Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

Applied Mathematics and Computation 218 (2011) 1735–1745

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations Claudio Cuevas a,⇑, Alex Sepúlveda b, Herme Soto b a b

Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, CEP 50540-740, Brazil Departamento de Matemática y Estadística, Universidad de La Frontera Casilla, 54-D Temuco, Chile

a r t i c l e

i n f o

C. Cuevas dedicates this work to Professor H. Henríquez who has been a guide and inspiration to the first author. Keywords: Fractional order differential equations Integro-differential equations Mild solutions Stepanov almost periodic functions Stepanov-like pseudo-almost periodic functions

a b s t r a c t We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions for fractional differential and integro-differential equations in the case when the forcing term belongs to the class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions.  2011 Elsevier Inc. All rights reserved.

1. Introduction The study of the existence of almost periodic and pseudo-almost periodic solutions for evolution equations is one of most attracting topics in the qualitative theory of evolution equations due to its mathematical interest and to the applications in physics and mathematical biology, among other areas. Related with this subject we refer the reader to extensive bibliography in [1–3,8,10,11,16,17,20,21,25,30,40,45]. In 1926, Stepanov found a wider class of almost periodic functions, for which continuity fails, and only measurability and integrability in the sense of Lebesgue are required. Those functions are known as Stepanov almost periodic functions (see, e.g. [4,43,44,47,48] and the references therein). Firstly, we study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions to autonomous fractional differential equation

Dat uðtÞ ¼ AuðtÞ þ Dta1 f ðt; uðtÞÞ;

t 2 R;

ð1:1Þ

where 1 < a < 2, A : D(A) # X ? X is a linear densely defined operator of sectorial type on a Banach space X and f : R  X ! X is Stepanov almost (resp., Stepanov-like pseudo-almost) periodic in t 2 R uniformly for x 2 X. The fractional derivative is understood in the Riemann–Liouville sense. The origin of fractional calculus goes back to Newton and Liebniz in the seventieth century. Fractional order differentiation consist in the generalization of classical integer differentiation to real or complex orders. We observe that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, biology and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [7,51] or to model activator-inhibitor dynamics with anomalous diffusion [29]. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (C. Cuevas), [email protected] (A. Sepúlveda), [email protected] (H. Soto). 0096-3003/$ - see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.054

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Secondly, we consider the following integro-differential equation

u0 ðtÞ ¼ AuðtÞ þ

Z

t

aðt  sÞAuðsÞds þ f ðt; uðtÞÞ;

t 2 R;

ð1:2Þ

1

where A is a closed linear operator defined in X and a 2 L1loc ðRþ Þ is an scalar-valued kernel. We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions to (1.2) in the case when the forcing term belongs to a class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions. This problem remains an untreated topic in the literature. The authors are filling this important gap. Since Stepanov almost periodic functions are not necessarily continuous this type of periodicity is interesting and difficult. We remark that equations of type (1.2) arise in the study of heat flow in material of fading memory type as well as some equations of population dynamics (see [12,16,26,38,42]). In [38] the authors have proved existence of various classes of mild solutions of Eq. (1.2) in spaces of vector-valued functions MðR; XÞ ranging between periodic functions and bounded continuous functions, where the perturbation f : R  X ! X in (1.2) is so that f ð; xÞ 2 MðR; XÞ uniformly for each x 2 K, where K is any bounded subset of X and satisfying a suitable Lipschitz type condition. We now turn to a summary of this work. The second section provides the definitions and preliminaries results to be used in theorems stated and proved in the article. In particular to facilitate access to the individual topics, we review some of the standard properties of Stepanov almost (resp., pseudo-almost) periodic functions; sectorial operators, solution operators and regularized families. The third (resp., fourth) section is devoted to obtain sufficient conditions to the existence of almost (resp., pseudo-almost) periodic mild solutions of (1.1) and (1.2) with Stepanov almost (resp., Stepanov-like pseudo-almost) periodic coefficients. From a more calculation point of view, throughout this article, we give a few applications. We believe that this work is a significant contribution to the study of qualitative properties of fractional differential and integro-differential equations. 2. Preliminaries 2.1. Sectorial operators We need to recall some definitions about sectorial operators. Definition 2.1. A closed linear operator (A, D(A)) with dense domain D(A) in the Banach space X is said to be sectorial of type x and angle h if there are constants x 2 R; h 20; p2 ½, and M > 0 such that its resolvent exists outside the sector

x þ Rh :¼ fx þ k : k 2 C; jargðkÞj < hg; kðk  AÞ1 k 6

M ; jk  xj

k R x þ Rh ;

ð2:3Þ ð2:4Þ

for more details on sectorial operators we refer to [39]. We denote by BðXÞ the space of all bounded linear operator from X into X endowed with the norm of operators. Let A be a closed linear operator with domain D(A) defined on a Banach space X, the notation q(A) stands for the resolvent set of A. To study the following fractional differential equation

Dat uðtÞ ¼ AuðtÞ þ Dta1 f ðtÞ;

t 2 R;

ð2:5Þ

where f is an X-valued function. We consider the following concept. Definition 2.2. We call A is the generator of a solution operator (or a-resolvent family) if there are x 2 R and a strongly conR1 tinuous function Sa : Rþ ! BðXÞ such that {ka : Rek > x} # q(A) and ka1 ðka  AÞ1 x ¼ 0 ekt Sa ðtÞxdt; Rek > x; x 2 X. In this case, Sa(t) is called the solution operator generated by A. We observe that the power function ka is uniquely defined as ka = jkjaeiargk, with  p < arg(k) < p.   We note that if A is sectorial of type x with 0 6 h < p 1  a2 , then A is the generator of a solution operator given by R 1 a 1 a Sa ðtÞ ¼ 21pi c ekt k ðk  AÞ dk, where c is a suitable path lying outside the sector x + Rh. Lemma 2.1 ([13], Theorem 1). Let A : D(A) # X ? X be a sectorial operator in a complex Banach space X, satisfying (2.3) and (2.4) for M > 0, x < 0 and 0 6 h < pð1  a2Þ. Then there is C(h, a) > 0 depending solely on h and a, such that

kSa ðtÞk 6

Cðh; aÞM ; 1 þ jxjt a

t P 0:

ð2:6Þ

2.2. Stepanov almost periodic functions Stepanov produced the first generalization of Bohr’s almost periodic functions. He found a wider class of almost periodic functions that are commonly known as Stepanov almost periodic functions, which need not be continuous. This notion plays an important role in discussing the solutions of evolution equations, both linear and nonlinear (see, e.g. [4,36,44,47,48]). Some of their properties were not established until recently (see [22,31,34,35,49,52]).

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Definition 2.3. A continuous function f : R ! X is called almost periodic if for each  > 0 there is a relatively dense set Pð; f Þ # R such that supt2R kf ðt þ sÞ  f ðtÞk < ; 8s 2 Pð; f Þ. We denote the set of all such functions by AP(X). Remark 2.1. Note that each almost periodic function is bounded and uniformly continuous. It is well known that the range Rf ¼ ff ðtÞ : t 2 Rg of an almost periodic function f is relatively compact. AP(X) endowed with the norm of uniform convergence on R is a Banach space. Definition 2.4. The space BSp(X) of all Stepanov functions, with the exponent p 2 [1, 1), consists of all measurable functions f on R with values in X such that

kf kSp :¼ sup

Z

t2R

tþ1

kf ðsÞkp ds

1p < þ1:

t

This is called the Stepanov norm. We observe that Lp ðR; XÞ # BSp ðXÞ # Lploc ðR; XÞ and BSp(X) # BSq(X) whenever p P q P 1. Note that for each p P 1, we have the following continuous inclusion: ðBCðXÞ; k  k1 Þ,!ðBSp ðXÞ; k  kSp Þ; see e.g. [19]. Remark 2.2. In the above definition, one may to replace the norm k  kSp , by the norm equivalent obtained varying the length of the integration interval as

jjjf jjj ¼ sup t2R

1 l

Z

tþl

!1p kf ðsÞkp ds

;

t

where l > 0. Definition 2.5. A function f 2 BSp(X) is called Stepanov almost periodic if for each Pð; f Þ # R such that

sup t2R

Z

1

1p

kf ðt þ s þ sÞ  f ðt þ sÞkp ds

< ;

 > 0,

there is a relatively dense set

8s 2 Pð; f Þ:

0

We denote the set of all such functions by APSp(X). We observe that AP(X) # APSp(X) # APSq(X) for p P q P 1. Remark 2.3. It should be noticed that a Stepanov almost periodic function, when uniformly continuous, it is also almost periodic in the classical sense; see for instance [4,36]. We also note that a Stepanov almost periodic function may be   support in  12 ; 12 and such that unbounded. In fact, we show the Tarallo’s example [52, p. 56]. Take H 2 C 1 0 ðRÞ, with R 12 P H P 0; Hð0Þ ¼ 1; 1 HðtÞdt ¼ 1, and define, for every integer n P 1, a function /n ðtÞ ¼ k2Pn Hðn2 ðt  kÞÞ by adding non overP 2 lapping, rescaled copies of H at any point of Pn ¼ 3n ð2Z þ 1Þ. The C1-function uðtÞ ¼ nP1 un ðtÞ is unbounded, since n u(3 ) = n. We can prove that u is almost periodic in the sense of Stepanov. Definition 2.6. A function f : R  X ! X with f ðt; Þ 2 CðR; XÞ and f ð; xÞ 2 BSp ðR; XÞ for each x 2 X, is said to be Stepanov almost periodic in t 2 R uniformly for x 2 X if for each  > 0 and each compact set K # X, there is a relatively dense set Pð; f ; KÞ # R such that

sup t2R

Z

1

1p kf ðt þ s þ s; xÞ  f ðt þ s; xÞkp ds < ;

0

for each s 2 P(, f, K) and each x 2 K. We denote by APSp ðR  X; XÞ the set of all such functions. It is easy to show that APSp ðR  X; XÞ # APSq ðR  X; XÞ for p P q P 1. We need the following composition theorem of Stepanov almost periodic functions. Lemma 2.2 ([22]). Assume that p > 2 and that the following conditions hold: (a) Let f 2 APSp ðR  X; XÞ be Sp-Lipschitzian in the second variable, that is, there is a function L 2 BSp(X) such that

kf ðt; uÞ  f ðt; v Þk 6 LðtÞku  v k; p

8t 2 R;

8u; v 2 X:

(b) x 2 APS (X), and there is a set E # R with mes E = 0 such that K ¼ fxðtÞ : t 2 R n Eg is compact in X. p

Then f ð; xðÞÞ 2 APS2 ðXÞ.

ð2:7Þ

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2.3. Stepanov-like pseudo-almost periodicity Next, we describe the set of pseudo-almost periodic functions: Definition 2.7 (The ergodic space). The set of bounded continuous function with vanishing mean value is defined as

Z

1 2T

P0 ðXÞ ¼ f/ 2 BCðR; XÞ : lim

T!1

T

k/ðrÞkdr ¼ 0g:

T

We define the set PAP(X) of pseudo-almost periodic functions as

PAPðXÞ ¼ ff ¼ g þ / 2 BCðR; XÞ; g 2 APðXÞ and / 2 P 0 ðR; XÞg: In this case g and / are called, respectively, the principal and the ergodic terms of f. In a recent paper of Diagana [18], the concept of Stepanov-like pseudo-almost periodicity (or Sp-pseudo almost periodicity) was introduced and several of their properties studied. Such a concept is a generalization of the notion of pseudo-almost periodicity due to Zhang [53–55]. Definition 2.8 ([18,19]). A function f 2 BSp(X) is called Sp-pseudo almost periodic (or Stepanov-like pseudo-almost periodic) if it can be expressed as f = h + u, where h 2 APSp(X) and u 2 BCðR; XÞ such that

lim

T!1

1 2T

Z

T T

Z

tþ1

kuðrÞkp dr

1p

dt ¼ 0:

ð2:8Þ

t

The collection of such functions will be denoted by PAPSp(X). Remark 2.4 ([18]). It is clear that if 1 6 q 6 p < +1, PAPSp(X) # PAPSq(X) and AP(X) # PAP(X) # PAPSp(X). Similarly one gets the following definition. Definition 2.9 ([18,19]). A function f : R  X ! X; ðt; xÞ ! f ðt; xÞ with f ð; xÞ 2 Lploc ðR; XÞ for each x 2 X, is said to be Sp-pseudo almost periodic in t 2 R uniformly in x 2 X if there are two functions H; U : R  X ! X such that f = H + U, where H is Stepanov almost periodic in t 2 R uniformly in x 2 X and U 2 BCðR  X; XÞ such that

1 lim T!1 2T

Z

T T

Z

tþ1

p

kUðr; xÞk dr

1p

dt ¼ 0;

t

uniformly in x 2 X. The collection of those Sp-pseudo almost periodic functions will be denoted by PAPSp ðR  X; XÞ. We have the following useful composition theorem. Theorem 2.1 ([18]). Let f : R  X ! X be a Sp-pseudo almost periodic in t 2 R, uniformly in x 2 X. Suppose that f(t, x) is Lipschitz in x 2 X uniformly in t 2 R, in the sense that there is L > 0 such that

kf ðt; xÞ  f ðt; yÞk 6 Lkx  yk;

ð2:9Þ

for all t 2 R; x; y 2 X. If /() 2 PAPSp(X), then f(, /()) 2 PAPSp(X). 2.4. Semilinear integro-differential equations Theory of integro-differential equations in infinite-dimensional spaces has been strongly promoted by the large number of applications in physics, engineering and biology. Questions like existence, continuous dependence, perturbations and asymptotic behavior are at present an active area of research. We consider the following integro-differential equation (see [38])

u0 ðtÞ ¼ AuðtÞ þ

Z

t

aðt  sÞAuðsÞds þ f ðtÞ;

t 2 R;

1

where A is a closed linear operator defined in X and a 2 L1loc ðRþ Þ is an scalar-valued kernel. We recall that the Laplace transform of a function g 2 L1loc ðRþ Þ is given by

LðgÞðkÞ :¼ b g ðkÞ :¼

Z

1

ekt gðtÞdt;

Re k > l;

0

where the integral is absolutely convergent for Rek > l. In order to give an operator theoretical approach to Eq. (2.10) we recall the following definition (cf. [38,45]).

ð2:10Þ

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Definition 2.10. Let b 2 L1loc ðRÞ be given and let A be a closed linear operator with domain D(A) defined on a Banach space X. We call A is the generator of a solution operator (or resolvent family) if there are l 2 R and a strongly continuous function   1 : Rek > l # qðAÞ and S : Rþ ! BðXÞ such that bbðkÞ

1 1 A kb bðkÞ b bðkÞ

!1 x¼

Z

1

ekt SðtÞxdt;

Re k > l;

x 2 X:

0

In this case, S(t) is called the solution operator (or (1, b)-regularized family according to [37]) generated by A. In the scalar case, where there is a large bibliography which studies the concept of resolvent, we refer the reader to Gripenberg et al. [27] and the references therein. We note that in the case b(t)  1 the family corresponds to a C0-semigroup (e.g. [38]). Remark 2.5 (See [38] for details). Assume that A generates an (1, 1 + (1⁄a))  regularized family S(t) on the Banach space X. Rt Rt Let bðtÞ ¼ 1 þ 0 aðsÞds and define uðtÞ :¼ 1 Sðt  sÞf ðsÞds. We can see that u is a mild solution of Eq. (2.10). In fact we note that S() satisfies the resolvent equations

SðtÞx ¼ x þ

Z

t

bðt  sÞASðsÞxds; x 2 X; t P 0; Z t aðt  sÞASðsÞxds; x 2 X; t P 0: S0 ðtÞx ¼ ASðtÞx þ 0

0

Using Fubini’ s theorem, we obtain

u0 ðtÞ ¼ Sð0Þf ðtÞ þ

Z

t 1

¼ f ðtÞ þ AuðtÞ þ

S0 ðt  sÞf ðsÞds ¼ f ðtÞ þ AuðtÞ þ Z

t

aðt  v Þ

1

Z

Z

t

1

Z

t

aðt  v ÞASðv  sÞf ðsÞdv ds

s

s

ASðv  sÞf ðsÞdsdv ¼ f ðtÞ þ AuðtÞ þ

1

Z

t

aðt  v ÞAuðv Þdv :

1

The idea behind of the above argument is the following: we take formally the Laplace transform and obtain

b ðkÞ ¼ bf ðkÞ þ initial conditions; FðkÞ u where FðkÞ ¼ k  A  b a ðkÞA: Then, we define an ad-hoc strongly continuous family S(t) that satisfies

FðkÞb SðkÞ ¼ I; b SðkÞFðkÞ ¼ I: Then, we directly prove that the mild solution of Eq. (2.10) has the convolution structure uðtÞ :¼ that S(t) should formally satisfy

b SðkÞ ¼ ðk  A  b a ðkÞAÞ

1

Rt 1

Sðt  sÞf ðsÞds. We find

0 11 1 1 @ ¼  AA : 1þb a ðkÞ ð1þbaðkÞÞ k

Choosing b0 (t) = a(t), b(0) = 1, we get

b SðkÞ ¼

1 1 A kb bðkÞ b bðkÞ

!1

and

bðtÞ ¼ 1 þ

Z

t

aðsÞds:

0

We finishes the discussion of remark. The definition of ‘‘mild’’ solution for the nonlinear case is then straightforward, see next definition. Definition 2.11 ([38]). Assume that A generates an (1, 1 + (1⁄a))-regularized family S(t) on the Banach space X. A function u : R ! X is said to be a mild solution to Eq. (1.2) if the function s ? S(t  s)f(s, u(s)) is integrable on (1, t) for each t 2 R and Rt uðtÞ ¼ 1 Sðt  sÞf ðs; uðsÞÞds; for each t 2 R. Recall that an strongly continuous family fSðtÞgtP0 # BðXÞ is said to be uniformly bounded (respectively, uniformly R1 integrable) if there is a constant M > 0 such that kS(t)k 6 M for all t P 0 (respectively, 0 kSðtÞkdt < þ1) (see [45]).

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Example 1 ([14]). Taking X ¼ R; A ¼ q; q > 0, and considering aðtÞ ¼ Ct ðaÞ ext , where x > 0 and 1 < a < 2, we can check that S(t) = ta1Ea,a(qt)ext, where Ea,a denotes the generalized Mittag–Leffer function [41].1 We have a explicit description to S(t)ext given by

1

SðtÞext ¼

p

sinðpaÞ

Z

ra

1

ert

0

r 2a

þ 2r a qa cosðpaÞ þ q2a

dr 

2

aqa1

 p p p etq cos ðaÞ cos t q sin þ :

a

a

1

In [6, Corollary 3.7], the authors have proven that kS(t)k 6 u(t), "t P 0, where u 2 L ðRþ Þ is given by

1

uðtÞ ¼ j sinðpaÞj p

Z

ra

1

ert

0

r 2a

þ 2r a qa cosðpaÞ þ q2a

dr þ

2

aqa1

p

etq cosðaÞ

and kuk1 ¼ aq2 a  q1a  aqa12cos p . ðaÞ Remark 2.6. (i) Let X be a Banach space. Suppose that a(t) is 1-regular [15, Definition 2.9],[45] and that A generates a parabolic [15, Definition 3.3] and uniformly integrable resolvent family {S(t)}tP0. Then {S(t)}tP0 is uniformly stable, that is, kS(t)k ? 0 as t ? 1. In particular {S(t)}tP0 is bounded. (ii) If A generates an analytic resolvent which is bounded on some sector R(0, h), then A generates a parabolic resolvent family. The converse is not true. A standard situation leading with generators of parabolic resolvents is the following: Let a(t) be of subexponential growth of positive type [45], and let A be generates a bounded analytic C0-semigroup in X, then A generate a parabolic resolvent (see [45, Proposition 3.1]).

Remark 2.7 ([9]). We set X ¼ Cn ; A ¼ qI; q 2 C, and a 2 L1 ðRþ Þ. Suppose that

q ba ðkÞ – 1 for all Rek P 0:

ð2:11Þ 1

nn

a

xt

We get that A generates an integral resolvent S 2 L ðRþ ; C Þ. As a concrete example, we can take a(t) = t e , x > 0, a P 0. Note that in case a = 0 we have S(t) = e(x+q)tI, t P 0. We observe that (2.11) is equivalent to say Re(q) < x, and hence, R1 pffiffiffiffi 1 p1ffiffiffi xt kSðsÞkds ¼ xþReð 0 qÞ. In case a = 1, we have for q > 0 : SðtÞ ¼ q e sinhð qtÞ; t P 0. 3. Almost periodic mild solutions to evolution equations with Stepa- nov almost periodic coefficients 3.1. Fractional differential equations We have the following result.   Theorem 3.2. Assume that A is sectorial of type x < 0 and angle h satisfying 0 6 h < p 1  a2 ; 1 < a < 2. If f is Stepanov almost periodic with exponent p > 1, then the Eq. (2.5) has an almost periodic (mild) solution given by

uðtÞ ¼

Z

t

Sa ðt  sÞf ðsÞds:

1

Proof. For each m 2 N and t 2 R, put

Um ðtÞ :¼

Z

tm

Sa ðt  sÞf ðsÞds ¼

tðmþ1Þ

Z

mþ1

Sa ðsÞf ðt  sÞds:

m

Since APSp(X) # APS1(X), we have that for each

sup t2R

Z

1

kf ðt þ s þ sÞ  f ðt þ sÞkds <

0

 > 0, there is a relatively dense set PðÞ # R such that

 Cðh; aÞM

;

ð3:12Þ

for all s 2 P(), where C(h, a) and M are the constants in (2.6). Let s 2 P() and t 2 R

kUm ðt þ sÞ  Um ðtÞk 6 Cðh; aÞ

Z

1

kf ðt  m  1 þ s þ sÞ  f ðt  m  1 þ sÞkds < :

0

1 In recent times the attention of mathematicians towards the Mittag–Leffer type functions has increased from both the analytical and numerical point of view, overall because of their relation with fractional calculus. A description of the most important properties of these functions can be found in [24]. For the interested readers we also recommend Dzherbashian [23]; and Sansone and Gerretsen [50].

C. Cuevas et al. / Applied Mathematics and Computation 218 (2011) 1735–1745

1741

From the following estimate

kUm ðtÞk 6

Z

tm

tðmþ1Þ

Cðh; aÞM Cðh; aÞM kf ðsÞkds 6 1 þ jxjma 1 þ jxjðt  sÞa

Z

tm

kf ðsÞkds 6

tðmþ1Þ

Cðh; aÞM kf kS1 : 1 þ jxjma

P We deduce from the Weierstrass test that 1 m¼0 Um ðtÞ is uniformly convergent on R. So by [40, Theorem 3.14, p. 52] we have Rt that 1 Sa ðt  sÞf ðsÞds 2 APðXÞ. This completes the proof. h The following example complements [6, Corollary 3.7, p. 3700]. Example 2. Let A = qa, where q > 0 and 1 < a < 2. Then we conclude that for each f 2 APSp(X), p > 1, the ordinary fractional differential equation

Dat uðtÞ ¼ qa uðtÞ þ Dta1 f ðtÞ;

t 2 R;

ð3:13Þ

has an almost periodic solution. We recall here the following definition that will be essential for us. Definition 3.12. Assume that A is sectorial of type x < 0 and angle h satisfying 0 6 h < pð1  a2Þ. A function u : R ! X is called a mild solution of (1.1) if the function s ? Sa(t  s)f(s, u(s)) is integrable on (1, t) for each t 2 R and

uðtÞ ¼

Z

t

Sa ðt  sÞf ðs; uðsÞÞds;

ð3:14Þ

1

for any t 2 R. Combining the composition result of Stepanov almost periodic functions (Lemma 2.2) and the previous theorem, we obtain the following result. Theorem 3.3. Assume that A is sectorial of type x < 0 and angle h satisfying 0 6 h < pð1  a2Þ; 1 < a < 2. Assume that asinðpaÞ f 2 APSp ðR  X; XÞ with p > 2 and that there is a function L 2 BSp(X) such that (2.7) holds. If kLkSp < 1 , where C(h, a) and a Cðh;aÞMjxj p M are the constants in (2.6), then the Eq. (1.1) has a unique almost periodic mild solution. Proof. Let u 2 AP(X), then u 2 APSp(X) and by Remark 2.1, we have that Ru is compact in X. We define the operator G : AP(X) ? AP(X) by

GðuÞðtÞ ¼

Z

t

Sa ðt  sÞf ðs; uðsÞÞds;

t 2 R:

ð3:15Þ

1 p

By Lemma 2.2, we have that s ? f(s, u(s)) belongs to APS2 ðXÞ. By Theorem 3.2, we obtain that G(u) 2 AP(X) and hence G is well defined. It suffices to show that the operator G has a unique fixed point in AP(X). For u, v 2 AP(X), by using the Hölder inequality, we can estimate

Z

t

LðsÞ kuðsÞ  v ðsÞkds 1 þ jxjðt  sÞa ! 1 Z tm X LðsÞ ds ku  v k1 6 Cðh; aÞM 1 þ jxjðt  sÞa m¼0 tðmþ1Þ 0 !1p 1 Z tm 1 X 1 p 6 Cðh; aÞM @ LðsÞ ds Aku  v k1 1 þ jxjma tðmþ1Þ m¼0 ! 1 X 1 kLkSp ku  v k1 ; 6 Cðh; aÞM 1 þ jxjma m¼0

kGðuÞðtÞ  Gðv ÞðtÞk 6 Cðh; aÞM

1

which finish the proof.

h

Remark 3.8. Similar results as the previous theorems were obtained by Ding et al. [22] for abstract semilinear equations. Agarwal et al. [3] have studied the existence and uniqueness of pseudo almost periodic mild solutions to the Eq. (1.1) where f : R  X ! X is a pseudo almost periodic function satisfying (2.7) where L() is an integrable bounded function. þ AuðtÞ 3 f ðtÞ, Remark 3.9. Haraux [28] has shown that if a C1-solution u of the linear first order differential inclusion du dt where A is maximal monotone and f is Stepanov almost periodic, which is bounded then it is in fact almost periodic (see also [5]).

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3.2. Integro-differential equations We have the following result. Theorem 3.4. Assume that A generates an (1, 1 + (1⁄a))-regularized family S(t) on the Banach space X such that

for all t P 0;

kSðtÞk 6 /ðtÞ;

ð3:16Þ P1

where / : Rþ ! Rþ is a decreasing function such that /0 ¼ m¼0 /ðmÞ < þ1. If f is Stepanov almost periodic with exponent p > 1, then the Eq. (2.10) has a unique almost periodic mild solution. Proof. For each m 2 N and t 2 R, put Um ðtÞ :¼ PðÞ # R such that

sup t2R

Z

1

kf ðt þ s þ sÞ  f ðt þ sÞkds <

0

 /ð0Þ

R tm

tðmþ1Þ

Sðt  sÞf ðsÞds. We have that for each

 > 0, there is a relatively dense ð3:17Þ

:

Let s 2 P() and t 2 R, we have that

kUm ðt þ sÞ  Um ðtÞk 6 /ð0Þ

Z

1

kf ðt  ðm þ 1Þ þ s þ sÞ  f ðt  ðm þ 1Þ þ sÞkds < :

0

From the estimate kUm ðtÞk 6 /ðmÞkf kS1 , we deduce that

P1

m¼0

Um ðtÞ is uniformly convergent on R, then u 2 AP(X).

h

Remark 3.10. We note that conditions of type (3.16) have been previously considered in the literature to study almost automorphicity of semilinear integral equations by Cuevas and Lizama [14] (see also [6,15]). The following consequence is now immediate which complements [38, Corollary 3.6]. Corollary 3.1. Assume that A generates an uniformly stable semigroup S(t) on the Banach space X. If f is Stepanov almost periodic with exponent p > 1, then there is a unique almost periodic mild solution of the equation

u0 ðtÞ ¼ AuðtÞ þ f ðtÞ;

t 2 R:

ð3:18Þ

Remark 3.11. As was emphasized by the referee, this result has been established by several authors (even in more general terms than those considered here) see [32,46]. The authors thanks the referee by providing these references. Example 3. Let A = q, where q > 0 and a(t)  0. Then S(t) = eqtI and we conclude that the equation 1 X X

u0 ðtÞ ¼ quðtÞ þ

Hðn2 ðt  kÞÞ;

n¼1 k2Pn

where H is given in Remark 2.3, has the unique mild solution

uðtÞ ¼

1 X Z X n¼1 k2P n

t

eqðtsÞ Hðn2 ðs  kÞÞds;

1

which belongs to AP(X). We have the following result. Theorem 3.5. Assume that A generates an (1, 1 + (1⁄a))-regularized family {S(t)}tP0 on the Banach space X such that (3.16) holds. Assume that f 2 APSp ðR  X; XÞ with p > 2 and that there is a function L 2 BSp(X) such that (2.7) holds. If /0 kLkSp < 1, where /0 is given in Theorem 3.4, then the Eq. (1.2) has a unique almost periodic mild solution. Proof. The proof makes use of a similar argument already done in the proof of Theorem 3.3. For the sake of brevity we leave the details as an exercise to the reader. h Example 4. We set X ¼ Cn ; A ¼ qI; q 2 C; aðtÞ ¼ ext with Re(q) < x. Let L : R ! Cn be a Stepanov almost periodic function with exponent p > 2 and let g : Cn ! Cn be a function so that kg(x)  g(y)k 6 Ckx  yk for all x; y 2 Cn . By Theorem 3.5 we conclude that if CkLkSp þ exþReðqÞ < 1, the equation

u0 ðtÞ ¼ quðtÞ þ

Z

t

exðtsÞ uðsÞdsÞ þ LðtÞgðuðtÞÞ;

1

has a unique almost periodic solution.

t 2 R;

C. Cuevas et al. / Applied Mathematics and Computation 218 (2011) 1735–1745

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4. Pseudo-almost periodic mild solutions to evolutions equations with Stepanov-like pseudo-almost periodic coefficients 4.1. Fractional differential equations In this subsection we make use of the previous properties of Sp-pseudo almost periodic functions to study the existence of pseudo-almost periodic solutions to fractional differential equation with Stepanov pseudo-almost periodic coefficients. We have the following result. Theorem 4.6. Assume that A is sectorial of type x < 0 and angle h satisfying 0 6 h < pð1  a2Þ; 1 < a < 2. If f is Sp-pseudo-almost periodic with p > 1, then Eq. (2.5) has a pseudo-almost periodic mild solution. Proof. Let f = h + u, where h 2 APSp(X) and u 2 BCðR; XÞ such that (2.8) holds. Consider for each m 2 N and t 2 R, the integral

mm ðtÞ :¼

Z

Z

mþ1

Sa ðsÞf ðt  sÞds ¼

m

mþ1

Sa ðsÞhðt  sÞds þ

m

Z

mþ1

Sa ðsÞuðt  sÞds :¼ Y m ðtÞ þ X m ðtÞ: m

We can deduce that Ym 2 AP(X) (the proof of this fact makes use of a similar argument already dome in the proof of Theorem Cðh;aÞM Cðh;aÞM p ; t 2 R (respectively, kX m ðtÞk 6 3.2). On the other hand, from the estimate kY m ðtÞk 6 1þj kukSp ; t 2 R) we dexjma khk 1þjxjma PN PNS duce that the sequence of functions m¼0 Y m ðtÞ(respectively, m¼0 X m ðtÞ) is uniformly convergent on R. The following estimate

kX m ðtÞk 6 Cðh; aÞM

Z

!1p

tm

p

kuðsÞk ds

tðmþ1Þ

is responsible for the fact that Xm 2 P0(X). Thus we conclude that each mm is a pseudo-almost periodic function and hence PN P1 m¼0 mm ðtÞ 2 PAPðXÞ. Consequently its uniform limit uðtÞ ¼ m¼0 mm ðtÞ 2 PAPðXÞ, by [18, Lemma 2.5, p. 11]. This completes the proof. h Remark 4.12. Diagana [18] has studied the existence and uniqueness of a pseudo-almost periodic solution to the semilinear equation

u0 ðtÞ ¼ BuðtÞ þ Fðt; uðtÞÞ;

t 2 R;

ð4:19Þ

where B : D(B) # X ? X is the infinitesimal generator of an exponentially stable C0-semigroup on a Banach space X and F : R  X ! X is Sp-pseudo almost periodic for p > 1 and jointly continuous. It is worth mentioning that the above-mentioned result generalizes most of existence results related to the pseudo-almost periodicity of solutions to the semilinear equation, Eq. (4.19), as the space of Sp-pseudo almost periodic functions contains the space of pseudo almost periodic functions. Hu and Jin [33] have obtained existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations

u0 ðtÞ ¼ AðtÞuðtÞ þ hðtÞ and

u0 ðtÞ ¼ AðtÞuðtÞ þ f ðt; BuðtÞÞ þ

Z

t

Cðt; sÞuðsÞds þ FðtÞ;

on R;

1

assuming that A(t) satisfy ‘‘Acquistapace-Terrini’’ conditions, that the evolution family generated by A(t) has exponential dichotomy, that R(k0, A()) is almost periodic, that B, (C(t, s))tP0 are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument, and that h, f, F are Stepanov-like pseudo almost periodic for p > 1 and continuous. Example 5. Let A = qa, where q > 0 and 1 < a < 2. Next, we use an example of Stepanov almost periodic function due to Hu P and Mingarelli [34]; we consider n(n = 1, 2, . . .) so that 0 < n < 1; 1 n¼1 n < þ1. For each n, define a function gn(t) as follows:

8 2ðtyn;k þn Þ2 > ; t 2 ðyn;k  n ; yn;k  2n ; > > 2n > > > > < 1  2ðtyn;k Þ2 ; t 2 ðy  n ; y þ n ; n;k n;k 2 2 2n g n ðtÞ :¼ > 2 > 2ðty   Þ n n;k  > > ; t 2 ðyn;k þ 2n ; yn;k þ n ; > 2n > > : 0; elsewhere;

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C. Cuevas et al. / Applied Mathematics and Computation 218 (2011) 1735–1745

P where yn, k = (2k + 1)n, k = 0, ± 1, ± 2, . . .. Then, for each n, gn(t) is a periodic function with period 2n. Let gðtÞ ¼ 1 n¼1 g n ðtÞ, Hu 1 and Mingarelli in [34, Section 5, p. 733] have proved that g is Stepanov almost periodic. Let f ðtÞ ¼ gðtÞ þ 1þt2 . By Theorem 4.6 we conclude that the fractional differential equation Dat uðtÞ ¼ qa uðtÞ þ Dta1 f ðtÞ; has a pseudo-almost periodic solution. Theorem 4.7. Assume that A is sectorial of type x < 0 and angle h satisfying 0 6 h < pð1  a2Þ; 1 < a < 2. Assume   that 1 f : R  X ! X is Sp-pseudo almost periodic in t 2 R uniformly in x 2 X such that (2.9) holds with Cðh; aÞMjxja pL < asin pa , then Eq. (1.1) has a unique pseudo-almost periodic mild solution. Proof. Consider the nonlinear operator G given by (3.15). Using Theorems 4.6 and 2.1 one can easily see that Gu 2 PAP(X) whenever u 2 PAP(X) # PAPSp(X). Thus G maps PAP(X) into itself. To complete the proof, one has to prove that G has a unique fixed-point. Now we get 1

kGu  Gv k1 6

Cðh; aÞMLjxj a p   ku  v k1 ; a sin pa 1

and hence G has unique fixed-point u whenever Cðh; aÞMLjxj a p < asinðpaÞ. Note that u is the only pseudo-almost periodic solution to Eq. (1.1). This completes the proof. h 4.2. Integro-differential equations In this subsection we study the existence of pseudo almost periodic solutions to Eq. (2.10) in the case when the semilinear forcing term is Stepanov-like pseudo-almost periodic. Theorem 4.8. Assume that A generates an (1, 1 + (1⁄a))-regularized family {S(t)}tP0 on the Banach space X such that (3.16) holds. If f is Sp-pseudo almost periodic for p > 1, then the Eq. (2.10) has a unique pseudo-almost periodic solution. Proof. Let f = H + W, where H 2 APSp(X) and W 2 BCðR; XÞ such that (2.8) holds. For each m 2 N and t 2 R, define a function R mþ1 R mþ1 fm(t) as follows: fm ðtÞ ¼ Y m ðtÞ þ X m ðtÞ, where Y m ðtÞ ¼ m SðsÞHðt  sÞds and X m ðtÞ ¼ m SðsÞWðt  sÞds. We note that Y m 2 APðXÞ. On the other hand, from the estimates

kY m ðtÞk 6 /ðmÞkHkSp ;

t 2 R;

kX m ðtÞk 6 /ðmÞkWkSp ;

t 2 R;

P1

P1

we deduce that m¼0 Y m ðtÞ and From the estimate

kX m ðtÞk 6 /ð0Þ

Z

tm

m¼0 X m ðtÞ

are uniformly convergent.

!1p p

kWðsÞk ds

;

tðmþ1Þ

we have that X m 2 P 0 ðXÞ. Thus we conclude that the solution uðtÞ ¼

P1

m¼0 fm ðtÞ

of (2.10) belongs to PAP(X).

h

Theorem 4.9. Assume that A generates an (1, 1 + (1⁄a))-regularized family {S(t)}tP0 on the Banach space X such that (3.16) holds. Assume that f : R  X ! X is Sp-pseudo almost periodic in t 2 R uniformly in x 2 X such that (2.9) holds with /0L < 1, where /0 is given in Theorem 3.4. Then Eq. (1.2) has a unique pseudo-almost periodic mild solution. Acknowledgments The first author thanks the Department of Mathematics of Universidad de La Frontera, where this project was started. We are grateful to the referees for their valuable comments and suggestions, which enabled us to improve this work. The first author is partially supported by CNPQ/Brazil. References [1] R.P. Agarwal, B. de Andrade, C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. RWA 11 (2010) 3532–3554. [2] R.P. Agarwal, B. de Andrade, C. Cuevas, On type of periodicity and ergodicity to a class of integral equations with infinite delay, J. Nonlinear Convex Anal. 11 (2) (2010) 309–335. [3] R.P. Agarwal, B. de Andrade, C. Cuevas, On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. Differ. Equ. (2010) 1–25. Article ID 179750. [4] L. Amerio, G. Prouse, Almost-periodic Functions and Functional Equations, Van Nostrand Reinhold Co., 1971.

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