Research ArticleExistence of Almost Periodic Solutions for Impulsive NeutralFunctional Differential Equations

Junwei Liu1 and Chuanyi Zhang2

1 College of Science, Yanshan University, Qinhuangdao 066004, China2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Junwei Liu; junweiliuhit@gmail.com

Received 8 May 2014; Accepted 20 July 2014; Published 12 August 2014

Academic Editor: Robert A. Van Gorder

Copyright © 2014 J. Liu and C. Zhang.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence of piecewise almost periodic solutions for impulsive neutral functional differential equations in Banach spaceis investigated. Our results are based on Krasnoselskii’s fixed-point theorem combined with an exponentially stable stronglycontinuous operator semigroup. An example is given to illustrate the theory.

1. Introduction

In this paper, we study the existence of piecewise almostperiodic solutions for a class of abstract impulsive neutralfunctional differential equationswith unbounded delaymod-eled in the form

𝑑

𝑑𝑡(𝑢 (𝑡) + 𝑔 (𝑡, 𝑢

𝑡)) = 𝐴𝑢 (𝑡) + 𝑓 (𝑡, 𝑢

𝑡) ,

𝑡 ∈ 𝑅, 𝑡 ̸= 𝑡𝑖, 𝑖 ∈ 𝑍,

Δ𝑢 (𝑡𝑖) = 𝐼

𝑖(𝑢

𝑡𝑖) , 𝑖 ∈ 𝑍,

(1)

where 𝐴 is the infinitesimal generator of an exponentiallystable strongly continuous semigroup of linear operators{𝑇(𝑡)}

𝑡≥0on a Banach space (𝑋, ‖ ⋅ ‖), the history 𝑢

𝑡:

(−∞, 0] → 𝑋, 𝑢𝑡(𝜃) = 𝑢(𝑡 + 𝜃) belongs to an abstract phase

space B defined axiomatically, 𝑓(⋅), 𝑔(⋅), 𝐼𝑖(⋅) (𝑖 ∈ 𝑍) are

appropriate functions, {𝑡𝑖}𝑖∈𝑍

is a discrete set of real numberssuch that 𝑡

𝑖< 𝑡

𝑗when 𝑖 < 𝑗, and the symbol Δ𝜉(𝑡) represents

the jump of the function 𝜉(⋅) at 𝑡, which is defined by Δ𝜉(𝑡) =𝜉(𝑡

+) − 𝜉(𝑡

−).

The existence of solutions to impulsive differential equa-tions is one of the most attracting topics in the qualita-tive theory of impulsive differential equations due to theirapplications in mechanical, electrical engineering, ecology,biology, and others; see, for instance, [1–6] and the referencestherein. Some recent contributions on mild solutions to

impulsive neutral functional differential equations have beenestablished in [7–14]. On the other hand, the existence ofalmost periodic solutions for impulsive differential equationshas been investigated by many authors; see, for example, [2–5, 15, 16]. However, the existence of almost periodic solutionsfor the impulsive neutral functional differential equations inthe form (1) is an untreated topic in the literature and this factis the motivation of the present work.

The paper is organized as follows: in Section 2, we recallsome notations, concepts, and useful lemmas which areused in this paper. In Section 3, some criteria ensuring theexistence of almost periodic solutions for impulsive neutralfunctional differential equations are obtained. In Section 4,we give an application.

2. Preliminaries

Let (𝑋, ‖ ⋅ ‖) be a Banach space. 𝐴 : 𝐷(𝐴) → 𝑋 is theinfinitesimal generator of a strongly continuous semigroup oflinear operators {𝑇(𝑡)}

𝑡≥0on the Banach space 𝑋 and 𝑀

1, 𝛿

are positive constants such that ‖𝑇(𝑡)‖ ≤ 𝑀1𝑒−𝛿𝑡 for 𝑡 ≥ 0.

Let 0 ∈ 𝜌(𝐴); it is possible to define the fractional power𝐴𝛼, 0 < 𝛼 < 1, as a closed linear operator with its domain

𝐷(𝐴𝛼). We denote by 𝑋

𝛼a Banach space between 𝐷(𝐴) and

𝑋 endowed with the norm ‖𝑥‖𝛼

= ‖𝐴𝛼𝑥‖, 𝑥 ∈ 𝐷(𝐴𝛼); the

following properties hold.

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 782018, 11 pageshttp://dx.doi.org/10.1155/2014/782018

2 Abstract and Applied Analysis

Lemma 1 (see [17, 18]). Let 0 < 𝛼 < 𝛽 < 1; then 𝑋𝛽is

continuously embedded into𝑋𝛼with bounded 𝐾; that is,

‖𝑥‖𝛼≤ 𝐾‖𝑥‖

𝛽. (2)

Moreover, the function 𝑠 → 𝐴𝛼𝑇(𝑠) is continuous in theuniform operator topology on (0,∞) and there exists 𝑀

2> 0

such that ‖𝐴𝛼𝑇(𝑡)‖ ≤ 𝑀2𝑒−𝛿𝑡

𝑡−𝛼 for every 𝑡 > 0.

Throughout this paper, let T be the set consisting of allreal sequences {𝑡

𝑖}𝑖∈𝑍

such that 𝛾 = inf𝑖∈𝑍

(𝑡𝑖+1

− 𝑡𝑖) > 0.

For {𝑡𝑖}𝑖∈𝑍

∈ T , let 𝑃𝐶(𝑅,𝑋𝛼) be the space formed by all

piecewise continuous functions 𝜙 : 𝑅 → 𝑋𝛼such that 𝜙(⋅)

is continuous at 𝑡 for any 𝑡 ∉ {𝑡𝑖}𝑖∈𝑍

and 𝜙(𝑡𝑖) = 𝜙(𝑡

−

𝑖) for all

𝑖 ∈ 𝑍; let𝑃𝐶(𝑅×𝑋𝛼, 𝑋

𝛼) be the space formed by all piecewise

continuous functions 𝜙 : 𝑅 × 𝑋𝛼

→ 𝑋𝛼such that, for any

𝑥 ∈ 𝑋𝛼, 𝜙(⋅, 𝑥) is continuous at 𝑡 for any 𝑡 ∉ {𝑡

𝑖}𝑖∈𝑍

and𝜙(𝑡

𝑖, 𝑥) = 𝜙(𝑡

−

𝑖, 𝑥) for all 𝑖 ∈ 𝑍 and for any 𝑡 ∈ 𝑅, 𝜙(𝑡, ⋅) is

continuous at 𝑥 ∈ 𝑋𝛼.

Definition 2. A number 𝜏 ∈ 𝑅 is called an 𝜖-translationnumber of the function 𝜙 ∈ 𝑃𝐶(𝑅,𝑋

𝛼) if

𝜙(𝑡 + 𝜏) − 𝜙(𝑡)𝛼 < 𝜖, (3)

for all 𝑡 ∈ 𝑅 which satisfies |𝑡 − 𝑡𝑖| > 𝜖, for all 𝑖 ∈ 𝑍. Denote

by 𝑇(𝜙, 𝜖) the set of all 𝜖-translation numbers of 𝜙.

Definition 3 (see [1]). A function 𝜙 ∈ 𝑃𝐶(𝑅,𝑋𝛼) is said to

be piecewise almost periodic if the following conditions arefulfilled.

(1) {𝑡𝑗𝑖

= 𝑡𝑖+𝑗

− 𝑡𝑖}, 𝑖 ∈ 𝑍, 𝑗 = 0, ±1, ±2, . . ., are

equipotentially almost periodic; that is, for any 𝜖 > 0,there exists a relatively dense set of 𝜖-almost periodsthat are common to all the sequences {𝑡𝑗

𝑖}.

(2) For any 𝜖 > 0, there exists a positive number 𝛿 = 𝛿(𝜖)such that if the points 𝑡 and 𝑡 belong to a sameinterval of continuity of 𝜙 and |𝑡 − 𝑡| < 𝛿, then‖𝜙(𝑡

) − 𝜙(𝑡

)‖𝛼< 𝜖.

(3) For every 𝜖 > 0, 𝑇(𝜙, 𝜖) is a relatively dense set in 𝑅.

We denote by 𝐴𝑃𝑇(𝑅,𝑋

𝛼) the space of all piecewise

almost periodic functions. 𝐴𝑃𝑇(𝑅,𝑋

𝛼) endowed with the

uniform convergence topology is a Banach space.

Definition 4. 𝑓 ∈ 𝑃𝐶(𝑅 × 𝑋𝛼, 𝑋

𝛼) is said to be piecewise

almost periodic in 𝑡 uniformly in 𝑥 ∈ 𝑋𝛼if for each compact

set 𝐾 ⊆ 𝑋𝛼, {𝑓(⋅, 𝑥) : 𝑥 ∈ 𝐾} is uniformly bounded and,

given 𝜖 > 0, there exists a relatively dense setΩ(𝜖) such that𝑓(𝑡 + 𝜏, 𝑥) − 𝑓(𝑡, 𝑥)

𝛼 < 𝜖, (4)

for all 𝑥 ∈ 𝐾, 𝜏 ∈ Ω(𝜖), and 𝑡 ∈ 𝑅, |𝑡 − 𝑡𝑖| > 𝜖 for all 𝑖 ∈ 𝑍.

Denote by 𝐴𝑃𝑇(𝑅 × 𝑋

𝛼, 𝑋

𝛼) the set of all such functions.

Lemma 5 (see [15]). Let 𝜙 ∈ 𝐴𝑃𝑇(𝑅,𝑋

𝛼); then the range of 𝜙,

𝑅(𝜙), is a relatively compact subset of𝑋𝛼.

Lemma 6. Suppose that 𝑓(𝑡, 𝑥) ∈ 𝐴𝑃𝑇(𝑅 × 𝑋

𝛼, 𝑋

𝛼) and

𝑓(𝑡, ⋅) is uniformly continuous on each compact subset𝐾 ⊆ 𝑋𝛼

uniformly for 𝑡 ∈ 𝑅; that is, for every 𝜖 > 0, there exists𝛿 > 0 such that 𝑥, 𝑦 ∈ 𝐾 and ‖𝑥 − 𝑦‖

𝛼< 𝛿 implies that

‖𝑓(𝑡, 𝑥) − 𝑓(𝑡, 𝑦)‖𝛼

< 𝜖 for all 𝑡 ∈ 𝑅. Then 𝑓(⋅, 𝑥(⋅)) ∈𝐴𝑃

𝑇(𝑅,𝑋

𝛼) for any 𝑥 ∈ 𝐴𝑃

𝑇(𝑅,𝑋

𝛼).

Proof. Since 𝑥 ∈ 𝐴𝑃(𝑅,𝑋𝛼), by Lemma 5, 𝑅(𝑥) is a relatively

compact subset of𝑋𝛼. Because𝑓(𝑡, ⋅) is uniformly continuous

on each compact subset𝐾 ⊆ 𝑋𝛼uniformly for 𝑡 ∈ 𝑅, then for

any 𝜖 > 0, there exist a number 𝛿 : 0 < 𝛿 ≤ 𝜖/2, such that

𝑓(𝑡, 𝑥1) − 𝑓(𝑡, 𝑥2)𝛼 <

𝜖

2, (5)

where 𝑥1, 𝑥

2∈ 𝑅(𝑥), and ‖𝑥

1− 𝑥

2‖𝛼< 𝛿, 𝑡 ∈ 𝑅. By piecewise

almost periodic of 𝑓 and 𝑥, there exists a relatively dense setΩ of 𝑅 such that the following conditions hold:

𝑓(𝑡 + 𝜏, 𝑥0) − 𝑓(𝑡, 𝑥0)𝛼 <

𝜖

2,

‖𝑥(𝑡 + 𝜏) − 𝑥(𝑡)‖𝛼<

𝜖

2,

(6)

for 𝑥0∈ 𝑅(𝑥) and 𝑡 ∈ 𝑅, |𝑡 − 𝑡

𝑖| > 𝜖, 𝑖 ∈ 𝑍, 𝜏 ∈ Ω. Note that

𝑓 (𝑡 + 𝜏, 𝑥 (𝑡 + 𝜏)) − 𝑓 (𝑡, 𝑥 (𝑡))

= 𝑓 (𝑡 + 𝜏, 𝑥 (𝑡 + 𝜏)) − 𝑓 (𝑡, 𝑥 (𝑡 + 𝜏))

+ 𝑓 (𝑡, 𝑥 (𝑡 + 𝜏)) − 𝑓 (𝑡, 𝑥 (𝑡)) .

(7)

We have𝑓(𝑡 + 𝜏, 𝑥(𝑡 + 𝜏)) − 𝑓(𝑡, 𝑥(𝑡))

𝛼

≤𝑓(𝑡 + 𝜏, 𝑥(𝑡 + 𝜏)) − 𝑓(𝑡, 𝑥(𝑡 + 𝜏))

𝛼

+𝑓 (𝑡, 𝑥 (𝑡 + 𝜏)) − 𝑓 (𝑡, 𝑥 (𝑡))

𝛼.

(8)

We deduce from (5) and (6) that the following formula holds:𝑓(𝑡 + 𝜏, 𝑥(𝑡 + 𝜏)) − 𝑓(𝑡, 𝑥(𝑡))

𝛼 ≤ 𝜖,

𝑡 ∈ 𝑅,𝑡 − 𝑡𝑖

> 𝜖, 𝑖 ∈ 𝑍, 𝜏 ∈ Ω.

(9)

That is, 𝑓(⋅, 𝑥(⋅)) is piecewise almost periodic. The proof iscomplete.

Since the uniform continuity is weaker than the Lipschitzcontinuity, we obtain the following lemma as an immediateconsequence of the previous lemma.

Lemma 7. Let 𝑓(𝑡, 𝑥) ∈ 𝐴𝑃𝑇(𝑅 × 𝑋

𝛼, 𝑋

𝛼) and 𝑓 is Lipschitz;

that is, there is a positive number 𝐿 such that𝑓(𝑡, 𝑥) − 𝑓(𝑡, 𝑦)

𝛼 ≤ 𝐿𝑥 − 𝑦

𝛼, (10)

for all 𝑡 ∈ 𝑅 and 𝑥, 𝑦 ∈ 𝑋𝛼; if for any 𝑥 ∈ 𝐴𝑃

𝑇(𝑅,𝑋

𝛼), then

𝑓(⋅, 𝑥(⋅)) ∈ 𝐴𝑃𝑇(𝑅,𝑋

𝛼).

In this paper, we assume that the phase space B is alinear space formed by functions mapping (−∞, 0] into 𝑋

𝛼

endowed with a norm ‖ ⋅ ‖B and satisfying the followingconditions.

Abstract and Applied Analysis 3

(1) If 𝑥 ∈ 𝑃𝐶(𝑅,𝑋𝛼), then 𝑥

𝑡∈ B for all 𝑡 ∈ 𝑅 and

‖𝑥𝑡‖B ≤ 𝐿 sup𝑠≤𝑡‖𝑥(𝑠)‖𝛼, where 𝐿 > 0 is a constant

independent of 𝑥(⋅) and 𝑡 ∈ 𝑅.

(2) The spaceB is complete.

(3) If {𝜙𝑛}𝑛∈𝑁

⊂ B is a uniformly bounded sequence in𝑃𝐶((−∞, 0], 𝑋

𝛼) formed by functions with compact

support and 𝜙𝑛 → 𝜙 in the compact open topology,then 𝜙 ∈ B and ‖𝜙𝑛 − 𝜙‖B → 0, as 𝑛 → ∞.

Lemma 8 (see [1, 15]). Assume that 𝑓 ∈ 𝐴𝑃𝑇(𝑅,𝑋

𝛼), the

sequence {𝑥𝑖: 𝑖 ∈ 𝑍} is almost periodic in 𝑋

𝛼, and {𝑡𝑗

𝑖=

𝑡𝑖+𝑗

− 𝑡𝑖}, 𝑖 ∈ 𝑍, 𝑗 = 0, ±1, ±2, . . ., are equipotentially almost

periodic. Then for each 𝜖 > 0, there are relatively dense setsΩ𝜖,𝑓,𝑥𝑖

of 𝑅 and 𝑄𝜖,𝑓,𝑥𝑖

of 𝑍 such that the following conditionshold.

(i) ‖𝑓(𝑡 + 𝜏) − 𝑓(𝑡)‖𝛼

< 𝜖 for all 𝑡 ∈ 𝑅, |𝑡 − 𝑡𝑖| > 𝜖, 𝜏 ∈

Ω𝜖,𝑓,𝑥𝑖

, and 𝑖 ∈ 𝑍.

(ii) ‖𝑥𝑖+𝑞

− 𝑥𝑖‖𝛼< 𝜖 for all 𝑞 ∈ 𝑄

𝜖,𝑓,𝑥𝑖and 𝑖 ∈ 𝑍.

(iii) For every 𝜏 ∈ Ω𝜖,𝑓,𝑥𝑖

, there exists at least one number𝑞 ∈ 𝑄

𝜖,𝑓,𝑥𝑖such that

𝑡𝑞

𝑖− 𝜏

< 𝜖, 𝑖 ∈ 𝑍. (11)

Definition 9. A bounded function 𝑢 ∈ 𝑃𝐶(𝑅,𝑋𝛼) is a mild

solution of (1) if the following holds:𝑢𝜎= 𝜙 ∈ B, the function

𝐴𝑇(𝑡 − 𝑠)𝑔(𝑠, 𝑢𝑠) is integrable, and for any 𝑡 ∈ 𝑅, 𝑡

𝑖< 𝑡 ≤ 𝑡

𝑖+1,

𝑢 (𝑡) = 𝑇 (𝑡 − 𝜎) [𝜙 (0) + 𝑔 (𝜎, 𝜙)] − 𝑔 (𝑡, 𝑢𝑡)

− ∫

𝑡

𝜎

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑢𝑠) 𝑑𝑠

+ ∫

𝑡

𝜎

𝑇 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑢𝑠) 𝑑𝑠

+ ∑

𝜎

4 Abstract and Applied Analysis

In order to obtain our results, we need to introduce someadditional notations. Letℎ : 𝑅 → 𝑅 be a continuous functionsuch that ℎ(𝑡) ≥ 1 for all 𝑡 ∈ 𝑅 and ℎ(𝑡) → ∞ as |𝑡| → ∞.We consider the space

(𝑃𝐶)0

ℎ(𝑅,𝑋

𝛼) = {𝑢 ∈ 𝑃𝐶 (𝑅,𝑋

𝛼) : lim

|𝑡|→∞

‖𝑢 (𝑡)‖𝛼

ℎ (𝑡)= 0} .

(16)

Endowed with the norm ‖𝑢‖ℎ= sup

𝑡∈𝑅(‖𝑢(𝑡)‖

𝛼/ℎ(𝑡)), it is a

Banach space.We recall here the following compactness criterion in

these spaces, which we can refer to [7, 19–23].

Lemma 10. A set 𝐵 ⊆ (𝑃𝐶)0ℎ(𝑅,𝑋

𝛼) is a relatively compact set

if and only if

(1) lim|𝑡|→∞

(‖𝑥(𝑡)‖𝛼/ℎ(𝑡)) = 0 uniformly for 𝑥 ∈ 𝐵;

(2) 𝐵(𝑡) = {𝑥(𝑡) : 𝑥 ∈ 𝐵} is relatively compact in 𝑋𝛼for

every 𝑡 ∈ 𝑅;(3) the set 𝐵 is equicontinuous on each interval (𝑡

𝑖, 𝑡𝑖+1

)

(𝑖 ∈ 𝑍).

Theorem 11 (Krasnoselskii’s fixed-point theorem [24]). Let𝑀 be a closed convex nonempty subset of a Banach space 𝑋;suppose that 𝐴 and 𝐵 map 𝑀 into𝑋 such that

(i) 𝐴𝑥 + 𝐵𝑦 ∈ 𝑀 (∀𝑥, 𝑦 ∈ 𝑀),(ii) 𝐴 is completely continuous,(iii) 𝐵 is a contraction with constant 𝐿 < 1.

Then there is a 𝑦 ∈ 𝑀 with 𝐴𝑦 + 𝐵𝑦 = 𝑦.

3. Main Results

In this section, we discuss the existence of piecewise almostperiodic solutions for impulsive neutral functional differen-tial equation (1). To begin, let us list the following hypotheses.

(A1) The operator 𝐴 is the infinitesimal generator of anexponentially stable strongly continuous semigroupof linear operators {𝑇(𝑡)}

𝑡≥0; that is, there exist con-

stants 𝑀1> 0, 𝛿 > 0 such that ‖𝑇(𝑡)‖ ≤ 𝑀

1𝑒−𝛿𝑡 for

𝑡 ≥ 0. Moreover, 𝑇(𝑡) is compact for 𝑡 > 0.(A2) 𝑓(𝑡, 𝑥) ∈ 𝐴𝑃

𝑇(𝑅 ×B, 𝑋

𝛼) is uniformly continuous in

𝑥 ∈ B uniformly in 𝑡 ∈ 𝑅; 𝐼𝑖(𝑥) is almost periodic

in 𝑖 ∈ 𝑍 uniformly in 𝑥 ∈ B and is a uniformlycontinuous function in 𝑥 for all 𝑖 ∈ 𝑍. For every𝑙 > 0, 𝐶

1𝑙= sup

𝑡∈𝑅,‖𝑥‖B≤𝑙𝐿‖𝑓(𝑡, 𝑥)‖

𝛼< ∞, 𝐶

2𝑙=

sup𝑖∈𝑍,‖𝑥‖B≤𝑙𝐿

‖𝐼𝑖(𝑥)‖

𝛼< ∞. Moreover, there exist a

number 𝐿0> 0, such that𝑀

1(𝐶

1/𝛿+𝐶

2/(1−𝑒

−𝛿𝛾)) ≤

𝐿0/2, where 𝐶

1= 𝐶

1𝐿0, 𝐶

2= 𝐶

2𝐿0.

(A3) 𝑔 ∈ 𝐴𝑃𝑇(𝑅 × B, 𝑋

𝛽), 𝑔(𝑡, 0) = 0, and there exist a

number 𝐿1> 0 such that

𝑔(𝑡, 𝜙1) − 𝑔(𝑡, 𝜙2)𝛽 ≤ 𝐿1

𝜙1 − 𝜙2B (17)

for all 𝑡 ∈ 𝑅, 𝜙1, 𝜙

2∈ B.

(A4) Let {𝑥𝑛} ⊆ 𝐴𝑃

𝑇(𝑅,𝑋

𝛼) be uniformly bounded in 𝑅

and uniformly convergent in each compact set of 𝑅;then {𝑓(⋅, 𝑥

𝑛(⋅))} is relatively compact in 𝑃𝐶(𝑅,𝑋

𝛼).

Theorem 12. Suppose that conditions (A1)–(A4) hold; then (1)has a piecewise almost periodic solution provided that𝐾𝐿𝐿

1+

(𝑀2𝜋/𝛿

𝛽−𝛼 sin(𝜋(1 + 𝛼 − 𝛽))Γ(1 + 𝛼 − 𝛽))𝐿1𝐿 ≤ 1/2.

Proof. Let

𝐵 = {𝑢 ∈ 𝐴𝑃𝑇(𝑅,𝑋

𝛼) : ‖𝑢‖

𝛼≤ 𝐿

0} . (18)

Note that 𝐵 is a closed convex set of𝐴𝑃𝑇(𝑅,𝑋

𝛼). By (A3) and

Lemma 1, we have

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑢𝑠)𝛼

≤𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)

𝑔(𝑠, 𝑢𝑠)𝛽

≤ 𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)𝑔(𝑠, 𝑢𝑠) − 𝑔(𝑠, 0)

𝛽

≤ 𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

𝐿1

𝑢𝑠B

≤ 𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

𝐿1𝐿 sup

𝑟≤𝑠

‖𝑢(𝑟)‖𝛼

≤ 𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

𝐿1𝐿𝐿

0,

(19)

and we infer that 𝑠 → 𝐴𝑇(𝑡 − 𝑠)𝑔(𝑠, 𝑢𝑠) is integrable on

(−∞, 𝑡].Define the operator Υ on (𝑃𝐶)0

ℎ(𝑅,𝑋

𝛼) by

Υ𝑢 (𝑡) = − 𝑔 (𝑡, 𝑢𝑡) − ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑢𝑠) 𝑑𝑠

+ ∫

𝑡

−∞

𝑇 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑢𝑠) 𝑑𝑠

+ ∑

𝑡𝑖

Abstract and Applied Analysis 5

For any 𝑥, 𝑦 ∈ 𝐵, by (A3) and Lemma 1, we have

Υ1𝑥 (𝑡)𝛼

=

−𝑔 (𝑡, 𝑥

𝑡) − ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑥𝑠) 𝑑𝑠

𝛼

≤𝑔 (𝑡, 𝑥𝑡)

𝛼

+ ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑥𝑠)𝛼𝑑𝑠

≤ 𝐾𝑔 (𝑡, 𝑥𝑡)

𝛽

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)

𝑔 (𝑠, 𝑥𝑠)𝛽𝑑𝑠

= 𝐾𝑔(𝑡, 𝑥𝑡) − 𝑔(𝑡, 0)

𝛽

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)

×𝑔 (𝑠, 𝑥𝑠) − 𝑔 (𝑠, 0)

𝛽𝑑𝑠

≤ 𝐾𝐿1

𝑥𝑡B

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)𝐿1

𝑥𝑠B𝑑𝑠

≤ 𝐾𝐿1𝐿 sup

𝑟≤𝑡

‖𝑥(𝑟)‖𝛼

+ ∫

𝑡

−∞

𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

× 𝐿1𝐿 sup

𝑟≤𝑠

‖𝑥 (𝑟)‖𝛼𝑑𝑠

≤ 𝐾𝐿1𝐿‖𝑥‖

𝛼

+ ∫

𝑡

−∞

𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

× 𝐿1𝐿‖𝑥‖

𝛼𝑑𝑠

≤ 𝐾𝐿1𝐿‖𝑥‖

𝛼

+𝑀

2𝜋

𝛿𝛽−𝛼 sin (𝜋 (1 + 𝛼 − 𝛽)) Γ (1 + 𝛼 − 𝛽)

× 𝐿1𝐿‖𝑥‖

𝛼,

Υ2𝑦 (𝑡)𝛼 =

∫

𝑡

−∞

𝑇 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑦𝑠) 𝑑𝑠

+∑

𝑡𝑖

6 Abstract and Applied Analysis

where 𝑡 ∈ 𝑅, |𝑡 − 𝑡𝑖| > 𝜖, 𝑖 ∈ 𝑍. So for 𝜏 ∈ Ω

𝜖,𝑓,𝑔,𝐼𝑖, we know

that

Υ1𝑥 (𝑡 + 𝜏) − Υ

1𝑥 (𝑡)

= −𝑔 (𝑡 + 𝜏, 𝑥𝑡+𝜏

) − ∫

𝑡+𝜏

−∞

𝐴𝑇 (𝑡 + 𝜏 − 𝑠) 𝑔 (𝑠, 𝑥𝑠) 𝑑𝑠

+ 𝑔 (𝑡, 𝑥𝑡) + ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑥𝑠) 𝑑𝑠

= −𝑔 (𝑡 + 𝜏, 𝑥𝑡+𝜏

) + 𝑔 (𝑡, 𝑥𝑡)

− ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) [𝑔 (𝑠 + 𝜏, 𝑥𝑠+𝜏

) − 𝑔 (𝑠, 𝑥𝑠)] 𝑑𝑠,

(28)

soΥ1𝑥(𝑡 + 𝜏) − Υ1𝑥(𝑡)

𝛼

≤𝑔(𝑡 + 𝜏, 𝑥𝑡+𝜏) − 𝑔(𝑡, 𝑥𝑡)

𝛼

+ ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) [𝑔 (𝑠 + 𝜏, 𝑥𝑠+𝜏) − 𝑔 (𝑠, 𝑥𝑠)]𝛼𝑑𝑠

≤ 𝐾𝑔(𝑡 + 𝜏, 𝑥𝑡+𝜏) − 𝑔(𝑡, 𝑥𝑡)

𝛽

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)

×𝑔 (𝑠 + 𝜏, 𝑥𝑠+𝜏) − 𝑔 (𝑠, 𝑥𝑠)

𝛽𝑑𝑠

≤ 𝐾𝜖 + ∫

𝑡

−∞

𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

𝜖𝑑𝑠

≤ 𝜖(𝐾 +𝑀

2𝜋

𝛿𝛽−𝛼 sin (𝜋 (1 + 𝛼 − 𝛽)) Γ (1 + 𝛼 − 𝛽)) .

(29)

That is,Υ1𝑥(𝑡 + 𝜏) − Υ1𝑥(𝑡)

𝛼

≤ 𝜖 (𝐾 + (𝑀2𝜋)

× (𝛿𝛽−𝛼 sin (𝜋 (1 + 𝛼 − 𝛽))

× Γ (1 + 𝛼 − 𝛽))−1

) ,

(30)

Υ2𝑦 (𝑡 + 𝜏) − Υ

2𝑦 (𝑡)

= ∫

𝑡+𝜏

−∞

𝑇 (𝑡 + 𝜏 − 𝑠) 𝑓 (𝑠, 𝑦𝑠) 𝑑𝑠

+ ∑

𝑡𝑖

Abstract and Applied Analysis 7

+ ∫

𝑡

−∞

𝐴𝑇 (𝑡 − 𝑠) [𝑔 (𝑠, 𝑥𝑠) − 𝑔 (𝑠, 𝑦𝑠)]𝛼𝑑𝑠

≤ 𝐾𝑔(𝑡, 𝑥𝑡) − 𝑔(𝑡, 𝑦𝑡)

𝛽

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)

𝑔(𝑠, 𝑥𝑠) − 𝑔(𝑠, 𝑦𝑠)𝛽𝑑𝑠

≤ 𝐾𝐿1

𝑥𝑡 − 𝑦𝑡B

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)𝐿1

𝑥𝑠 − 𝑦𝑠B𝑑𝑠

≤ 𝐾𝐿1𝐿 sup

𝑟≤𝑡

𝑥 (𝑟) − 𝑦 (𝑟)𝛼

+ ∫

𝑡

−∞

𝐴1+𝛼−𝛽

𝑇 (𝑡 − 𝑠)𝐿1𝐿 sup

𝑟≤𝑠

𝑥(𝑟) − 𝑦(𝑟)𝛼𝑑𝑠

≤ 𝐾𝐿1𝐿𝑥 − 𝑦

𝛼

+ ∫

𝑡

−∞

𝑀2𝑒−𝛿(𝑡−𝑠)

(𝑡 − 𝑠)−(1+𝛼−𝛽)

𝐿1𝐿𝑥 − 𝑦

𝛼𝑑𝑠

≤ 𝐾𝐿1𝐿𝑥 − 𝑦

𝛼

+𝑀

2𝜋

𝛿𝛽−𝛼 sin (𝜋 (1 + 𝛼 − 𝛽)) Γ (1 + 𝛼 − 𝛽)

× 𝐿1𝐿𝑥 − 𝑦

𝛼.

(34)

Therefore,Υ1𝑥 − Υ1𝑦

𝛼

≤ [𝐾𝐿1𝐿 +

𝑀2𝜋

𝛿𝛽−𝛼 sin (𝜋 (1 + 𝛼 − 𝛽)) Γ (1 + 𝛼 − 𝛽)𝐿1𝐿]

×𝑥 − 𝑦

𝛼.

(35)

Since𝐾𝐿1𝐿+ (𝑀

2𝜋/𝛿

𝛽−𝛼 sin(𝜋(1+𝛼−𝛽))Γ(1+𝛼−𝛽))𝐿1𝐿 ≤

1/2 < 1, it follows that Υ1is a contraction.

Step 3. Υ2is completely continuous.

Claim 1. Υ2is continuous.

Let {𝑥𝑛} ⊆ 𝐴𝑃

𝑇(𝑅,𝑋

𝛼), 𝑥

𝑛→ 𝑥 in 𝐴𝑃

𝑇(𝑅,𝑋

𝛼) as 𝑛 →

∞; by Lemma 5, we may find a compact subset 𝐵0⊆ 𝑋

𝛼such

that 𝑥𝑛(𝑡), 𝑥(𝑡) ∈ 𝐵

0for all 𝑡 ∈ 𝑅, 𝑛 ∈ 𝑁; here we assume

𝐵 ⊆ 𝐵0. By (A2), for the given 𝜖, there exist 𝛿 > 0 such that

𝑥, 𝑦 ∈ B, ‖𝑥 − 𝑦‖B ≤ 𝛿, implies that

𝑓 (𝑡, 𝑥𝑡) − 𝑓 (𝑡, 𝑦𝑡)𝛼 ≤ 𝜖,

𝐼𝑖(𝑥

𝑡𝑖) − 𝐼

𝑖(𝑦

𝑡𝑖)𝛼

≤ 𝜖.

(36)

For the above 𝛿, there exists 𝑛0∈ 𝑁 such that

𝑥𝑛(𝑡) − 𝑥(𝑡)𝛼 ≤

𝛿

𝐿(37)

for 𝑛 > 𝑛0and 𝑡 ∈ 𝑅; then

(𝑥𝑛)𝑡 − 𝑥𝑡B ≤ 𝛿,

𝑓(𝑡, (𝑥𝑛)𝑡) − 𝑓 (𝑡, 𝑥𝑡)𝛼 ≤ 𝜖,

𝐼𝑖((𝑥

𝑛)𝑡𝑖) − 𝐼

𝑖(𝑥

𝑡𝑖)𝛼

≤ 𝜖,

(38)

for 𝑛 > 𝑛0and 𝑡 ∈ 𝑅. Hence,

Υ2(𝑥𝑛)(𝑡) − Υ2𝑥(𝑡)𝛼

=

∫

𝑡

−∞

𝑇 (𝑡 − 𝑠) 𝑓 (𝑠, (𝑥𝑛)𝑠) 𝑑𝑠

+ ∑

𝑡𝑖

8 Abstract and Applied Analysis

where {Υ2𝑢(𝑡 − 𝜖) : 𝑢 ∈ 𝐵} is uniformly bounded in 𝑋

𝛼and

𝑇(𝜖) is compact, so {Υ𝜖2𝑢(𝑡) : 𝑢 ∈ 𝐵} is relatively compact in

𝑋𝛼. Moreover,

Υ2𝑢 (𝑡) − Υ𝜖

2𝑢 (𝑡)

𝛼

=

∫

𝑡

𝑡−𝜖

𝑇 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠) 𝑑𝑠

+ ∑

𝑡−𝜖

Abstract and Applied Analysis 9

≤𝜖

4,

∑

𝑡

10 Abstract and Applied Analysis

Let 𝑋 = 𝐿2([0, 𝜋]) and 𝐴 be the infinitesimal generatorof an analytic semigroup {𝑇(𝑡)}

𝑡≥0which is compact for 𝑡 > 0

and given by 𝐴𝑢 = 𝑢 with domain 𝐷(𝐴) = {𝑢 ∈ 𝐿2([0, 𝜋]) :𝑢

∈ 𝐿2([0, 𝜋]), 𝑢(0) = 𝑢(𝜋) = 0}. The semigroup {𝑇(𝑡)}

𝑡≥0is

defined for 𝑢 ∈ 𝐿2([0, 𝜋]) by

𝑇 (𝑡) 𝑢 =

∞

∑

𝑛=1

𝑒−𝑛2𝑡⟨𝑢, 𝜙

𝑛⟩ 𝜙

𝑛, (52)

where {𝜙𝑛, 𝑛 ∈ 𝑍} is an orthonormal basis of 𝑋; then

|𝑇(𝑡)𝑢| ≤ 𝑒−𝑡|𝑢|, 0 ̸= 𝑢 ∈ 𝐿2([0, 𝜋]). For 𝑢 ∈ 𝐿2([0, 𝜋]),

𝛼 ∈ (0, 1), (−𝐴)−𝛼𝑢 = ∑∞𝑛=1

𝑛−2𝛼

⟨𝑢, 𝜙𝑛⟩𝜙

𝑛, 𝑋

𝛼is the Banach

space endowed with the norm ‖ ⋅ ‖𝛼between𝑋 and𝐷(𝐴).

In this paper, we assume 𝛼 = 1/4, 𝛽 = 3/4.To study the system (51), we make the following assump-

tions.(i) The functions (𝜕𝑖/𝜕𝜉𝑖)𝑏

1(𝜏, 𝜂, 𝜉), 𝑖 = 0, 1 are Lebesgue

measurable, 𝑏1(𝜏, 𝜂, 0) = 𝑏

1(𝜏, 𝜂, 𝜋) = 0 and

𝐿1

:= max{∫𝜋

0

∫

0

−∞

∫

𝜋

0

(𝜕𝑖

𝜕𝜉𝑖𝑏1(𝜏, 𝜂, 𝜉))

2

𝑑𝜂 𝑑𝜏 𝑑𝜉 : 𝑖 = 0, 1}

< ∞.

(53)

(ii) The functions 𝑏𝑖(𝑖 = 2, 3, 4), 𝑎

𝑖(𝑖 ∈ 𝑍) are

continuous, and the sequence of functions {𝑎𝑖, 𝑖 ∈ 𝑍} is

almost periodic.

Under these conditions, we define 𝑓, 𝑔 : 𝑅 ×B → 𝑋, 𝐼𝑖:

B → 𝑋 (𝑖 ∈ 𝑍) by

𝑔 (𝑡, 𝜙) = ∫

0

−∞

∫

𝜋

0

𝑏1(𝑠, 𝜂, 𝜉) 𝜙 (𝑡 + 𝑠, 𝜂) 𝑑𝜂 𝑑𝑠,

𝑡 ∈ 𝑅, 𝜉 ∈ [0, 𝜋] ,

𝑓 (𝑡, 𝜙) = 𝑏2(𝜉) 𝜙 (𝑡, 𝜉) + ∫

0

−∞

𝑏3(𝑠) 𝜙 (𝑡 + 𝑠, 𝜉) 𝑑𝑠 + 𝑏

4(𝑡, 𝜉) ,

𝑡 ∈ 𝑅, 𝜉 ∈ [0, 𝜋] ,

𝐼𝑖(𝜙) = ∫

𝜋

0

𝑎𝑖(𝑡𝑖− 𝑠) 𝜙 (𝑠, 𝜉) 𝑑𝑠, 𝑖 ∈ 𝑍, 𝜉 ∈ [0, 𝜋] .

(54)

We assume further that 𝑓, 𝑔 satisfy the following condi-tion.

(iii) 𝑔 ∈ 𝐴𝑃𝑇(𝑅 × B, 𝑋

𝛽) and 𝑓 ∈ 𝐴𝑃

𝑇(𝑅 × B, 𝑋

𝛼) are

uniformly continuous in 𝑥 ∈ B uniformly in 𝑡 ∈ 𝑅.Under the above assumptions, we can rewrite (51) as

the abstract form (1) and verify that the assumptions ofTheorem 12 hold; then we can get the next result, which is aconsequence of Theorem 12.

Proposition 14. Assume that the previous conditions areverified, if

𝐿1𝐿 (𝐾 + √𝜋) < 1. (55)

The system (51) has a piecewise almost periodic solution.

Conflict of Interests

The authors declare that they have no conflict of interests.

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