Nonlinear Analysis 72 (2010) 1683–1689

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Existence of S-asymptotically ω-periodic solutions for fractional orderfunctional integro-differential equations with infinite delayClaudio Cuevas ∗, Julio César de SouzaUniversidade Federal de Pernambuco, Departamento de Matemática, Av. Prof. Luiz Freire, S/N, Recife-PE, CEP. 50540-740, Brazil

a r t i c l e i n f o

Article history:Received 4 June 2009Accepted 1 September 2009

MSC:primary 47A60secondary 34G2026A33

Keywords:S-asymptotically ω-periodic functionSolution operatorFractional integro-differential equationsPhase space

a b s t r a c t

We study S-asymptotically ω-periodic solutions of the abstract fractional equation u′ =∂−α+1Au+ f (t, ut), 1 < α < 2, where A is a linear operator of sectorial type µ < 0.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

We study in this work the S-asymptotic ω-periodicity of the abstract integro-differential equation of fractional order

v′(t) =∫ t

0

(t − s)α−2

0(α − 1)Av(s)ds+ f (t, vt), t ≥ 0, (1.1)

with initial condition

v0 = ϕ0 ∈ B, (1.2)

where 1 < α < 2, A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X ,the history vt : (−∞, 0] → X defined by vt(θ) = v(t + θ), belongs to some abstract phase spaceB defined axiomatically,and f : [0,∞) × B → X is an appropriate function. The convolution integral in (1.1) is known as the Riemann–Liouvillefractional integral (see [1,2]).The literature concerning S-asymptotically ω-periodic functions with values in Banach spaces is very new. Recently

four interesting articles were published, the first by Henríquez et al. [3], concerning to develop a theory for these typesof functions in Banach space setting. In particular, the authors have established a relationship between S-asymptoticallyω-periodic functions and the class of asymptotically ω-periodic functions. In [4, Lemma 2.1], it is established that a scalarS-asymptoticallyω-periodic function is asymptoticallyω-periodic. In the second article, by Nicola and Pierri [5], the authorsprovided two examples which show that the above assertion is false. The third paper, by Henríquez et al. [6], concerning tothe existence and qualitative properties of S-asymptotically ω-periodic mild solutions for some class of abstract neutral

∗ Corresponding author.E-mail addresses: cch@dmat.ufpe.br (C. Cuevas), jcmath.es@hotmail.com (J. César de Souza).

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.09.007

1684 C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689

functional differential equations with infinite delay. They have applied the results to partial differential equations. Thefourth paper, by Cuevas and de Souza [7], concerning to the existence of S-asymptotically ω-periodic mild solutions offractional integro-differential equations. These types of equations arise in many areas of applied mathematics and theyhave receivemuch attention in recent years, we refer the reader to the extensive bibliography in [8,9,2,10–13]. Very recently,some basic theory for initial value problems of fractional differential equations involving the Riemann–Liouville differentialoperators was discussed by Benchohra et al. [14] and Lakshmikantham and Vatsala [15,16]. El-Sayed and Ibrahin [17] andBenchohra et al. [18], initiated the study of fractional multivalued differential inclusions. In this direction, we refer thearticle byHenderson andOuahab [19], concerning to the existence of solutions to fractional functional differential inclusionswith finite delay; to existence of solutions for these type of equations in the infinite delay framework (see e.g. [18]). In thecase where α ∈ (1, 2], existence results for fractional boundary value problems of differential inclusions were studied byOuahab [20].To the best of the authors’ knowledge, the existence of S-asymptoticallyω-periodic mild solutions for abstract fractional

integro-differential equations is a subject that has not been treated in the literature. In particular, to illustrate our mainresults, we examine sufficient conditions for the existence and uniqueness of an S-asymptotically ω-periodic mild solutionto a fractional oscillation equation (see Example 3.8).

2. Preliminaries

A closed and linear operator A is said to be sectorial of type µ if there exist 0 < θ < π/2,M > 0 and µ ∈ R such that itsresolvent exists outside the sector µ+ Sθ := {µ+ λ : λ ∈ C, |arg(−λ)| < θ}, and ‖(λ− A)−1‖ ≤ M

|λ−µ|, λ 6∈ µ+ Sθ .

Definition 2.1 ([7,21]). Let A be a closed and linear operator with domain D(A) defined on a Banach space X . We call Athe generator of a solution operator if there exist µ ∈ R and a strongly continuous function Sα : R+ → B(X) such that{λα : Re λ > µ} ⊂ ρ(A) and λα−1(λα − A)−1x =

∫∞

0 e−λtSα(t)xdt, Re λ > µ, x ∈ X . In this case, Sα(t) is called the

solution operator generated by A.

We note that if A is sectorial of type µwith 0 ≤ θ < π(1− α/2), then A is the generator of a solution operator given bySα(t) := 1

2π i

∫γeλtλα−1(λα − A)−1dλ, where γ is a suitable path lying outside the sector µ + Sθ (cf. [1,2]). Very recently,

Cuesta [2, Theorem 1] has proved that if A is a sectorial operator of type µ < 0 for someM > 0 and 0 ≤ θ < π(1− α/2),then there exists C > 0 such that

‖Sα(t)‖B(X) ≤CM

1+ |µ|tα, (2.1)

for t ≥ 0. Note that∫∞

01

1+|µ|tα dt =|µ|−1/απα sin(π/α) for 1 < α < 2 and therefore Sα(t) is, in fact, integrable. We note that solution

operators, as well as resolvent families, are a particular case of (a, k)-regularized families introduced in [22]. Accordingto [22] a solution operator Sα(t) corresponds to a (1, t

α−1

0(α))-regularized family. As in the situation of C0-semigroups we

have diverse relations between a solution operator and its generator. The following result is a direct consequence of [22,Proposition 3.1 and Lemma 2.2].

Proposition 2.2. Let Sα(t) be a solution operator on X with generator A. Then, we have

(a) Sα(t)D(A) ⊂ D(A) and ASα(t)x = Sα(t)Ax for all x ∈ D(A), t ≥ 0;(b) Let x ∈ D(A) and t ≥ 0. Then Sα(t)x = x+

∫ t0(t−s)α−1

0(α)ASα(s)xds.

(c) Let x ∈ X and t ≥ 0. Then∫ t0(t−s)α−1

0(α)Sα(s)xds ∈ D(A) and

Sα(t)x = x+ A∫ t

0

(t − s)α−1

0(α)Sα(s)xds.

A characterization of generators of solution operators, analogous to the Hille–Yosida Theorem for C0 semigroups, can bedirectly deduced from [22, Theorem 3.4]. Results on perturbation, approximation, representation as well as ergodic typetheorems can be also deduced from the more general context of (a, k) regularized resolvents (see [23–26]). We note thatsome of the results contained in the above cited papers were obtained independently in the work [1] for the particular caseof solution operators.In this work, we employ the axiomatic definition of the phase spaceB introduced in Hino et al. [27]. Specifically,B is a

linear space of functions mapping (−∞, 0] into X endowed with a seminorm ‖ · ‖B and verifying the following axioms:

(A) If x : (−∞, σ + a) → X, a > 0, σ ∈ R, is continuous in [σ , σ + a) and xσ ∈ B, then for every t ∈ [σ , σ + a) thefollowing hold:(i) xt is inB;(ii) ‖x(t)‖X ≤ H‖xt‖B;

C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689 1685

(iii) ‖xt‖B ≤ K(t − σ) sup {‖x(s)‖X : σ ≤ s ≤ t} + M(t − σ)‖xσ‖B; where H > 0 is a constant, K ,M : [0,∞) →[1,∞), K is continuous,M is locally bounded and H, K ,M are independent of x(·).

(A1) For function x(·) in (A), the function t → xt is continuous from [σ , σ + a) intoB.(B) The spaceB is complete.(C) If (ψn)n is a uniformly bounded sequence of continuous functions with compact support andψn → ψ , n→∞, in thecompact-open topology, then ψ ∈ B and ‖ψn − ψ‖B → 0 as n→∞.

We introduce the spaceB0 := {φ ∈ B : ψ(0) = 0} and the operator S(t) : B → B defined by

S(t)φ(θ) ={φ(0), θ ∈ [−t, 0],φ(t + θ), θ ∈ (−∞,−t).

It is well known that (S(t))t≥0 is a C0-semigroup [27].

Definition 2.3. The phase spaceB is called a fading memory space if ‖S(t)φ‖B → 0, as t →∞ for every φ ∈ B0.

Remark 2.4. Since B satisfies axiom (C), the space Cb((−∞, 0], X) consisting of all continuous and bounded functionsψ : (−∞, 0] → X , is continuously included inB. Thus, there exists a constant L ≥ 0 such that ‖ψ‖B ≤ L‖ψ‖∞, for everyψ ∈ Cb((−∞, 0], X) [27, Proposition 7.1.1]. Moreover, ifB is a fading memory space, then K ,M are bounded functions [27,Proposition 7.1.5].

Example 2.5 (The Phase Space Cr × Lp(ρ, X)). Let r ≥ 0, 1 ≤ p < ∞ and let ρ : (−∞,−r] → R be a non-negative mensurable function which satisfies the conditions (g-5)–(g-6) in the terminology of Hino et al. [27]. Briefly, thismeans that ρ is locally integrable and there exists a non-negative locally bounded function γ (·) on (−∞, 0] such thatρ(ξ + θ) ≤ γ (ξ)ρ(θ) for all ξ ≤ 0 and θ ∈ (−∞,−r] \ Nξ , where Nξ ⊂ (−∞,−r] is a set whose Lebesgue measurezero. We denote by B = Cr × Lp(g, X) the set of all functions ϕ : (−∞, 0] → X such that ϕ is continuous in [−r, 0],Lebesgue measurable and ρ‖ϕ‖p is Lebesgue integrable in (−∞,−r). The seminorm in Cr × Lp(ρ, X) is defined as follows:

‖ϕ‖B := supθ∈[−r,0] ‖ϕ(θ)‖X+(∫−r−∞

ρ(θ)‖ϕ(θ)‖pXdθ

)1/p. Frompreceding conditions, the spaceB = Cr×Lp(ρ, X) satisfies

axioms (A), (A1) and (B). Moreover, when r = 0, and p = 2, it is possible to choose H = 1, K(t) = 1 + (∫ 0−t ρ(θ))

1/2 andM(t) = γ (−t)1/2 for t ≥ 0 (see [27, Theorem 1.3.8]). Note that if conditions (g-6)–(g-7) of [27] hold, then B is a fadingmemory space (see [27, Example 7.1.8]).

In the rest of this work Cb([0,∞), X) denotes the space consisting of the continuous and bounded functions from [0,∞)into X , endowed with the norm of the uniform convergence which is denoted by ‖ • ‖∞. C0([0,∞), X) denotes the space ofall functions x ∈ Cb([0,∞), X) such that limt→∞ ‖x(t)‖X = 0, endowed with the norm ‖ • ‖∞.

Definition 2.6 ([6]). A function f ∈ Cb([0,∞), X) is called S-asymptotically ω-periodic if limt→∞(f (t + ω)− f (t)) = 0.

Definition 2.7 ([3]). A function f ∈ Cb([0,∞), X) is called asymptotically ω-periodic if there exists an ω-periodic functiong and ϕ ∈ C0([0,∞), X) such that f = g + ϕ.

The notation SAPω(X) stands for the space formed by the X-valued S-asymptoticallyω-periodic functions endowed withthe norm of the uniform convergence. It is clear that SAPω(X) is a Banach space (see [3, Proposition 3.5]). In the followingstatements (W , ‖ • ‖W ) and (Z, ‖ • ‖Z ) are Banach spaces.

Definition 2.8 ([6]). A continuous function F : [0,∞)×Z → W is called uniformly S-asymptoticallyω-periodic on boundedsets if F(·, x) is bounded for each x ∈ Z , and for every ε > 0 and all bounded subset K of Z , there exists Lε,K ≥ 0 such that‖F(t, x)− F(t + ω, x)‖W ≤ ε, for every t ≥ Lε,K and all x ∈ K .

Definition 2.9 ([6]). A continuous function F : [0,∞) × Z → W is called asymptotically uniformly continuous onbounded sets, if for every ε > 0 and every bounded subset K of Z , there exist constants Lε,K ≥ 0 and δε,K > 0 suchthat ‖F(t, x)− F(t, y)‖W ≤ ε, for all t ≥ Lε,K and all x, y ∈ K with ‖x− y‖Z ≤ δε,K .

Lemma 2.10 ([6]). Let F : [0,∞) × Z → W be uniformly S-asymptotically ω-periodic on bounded sets and asymptoticallyuniformly continuous on bounded sets and let u be in SAPω(X). Then, limt→∞(F(t + ω, u(t + ω))− F(t, u(t))) = 0.

Lemma 2.11 ([6]). If u ∈ Cb([0,∞), X) is a function such that limt→∞(u(t + nω)− u(t)) = 0, uniformly for n ∈ N, then u(·)is asymptotically ω-periodic.

Lemma 2.12 ([6]). Assume that B is a fading memory space. Let u : R→ X be a function with u0 ∈ B and u|[0,∞) ∈ SAPω(X).Then the function t → ut belongs to SAPω(B).

1686 C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689

3. S-asymptotically ω-periodic mild solutions

In this section, we discuss the existence and uniqueness of an S-asymptoticallyω-periodic mild solutions of the problem(1.1)–(1.2). We recall the concept of mild solution (see, e.g. [21,7,6]).

Definition 3.1. Suppose A generates an integrable solution operator Sα(t). A function v : R → X is called a mild solutionof the system (1.1)–(1.2) if v0 = ϕ, v(·) is continuous on [0,∞), the function s→ Sα(t − s)f (s, vs) is integrable in [0, t) forevery t ≥ 0 and

u(t) = Sα(t)ϕ(0)+∫ t

0Sα(t − s)f (s, vs)ds, for all t ≥ 0. (3.1)

We need the following lemma.

Lemma 3.2. Assume that A is sectorial of type µ < 0. Let u ∈ SAPω(X) and let vα : [0,∞)→ X be the function defined by

vα(t) =∫ t

0Sα(t − s)u(s)ds. (3.2)

Then vα ∈ SAPω(X).

Proof. We can estimate ‖vα(t)‖X by∫ t

0‖Sα(t − s)‖B(X)‖u(s)‖Xds ≤ CM‖u‖∞

(∫ t

0

11+ |µ|sα

ds)≤CM‖u‖∞|µ|−1/απα sin(π/α)

,

whence vα ∈ Cb([0,∞), X). Furthermore, for t ≥ L1, we can estimate ‖vα(t + ω)− vα(t)‖X by∫ ω

0‖Sα(t + ω − s)‖B(X)‖u(s)‖Xds+

∫ L1

0‖Sα(t − s)‖B(X)‖u(s+ ω)− u(s)‖Xds

+

∫ t

L1‖Sα(t − s)‖B(X)‖u(s+ ω)− u(s)‖Xds.

For ε > 0, we select L1 > 0 such that ‖u(s + ω) − u(s)‖X ≤ ε for all s ≥ L1 and∫∞

L11

1+|µ|sα ds < ε. Hence, for t ≥ 2L1, weobtain

‖vα(t + ω)− vα(t)‖X ≤ CM‖u‖∞

∫ t+ω

t

11+ |µ|sα

ds+ 2CM‖u‖∞

∫ t

L1

11+ |µ|sα

ds+ CMε∫ t−L1

0

11+ |µ|sα

ds

≤ CM(3‖u‖∞ +

|µ|−1/απ

α sin(π/α)

)ε. �

We can state the following result.

Theorem 3.3. Assume that A is sectorial of type µ < 0 and that B is a fading memory space. Let f : [0,∞) × B → X be afunction uniformly S-asymptotically ω-periodic on bounded sets and there exists Lf > 0 such that

‖f (t, ψ1)− f (t, ψ2)‖X ≤ Lf ‖ψ1 − ψ2‖B, for all (t, ψi) ∈ [0,∞)×B, i = 1, 2, (3.3)

where LLf CM|µ|−1/απ < α sin(π/α), where C and M are the constants given by (2.1) and L is the constant introduced inRemark 2.4, then the problem (1.1)–(1.2) has a unique S-asymptotically ω-periodic mild solution.

Proof. We set SAPω,0(X) = {x ∈ SAPω(X) : x(0) = 0}. It is clear that SAPω,0(X) is a closed subspace of SAPω(X). We nextidentify the elements x ∈ SAPω,0(X) with its extension to R given by x(θ) = 0 for θ ≤ 0. Moreover, we denote by y(·) thefunction defined by y0 = ϕ and y(t) = Sα(t)ϕ(0) for t ≥ 0. Keeping in mind that supt≥0 ‖Sα(t)‖B(X) < ∞, we have thaty ∈ Cb([0,∞), X). On the other hand, the estimate (2.1) implies that y|[0,∞) ∈ SAPω(X). So by Lemma 2.12, the functiont → yt belongs to SAPω(B). Next, we define the map 0α on SAPω,0(X) by (0αx)0 = 0 and

(0αx)(t) :=∫ t

0Sα(t − s)f (s, xs + ys)ds, t ≥ 0. (3.4)

Taking into account that B is a fading memory space and Lemma 2.12, we have that the function s → xs + ys belongsto SAPω(B). In view of f being asymptotically uniformly continuous on bounded sets. By Lemma 2.10, we conclude thatthe function s → f (s, xs + ys) is in SAPω(X). From Lemma 3.2, we infer that 0α is a map from SAPω,0(X) into SAPω,0(X).Furthermore, we have the estimate

C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689 1687

‖(0αx)(t)− (0αz)(t)‖X ≤ CMLf

∫ t

0

11+ |µ|(t − s)α

‖xs − zs‖Bds

≤ CMLf L(∫ t

0

11+ |µ|sα

ds)‖x− z‖∞ ≤

LLf CM|µ|−1/απα sin(π/α)

‖x− z‖∞,

which proves that0α is a contraction,we conclude that0α has a unique fixed point x ∈ SAPω,0(X). Defining u(t) = y(t)+x(t)for t ∈ R, we can confirm that u ∈ SAPω(X) is a mild solution of problem (1.1)–(1.2). The proof is complete. �

Theorem 3.4. Assume that the hypotheses of Theorem 3.3 hold, and every bounded subset K of B , limt→∞(f (t, ψ) − f (t +nω,ψ)) = 0 uniformly inψ ∈ K and n ∈ N. Then the problem (1.1)–(1.2) has a unique asymptoticallyω-periodic mild solution.

We derive the preceding result from the following lemma.

Lemma 3.5. Assume that A is sectorial of type µ < 0. Let u ∈ SAPω(X) and let vα : [0,∞) → X be the function defined by(3.2). If limt→∞(u(t)− u(t + nω)) = 0 uniformly in n ∈ N, then limt→∞(vα(t)− vα(t + nω)) = 0 uniformly in n ∈ N.

Proof. For ε > 0, we select L1 > 0 which is large enough and t ≥ 2L1, we have the following estimate to ‖vα(t + nω) −vα(t)‖X∫ nω

0‖Sα(t + ω − s)‖B(X)‖u(s)‖Xds+

∫ L1

0‖Sα(t − s)‖B(X)‖u(s+ nω)− u(s)‖Xds

+

∫ t

L1‖Sα(t − s)‖B(X)‖u(s+ nω)− u(s)‖Xds

≤ CM‖u‖∞

∫ t+nω

t

11+ |µ|sα

ds+ 2CM‖u‖∞

∫ t

t−L1

11+ |µ|sα

ds+ CMε∫∞

0

11+ |µ|sα

ds,

which finishes the proof. �

Proof of Theorem 3.4. Let Sω,0(X)be the space consisting of functions x ∈ SAPω,0(X) such that limt→∞(x(t)−x(t+nω)) = 0uniformly in n ∈ N. It is easy to see that Sω,0(X) is a closed subspace of SAPω,0(X). We next identify the elements x ∈ Sω,0(X)with its extension to R given by x(θ) = 0 for θ ≤ 0. We keep the notation introduced in the proof of Theorem 3.3. Weconsider the map 0α defined on Sω,0(X), since the function s → f (s, xs + ys) is in SAPω(X). From the preceding lemma,we infer that 0α is Sω,0(X)−valued. Therefore the fixed point of 0α belongs to Sω,0(X) and the assertion is consequence ofLemma 2.11. The proof is complete. �

We have the following complementary result.

Theorem 3.6. Assume that A is sectorial of type µ < 0 and that B is a fading memory space. Let f : [0,∞) × B → X be afunction uniformly S-asymptotically ω-periodic on bounded sets and satisfies the Lipschitz condition

‖f (t, ψ1)− f (t, ψ2)‖X ≤ L(t)‖ψ1 − ψ2‖B, for all (t, ψi) ∈ [0,∞)×B, i = 1, 2, (3.5)

where L ∈ L1(R), then the problem (1.1)–(1.2) has a unique S-asymptotically ω-periodic mild solution.

Proof. Using the notation introduced in the proof of Theorem 3.3, we consider the map 0α defined on SAPω,0(X). It followsfrom our assumptions that 0α is well defined. Let x, z be in SAPω,0(X) and define Cα := supt≥0 ‖Sα(t)‖B(X). We have

‖(0αx)(t)− (0αz)(t)‖X =∥∥∥∥∫ t

0Sα(t − s)[f (s, xs + ys)− f (s, zs + ys)]

∥∥∥∥Xds

≤ Cα

∫ t

0L(s)‖xs − zs‖Bds ≤ LCα‖L‖1‖x− z‖∞.

We define K∞ := supt≥0 K(t), from axiom (A)(iii) we obtain the following estimate

‖(0αx)s − (0αz)s‖B ≤ K∞Cα sup0≤τ≤s

∫ τ

0L(u)‖xu − zu‖Bdu ≤ LK∞Cα

(∫ s

0L(u)du

)‖x− z‖∞,

whence ‖(02αx)(t)− (02αz)(t)‖X is estimated as follows

Cα

∫ t

0L(s)‖(0αx)s − (0αz)s‖Bds ≤ LK∞C2α

(∫ t

0L(s)

(∫ s

0L(u)du

)ds)‖x− z‖∞

≤LK∞C2α2

(∫ t

0L(s)ds

)2‖x− z‖∞ ≤

L(K∞Cα‖L‖1)2

2K∞‖x− z‖∞.

1688 C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689

In general, we get

‖(0nαx)(t)− (0nαz)(t)‖X ≤

L(K∞Cα)n

K∞(n− 1)!

(∫ t

0L(s)

(∫ s

0L(u)du

)n−1ds)‖x− z‖∞

≤L(K∞Cα)n

K∞n!

(∫ t

0L(s)ds

)n‖x− z‖∞

≤L(K∞Cα‖L‖1)n

K∞n!‖x− z‖∞.

Hence, since L(K∞Cα‖L‖1)n

K∞n!< 1 for n sufficiently large, by the fixed point method 0α has a unique fixed point x ∈ SAPω,0(X).

This completes the proof of the theorem. �

We note that condition of type (3.5) have been previously considered in the literature (see e.g. [28,29,21]). The followingresult ensure the existence of mild solutions in C0([0,∞), X).

Theorem 3.7. Assume that A is sectorial of type µ < 0 and that B is a fading memory space. Let f : [0,∞) × B → X be acontinuous function such that f (·, 0) is integrable in [0,∞) and satisfies the Lipschitz condition (3.5)with L ∈ L1(R), then thereexist a unique u(·)mild solution of (1.1)–(1.2) such that limt→∞ u(t) = 0.

Proof. Let C0,0(X) be the space consisting of functions x ∈ C0([0,∞), X) such that x(0) = 0. We observe that C0,0(X) is aclosed subspace of C0([0,∞), X). We next identify the elements x ∈ C0,0(X) with its extension to R given by x(θ) = 0 forθ ≤ 0. Next, we define the map 0α on C0,0(X) by (0αx)0 = 0 and (3.4). It follows from f (·, 0) ∈ L1([0,∞), X) and (3.5) thats→ f (s, xs + ys) is in L1([0,∞), X). Furthermore, we have the estimate

‖(0αx)(t)‖X ≤ CM∫∞

0

χ[0,t](s)1+ |µ|(t − s)α

‖f (s, xs + ys)‖Xds.

Using the Lebesgue Dominated Convergence Theorem, we conclude that limt→∞(0αx)(t) = 0. Proceeding as in the proofof Theorem 3.6, we infer that

‖0nαx− 0nαz‖∞ ≤

L(K∞Cα‖L‖1)n

K∞n!‖x− z‖∞.

This completes the proof. �

Example 3.8. To illustrate our results, we examine the existence and uniqueness of an S-asymptotically ω-periodic mildsolution to the fractional oscillation equation

∂α

∂tαu(t, ξ) =

∂2

∂ξ 2u(t, ξ)− νu(t, ξ)+

∂α−1

∂tα−1

[∫ t

−∞

b(s− t)u(s, ξ)ds+ c(t)F(u(t, ξ))], t ≥ 0, ξ ∈ [0, π], (3.6)

u(t, 0) = u(t, π) = 0, t ≥ 0, (3.7)u(0, ξ) = ϕ(θ, ξ), ξ ∈ [0, π], (3.8)

Inwhat followswe consider the space X = L2[0, π] and A : D(A) ⊂ X → X is the operator defined by Au = u′′−νu, (ν > 0)with domain D(A) = {u ∈ L2[0, π] : u′′ ∈ L2[0, π], u(0) = u(π) = 0}. It is well known that A is sectorial of typeµ = −ν < 0. As phase space we choose the space B := C0 × L2(ρ, X) defined in Example 2.5, and we assume that theconditions (g-6)–(g-7) in the nomenclature of [27] are satisfied. We note that, under these conditions,B is a fadingmemoryspace and (see Remark 2.4)

L = 1+(∫ 0

−∞

ρ(θ)dθ)1/2

. (3.9)

To treat this system, we assume that the function ϕ(θ)(ξ) = ϕ(θ, ξ) belongs to B, the functions b : (−∞, 0] → R andc : [0,∞)→ R are continuous, the function c(·) is S-asymptoticallyω-periodic, F : R→ R is globally Lipschitz continuous

with Lipschitz constant LF > 0 and L1f =(∫ 0−∞

b2(s)ρ(s) ds

)1/2< ∞. Under these conditions, we can define the function

f : [0,∞) × B → X by f (t, ψ)(ξ) =∫ t−∞b(s)ψ(s, ξ)ds + c(t)F(ψ(0, ξ)).With this notation the system (3.6)–(3.8) is

reduced to the abstract form (1.1)–(1.2). Moreover, we observe that f (t, ·) is globally Lipschitz continuous with Lipschitzconstant

Lf := L1f + ‖c‖∞LF . (3.10)

On the other hand, for everyψ ∈ B, the function f (·, ψ) is bounded with supt≥0 ‖f (t, ψ)‖X ≤ L1f ‖ψ‖B +‖c‖∞[LF‖ψ‖B +|F(0)|

√π ]. The next result is a consequence of Theorems 3.3 and 3.4.

C. Cuevas, J. César de Souza / Nonlinear Analysis 72 (2010) 1683–1689 1689

Proposition 3.9. Assume that LLf CM|ν|−1/απ < α sin(π/α), where L and Lf are the constants given by (3.9) and (3.10)respectively. Then there exists a unique S-asymptotically ω-periodic mild solution u(·) of the problem (3.6)–(3.8). If, in addition,limt→∞(c(t + nω)− c(t)) = 0 uniformly in n ∈ N, then u(·) is asymptotically ω-periodic.

Acknowledgement

The first author is partially supported by CNPQ/Brazil under Grant 300365/2008-0.

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