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Volume 2011, Article ID 654695, 12 pagesdoi:10.1155/2011/654695 Research Article Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Funct

Volume 2011, Article ID 654695, 12 pages

doi:10.1155/2011/654695

Research Article

Composition Theorems of Stepanov

Almost Periodic Functions and Stepanov-Like

Pseudo-Almost Periodic Functions

Wei Long and Hui-Sheng Ding

College of Mathematics and Information Science, Jiangxi Normal University Nanchang,

Jiangxi 330022, China

Correspondence should be addressed to Hui-Sheng Ding,dinghs@mail.ustc.edu.cn

Received 31 December 2010; Accepted 20 February 2011

Academic Editor: Toka Diagana

Copyrightq 2011 W Long and H.-S Ding This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We establish a composition theorem of Stepanov almost periodic functions, and, with its help, a composition theorem of Stepanov-like pseudo almost periodic functions is obtained In addition,

we apply our composition theorem to study the existence and uniqueness of pseudo-almost periodic solutions to a class of abstract semilinear evolution equation in a Banach space Our results complement a recent work due to Diagana2008

1 Introduction

Recently, in 1, 2, Diagana introduced the concept of Stepanov-like pseudo-almost periodicity, which is a generalization of the classical notion of pseudo-almost periodicity, and established some properties for Stepanov-like pseudo-almost periodic functions Moreover, Diagana studied the existence of pseudo-almost periodic solutions to the abstract semilinear

evolution equation ut  Atut  ft, ut The existence theorems obtained in 1,2 are

interesting since f ·, u is only Stepanov-like pseudo-almost periodic, which is different from

earlier works In addition, Diagana et al.3 introduced and studied Stepanov-like weighted pseudo-almost periodic functions and their applications to abstract evolution equations

On the other hand, due to the work of4 by N’Gu´er´ekata and Pankov, Stepanov-like almost automorphic problems have widely been investigated We refer the reader to5 11 for some recent developments on this topic

Since Stepanov-like almost-periodic almost automorphic type functions are not necessarily continuous, the study of such functions will be more difficult considering complexity and more interesting in terms of applications

Very recently, in12, Li and Zhang obtained a new composition theorem of Stepanov-like pseudo-almost periodic functions; the authors in13 established a composition theorem

of vector-valued Stepanov almost-periodic functions Motivated by2,12,13, in this paper,

we will make further study on the composition theorems of Stepanov almost-periodic functions and Stepanov-like pseudo-almost periodic functions As one will see, our main results extend and complement some results in2,13

Throughout this paper, letR be the set of real numbers, let mesE be the Lebesgue measure for any subset E ⊂ R, and X, Y be two arbitrary real Banach spaces Moreover, we

assume that 1≤ p < ∞ if there is no special statement First, let us recall some definitions and

basic results of almost periodic functions, Stepanov almost periodic functions, pseudo-almost periodic functions, and Stepanov-like pseudo-almost periodic functionsfor more details, see

2,14,15

Definition 1.1 A set E ⊂ R is called relatively dense if there exists a number l > 0 such that

Definition 1.2 A continuous function f : R → X is called almost periodic if for each ε > 0 there exists a relatively dense set P ε, f ⊂ R such that

sup

t∈R

f t  τ − ft< ε, ∀τ ∈ P

ε, f

We denote the set of all such functions by APR, X or APX.

Definition 1.3 A continuous function f : R × X → Y is called almost periodic in t uniformly for x ∈ X if, for each ε > 0 and each compact subset K ⊂ X, there exists a relatively dense set

P ε, f, K ⊂ R

sup

t∈R

f t  τ, x − ft, x< ε, ∀τ ∈ P

ε, f, K

We denote by AP R × X, Y the set of all such functions.

Definition 1.4 The Bochner transform f b t, s, t ∈ R, s ∈ 0, 1, of a function ft on R, with values in X, is defined by

Definition 1.5 The space BS p X of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f on R with values in X such that

f S p : sup

t∈R

t1

t fτ p dτ

1/p

It is obvious that L p R; X ⊂ BS p X ⊂ L p

locR; X and BS p X ⊂ BS q X whenever

p ≥ q ≥ 1.

Definition 1.6 A function f ∈ BS p X is called Stepanov almost periodic if f b ∈

AP L p 0, 1; X; that is, for all ε > 0, there exists a relatively dense set Pε, f ⊂ R such

that

sup

t∈R

1 0

ft  s  τ − ft  s p ds

1/p

< ε, ∀τ ∈ Pε, f

We denote the set of all such functions by AP S p R, X or APS p X.

Remark 1.7 It is clear that AP X ⊂ APS p X ⊂ APS q X for p ≥ q ≥ 1.

Definition 1.8 A function f : p Y, for each

u ∈ X, is called Stepanov almost periodic in t ∈ R uniformly for u ∈ X if, for each ε > 0 and each compact set K ⊂ X, there exists a relatively dense set Pε, f, K ⊂ R such that

sup

t∈R

1 0

f t  s  τ, u − ft  s, up

ds

1/p

< ε, 1.7

for each τ ∈ Pε, f, K and each u ∈ K We denote by APS p R × X, Y the set of all such

functions

It is also easy to show that AP S p R × X, Y ⊂ APS q R × X, Y for p ≥ q ≥ 1.

Throughout the rest of this paper, let C b R, X resp., C b R × X, Y be the space of

bounded continuousresp., jointly bounded continuous functions with supremum norm, and

P AP0R, X 



ϕ ∈ C b R, X : lim

T→ ∞

1

2T

T

−T

ϕ tdt 0 . 1.8

We also denote by P AP0R × X, Y the space of all functions ϕ ∈ C b R × X, Y such that

lim

T→ ∞

1

2T

T

−T

uniformly for x in any compact set K ⊂ X.

Definition 1.9 A function f ∈ C b R, X C b R × X, Y is called pseudo-almost periodic if

with g ∈ APXAPR × X, Y and ϕ ∈ PAP0R, XPAP0R × X, Y We denote by

P AP XPAPR × X, Y the set of all such functions.

It is well-known that P APX is a closed subspace of C b R, X, and thus PAPX is a

Banach space under the supremum norm

Definition 1.10 A function f ∈ BS p X is called Stepanov-like pseudo-almost periodic if it can be decomposed as f  g  h with g b ∈ APR, L p 0, 1; X and h b ∈ PAP0R, L p 0, 1; X.

We denote the set of all such functions by P AP S p R, X or PAPS p X.

It follows from2 that PAPX ⊂ PAPS p X for all 1 ≤ p < ∞.

Definition 1.11 A function F : p Y, for each

u ∈ X, is called Stepanov-like pseud-almost periodic in t ∈ R uniformly for u ∈ X if it can be decomposed as F  GH with G b ∈ APR×X, L p 0, 1; Y and H b ∈ PAP0R×X, L p 0, 1; Y.

We denote by P AP S p R × X, Y the set of all such functions.

Next, let us recall some notations about evolution family and exponential dichotomy For more details, we refer the reader to16

Definition 1.12 A set {Ut, s : t ≥ s, t, s ∈ R} of bounded linear operator on X is called an

evolution family if

a Us, s  I, Ut, s  Ut, rUr, s for t ≥ r ≥ s and t, r, s ∈ R,

b {τ, σ ∈ R2: τ

Definition 1.13 An evolution family U t, s is called hyperbolic or has exponential

dichotomy if there are projections Pt, t ∈ R, being uniformly bounded and strongly

continuous in t, and constants M, ω > 0 such that

a Ut, sPs  PtUt, s for all t ≥ s,

b the restriction U Q t, s : QsX → QtX is invertible for all t ≥ s and we set

U Q s, t  U Q t, s−1,

c Ut, sPs ≤ Me −ωt−sandU Q s, tQt ≤ Me −ωt−s for all t ≥ s,

where Q : I − P We call that

Γt, s :

⎧

⎨

⎩

U t, sPs, t ≥ s, t, s ∈ R,

is the Green’s function corresponding to Ut, s and P·.

Remark 1.14 Exponential dichotomy is a classical concept in the study of long-term behaviour

of evolution equations; see, for example,16 It is easy to see that

Γt,s ≤⎧⎨⎩Me −ωt−s , t ≥ s, t, s ∈ R,

Me −ωs−t , t < s, t, s ∈ R. 1.12

2 Main Results

Throughout the rest of this paper, for r ≥ 1, we denote by Lr R × X, X the set of all the functions f : R × X → X satisfying that there exists a function L f ∈ BS rR such that

f t, u − ft, v ≤ L f tu − v, ∀t ∈ R, ∀u, v ∈ X, 2.1

and, for any compact set K ⊂ X, we denote by APS p

K R × X, Y the set of all the functions

f ∈ APS p R × X, Y such that 1.7 is replaced by

sup

t∈R

1 0

 sup

u ∈K

f t  s  τ, u − ft  s, up ds1/p < ε. 2.2

In addition, we denote by · p the norm of L p 0, 1; X and L p 0, 1; R.

f ∈ APS p

K R × X, X.

Proof For all ε > 0, there exist x1, , x k ∈ K such that

K⊂k

i1

Since f ∈ APS p R × X, X, for the above ε > 0, there exists a relatively dense set Pε ⊂ R

such that

f t  τ  ·, u − ft  ·, u

p < ε

for all τ ∈ Pε, t ∈ R, and u ∈ K On the other hand, since f ∈ L p R × X, X, there exists a function L f ∈ BS pR such that 2.1 holds

Fix t ∈ R, τ ∈ Pε For each u ∈ K, there exists iu ∈ {1, 2, , k} such that u−x i u <

ε Thus, we have

f t  s  τ, u − ft  s, u

≤ L f t  s  τε f

t  s  τ, x i u

− ft  s, x i u   L f t  sε, 2.5

for each u ∈ K and s ∈ 0, 1, which gives that

sup

u ∈K

f t  s  τ, u − ft  s, u

≤L f t  s  τ  L f t  sεk

i1

f t  s  τ, x i  − ft  s, x i, ∀s ∈ 0, 1. 2.6

Now, by Minkowski’s inequality and2.4, we get

1 0

 sup

u ∈K

f t  s  τ, u − ft  s, up

ds

1/p

0

L p f t  s  τds

1/p

· ε  1

0

L p f t  sds

1/p

· ε

k

i1

1 0

f t  s  τ, x i  − ft  s, x ip

ds

1/p

≤2L f

S p 1ε,

2.7

which means that f ∈ APS p

K R × X, X.

Theorem 2.2 Assume that the following conditions hold:

a f ∈ APS p R × X, X with p > 1, and f ∈ L r R × X, X with r ≥ max{p, p/p − 1}.

b x ∈ APS p X, and there exists a set E ⊂ R with mes E  0 such that

is compact in X.

Then there exists q ∈ 1, p such that f·, x· ∈ APS q X.

Proof Since r ≥ p/p − 1, there exists q ∈ 1, p such that r  pq/p − q Let

p p

p − q , q

p

Then p, q > 1 and 1/p 1/q  1 On the other hand, since f ∈ L r R × X, X, there is a function L f ∈ BS rR such that 2.1 holds

It is easy to see that f·, x· is measurable By using 2.1, for each t ∈ R, we have

t1

t fs, xs q ds

1/q

≤

t1

t

f s, xs − fs, 0q

ds

1/q

f ·, 0

S q

≤

t1

t

L q f sx sq

ds

1/q

f ·, 0

S q

≤

t1

t

L r f sds

1/r

·

t1

t

x sp

dt

1/p

f ·, 0

S q

≤L f

S r · x S p  f·, 0 S q < ∞.

2.10

Thus, f ·, x· ∈ BS q X.

Next, let us show that f ·, x· ∈ APS q X ByLemma 2.1, f ∈ APS p

K R × X, X In addition, we have x ∈ APS p X Thus, for all ε > 0, there exists a relatively dense set Pε ⊂ R

such that

1 0

 sup

u ∈K

f t  s  τ, u − ft  s, up

ds

1/p

< ε,

xt  τ  · − xt  · p < ε

2.11

for all τ ∈ Pε and t ∈ R By using 2.11, we deduce that

1

0

f t  s  τ, xt  s  τ − ft  s, xt  sq

1/q

≤

1

0

L q f t  s  τx t  s  τ − xt  sq

1/q



1

0

f t  s  τ, xt  s − ft  s, xt  sq

1/q

≤

1

0

L r f t  s  τdt

1/r

·

1 0

x t  s  τ − xt  sp

dt

1/p



1

0

ft  s  τ, xt  s − ft  s, xt  s p

1/p

≤ L f S r · xt  τ  · − xt  · p 1

0

 sup

u ∈K

f t  s  τ, u − ft  s, up ds1/p

≤L f S r 1ε

2.12

for all τ ∈ Pε and t ∈ R Thus, f·, x· ∈ APS q X.



f ∈ PAP0R, R, where



f t 



supu ∈Kf t  ·, u



p

Proof Noticing that K is a compact set, for all ε > 0, there exist x1, , x k ∈ K such that

K⊂k

i1

Combining this with f ∈ Lp R × X, X, for all u ∈ K, there exists x isuch that

f t  s, u ≤ ft  s,u − ft  s,x i  ft  s,xi ≤ Lf t  sε f t  s, x i

2.15

for all t ∈ R and s ∈ 0, 1 Thus, we get

sup

u ∈K

f t  s, u ≤ L f t  sε k

i1

f t  s, x i, ∀t ∈ R, ∀s ∈ 0, 1, 2.16 which yields that



f t 



supu ∈Kf t  ·, u



p

≤ L S p · ε k

i1

f b t, x ip , ∀t ∈ R. 2.17

On the other hand, since f b ∈ PAP0R × X, L p 0, 1; X, for the above ε > 0, there exists T0> 0

such that, for all T > T0,

1

2T

T

−T f b t, x ip dt < ε

This together with2.17 implies that

1

2T

T

−T



Hence, f ∈ PAP0R, R.

Theorem 2.4 Assume that p > 1 and the following conditions hold:

a f  g  h ∈ PAPS p R × X, X with g b ∈ APR × X, L p 0, 1; X and h b ∈ PAP0R ×

X, L p 0, 1; X Moreover, f, g ∈ L r R × X, X with r ≥ max{p, p/p − 1};

b x  y  z ∈ PAPS p X with y b ∈ APR, L p 0, 1; X and z b ∈ PAP0R, L p 0, 1; X,

and there exists a set E ⊂ R with mes E  0 such that

is compact in X.

Then there exists q ∈ 1, p such that f·, x· ∈ PAPS q X.

Proof Let p, p, and qbe as in the proof ofTheorem 2.2 In addition, let ft, xt  Ht 

I t  Jt, where

H t  gt, y t, I t  ft, xt − ft, y t, J t  ht, y t. 2.21

It follows fromTheorem 2.2that H ∈ APS q X, that is, H b ∈ APR, L q 0, 1; X.

Next, let us show that I b , J b ∈ PAP0R, L q 0, 1; X For I b, we have

1

2T

T

−T I b t q dt 1

2T

T

−T

1 0

It  s q ds

1/q

dt

2T

T

−T

1 0

L q f t  szt  s q ds

1/q

dt

≤ L f S r

1

2T

T

−T z b t p dt → 0, T → ∞,

2.22

where z b ∈ PAP0R, L p 0, 1; X was used For J b , since h  f − g ∈ L r R × X, X ⊂ L pR ×

X, X, byLemma 2.3, we know that

lim

T→ ∞

1

2T

T

−T





supu ∈Kh t  ·, u



p

which yields

1

2T

T

−T J b t q dt≤ 1

2T

T

−T J b t p dt

2T

T

−T

1 0

ht  s, yt  sp

ds

1/p

dt

2T

T

−T

1 0

 sup

u ∈K

h t  s, up

ds

1/p

2.24

that is, J b ∈ PAP0R, L q 0, 1; X Now, we get f·, x· ∈ PAPS q X.

Next, let us discuss the existence and uniqueness of pseudo-almost periodic solutions

for the following abstract semilinear evolution equation in X:

Theorem 2.5 Assume that p > 1 and the following conditions hold:

a f  g  h ∈ PAPS p R × X, X with g b ∈ APR × X, L p 0, 1; X and h b ∈ PAP0R ×

X, L p 0, 1; X Moreover, f, g ∈ L r R × X, X with

r≥ max



p, p

p− 1



, r > p

b the evolution family Ut, s generated by At has an exponential dichotomy with constants

M, ω > 0, dichotomy projections P t, t ∈ R, and Green’s function Γ;

c for all ε > 0, for all h > 0, and for all F ∈ APS1X there exists a relatively dense set

P ε ⊂ R such that sup r∈RFr  ·  τ − fr  · < ε and

sup

r∈RΓt  r  τ,s  r  τ − Γt  r,s  r < ε, 2.27

for all τ ∈ Pε and t, s ∈ R with |t − s| ≥ h.

Then2.25 has a unique pseudo-almost periodic mild solution provided that

L f

S r < 1− e −ω



ωr

1− e −ωr

1/r

, where 1/r 1/r

Proof Let u  v  w ∈ PAPX, where v ∈ APX and w ∈ PAP0X Then u ∈ PAPS p X and K : {vt : t ∈ R} is compact in X By the proof of Theorem 2.4, there exists q ∈ 1, p such that f·, u· ∈ PAPS q X.

Let

where f b

1 ∈ APR, L q 0, 1; X and f b

2 ∈ PAP0R, L q 0, 1; X Denote

F ut :



RΓt, sfs, usds  F1ut  F2ut, t ∈ R, 2.30 where

F1ut 



RΓt, sf1sds, F2ut 



RΓt, sf2sds. 2.31