t) lim T{-Sn)Fg(t = T(i)GF/(i)

=mm, since / ( t ) G B, The theorem is proved. D

- Sn)

22

1 Introduction and Preliminaries

Theorem 1.37. Assume that (T{t))t>o is a Co-group of bounded linear operators on X and let x{t) = T{t)xo is almost automorphic for some XQ € X . Then inf \\x{t)\\ > 0, or x(t) = 0 for every t € M. Proof: Assume that infteu ||^(*) || = 0 and let (s!^) be a minimizing sequence of real numbers, that is limn_foo lk(^n)l| = 0. We can extract a subsequence (sn) Q (s!^) such that y{t) := lim x(t + Sn) n->oo

is well defined for each f € R, and lim y(t - Sn) = x{t) n-4oo

for each « € M. We also have X{t + Sn) = T{t + Sn)Xo = T{t)T{Sn)Xo

=

T{t)x{Sn).

Thus y{t) = Um a:(£ + Sn) = T{t) lim x{sn) = 0, n-4oo

n—¥oo

for each t e M; which shows that y(t) = 0 for each t € E and consequently x{t) = 0 identically on R. D Differentiability and integration of almost automorphic functions are presented in [80]. We recall the following important BohrAmerio type result:

1.6 Almost Automorphic Functions

23

If f : R -^ X is almost aumorphic and F : R -^ X defined by F{t) := JQ f{t)dt has a relatively compact range in X, then F is also almost automorphic. In the cose X is a uniformly convex Banach space, the conclusion holds true if the range of F is bounded in X, In conclusion to this section, let ns note that the set

AA{X)

of all almost automorphic functions M -^ X, {{X, \\ ||) a Banach space), is a linear vector space in view of Theorem LSI. Equipped with the norm

||/IU^W=sup||/(i)||, AA{X) turns out to be a Banach space in view of the above remarks. 1.6*1 Asymptotically Almost Automorphic Functions Definition 1.38* Let {X, \\ (|) be a (real or complex) Banach space. A continuous function f : R^ -> X is said to be asymptotically almost automorphic if it admits a decomposition

fit) = git) + hit), where g :R-^

teR-^

X is an almost automorphic function, h : R'^ -^ X

is a continuous function with limt^oo ||^(*)|| = 0g and h are called respectively the principal and corrective terms of the function

f.

We have the following immediate facts (see [80] for details):

24

1 Introduction and Preliminaxies

Theorem 1.39. If f,f 1^/2 are asymptotically almost automorphic, then / i + /2 and Xf, A an arbitrary scalar, are also asymptotically almost automorphic. We also have the important restilt: Theorem 1.40. The decomposition of an asymptotically almost automorphic function is unique. Denote AAA{X)

the linear vector space of all assonptotically

almost automorphic functions / : R~*" H> X. It is clear that the formula: WfWAAAW = Mt)\\AA(X) + sup \\h{t)\\

(1.5)

where g and h are the principal and corrective terms of / , respectively, defines a norm on the space

AAA{X),

The following holds true. Theorem 1.41. AAA{X)

is a Banach space.

Proof: Let (/n) be a Cauchy sequence in AAA{X),

with (p^)

and {hn) as respective principal and corrective terms. It is clear that {Qn) is a Cauchy sequence in the Banach space of all almost automorphic functions AA{X), Thus there exists g € AA{X) such that gn -^ g uniformly on R. Moreover the corrective terms (hn) also form a Cauchy sequence of continuous functions with respect to the norm sup. We then deduce that there exists a function h €

C{R^yX),

such that hn ^ h imiformly on M^.. Using the fact that for each

1.6 Almost Automorphic Functions

25

n = 1,2,..., liint_yoo ||/in(OII = ^J ^^^ the equality h{t) — (h(t) — hn{t)) + hnit) for < € R+, we obtain

lim |lMf)|| = 0. t->oo

This implies that the function / defined as f := g + h E and limn->oo \\fn — f\\ = 0, thus AAA{X)

AAA{X)

is a Banach space. D

1.6.2 Applications to Abstract Dynamical Systems In this section, we will study the behavior of asymptotically almost automorphic semigroups of linear operators T = (T'(t))teR+ as t tends to infinity. We will present some topological and asymptotic properties based on the classical Nemytskii-Stepanov theory of dynamical systems. First of all we present a connection between the so-called abstract dynamical systems and Co-semigroups of linear operators. (X, II II) will denote a Banach space (over K or C). Definition 1.42. A mapping u : R"^ x X -^ X is called an (abstract) dynamical system if i) u{Q,x) = Xy for every x E X. ii)u(-^x)

: R"*" -^ X is continuous for any t > 0 and right-

continuous att = 0, for each x E X. iii)u{t^ *) : X -^ X is continuous for each i € R"^. iv)u{t + SjX) = u{t, u{s, x))y for all <, 5 e R"*" and x E X, Ifu : R"^ xX is a dynamical system, the mapping u{'^ x) : R"^ ~> X will be called a motion originating at x E X.

26

1 Introduction and Preliminaries

Now we are ready to state and prove the following basic result: Theorem 1.43. Every Co-semigroup iT{t))t^^+ determines a dynamical system and conversely by defining u{t,x) = T{t)x, t G R.'^, XGX,

Proof: Let u{t, x) be a dynamical system in the sense of Definition 1.4^ above and consider T{t)x = u{t,x),

f G M^, X G X

Then obviously T(0) = / , the identity operator on X since for every x E X, T{0)x = u(0, x) = x. Let t, 5 G K"*" and x G X\ then we have T{t + s)x = u{t^ s, x) = u{t, u{Sj x)) by property iv) of Definition 1.4^. But we have also T{t)T{s)x

= T{t)u{s,x)

=

T{t,u{s,x))

using the definition of T{t)x, Therefore, T{t + s)x =

T{t)T{s)x,

for every t, 5 G R"^, x G X, which proves the semigroup property T(t + s)x =

T{t)T{s)x,

for alH,5GK"^. Continuity of T{t)x : X -> X follows readily from property iii) of Definition 14^, for every f G M"^.

1.6 Almost Automorphic Functions

27

Now we have lim T(t)x = lim u(t, x) = ix(0, x) = x using properties ii) then i) in the above Definition 1.42. We have proved that {T{t))t^^+ is a Co-semigroup. Conversely, suppose we have a Co-semigroup (T(i))tGR+ and

define uiR-^

xX^Xhy u{t, x) = T{t)x,

t € R+,

xeX.

Then all properties i)-iv) in Definition 1.42 are obviously true. The mapping u is then a dynamical system.

D

Theorem 1.43 tells us that the notions of abstract dynamical systems and Co-semigroups are equivalent. This fact provides a solid groimd to study Co-semigroups of linear operators as an independent topic. In the rest of the section, we will consider a Co-semigroup of linear operators T — {T{t))teR+ such that the motion T{t)xo :R-^ is in AAA(X)

^X

with principal term f{t).

Let us now introduce some notations and definitions. We let XQ be some fixed element of X. Definition 1.44. A function cp :R --^ X is said to be a complete trajectory of T if it satisfies the functional equation
and all t> a.

28

1 Introduction and Preliminaries

We have also the following properties. Theorem 1A5. The principal term ofT{t)xo

is a complete tra-

jectory for T. Proof: We have T(t)xo = f(t) + h(t), t e R"^. Since / is ahnost automorphic, there exists a subsequence (uk) C (n) = N such that g{t) := lim f{t + Uk) is well-defined for each t 6 ]R and lim g{t ~ Tik) = fit) k—^oo

pointwise on R. Put ip{t) = T{t)xo. Then (p{0) = XQ. Let us fix a € R and choose k large enough so that a + n^ > 0. If 5 > 0, then (p{a + s + nk) = T{a + s + nk)(p{0) = T{s)T{a + nkMO) = T{s) 0 and a + n^ > 0. But we have lim f{a + s + rifc) = g{a + s), lim h{a + 5 + TIA;) = 0, SO

1.6 Almost Automorphic Functions

lim (p{a + 5 + Uk) = lim T{s)ip{a + Uk) = g{a + 5). We also have lim (p{a + nk) = g{a).

k-^00

Using continuity of T{t),we get lim T{s)(p{a + Uk) = T{s)g{a). k-¥oo

We can now establish the following equality T(s)g{a) = g{a + s),

Va e R, V5 > 0.

But we have lim g{t - Uk) = f{t) , for each

t

eR

and g{a -rik+s)

= T{s)g{a - Uk) ,

Va G R,

V5 > 0.

Va G R,

V5 > 0

Therefore lim g(a -nk+s)

= T(s)f{a)

,

k-¥oo

so that f{a + 5) = T(s)f(a)

,

Va G R,

V^ > 0.

Finally let us put s = t — a with t >0. Then /(t) = T ( t - a ) / ( a ) , The proof is complete. D

VaGR,

Vt > a.

29

30

1 Introduction and Preliminaxies

Definition 1.46. The

(^^{xo) = {y€X/30
lim r(t„)xo = y}, n-foo

is called the uj-limit set ofT{t)xQ. ) = {?/ € X / 3 0 < in -^ 00 :

lim /(in) = y}

is called the u>-limit set of f(t), the principal term y+ixo) =

ofT(t)xo.

{T{t)xo/teR-^}

is the trajectory ofT{t)xQ. We have the following properties. Theorem 1.47. a;+(a;o) ^ 0. Proof: We let tn = n, n = 1,2,

. Since / E AA{X)

, there

exists a subsequence {tnk) C (<„) with tn^. = n* such that

lim /(t„J = ^(0). But lim T{tnk)xo = lim / ( i n j k-^oo

k—^cx>

We then get lim T(tnJxo = g{0). k-^oo

Consequently, 5(0) G a;"^(xo), since tn^ -^ 00 as k -> 00, So a;"^(xo) is not empty. The proof is complete. D Theorem 1A8. a;'^(rco) = ^/(^o)-

1.6 Almost Automorphic Functions

31

Proof: To see that T{t)xo and its principal term have the same a;-Umit set, it suffices to observe that lim T(t)xo = lim f(t). The proof is complete. D Definition 1.49, A subset B C X is said to be invariant set under the semigroup T = {T{t))t^^+

if T{t)y e B for every y e B

andteR-^, Theorem 1.50. a;"^(xo) is invariant under T. Proof: Let y € u;"^(xo); then there exists 0 < tn —> oo such that limt->oo T{tn)xo = y. Consider the sequence (sn) where Sn = t + tn,n = 1,2,

for a

given t G R"*". Then 5^ —> oo as n -^ oo. We have T{Sn)Xo = T{t)T{tn)xo,

71 = 1, 2

and limn-^ooT{sn)xo = T(t)y, using continuity of T{t), Therefore T{t)yeuj-^{xo). This completes the proof. D Theorem 1.51* UJ'^{XQ) is closed in X. Proof:

Let y € ct;^"(xo) be the closure of a;^(xo); then there

exists a sequence of elements ym G a;"^(xo), m = 1,2,... such that Vm —^ y- For each ym, there exists 0 < tm,n -^ +oc, as n -^ +oo such that lim^^oo ^(^m,n)^o = Z/m- Recursively choose

32

1 Introduction and Preliminaries

ti,ni > 1 such that ||yi ~ r(ti,nJxo|| < | *2,n2 > max(2,£i,ni) such that \\y2 ~ T(<2,n2)^o|| < ^ h,ns > max(3,t2,n2) such that ||t/3 - r(t3,n3)a:o|| < ^ tk^nk > niax(A:,tjb^i,nfc.J such that \\yk - r(tjb,nJ^o|| < ^ Let Sk = tk^uk^k — 1^2,

. Clearly 0 < 5^ —>^ +oo as A; -^ H-oo,

and we have ||T(5it)xo - y|| < \\T{sk)xo - VkW + |2/fc - 2/||

Since limjfe_^_,.oo Vk = y^ we have y euj'^{xo)> This achieves the proof. D Theorem 1.52. uj^{xo) is compact if^^{xo) is relatively compact Proof: It is obvious that iV^{xo) C 7"^(xo) the closure of 7"^(xo). But 7'*"(a:o) is a compact set and uJ'^{xo) is a closed set (see Theorem 1.51). Therefore a;"^(xo) is itself compact.

D

Theorem 1.53. 7/(a:o) = {f{t) /t e M} is invariant under the semigroup T. We recall also that 7/(a:o) is relatively compact, since f{t) is almost automorphic. Proof: Let y G 7/(xo). So there exists a G R such that y = f{cr). For arbitrary a 6 M such that a > a, we can write

1.6 Almost Automorphic Functions

V = Sip) = Tie -

33

a)f(a),

since / is a complete trajectory Theorem 1.45. Now let < > 0. Then T(t)y = T(t + a-

a)f{a)

i.e., T(t)y € 7/(a:o), V< > 0. 1ff(xo) is indeed invariant under the semigroup T.

D

Theorem 1.54. Let u(t) = mfy^^+^xo) \\T{t)xo - y\\. Then lim i^(t) = 0. Proof: Suppose not, that is limt_>_,_oo ^(t) 7^ 0. Then there exists 6 > 0 such that for every n = 1,2,

3 i ; > n , \\T{Oxo-y\\>e

there exists t!^> n such that

V^/€ a;+(xo), Vn = 1,2,-...

Let ( t n ) ^ i be a subsequence of ( t ^ ) ^ i such that {f{tn)) converges, say to ^, as is guaranteed by the relative compactness of yf(xo). Now since i^ ~> 00 as n -> 00, we get lim T(tn)xo = lim f(tn) = y. Therefore y € a;"^(a:o), which is a contradiction.

D

Remark 1.55, This minimality property shows that the a;-liniit set UJ'^{XQ)

is the smallest closed set towards which the asymptotically

almost automorphic function T{t)xQ tends as t goes to infinity.

34

1 Introduction and Preliminaries

Definition 1.56. e E X is called a rest-point for the semigroup T ifT(t)e

= e, Vt > 0.

T h e o r e m 1.57. If XQ is a rest-point of the semigroup T, then

Proof: Since T{t)xQ = xo, Vt > 0, then for every sequence of real ninnbers ( t n ) ^ i such that 0 < t^ -> +oo, we get lim T{tn)xQ = xo, i.e., Xo € U;~^(XQ).

Now let be y € a;"^(xo). There exists 0 < ^n -^ oo, such that \inin-^ooT{sn)xo = y. But T(5n)xo =

XQJU

= 1,2,--. Therefore

Xo = 2 / .

The proof is now complete. D

1.7 Almost Periodic Functions Definition 1.58. Let E — E{T) he a complete Hausdorff locally convex space. A continuous function f :R -^ E is said to be almost periodic if for each neighborhood of the origin U there exists a real number I > 0 such that every interval [a, a + I] contains at least one point s such that fit + s)- f{t) e U, for every

t e R.

The numbers s depend onU and are called U-translation numbers or U-almost periods of the function

f.

1.7 Almost Periodic Functions

35

Remark 1.59, Prom Definition 1.58, we observe that for each neighborhood of the origin U, the set of C/-translation numbers is relatively dense in R. Theorem 1.60* (i) If f : Ri-¥ E is an almost periodic function, then f is uniformly continuous, (ii) V ifn) is a sequence of almost periodic functions, /„ : M »-^ E, n ^ 1,2,3,... such that (fn) converges uniformly to f on R, then f is also almost periodic. The following Criterion due to Bochner is a key result. Theorem 1.61. ^ o c h n e r ' s Criterion). Let E be a Frechet space, that is a Hausdorff locally convex space whose topology is induced by a complete and invariant metric. Then f € C(R, E) is almost periodic if and only if for every sequence of real numbers {s'^, there exists a subsequence (sn) such that {f(t + Sn)) converges uniformly m t € M. Now we denote AP(E) the set of all almost periodic functions R -> £*, where £* is a Frechet space. By Theorem 1.60 axid Theorem 3.1.3 and Theorem 3.1.9 i) in [80], AP{E) is a linear space. We also have the following result (see [16] for details). Theorem 1.62. AP{E) is a Frechet space. Proof: Denote by C{R,E)

the linear space of all continuous

bounded functions R -^ E and by (9^), n € N, the family of seminorms which generates the topology r od E. Without loss of a generality we may assume that Qn-ti ^ Qm pointwise, for n € N.

36

1 Introduction and Preliminaries

Define q^{f):=swpqn{f{x)),

n € N.

Obviotisly (q^) form a family of seminorms of C(M, E). Moreover, it is clear that q^_^i > q^ for n G N. Define the pseudo-norm

ifi't^TTm ""

-

Obviously C{R^E) with the above defined pseudo-norm is a Prechet space. Now it is clear that AP(E) is a linear subspace of C(R, E). In view of Theorem 1,60 ii) it is closed. This completes the proof.

D

Corollary 1.63. If E is a Banach space, then the linear space of all almost periodic functions R ^ E is a Banach space with the norm sup. We have also the following simple fact. Proposition 1.64. Let E be a Prechet space over the field K (K = R or C) and assume f G AP{E) and v G AP{K). Thenuf

eAP(E).

Proof: It is a simple consequence of the Bochner's criterion Theorem 1.61. Definition 1.65. A Prechet space E is said to be perfect if every bounded function f :R -^ E with an almost periodic derivative f is necesssarily almost periodic.

1.7 Almost Periodic Functions

37

Does there exist a perfect Prechet space which is not a Banach space? The answer to this question is positive what we illustrate by the following. Example 1.66. Denote s the linear space of all real sequences: s = {x = {xn) : Xn € N forn e N}. For each n e N , define Pn{x) =^ \xn\^ x E s. Obviously Pn is a seminorm defined on s. Define qn := Pi Vp2 V . . . Vpn for n G N. We have gn+i > Qn for neN. The space s considered with the family of seminorms (qn) is a Prechet space. Moreover, it can be proved (see [1] Theorem 17.7, p.210) that each closed and bounded subset of s is compact. Thus, in particular, s is not a Banach space. Finally, in view of Theorem 3.2.6 [80], s is perfect. It is also possible to enlarge Definition 1.58 to functions of two variables of the form f{t,x)

(see for instance [16]) as follows.

Definition 1.67. A continuous function f :Rx E -^ E is said to be almost periodic in t for each x E E, if for each neighbourhood of the origin U, there exists a real number I > 0, such that every compact interval of the real line contains at least a point r such that f(t + r, x) - / ( t , x) e U,

for

each

t € M and

x e E.

38

1 Introduction and Preliminaries

This definition is equivalent to the following, in view of the Bochner's Criterion. Definition 1.68. A continuous function f{t,x)

: R x E -^ E

is almost periodic in t for each x E E if for every sequence of real numbers {s!^^), there exists a subsequence (sn) such that the sequence (f{t+Sn,x))

is uniformly convergent int eR andx G E.

We finally recall the useful result [16] Lemma 3.8. Theorem 1.69. Let f : R x E —^ E be almost periodic in t for each t G E, and assume that f satisfies a Lipschitz condition in x uniformly int E:R, that is p{f{t^x)J{t,y)) for all t E R and x, (j> :R-^


y E E, where p is a metric on E. Let

E be almost periodic.

Then the function F : R -^ E defined by F{t) = f{t,({){t)) is almost periodic.

1.8 Bibliographical Remarks and Open Problems

39

1.8 Bibliographical Remarks and Open Problems The study of almost automorphic functions with values in a Banach space was initiated by M. Zaki in his Ph. D. Thesis conducted imder the supervision of S. Zaidman and published in the important paper [101]. After various subsequent contributions by S. Zaidman, B. Basit, G. M. N'Guerekata and many others, a systematic presentation has been made for the first time in [80]. To complete the study of vector-valued almost automorphic , it would be of great interest to establish the harmonic analysis of vector-valued almost automorphic functions. To this end we like to indicate the introduction of the notion of imiform spectriun by T. Diagana , M.V. Nguyen and G. M. NGuerekata in their recent and very important paper [32] that applies to almost autmorphic functions as well. The Memoirs [88] from W. Shen and Y, Yi are to be cited among the most important contributions to the study of almost autmorphic functions and their applications to ordinary differential equations and dynamical systems. Theorem 1.37 has a weak version. Indeed, the conclusion holds true even if x{t) is weakly almost automorphic (see [79]). It would be interesting to pursue the study in [80] Section 2.2 in connection with the equation of vibrating string and Probability theory. Section 1.7^ including the notion of asymptotically almost periodic functions with values in a Prechet space, is contained in a

40

1 Introduction and Preliminaries

broader work by D. Bugajewski and G. M. N'Guerekata presented for the first time in [16]. This paper contains also many interesting applications to nonlinear differential and integro differential equations in Prechet spaces. The definition of asymptotically almost automorphic function used here might be generalized in taking for instance the corrective term as h £ C(R, X) and limr^oo ^ fir II^^WIM* = 0-

Almost Automorphic Evolution Equations

This chapter presents some of the most recent developments of the apphcations of almost automorphy to evolution equations. Some existence theorems are presented along with new methods to study almost automorphic solutions to linear and nonlinear evolution equations.

2.1 Linear Equations 2,1.1 The inhomogeneous equation x' = Ax + / We consider in a Banach space {X, ||.||) the differential equation x\t) = Ax{t) + f{t),

t e R.

(2.1)

We will present various conditions for ensuring almost automorphy of the classical and/or mild solutions. We start with the simplest case ^ = A G C. Theorem 2.1. Let X be a uniformly convex (complex) Banach space. Suppose f G AA{X). Then every bounded solution of (2.1) is in

AA{X).

42

2 Almost Automorphic Evolution Equations

Proof: It is easy to check that equation (2.1) admits solutions of the form e^<*-'")/(r)dr, if R e A > 0 and X2it) = f

e^^*-^^f(r)dr, if ReA < 0.

J —OO

Let us prove that they are almost automorphic. We start with Let s = t — r; then we can write x^(t) = - I

e^'f(t-s)ds.

J —CX)

Let (5^) be an arbitrary sequence of real numbers. Since / is almost automorphic, there exists a subsequence (sn) Q (5^) such that lim f{t + Sn) = g{t) n—>>oo

and lim g{t - Sn) = fit) o

pointwise on M. So that, if we fix f G R, we can say that lim fit-

s + Sn) = g(t - s)

o

for each 5 G R. We have xi{t + sn) = -

e^'f{t -s + Sn) ds. J—00

Let us observe that | | e ^ V ( t - s + s„)||<e(^«^)^-sup||/(t)|| ten

2.1 Linear £}quations

43

and note that the right hand side of the above inequaUty is in L^(-~oo,0) since for x, we consider Re A > 0. Now if we apply the Lebesgue's Dominated Convergence theorem (Theorem 1.9), since g is boimded and measurable over M, we get lim xi{t + Sn) = -

e^^g(t - s) ds := y{t)

for each < € K. We can apply the same reasoning to obtain lim y{t-

Sn) = xi{t),

n-foo

for each t, which proves almost automorphy of Xi{t). The proof of almost automorphy of X2{t) is analogous. Now if A = i^ a pure imagin£u*y nimiber, then we get x{t) = e'^^xiO) + f e''^f(s)ds, Jo

te R

Clearly e*^ is C- almost automorphic since it is almost periodic. Therefore e^^^f{s) is almost automorphic too {Proposition i.55). ^ ^ ^ /o ^^^^f{^)ds is boimded in X for x{t) is bounded. And since X is uniformly convex, we deduce that it is almost automorphic. So x € AA{X) as the sum of two almost automorphic functions. The proof is now complete. D Before we state our next result, let us recall the following definition: Definition 2*2* A linear operator A : D{A) C X -^ X where X is a (complex) Banach space is said to be of simplest type if A e L{X)

the space of bounded linear operators in X, and

44

2 Almost Automorphic Evolution Equations

A = Y!ik=i ^kPk) where the complex numbers X^ are mutually distinct, and (Pk)i
where X is a uniformly con-

vex Banach space and A is of simplest type. Then every bounded solution of (2.1) is in

AA{X).

Proof. Take the projection Pk and apply it to equation (2.1). We get

Pkx\t) =

j^{Pkxm n

=PkC^\jPMt)+Pkm = Xk(Pkx)it) + iPkf)(t). Clearly Pkf G AA{X) by Theorem 2.3.6 [80], since Pk € L{X). Therefore by Theorem 2.1, P^x € AA{X). We conclude that n

^(*) = E^^^(*)^^^w A;=l

as a finite sum of ahnost automorphic functions.

D.

We now present the following reduction method: Theorem 2.4. Let A be a linear operator C^ -> C^ and f{t) : R -> C^ an almost automorphic function. Then every bounded solution x{t) of (2.1) is almost automorphic.

2.1 Linear Equations

45

Proof: Using Proposition 1.3.7 in [80], we let B be an invertible linear operator C^ -> C^ such that B~^AB is triangular with the representation / Ai Ci2

Cin

0 A2

C2n

B^^AB =

\0

0 ...An/

where A i , . . . , A^ are the eigenvalues of A. Let x{t) be a bounded solution of (2.1) and put y{t) =

B~^x(t).

Then t/(i) is also bounded and it satisfies the equation y\t) =

B-'x'{t)

= B-^Axit) + 5-V(<)

It is observed that B''^f{t)

: R —) C^ is almost automorphic since

B^^ is a bounded linear operator (Theorem 2.3.6 [80]). We can write the above equation as follows: y'l{t) = All/1 (t) + Ci22/2(t) + 2/2(0 =

J/n-1 = 2/;(«) =

A22/2(t) +

+ CinVnit) + P l ( 0 + C2nyn{t) + 52(t)

An-l2/n-l(0 + <^-l n2/nW + 5n~l(0 An2/n(t)+5n(t)

where y(0 = (2/l(t),...,2/nW)€C^

46

2 Almost Automorphic Evolution Equations

and {gi{t),.,.,gn{t))

= B-'f{t),

teR.

Now, yn{t) is an almost automorphic solution to the last equation {Theorem 2.1). We can say that yn-i(<) is also almost automorphic and proceed until y\{t), which proves that y G AA{X)

and

consequently x = By € AA{X) too. D 2»1.2 Method of Invariant Subspaces In a Hilbert space (if, || ||), we consider the differential equation x\t) = (A + B)x{t),

t e M,

where both A : D(A) C H h^ H axid B : D(B) C H ^

(2.2) H are

densely defined closed linear operators. In Section 4-4 [80], equation (2.2) is studied in the case where the operator B is boimded, using the so-called method of "decomposition of spaces". Now, when B is unboimded, (that is we have im imboimded perttirbation of the almost automorphic equation x^ = Ax), this method is no longer efficient. We note that in general the algebraic sum A + B is not always well-defined. Let's present some examples (see [26, 27]): Example: Let H = L^(K) and consider the operators A and B defined by Au = -u'\

D{A)=:H''{R)

and

Bu = Qu, D{B) =

{ueL^R):QueL^{R)}.

2.1 Linear Equations

47

Assume also that (5 : K —) M is a measurable function satisfying the assumption: (H)

Q{x) > 0, Q € LHK), and g ^ L^^)

(ie., f^ \f\^dx =

DO for every compact [a, 6] C M). Then we have: Proposition 2*5. f26j Under assumption (H), D(A) D D{B) = {0}. Proof. Let u G D{A) D D{B) and suppose that u ^ 0. u is continuous since it is in H^{R). Therefore there exist / C R an open subset and S > 0 such that \u{x)\ > S for all x e L Now let / ' C / be a compact subset equipped with the induced topology on / . Clearly: Q,,, = ^ ^ \ ^

^ L^V),

As a consequence Q G L^il')-

since (Q|^|)|^, G L\r)

and

This is impossible since Q ^

Therefore u = 0. Example of Potential Q satisfying (H): Assume that the fimction / satisfies: f >0, f e L^{R) with /^ not integrable near x = 0. Let ((Tn)^i be an enimieration of the rational ntimbers, set oo

satisfies the assumption (H).

48

2 Almost Automorphic Evolution Equations

Generalization. Let H = Z/^(E^) and let A, B be the operators given by Au = -Au,

D{A) = H''(W)

and Bu = Q% D{B) = {ue L 2 ( R ^ ) : Qu e ^^(R^)}, where Q : R^ H-> C is a measurable function satisfying (H)

Re Q{x) > 0, Q € L\W),

and Q ^ ^LCR"") 0^-.

/ ^ l / P ^ = oo for every compact Q C R^). Then: 1. If n < 3, the previous proposition is still valid. One can prove it using a similar method as in the proof of Proposition 2.5 . 2. If n > 4, the previous proposition is still valid (the Sobolev theorem impUcitly used in the proof of Proposition 2.5 cannot be used here anymore). This can be proved using the boimdedness of the fractional operator I^u := {—A)^^'^Uy when a. la : 1/2 (R^) h-> BMO(R^) and b. la : I/^(R^) »-> LP{W), where 1/q = 1/p - a/n with 1/p > a/n. Now, to overcome this difficulty, we will use the so-called "method of invariant spaces" introduced in [30]. But first, we will recall some preliminary facts:

2.1 Linear Equations

49

Let 5 C if be a closed subset and Ps, the orthogonal projection onto the subspace S. The operator is still a densely defined closed (possibly imboiinded) linear operator in H, Definition 2.6* S is said to be an invariant subspace for A if we have the inclusion A{D{A) DS) G S. Example 2.7. Let us mention the following classical invariant subspaces for the closed imbounded linear operator A defined into the Hilbert space H, 1. 5 == N{A) = {x € D{A) : Ax = 0} is an invariant subspace for A. 2. If i4 is a self-adjoint linear operator, then any eigenspace Sx = N(XI — >l) is an invariant for A. In iact it can be easily shown that 5A reduces A. Theorem 2.8. The equality PsAPs = APs is a necessary and sufficient condition for a subspace S to be invariant for a linear operator A. Proof Assume PsAPs = APs and if x € D{A) n 5, then x = Psx e D{A) and Ax = APsx = PsAPsx e S. Conversely, if 5 is invariant for A; letx £ H be such that Psx € D(A). Then APsx € 5 and then PsAPsX = APsx. APs C PsAPs. Since D{APs) = D{PsAPs),

Therefore

it turns out that

APs = PsAPs. Definition 2.9* A closed proper subspace S of the Hilbert space H is said to reduce an operator A if PsD{A) C D{A) and both S and H Q S, the orthogonal complement of S, are invariant for A.

50

2 Almost Automorphic Evolution Equations

Using the above Theorem 2.8, the following key result can be proved. Theorem 2.10. A closed subspace S of H reduces an operator A if and only if PsA C APsProof See the proof in [59] Theorem 4.1L, p. 29. Remark 2.11. In fact the meaning of the inclusion PsA C APs is that: if a: e D{A) , then Psx e D{A) and PsAx = APsx . Recall that their algebraic sum of A and B is defined by D{A + B) = D{A) n D{B) and {A + B)u = Au + Bu, VueD(A)nD(B). We assume that D{A) n D{B) = H and the operators A and B are infinitesimal generators of Co-groups of boimded linear operators {T{t))ten, (i?(t))t€Rj respectively, such that (i) T{t)x : t H-> T{t)x

is almost automorphic for each x e H,

R{t)y : t !-> R{t)y

is almost automorphic for each y E H,

respectively; (ii) there exists 5 C if, a closed subspace that reduces>l and B. We denote by P5, Q^ = (J - Ps) = PHQS^ the orthogonal projections onto S and H Q S, respectively; (iii) R{A) C R{Ps) = NiQsY, (iv) R{B) C R{Qs) = N{Ps). Remark 2.12. (1) Recall that if A, B generate Co-groups, their simi A + B need not be a Co-group generator.

2.1 Linear Equations

51

(2) The assumption (ii) above implies that both S and H Q S are invariant for the algebraic sum (it is well-defined as stated above)>l + B. Theorem 2*13. Under assumptions (i)'(ii)-(iii)-(iv),

every solu-

tion to the differential equation (2,2) is almost automorphic. Proof. Let x(s) he a solution to (2.2). Clearly x{s) 6

D(A)nD{B)

(notice that the algebraic sum A + B does exist since A + B is assumed to be densely defined). Now decompose x{s) as follows x(s) = Xi(s)+X2is),

(2.3)

where xi{s) = Psxis) e R{Ps) = N{Qs) and X2(s) = Qsx(s) e N{Ps) = RiQs^ We have

1(.,W) = P 4 . W = PsAx(s) + PsBx(s) = APsx{s) + PsBx{s)

(according to(ii))

= Axi{s) + PsBx{s) = ^0:1(5) (according to{iv)). Prom —(xi{s)) = Axi{s)y it follows that do

Xiis)=T{s)xx{0).

(2.4)

52

2 Almost Automorphic Evolution Equations

Now according to (i), the vector-valued function s ^

T{s)xi{0)

is almost automorphic. In the same way, since H G {H Q S) = S. It follows that the closed subspace S reduces A and B if and only if if 0 5 does. In other words, HQS reduces A and B. That is, a similar remark as Remark 2.11 holds when S is replaced hy H Q S. Thus, we have

d-sMs))=Qs-x{s) = QsAx{s) + QsBx(s) = QsAx{s) + BQsx{s)

(according to(n))

= QsAx{s) + Bx2{s) = Bx2 (s) (according to(m)). Prom the equation 'T{X2{S)) = Bx2{s), it follows that s h-> as R{s)x2{0) is almost automorphic (according to (i)). Therefore x{s) = xi{s) + X2{s) is also almost automorphic as the smn of almost automorphic vector-valued ftmctions. Corollary 2.14. Let B : H \-^ H be a bounded linear operator in the Hilbert space H. Under assumptions (i)-(ii)'(iii)-(iv),

every

solution to the equation (2.2) is almost automorphic. Proof. This an immediate consequence of the previous Theorem 2.13 to the case where B is a boimded linear operator, it is straightforward.

2.1 Linear Equations

53

2* 1.3 Almost Automorphic Solutions to Some Second-Order Hyperbolic Equations Consider now as in [28], the homogeneous second-order hyperboUc differential equation of the form ^ d -—ii(5) + 2B —u(s) + A u{s) = 0, as^ as

(2.5)

and the associated nonhomogeneous differential equation — n ( 5 ) + 2 5 ^uis)

+ A u(s) = f(s),

(2.6)

where A^ B are densely defined closed linear operators acting in a Hilbert space H and / : R »-> i / is an almost automorphic vector-valued function. The method of invariant subspaces described in Section 2.1.2 above can be used to deal with (2.5)-(2.6). For that, the idea is to reduce (2.5)-(2.6) into differential equations of first-order. Indeed, setting v{s) = -7-1/(5), the problem (2.5)-(2.6) can be as rewritten in H x H of the form ^U{s) = {A + B)Uis), (2.7) as and ^U{s) as

= {A + B) U{s) + F{s),

where U{s) = {u{s),v{s)),

F{s) = (0,/(s)) and A^B are the

operator matrices of the form A = I

(2.8)

and

B =

54

2 Almost Automorphic Evolution Equations

onHxH

with D{A) = D{A) x H, D{B) = Hx D(B), and O, /

denote the zero and identity operators on AT, respectively. Since (2.5)-(2.6) is equivalent to (2.7)-(2.8), instead of studying (2.5)-(2.6), we will focus on the characterization of almost automorphic solutions to (2.7)-(2.8). In this book, we will treat only the homogeneous case (2.7). We refer the reader to [28] for the nonhomogeneous case (2.8). As in the previous section, we will make the following assumptions: The operators A and B are infinitesimal generators of Co-groups of boimded operators (T(t))aGR) ('^(0)«€RJ respectively, such that (i) T{s)U : 3 »-> T{s)U is almost automorphic for each U e H x H, 7^(5) V : s »-> 7^(5) V is almost automorphic for each V G H X H^ respectively; (ii) There exists S C H x H, a. closed subspace that reduces both A and B. We denote by P5, Q5 = (/ x I-Ps)

= P[HxH]esj the orthogonal

projections onto S and [H x H]0 S, respectively; (iii) R{A) C R(Ps) = N{Qs); (iv) RiB) C R(Qs) = N{Ps). We have Theorem 2,15, Under assumptions (i)-(it)'(Hi)'(iv),

every solu-

tion to the differential equation (2.7) is almost automorphic. Proof. Let X{s) be a solution to (2.7).

2.1 Lineax Equations

55

Clearly X{s) e D(A) D D{B) = D(A) x D{B) (observe that the algebraic sum A + B exists since D(A + B) = H x H). Now decompose X{s) as follows X{s) = PsX{s) + {1x1where PsX{s)

e R{Ps) = NiQs),

Ps)Xis), and QsX{s)

€ N{Ps)

=

R(Qs). We have =

Pslxis)

= PsAXis)

+ PsBX{s)

= APsX{s) + PsBX{s) = APsX{s) Prom —(PsX(s)) as

(by

(by

(it))

{w))

= APsX(5), it follows that

PsX{s) = r{t)PsXio). Now ax^cording to (i), the vector-valued function s H-> PSX{S) T{t)PsX{0)

=

is ahnost automorphic.

In the same way, since we have [H x H] Q ([H x H] Q S) = Sy then it follows that the closed subspace S reduces A and B if and only ii[H X H]eS

does.

Hence

^iQsXis))^Qslxis) =

QsAXis)+QsBX{s)

= QsAXis)+BQsXis) = BQsX{s)

(by

(by (in))

(u))

56

2 Almost Automorphic Evolution Equations

Prom the equation —(QsX(s)) = BQsXis), it follows that at s i-> QsX{s) = 7l{s)QsX{0) is almost automorphic (according to (i))Therefore X{s) = PsX{s)+QsX{s)

is also almost automorphic

as the sum of almost automorphic vector-valued functions. Corollary 2.16. Let B : H \-^ H be a bounded linear operator in the Hilbert space H. Under assumptions (i)-(ii)'(iii)'(iv).

Then

every solution to the equation (2.7) is almost automorphic. Proof. This an inmiediate consequence of the Theorem 2.15 to the case where B is a, boimded linear operator, it is straightforward.

2.2 Nonlinear Equations 2.2.1 Existence of Almost Automorphic Mild Solutions-Case I We first begin with the following semilinear evolution equation in a Banach space (X, ||

||):

x\t) = Ax{t) + f{t, x{t)),

te R

(2.9)

where A is the infinitesimal generator of an exponentially stable Co-semigroup (T(t))t>o; that is, there exist K > 0, u < 0 such that \\T{t)\\ < Ke^',

for all t>0.

(2.10)

2.2 Nonlinear Equations

57

We assume that / : M x X »-> X satisfies a Lipschitz condition in X uniformly in t, that is, there exists L > 0 such that \\f{t,x)-f{t,y)\\
t e R,

(2.11)

where / G AA{X). In fact we have. Theorem 2.17. Let f E AA{X)

and A be the generator of an

exponentially stable Co-semigroup as above . Then equation (2,11) has a unique almost automorphic mild solution. Proof. We first prove existence of an almost automorhic solution. Let x{t) = T(t ~ a)x(a) + f T{t - s)f{s)ds,

for all a G R, t > a,

Ja

be a mild solution. It remains to prove that x €

AA{X).

First, we consider u{t) = /__^ T{t — s)f{s)ds, defined as lim / T{t -

s)f{s)ds.

Clearly for each r
exists. More-

58

2 Almost Automorphic Evolution Equations

II fnt Jr

- s)f{s)ds\\ < j^ll/lloo, for all r < t M

which shows /__^ T{t — s)f{s)ds is absolutely convergent. Now let (5^) be an arbitrary sequence of real numbers. Since / € AA{X)^ there exists a subsequence (sn) of (3^) such that g{t)—

lim f{t + Sn) n-4+00

is well-defined for each t G R and f{t) = lim g{t - Sn) for each

teR

Now consider r ( t + Sn -

S)f{s)ds

00

t /

= f T{t- T{t — a)g{cr), as n —> 00 for each cr e M fixed and any t> a. If we let v{t) = J^^T{t

— s)g{s)ds, we observe that the in-

tegral is absolutely convergent for each t. So, by the Lebesgue's Dominated Convergent Theorem {Theorem i.P),

2.2 Nonlinear Equations

^(* + Sn) -> v{t)j for each

59

as n —^ oo

teR.

We can show in a sunilar way that ^(* — 5n) —^ u(t) as n —^ oo for each t G M. This shows that u e Now let u{a) = / ! ^ r ( a -

AA{X).

s)f{s)ds.

So T(f - a)ix(a) = / ! ^ T{t -

s)f{s)ds.

If i > a, then /* r(< - 3)/(5)d5 = / Ja

T(t-

s)f(s)ds

- f

J—oo

=

T{t-

s)f(s)ds

J—oo

u{t)-T{t'-a)u{a).

so that, u{t) = T{t - a)u{a) + jl T{t - s)f{s)ds. u{a), then x{t) = u{t), that is a: €

If we fix x{a) =

AA(X).

We finally prove the uniqueness of the almost automorphic solution. Assume x and y are two such solutions and we let z = a: — y. Then z € AA{X) and satisfies the equation z\t) = Az{t),

teR.

Note that z is boimded and satisfies also the equation z{t) = T{t - s)z{s)

for all 5 G K, and t > s

We also have the inequality \\z{t)\\< Ke^'^'-'K

60

2 Almost Automorphic Evolution Equations

Take a sequence of real numbers (sn) such that Sn —> —oo. For any fixed £ € M, we then can find a subsequence (^n^) of (Sn) with Snj^ < t for all k = 1,2,.... Usmg the fact that a; < 0, we obtain This shows uniqueness of the solution and ends the proof. D We now state and prove: T h e o r e m 2.18. Assume that f :RxX

^ X satisfies a Lipschitz

condition in x uniformly in t, that is, \\f{t,x)-f{t,y)\\
for all x,y e X, be almost automorphic in t for

Then equation (2.9) has a unique almost automorphic mild solution. Proof. Let x be a mild solution. It is continuous and satisfies the integral equation x{t) = T{t - a)x{a) + f T{t - s)f{s, x{s))ds, Ja Consider now / T{t — s)f{s,x{s))ds

Va G R, V t > a.

and the nonlinear operator

G : AA{X) K4 AA{X) given by {Gm)-=

f

T{t-s)f(s,{s))ds.

In view of Theorem 2.2.6 in [80], if 0 G AA{X), then f{s, (t>{s)) is almost automorphic, thus G(t> G AA{X)^ so that G is well-defined. Now for 01, 02 G AA{X), we have:

2.2 Nonlinear Equations

\\Gi-G
61

T{t-s){f{sAi{s))-f{sMs)))ds\\ \\T{t -

- 2{s)\\ds

S)\\B(X)LUI{S)

< L\\(t>i - (h\\oo.s\v^ I

\\T{t - s)\\Bix)ds

t€K J-oo

=-rj\\i -
J-oo

So LK LK which proves that G is continuous. And since -;—:- < 1, then G M is a contraction. So there exists a imique u G AA{X)^ such that Gu = u, that is u{t) = f_^ T{t — s)f{s,

u{s))ds.

If we let u{a) = / ^ ^ T{a — s)f{s, u{s))ds, then T{t - a)u{a) = f

T{t-

s)f{s,

u(s))ds.

But for t > a, I T(t - s)f{s, u{s))ds = f Ja

T{t-

s)f{s,

u{s))ds

J—oo

= f

T{t-s)fis,u{s))ds

J—OO

^u{t)-T{t-a)u{a). So u{t) = T(t-a)u{a)

+ J^^ T{t-s)f{s,

u{s))ds is a mild solution

of equation (2.9) and u € AA{X). The proof is now complete. D

62

2 Almost Automorphic Evolution Equations

2.2.2 Existence of Almost Automorphic Mild Solutions-Case II We still consider equation (2.9) where A is described as above, but this time / ( t , x) does not satisfy necessarily a Lipschitz condition. In fact we wish to consider another class of functions f{t^x)

as

described below. First, let (Y, | |) denote a Banach space algebraically contained in (X, II II) such that the canonical injection Y —f X is compact. An example of such a space Y is an abstract Sobolev space that we construct as follows: Let i4 be as above. Then clearly 0 € p{A)^ so that the fractional powers (—>l)^, 0 < a < 1, are well defined. Also, since 0 G p{A)^ the norm

i/i = ii(-^r/ii

(2.12)

is equivalent to the graph norm

U = ||(-A)VII + II/||. Now we take X = I^(/2), where 1 < p < oo and i? C M^ is a smooth boimded domain in R"^. Let A be a linear uniformly elliptic operator (with suitable boundary conditions), of order 2m. Then let Y be the domain of {—A)^ with norm (2.12); we have Wo^^"^^(/2) C Y C W^2^^'^(I2) and the norm | | in Y is equivalent to the usual norm in W'^'^^'^{Q). Also, the injection Y ~> X is compact in this case, by Sobolev embedding.

2.2 Nonlinear Equations

Now let Y = D{{-A)°),

the domain of {-A)°',with

\y\ = \\i-Aryl

63

norm

yeD{i-Ar),

where 0 < a < 1 is fixed. We get \T{t)y\ = \\Tit){-Ary\\

<

Ke-'^\[(-Ary\\,

and since

Ke-'^'\\i-Ary\\ = Ke-'^\yl we obtain \T{t)y\ < Ke-'^\y\ for each j / € Y and every t>0,hy

(2.13)

(2.10).

We also make the following assiunptions: F{t,x) = P{t)Q{x),

far all

teR,xe

X,

(2.14)

where P{t) e AA(Z) for each t e R with Z = 5(X,Y); P is continuous from R to AA(Z), and Q : BC(R,X)

-^ BC{R,X)

is

continuous and satisfies the estimate \\Q
(2.15)

where ||/||oo := supfgR ||/(t)|| and X G C(R+,R+) satisfies lim : ^ ^ = 0. r—>oo

(2.16)

T

Note that M can be unbounded but must grow slower than a Imear function. Let

64

2 Almost Automorphic Evolution Equations

[P]:=sup||P(t)||z
(2.17)

teR

Define G : BC(R,X) -> BC(R, Y) by (G
T{t-s)F{s,(p{s))ds.

(2.18)

For (f € -BC(R,X), this integral exists. Indeed, we have \(Gcp){t)\< f

\T{t-s)\\P{t)Q{cp{s))\d^

J —OO

J —OO

using (2.13), (2.14) and (2.17). Consequently \G
(2.19)

Continuity of G is straightforward in virtue of continuity of both P and Q. Thus we have G(BC7(R,X))cBC(R,Y). Finally, for 0 < <5 < 1, let 5C*(R,Y) = {fe

BC{R,Y)

: \f\s,Y < oo},

where \f\s,y ^snp\f(t)\+S

sup ^^^^}~il'^^-

With the norm \'\S,YJ BC^{R, Y ) turns out to be a Banach space of all bounded Holder continuous Y-valued functions on R of Holder exponent S.

2.2 Nonlinear Equations

65

Proposition 2.19. The function G defined above maps bounded sets ofBC{R,X)

into bounded sets of BC\R,Y)

for any S > 0

satisfying S < a, where 0 < a < 1 is the exponent defining Y =

Proof. The proof is basically a modification of the above remarks. Let 0 < /? < a. Then \(G
T{t-s)i-An-A)-<'F(sMs))ds\

\Tit-

s)(-Am-A)-^P{s)\\Q{^{s))\ds

(2.20)

J — OO

Now, by Proposition 1.19, there exists a constant Ki such that

\\T(r)(-An < ^ ^ for all r > 0. Thus we obtain, as previously, \T{r){-Af\

< Kie-'^r-^,

Next, we observe that the function s

r > 0.

> {—A)~^P{s) is a uni-

formly bounded function R -> B(X, D{{-AY-^). composition of P(-) : R -^ B(X, D{{-AY)) [P], with i-A)-^,

(2.21)

Indeed, it is the

which is bounded by

an isometry from D{{-AY)

onto £>((-^)'*-^).

Thus SUp||P(i)||B(X,D((-^)'»-/')) < [^]-

Now combining the estimates in (2.20) and (2.21), we deduce \{G'p){t)\< f

K,e-'^^'-^\t

-

J —<X>

Letting r = i — 5 in the integral gives

s)-^[P]M{\\P\Uds.

66

2 Almost Automorphic Evolution Equations

\G
Kre--W-^[P]M{M\)dr,

that is, \(G
(2.22)

where Ci(/3) depends on P.Ki^oj and [P]. Next, for ^2 > ti, we have

\(Gcp)it2) - {G
KT

+1 r

- r

)nh-

s){-AY{-A)-^P{s)Q{
(r(<2 - 5) - r(
J —oo

< r

s)){-AY{-A)-^P{s)Q{
\T{t2 - s)(-An-A)-^

X Pis)Q(
Jti

+ r

l(n*2 - U) - /) X T{t, -

s){-AY{-A)-^P{s)Q{
J —OO

= Ji + J2. By the same argument leading to (2.22) we get

0


2.2 Nonlinear Equations

J2 < r

\{nh - h) - I){-A)-^T(h

-

67

S)(-AY^-^\-A)-^

J 00

X P{s)Q{ip{s))\ds < r

\{T{t,-t,)-I){-A)-'^\-\{T{h-s)

J 00

X {-A)^P-'^\-A)-^P{s)Q{
< mh - h) - /)(-^)-^| Xp

\T{t, -

s){-Af^-'^\-A)-^P{s)Q{
< \{T{h - fO - /)(-A)-^|C73(/3,7)^(||<^||oo) provided 0 < 7 < /3. Next recall that (T(r) ~ I)g = /J* T{s)Agds for g e D{A), by the fundamental theorem of calculus. Thus, for / G Y, |(T(r) - I){-A)-y\

= II r

TisX-AY-^-^i-ArfdaW

Jo

< \\{-Arf\\ r

Mre-^'s'-^-'^ds

= 0,(7,0;, i^i)r2-^-°|/|, since 1 — 7 — o; > —1, because 0 < 7 < / ? < a < l . In other words, we have \iT(r) - I)(-A)-^\

< Cy-'^-'';

consequently, J2 < C^ih For 6 = min{2

- a,l-

hf-''-''CzM(y\U. (3) > 0, it follows that

\iG
(2.23)

68

2 Almost Automorphic Evolution Equations

where C5 depends on a;, Ki^ [P]jCty /?, 7 and Y, that is, on parameters of the problem. It follows that, for (p G SC(R,X) with \\(p{t)\\ < i? for alH € M, then G(p € BC^{R, Y) with i|G<^(OII < ^1 for all « € E and some Ri that depends on R. This completes the proof. D Proposition 2.20. The function G maps bounded sets of AA{X) into bounded sets of BC^(R, Y) n AA(X) forO

Proof We just need to check that GiAA{X)) c

AA{X).

To this end, let (p G AA{X). Then given a sequence (5^) C M, there exists a subsequence (sn) C {s'^) ^^^^ *h^* r/j{t) = lim (p{t + Sn)y n->oo

is well defined for each t GR and lim V^(t - Sn) = (fit) for each t e R. Since ^ € BC(R,X), then T{t + sn-

s)Pis)Q{
0

Let a = s — Sn- Then {Gip){t + Sn)= f = / J—CX)

T{t - a)P{a + Sn)Q{
2.2 Nonlinear Equations

69

where P„(o-) = P{(T + Sr,),Qr,{a) = Q{{
exists for each cr € M and lim P{a - 5n) = P(^) n-¥oo

for each cr G M. Clearly we also have, by passing to a subsequence if necessary, lim ^{t + Sn) = il^{t) o

and lim i){t-Sn)

= ^{t),

n-¥oo

for each f G M. By the Bochner's integral version of the Lebesgue's Dominated Convergence theorem, we get {G^){t + Sn)= f -> f

T(t - a)Pn{a)Qn{p)dG

T{t - a)P{a)Q{^{a))da

= x{t)

J—oo

for each t € R, and X{t - Sn) = f

'^ T{t - 5n -

a)P{a)Q{^lj{a))da

J—oo

= f

Tit-

r)P{r - Sn)Q(ipir - 5„))dr

,

70

2 Almost Automorphic Evolution Equations

by letting r = <J + 5n, thus we obtain X(t-sn)—> r

T{t-r)P{r)Qi
= {G
J —OO

again by the Lebesgue's Dominated Convergence theorem. This shows that G{AA{X)) C ^ ^ ( X ) and the proof is now complete. Proposition 2.21. The canonical injection id : BC^{R,Y)

->

B<7(R, X) is compact, which implies that id : BC\R, Y) n AA{X) -^ AA(K) is compact too. Proof. We wish to show that id maps bounded sets of BC^(R, Y) into relatively compact sets of J3C(M,X). To this end, let (ipi,) be a boimded sequence in J3C^(R, Y). Let Q = {rn} be the set of all rational nimibers. Then {(piyivn)) is a boimded sequence in Y, for each n. By the well known Cantor diagonalization process, there exists a subsequence (^uf^) such that

as A; —) OO in X, for each n, and some cp :Q—^X. But the sequence {(pn) is an equicontinuous family of functions in BUC{R,Y) dition.

C 5t/C7(M,X), because of the uniform Holder con-

2.2 Nonlinear Equations

71

Thus, as in the proof of the Arzela-AscoU theorem, there is a further subsequence (which we still denote by ( (p{t), as A; -f oo

(2.24)

in X, for alH € M. In addition the convergence is uniform in t e M. Note that BUC{R, X) can be identified with C(r, X) for a suitable Hausdorff compactification T of R (see for instance [33]). Thus the convergence (p„^ -^

and \iG
ti\6M{M\oo),

hold for all ¥? € BC(R,Y) and all ti,t2 € R with t2 not equal to ti.

It follows that there exists a constant Ce = C6(u},K,Ki^a, /?, 7) such that (p e BC{R,X) Gip e BC\R,Y) where Ri = CeM{R).

and and \G(p\ < Ri,

\\ip\\oo < R implies

72

2 Almost Automorphic Evolution Equations

Since M(R)/R

-^ 0 as R^

0, said since \\y\\ < Crlvl holds for

some constant C7 and all y 6 Y, it follows that there exists p> 0 such that for all i2 > p, we have G{BAA(X)(0,

R)) C SBC^(R,Y)(0, R)

n SAA(X)(0, iJ).

(2.25)

Since G leaves i4A(X) C J5C(R, X) invariant, the estimate (2.25) along with the continuity properties of G imply that G is a continuous, compact mapping S -^ S^ where S is the ball of radius R in AA{X) and

R>p.

By the Schauder fixed point theorem G has a fixed point in 5, Obviously, ^0 is ^ ii^ild solution of (2.9). Finally, the above results can be summarized as follows. Theorem 2.23. Consider the evolution equation (2.9) where A generates an exponentially stable Co-semigroup T in B{X). Assume assumptions (2.10) and (2.14)-(2.17). Then (2.9) has a mild solution in i4A(X). Now we give the following example. Example of Nonuniqueness of Almost Automorphic Solutions LetX = R, A = ~ 1 and ^3/2gi-t

u{t) = {

0

Then for t e [0, |] we have

fortG[0,|],

forte h|,0].

2.3 Optimal weak-almost periodic solutions U'{t) = -U{t)

+ ^i^/2g(l-t) ^ _ ^ ^ ) + 3

73

(^)l/3g§(l-t)

which can be written as

where ..f

.

.'Iv^'^'eK^-*^

forf€lO,|]xR,

Note that «'(|) = Oaiidt.(|) = ( | ) t e - § . Now let on [|,3] x R , /(*,V') = <

on [3, | ] X R.

u{\) u{t) = {u{\-t) 0,

on [|,3], on[3,|], for t = 0.

Extend u to be a periodic function of period 6 (hence an almost automorphic function). Then u and v = 0, both satisfy ^

= - x + /(t,a;), x(0) = 0.

2.3 Optimal weak-almost periodic solutions In this section we still consider equation (2.1) in a imiformly convex Banach space (X, || ||). We will prove existence and uniqueness

74

2 Almost Automorphic Evolution Equations

of the so-called "optimal" almost periodic solutions in the weak sense, a result contained in [77]. We make the following assumptions: Al: A: D{A) C X »-> X is a linear operator (generally imbounded) that generates a Co-semigroup of uniformly bounded linear operators T{t), t e R"*"., i.e. there exists M > 0 such that s^Pt€M+ ll^(*)ll = M' For each T(t), t € R+, T*{t) will denote the adjoint operator of T{t). A2: / : E H-> X is a nontrivial strongly continuous function. Now we denote by i?/, the set of all mild solutions x(t) of equation (2.1) which are bounded over R, that is //(x) = sup ||x(<)|| < oo. We assume here that 1?/ ^ 0. Definition 2.24. A bounded mild solution x{t) of (2.1) will be called an optimal mild solution of (2.1) if /x(x) = /i* = inf /i(x). x£Qf

Theorem 2.25. Under assumptions A1-A2 and assuming that Qf ^ 0, equation (2.1) has a unique optimal mild solution. The proof is based on the following well-known fact (cf. [55], Corollary 8.2.1) Lemma 2.26. If K is a non-empty convex and closed subset of a uniformly convex Banach space X and v ^ K, then there exists a unique ko G K such that \\v — ko\\ = i^keK \\v — ^||-

2,3 Optimal weak-almost periodic solutions

75

Now we begin the proof of the theorem: Proof. Since the trivial solution 0 ^ i?/, it suffices to prove that i7/ is a convex and closed set; we then use Lemma 2.26 to deduce uniqueness of the optimal mild solution to equation (2,1). Let us first obtain convexity of Qf. Consider two distinct boimded mild solutions Xi{t) and X2{t), and a real number 0 < A < 1 and let x{t) = Axi(t) + (1 — A)x2(t), t G R. x{t) is continuous and has the integral representation x{t) = T{t - io)a:(to) + f T{tJto for every to ER and t>to. Here we have

s)f{s)ds

x{to) = Axi(to) + (1 - A)x2(to). x{t) is then a mild solution of equation (2.1). It is easy to see that x{t) is bounded over R since fi{x) = snp\\x(t)\\ < A/i(j;i) + (1 — \)fi{x2) < oo. We conclude that x{t) € i?/. Let us now show that i?/ is closed: Consider a sequence (xn) in ]?/ such that limn->oo^n(*) = ^(*)? t G R. We have for all to 6 R and t > to : Xn{t) = T{t ~ to)Xnito)

+

f

T{t ^

Jto

n = 1,2,... For every fixed t and to with t>tQ,we

have

s)f(s)ds

76

2 Almost Automorphic Evolution Equations

liinn->oo T{t - to)Xn{to)

= T{t - to) linin-^oo Xn{to)

= T{t - to)x(to) using contintdty of the operator T(t — to). We then deduce that x{t) = T{t - to)x{to) + f T{tJto

s)f{s)ds

for all to € R, t > to. This shows that x{t) is a mild solution of (2.1). Finally we claim that x{t) is bounded over R. Indeed, let us write x{t) as follows: X{t) = T(t - to)x{to) = T{t -- to){x{to)

+ fl T{t - S)f{s)ds

- Xn{t) + Xn{t)

- Xnito)) + Xn{t)

for every n = 1,2,.,., and every to € R and t > to* This gives \\x{t)\\<M\\xito)-Xr.{to)\\+\\Xnm

<M||x(to)-a:n(to)||+/i(xn). Choose n large enough such that ||x(
D

Theorem 2.27. Let f e AP{X) and assume that Al and A2 hold true. Then the optimal mild solution of equation (2.1) is weakly almost periodic.

2.3 Optimal weak-almost periodic solutions

77

Proof. Consider x{t) the optimal mild solution of equation (2.1). Such x{t) is unique by the previous Theorem. We have x{t) = T{t - to)x{to) + f T{tJto

x)f{s)ds

for all to ^ K, f > to- Let {s'^) be an arbitrary sequence of real numbers. Since / is almost periodic, we can extract a subsequence (sn) C {s'^) such that lim / ( t + Sn) = 9{t) imiformly in t € M. This fact is assured by Bochner's Criterion. We know g(t) is also strongly continuous. Now, for fixed to € M, we can obtain a subsequence of (5^), which again we will denote (5n), such that w - lim x(to + Sn) =

vo^X,

n->oo

since X is a reflexive Banach space. The function 2/(t) = T(t - to)i;o + f Tit--

s)g{s)ds

Jto

is then strongly continuous. It is a mild solution of

x\t)=Ax{t)-tg{t),

teM.

We can prove. Lemma 2 . w ~ lim x(t + Sn) = yit) for each t e R.

7S

2 Almost Automorphic Evolution Equations

Proof. Let us write X{t + Sn) = T{t - to)x{to

+ Sn)+

f

T{t - s)f{s

+

Sn)ds

Jto

n = 1,2,

Take x* in X* and let

< x\ T{t - to)x(to + Sn)> - < x\ T{t - to)vo >

= < T*(t - to)x*,x{to

+ Sn)

-Vo>

for every n = 1,2,..., from which we observe that the sequence {T{t — to)x{to + Sn)) converges to T{t — to)vo in the weak sense. Also we have the following inequalities f T{t - s)f{s + Sn)ds - f T{tJto Jto = " / ''T{t-s){f{s Jto

< f\\T{t-s)\\-\\f{s

+

s)g{s)ds

sn)-g{s))ds\\ +

s,,)-9{s)\\ds

Jto

<M

f \\f{s +

sn)-gis)\\ds.

Jto

Therefore lim / T{t - s)f(s + Sn)ds = [ T(t^^«^ Jto

s)g(s)ds

Jto

in the strong sense and consequently in the weak sense in X. This proves the Lemma D We also have: Lemma 2.29* fi{y) = ^{x) = /x*

2.3 Optimal weak-almost periodic solutions

79

Proof, Since x{t) is an optimal mild solution of equation (2.1), we have /i*=/i(a:) = sup||a:(i)||. Take an arbitrary x* in X*\ then by Lemma 2,28 we obtain lim < X*, x(t + Sn) >=< X*, y(t) > for every < e R. Now for each n = 1,2,..., we have \<x\x{t

+ Sn) > I < \\x*\\\\x{t + 8n)\\ < ||x*||/i*.

Therefore, | < x*yy{t) > \ < \\x*\\fjL*, for every t G M, and consequently ||j/(t))|| < /i*, for every t G M, so that /i(y) < //*. Suppose that fji{y) < //*. Plemark that limn--^oop(* — ^n) = /(*) uniformly in i G M since / G AP{X).

Also since X is a reflexive

Banach space, we can extract from the sequence (5^), a subsequence which we still denote (sn) such that (y{to — Sn)) is weakly convergent, say to z G A'. Now we have lun y{t ~ Sn) = T{t - to)z + f T{t ^-^«^ Jto

s)f{s)ds

in the weak sense, for every t G R. Let us consider the function z{t) = T(t - to)z + f T{t-

s)f{s)ds.

Jto

It is a bounded mild solution of equation (2.1). For the same reasons stated above, we have fJ'iz) < /i(t/), therefore /i(z) < /i*, which is absurd by definition of //*.

80

2 Almost Automorphic Evolution Equations

We also need the following: Lemma 2.30* /i(j/) = mf fiiv) i.e. y(i) is an optimal mild solution of the equation x'(t) = Ax(t)+g(t),

teR.

Proof. By Lemma 2.29^ y(t) is bounded over R. We know also that y{t) is a mild solution of x'{t) = Ax{t) +g{t), teR.

So y{t) € Og.

It remains to prove that y{t) is optimal. Suppose it is not. Since fig ^ 0, there exists a unique optimal solution v(t) of x\t) = Ax{t) + g{t) by Theorem 2.25. And ti{v) < fi{y) and v(t) = T{t - to)v{to) + f T{t-

s)g{s)ds

JtQ

for alHo e R, i > toWe can find a subsequence {snf,) C {sn) such that weak - limjb^oo v{t - Sn^) = T{t - £o)^ + £ T{t := V{t) Observe that V{t) € i?/ and f^{V) < tx{v) < fx{y) which is absurd. Therefore y(t) is an optimal mild solution of x\t) =Ax{t)+g{t),

teR,

and in fact the only one by Theorem 2.25. D

s)f{s)ds

2.3 Optimal weak-almost periodic solutions

81

Proof of Theorem 2.27(coiitinued): To show that x(t) is weakly almost periodic, it suffices to prove now that weak — lim x{t + Sn) = y{t) n—¥oo

miiformly in t e R. Suppose that it is not the case; then there exists x* € X* such that lim < x*,j:(i + 5n) > = < x*,y(t) > n->oo

is not uniform in t € M. Consequently, we can find a number a > 0, a sequence (tk) with two subsequences (5]^) and (sk^) of {sn) such that \<x\x{t

+ 4 ) - x{t + 5fc") >\>a

(2.26)

for all A:= 1,2,... Let us again extract two subsequences of (5^) and (s!^) respectively, without changing the notation, such that \mif{t

+ tk +

4)=gi{t)

limf{t

+ tk +

4)=g2{t)

and

both uniformly in i G M, since / € AP{X), As we did previously, we may obtain weak- lim / ( t + t f c + 4 ) = T{t-to)zi+ ^-^<»

and

f T{t-s)giis)ds Jto

= yi{t)

82

2 Almost Automorphic Evolution £k|uations

weak- lim f{t+tk+4) = T{t-to)z2+ f T{t-s)g2{s)ds = y2{t) ^-^~ Jto for each t e R, where y\{t) and y2{t) are optimal mild solutions in Qg^ and Qg^^ respectively. Now we can show that 5^1(5) = g2{s)^ s E R; indeed since lmiA;-^oo/(* + *A; + ^A:) exists Uniformly in i € M and (5J^), {s'l^) are two subsequences of {sk)^ we will get

sup||/(5 + 4 ) - / ( 5 + 4')ll<e if A; > A:o(^) and consequently

sup ||/(t + tife + 4) - f{t +1^ + 4011 < ^ for k > koie)^ which shows that gi(s) = 52(^) for all 5 G M. By the imiqueness of the optimal mild solution we get yi{t) = 2/2(0? t e E. But yi{0) = weak - lim x{tk + 4 ) and ^2(0) = weak - lim x{tk + 4 ) It is then clear that the equality yi{0) = 2/2(0) contradicts the inequality (2,24) above and establishes the proof of the Theorem. D

2.4 Existence of Weakly Almost Automorphic Solutions We give in this section a result on the existence of a weakly almost automorphic solution to the equation (2.1), It is a slightly different version of a result in [101].

2.4 Existence of WeaMy Almost Automorphic Solutions

83

Theorem 2,31. Let {X, \\ ||) be a reflexive, separable Banach space, and assume that A is the infinitesimal generator of a Cosemigroup {T{t))t>0' Let X* be the dual space of X and T*{t) € L{X*) the adjoint operator ofT{t), for each t >0 with the property that lim T*(t)(p = 0 for every t

(p e X*

>'O0

in the uniform operator topology. Assume also that f is weakly almost automorphic. Then every bounded mild solution of (2.1) is weakly almost automorphic. We first state and prove the following: Lemma 2.32. Under assumptions of the theorem, we claim that the functions T{t — s)f{s),

T{t — s)g{s) : [a, t] ^-^ X are strongly

measurable and \\T{t — 5)7(5) ||, \\T{t — 5)5'(5)|| are Lebesgue integrable. Proof. By strong continuity of T{t—s) and weak continuity of f{s), it is clear that T{t — s)f{s) is weakly continuous, thus strongly measurable. Moreover the set B = {T{t — s)f{s)/

s G [a, t]} is contained in

the least closed subspace spanned by the set

{T{t-s)f{s)/seQf)[a,t]} (Q denotes the set of rational numbers). Hence {T{t — s)f{s)/

s G [a,t]} is separable.

84

2 Almost Automorphic Evolution Equations

Also note that T{t — s)g{s) is weakly measurable as pointwise limit of the following sequence of strongly measurable functions T{t-S)f{s

+ Sn).

And since the Banach space X is assumed to be separable, strong and weak measurabilty are equivalent. Measurabilty of both numerical functions \\T{t — s)f{s)\\ and \\T{t — s)g{s)\\ is also easy to establish. As a result of the lemma, the functions T(t - s)f{s),

T{t - s)g{s) : [a,

t]^X

are integrable in Bochner's sense. We are now ready to prove the theorem. Proof. Let x{t) = T(t-a)x(a)+/JT{t-s)f{s)ds

t>aheamild

solution of the equation (2.1) such that 5ixpt€R||^(*)ll = Af < oo. Given an arbitrary sequence of real numbers (5^), consider the functions Xn{t) defined by Xn{t) := x(t + s^)

teR.

Since for each t € R, the sequence {Xn{t)) is bounded, there exists a subsequence {sn,o) of (s^) such that w - lim Xn,o(0) =W'-

lim x{sn,o) = yo

exists in X by Proposition 1.2.18 in [80]. Prom the sequence (5^,0), we can extract a subseqence (5^,1) such that w — lim a:n,i(-l) = i/; — lim x(—1 + Sn,i) = 2/1

2.4 Existence of Weakly Almost Automorphic Solutions

85

exists in X. We continue the process inductively and we take the diagonal sequence (sn) to obtain w - lim xJ-N)

= w-

lim xl-N

+ Sn) =yNj

^ = 0,1,2,...

Now using weak almost automorphy of the function / , we can find a subsequence of (sn) which denote again by (s^) such that w-

hm x{-N+

Sn) = yNi iV = 0,1,2,...

ty - lim f{t + Sn) = git) for each w — lim g(t — Sn) = fit)

teR

for each i € R.

rn-¥oo

We now need to prove that w — lim xCt + Sn) exists for each i G M. Fix teR

and choose N such that —A^ < t. Then it is

X{t + Sn) = T{t + N)x{-N

+ Sn)+

I T{t - S)f{s + Sn)ds. J-N

Take an arbitrary

) between

X and X*, we get (
T{t ~ 5)/(5 + Sn) ~ 5(5)d5) =

/ {T''{t-s)^J{s J-N

+

Sn)-g{s))ds.

Now we put Fn{s)=^{
+ Sn)-9is)),

Tl = 1,2, ...

86

2 Almost Automorphic Evolution Equations

Prom the lemma above it follows immediately that {Fn(s)) is a sequence of measurable fimctions defined on the compact interval [—iV,t]. Moreover we have the inequality \Fn{s)\<M\\T{t-sm\\f(s

+ Sn) +

Since ||T(i)|| < Ke^^ for every t > 0, the bound \\T{t - s)\\ < L holds true on [—N^t] for some 0 < L < oo. Note also that both f{t) and g{t) are bounded (cf Proposition 1.32). Hence {Fn{s)) is a boimded sequence of measurable functions on [—iV, t]. Also it is clear that w — limnH^oo F{s) = 0 everywhere on [—N,t]; thus lim /

F{s)ds = Oy

that is tt; ~ lim /

T{t - s)f{s + Sn)ds = /

T(f -

s)g{s)ds.

We infer that w — liinn»->oo ^(t + Sn) = y{t) where y{t) = T{t + N)y{-N)

+ f

T{t^

s)g{s)ds.

J-N

Now take any pair of real numbers a, t such that t> a and choose a positive integer N such that —N
Then we get s)gis)ds.

Ja

We have also \\y{t)\\ < liminf \\x{t + Sn)\\ < M for every t G R. Hence

2.4 Existence of Weakly Almost Automorphic Solutions

87

sup 11^(011 <M. Repeating the argument that we used to show the existence oiw — Um^H-xx) 00{t + 5n), we can show that there exists a subsequence of (Sn) which we still designate by (sn) such that w—limn^oo vit — Sn) exists for every t ER. Call z(t) this limit, then we have z{t) = Tit - a)z(a) + / T(t -

s)f{s)ds,

Ja

with sup|Ki)||<sup||2/(£)||<M. Finally let us show that z{t) = x(t) for every

teR.We&xteR

and choose arbitrary e > 0 and (p e X*. We have \{oo^*(*)v^ = 0 for every ^ G A'*, we can choose a negative number a sufficiently large so that ||T'*(t — a)y?|| < -^^ which gives |(V?, z{t) - a;(t))| = 0 for every ^ G X*, that means 2:(t) = x{t) for each t € R. The proof is now established.

D

88

2 Almost Automorphic Evolution Equations

2.5 A Correspondence Between Linear and Nonlinear Equations Consider in a Banach space (X^ ||.||) the differential equation x\t) = Ax{t) + fit),

t € R,

(2.27)

and the nonlinear differential equation x\t) = Ax{t) + f{t) + g{t,x{t)),

t e R,

(2.28)

where A is the infinitesimal generator of a Co-group T = (T{t))teR of boimded linear operators on X, such that sup^^jj^ 11^(011 = M < GO. Let i7i:= the spsice of all bounded mild solutions of (2.27) and 1?2^= the space of all bounded mild solutions of (2.28). Let Cb{R;X) denote the Banach space of continuous bounded functions u:R^

X equipped with the norm ||t6||o = sup^^^ ||u(*)ll-

Then obviously Oi C Cb{R] X), for each 2 = 1,2. We begin with the following: Theorem 2*33. Assume that the function f :R —> X is (strongly) continuous andg :Rx

X -^ X is (strongly) jointly continuous in

t and X. Let \\9{t,x)-git,y)\\
for every

teR

and, x^y £ X where the numerical function h G L^(]R) and

lmdt<j^. We assume also that

2.5 A Correspondence Between Linear and Nonlinear Equations

89

f \\9{t,0)\\dt
Then i7i and Q2 o.re homeomorphic. Proof: Consider the mapping F : Ct{R] X) —> Cb{R] X) defined by Fx{t) = f T{tJo

s)g{s, x{s)) ds,

t eM

for each continuous bounded function x(i) : M --)' X . It is easy to observe that F is well-defined on Cb{^\X). \\Fx{t)\\<M

Jo

Indeed we have

f\\9{s,x{s))\\ds

< M [ ^ ' \\g{s, x{s)) ~ g{s, 0) || ds 4- f^ \\g{s, 0) || dsj < M \^j\{s)\\x{s)\\ <M\K I

ds + ^ ' \\g{sM\ d^

f h{s) ds-^ f \\gis,0)|| ds] JR J

JR

< 00

for each t G K, where K = ||x||o = sup^^j^ lk(OIIAlso for each pair, u,v e Cbi^j X), we have \\Fu{t) - Fv{t)\\ = Ifnt

^ s)(^g{s,u{s)) - g{sMs)))

<M\\u-v\\o

ds

f h{s)ds Jo


where the constant c = M J^ h{t) dt. So the mapping F is a strict contraction. Now let z{t) be a bounded mild solution of (2.27) with the representation

90

2 Almost Automorphic Evolution Equations

z{t) = T{t)z{0) + f T{tJo

s)f{s) ds,

t e M.

Consider the mapping S : Cb{R\ X) —y Cb{R\ X) defined by Su{t) = z(t) + Fu{t) for each ^ t ) eC7fe(R;X). 5 is also a strict contraction on Ch(^\ X). Hence it possesses a unique fixed point, say w{t)^ which satisfies the equation w{t) = Sw{t) = z(t) + Fw{t) for each t € M. That is, w{t) = T{t)ziO) + / r ( * - s) (/(^) + 9{s. ^i^))) = T{t)w(0) + f Tit-

ds

s) (/(5) + gis, w{s))^ ds

for each t e R, since z{Qi) = w{0). Obviously, w{t) is a mild solution of (2.28). It is bounded as the stun of two bounded functions and is obviously continuous. On the other hand, let x{t) = T{t)x{0) + I Tit-

s)(jis)+gisMs)))

ds

be a given boxmded mild solution of (2.28), and consider zit) = Tit)xiO) + f TitJo

s)fis)ds.

Then zit) is a boimded mild solution of (2.27) and zit) = a:(0), zit) = xit) - Fxit),

t €R. The fact that the mapping z i—> x is

a homeomorphism follows from the inequalities

2.5 A Correspondence Between Linear and Nonlinear Equations

91

l k i - X 2 | | o < :; -|ki-Z2||o 1 —c

and l k i - ^ 2 | | < (l + c)||a:i-X2||o. The theorem is proved. D T h e o r e m 2.34. Let the functions f and g have the properties described in Theorem 2.33. Assume in addition that / €

AA{X).

Then every mild solution of (2.28) restricted to R"*" is asymptotically almost automorphic. Proof: Let x{t) be a mild solution of (2.28) restricted to R^. So x{t) = T(t)x(0) + f T{t-

s) (/(5) + g{s, x(s))^ ds

for each t G M"^. Observe that the fmiction t{t) = T{t)x{0) + f T{t-

s)fis) ds,

t €

is almost automorphic. Writing g(s, x{s)) = g{s, x{s)) - g{s, 0) + g{s, 0), we obtain the inequality

y <M(K

T{-s)g{s,x{s))dsj

rhis)ds

+ r\\gis,0)\\dsj

< oo,

which shows that the improper integral /»00

/ Jo

T{—s)g{SyX{s))ds exists in X.

Consequently the function R -^ X defined by

92

2 Almost Automorphic Evolution Equations /»oo

o

T{t) / T{-s)gis, Jo

x(s))ds =

Jo

T{t-

s)g(s, x{s)) ds

is almost automorphic. On the other hand, for each t G M"^ we have oo

/»oo

/

T{t - s)g(s, xis))ds\\ < M(K

/

h{s)ds

+1^°° \\gis,0)\\ds which shows t h a t

I

I f^^

II

/

T{t-

s)g{s, x{s))ds\\ = 0.

I 1+

I

Finally, if we write o

a:(0 = z{t) +

o

T{t-

s)g{s, x{s))ds - /

T{t - s)g{s, x{s))ds

for each t € R"^, we see that x{t) is indeed asymptotically almost automorphic,

D

We now state the following theorem whose proof is a combination of the above results: Theorem 2.35. Under the assumption of Theorem 2.33 and Theorem 2.34, ihe spaces AA{X)f)S^i omorphic.

and AAA{X)f)Q2

CL're home-

2.5 A Correspondence Between Linear and Nonlinear Equations

93

Bibliographical Remarks and Open Problems Among important results in this chapter, we like to mention N'Guerekata's contribution (Theorem 2.1) and the so-called method of reduction (Theorem 2.4) also refered as a result of BohrNeugebauer type which has various generalizations for instance in [62], and other cases in [76] where the use of a method of decomposition of the space has been successful. In perturbed equations of the form x\t) = (A + B)x{t) and its purtiu:bation x\t)^{A

+ B)x{t) + f{t),

where both operators A and B are unboimded, the use of the invariant subspaces theory in [31] has successfully produced almost automorphy of solutions. Theorem 2.15 is contanied in [28]. For nonlinear equations, a partial result on the existence and the imiqueness of almost automorphic solutions was obtained in Theorem 2.17 (see also [75]) when A generates an exponentially stable semigroup. Its variant, Theorem 2.23, is a contribution by J. A. Goldstein and G.M. N'Guerekata (see [46]).This result is new, even for the almost periodic case. Another important result is contained in [32], in the case of holomorphic semigroup, using the method of sums of conmauting operators. The same paper introduces the notion of imiform spectrum of the the forced term / . The ultimate aim of this pioneering work is to compare the spectrum of / and the solution's

94

2 Almost Automorphic Evolution Equations

one as in almost periodic case. This work has generated several other papers, and the method introduced there can be applied to higher order evolution equations (see for instance [58]). It is also of great interest to study linear inhomogeneous evolution equations of the for x' = Ax + f where the operator A generates instead a C-semigroup. For linear functional differential equations, Y. Hino and S. Murakami studied the existence of almost aimorphic solutions in their interesting paper [50]. One might obtain an easier proof of their results using the methods presented in this chapter. Theorem 2.35 is SL variant of Theorem 6.1.3 in [80].

Almost Periodicity in Fuzzy Setting

In this chapter, the theory of ahnost periodicity as known in Banach spaces (see for instance [2], [23], [80]), is studied in fuzzy setting. It is based on a work by B. Bede and S. G. Gal [40]. We start with a brief overview of basic properties of the so-called fuzzy sets in Section 3.L Then we introduce the notion of almost periodic fuzzy functions in Section 3.2, and study their harmonics in Section 3.3. Integration and differentiability of fuzzy functions are also introduced. Finally we apply the results obtained to fuzzy differential equations in Section 3.4.

3.1 Fuzzy Sets Definition 3.1, Given a set X ^ 0, a fuzzy subset of X is a mapping it: X -> [0,1] and obviously any classical subset A of X can be considered as a fuzzy subset of X defined by XA * X -^ [0,1], XA {X) =

l,ifxeA,XA{x)=OifxeX\A.

96

3 Almost Periodicity in Fuzzy Setting

Definition 3.2. Let us denote by R^ the class of fuzzy subsets of the real axis E (^ie. r/ : R -4 [0,1]^, satisfying the following properties: (i)^u G R^, u is normal i.e. 3xu € R with u {xu) = 1; {ii)\/u € R^, u is convex fuzzy set, i.e. u{tx + {l-t)y)

> min{u(x)

,u(y)},

Vt€[0,l],a:,y€R; (Hi) VIA 6 R^, u is upper semi-continuous on R; (iv) {x eR:

u{x) > 0} is compact.

Then R^ is called the space of fuzzy real numbers. Remark 3.3. It is clear that R C R^, because any real number Xo € R, can be described as the fuzzy number whose value is 1 for x = xo and 0 otherwise. We will collect some other definitions and notations needed in the sequel. For 0 < r < 1 and u e Rjr, we define [uY :={ar€R;n(a:) > r}

[uf := {xeR\u{x)

>0}.

Now it is well known that for each r € [0,1], [ix]^ is a boimded closed interval. For ix,v G R^ and A € R, we have the sum and the product A © tx defined by [u®vY = [uX + [vY , [X(duY = \[uY,

Vr€[0,l],

u®v

3.1 Fuzzy Sets

97

where [uY + [vY means the usual addition of two intervals (as subsets of M) and A [uY means the usual product between a scalar and a subset of R. (see e.g. [20]) Now we define D :Rjr xRj:-^R^U

{0} by

D (u, v) = sup max {|t/!l — t;l | , |Ti!j_ — i;!j. |} , r€[0,l]

where [i/f = [n!l,tx!i.], [vY = [^->^!I-]We also have the following well-known properties ([20]): (a) D{u®w,v®w) (b) D{kQu,kQv)== (c) D(u®v,w®e)

= D(iz,v), \fu,v,w e Rjr; \k\D{u,v),

Vu.v eRj^^k

< D{u,w) + D{v,e) ,^u,v,w,e

eR] € R^ and

(R^, D) is a complete metric space. Also, the following is known: Theorem 3.4. (i) If we denote 6 = X{o} then 0 e R^ is neutral element with respect to ®, i.e. u^O = 0®u = u, for allu € R^. (ii) With respect to 0, none ofu£

R ^ \ R has opposite in R^ (with

respect to ^). (Hi) For any a, 6 € R with a, 6 > 0 or a, 6 < 0^ and any u G R^, we have {a + h)Qu =

aQu®hQ)u.

For general a, 6 G R, the above property does not hold, (iv) For any A € R and any u^v E R^, we have X(Z){U®V)

=

XQU®XQIV.

98

3 Almost Periodicity in Fuzzy Setting

(v) For any A, /x € R and any u 6 M^; we have

(vi) If we denote \\u\\jr = D(u,0),

"iu 6 R^, then ||-||^ has the

properties of a usual norm on R^, i.e. \\u\\j: = 0 if and only if u = 0, IIA 0 u\\jr = |A| llt^ll-^ and \\u e t;||^ < ||n||^ + ||t;||^,

I jjujl^ - ||t;||^| < D (u, v).

Remark 3.5. The properties (ii) and (iii) in Theorem 3.4 show us that ( R ^ , ® , 0 ) is not a Hneax space over R and consequently (R^, IHJ^) cannot be a normed space. However, the properties of D and those in Theorem 3.4^ (iv)-(vi), have as an effect that most of the metric properties of a ftmction defined on R with values in a Banach space, can be extended to functions / : R ^ R^, called fuzzy functions.

3.2 Almost Periodicity in Fuzzy Setting In this section , we present a fuzzy version of the theory of almost periodic functions as known in Banach spaces (see for instance [2], [23]), or Prechet spaces (see [80]). Definition 3.6. A generalized fuzzy trigonometric polynomial of degree
af\hf

e R.

3.2 Almost Periodicity in Fuzzy Setting

99

Here, the polynomials Tf^ (x), are not necessarily supposed to be linearly independent on R and ^ * denotes addition (sum) with respect to ® in R^. Note that the usual fuzzy trigonometric polynomials could naturally be defined of the form *

/^^[dj 0 cos j x ® bj 0 sin jar], with ajybj € R, which because of the lack of distributivity of 0 with respect to ©, obviously are only particular cases of the above concept of generalized fuzzy trigonometric pol3aiomials. But, as it was pointed out in e.g. [3], the suitable fuzzy polynomials in approximation are of generalized type and not of usual type. We also recall the following definition. Definition 3.7. / : R -> R^ is said to be continuous at XQ € R if: \/e > 0,36 > 0 such that D (/ (x), / (XQ)) < e, whenever x e R, |x — xo| < S. Definition 3.8. Let / : R -^ R^ be continuous on R. (i) We say that / is B-almost periodic if : Ve > 0, 3 Z > 0 such that any interval of the form [a, a+1] contains at least a point T with

DU{t +

r)J{t))<e,^teR.

(ii) We say that / is normal if for any sequence Fn : R —>" R^ of the form Fn {x) = f {x + hn), n eN, where {hn)^ is a sequence of real nmnbers, one can extract a subsequence of (Fn)^, converging uniformly on R (i.e. V {hn)^, 3 (hn,), 3 F : R -> Rjr (which

100

3 Almost Periodicity in Fuzzy Setting

may depend on {hn)^), such that D{Fn^ (x) , F ( x ) ) -> 0, as k —^ oo, uniformly with respect to x € M) (iii) We say that / has the approximation property, if Ve > 0, exists some generalized fuzzy trigonometric polynomial T with

D{f{x),T{x))

<£, VxeR.

Remark 3.9. We observe that Definition 3.8 i) is compatible with Definition 3.1.1 [80]. Also property ii) in Definition 3.8 is the so-called Bochner^s criterion {Theorem 1.61). Let us denote AP (Rjr) = {/ : R -> R^; /

is B-almost periodic}.

We will show in the next two theorems that AP (R^) is a subclass of uniformly continuous boimded functions. Theorem 3.10, / / / : R —> Rjr is (continuous) B-almost periodic then f is bounded (i.e. 3M > 0 with D{f(x),f{y))

< M, Vx,j/ €

R;. Proof. We follow the proof in [23, Theorem 6.1, p.l54]. Because

D{f(x)J{y))
= \\fix)\\^M\^

it is sufficient to prove that 3Mi > 0 with ||/ (x)||^ < Mi. Let £ = 1 and / (1) be as in Definition 3.8, (i). As in [23, Theorem 6.1, p.154] it follows 11/ (x)ll^ < Ml, Vx G [0,/ (1)]. Now, if t € R is arbitrary, then in [—t, —t + / (1)] there is at least a point ^ = ^ (1) in Definition 3.8 i). Hence

||/(t)||^ = p(/(<),o) = z?(/(Oe/(f + 0,/(< + Oeo)

3.2 Almost Periodicity in Fuzzy Setting


+ C))+D{f(t + 0,b)
101

+ Mu

because t + ^ € [0, / (1)], which proves the theorem.

D

Theorem 3.11. / / / : M ~> Mjr is B-almost periodic then f is uniformly continuous on R. Proof. FoUowmg Theorem 6.2, p. i5^in [23], and in view of the properties of Z?, we have D if (t2), / (tx)) = D{f (t2) e / (t2 + r ) , / (ti) e / (t2 + r)) = D{f (<2) e / (t2 + r ) , / (<2 + r ) e / ih))


+ T))

Dif{h),fit2+T))

= D{f{h)J{t2+r))

+ D{fih)

e / ( t i + r ) , / ( t 2 + T)®/(fi+T))
+ T)) +

+ D{f(ti+T),f{t2

D(f(t,)J{h+T))

+ r))n

We also have: Theorem 3.12. / / / : R -^ R^ is B-almost periodic, then A © / ( A e R ) , Ffc(x) = f{x^h)

( x € R ) anrf G(a;) =

||/(x)||^,

(ar e R) one B-almost periodic. Proof. Because

D{\Q f {t + i) ,\Q) f {t)) = \\\DU it + i) J (t)), for all A 6 R, it easily follows that A © / is B-ahnost periodic.

102

3 Almost Periodicity in Fuzzy Setting

And since

\\\f(t + 0\\r-\\fm^\
+ OJit)),

then it is immediate that G(x) = ||/(a:)||^, a: € M, is B-almost periodic (in the usual sense of function G

:R—^R).

Let /i G R be fixed and for e > 0, let / > 0 and r attached to / in Definition 3.8 i). By DU\t we get (by taking

+ ^)J{t))<e,

Vt€R,

t=^u-\-h)

D{f(u + h + ^)J(u + h))<e,

V<€R,

which immediately imphes F^ is B-almost periodic. D The next result shows that AP (Rjr) is closed with respect to uniform convergence on R. Theorem 3.13. / / / ^ : R -> R^, n€:N are B-almost periodic and fn -^ f as n -^ oo uniformly on R (i.e. Ve > 0, 3no € N, such that D{fn{x)J{x))<e,

V n > n o , Vx G R),

then f is B-almost periodic. Proof. It is similar to the proof of Theorem 3.3 [23], or Theorem 3.1.4 [80], by taking into accoimt the properties of D in Section 3.1. U Theorem 3.14. The set of values of f : R -> Rjr supposed to be B-almost periodic, is relatively compact in the complete metric space

{RT,D).

3.2 Almost Periodicity in Fuzzy Setting

103

Proof. We just follow the Proof of Theorem 3.4 [23], or Theorem 3.1.5 [80], with in mind the fact that in complete metric spaces, the relatively compact sets coincide with precompact sets, it is sufficient to show that for any e > 0, the set of values of the ftmction can be embedded in a finite ntmiber of spheres of radius e. D Remark 3.15. Let / : R -> R^ be B-almost periodic and let us consider the sequence of values ( / (*n))n€NDenote A = {f (tn); n € N} and take the closure A C / ( R ) C RjTy it follows that A is compact, so A is sequentially compact too, which by >1 C ^ implies that the sequence ( / (tn))^ has convergent subsequence in R^. The following result shows that the concepts in Definition 3.8 i), and (ii), in fact are equivalent. Theorem 3.16. A function f : R -¥ Rjr is B-almost periodic if and only if it is normal. Proof. It is similar to the proof of Theorem 6.6, p. 156 in [23]. D Also, we have Theorem 3.17. The sum ®, of two B-almost periodic functions is B-almost periodic. Proof. Similar to the proof of Theorem 6.7 m [23]. D Theorem 3.18. If f :R^

R^ has the approximation property in

Definition 3.8 (Hi), then f is B-almost periodic.

104

3 Almost Periodicity in Fuzzy Setting

Proof. A function / : R -> R^ is called 5o-periodic if / (t + SQ) = / ( t ) , Vt G R. Obviously a 5o-periodic function is B-almost periodic. It follows by Theorem 3.12 and Theorem 3.17 above that any generalized fuzzy trigonometric polynomial is B-almost periodic, which combined with Theorem 3.13 completes the proof

D

Remark 3.19. Let us denote AP (R:F) = {/ : R -^ R:r; / is B-abnost periodic} , and for / G AP{Rjr), let us define ||/|| = sup{||/(«)||^; t G R}. By the proof of Theorem 3.10 we get ||/|| < +oo. Also by Theorem 3.4 and Theorems 3.12, 3.17, {AP (R^), 0 , 0 ) is not a linear space, and consequently {AP{Rjr), |H|^) is not a normed space. However, endowed with D* : AP (Rj:) x AP (R^) -> R+, where D^{f,g)=

sup

D{f{t),g{t)),

AP (R^) becomes a complete metric space. Indeed denoting Cb (R^) = {/ : R -^ R^; / is continuous and boimded on R}, by standard reasonings (taking into account that (R^, D) is a complete metric space) it follows that (^^(R^), /)*) is a complete metric space. Then, Theorems 3.10 and 3.13 show that AP (R^) is a closed subset of Cb (R:F)J i-e. (AP ( R ^ ) , Z>*) is a complete metric space. By similar reasonings with those in the proofs of Theorems 6.9 and 6.10 in [23, p. 158-160], where we define on R^ the metric

3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting

105

m

Dm{x,y) =

^D(xi,yi) i=l

for all a: = (xi,..., Xm), and y = {yi^..., t/^) e M^, we can state the following compactness criterion. Theorem 3.20. The necessary and sufficient condition that a family A C AP{Rjr) be relatively compact is that the following properties hold true: (i) A is equi-continuous; (ii) A is equi-almost periodic; (Hi) for any t G M^ the set of values of functions from A he relatively compact in R^.

3*3 Harmonics of Almost Periodic Functions in Fuzzy Setting We start with the concept of integrals of fuzzy functions compatible with the operations introduced in Section 3.1 Definition 3-21. (see [20]) A function / : [a,b] -> Rjr, [a,b] C R is said to be Riemann integrable on [a, 6], if there exists / € M^, with the property: Ve > 0, 3<J > 0, such that for any partition of [a,6], d : a = Xo < ... < Xn = 6 of mesh u{d) < S, and for any points ^i e [xi.Xi^i], 0 < i < n — Ij we have

where ^ * means sum with respect to ®.

106

3 Almost Periodicity in Fuzzy Setting

In this case we denote

Ja

In order to introduce Fourier series attached to a given function / € AP (K^), we need the concept of mean value of / , as follows Theorem 3.22* For any f € AP(Rjr),

there exists the mean

value M(/)=

lim i © /

fit)dteR:r,

where the limit is considered in the metric space (Rjr^D), i.e. 3M{f)eRjr

with Jhn^D(M{f),^ef

f{t)dt)=0.

Proof. We follow the ideas in the proof of Theorem 6.11 in [23, p. 161]. We get / f{t)dt= Ja which implies

I f{t)dt® Ja Ji

f{t)dte

f{t)dt, J^^T


+D(^^Qj''^Jf{t)dt,OJ

3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting

107

Denote h = J^'^'^ f (t) dt and h = JQ f{t + ^)dt, where by definition n-l'

^ M l ' E fi^OixHi-^i)]

<^. iil^(d')<S,

with d' : ^ = XQ < ... < Xn = ^ + T, ^[ € [xi,x.+i], 0 < i < n - 1, d" :0 = yo < ...
[yi,Vi+i], 0 < i < n - 1. Let us

make the choice Xi = yi+ ^, 4' = C,'' + C) 0 < i < n - 1. We get: *

*n—1

*n—1

E = E / (^i)® (^^+1 - ^i) = E / (^+^)® (^^+1 - 2'^) = E ' 1

t=0

t=0

2

and therefore

Dih,i2)
1+1=6,

for any e > 0, that is /i = /2. It follows (see the properties of D) that

D[^ej^f{t)dt,^Q)j

Similarly, by c © 0 = 6, Vc > 1, we get

f{t)dt^

108

3 Almost Periodicity in Fuzzy Setting

^(^®/

^^*^'^*'^) " ? ^ ( / <|;^

)

D{f{t),b)dt

and

^ f(t)dt.0j<^j

^[f^J

^ J9(/(i),0)dt, forr>l.

Reasoning exactly as in the proof of Theorem 6.11 in [23], we obtain /I

r^

1

n-^'^

\

p

A

and

)

< | + 2^.z,r>i

The function <^ : [l,+oo) -) Rjr, (^(r) = ^ 0 J^ f(t)dt,

where

/ G AP (K^), is continuous (in the metric D), as product of two continuous functions. This is a property which can be derived by using the properties of D. Indeed, firstly let / G AP (Rjr) be and define F(T) = fQf(t)dt,

T e [l,+oo). Let r„ \

T, when

n -^ +00. We get DiF(Tr,),F(T)) =

off

fit)dte f " f{t)dt,f /(t)dteoj

< D (p f it) dtA< IJ D (/ (t) ,0) dt < 11/11 (T„ - T).

3.3 Haxmonics of Almost Periodic Functions in Fuzzy Setting

109

If r„ /« T, similarly we get Z) (F (T„), F (T)) < ||/|| ( T - T „ ) , which proves the continuity of F. Then, for T„ -^ T, r„, T € [1, +00), we get

<^D{F{T^),F(T)) J-n.

n

+ J__i D{F(T),b)-^0, r„ T

when n —>^ +00 . As a conclusion, cp is continuous on [1, +00). As in the proof of Theorem 6.11 in [23], take Ti, Ta € [1, +00) such that miTi = mal^j, where mi and m2 are two real numbers. The properties of D allow us to arrive at the inequality (as in the above mentioned proof)

Contintiing the reasoning in [23], we finally arrive at

for Ti, T2 sufficiently large {Ti,T2 > 4Al/e), which proves the theorem, D Remark 3.23. We can also show that for / e AP (M^), we have

lmi^D(M(f),^Q

r

f(t)dt\=0,

for all a

110

3 Almost Periodicity in Fuzzy Setting

In what follows we will attach to any function / e AP (Rjr) a Fourier series. We start with the following (see for instance [20]). Theorem 3.24. R^ can be embedded in M = (7[0,1] x C[Q,1], where (7[0,1] is the class of all real valued bounded functions f : [0,1] -^ R such that f is left continuous for any x e (0,1], / has right limit for any x G [0,1) and f is right continuous at 0. With the norm ||-|| = sup^^^ jj \f{x)\, C[Oy 1] is a Banach space. Denote \\*\\^ the usual product norm i.e.

||(/,5)||3 = max{||/||,|M|}. Also denote the embedding by j : Rjr -> B, j{u) = (t6_,n_|_). Then j(R^) is a closed convex cone in B and j satisfies the following properties: (i) j{s Q u®t s^t>0

Q v) = s ' j{u) + t

j{v) for all u,v £ R^ and

(here "-" and ^M-'' denote the scalar multiplication and

addition in M); (a)D{UyV) = \\j(u) — j{v)\\^ (i.e. j embedsRjr inB

isometrically).

Remark 3.25. Let us denote Cc[0,l] = {F : [0,1] ^

C;F

=

Fi +iF2, Fi,F2 e C[0,1]} and Be = Cc[0,1] x Cc[0,l], where C represents the set of complex numbers. It is obvious that C[0,1] C Cc[0,1], Cc[0,1] is a complex Banach space endowed with the norm ||/|| = sup {|/ (x)|; x G [0,1]} and Be is a complex banach space endowed with the norm \\F\\^ = m a x { | H | , | | t ; | | } , VF = (u,v) ^€[0,1].

e Be = Cc[0,l] x

3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting

111

It is easy to see that B = (7[0,1] x (7[0,1] C Be can be isometrically embedded into the complex Banach space Be x Be endowed with the product norai \\{F,G)\\j^^^^

= max {||F||jj^ , HCHu^},

where the isometry is defined by / : B -> Be X Be, I[{f,g)] = [(/,^), (0,0)], with 0 representing the identical zero function. Now for the proof of approximation result, we need the following two auxiliary lemmas. L e m m a 3.26* Let f : R -¥ Rjr be a B-almost periodic function and Tra be a positive valued trigonometric polynomial Then T^ © / : R -^ R:F defined by {Tm 0 / ) {t) = T^ (t) © f(t) for all t € R, is B-almost periodic. Proof. By Theorem 3.16, it is enough to prove that T^ © / is normal. Since / is normal, for any sequence of translates {/(t + /in)}n€N we have a uniformly convergent subsequence, which we denote Since T^ is a real valued trigonometric polynomial, it is also normal. It follows that {Tm{t + hk^)}^^^ has a convergent subsequence, denoted {f{t + /iz^)}n€NThen for n,p € N we have:

D {{Tm ef){t + hij , (T^ © /) (t + hi^J) D{Tmit + hijQf{t

+ hiJ,Tm{t + hijQf{t

+ hi^J)

112

3 Almost Periodicity in Fuzzy Setting

D {Tm (t + hj

Qf{t

+ hi„,^) ,T,n(t + hi„,„) efit

+ hu^„)).

But it is known (see Lemma 2.2 in [40]) that D (a 0 X, 6 © x) = |6 ~ a| ||x||^, for any a, 6 € K of the same sign and x € MF- Then we obtain

D {(Tm ef){t + hj, (Trn ef){t

+ hi^J) <

\Tm {t + hij\ ^D{f{t + hj J{t + hi^J) + + \Tm {t + hiJ - Tm {t + hi^J I . 11/ {t + hi^J 11^ , which proves the lemma. D Let us now define Pi : C[0,l]xC[0,1] -> C[0,l],bypi((/i,/2)) = fu « = 1,2, for aU (/i,/z) € C[0,1] x C[0,1]. In what follows, the following lemma will be helpful. Lemma 3.27. Let / : R ~> Rjr, Then f is B-almost periodic (in the sense of Definition 3.8 if and only ifjof:R-^

C[0,1] x C[0,1]

is almost periodic in Bochner's sense (see [23], or [80]), if and only ifpioj

o f iR-^

C[Oj 1], i = 1,2 are almost periodic in Bochner^s

sense. Proof. Let us suppose / is almost periodic in the sense of Definition 3.8 i). It follows: Ve > 0, 3/(6:) > 0 such that any interval of length l{e) contains (at least) one point ^ with D(f(t + ^),f(t)) e, Vt G R.

<

3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting

113

Because

Dif(t+T)j{t)) = Ujof){t+o-uof)mB^ we obtain the first equivalence. To prove the second equivalence, let us first assume that j o f is almost periodic in Bochner's sense. Because lia o f)(t + r ) - (j o / ) ( < ) | | B = max{||(pi ojof)(t

+ ^

-{piojof){t)l UP2ojof)(t + 0

-iP20Jof)it)\\}, it easily follows that Pi o j o / : R -> CfO, 1], z = 1,2, are also almost periodic in Bochner's sense. The converse implication is immediate by the above relation, which proves the lemma. D The next result called approximation property, is in fact the converse of Theorem 3.18, and represents one of the most important property of / € AP{R:p). Theorem 3.28. Let / : M -> R^ a B-almost periodic function. Then there exists a sequence of generalized trigonometric polynomials Tm such that Tm -^ f uniformly on R. Proof. Let j o / : R -^ B be the embedding defined in Theorem 3.24. By Theorem 6.13 [23], the mean values ai(A) = M{cos At j o f{t)}

114

3 Almost Periodicity in Fuzzy Setting

and a2iX) = M{smXt ^ j o f{t)} exist. Indeed let ji : Rj- -> Be x Be, where j i = / o j with / the natural embedding / : B -> Be x BeFor ji o / we apply again Theorem 6.13 [23] and we obtain that a (A) = M{ji o f{t)

e""*-^*} ^ 0 only for a set at most comitable

of complex numbers Ai, A2,... In what follows we use the ideas in the proof of Theorem 6.15 [23]. Let us consider the Fejer kernel K^ (*) = ^ ~ ^ ^ - I* ^ ^^^1 known that K^, {t) = YlZ=-n \^^n)

^^^ ^^ ^^ ^^ ^^^^ trigono-

metric polynomial. Let ^i,...,/?n,-- be a basis of the Fourier exponents Ai,...,Ajt,... (i.e. there exist ri,...,rn € Q such that A^ = r i A + ... + r^PnY Let /C^ (t) = K^r^^ ( f f ) ...K^r,^ ( ^ ) > 0. Observe that Km (*) is also an even trigonometric polynomial. Let /Cm (/ o j o / ) : K -> Be X Be. By Lemma 3.27 we obtain jofis

almost periodic and it is easy to see that lojof

is almost

periodic too. By the proof of Theorem 6.15 [23] we obtain 1 f^ (jm (t) = lim - K m {u) {lojo

f){u + t)du

T-^00 I Jo

converges uniformly to

lojof.

By the form of the embedding / , it follows that a!^ (t) = limT->oo f^ So ^m (^) ( i o / ) (* + u) du converges uniformly

tojof.

Let us consider a^ (t) = M{Km{u) Qf{t + u)} which exists by Theorem 3.12 and Lemma 3.26. Then we obtain:

3.3 Harmonics of Almost Periodic Fmictions in Fuzzy Setting

D K (*)' m) = 110' ° ^ ) it) - u o m)\i

115

<

U O <^D (t) -
j (imx^ i y

)Cm{u)ef{u

+1) duj

1 r^

11

It is easy to see that JU^ff

ICm{u)ef{u

= ^lim^ i j U

+ t)du\

Km (u) Qf{u

=

+ t) duj .

By Theorem 3.2 [20] and since Km is positive it follows: iJoaQit)=\mx^-J

= ^ f l

1 /"^

j{Km{u)Of{u

Kmiu)-Uof){u

+ t))du =

+ t)du = a'^{t).

This leads us to i? [a^ {t) ,/(<)) < f for sufficiently large m and any t € M. Observe that a^ (t) is not necessarily a fuzzy trigonometric polynomial, since by Theorem 3.4, (iii), for general coefficients, the integral does not necessarily commute with the sum related to " e " .

116

3 Almost Periodicity in Fuzzy Setting

However a^ (t) is a periodic function (since cr^ (t) is periodic). By Theorem 3.1 [3] it follows that there exists a generalized fuzzy trigonometric pol3aiomial Pm such that

PK(t),F^(t))<|. Finally we obtain D (Pm (t), fit)) < D [Pm (t), a^ {t)) +D{ai

(t), /(<)) < e.

Which completes the proof. D In conclusion we infer that , Theorems 3.16, 3.18 and 3.28 lead to the equivalence of all three concepts in Definition 3.8, in other words we have: Corollary 3*29« The set of all B-almost periodic fuzzy-numbervalued functions coincides with the set of all continuous normal functions and with the set of all functions with approximation property.

3.4 Applications to Fuzzy Differential Equations We start this section by the illustration of the idea of propagation of almost periodicity from the fuzzy input data to the solutions of a fuzzy differential equation as follows: Theorem 3-30. Let f € i4P(R) that is f is (Bochner) almost periodic assume c G R^ is a fuzzy real number. If f{t) > 0, for all t eR,

then the function y : M -) Rjr given by

3.4 Applications to Fuzzy Differential Equations

y{t) = cQ f

e*-''f{u)du,

t e R

117

(3.1)

is B-almost periodic and satisfies the fuzzy differential equation y\t)®y(t)=cQf{t), for all t e Q, where I? = {i € R; f{t) > Jl^ e ^ - 7 ( n ) d n } . Here y'{t) is defined as the common value of the following limits in the metric D supposed that exist, together with the H-differences y{t + h) — y(t), y{t) —y{t — h),h>Q yit+h)'-y{t)

(recall that the H-difference

exists, if exists a G Rjr such thaty{t+h)

h\0

h\0

h

=

y{t)®a),

h

(see e.g. Definition 3.3 [20]). Proof. Let us denote F{t) = / _ ^ e""-^ f {u)du, t G R. Then cleaxly F is (Bochner) almost periodic with the same e-period than the function / . Now since D{cQF{t),cQF{t

+ T))<\F{t)-F{t

+

i)\^D{c,0),

where Oe Rjr , it is immediate that y{t) = cQ F{t) is B-ahnost periodic ( i.e. in the sense of Definition 3.8y (i)). It remains to prove that y{t) is differentiable and satisfies the fuzzy differential equation. Indeed, let h > 0. By hypothesis we have F{t + h), F(t) > 0, for all

teR.

On the other hand F\t) = fit) - I J —oo

e''''f{u)du

> 0, Vt e n,

118

3 Almost Periodicity in Fuzzy Setting

which implies t+h

pt

o

J —oo

/

=

Hit,h)>0,

for i e i? and h> 0, sufficiently small. By Theorem 3.4, (iii), we get cGF{t + h) = ce F(t) e c ©

H(t,h),

that is cQF{t + h)-cQFit)

= y{t + h)- y(t) =cQ

Hit,h).

Multiplying by ^, in view of Theorem 3.4, (v), gives

yit + h)-yit) h

_„^Hit,h) ~ ^

Passing to the limit as /i \

h

0 in the metric speice (R,Z)), we

easily obtain that lim h\fi

yit + h)-yit) h

= c © /(f) - f

e"-V(n)dJ .

Similarly we obtain

^:^y{t)-yit-h) h\0

h

= cQ\fit)-f

e^-'fiu)du

that is y'it) = cQ fit) - f e^-'fiu)du J —OO

Then, again in view of Theorem 3,4J (hi)) we get

3.4 Applications to Fuzzy Differential Equations

y{t)ey'{t) = CQ f

e^-7(«)
= c0 I /

e''-'f(u)du

L

+ f{t) - I

119

e"-'/(tx)rfJ

J—oo

J

e"-7(w)dJ = c © /(t),

for all t € O, which proves the theorem. D Example 3.31. A simple example satisfying the above Theorem 3.30 is f{t) = 3 + cost + cos(t\/2) > 0, Vt € K, which is almost periodic (but it is not periodic) on R. Then according to Theorem 5.^,we get y(t) = c © F{t), y : R -^ Mjr,with c G Mjr and F(t)

=

/_gQ e"~* [3 + cos u + cos{u^/2)] du, is B-almost periodic. On the other hand, we have F{t) = f

_

-e

-tjot

i6e+

e"-* [3 + cos w + cos(u>/2)] du =

e*(cost + sin<)

-

COS t + sint

__ +

2

+

e*[cos(t\/2) + y/2sm(tV2)]\

-

_

|_

cos{ty/2) + y/2 sin(i\/2) "^

3

The condition f{t) > / _ ^ e^~'^f{u)du becomes ,^ /r. ^ cost + sin< cos(tv/2) + V^sin(i\/2) 3 + cost + cos(tv2) > 3 + + ^-^-y—!-—~ ^^ ^, which is equivalent to 3(cost-sinf)+2\^[cos(tv^)-sin(i\/2)]+(4-2\/2)cos(t\/2) > 0,

120

3 Almost Periodicity in Fuzzy Setting

Simple considerations prove that for t € ( ~ ^ > ^ ) ) the above inequality holds, that is ( ~ ^ 5 ^ )

^ 17.

Denoting E = Ufcez ( ~ 5 ^ + 2A:7r, ^

+ 2A;7rj , actually we

have E d Q and therefore for all t G -E, y{t) = c©

cosi + sin*

cos{ty/2) + v^sin(^\/2)

satisfies the fuzzy differential equation y\t) e y{t) = c © [3 + cos i + cos{tV2)],

t 6 E.D

3.5 Bibliographical Remarks and Open Problems

121

3.5 Bibliographical Remarks and Open Problems In this chapter the main properties of real-valued almost periodic functions were extended to fuzzy-number-valued almost periodic functions as in [23], [80]. Apphcations to dynamical systems as in Section 1.6.2 are also possible (see B. Bede and S. G, Gal [40] for details). It would be interesting to use other concepts of differentiabilty in fuzzy settings in the study of fuzzy differential equations as indicated by B. Bede and S. Gal ([9, 40]).

4 Almost Automorphy in Fuzzy Setting

4.1 Introduction The purpose of this chapter is to extend the main properties of Banach-space-valued almost automorphic functions as presented in [80], to fuzzy-number-valued almost automorphic functions. This is done in Section 4-3 below. Although majority of proofs follow standard ideas of proofs in Chapter 2 of [80], however, their adaptation requires a careful manipulation of the properties in the complete metric spaces (R^, D) and ( X , ® , 0 , d ) (see Section 3,1 for details). Also the facts that R^ (and X) with respect to addition ® is not a group and that with respect to real scalars multiphcation © too is not a linear space (the distributivity of stun + of reals with respect to © does not hold in general, it holds only if the real scalars are all > 0 or all < 0), require changes of some concepts and proofs.

124

4 Almost Automorphy in Fuzzy Setting

Section 4*^ contains new spaces constructed with the aid of (Rjr.D)^ with properties similar to those of (Rjr,!?), fact which permits to enlarge considerably the applicabiUty of the theory. In Section 4*4^^ present some applications to fuzzy differential equations.

4.2 Preliminaries With the aid of (R^^r, ®, 0 , D) introduced in Chapter 3, let us define new spaces, as follows. (1) (7([a, 6];lR^)-the space of all continuous functions / : [a, 6] ~> R^, endowed with the metric -D*(/, g) = sup{Z>(/(x), ^(x)); x € [a, 6]} (and the natural operations induced by those in R^;) (2) For 1 < p < +00, i7([a,6];R^) the space of strongly measinrable functions on [a,6], / : [a, 6] —> RJF, such that (L)

f Ja endowed with the metric

D^{QJ{x))dx<+oo,

D,{f,9) = [{L)£iy>{f{x),g{x))d:^ ; (3) For 1 < p < +00, OO

11^ = {x = {xn)\Xn € R:F,Vn e N , 5 3 ||a;n||?r < +00}, n=l

endowed with the metric

4.2 Preliminaries

125

(4) rriR^-the space of all sequences of fuzzy numbers x = {Xn)n^ bounded in the "norm" ||.||^, i.e. there exists M > 0 (depending on x) such that ||a:n||:r < M, for all n € N, endowed with the metric li{x^y) = sup{Z?(xn,2/n);^ € N}, for all X = {Xn)n^y

= {yn)n

^ RjF \

(5) CR^-the space of all convergent sequences (in the metric D) of fuzzy nimibers and c^^-the space of all convergent to 0 sequences of fuzzy numbers, both endowed with the metric /i from the above case; (6) First we need the following known definition : / : [a, h] -^ Rj: is called Hukuhara differentiable on x G (^,6), if there is 5 > 0 such that for all 0 < /i < J there exist the quantities f{x + h)Qf{x),

f{x)Of{x-h)

and / € Kjr denoted

by f{x)^ such that Irni P ( i © (fix + h)e m),fix)) lim D ( l © (fix) e fix - h)), fix))

= = 0.

For p G N, one considers the space C^([a,6];E^) = {/ : [a,6] -> lR^;3/(^) G C([a,6];R^)}, endowed with the metric D*{f,g) = X^f^^ D*{f^\g^^),

where the

derivative is in Hukuhara sense. The class of Hukuhara differentiable fuzzy-number-valued functions can considerably be enlarged, with the aid of the following more general definition of differentiability introduced in [40] : A function / : (a, 6) —> M^ is called generalized differentiable on t G (a, 6) if:

126

4 Almost Automorphy in Fuzzy Setting

(i) There exist f{t + h) e f(t), f(t) Q f{t - h), for oil h > 0 sufficiently small aaid there exist

„„/« + /.)e/W ^,^/we/ft-z.)^ /i\o

h

h\o

h

J \ J

j-

or (ii) There exist f{t) e fit + /i), f{t - h) e f{t), for all /i > 0 sufficiently small and there exist

,^/We/ft + /.)^,^/(e->.)e/W^ h\o

-h

h\o

-h

or (iii) There exist f{t + h) e f{t), f{t - h) e / ( t ) , for all /i > 0 sufficiently small and there exist

^ /(t+ft) e m ^^f(th\o

h

h\o

k)B m ^

^

-h

or (iv) There exist f(t) Q f(t - /i), f{t) 0 f{t + h), for all /i > 0 sufficiently small and there exist

^ m 9 Kt - ») , , ^ /(t) e fit + >.) ^ , /i\0

/l

/i\0

-/l

^ ^ ^

(Here all the limits are considered in the metric D and h or —/i at denominators, in fact means ^ 0 or —^0, respectively). It is evident that Hukuhara differentiabihty implies the generalized differentiability but the converse implication does not hold. Also, the space C^([a,6];R^) = {/ : [a, 6] -^ M ^ ; 3 / ( P > € C([a,6];R^)}, can be considered for the generalized differentiability too.

4.2 Preliminaries

127

Remark 4-1- AH these spaces have been studied m [38], where as a conclusion it is derived that if we denote by {X, ®, ©, d) any from the spaces considered by the previous points l)-6), endowed of course with the natural operations ©, 0 induced by those © and © in Mjr, then it has all the properties of (K^, ®, ©, D), presented in Chapter 3. Also, any finite Cartesian product of the spaces considered above (including (R^, D) too) endowed with the "box metric" (i.e. d = max{pi;i}) and with the nattual induced operations © and ©, has all the above mentioned properties of (R^, ®, ©, Z?). Finally, let us note that the definitions of Hukuhara differentiability and of generalized differentiability, can similarly be considered if the function / : (a, 6) ~> R^ is replaced by / : (a, b) —> X , where (X, ®, ©, d) is any from the above mentioned spaces. Let us recall now some elements of operator theory and semigroup of operators on (A',®,©,^) in [38], where (X,©,©,(i) denotes any from the above mentioned spaces (including the case X = R^). Definition 4«2* (i) A: X -> X is called linear operator if A{X©x©/x©t/)

= A© A{x) ® /i© A(y),

for all A,/i 6 R and all x^y € X. (ii) The family T = {T{t)),t G R+} of continuous linear operators on X is called Co-semigroup if : 1) For allx G X, the mapping T{t){x) :R^ -^ X is continuous with respect to t >0 ;

128

4 Almost Automorphy in Fuzzy Setting

; ^

2) T{t + s) = T{t)[T{s)], for all t.seR^

3) T(0) = / , where I is the identity operator on X ; (Hi) If A : X -^ X is a linear operator, then it is called generator of the Co-semigroup, if for all x E X, there exists T(t){x) 0 x and lim/,N^o d{A{x), \ © [T{t){x) Q x]) = 0. Theorem 4.3. ([38]) (i) If A : X -^ X is linear and continuous on Ox, then for all x £ X we have

||A(a:)||^<|||^||W|x||^, where \\\A\\\:r = sup{||>l(a:)||:F;x € X, \\x\\jr < 1} € R. If A is linear on X and continuous on Ox, then it does not follow the continuity of A on the whole space X. All these considerations remain valid if instead to be linear, A is supposed to be only additive (i.e. A{x ®y)=

A{x) © A{y))

and positive homogeneous (i.e A(\ © x) = A 0 A{x), for all \>0). (a) (Uniform boundedness principle) Let {Aj^j G J } be a family of additive, positive homogeneous and continuous operators on X. If {Aj^j G J } is pointwise bounded (i.e. for any x e X, there exists ME G K such that \\Aj{x)\\jr < M^.'ij G J), then there exists a real number M > 0 such that \\\Aj\\\j: < M,^j

G J.

(Hi) For any A, linear and continuous operator on X, can be defined the linear and continuous operators T{t) = e^^^,t G M 61/

lim d{T{t),y2-QA^) p=0 ^

= 0,

4.2 Preliminaries

where ^ * is the sum with respect to ® and A^ = I,A^

129

=

A^~^ o A,p = 2,3, ...„ satisfying the following properties: 1) The family T = {T{t)),t € R+} is Co-semigroup on X (as in the above Definition 4-2, (H)) d^id in addition, T{t) is continuous for t <0 too. Also, the property T(t + s) = T(t)[T(s)] holds for all t^s <0, but does not hold ift and s are of contrary signs. 2) T(t) is generalized differentiable with respect to all t € M, with the derivative equal to A[T{t)]. More exactly, it is Hukuhara differentiable with respect tot^B.^,

i.e.

lim rf(i 0 (Tit + h){x) e T{t){x)), A[T{t){x)]) = ][imd(i 0 {T{t){x)ent

- h){x)%A[T{t){x)]) = 0,

and generalized differentiable with respect to t <0, i.e. ^d{~Q{T{t){x)eTit

+ h)ix)),A[Tit)(x)])

lini d ( - l 0 {T{t - h){x) e T{t){x)),A[T{t){x)])

= = 0,

for all X e X. 3) limt\o d{\ © [T{t){x) 0 x],A{x))\ = 0, for all 4)IfuQ^X

xeX.

and g :R-> X is continuous on R, then u{t) = T(t)(txo) ® f TitJo

s)g{s)ds

is generalized differentiable on M (more exactly it is Hukuhara differentiable on M+ and generalized differentiable for t <0 as in the above point 2) ) and satisfies u{0) = UQ, v!{t) = A[u{t)]® g{t)^'it e R, where u'{t) denotes the generalized derivative.

130

4 Almost Automorphy in Fuzzy Setting

Here the integral for functions defined on a compact interval with values in X is considered in the Riemann (classical) sense (see for instance [80]).

4.3 Basic Definitions and Properties Everywhere in the rest of the chapter, (X, ®, 0 , d ) will denote any from the spaces (including (R^^r,®, ©,£))) considered by the previous section. Starting from the Bochner-kind definition for the almost automorphy, in this section we develop a theory of almost automorphic functions with values in (X, ©, ©, d), similar to that for Banachspace valued functions (see [80]). We rewrite Definition 1.29 as follows. Definition 4.4. . We say that a continuous function f : R —^ X, is almost automorphic, if every sequence of real numbers (r^), contains a subsequence (sn), such that there exists g{t) G X with the property lim d(g{t)j{t

+ sn))=

lim d(g{t - Sn)J(t))

= 0.

for each t G R Remark 4-5. As in the classical theory, the above convergence on R is pointwise. The concept of almost automorphy in Definition 4.4, is more general than almost periodicity in Bochner's sense. Indeed, if the convergence in Definition 4-4 is imiform on R, then

4.3 Basic Definitions and Properties

131

according to the theory developped in Chapter
=

cQg{x), X € M, where c € M^ and ^ : R —> R is an example in e.g. [89], of almost automorphic function which is not almost periodic. Then it easily follows that / is almost automorphic in the sense of the above Definition 4-4 but it is not almost periodic in Bochner's sense, i.e. as in Chapter 3 Definition 3.8. The following elementary properties hold. Theorem 4*6. . / / / , / i , /2 : R -> X are almost automorphic functions then we have : (i) / i ® /2 is almost automorphic ; (a) cQ f is almost automorphic for every scalar c € R ; (Hi) fait) := f{t + a)yt

€ R is almost automorphic for each

fixed a e R ; (iv) f is bounded, i.e. sup{||/(f)||^;t € R} < +oo ; (v) The range Rj = {/(t);t € R} is relatively compact in the complete metric space (X^d); (vi) The function h defined by h{t) := f{—t),t automorphic;

e R is almost

132

4 Almost Automorphy in Fuzzy Setting

(vii) If f(t) = Ox for all t > a for some real number a, then f(t) = Ox for all t e R; (viii) If A : X -^ Y is continuous, where Y also is any from the spaces considered in Section 4-2, then A(f) :R^Y

is almost

automorphic. (ix) Assume that A : X -^ X is a continuous linear operator on X and x{t) = e^^'^(xo)^ t e R is almost automorphic for some Xo € X. If there exists a bounded subset K ofR^ such that iiif{||a:(t)||:r;t eK} (x) Let hn :R-^

= 0, then x{t) = Ox, for all t G R; X^neN

be a sequence of almost automorphic

functions such that hn{t) -^ h{t) when n -> +oO; uniformly in f e E. Then h is almost automorphic. Proof, (i) It is immediate from the property, d{u®v^w®e)

< d{u^w) +d(v,e),

\/ u,v,w^e E X,

and from Definition 4-4(ii) It is immediate from the property, d{cQu,cQv)

= \c\d{u,v)y

"i u,v e X,\/c

eR,

€md from Definition 4-4(iii) It is immediate by Definition 4-4* (iv) We follow the lines of proof of Theorem 1.31 iv). Indeed, let us suppose that sup{||/(t)||^;t € R} == +oo, i.e. there exists a sequence of real numbers {rn)n such that ||/(rn)||^ ~> +oo, when n -^ +00. Since / is almost automorphic, by Definition 4-4 for t = 0, there exists a subsequence (sn) of (rn) such that

4.3 Basic Definitions and Properties

lim d{g{0),fisn))

133

= 0,

n-)-hoo

where ^(0) € X, By passing to limit with n —> +00 in the relations

WfMWj^ = d(OJ(sn)) < d(0,g(Q)) + d(g(0)J(s,,)), we get the contradiction d{6yg{0)) = +00. (v) Let {/{vn)) be an arbitrary sequence in X. Prom Definition 44J there exists a subsequence (Sn) of (vn) such that lim d(g(0)Jisn))

= 0.

i.e. {f{sn))n is a convergent subsequence of {f{rn)) in the complete metric space {X^ d), which proves that Rf is relatively compact in (X,d). (vi) The proof is similar to the proof of Theorem 2.L4^^ [80]. (vii) The proof is identical to the proof of Theorem 2.1.8 in [80]. (viii) It is an immediate consequence of Definition 4-4 ^^^ continuity of A. (ix) We follow some ideas in the proof of Theorem 2.L9 [80], but adapted to our case. Suppose that /C is a bounded subset of R.^ such that inf{||x(t)||^;tGi^} = 0. We can find a sequence 5n G AT, n € N and y :B.-^ X, such that limn->4-oo lk(5n)||:F = 0 and lim d{y{t),x{t + Sn))=

lim d{y{t - Sn),x{t)) = 0,

134

4 Almost Automorphy in Fuzzy Setting

pointwise on R. Because K is bounded, there exists M > 0 such that 0<s<M,Vs€A'. Prom the above Theorem 4-3, we get

= e*®^(e*"®^(xo)) = e*®^(x(5n)), Vt > 0. It follows that 0=

lim d(y{t),x(t + sn))= n-f+oo

lim 0. n—>-|-oo

On the other hand, by Theorem 4-3, (i) we have ||e*®^(x(s„))||^<|||e'®^|||H|x(s„)||^,

which implies lim ||e'®^(x(s„))||^ = 0. But %(t),Ox) < rf(y(t),e*®^(5„)) +d(e*®^(5„),0x). Passing to limit with n -> +00, from the above relation we obtain

d{y{t),Ox)=0,VteR+, which immediately implies 2/W=6x,VtGR+. Now, ioT t> M we get t — s„ > 0, which combined with

4.3 Basic DeiinJtions and Properties

135

lim d{y{t),x{t + s„)) = lim d{y{t - s„), x{t)) = 0, t € R and with y(t) = Ox, Vi G R4., immediately proves that x{t) = 6x,Vt>

M.

Prom the above point (vii), it follows that

x(t) =

bx,^teR.

(x) The proof is identical to the proof of Theorem 2.1.10 [80], by using the fact that (X, d) is a complete metric space as well as the properties of d as a metric. D iZemarA-^.Z The hypothesis in the above Theorem 4-^ (ix) is stronger than that in the case of Banach-space valued functions, where it is inf{||a:(t)||:F;t G M} = 0. This happens because in the case of (X, ®, ©, d) the property T{t + 5) = T(t)[T{s)] does not hold for all t,5 € R (it holds only for all tjS>OoT

for all tyS <0 and does not hold if ts < 0).

Regarding the Hukuhara derivative of almost automorphic functions, we present the following result. Theorem 4.8. Assume that / : R -> R^ is almost automorphic and the Hukuhara derivative f :R —^ Rjr exists and is uniformly continuous on R. Then f is almost automorphic. Proof. Observe that for any a,6 e R, with a < b, the LeibnizNewton formula holds, i.e. f{b) = f{a) ® /^ f{t)dt, exists /(6) e / ( a ) = jl f{t)dt.

This implies

that is there

136

4 Almost Automorphy in Fuzzy Setting

nQ [ Jo

f'{t)d.S

f'(t + s)ds = ne [/{t + l/n) G / ( t ) ] , / J9 n

We get ^l/n

"" f\t + s)ds,

D{n 0 [fit + l/n) e f{t)l f) = D{nQ f Jo pl/n

nQ

f{t)ds) Jo

= nD{ / Jo

f'{t + s)ds,

fl/n

mds)

/ Jo


The last inequality follows because the continuity of f{s) [0, ^] implies the continuity of F{s) = D{f{t+s)jf{t))

on

as function

of s. Prom the uniform continuity of f, for any £ > 0, there exists ^ > 0, such that for all \s\ < <J, t € M, we have

D{f'{t +

s),f'{t))<e,

that is there exists no such that for all n > no we have ^ < S and therefore D{n © [fit + l/n) e fit)]J\t))

< e, Vt e R.

As a conclusion, the sequence Fnit) := n © [fit + l/n) © fit)], n = 1,2...,

4.3 Basic Definitions and Properties

137

converges uniformly on M to f{t). If we prove that each function Fn{t) is almost automorphic, then according to Theorem 4-6 (x), will follow that f is almost automorphic. For that, we need to prove the following helpful result : If / I J / 2 are almost automorphic and for all t € M there exists /aW = /i(*) © h{t)^ then /a is almost automorphic. Indeed, from hypothesis we have fi{t) = /2(t) ® hit),"it

€ R,

which according the definition of ® means

[fiitW = [f2it)Y + [Mt)Y,r e [0,i],teR i.e. [/i(t)_(r),/i(t)+(r)] = [[/2(t)-(r),/2(t)+(r)] + [/3(t)_(r),/3(0+(r)],Vr€[0,l]. This last formula gives for all r € [0,1], < € R

fsitUr) = MtUr) -

f2(tUr),Mt)M

= MtUr) - f2{tUir). But from the definition of D, it easily follows that fi{t) is almost automorphic, if and only if all real-valued functions, /x(0-(r),/i(0+(r),r€[O,l] are almost automorphic (as functions of t). Similar result holds for f2{t).

138

4 Almost Automorphy in Fuzzy Setting

Then from the classical theory we immediately get that all h{t)-(r),fs(t)^{r),r

€ [0,1] are almost automorphic as fimctions

of t, which finally gives that fs{t) is almost automorphic. The theorem is proved.

D

Regarding the integral of almost automorphic fmictions, we present T h e o r e m 4.9. Let f : R -^ X be almost automorphic and consider the function F :R-^ {Xj^jQjd)

X defined by F{t) = J^ f{s)ds,

where

is any of the spaces considered in Section 4-^-

Then F is almost automorphic if and only if its range Rp = {F{t); t eR}

is relatively compact in X.

Proof. We adapt the proof of Theorem 2.4-4 hi [80] to our case. According to Theorem 4-6 (v)j it suiRces to prove that if Rp is relatively compact, then F is almost automorphic. Since / is almost automorphic and Rp is relatively compact in X, given (sJO a sequence of real numbers, there exists a subsequence s'^) and ai & X such that lim d(f(t + s',),g{t))=

lim

d(m,git-s'J)

= lim d ( F ( 4 ) , a i ) ) = 0. n—>-foo

Then, as in [80], we get Fit + s',) = F{s'J e f fir + s'Jdr. Jo We have lim d{F{t + < ) , aie

f 9{r)dr) = 0.

-^+<»

Jo

4.3 Basic Definitions and Properties

139

Indeed, denoting 5n(^) = / ( ^ + ^n) obviously lim d{gn{r),g{r)) = 0 , n—f+oo

pointwise with respect to r and becaiise / is almost automorphic, by Theorem 4-6 (iv), f is bounded, i.e. sup{||/(0||^;*GR}<M. Denoting the real functions of real variable hn(r) =

d(gn(r),g(r)),

we have lini„^+oo hn{t) = 0, for all r € K and Kir) < dignir), Ox) e d(Ox,g(r))

= IMr)|Uel|^(r)||^<2||/||^ = 2M. But d(Fit + s'J,ax ® rg{r)dr) = d ( F « ) ® / y„(r)dr,ai® / g(r)dr) Jo Jo
gn(r)dr, / g(r)dr) Jo


+

/ Jo

d{gn{r),g{r))dr,

which, by the Lebesgue's Dominated Convergence theorem, impUes the convergence to zero of the last expression and therefore we get the required relation

lim d{F{t + O , ^ 1 0 / gir)dr) = 0. n-^-hoo

JQ

140

4 Almost Automorphy in Fuzzy Setting

Note that the inequaUty /»t

pi

/»t

d{l gn{r)dr, g{r)dr) < I Jo Jo Jo

d{gn{r),g{r))dr

used above is well-known in the case of fuzzy-number-valued functions. But it can similarly be extended to functions with values in (X, ©, ©, d), taking also into account that all the functions are continuous (actually it follows in an easy way from the definition of Riemann integral as limit of Riemann integral smns and the properties of metric din X). Now, denoting G{t) := a i ® f^ g{r)dr, from the relation lim d(F{t + O , Git)) = 0,

far all t e R,

we get that the range of G also is relatively compact and the the following inequality sup{\\Git)\\:F\te

R} < sup{||F(t)||^;t € R}

holds. So there is a subsequence {sn)n of {Sn)n and a2 € X, such that lim

d(G('-Sn)jCt2)

= 0.

Then reasoning exactly as in [80], we get lim d(G{t - Sn),ot2 0 Fit)) = 0. n-^-f-oo

It remains to prove that 0^2 = OxAs in Theorem 2.4-4 [80], we get As{F){t) = ^2 0 F{t), for all t e R,

4.3 Basic Definitions and Properties

141

where we use the notations s = {sn)j As(F) = Ts[T-s{F)] and Ts is defined by Ts(F) = H, with H given by the relation linin->+oo F(t + Sn) = if (t), Vt e M. Denoting A^ := AfAj""^], we get A'^{F){t) = n © a2 0 ^ ( 0 Firstly, let us prove sup{|K(F)(i)||^;< e R}sup{||F(<)|k,< € R}. For that, it suffices to prove the inequality snp{\\As(F){t)\\jr;t

eR}<

sup{||F(Olk,t € R}.

In this sense we need the following result in (X®, ©, d): if lim d(xnJ) = 0, n->-foo

then = d{bxJ) < d{l,Xn) + d{xn,Ox), where from passing to limit, we get \\1\\:F

||/||^< lim llxnll^. Now, applying this result for Xn = G{t—Sn) and / = a2®F(t) = As{t), we obtain \\AiF){t)\\:r

< lim | | G ( t - 5 n ) | U < ||F(0||^,Vi € M.

Passing to supremum with i € M, we obtain the desired inequality. Note here that because the range of F is relatively compact, it follows that sup{||F(i)||:F;* ^ M} < +oo. Indeed, since Rp is boimded in the metric space, there exists M > 0 such that d{x,y) < M,\/x,y

€ Rp- Then, we get

142

4 Almost Automorphy in Fuzzy Setting

mt)\\:r

= d{Ox, F{t)) < d{bx, F{to)) + d{F{to), F{t)) <||F(to)||^ + M < +00,

Finally, from n 0 aa = [A'^{F)(t) e F(t)] 0 F(t), we have ||n0a2||^ = IK(F)(t)eF(f)||^

<|K(F)(t)|(^ + ||F(0||^ <2||F(t)|U, where from passing to limit with n —^ +00 we obtain a contradiction if a2 7^ 0. Note that we used here the following inequality in (X, 0 , 0 , rf) : if there exists a 0 6, then \\aeb\\jr = d{bx,aeb)

= d(Ox 0 ^ (a©6) 0 6 ) =

d{b,a) < d(b,Ox) + d{bx.a) = ||a||^ + ||6||^. The theorem is thus completely proved.

D

Another useful result concerning the semigroups of operators is the following. T h e o r e m 4.10. . Let x : R^ -^ X and f : R -^ X be two continuous functions and T = (T(t))teiR^. be a Co-semigroup of linear operators on ( ^ , 0 , ©,d). Suppose that x{t) = T{t){x{0)) 0 / r ( t - s)if{s))ds,t e R+. Jo Then for t given in R and b> a> 0, a + t> 0, we have :{t + XI

6) = T{t + a){x{b - a)) 0 f T{t - s){f{s + b))ds. J—a

4.3 Basic Definitions and Properties

143

Proof. As in the proof of Theorem 24- 7 [80], we get r»6—a

x{t + 6) = T{t + a) \x{h-a)e e / Jo

T{t +

I

'^Tib-a-

s){f{s))ds]

b--s){f{s))ds.

But if O is a linear operator on X and if there exists the quantity xey,

then 0(x Qy) = 0{x) e 0{y).

Indeed, if we denote x Q y = a^ it follows x = y ® a and 0{x) = 0{y) e 0{a), which means 0(a) = 0{x) 0 0{y). Then from the above relation we get x(t+b)eT{t+a)

\f

Tib-a-

s){f{s))ds

=

T{t+a)[x{b-a)]e

/ T{t^b-s){f[s))ds. Jo Taking into account that T commutes with the integral (since it is linear and continuous operator), by the property T{u + v) = T{u)[T{v)]^yUyV e R-H and making the substitution u = s — b, we obtain x{t + b)e I

T{t-

u)[f{u + b)]du = T{t + a)[x{b ~ a)] 0 I

T{t-u)[f{u

+ b)]du.

But because t > —a, we can write / T{t-u)[f{u J^b

+ b)]du= f V ( t - n ) [ / ( i / + 6)]dix J-b 0 / T(t-u)[f{u + b)]du. J—a

144

4 Almost Automorphy in Fuzzy Setting

We then immediately get the required relation in the statement of theorem. Note that we used the following relation :

//

AQE = BeE,

then A=^B.

This is a trivial consequence of the relations d(A, B) = d{A ® E^B ^ E) =0j which implies A = B, since d is a metric. The proof is achieved. D In the study of almost automorphic solutions of nonlinear fuzzy differential equations , the following concepts and results can be useful. We follow here the ideas in Section 1.6.2. Actually the results in the case of Banach spaces remain the same for the case of oiu" metric spaces (Jf, ®, 0 , rf). Definition 4.11. ^4 continuous function f : R x X —> X is said to be almost automorphic in t € M for each x & X, if for every sequence of real numbers {rn), there exists a subsequence {sn) such that for aW t € R and x £ X, there exists g{t^ x) with the property lim d{f{t + Sn,x),g{t,x))

= lim d{g{t - Sn,x),f{t,x))

= 0.

The following simple properties hold. T h e o r e m 4.12. (i) If fi^f2 : M x X -> X ane almost automorphic in t for each x E X, then / i ® /2 and cQ fi, where ceR

are also

almost automorphic in t for each x E X. (a) If f{ty x) is almost automorphic in t for each x & X then sup{||/(t,a:)||^;t € M} < +oo. Also, for the corresponding function g in Definition 4-^1, ^^ have sup{||p(i,x)||^; f € R} < +oo.

4.3 Basic Defmitions and Properties

145

(in) If f{tj x) is almost automorphic in t for each x G X and if d{f{t,x)J{t,y))

< L d{x,y),yx,y

€ X,

and t € Ry where L is independent of x^y and t, then for the corresponding g in Definition -^.ii; we have d{g{t,x),g{t,y))

< L d{x,y),'ix,y

€ X

and < e M. (iv) Let f{t^ x) be almost automorphic in t for each x E X such that d{f{t,x)yf{t,y))

< L
where L

is independent of x^y and t. If (p :R --> X is almost automorphic then the function F iR—^ X defined by F{t) = f{t,
ieR+,

where g : R -^ X is almost automorphic and h : R-^. -^ X is a continuous function with limt^-foo l|/^(*)ll^ = 0-

146

4 Almost Automorphy in Fuzzy Setting

Thus g and h are called the principal and the corrective terms of f, respectively. Remark 4'M* Every almost automorphic function restricted to R+ is asymptotically almost automorphic, by taking h{t) = Ox,Vi G R+. Regarding this new concept, the following results in the classical case hold. Theorem 4.15. . Let / , / i , / 2 be asymptotically almost automorphic. Then we have : (i) h®f2

o.Tid cQfjCeR

are asymptotically almost automor-

phic ; (a) For fixed a eR^,

the function fa{t) = f{t + a) is asymp-

totically almost automorphic ; (Hi) f is bounded, i.e. snp{\\f{t)\\jr]t

G R4-} < +00.

(iv) Let (A',®,©,d),(y,©,0,p) be any from the spaces considered in Section 4-^ 0,'^d f : R4. -^ X be an almost automorphic function, f = g ® h. Let (j> : X —^ Y be continuous and assume there is a compact set B in (A', d) which contains the closures of {f{t)\t

e R^} and {g{t);t G R+}. If, in addition, for

all i G R4. there exists continuous r(t) = ^{f{t)) 0 (t>{g{t)), then (/>o f :R^ -^Y

is asymptotically almost periodic ;

(v) In general, the decomposition of an asymptotically almost automorphic function is not unique. Proof, (i) and (ii) are immediate from Definition 4-i3 dnid Theorem 4 6, (i), (ii), (Hi).

4.3 Basic Definitions and Properties

147

(iv) We have <^(/(0) = Hdi*)) ® ^(*)> where by Theorem 4.6, (via) , (g{t)) is almost automorphic. Therefore, it remains to prove that lim^^+oo ll-^C^)!!^ = 0We get

\\ry = PiOYMm)eH9m = p{{gm{f{t)) e (g(t)) e

Hgm

=p{i9(t)Uim)). For the rest of the proof, we follow the lines as in the proof of Theorem 2.5.7in [80]. Here we have used the property that //

Ujb E X are such that aQb exists, then (a© 6) © 6 = a.

This is atraightforward. Indeed, denoting rr = a © 6, by definition we get a = a:®6, which implies (a©6) ®6 = [(a:©6) ©6] ® 6 = X ©6 = a, because it is evident that {x®b)Qb

= x.

(v) First, we prove the following relation : if a,6, c G X are such that there exists 6 © c, then a © (6 © c) = (a © 6) © c. Indeed, denoting x = 6 © c, by definition we get 6 = x © c and therefore (a © 6) © c) = {a® X ® c) Q c = a © x = a © (6 © c), because, in general it is immediate that {A® B) Q B = A. Note that this relation does not hold if 6 © c does not exist. Let us suppose that f :R^ -^ X admits two decompositions

f{t) = giit)ehi{t),teR^,i We get

gi{t)®hi{t)=g2{t)®h2{t),

= h2.

148

4 Almost Automorphy in Fuzzy Setting

which imphes 9i(t) = and /i2(t) = [hi{t)®gi{t)]eg2{t),

[g2(t)eh2{t)]ehi{t) for all t G E+.

Now, let us suppose that for each t € M-i- there exists /i2(*) © Then by the above relation we get gi{t) = g2(t) ® [h2{t) Qhi{t)] or /i2(0 = hi{t)®[gi{t)Qg2{t)]^ respectively. Both cases imply the same relation gi{t) Q g2{t) = h2{t) © /ii(t), Vt € R+. In this case, we obtain d(gi(t) ©g2{t),Ox) = d{h2{t) ©hi{t),Qx) <

d{h2{t),Qx) + d{QxMi)) = 11/^2(011^ + l|/^i(*)ll^Passing to limit with t -4 -f 00, we get lim d{gi{t)eg2{t),0x)

= 0.

According to the proof of Theorem 4-6, gi{t) 0 g2{t) is almost automorphic. Considering the sequence n, n = 1,2, ...„ there exists a subsequence {rik) such that lim d{gi{t + rik) © g2it + n^), F(t)) = AC->-fOO

lim d(gi(t) e 92it), F{t - n^)) = 0, pointwise on R. Using the inequality d(F(t),Ox) < d{gi{t)eg2it),bx)+d(gi(t

+ nk)eg2{t + nk),F{t)),

4.3 Basic Definitions and Properties

149

and passing to limit with k -> +00, it follows that F{t) = Ox, Vi € M and consequently gi{t)Qg2{t) == Ox, V< e E. Therefore, gi = g2 and hi = /i2-

As a conclusion, the uniqueness follows only in the special case when for each t € R-|. there exists h2{t) Q hi(t) or gi(t) Q g2{t). But in general, this condition does not hold, which implies that the imiqueness of decomposition does not hold.

D

Remark 4-16. 1) In comparison with the case of Banach-space valued functions, in Theorem 4*^5, (iv) we need the additional hypothesis that for all t € M-i-, there exists (continuous) r(t)

=

{m) e {9{t)). 2) The effect of Theorem 4-15, (v), is that it produces less properties than in the classical case. In what follows, we are concerned with the asymptotical behavior of asymptotically almost automorphic semigroups of linear operators T = T(t)yt € M-f on (X,©,0,rf). We present some topological and asymptotic properties based on the Nemytskii and Stepanov theory of dynamical systems as presented in Section 1.6.2. Definition 4.17. . A mapping u : R^ x X -^ X is called a dynamical system if : (i) ui^xjx)

= a:,Vx G X ;

(ii) it(.,x) : M4. —» X is continuous for any t > 0 and rightcontinuous att = 0, for each x £ X. (The mapping u{., x) is called a motion originating at x £ X).

150

4 Almost Automorphy in Fuzzy Setting

(Hi) u{tj.) : X --> X is continuous for each (iv) u{t + s,x) = u{t^u{s,x)),Wx

t>0;

E X^t^s E M+.

We now present the following important correspondence. Theorem 4»18. . Every Co-semigroup {T{t))t>o on (X,®, ©,d) determines a dynamical system and conversely, by defining u{t^ x) = T{t){x),teR^,xeX. Proof, Similar to Theorem 1.43. In the rest of this section, T = {T{t))t^^_^ will be a Cosemigroup of linear operators on (X, ®, ©, d) such that for fixed Xo € X, the motion T{t){xo) : M-^ —> A' is an asymptotically almost automorphic function in the sense of Definition ^ . i 5 , with principal term / and corrective term h. Definition 4.19, . A function (p :R^

X is said to be a complete

trajectory of T if it satisfies the functional equation ip{t)=T{t-a){

a.

We have the following Theorem 4.20, . The principal term f ofT{t){xo) trajectory for T. Proof Similar to the proof of Theorem 1.45*

is a complete

4.3 Basic Definitions and Properties

151

Definition 4.21. u;+{xo) = {yeX;30
lim d{T{t){xo),y) = 0} n—>-|-oo

is called the u;-limit set

ofT(t){xo).

^t{xo) = {yeX;30
lim

d{f{tn),y)=0}

is called the uj4imit set of f, the principal term ofT(t){xo)* 7+(xo) = {T(t){xo);te

R + } is the trajectory

ofT(t){xo).

A set B C X is said to be invariant under the semigroup T = {T{t))teR^, if T{t){y)eB,WyeB,teR^. e E X is called a rest-point for the semigroup T if r(t)(e) = e , V t > 0 . The following properties hold. Theorem 4.22. » (i) a;'*"(xo) is not empty, LJ'^{XO) =

uj{xo),

a;'^(xo) is invariant under T and is closed in X, uj'^{xo) is compact ify^{xo)

is relatively compact Also, if XQ is a rest-point of

the semigroup T then u)'^{xo) = {xo}. (a) If we denote 7/(xo) = {/(*);* ^ R} then 7/(a:o) is relatively compact and invariant under the semigroup T, (Hi) If we denote v{t) = iiif{d(T(t(xo),?/);?/ G a;"^(xo)}; then limt_,+oo ^{t) = 0. Remark 4-23. All the above results show us that the properties in the cases of Banach spaces, remain valid for the case of the more general (X, ®, 0 , d) spaces.

152

4 Almost Automorphy in Fuzzy Setting

4.4 Applications to Fuzzy Differential Equations It is known that the classical (abstract) differential equations ,i.e. whose solutions are real-valued functions (or Banach-space valued functions, respectively) often represent an idealization of real situations, where imprecision may in fact play a significant role. A way to solve this shortcoming is to consider random differential equa^ tions (i.e. whose solutions are random-variable-valued functions), which have been used to incorporate the effects of statistical fluctuations. On the other hand, imprecision due to uncertainty or vagueness suggests the introduction of so-called fuzzy differential equations, i.e whose solutions represent functions with values in R^ or more general, with values in X, where (X, ®, O, d) represents any from the spaces introduced in Section 4-2. Applications of the semigroup of operators in solving fuzzy partial differential equations have been done in the recent paper [38]. Now we like to illustrate the idea of propagation of almost automorphy from the fuzzy input data to the solutions of fuzzy differential equations. The first result in this sense is the following. Theorem 4.24. . Let us consider the fuzzy differential equation y\t)ey{t)=g{t),

4.4 Applications to Fuzzy Differential Equations

where y\t)

153

means the Hukuhara derivative , p : R -> E ^ is of

the particular form g(t) = c 0 / ( t ) , Vi € M^ with c € Mjr a fuzzy number and / : R -> R a usual almost automorphic function. In addition suppose that f satisfies the condition /(<) > 0, Vi € R. Then the function defined by y :R-^ y(t)=cQ

f

R^ defined by

e'''^f{u)du,

J—oo

is a almost automorphic function on R and satisfies the above fuzzy differential equation for all t e O, where n = {te R; fit) > f

e''--'f{u)du}.

J—oo

Proof* First we notice that by the hypothesis on / and by Definition 4-4^ it immediately follows that ^ : R -^ R^ is almost automorphic. Now, according to [38], Section 4, Theorem 14, y{t) satisfies the fuzzy equations for alH G 1?. Then the function F{t) = J_^ e^^^f{u)du is almost automorphic on R, which by the Definition 4-4 and the properties of D immediately implies that y{t) = c © F(t) is almost automorphic. Remark 4-25. Theorem 4-^4 remains valid for the more general differential equation

y\t)eyit)

= git),

where y\t) means the Hukuhara derivative , ^ : R —> X is of the particular form g{t) = c©/(£), Vt € R, with c e X a n d / : R - ^ R a usual almost automorphic function, and (X, ®, ©, d) is any from the spaces considered by the previous sections.

154

4 Almost Automorphy in Fuzzy Setting

Before to consider the next result, let us make some remarks on the concepts of differentiability presented in Section 4-^- Firstly, note that the Hukuhara differentiability has the following shortcoming : if c is a fuzzy number, / is a real valued function (of real variable) differentiable on t and g{t) = cQ f{t) then for f{t) we have g'{t) = cQf{t),

>0

while for f'(t) < 0 , the function g is not

Hukuhara differentiable on t. This shortcoming is solved by the generalized differentiability, which at its turn has another shortcoming : if f{t) = 0, then g is not necessarily generalized differentiable on t. In the paper [9] these kinds of shortcomings are completely solved in such a way that g is always differentiable (in the new sense), if / is differentiable. The new concept of differentiabiUty can be stated as follows (see [9]) : Let / : (a, 6) —^ Rjr be and t 6 (a, 6). For a sequence of real nimabers hn \ 0, let us consider the sets

4^) = {n> p; 3Ei'^ = /{t + K) 0 /(t)}, 42) = {n>p- 342) = /(t) e f{t + hn)}, 43) = {n>p; 3^(3) = m

e fit - hn)},

Ajf) = {n > p; 3£W = fit - hn) e /(*)}. We say that / is weakly generaUzed differentiable on £, if for any sequence hn \ 0, there exists p € N such that

4>U4'^U4'^U^?^ = {n € N',n>p} and moreover, there exists an element in Rjr denoted by f{t), such that if for some j € {1,2,3,4} we have card{A^p) = +oo, then

4.4 Applications to Fuzzy Differential Equations

lim

155

,D{{-iy+'QEi,f'(t))=0.

Remark 4*26. Obviously the generalized differentiability implies the weakly generalized differentiability but the converse is not true. Now we are in position to present the following. Theorem 4.27* . Let us consider the fuzzy wave equation &^u{x,t)

1 &^u{x,t) -,t>0,a:GR,

with the boundary conditions duix 0) u{x, 0) = a © f{x), — ^ 7 - ^ = kg{x) © a, x G R dt

where a € R^, c,k G R,c > 0 and f,g :R -^ R, with f

ofC^-

class and g of C^-class. Here the differentiability is considered in the weak generalized sense. Then u

rx-^ct

u{x, i) = a © {[fix - ct) + f{x + ct)]/2 + —

g{s)ds}

^^ Jx-ct

satisfies the above fuzzy differential equation, and if in addition, f and g are almost automorphic, and F{x) = f^ g{s)ds is bounded on R, then for each fixed t>0,

the solution u{., t) is almost auto-

morphic. Proof* It is evident from the classical theory that the fimction between the brackets in the above expression of u{x^ t) is almost automorphic with respect to the variable x (for each fixed t), which by Definition 4-4 immediately implies that u{x^ t) also is almost automorphic with respect to x.

156

4 Almost Automorphy in Fuzzy Setting

Also, the fact that u{x, t) satisfies the above fuzzy wave equation is immediate from the classical theory and from the property of the above weak generalized differentiability. Remark 4-^8. According to the above Theorem ^.5, the so-called mild solution u{t) = T{t)(uo) ® I T{tJo

s)g{s)ds,

where T{t) represents the exponential operators in the same Theorem 4^3, (Hi), satisfies the abstract "fuzzy" differential equation u\t) =

A[u{t)]®g{t),

where u' is considered in the generalized sense.

4.5 Bibliographical Remarks and Open Problems

157

4,5 Bibliographical Remarks and Open Problems This chapter is based on a study by S. G. Gal and G. M. N'Guerekata ([41]). It would be of interest to see for a Bohr-Neugebauer-N'Guerekatatype result (see e.g. [30]) in the case discussed in the last section above, i.e. how almost automorphy of the forced term along with other conditions, on the operator for instance, would produce almost automorphy of the solution to the fuzzy differential equation. Also extensions to fuzzy settings of the results in Theorems 2.17 and Theorem 2.18 were obtained by C. S. Gal, S. G. Gal and G. N. N'Guerekata ([42]).

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(2002), 600-611. 28. T. Diagana, Some Remarks on some Second-Order Hyperbolic Differential Equations, Semigroup Forum, 68, (2004), pp. 357-364. 29. T. Diagana and G. M. N'Gu^rekata, Some Remarks on Almost Automorphic Soutions of some Abstract Differential Equations, Far East J. Math. 8 (3) (2003), pp. 313-322. 30. T. Diagana and G. M. N'Guerekata,

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Index

Co-semigroup, 7 ZZ-almost periods, 34 [/-translation numbers, 34

corrective term, 23, 40, 150 dynamical system, 25

a;-limit set, 30, 31

Egorov's theorem, 2

Abstract Cauchy Problem, 10

evolution equation, 9

algebraic sum, 51

exponentially stable, 56, 72

eigenspace, 49

almost automorphic, 12, 41, 57 Almost Automorphy, 1 almost periodic, 34, 35, 98 analytic semigroup, 8, 9 ArzelarAscoli theorem, 71 asymptotically almost automorphic, 23, 150

Fejer kernel, 114 fixed point, 71 Fourier exponents, 114 Fourier series, 106 Frechet space, 35 fractional power, 8, 62 fuzzy function, 95

B-almost periodic, 101, 112, 117

fuzzy number, 96

Bochner's Criterion, 35, 77

fiizzy set, 95

Bochner's theorem, 3

fuzzy trigonometric polynomial, 98, 116

Bohr-Amerio, 22 Cantor diagonalization process, 18, 70

Ganmia function, 8 generator, 7, 8, 10

classical solution, 10, 11

Holder continuous, 64

compact support, 1

Holder exponent, 64

complete trajectory, 27

Hausdorff compactification, 71

contraction, 90

homeomorphic, 92

168

Index

homeomorphism, 90

principal term, 23, 28, 31, 151 pseudo-norm, 36

infinitesimal generator, 7 invariant, 31

relatively compact, 14, 70

invariant set, 31

relatively dense, 35

invariant subspace, 49

rest-point, 34, 151

Kronecker*s symbol, 44 Lebesgue Dominated Convergence theorem, 43 Lebesgue's Dominated Convergence theorem, 4, 58, 70 Lipschitz condition, 57 locally convex space, 34

Schauder fixed point theorem, 72 second-order hyperbolic equations, 53 self-adjoint, 49 semigroup, 7, 26 simplest type, 43 Sobolev space, 62 trajectory, 30, 151

measurable functions, 1, 5, 84 method of invariant subspaces, 46 mild solution, 10, 11, 57 motion, 150

uniformly convex, 23, 43, 74 upper semi-continuous, 96 Variation of Constants Formula, 11

optimal mild solution, 74, 76, 79 Optimal weak-almost periodic solutions, 73 orthogonal complement, 49

weak almost automorphy, 13 weak derivative, 5 weakly almost automorphic, 82 weakly almost periodic, 76

perfect Frechet space, 36

weakly measurable, 2

Pettis' theorem, 2

well-posed, 10