Topics in Almost Automorphy

TOPICS IN ALMOST AUTOMORPHY

TOPICS IN ALMOST AUTOMORPHY Gaston M. N'Guerekata Morgan State University Baltimore, Maryland

Springer

Library of Congress Cataloging-in-Publication Data N'Gu^r^kata, Gaston M., 1953Topics in almost automorphy/Gaston M. N'Guerekata. p. cm. Includes bibliographical references and index. ISBN 0-387-22846-2 1. Automorphic functions. I. Title QA353.A9N52 2004 515'.9—dc22 2004059527

©2005 Springer Science+Business Media, Inc. New York, Boston, Dordrecht, London, Moscow ISBN 0-387-22846-2 (Hardbound)

Printed on acid-free paper.

©2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

(BS/DH) SPIN 11305217

In memory of Therese N^Guerekata, my mother

Preface

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant subspaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the theory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces.

VIII

Preface

Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It contains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40]. Finally in Chapter 4 the classical theory of almost automorphic vector-valued functions is extended to fuzzy settings. This chapter begins with the presentation of several "new" spaces in which the theory holds, called fuzzy-number type spaces. These spaces are more general than the Banach and Frechet spaces, since they are not linear structures although they present nice metric properties. Their importance consists in the fact that they are very appropriate for situations where imprecision which appears in the modelization of real world problems by differential equations is due to imcertainty or vagueness (and not randomness). Applications to some fuzzy differential equations are also given. It is based on S. G. Gal and G. M. N'Guerekata's recent work [41]. At the end of each chapter, we recall some relevant bibliographical remarks and raise some open problems and/or potential research subjects for graduate students and begining researchers in the area. It is our hope that this monograhp be used to stimulate some seminars and graduate courses in Analysis, D3niamical Systems, Fuzzy Mathematics and other branches of Mathematics. Ackowledgements. I like to express my deepest gratitude to my colleagues and friends Professors D. Bugajewski and S. G. Gal who gave the entire manuscript a careful proofreading. Their com-

Preface

IX

ments and valuable suggestions have been helpful while I have been selecting the topics of this monograph. I also appreciate collaborating with Professors Nguyen Van Minh, Jerome A. Goldstein and James Liu over the past 2 years. I would like to express my appreciation for the editorial assistance I received from Kluwer, especially from Ana Bozicevic. Many thanks to Morgan State University officials for granting me the necessary financial support during the preparation of the manuscript. Finally, this book would hardly have been possible without the emotional support and encouragement of my wife Beatrice.

Baltimore, MD- USA

Gaston M. N'Guerekata May 2004

Contents

1

Introduction and Preliminaries

1

1.1 Measurable Functions

1

1.2 Sobolev Spaces

5

1.3 Semigroups of Linear Operators

7

1.4 Fractional Powers of Operators

8

1.5 Evolution Equations

9

1.6 Almost Automorphic Functions

12

1.6.1 Asymptotically Almost Automorphic Functions 23 1.6.2 Applications to Abstract Dynamical Systems . 25

2

1.7 Almost Periodic Functions

34

1.8 Bibliographical Remarks and Open Problems

39

Almost Automorphic Evolution Equations

41

2.1 Linear Equations

41

2.1.1 The inhomogeneous equation x' = Ax + f .,..

41

2.1.2 Method of Invariant Subspaces

46

2.1.3 Almost Automorphic Solutions to Some Second-Order Hyperbolic Equations

53

XII

Contents

2.2 Nonlinear Equations

56

2.2.1 Existence of Almost Automorphic Mild Solutions-Case I

56

2.2.2 Existence of Almost Automorphic Mild Solutions-Case II

62

2.3 Optimal weak-almost periodic solutions

73

2.4 Existence of Weakly Almost Automorphic Solutions

82

2.5 A Correspondence Between Linear and Nonlinear Equations 3

88

Almost Periodicity in Fuzzy Setting

95

3.1 Fuzzy Sets

95

3.2 Almost Periodicity in Fuzzy Setting

98

3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting

4

105

3.4 Applications to Fuzzy Differential Equations

116

3.5 Bibliographical Remarks and Open Problems

121

Almost Automorphy in Fuzzy Setting

123

4.1 Introduction

123

4.2 Preliminaries

124

4.3 Basic Definitions and Properties

130

4.4 Apphcations to Fuzzy Differential Equations

152

4.5 Bibliographical Remarks and Open Problems

157

References

159

Index

167

Introduction and Preliminaries

This chapter has an introductory character to this monograph. We wish to recall briefly some concepts, results, methods and notations that will be used in the sequel. We will indicate in general some references where the reader can find more informations if necessary. Although for almost automorphy, our book [80] remains the main sotu-ce of information, we give detailed proofs to some new results.

1.1 Measurable Functions In this section we will recall some facts about measurable vectorvalued functions and their integrals. We consider (X, ||.||) a Banach space and / an open interval in E. We denote by Cc(/; X) the Banach space of continuous functions f : I -^ X with compact support in / . Definition lA. A function f : I -^ X is said to be measurable if there exists a set S C I of measure 0 and a sequence (fn) C Cc{I]X) such that fn{t) -^ f{t) asn-^oo,

for all t G I\S.

2

1 Introduction and Preliminaries

We observe that if / : / -> X is measurable, then ||/|| : I —^R is measurable too. Theorem 1.2. Let /^ : / -^ X , n = 1,2... be a sequence of measurable functions and suppose that f : I -¥ X and fn{t) —>/( GO, for almost all t e I. Then f is measurable. Proof. We have /n -> / on / \ 5 ' , where 5 is a set of measure 0. Let {fn,k)k€N be a sequence of functions in Cc{I] X) such that /n^jt -^ fn almost everywhere on / as A; —> oo. By Egorov's Theorem (see [90, p. 16]) applied to the sequence of functions ||/n,A; —/n||, there exists a set Sn C I o{ measure less than ^ such that fn,k ~> fn uniformly on I\Sny as A: -> CO. Now let k(n) be such that \\fn,k{n) " /nil < ^ on I\Sn

and

Fn = fnMny Also let B = 5'U(nm>iUn>m^n). Then it is dear that B is a subset of / of measure 0. Take t G I\B.

So we get

fn(t) -^ f{t)y OS n --> oo. On the other hand if n is large enough, t e / \ 5 n . It follows that jjFn - /nil < ^. Which means Fn{t) -> / ( t ) , as n —> oo, and consequently, / is measurable. D Remark 1.3. It is easy to observe that if 0 : / -> R and f : I -^ X are measurable, the (/>f : I —> X is measurable too. Theorem 1.4. (Pettis' Theorem) A function f : I -^ X is measurable if and only if the following two conditions are satisfied: (a) f is weakly measurable (i.e. for every x* E X"", the dual space of X, the function (x*/)( X is weakly continuous, then it is measurable. Definition 1.6. A measurable function f : I --^ X is said to be integrable on I if there exists a sequence of functions fn € Cc{I\X),

n = 1,2,... such that f\\fn{t)-f{t)\\dt^O,

as n - > o o .

Remark 1.7A{ f : I —> X is integrable, it can be shown that there exists a vector x G X, such that if fn € Cc{I] X), n = 1,2,.. and / / l!/n(*) — /(*)IM* -> 0 as n ^ cx), then / ^ / n -> x as n -^ oo. Such X is called the integral of / on / and denoted x := fj f. Moreover if / = (a, 6), then we denote ^ -= f^ f* Theorem 1.8. (Bochner's Theorem). Assume f : I -^ X is measurable. Then f is integrable if and only if\\f\\ is integrable. Moreover we have

Proof. Let / : / -> X be integrable. Then by the definition, there exist fn e CciI;X),

n = 1,2,... such that fj \\f(t) - f{t)\\dt -> 0

as n —>^ oo. We have ||/|| < ||/n|| + | | / n - / | | , for each n, so ||/|| is integrable. Conversely assume now ||/|| is integrable. Let Fn G Cc{I] M), n = 1,2,... be a sequence of functions such that jj \Fn{t) — \\f{t) \\ \dt ->

4

1 Introduction and Preliminaries

0 as n —> 00 and \Fn\ < F almost everywhere for some F : / —> R, with fj\\F{t)\dt
0 as n -> oo and so / is integrable. Using Lebesgue-Fatou's Lemma (see [90]), we get ||//|| ||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 < aoo

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 + Sn)f(t + S„) - V(t)g(t) = {t + Sn)f(t + S„) - 4>(t + 5„)p( 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

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 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 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-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 (

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( 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-^

=< 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+ ^-^ 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

a and choose a positive integer N such that —Ni

The function 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)) 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(«) / _ ^ 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{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:)||^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