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Asymptotically periodic solutions of semilinear fractional integro-differential equations

Abstract

In this paper, we study the existence of an -asymptotically ω-periodic mild solution of semilinear fractional integro-differential equations in Banach space, where the nonlinear perturbation is -asymptotically ω-periodic or -asymptotically ω-periodic in the Stepanov sense. A fixed point theorem and the nonlinear Leray-Schauder alternative theorem are the main tools in carrying out our proof. Some examples are given to show the efficiency and usefulness of the main findings.

MSC:65R05, 35B40.

1 Introduction

The study of the existence of periodic solutions is one of the most interesting and important topics in the qualitative theory of differential equations, due to its mathematical interest as well as their applications in physics, control theory, mathematical biology, among other areas. Some contributions on the existence of periodic solutions for differential equations have been made. Mostly, the environmental change in the real word is not periodic, but approximately periodic. For this reason, in the past decades many authors studied several extensions of the concept of periodicity, such as asymptotic periodicity, almost periodicity, almost automorphy, pseudo almost periodicity, pseudo almost automorphy, etc. and the same concept in the Stepanov sense, one can see [14] for more details.

The notion of -asymptotic ω-periodicity, introduced by Henríquez et al. in [5, 6], is related to and more general than that of asymptotic periodicity. Since then, it has attracted the attention of many researchers [713]. Recently, in [14], the concept of -asymptotic ω-periodicity in the Stepanov sense, which generalizes the notion of -asymptotic ω-periodicity, was introduced and the applications to semilinear first-order abstract differential equations were studied.

Due to their numerous applications in several branches of science, fractional integro-differential equations have received much attention in recent years [1519]. The properties of solutions of fractional integro-differential equations have been studied from a different point of view, e.g., maximal regularity [17], positivity and contractivity [20], asymptotic equivalence [21], asymptotic periodicity [2225], almost periodicity [26, 27], almost automorphy [28, 29] and so on. To the best of our knowledge, there is no work reported in literature on -asymptotic ω-periodicity for fractional integro-differential equations if the nonlinear perturbation is -asymptotically ω-periodic in the Stepanov sense. This is one of the key motivations of this study.

The paper is organized as follows. In Section 2, some notations and preliminary results are presented. Section 3 is divided into two parts. In the first one, Section 3.1, we investigate the existence and uniqueness of an -asymptotically ω-periodic mild solution of semilinear fraction integro-differential equations when the nonlinear perturbation f satisfies the Lipschitz condition. In the second part, Section 3.2, when f is a non-Lipschitz case, we explore the properties of solutions for the same equation. In Section 4, we provide some examples to illustrate the main results.

2 Preliminaries and basic results

Let (X,), (Y, Y ) be two Banach spaces and , , R + , and stand for the set of natural numbers, real numbers, nonnegative real numbers, and complex numbers, respectively. In order to facilitate the discussion below, we further introduce the following notations:

  • BC( R + ,X) (resp. BC( R + ×Y,X)): the Banach space of bounded continuous functions from R + to X (resp. from R + ×Y to X) with the supremum norm.

  • C( R + ,X) (resp. C( R + ×Y,X)): the set of continuous functions from R + to X (resp. from R + ×Y to X).

  • L(X,Y): the Banach space of bounded linear operators from X to Y endowed with the operator topology. In particular, we write L(X) when X=Y.

  • L p ( R + ,X): the space of all classes of equivalence (with respect to the equality almost everywhere on R + ) of measurable functions f:RX such that f L p ( R + , R + ).

  • L l o c p ( R + ,X): stand for the space of all classes of equivalence of measurable functions f: R + X such that the restriction of f to every bounded subinterval of R + is in L p ( R + ,X).

2.1 Sectorial operators and Riemann-Liouville fractional derivative

Definition 2.1 [30]

A closed and densely defined linear operator A is said to be sectorial of type ω ˜ if there exist 0<θ<π/2, M>0, and ω ˜ R such that its resolvent exists outside the sector

ω ˜ + S θ : = { ω ˜ + λ : λ C , | arg ( λ ) | < θ } , ( λ I A ) 1 M | λ ω ˜ | , λ ω ˜ + S θ .

The sectorial operators are well studied in the literature, we refer to [30] for more details.

Definition 2.2 [31]

Let A be a closed and linear operator with domain D(A) defined on a Banach space X. We call A the generator of a solution operator if there exist ω ˜ R and a strong continuous function S α : R + L(X) such that { λ α :Reλ> ω ˜ }ρ(A) and

λ α 1 ( λ α A ) 1 x= 0 e λ t S α (t)xdt,Reλ> ω ˜ ,xX.

In this case, S α (t) is called the solution operator generated by A.

Note that if A is sectorial of type ω ˜ with 0<θ<π(1α/2), then A is the generator of a solution operator given by

S α (t):= 1 2 π i γ e λ t λ α 1 ( λ α A ) 1 dλ,

where γ is a suitable path lying outside the sector ω ˜ + S θ [32]. Recently, Cuesta [32] proved that if A is a sectorial operator of type ω ˜ <0 for some 0<θ<π(1α/2) (1<α<2), M>0, then there exists a constant C>0 such that

S α ( t ) CM 1 + | ω ˜ | t α ,t0.
(2.1)

Note that

0 1 1 + | ω ˜ | t α dt= | ω ˜ | 1 / α π α sin ( π / α )

for 1<α<2, therefore S α (t) is integrable on (0,).

In the rest of this subsection, we list some necessary basic definitions in the theory of fractional calculus.

Definition 2.3 [19]

The fractional order integral of order α>0 with the low limit t 0 >0 for a function f is defined as

I α f(t)= 1 Γ ( α ) t 0 t ( t s ) α 1 f(s)ds,t> t 0 ,α>0,

provided the right-hand side is pointwise defined on [ t 0 ,), where Γ is the gamma function.

Definition 2.4 [19]

Riemann-Liouville derivative of order α>0 with the low limit t 0 >0 for a function f:[ t 0 ,)R can be written as

D t α f(t)= 1 Γ ( n α ) d n d t n t 0 t ( t s ) n α 1 f(s)ds,t> t 0 ,n1<α<n.

2.2 Compactness criterion and fixed point theorem

First, we recall two useful compactness criteria.

Let h:[0,)[1,) be a continuous nondecreasing function such that h(t) as t. Define

C h ( R + , X ) := { u C ( R + , X ) : lim t ( u ( t ) / h ( t ) ) = 0 }

endowed with the norm u h = sup t 0 (u(t)/h(t)).

Lemma 2.1 [33]

A set K C h ( R + ,X) is relatively compact in C h ( R + ,X) if it verifies the following conditions:

(c1) For all b>0, the set K b (t):={u | [ 0 , b ] :uK} is relatively compact in C([0,b],X).

(c2) lim t (u(t)/h(t))=0 uniformly for uK.

Lemma 2.2 (Simon’s theorem [34])

Let F L p ([0,T],X), F is relatively compact in L p ([0,T],X) for 1p< if and only if

  1. (1)

    { t 1 t 2 f(t)dt:fF,0< t 1 < t 2 <T} is relatively compact in X.

  2. (2)

    τ h f f L p ( [ 0 , T h ] , X ) 0 as h0 uniformly for fF, where ( τ h f)(t)=f(t+h).

Now, we recall the so-called Zima’s fixed point theorem [35] and the Leray-Schauder alternative theorem [36] which will be used in the sequel.

Let (Y, Y ,,m) denote a Banach space of elements yY with a binary relation ‘’ and a mapping m:YY such that

  1. (i)

    the relation is transitive;

  2. (ii)

    0m(u) and m ( u ) Y = u Y for all uY;

  3. (iii)

    the norm Y is monotonic, that is, if 0uv, then u Y v Y for all u,vY.

Theorem 2.1 ([35] Zima’s fixed point theorem)

In the Banach space considered above, let the operators Γ:YY and B:YY be given with the following properties:

  1. (iv)

    B is a bounded linear operator with spectral radius r(B)<1.

  2. (v)

    B is increasing, that is, if 0uv, then BuBv for all u,vY.

  3. (vi)

    m(ΓuΓv)Bm(uv) for all u,vY.

Then the equation Γu=u has a unique solution in Y.

Theorem 2.2 ([36] Leray-Schauder alternative theorem)

Let D be a closed convex subset of a Banach space X such that 0D. Let F:DD be a completely continuous map. Then the set {xD:x=λF(x),0<λ<1} is unbounded or the map F has a fixed point in D.

2.3 -Asymptotic ω-periodicity in the Stepanov sense

For ω>0, define

C 0 ( R + , X ) = { x BC ( R + , X ) : lim t x ( t ) = 0 } . C ω ( R + , X ) = { x BC ( R + , X ) : x  is  ω -periodic } .

Definition 2.5 [37]

A function fBC( R + ,X) is called asymptotically ω-periodic if there exist g C ω ( R + ,X), φ C 0 ( R + ,X) such that f=g+φ. The collection of those functions is denoted by A P ω ( R + ,X).

Definition 2.6 [5]

A function fBC( R + ,X) is said to be -asymptotically periodic if there exists ω>0 such that lim t (f(t+ω)f(t))=0. In this case, we say that f is -asymptotically ω-periodic. The collection of those functions is denoted by SA P ω ( R + ,X).

Definition 2.7 [5]

A continuous function f: R + ×XX is said to be uniformly -asymptotically ω-periodic on bounded sets if for every bounded set K of X, the set {f(t,x):t0,xK} is bounded and lim t (f(t+ω,x)f(t,x))=0 uniformly in xK. Denote by SA P ω ( R + ×X,X) the set of such functions.

Definition 2.8 [5]

A continuous function f: R + ×XX is said to be asymptotically uniformly continuous on bounded sets if for every ε>0 and every bounded set KX, there exist L ε 0 and δ ε >0 such that f(t,x)f(t,y)ε for all t t ε and all x,yK with xy δ ε .

We introduce the following composition theorem for an -asymptotically ω-periodic function.

Lemma 2.3 [5]

Assume that fSA P ω ( R + ×X,X) is an asymptotically uniformly continuous on bounded sets function. Let uSA P ω ( R + ,X), then υ()=f(,u())SA P ω ( R + ,X).

Let p[1,). The space B S p (R,X) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f:RX such that f b L (R, L p ([0,1];X)), where f b is the Bochner transform of f defined by f b (t,s):=f(t+s), tR, s[0,1]. B S p (R,X) is a Banach space with the norm [38]

f S p = f b L ( R , L p ) = sup t R ( t t + 1 f ( τ ) p d τ ) 1 / p .

It is obvious that L p (R,X)B S p (R,X) L l o c p (R,X) and B S p (R,X)B S q (R,X) for pq1. We denote by B S 0 p (R,X) the subspace of B S p (R,X) consisting of functions f such that t t + 1 f ( s ) p ds0 as t.

Definition 2.9 [14]

A function fB S p ( R + ,X) is called -asymptotically ω-periodic in the Stepanov sense (or S p --asymptotically ω-periodic) if

lim t t t + 1 f ( s + ω ) f ( s ) p ds=0.

Denote by S p SA P ω ( R + ,X) the set of such functions.

It is easy to see that

C 0 ( R + , X ) A P ω ( R + , X ) SA P ω ( R + , X ) S p SA P ω ( R + , X ) .

Definition 2.10 [14]

A function f: R + ×XX is said to be uniformly -asymptotically ω-periodic on bounded sets in the Stepanov sense if for every bounded set BX, there exist positive functions g B B S p ( R + ,R) and h B B S 0 p ( R + ,R) such that f(t,x) g B (t) for all t R + , xB and

f ( t + ω , x ) f ( t , x ) h B (s)for all s0,xB.

Denote by S p SA P ω ( R + ×X,X) the set of such functions.

Definition 2.11 [14]

A function f: R + ×XX is said to be asymptotically uniformly continuous on bounded sets in the Stepanov sense if for every ε>0 and every bounded set BX, there exist t ε 0 and δ ε >0 such that

t t + 1 f ( s , x ) f ( s , y ) p ds ε p

for all t t ε and all x,yB with xy δ ε .

Lemma 2.4 [14]

Assume that f S p SA P ω ( R + ×X,X) is an asymptotically uniformly continuous on bounded sets in the Stepanov sense function. Let uSA P ω ( R + ,X), then υ()=f(,u()) S p SA P ω ( R + ,X).

Lemma 2.5 Let { S ( t ) } t 0 L(X) be a strongly continuous family of bounded and linear operators such that S(t)ϕ(t), t R + , where ϕ L 1 ( R + ) is nonincreasing. If f S p SA P ω ( R + ,X), then

(Λf)(t):= 0 t S(ts)f(s)dsSA P ω ( R + , X ) ,t R + .

Proof For ntn+1, nN, one has

( Λ f ) ( t ) = 0 t S ( s ) f ( t s ) d s 0 n + 1 ϕ ( s ) f ( t s ) d s = k = 0 n k k + 1 ϕ ( s ) f ( t s ) d s k = 0 n ϕ ( k ) ( k k + 1 f ( t s ) p d s ) 1 / p ( ϕ ( 0 ) + ϕ ( 1 ) + + ϕ ( n ) ) f S p ( ϕ ( 0 ) + 0 1 ϕ ( t ) d t + + n 1 n ϕ ( t ) d t ) f S p ( ϕ ( 0 ) + ϕ L 1 ) f S p ,

that is, Λf is bounded. It is clear that Λf is continuous for each t R + , whence ΛfBC( R + ,X). Moreover, note that

( Λ f ) ( t + ω ) ( Λ f ) ( t ) = 0 t + ω S ( t + ω s ) f ( s ) d s 0 t S ( t s ) f ( s ) d s = 0 ω S ( t + ω s ) f ( s ) d s + ω t + ω S ( t + ω s ) f ( s ) d s 0 t S ( t s ) f ( s ) d s = 0 ω S ( t + ω s ) f ( s ) d s + 0 t S ( t s ) [ f ( s + ω ) f ( s ) ] d s , : = I ( t ) + J ( t ) ,

where

I(t)= 0 ω S(t+ωs)f(s)ds,J(t)= 0 t S(ts) [ f ( s + ω ) f ( s ) ] ds.

By the hypothesis of ϕ, one has

I ( t ) 0 ω ϕ(t+ωs) f ( s ) dsϕ(t) 0 ω f ( s ) ds0,t,

then

lim t I ( t ) dt=0.

On the other hand, since f S p SA P ω ( R + ,X), there exists mN such that

( t t + 1 f ( s + ω ) f ( s ) p d s ) 1 / p <εfor tm.

For mntn+1, one has

J ( t ) 0 t S ( t s ) f ( s + ω ) f ( s ) d s 0 n ϕ ( t s ) f ( s + ω ) f ( s ) d s + n t ϕ ( t s ) f ( s + ω ) f ( s ) d s 0 n ϕ ( n s ) f ( s + ω ) f ( s ) d s + ϕ ( 0 ) n t f ( s + ω ) f ( s ) d s k = 0 n 1 k k + 1 ϕ ( n s ) f ( s + ω ) f ( s ) d s + ϕ ( 0 ) n n + 1 f ( s + ω ) f ( s ) d s k = 0 n 1 ϕ ( n k 1 ) k k + 1 f ( s + ω ) f ( s ) d s + ϕ ( 0 ) n n + 1 f ( s + ω ) f ( s ) d s k = 0 n 1 ϕ ( n k 1 ) ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ϕ ( 0 ) ( n n + 1 f ( s + ω ) f ( s ) p d s ) 1 / p = k = 0 m ϕ ( n k 1 ) ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ϕ ( 0 ) ( n n + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + k = m + 1 n 1 ϕ ( n k 1 ) ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p ( ϕ ( n 1 ) + ϕ ( n 2 ) + + ϕ ( n m 1 ) ) max 0 k m ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ϕ ( 0 ) ( n n + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ( ϕ ( n m 2 ) + ϕ ( n m 3 ) + + ϕ ( 0 ) ) ε n m 2 n 1 ϕ ( t ) d t max 0 k m ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ϕ ( 0 ) ( n n + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ( ϕ ( 0 ) + 0 n m 2 ϕ ( t ) d t ) ε n m 2 n 1 ϕ ( t ) d t max 0 k m ( k k + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ϕ ( 0 ) ( n n + 1 f ( s + ω ) f ( s ) p d s ) 1 / p + ( ϕ ( 0 ) + ϕ L 1 ) ε ,

which implies that J(t)0 as t. So

lim t ( Λ f ) ( t + ω ) ( Λ f ) ( t ) =0.

The proof is complete. □

3 Semilinear fractional integro-differential equation

Consider the semilinear fractional integro-differential equation

{ u ( t ) = 0 t ( t s ) α 2 Γ ( α 1 ) Au ( s ) d s + f ( t , u ( t ) ) , t R + , u ( 0 ) = u 0 X ,
(3.1)

where 1<α<2, A:D(A)XX is a linear densely defined operator of sectorial type on a complex Banach space X and f: R + ×XX is an appropriate function.

Before starting our main results, we recall the definition of the mild solution to (3.1).

Definition 3.1 [23]

Assume that A generates a solution operator S α (t). A function uBC( R + ,X) is called a mild solution of (3.1) if

u(t)= S α (t) u 0 + 0 t S α (ts)f ( s , u ( s ) ) ds,t R + .

To study (3.1), we require the following assumptions:

(H1) A is a sectorial operator of type ω ˜ <0 with 0<θ<π(1α/2).

(H2) fSA P ω ( R + ×X,X).

( H 2 ) f S p SA P ω ( R + ×X,X), p1.

(H31) f satisfies the Lipschitz condition

f ( t , u ) f ( t , v ) L f uv,u,vX,t R + .

(H32) f satisfies the Lipschitz condition

f ( t , u ) f ( t , v ) L f (t)uv,u,vX,t R + ,

where L f B S p ( R + , R + ).

(H33) f satisfies the Lipschitz condition

f ( t , u ) f ( t , v ) L f (t)uv,u,vX,t R + ,

where L f B S 0 p ( R + , R + ).

(H4) f is asymptotically uniformly continuous on bounded sets.

( H 4 ) f is asymptotically uniformly continuous on bounded sets in the Stepanov sense.

3.1 Lipschitz case

In this subsection, we study the existence and uniqueness of -asymptotically ω-periodic mild solution of (3.1) when f satisfies the Lipschitz condition.

If f(t,u) is uniformly Lipschitz continuous at u, i.e., (H31) holds, we reach the following claim.

Theorem 3.1 Assume that (H1), (H2) (or ( H 2 )), (H31) hold, then (3.1) has a unique mild solution u(t)SA P ω ( R + ,X) if CM| ω ˜ | 1 / α π L f <αsin(π/α).

Proof Define the operator F:SA P ω ( R + ,X)SA P ω ( R + ,X) by

(Fu)(t)= S α (t) u 0 + 0 t S α (ts)f ( s , u ( s ) ) ds,t R + .
(3.2)

By (2.1), one has lim t S α (t) u 0 =0, so S α (t) u 0 C 0 ( R + ,X)SA P ω ( R + ,X). By (H31), if (H2) holds, f(,u())SA P ω ( R + ,X) S p SA P ω ( R + ,X) by Lemma 2.3, and if ( H 2 ) holds, f(,u()) S p SA P ω ( R + ,X) by Lemma 2.4. Hence is well defined by Lemma 2.5.

Moreover, let u,vSA P ω ( R + ,X), one has

( F u ) ( t ) ( F v ) ( t ) 0 t S α ( t s ) f ( s , u ( s ) ) f ( s , v ( s ) ) d s L f 0 t S α ( t s ) u ( s ) v ( s ) d s L f u v 0 t S α ( s ) d s L f u v 0 t CM 1 + | ω ˜ | s α d s CM | ω ˜ | 1 / α π L f α sin ( π / α ) u v ,

by the Banach contraction mapping principle, has a unique fixed point in SA P ω ( R + ,X), which is the unique SA P ω mild solution to (3.1). □

Theorem 3.2 Assume that (H1), (H2) (or ( H 2 )), (H32) hold and

CM ( 1 + | ω ˜ | 1 / α π α sin ( π / α ) ) L f S p <1,
(3.3)

then (3.1) has a unique mild solution u(t)SA P ω ( R + ,X).

Proof Define the operator as in (3.2). If (H2) holds, then fSA P ω ( R + ×X,X) S p SA P ω ( R + ×X,X). Since (H32) holds, f is asymptotically uniformly continuous on bounded sets in the Stepanov sense, so f(,u()) S p SA P ω ( R + ,X) by Lemma 2.4. If ( H 2 ) holds, f(,u()) S p SA P ω ( R + ,X) by Lemma 2.4. Hence is well defined by Lemma 2.5.

For u,vSA P ω ( R + ,X), one has

( F u ) ( t ) ( F v ) ( t ) 0 t S α ( t s ) f ( s , u ( s ) ) f ( s , v ( s ) ) d s 0 t CM 1 + | ω ˜ | ( t s ) α L f ( s ) d s u v .
  • If t=mN, in this case

    0 t 1 1 + | ω ˜ | ( t s ) α L f ( s ) d s = 0 m 1 1 + | ω ˜ | ( m s ) α L f ( s ) d s = k = 0 m 1 k k + 1 1 1 + | ω ˜ | ( m s ) α L f ( s ) d s k = 0 m 1 1 1 + | ω ˜ | ( m k 1 ) α k k + 1 L f ( s ) d s k = 0 m 1 1 1 + | ω ˜ | ( m k 1 ) α ( k k + 1 L f ( s ) p d s ) 1 / p [ 1 + ( 0 1 + 1 2 + + m 2 m 1 ) 1 1 + | ω ˜ | t α d t ] L f S p ( 1 + 0 1 1 + | ω ˜ | t α d t ) L f S p ( 1 + | ω ˜ | 1 / α π α sin ( π / α ) ) L f S p .
    (3.4)
  • If t=mh, where 0<h<1. In this general case,

    0 t 1 1 + | ω ˜ | ( t s ) α L f ( s ) d s = 0 m h 1 1 + | ω ˜ | ( m h s ) α L f ( s ) d s = h m 1 1 + | ω ˜ | ( m s ) α L f ( s h ) d s = 0 m 1 1 + | ω ˜ | ( m s ) α L ˜ f ( s ) d s ( 1 + | ω ˜ | 1 / α π α sin ( π / α ) ) L ˜ f S p ,

where L ˜ f is defined by

L ˜ f (s)= { 0 , 0 s < h , L f ( s h ) , s h ,

then L ˜ f S p = L f S p . So we infer that

0 t 1 1 + | ω ˜ | ( t s ) α L f (s)ds ( 1 + | ω ˜ | 1 / α π α sin ( π / α ) ) L f S p .
(3.5)

By (3.4), (3.5), one has

( F u ) ( t ) ( F v ) ( t ) CM ( 1 + | ω ˜ | 1 / α π α sin ( π / α ) ) L f S p uv.
(3.6)

By the Banach contraction mapping principle, has a unique fixed point in SA P ω ( R + ,X), which is the unique SA P ω mild solution to (3.1). □

In next results, we relax condition (3.3) to study the existence and uniqueness of SA P ω mild solution of (3.1).

Theorem 3.3 Assume that (H1), (H2) (or ( H 2 )), (H32) hold and the integral 0 t L f (s)ds exists for all t R + . Then (3.1) has a unique mild solution u(t)SA P ω ( R + ,X).

Proof Define an equivalent norm on SA P ω ( R + ,X) as f c = sup t R + { e c λ ( t ) f}, where c>MC and λ(t)= 0 t L f (τ)dτ. Define the operator as in (3.2). Let u,vSA P ω ( R + ,X), one has

( F u ) ( t ) ( F v ) ( t ) 0 t S α ( t s ) f ( s , u ( s ) ) f ( s , v ( s ) ) d s 0 t CM 1 + | ω ˜ | ( t s ) α L f ( s ) u ( s ) v ( s ) d s CM u v c 0 t L f ( s ) e c λ ( s ) d s = CM u v c 0 t λ ( s ) e c λ ( s ) d s CM c u v c e c λ ( t ) ,

consequently,

F u F v c CM c u v c .

Since c>MC, is a contraction and then it has a unique fixed point u(t), which is the unique SA P ω mild solution to (3.1). □

Theorem 3.4 Assume that (H1), (H2) (or ( H 2 )), (H33) hold, then (3.1) has a unique mild solution u(t)SA P ω ( R + ,X).

Proof Define the operator as in (3.2), then is a map from SA P ω ( R + ,X) into SA P ω ( R + ,X). Moreover, is continuous by (3.6). Define the map B on BC( R + ,R) by

(Bα)(t)=CM 0 t 1 1 + | ω ˜ | ( t s ) α L f (s)α(s)ds,t R + .
(3.7)

It is clear that B is a bounded linear operator from BC( R + ,R) into BC( R + ,R).

First, we will show that B is a compact operator. For each a0 and each αBC( R + ,R) with α1, define the functions

w 1 (α)(t)= { CM 0 t 1 1 + | ω ˜ | ( t s ) α L f ( s ) α ( s ) d s , 0 t a , CM 0 a 1 1 + | ω ˜ | ( t s ) α L f ( s ) α ( s ) d s , t a ,

and

w 2 (α)(t)= { 0 , 0 t a , CM a t 1 1 + | ω ˜ | ( t s ) α L f ( s ) α ( s ) d s , t a .

It follows from the Ascoli-Arzelá theorem in the space C 0 ( R + ,R) that the set K a ={ w 1 (α):α1} is relatively compact in C 0 ( R + ,R), and therefore in BC( R + ,R).

Since L f B S 0 p ( R + ,R), for each ε>0, take a0 such that for ta,

sup r a ( r r + 1 L f ( s ) p d s ) 1 / p <ε.

For a+mt<a+m+1, mN, one has

| w 2 ( α ) ( t ) | = CM a a + m 1 1 + | ω ˜ | ( t s ) α L f ( s ) | α ( s ) | d s + CM a + m t 1 1 + | ω ˜ | ( t s ) α L f ( s ) | α ( s ) | d s CM k = 0 m 1 a + k a + k + 1 1 1 + | ω ˜ | ( t s ) α L f ( s ) | α ( s ) | d s + CM a + m t L f ( s ) | α ( s ) | d s CM k = 0 m 1 a + k a + k + 1 1 1 + | ω ˜ | ( a + m s ) α L f ( s ) d s + CM a + m a + m + 1 L f ( s ) d s CM k = 0 m 1 1 1 + | ω ˜ | ( m k 1 ) α a + k a + k + 1 L f ( s ) d s + CM a + m a + m + 1 L f ( s ) d s CM k = 0 m 1 1 1 + | ω ˜ | ( m k 1 ) α ( a + k a + k + 1 L f ( s ) p d s ) 1 / p + CM ( a + m a + m + 1 L f ( s ) p d s ) 1 / p CM 0 m 1 1 1 + | ω ˜ | t α d t sup r a ( r r + 1 L f ( s ) p d s ) 1 / p + CM ( a + m a + m + 1 L f ( s ) p d s ) 1 / p CM | ω ˜ | 1 / α π α sin ( π / α ) sup r a ( r r + 1 L f ( s ) p d s ) 1 / p + CM ( a + m a + m + 1 L f ( s ) p d s ) 1 / p ,

then | w 2 (α)t|ε. Since Bα(t)= w 1 (α)(t)+ w 2 (α)(t) for t R + , one has

{ B ( α ) : α 1 } K a + { φ : φ BC ( R + , R ) , φ ε } ,

which implies that {B(α):α1} is relatively compact, so B is a compact operator. Moreover, it follows from the Gronwall-Bellman lemma that the point spectrum σ p (B)={0}, which implies that the spectral radius of B is equal to zero since B is a compact operator.

Consider the Banach space Y=BC( R + ,R) equipped with both the relation and the mapping m:BC( R + ,R)BC( R + ,R) defined by: if u,vBC( R + ,R)

uvif and only if u ( t ) v ( t ) t R + ,

and (m(u))(t)= sup 0 s t u(s). It is easy to check that conditions (i), (ii), (iii) are satisfied.

Let u,vBC( R + ,R), one has

( F u ) ( t ) ( F v ) ( t ) CM 0 t 1 1 + | ω ˜ | ( t s ) α L f (s) u ( s ) v ( s ) ds,

hence m(F(u)F(v))Bm(uv), and B is increasing with spectral radius r(B)<1. By Theorem 2.1, has a unique fixed point in BC( R + ,R), which is the unique SA P ω mild solution to (3.1). □

3.2 Non-Lipschitz case

In this subsection, we study the existence of -asymptotically ω-periodic mild solution of (3.1) when f does not satisfy the Lipschitz condition.

The following existence result is based upon the nonlinear Leray-Schauder alternative theorem.

Theorem 3.5 Assume that (H1), (H2), (H4) hold (or (H1), ( H 2 ), ( H 4 ) hold) and satisfy the following conditions:

(A1) There exists a continuous nondecreasing function W:[0,+)[0,+) such that f(t,u)W(u) for all t R + , uX.

(A2) For each ν>0, lim t 1 h ( t ) 0 t W ( ν h ( s ) ) 1 + | ω ˜ | ( t s ) α ds=0.

(A3) For each ε>0, there exists δ>0 such that for u,v C h ( R + ,X), u v h δ implies that

0 t f ( s , u ( s ) ) f ( s , v ( s ) ) 1 + | ω ˜ | ( t s ) α dsεfor all t R + .

(A4) For all a,b R + , ab and r0, the set {f(s,u):asb,uX,ur} is relatively compact in X.

(A5) lim inf ξ ξ β ( ξ ) >1, where β(ν)= σ ν h and

σ ν (t):= S α ( ) u 0 +CM 0 t W ( ν h ( s ) ) 1 + | ω ˜ | ( t s ) α ds,t0,

C, M are constants given in (2.1).

Then (3.1) has a mild solution u(t)SA P ω ( R + ,X).

Proof Define Γ: C h ( R + ,X)C( R + ,X) by

(Γu)(t)= S α (t) u 0 + 0 t S α (ts)f ( s , u ( s ) ) ds,t R + .

Next, we prove that Γ has a fixed point in SA P ω ( R + ,X). We divide the proof into several steps.

  1. (i)

    For x C h ( R + ,X), by (A1), one has

    Γ u ( t ) h ( t ) CM h ( t ) u 0 + CM h ( t ) 0 t f ( s , u ) 1 + | ω ˜ | ( t s ) α d s CM h ( t ) u 0 + CM h ( t ) 0 t W ( u h h ( s ) ) 1 + | ω ˜ | ( t s ) α d s .

It follows from (A2) that Γ: C h ( R + ,X) C h ( R + ,X).

  1. (ii)

    Γ is continuous. In fact, for each ε>0, by (A3), there exits δ>0, for u,v C h ( R + ,X) and u v h δ, one has

    Γ u Γ v 0 t S α ( t s ) f ( s , u ( s ) ) f ( s , v ( s ) ) d s CM 0 t f ( s , u ( s ) ) f ( s , v ( s ) ) 1 + | ω ˜ | ( t s ) α d s .

Take into account that h(t)1, by (A3)

Γ u Γ v h ( t ) CMε,

which implies that Γ u Γ v h CMε, so Γ is continuous.

  1. (iii)

    Γ is completely continuous. Set B r (Z) for the closed ball with center at 0 and radius r in the space Z. Let V=Γ( B r ( C h ( R + ,X))) and v=Γ(u) for u B r ( C h ( R + ,X)).

Initially, we prove that V b (t) is a relatively compact subset of X for each t[0,b], here V b (t)={v(t),vV,t[0,b]}. Since

v(t)= S α (t) u 0 + 0 t S α (s)f ( t s , u ( t s ) ) ds S α (t) u 0 +t c ( K ) ¯ ,

where c(K) denotes the convex hull of K and K={ S α (s)f(ξ,u):0st,0ξt,ur}. Using the fact that S α () is strong continuous and (A4), we infer that K is a relatively compact set, and V b (t) S α (t) u 0 +t c ( K ) ¯ is also a relatively compact set.

Next, we show that V b is equicontinuous. In fact,

v ( t + s ) v ( t ) = ( S α ( t + s ) S α ( t ) ) u 0 + t t + s S α ( t + s ξ ) f ( ξ , u ( ξ ) ) d ξ + 0 t ( S α ( ξ + s ) S α ( ξ ) ) f ( t ξ , u ( t ξ ) ) d ξ .

For each ε>0, we can choose δ 1 >0 such that

t t + s S α ( t + s ξ ) f ( ξ , u ( ξ ) ) d ξ CM t t + s W ( r h ( ξ ) ) 1 + | ω ˜ | ( t + s ξ ) α dξ ε 3 for s δ 1 .

Moreover, since {f(tξ,u(tξ)):0ξt,u B r ( C h ( R + ,X))} is a relatively compact set and S α () is strong continuous, we can choose δ 2 >0, δ 3 >0 such that

( S α ( t + s ) S α ( t ) ) u 0 ε 3 for s δ 2

and

( S α ( ξ + s ) S α ( ξ ) ) f ( t ξ , u ( t ξ ) ) ε 3 ( t + 1 ) for s δ 3 .

So, v(t+s)v(t)ε for |s|min{ δ 1 , δ 2 , δ 3 } with t+s0 and for all u B r ( C h ( R + ,X)).

Finally, by (A2), one has

v ( t ) h ( t ) CM h ( t ) u 0 + CM h ( t ) 0 t W ( r h ( s ) ) 1 + | ω ˜ | ( t s ) α ds0,t,

and this convergence is independent of u B r ( C h ( R + ,X)). Hence V satisfies (c1), (c2) of Lemma 2.1, which completes the proof that V is a relatively compact set in C h ( R + ,X).

  1. (iv)

    If u λ is a solution of the equation u λ =λΓ( u λ ) for some 0<λ<1, then

    u λ = λ S α ( t ) u 0 + 0 t S α ( t s ) f ( s , u λ ) d s S α ( ) u 0 + CM 0 t W ( u λ h h ( s ) ) 1 + | ω ˜ | ( t s ) α d s β ( u λ h ) h ( t ) .

Hence, one has

u λ h β ( u λ h ) 1

and by (A5), we conclude that the set { u λ : u λ =λΓ( u λ ),λ(0,1)} is bounded.

  1. (v)

    If follows from Lemmas 2.3, 2.4 and 2.5 that Γ(SA P ω ( R + ,X))SA P ω ( R + ,X); consequently, we consider Γ: S A P ω ( R + , X ) ¯ S A P ω ( R + , X ) ¯ . Using (i)-(iii), we have that the map is completely continuous. By (iv) and Theorem 2.2, we deduce that Γ has a fixed point u S A P ω ( R + , X ) ¯ .

Let u n be a sequence in SA P ω ( R + ,X) such that it converges to u in the norm C h ( R + ,X). For ε>0, let δ>0 be the constant in (A3), there exists n 0 N such that u n u h δ for all n n 0 . For n n 0 ,

Γ u n Γ u 0 t S α ( t s ) f ( s , u n ( s ) ) f ( s , u ( s ) ) d s CM 0 t f ( s , u n ( s ) ) f ( s , u ( s ) ) 1 + | ω ˜ | ( t s ) α d s CM ε .

Hence, ( Γ u n ) n converges to Γu=u uniformly in [0,). This implies that uSA P ω ( R + ,X) and completes the proof. □

Corollary 3.1 Assume that (H1), (H2) (or ( H 2 )) hold and satisfy the following conditions:

  1. (a)

    f(t,0)=q(t).

  2. (b)

    f satisfies the Hölder-type condition

    f ( t , u ) f ( t , v ) C 1 u v α ,u,vX,t R + ,

where 0<α<1, C 1 >0 is a constant.

  1. (c)

    For all a,b R + , ab and r0, the set {f(s,u):asb,uX,ur} is relatively compact in X.

Then (3.1) has a mild solution u(t)SA P ω ( R + ,X).

Proof By (b), it is easy to see that (H4), ( H 4 ) hold. Let C 0 =q and W(ξ)= C 0 + C 1 ξ α , then (A1) is satisfied. Take a function h such that sup t R + 0 t h ( s ) α 1 + | ω ˜ | ( t s ) α ds:= C 2 <, it is not difficult to see that (A2) is satisfied. To verify (A3), note that for each ε>0, there exists 0<δ< ( ε C 1 C 2 ) 1 / α such that for every u,v C h ( R + ,X), u v h δ implies that

0 t f ( s , u ( s ) ) f ( s , v ( s ) ) 1 + | ω ˜ | ( t s ) α ds 0 t C 1 h ( s ) α u v h α 1 + | ω ˜ | ( t s ) α ds C 1 C 2 δ α εfor all t R + .

On the other hand, (A5) can be easily verified using the definition of W. By Theorem 3.5, (3.1) has a mild solution u(t)SA P ω ( R + ,X). □

4 Examples

In this section, we provide some examples to illustrate our main results.

Example 4.1 Consider the following fractional differential equation:

{ t α u ( t , x ) = x 2 u ( t , x ) μ u ( t , x ) + t α 1 ( 0 x η a ( t ) u ( t , ξ ) d ξ ) , t R + , x [ 0 , π ] , u ( t , 0 ) = u ( t , π ) = 0 , t 0 , u ( 0 , x ) = u 0 ( x ) , x [ 0 , π ] ,
(4.1)

where μ>0, u 0 L 2 [0,π], aSA P ω ( R + ,R). In what follows we consider X= L 2 [0,π] and let A be the operator given by

Au= u μu

with domain

D(A)= { u X , u X , u ( 0 ) = u ( π ) = 0 } .

It is well know that A is sectorial of type ω ˜ =μ<0 [30]. Equation (4.1) can be expressed as an abstract system of the form (3.1), where u(t)(x)=u(t,x) for t R + , x[0,π], and f(t,ϕ)(ξ)=ηa(t) 0 ξ ϕ(τ)dτ for t R + , ξ[0,π]. Moreover, one has

f ( t , ϕ ) L 2 π | η | | a ( t ) | ϕ L 2 , t 0 , ϕ X , f ( t + ω , ϕ ) f ( t , ϕ ) L 2 π | η | | a ( t + ω ) a ( t ) | ϕ L 2 , t 0 , ϕ X ,

since aSA P ω ( R + ,R), we deduce that fSA P ω ( R + ×X,X). From

f ( t , ϕ 1 ) f ( t , ϕ 2 ) L 2 π | η | | a ( t ) | ϕ 1 ϕ 2 L 2 π | η | a ϕ 1 ϕ 2 L 2 , t 0 , ϕ 1 , ϕ 2 X ,

so (H31) holds with L f =π|η|a. If |η| is small enough, (4.1) has a unique mild solution uSA P ω ( R + ,X) by Theorem 3.1.

Example 4.2 Consider the following fractional differential equation:

{ t α u ( t , x ) = x 2 u ( t , x ) μ u ( t , x ) + t α 1 F ( t , u ) ( x ) , t R + , x [ 0 , π ] , u ( t , 0 ) = u ( t , π ) = 0 , t 0 , u ( 0 , x ) = u 0 ( x ) , x [ 0 , π ] ,
(4.2)

where μ>0, u 0 L 2 [0,π], F(t,u)(x)= e λ t | 0 x u(t,τ)dτ | ϑ sinx, ϑ(0,1). Let X= L 2 [0,π], Au= u μu with domain D(A)={uX, u X,u(0)=u(π)=0}, so A is sectorial of type ω ˜ =μ<0. Equation (4.2) can be rewritten as the abstract form (3.1), where

f(t,ϕ)(ξ)= e λ t | 0 ξ ϕ(τ)dτ | ϑ sinξ,ϑ(0,1).

Moreover, one has

f ( t , ϕ ) L 2 e λ t π ϑ + 1 2 ϑ + 1 ϕ L 2 ϑ π ϑ + 1 2 ϑ + 1 ϕ L 2 ϑ ,t0,ϕX,
(4.3)
f ( t + ω , ϕ ) f ( t , ϕ ) L 2 ( e λ ( t + ω ) + e λ t ) π ϑ + 1 2 ϑ + 1 ϕ L 2 ϑ ,t0,ϕX,
(4.4)
f ( t , ϕ 1 ) f ( t , ϕ 2 ) L 2 e λ t π ϑ + 1 2 ϑ + 1 ϕ 1 ϕ 2 L 2 ϑ ,t0, ϕ 1 , ϕ 2 X,
(4.5)

so fSA P ω ( R + ×X,X) and f is asymptotically uniformly continuous on bounded sets by (4.5). By (4.3), we define W by W(ξ)= π ϑ + 1 2 ϑ + 1 ξ ϑ . Let h(t)= e λ t , λ>0, u,v C h ( R + ,X), one has

1 h ( t ) 0 t W ( ν h ( s ) ) 1 + | μ | ( t s ) α d s ν ϑ π ϑ + 3 2 | μ | 1 / α ϑ + 1 α sin ( π / α ) 1 e λ ( 1 ϑ ) t 0 , t , 0 t f ( s , u ( s ) ) f ( s , v ( s ) ) L 2 1 + | μ | ( t s ) α d s π ϑ + 3 2 | μ | 1 / α ϑ + 1 α sin ( π / α ) u v h ϑ .

Hence (A1)-(A3) hold.

Next, we prove that the set {f(s, e λ s ϕ):asb,ϕX, ϕ L 2 r} is relatively compact in L 2 [0,T] by Simon’s theorem. In fact, one has

f ( s , e λ s ϕ ) L 2 π ϑ + 1 2 r ϑ ϑ + 1 ,ϕ L 2 [0,π], ϕ L 2 r.

Hence, for a 1 < a 2 , a 1 a 2 f(s, e λ s ϕ)(ξ)dξ is bounded uniformly for asb and ϕ L 2 [0,π], ϕ L 2 r. On the other hand,

f ( s , e λ s ϕ ) ( ξ ) f ( s , e λ s ϕ ) ( ξ ) r ϑ / 2 |ξ ξ | ϑ / 2 + π ϑ / 2 r ϑ |ξ ξ |,

therefore,

0 π h |f ( s , e λ s ϕ ) (ξ+h)f ( s , e λ s ϕ ) (ξ) | 2 dξ0,as h0

uniformly for asb, ϕ L 2 [0,π], ϕ L 2 r. So (A4) holds by Lemma 2.2. It is not difficult to see that (A5) holds. Whence (4.2) has a mild solution uSA P ω ( R + ,X) by Theorem 3.5.

Author’s contributions

The author has made this manuscript independently. The author read and approved the final version.

References

  1. 1.

    Agarwal RP, Cuevas C, Soto H, El-Gebeily M: Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Anal. 2011, 74: 1769–1798. 10.1016/j.na.2010.10.051

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Hino Y, Naito T, Van Minh N, Son SJ: Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor and Francis, London; 2002.

    MATH  Google Scholar 

  3. 3.

    Lizama C, Cuevas C, Soto H: Asymptotic periodicity for strongly damped wave equations. Abstr. Appl. Anal. 2013., 2013: Article ID 308616

    Google Scholar 

  4. 4.

    Xia ZN, Fan M: Weighted Stepanov-like pseudo almost automorphy and applications. Nonlinear Anal. 2012, 75: 2378–2397. 10.1016/j.na.2011.10.036

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Henríquez HR, Pierri M, Táboas P: On -asymptotically ω -periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 2008, 343: 1119–1130. 10.1016/j.jmaa.2008.02.023

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Henríquez HR, Pierri M, Táboas P: Existence of -asymptotically ω -periodic solutions for abstract neutral functional-differential equations. Bull. Aust. Math. Soc. 2008, 78: 365–382. 10.1017/S0004972708000713

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Blot J, Cieutat P, N’Guérékata GM: -Asymptotically ω -periodic functions and applications to evolution equations. Afr. Diaspora J. Math. 2011, 12: 113–121.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cuevas C, Lizama C: -Asymptotically ω -periodic solutions for semilinear Volterra equations. Math. Methods Appl. Sci. 2010, 33: 1628–1636.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    de Andrade B, Cuevas C: -Asymptotically ω -periodic and asymptotically ω -periodic solutions to semi-linear Cauchy problems with non-dense domain. Nonlinear Anal. 2010, 72: 3190–3208. 10.1016/j.na.2009.12.016

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Dimbour W, N’Guérékata GM: -Asymptotically ω -periodic solutions to some classes of partial evolution equations. Appl. Math. Comput. 2012, 218: 7622–7628. 10.1016/j.amc.2012.01.029

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Henríquez HR, Cuevas C, Caicedo A: Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Commun. Pure Appl. Anal. 2013, 12: 2031–2068.

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Pierri M: On -asymptotically ω -periodic functions and applications. Nonlinear Anal. 2012, 75: 651–661. 10.1016/j.na.2011.08.059

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Pierri M, Rolnik V: On pseudo -asymptotically periodic functions. Bull. Aust. Math. Soc. 2013, 87: 238–254. 10.1017/S0004972712000950

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Henríquez HR: Asymptotically periodic solutions of abstract differential equations. Nonlinear Anal. 2013, 80: 135–149.

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Agarwal RP, Cuevas C, Soto H: Pseudo-almost periodic solutions of a class of semilinear fractional differential equations. J. Appl. Math. Comput. 2011, 37: 625–634. 10.1007/s12190-010-0455-y

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Cuevas C, Sepúlveda A, Soto H: Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations. Appl. Math. Comput. 2011, 218: 1735–1745. 10.1016/j.amc.2011.06.054

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Hilfer H: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.

    Book  MATH  Google Scholar 

  18. 18.

    Li F: Existence of the mild solutions for delay fractional integrodifferential equations with almost sectorial operators. Abstr. Appl. Anal. 2012., 2012: Article ID 729615

    Google Scholar 

  19. 19.

    Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.

    MATH  Google Scholar 

  20. 20.

    Cuesta E, Palencia C: A numerical method for an integro-differential equation with memory in Banach spaces: qualitative properties. SIAM J. Numer. Anal. 2003, 41: 1232–1241. 10.1137/S0036142902402481

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Prüss J Monographs in Mathematics 87. In Evolutionary Integral Equations and Applications. Birkhäuser, Basel; 1993.

    Chapter  Google Scholar 

  22. 22.

    Caicedo A, Cuevas C: -Asymptotically ω -periodic solutions of abstract partial neutral integro-differential equations. Funct. Differ. Equ. 2010, 17: 56–77.

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Cuevas C, de Souza JC: -Asymptotically ω -periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 2009, 22: 865–870. 10.1016/j.aml.2008.07.013

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Cuevas C, de Souza JC: Existence of -asymptotically ω -periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 2010, 72: 1683–1689. 10.1016/j.na.2009.09.007

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Cuevas C, Pierri M, Sepulveda A: Weighted -asymptotically ω -periodic solutions of a class of fractional differential equations. Adv. Differ. Equ. 2011., 2011: Article ID 584874

    Google Scholar 

  26. 26.

    Agarwal RP, de Andrade B, Cuevas C: On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differ. Equ. 2010., 2010: Article ID 179750

    Google Scholar 

  27. 27.

    Lizama C, N’Guérékata GM: Bounded mild solutions for semilinear integro differential equations in Banach spaces. Integral Equ. Oper. Theory 2010, 68: 207–227. 10.1007/s00020-010-1799-2

    Article  MathSciNet  MATH  Google Scholar 

  28. 28.

    Abbas S: Pseudo almost automorphic solutions of some nonlinear integro-differential equations. Comput. Math. Appl. 2011, 62: 2259–2272. 10.1016/j.camwa.2011.07.013

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Mishra I, Bahuguna D: Weighted pseudo almost automorphic solution of an integro-differential equation, with weighted Stepanov-like pseudo almost automorphic forcing term. Appl. Math. Comput. 2013, 219: 5345–5355. 10.1016/j.amc.2012.11.011

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Lunardi A Progress in Nonlinear Differential Equations and Their Applications 16. In Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel; 1995.

    Google Scholar 

  31. 31.

    Bazhlekova, E: Fractional evolution equation in Banach spaces. Ph.D. Thesis, Eindhoven University of Technology (2001)

    Google Scholar 

  32. 32.

    Cuesta E: Asymptotic behaviour of the solutions fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst., Ser. B 2007, 2007: 277–285.

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Cuevas C, Henríquez HR: Solutions of second order abstract retarded functional differential equations on the line. J. Nonlinear Convex Anal. 2011, 12: 225–240.

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Simon J:Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 1986, 146: 65–96. 10.1007/BF01762360

    Article  MathSciNet  MATH  Google Scholar 

  35. 35.

    Zima M: A certain fixed point theorem and its application to integral-functional equations. Bull. Aust. Math. Soc. 1992, 46: 179–186. 10.1017/S0004972700011813

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.

    Book  MATH  Google Scholar 

  37. 37.

    Wei FY, Wang K: Asymptotically periodic logistic equation. J. Biomath. 2005, 20: 399–405.

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Pankov A: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. Kluwer, Dordrecht; 1990.

    Book  MATH  Google Scholar 

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Acknowledgements

The author is grateful to the referees for their valuable suggestions. This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13A010015.

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Xia, Z. Asymptotically periodic solutions of semilinear fractional integro-differential equations. Adv Differ Equ 2014, 9 (2014). https://doi.org/10.1186/1687-1847-2014-9

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Keywords

  • -asymptotically ω-periodic function
  • fractional integro-differential equations
  • sectorial operator
  • Leray-Schauder alternative theorem