Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2011, Article ID 793023, 20 pages
http://dx.doi.org/10.1155/2011/793023
Research Article

Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

1Department of Paper Technology, IIT Roorkee, Saharanpur Campus, Saharanpur 247001, India
2Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, India

Received 25 May 2011; Revised 22 July 2011; Accepted 6 August 2011

Academic Editor: D. D. Ganji

Copyright © 2011 Jaydev Dabas et al. 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.

Abstract

This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.

1. Introduction

This paper is concerned with the existence and uniqueness of a mild solution of an impulsive fractional-order functional differential equation with the infinite delay of the form𝐷𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥𝑡[],𝐵𝑥(𝑡),𝑡𝐽=0,𝑇,𝑡𝑡𝑘,𝑡Δ𝑥𝑘=𝐼𝑘𝑥𝑡𝑘,𝑘=1,2,,𝑚,𝑥(𝑡)=𝜙(𝑡),𝜙(𝑡)𝔅,(1.1) where 𝑇>0,0<𝛼<1,𝐴𝐷(𝐴)𝑋𝑋 is the infinitesimal generator of an 𝛼-resolvent family 𝑆𝛼(𝑡)𝑡0, the solution operator 𝑇𝛼(𝑡)𝑡0 is defined on a complex Banach space 𝑋, 𝐷𝛼 is the Caputo fractional derivative, 𝑓𝐽×𝔅×𝑋𝑋 is a given function, and 𝔅 is a phase space defined in Section 2. Here, 0=𝑡0<𝑡1<<𝑡𝑚<𝑡𝑚+1=𝑇, 𝐼𝑘𝐶(𝑋,𝑋), (𝑘=1,2,,𝑚), are bounded functions, Δ𝑥(𝑡𝑘)=𝑥(𝑡+𝑘)𝑥(𝑡𝑘),𝑥(𝑡+𝑘)=lim0𝑥(𝑡𝑘+) and 𝑥(𝑡𝑘)=lim0𝑥(𝑡𝑘) represent the right and left limits of 𝑥(𝑡) at 𝑡=𝑡𝑘, respectively.

We assume that 𝑥𝑡(,0]𝑋, 𝑥𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑠0, belongs to an abstract phase space 𝔅. The term 𝐵𝑥(𝑡) is given by 𝐵𝑥(𝑡)=𝑡0𝐾(𝑡,𝑠)𝑥(𝑠)𝑑𝑠, where 𝐾𝐶(𝐷,+) is the set of all positive continuous functions on 𝐷={(𝑡,𝑠)20𝑠𝑡𝑇}.

Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science. Many physical phenomena in evolution processes are modelled as impulsive differential equations and have been studied extensively by several authors, for instance, see [13], for more information on these topics. Impulsive integro-differential equations with delays represent mathematical models for problems in the areas such as population dynamics, biology, ecology, and epidemic and have been studied by many authors [27]. The study of fractional differential equations has emerged as a new branch of applied mathematics, which has been used for construction and analysis of mathematical models in science and engineering. In fact, the fractional differential equations are considered as models alternative to nonlinear differential equations. Many physical systems can be represented more accurately through fractional derivative formulation. For more detail, see, for instance, the papers [1, 35, 712] and references therein.

Recently, in [4], the author has established sufficient conditions for the existence of a mild solution for a fractional integro-differential equation with a state-dependent delay. Mophou and N’Guérékata [7] have investigated the existence and uniqueness of a mild solution for the fractional differential equation (1.1) without impulsive conditions. Authors of [7] have established the results assuming that 𝐴 generates an 𝛼-resolvent family (𝑆𝛼(𝑡))𝑡0 on a complex Banach space 𝑋 by means of classical fixed-point methods.

In [5], Benchohra et al. have considered the following nonlinear functional differential equation with infinite delay𝐷𝑞𝑥(𝑡)=𝑓𝑡,𝑥𝑡[]]],𝑡0,𝑇,0<𝑞<1,𝑥(𝑡)=𝜙(𝑡),𝑡,0,(1.2) where 𝐷𝑞 is Riemann-Liouville fractional derivative, 𝜙𝔅, with 𝜙(0)=0, and established the existence of a mild solution for the considered problem using the Banach fixed-point and the nonlinear alternative of Leray-Schauder theorems.

Motivated by the above-mentioned works, we consider the problem (1.1) to study the existence and uniqueness of a mild solution using the solution operator and fixed-point theorems. The paper is organized as follows: in Section 2, we introduce some function spaces and notations and present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. In the last section one example is presented.

2. Preliminaries

In this section, we mention some definitions and properties required for establishing our results. Let 𝑋 be a complex Banach space with its norm denoted as 𝑋, and 𝐿(𝑋) represents the Banach space of all bounded linear operators from 𝑋 into 𝑋, and the corresponding norm is denoted by 𝐿(𝑋). Let 𝐶(𝐽,𝑋) denote the space of all continuous functions from 𝐽 into 𝑋 with supremum norm denoted by 𝐶(𝐽,𝑋). In addition, 𝐵𝑟(𝑥,𝑋) represents the closed ball in 𝑋 with the center at 𝑥 and the radius 𝑟.

To describe a fractional-order functional differential equation with the infinite delay, we need to discuss the abstract phase space 𝔅 in a convenient way (for details see [3]). Let (,0](0,) be a continuous function with 𝑙=0(𝑡)𝑑𝑡<. For any 𝑎>0, we define

𝔅={𝜓[𝑎,0]𝑋 such that 𝜓(𝑡) is bounded and measurable} and equip the space 𝔅 with the norm 𝜓[𝑎,0]=sup[]𝑠𝑎,0||||𝜓(𝑠),𝜓𝔅.(2.1) Let us define by 𝔅=]𝜓(,0𝑋,suchthatforany𝑐>0,𝜓[𝑐,0]𝔅with𝜓(0)=0and0(𝑠)𝜓[𝑠,0].𝑑𝑠<(2.2)

If 𝔅 is endowed with the norm 𝜓𝔅=0(𝑠)𝜓[𝑠,0]𝑑𝑠,𝜓𝔅,(2.3) then it is known that (𝔅,𝔅) is a Banach space.

Now, we consider the space 𝔅=]𝑥(,𝑇𝑋suchthat𝑥𝐽𝑘𝐽𝐶𝑘𝑥𝑡,𝑋andthereexist+𝑘𝑡and𝑥𝑘𝑡with𝑥𝑘𝑡=𝑥𝑘,𝑥0=𝜙𝔅,,𝑘=1,,𝑚(2.4) where 𝑥𝐽𝑘 is the restriction of 𝑥 to 𝐽𝑘=(𝑡𝑘,𝑡𝑘+1],𝑘=0,1,2,,𝑚. The function 𝔅 to be a seminorm in 𝔅, it is defined by 𝑥𝔅||𝑥||[]=sup(𝑠)𝑠0,𝑇+𝜙𝔅,𝑥𝔅.(2.5) If 𝑥],𝑇]𝑋,𝑇>0, is such that 𝑥0𝔅, then for all 𝑡𝐽, the following conditions hold:(1)𝑥𝑡𝔅, (2)𝑥𝑡𝔅𝐶1(𝑡)sup0<𝑠<𝑡𝑥(𝑠)+𝐶2(𝑡)𝑥0𝔅, (3)𝑥(𝑡)𝐻𝑥𝑡𝔅, where 𝐻>0 is a constant and 𝐶1:[0,)[0,) is continuous, 𝐶2:[0,)[0,) is locally bounded, and 𝐶1,𝐶2 are independent of 𝑥(). For more details, see [6].

A two parameter function of the Mittag-Lefller type is defined by the series expansion𝐸𝛼,𝛽(𝑧)=𝑘=0𝑧𝑘=1Γ(𝛼𝑘+𝛽)2𝜋𝑖𝐶𝜇𝛼𝛽𝑒𝜇𝜇𝛼𝑧𝑑𝜇,𝛼,𝛽>0,𝑧,(2.6) where 𝐶 is a contour which starts and ends at and encircles the disc |𝜇||𝑧|1/2 counter clockwise. For short, 𝐸𝛼(𝑧)=𝐸𝛼,1(𝑧). It is an entire function which provides a simple generalization of the exponent function: 𝐸1(𝑧)=𝑒𝑧 and the cosine function: 𝐸2(𝑧2)=cosh(𝑧),𝐸2(𝑧2)=cos(𝑧), and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Lefller functions are associated with their Laplace integral0𝑒𝜆𝑡𝑡𝛽1𝐸𝛼,𝛽(𝜔𝑡𝛼𝜆)𝑑𝑡=𝛼𝛽𝜆𝛼𝜔,Re𝜆>𝜔1/𝛼,𝜔>0,(2.7) see [12] for more details.

Definition 2.1. A closed and linear operator 𝐴 is said to be sectorial if there are constants 𝜔𝑅,𝜃[𝜋/2,𝜋],𝑀>0, such that the following two conditions are satisfied: (1)𝜌(𝐴)(𝜃,𝜔)=||||,𝜆𝐶𝜆𝜔,arg(𝜆𝜔)<𝜃(2)𝑅(𝜆,𝐴)𝐿(𝑋)𝑀||||𝜆𝜔,𝜆(𝜃,𝜔).(2.8) Sectorial operators are well studied in the literature. For details see [13].

Definition 2.2 (see Definition  2.3 in [10]). Let 𝐴 be a closed and linear operator with the domain 𝐷(𝐴) defined in a Banach space 𝑋. Let 𝜌(𝐴) be the resolvent set of 𝐴. We say that 𝐴 is the generator of an 𝛼-resolvent family if there exist 𝜔0 and a strongly continuous function 𝑆𝛼𝑅+𝐿(𝑋) such that {𝜆𝛼Re𝜆>𝜔}𝜌(𝐴) and (𝜆𝛼𝐼𝐴)1𝑥=0𝑒𝜆𝑡𝑆𝛼(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥𝑋,(2.9) in this case, 𝑆𝛼(𝑡) is called the 𝛼-resolvent family generated by 𝐴.

Definition 2.3 (see Definition  2.1 in [4]). Let 𝐴 be a closed linear operator with the domain 𝐷(𝐴) defined in a Banach space 𝑋 and 𝛼>0. We say that 𝐴 is the generator of a solution operator if there exist 𝜔0 and a strongly continuous function 𝑆𝛼𝑅+𝐿(𝑋) such that {𝜆𝛼Re𝜆>𝜔}𝜌(𝐴) and 𝜆𝛼1(𝜆𝛼𝐼𝐴)1𝑥=0𝑒𝜆𝑡𝑆𝛼(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥𝑋,(2.10) in this case, 𝑆𝛼(𝑡) is called the solution operator generated by 𝐴.

The concept of the solution operator is closely related to the concept of a resolvent family (see [14] Chapter 1). For more details on 𝛼-resolvent family and solution operators, we refer to [14, 15] and the references therein.

Definition 2.4. The Riemann-Liouville fractional integral operator for order 𝛼>0, of a function 𝑓+ and 𝑓𝐿1(+,𝑋), is defined by 𝐼0𝑓(𝑡)=𝑓(𝑡),𝐼𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0(𝑡𝑠)𝛼1𝑓(𝑠)𝑑𝑠,𝛼>0,𝑡>0,(2.11) where Γ() is the Euler gamma function. The Laplace transform of a function 𝑓𝐿1(+,𝑋) is defined by 𝑓(𝜆)=0𝑒𝜆𝑡𝑓(𝑡)𝑑𝑡,Re(𝜆)>𝜔,(2.12) provided the integral is absolutely convergent for Re(𝜆)>𝜔.

Definition 2.5. Caputo’s derivative of order 𝛼 for a function 𝑓[0,) is defined as 𝐷𝛼𝑡1𝑓(𝑡)=Γ(𝑛𝛼)𝑡0(𝑡𝑠)𝑛𝛼1𝑓(𝑛)(𝑠)𝑑𝑠=𝐼𝑛𝛼𝑓(𝑛)(𝑡),(2.13) for 𝑛1𝛼<𝑛,𝑛𝑁. If 0<𝛼1, then 𝐷𝛼𝑡1𝑓(𝑡)=Γ(1𝛼)𝑡0(𝑡𝑠)𝛼𝑓(1)(𝑠)𝑑𝑠.(2.14) Obviously, Caputo’s derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order 𝛼>0 is given as 𝐿𝐷𝛼𝑡𝑓(𝑡);𝜆=𝜆𝛼𝑓(𝜆)𝑛1𝑘=0𝜆𝛼𝑘1𝑓(𝑘)(0);𝑛1𝛼<𝑛.(2.15)

Lemma 2.6. If 𝑓 satisfies the uniform Holder condition with the exponent 𝛽(0,1] and 𝐴 is a sectorial operator, then the unique solution of the Cauchy problem 𝐷𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥𝑡,𝐵𝑥(𝑡),𝑡>𝑡0,𝑡0𝑅,0<𝛼<1,𝑥(𝑡)=𝜙(𝑡),𝑡𝑡0,(2.16) is given by 𝑥(𝑡)=𝑇𝛼𝑡𝑡0𝑥𝑡+0+𝑡𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠,𝐵𝑥(𝑠)𝑑𝑠,(2.17) where 𝑇𝛼(𝑡)=𝐸𝛼,1(𝐴𝑡𝛼1)=𝐵2𝜋𝑖𝑟𝑒𝜆𝑡𝜆𝛼1𝜆𝛼𝑆𝐴𝑑𝜆,𝛼(𝑡)=𝑡𝛼1𝐸𝛼,𝛼(𝐴𝑡𝛼1)=𝐵2𝜋𝑖𝑟𝑒𝜆𝑡1𝜆𝛼𝐴𝑑𝜆,(2.18)𝐵𝑟 denotes the Bromwich path. 𝑆𝛼(𝑡) is called the 𝛼-resolvent family, and 𝑇𝛼(𝑡) is the solution operator, generated by 𝐴.

Proof. Let 𝑡𝑡0=𝑢, then we get 𝐷𝛼𝑢𝑥𝑢+𝑡0=𝐴𝑥𝑢+𝑡0+𝑓𝑢+𝑡0,𝑥𝑢+𝑡0,𝐵𝑥𝑢+𝑡0,𝑢>0.(2.19) Taking the Laplace transform of (2.19), we have 𝜆𝛼𝐿𝑥𝑢+𝑡0𝜆𝛼1𝑥𝑡+0𝑥=𝐴𝐿𝑢+𝑡0𝑓+𝐿𝑢+𝑡0,𝑥𝑢+𝑡0,𝐵𝑥𝑢+𝑡0.(2.20) Since (𝜆𝛼𝐼𝐴)1 exists, that is, 𝜆𝛼𝜌(𝐴), from (2.20), we obtain 𝐿𝑥𝑢+𝑡0=𝜆𝛼1(𝜆𝛼𝐼𝐴)1𝑥𝑡+0+(𝜆𝛼𝐼𝐴)1𝐿𝑓𝑢+𝑡0,𝑥𝑢+𝑡0,𝐵𝑥𝑢+𝑡0.(2.21) By the inverse Laplace transform of (2.21), we get 𝑥𝑢+𝑡0=𝐸𝛼,1(𝐴𝑢𝛼𝑡)𝑥+0+𝑢0(𝑢𝑠)𝛼1𝐸𝛼,𝛼(𝐴(𝑢𝑠)𝛼)𝑓𝑠+𝑡0,𝑥𝑠+𝑡0,𝐵𝑥𝑠+𝑡0𝑑𝑠.(2.22) Set 𝑢+𝑡0=𝑡, in (2.22), we have 𝑥(𝑡)=𝐸𝛼,1𝐴𝑡𝑡0𝛼𝑥𝑡+0+𝑡𝑡00𝑡𝑡0𝑠𝛼1𝐸𝛼,𝛼𝐴𝑡𝑡0𝑠𝛼𝑓𝑠+𝑡0,𝑥𝑠+𝑡0,𝐵𝑥𝑠+𝑡0𝑑𝑠.(2.23) On simplification, we obtain 𝑥(𝑡)=𝐸𝛼,1𝐴𝑡𝑡0𝛼𝑥𝑡+0+𝑡𝑡0(𝑡𝜃)𝛼1𝐸𝛼,𝛼(𝐴(𝑡𝜃)𝛼)𝑓𝜃,𝑥𝜃,𝐵𝑥(𝜃)𝑑𝜃.(2.24) Set 𝑇𝛼(𝑡)=𝐸𝛼,1(𝐴𝑡𝛼) and 𝑆𝛼(𝑡)=𝑡𝛼1𝐸𝛼,𝛼(𝐴𝑡𝛼) in (2.24). We have 𝑥(𝑡)=𝑇𝛼𝑡𝑡0𝑥𝑡+0+𝑡𝑡0𝑆𝛼(𝑡𝜃)𝑓𝜃,𝑥𝜃,𝐵𝑥(𝜃)𝑑𝜃.(2.25) This completes the proof of the lemma.

Now, we give the definition of a mild solution of the system (1.1) by investigating the classical solution of the system (1.1).

Definition 2.7. A function 𝑥(,𝑇]𝑋 is called a mild solution of (1.1) if the following holds: 𝑥0=𝜙𝔅 on (,0] with 𝜙(0)=0;Δ𝑥𝑡=𝑡𝑘=𝐼𝑘(𝑥(𝑡𝑘)),𝑘=1,,𝑚, the restriction of 𝑥() to the interval [0,𝑇){𝑡1,,𝑡𝑚} is continuous and satisfies the following integral equation: ],𝑥(𝑡)=𝜙(𝑡),𝑡(,0𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡0,𝑡1,𝑇𝛼𝑡𝑡1𝑥𝑡1+𝐼1𝑥𝑡1+𝑡𝑡1𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠𝑡,𝐵𝑥(𝑠)𝑑𝑠,𝑡1,𝑡2,𝑇𝛼𝑡𝑡𝑚𝑥𝑡𝑚+𝐼𝑚𝑥𝑡𝑚+𝑡𝑡𝑚𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠𝑡,𝐵𝑥(𝑠)𝑑𝑠,𝑡𝑚.,𝑇(2.26)

Now, we introduce the following assumptions: (H1)there exist 𝜇1,𝜇2>0 such that 𝑓(𝑡,𝜑,𝑥)𝑓(𝑡,𝜓,𝑦)𝑋𝜇1𝜑𝜓𝔅+𝜇2𝑥𝑦𝑋,𝑡𝐼,(𝜑,𝜓)𝔅2,𝑥,𝑦𝑋.(2.27)(H2)for each 𝑘=1,,𝑚, there exists 𝜌𝑘>0 such that 𝐼𝑘(𝑥)𝐼𝑘(𝑦)𝑋𝜌𝑘𝑥𝑦𝑋,𝑥,𝑦𝑋.(2.28)(H3)max1𝑖𝑚𝑀𝑇1+𝜌𝑖+𝑀𝑆𝑇𝛼𝛼𝜇1𝐶1+𝜇2𝐵<1,(2.29) where 𝐶1=sup0<𝜏<𝑇𝐶1(𝜏) and 𝐵=sup𝑡[0,𝑡]𝑡0𝐾(𝑡,𝑠)𝑑𝑠< and 𝑀𝑇=sup0𝑡𝑇𝑇𝛼(𝑡)𝐿(𝑋),𝑀𝑆=sup0𝑡𝑇𝐶𝑒𝜔𝑡1+𝑡1𝛼.(2.30)

If 𝛼(0,1) and 𝐴𝐴𝛼(𝜃0,𝜔0), then for any 𝑥𝑋 and 𝑡>0, we have 𝑇𝛼(𝑡)𝐿(𝑋)𝑀𝑒𝜔𝑡 and 𝑆𝛼(𝑡)𝐿(𝑋)𝐶𝑒𝜔𝑡(1+𝑡𝛼1),𝑡>0,𝜔>𝜔0. Hence, we have 𝑇𝛼(𝑡)𝐿(𝑋)𝑀𝑇,𝑆𝛼(𝑡)𝐿(𝑋)𝑡𝛼1𝑀𝑆. See [1] for details.

3. The Main Results

Our first result is based on the Banach contraction principle.

Theorem 3.1. Assume that the assumptions (H1)–(H3) are satisfied. If 𝐴𝔸𝛼(𝜃0,𝜔0), then the system (1.1) has a unique mild solution.

Proof. Consider the operator 𝑁𝔅𝔅 defined by ],(𝑁𝑥)(𝑡)=𝜙(𝑡),𝑡(,0𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡0,𝑡1,𝑇𝛼𝑡𝑡1𝑥𝑡1+𝐼1𝑥𝑡1+𝑡𝑡1𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠𝑡,𝐵𝑥(𝑠)𝑑𝑠,𝑡1,𝑡2,𝑇𝛼𝑡𝑡𝑚𝑥𝑡𝑚+𝐼𝑚𝑥𝑡𝑚+𝑡𝑡𝑚𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑥𝑠𝑡,𝐵𝑥(𝑠)𝑑𝑠,𝑡𝑚.,𝑇(3.1) Let 𝑦()(,𝑇]𝑋 be the function defined by ]𝑦(𝑡)=𝜙(𝑡),𝑡(,00,𝑡𝐽,(3.2) then 𝑦0=𝜙. For each 𝑧𝐶(𝐽,) with 𝑧(0)=0, we denote by 𝑧 the function defined by ];𝑧(𝑡)=0,𝑡(,0𝑧(𝑡),𝑡𝐽.(3.3) If 𝑥() satisfies (2.26), then we can decompose 𝑥() as 𝑥(𝑡)=𝑦(𝑡)+𝑧(𝑡) for 𝑡𝐽, which implies 𝑥𝑡=𝑦𝑡+𝑧𝑡 for 𝑡𝐽, and the function 𝑧() satisfies 𝑧(𝑡)=𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡0,𝑡1,𝑇𝛼𝑡𝑡1𝑦𝑡1+𝑧𝑡1+𝐼1𝑦𝑡1+𝑧𝑡1+𝑡𝑡1𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑡𝑧(𝑠)𝑑𝑠,𝑡1,𝑡2,𝑇𝛼𝑡𝑡𝑚𝑦𝑡𝑚+𝑧𝑡𝑚+𝐼𝑚𝑦𝑡𝑚+𝑧𝑡𝑚+𝑡𝑡𝑚𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑡𝑧(𝑠)𝑑𝑠,𝑡𝑚.,𝑇(3.4) Set 𝔅={𝑧𝔅 such that 𝑧0=0} and let 𝔅 be the seminorm in 𝔅 defined by 𝑧𝔅=sup𝑡𝐽𝑧(𝑡)𝑋+𝑧0𝔅=sup𝑡𝐽𝑧(𝑡)𝑋,𝑧𝔅,(3.5) thus (𝔅,𝔅) is a Banach space. We define the operator 𝑃𝔅𝔅 by (𝑃𝑧)(𝑡)=𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡0,𝑡1,𝑇𝛼𝑡𝑡1𝑧𝑡1+𝐼1𝑧𝑡1+𝑡𝑡1𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑡𝑧(𝑠)𝑑𝑠,𝑡1,𝑡2,𝑇𝛼𝑡𝑡𝑚𝑧𝑡𝑚+𝐼𝑚𝑧𝑡𝑚+𝑡𝑡𝑚𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑡𝑧(𝑠)𝑑𝑠,𝑡𝑚.,𝑇(3.6) It is clear that the operator 𝑁 has a unique fixed-point if and only if 𝑃 has a unique fixed-point. To prove that 𝑃 has a unique fixed-point, let 𝑧,𝑧𝔅, then for all 𝑡[0,𝑡1]. We have 𝑃(𝑧)(𝑡)𝑃(𝑧)(𝑡)𝑋𝑡0𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑆𝑡0(𝑡𝑠)𝛼1𝜇1𝑧𝑠𝑧𝑠𝔅+𝜇2𝐵𝑦(𝑠)+𝑧(𝑠)𝐵𝑦(𝑠)+𝑧𝑋𝑀𝑑𝑠𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼𝑧𝑧𝔅.(3.7) For 𝑡(𝑡1,𝑡2], we have 𝑃(𝑧)(𝑡)𝑃(𝑧)(𝑡)𝑋𝑇𝛼(𝑡𝑡1)𝐿(𝑋)𝑧(𝑡1)𝑧(𝑡1)𝑋+𝐼1(𝑧(𝑡1))𝐼1(𝑧(𝑡1))𝑋+𝑡𝑡1𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑇𝑧(𝑡1)𝑧(𝑡1)𝑋+𝜌1𝑧(𝑡1)𝑧(𝑡1)𝑋+𝑀𝑆𝑡𝑡1(𝑡𝑠)𝛼1𝜇1𝑧𝑠𝑧𝑠𝔅+𝜇2𝐵𝑦(𝑠)+𝑧𝑦(𝑠)𝐵(𝑠)+𝑧𝑋𝑀𝑑𝑠𝑇1+𝜌1𝑧𝑧𝔅+𝑀𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼𝑧𝑧𝔅.(3.8) Similarly, when 𝑡(𝑡𝑖,𝑡𝑖+1],𝑖=2,,𝑚, we get 𝑃(𝑧)(𝑡)𝑃(𝑧)(𝑡)𝑋𝑀𝑇1+𝜌𝑖𝑧𝑧𝔅+𝑀𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼𝑧𝑧𝔅.(3.9) Thus, for all 𝑡[0,𝑇], we have 𝑃(𝑧)𝑃(𝑧)𝔅max1𝑖𝑚𝑀𝑇1+𝜌𝑖+𝑀𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼𝑧𝑧𝔅.(3.10) Hence, 𝑃 is a contraction map, and therefore it has an unique fixed-point 𝑧𝔅, which is a mild solution of (1.1) on (,𝑇]. This completes the proof of the theorem.

The second result is established using the following Krasnoselkii’s fixed-point theorem.

Theorem 3.2. Let 𝐵 be a closed-convex and nonempty subset of a Banach space 𝑋. Let 𝑃 and 𝑄 be two operators such that (𝑖)𝑃𝑥+𝑄𝑦𝐵 whenever 𝑥,𝑦𝐵, (𝑖𝑖)𝑃 is compact and continuous; (𝑖𝑖𝑖)𝑄 is a contraction mapping, then there exists 𝑧𝐵 such that 𝑧=𝑃𝑧+𝑄𝑧.

Now, we make the following assumptions: (H4)𝑓𝐽×𝔅×𝑋𝑋 is continuous, and there exist two continuous functions 𝜇1,𝜇2𝐽(0,) such that 𝑓(𝑡,𝜓,𝑥)𝑋𝜇1(𝑡)𝜓𝔅+𝜇2(𝑡)𝑥𝑋,(𝑡,𝜓,𝑥)𝐽×𝔅×𝑋.(3.11)(H5)the function 𝐼𝑘𝑋𝑋 is continuous, and there exists Ω>0 such that Ω=max1𝑘𝑚,𝑥𝐵𝑟𝐼𝑘(𝑥)𝑋.(3.12)

Before going further, we need the following lemma.

Lemma 3.3 (see Lemma  3.2 in [7]). Let 𝐶1=sup0<𝜏<𝑇𝐶1(𝜏),𝐶2=sup0<𝜏<𝑇𝐶2(𝜏),𝜇1=sup0<𝜏<𝑇𝜇1(𝜏),𝜇2=sup0<𝜏<𝑇𝜇2(𝜏)(3.13) then for any 𝑠𝐽, 𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝜇1𝐶2𝜙𝔅+𝐶1sup0𝜏𝑠𝑧(𝜏)𝑋+𝜇2𝑠0𝐾(𝑠,𝜏)𝑧(𝜏)𝑋𝑑𝜏.(3.14) If 𝑧𝑋<𝑟,𝑟>0, then 𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝜇1𝐶2𝜙𝔅+𝐶1𝑟+𝜇2𝑟𝐵=𝜆.(3.15)

Theorem 3.4. Suppose that the assumptions (H1), (H4), (H5) are satisfied with 𝑀𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼<1,(3.16) then the impulsive problem (1.1) has at least one mild solution on (,𝑇].

Proof. Choose 𝑀𝑟[𝑇𝑀(𝑟+Ω)+(𝑆𝑇𝛼𝜆/𝛼)] and consider 𝐵𝑟={𝑧𝔅𝑧𝔅𝑟}, then 𝐵𝑟 is a bounded, closed-convex subset in 𝔅.
Let Γ1𝐵𝑟𝐵𝑟 and Γ2𝐵𝑟𝐵𝑟 be defined as Γ1𝑧(𝑡)=0,𝑡0,𝑡1,𝑇𝛼𝑡𝑡1𝑧𝑡1+𝐼1𝑧𝑡1𝑡,𝑡1,𝑡2,𝑇𝛼𝑡𝑡𝑚𝑧𝑡𝑚+𝐼𝑚𝑧𝑡𝑚𝑡,𝑡𝑚,Γ,𝑇(3.17)2𝑧(𝑡)=𝑡0𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡0,𝑡1,𝑡𝑡1𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑡𝑧(𝑠)𝑑𝑠,𝑡1,𝑡2,𝑡𝑡𝑚𝑆𝛼(𝑡𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠𝑦,𝐵(𝑠)+𝑧𝑡(𝑠)𝑑𝑠,𝑡𝑚.,𝑇(3.18)
Step 1. Let 𝑧,𝑧𝐵𝑟, then show that Γ1𝑧+Γ2𝑧𝐵𝑟, for 𝑡[0,𝑡1], we have (Γ1𝑧)(𝑡)+(Γ2𝑧)(𝑡)𝑋𝑡0𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑆𝑡0(𝑡𝑠)𝛼1𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝑑𝑠,(3.19) and by using Lemma 3.3, we conclude that (Γ1𝑧)+(Γ2𝑧)𝔅𝑀𝑆𝜆𝑇𝛼𝛼<𝑟.(3.20) Similarly, when 𝑡(𝑡𝑖,𝑡𝑖+1], 𝑖=1,,𝑚, we have the estimate (Γ1𝑧)(𝑡)+(Γ2𝑧)(𝑡)𝑋𝑇𝛼(𝑡𝑡𝑖)[𝑧(𝑡𝑖)+𝐼𝑖(𝑧(𝑡𝑖))]𝑋+𝑡𝑡𝑖𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑇𝑧𝔅+𝐼𝑖(𝑧(𝑡𝑖))𝑋+𝑡𝑡𝑖𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝑀𝑑𝑠𝑇𝑀(𝑟+Ω)+𝑆𝑇𝛼𝜆𝛼<𝑟,(3.21) which implies that Γ1𝑧+Γ2𝑧𝔅𝑟.Step 2. We will show that the mapping (Γ1𝑧)(𝑡) is continuous on 𝐵𝑟. For this purpose, let {𝑧𝑛}𝑛=1 be a sequence in 𝐵𝑟 with lim𝑧𝑛𝑧𝐵𝑟, then for 𝑡(𝑡𝑖,𝑡𝑖+1], 𝑖=0,1,,𝑚, we have (Γ1𝑧𝑛)(𝑡)(Γ1𝑧)(𝑡)𝑋𝑇𝛼(𝑡𝑡𝑖)𝐿(𝑋)𝑧𝑛(𝑡𝑖)𝑧(𝑡𝑖)𝑋+𝐼𝑖(𝑧𝑛(𝑡𝑖))𝐼𝑖(𝑧(𝑡𝑖))𝑋.(3.22) Since the functions 𝐼𝑖,𝑖=1,2,,𝑚 are continuous, hence lim𝑛Γ1𝑧𝑛=Γ1𝑧 in 𝐵𝑟 which implies that the mapping Γ1 is continuous on 𝐵𝑟.Step 3. Uniform boundedness of the map (Γ1𝑧)(𝑡) is an implication of the following inequality: for 𝑡(𝑡𝑖,𝑡𝑖+1], 𝑖=0,1,,𝑚, we have (Γ1𝑧)(𝑡)𝑋𝑇𝛼(𝑡𝑡𝑖)𝐿(𝑋)𝑧(𝑡𝑖)𝑋+𝐼𝑖(𝑧(𝑡𝑖))𝑋𝑀𝑇(𝑟+Ω).(3.23)Step 4. To show that the map (3.17) is equicontinuous, we proceed as follows. Let 𝑢,𝑣(𝑡𝑖,𝑡𝑖+1], 𝑡𝑖𝑢<𝑣𝑡𝑖+1, 𝑖=0,1,,𝑚, 𝑧𝐵𝑟, then we obtain (Γ1𝑧)(𝑣)(Γ1𝑧)(𝑢)𝑋𝑇𝛼(𝑣𝑡𝑖)𝑇𝛼(𝑢𝑡𝑖)𝐿(𝑋)𝑧(𝑡𝑖)+𝐼𝑖(𝑧(𝑡𝑖))𝑋𝑇(𝑟+Ω)𝛼(𝑣𝑡𝑖)𝑇𝛼(𝑢𝑡𝑖)𝐿(𝑋).(3.24) Since 𝑇𝛼 is strongly continuous, the continuity of the function 𝑡𝑇(𝑡) allows us to conclude that lim𝑢𝑣𝑇𝛼(𝑣𝑡𝑖)𝑇𝛼(𝑢𝑡𝑖)𝐿(𝑋)=0, which implies that Γ1(𝐵𝑟) is equicontinuous. Finally, combining Step 1 to Step 4 together with Ascoli’s theorem, we conclude that the operator Γ1 is compact.
Now, it only remains to show that the map Γ2 is a contraction mapping. Let 𝑧,𝑧𝐵𝑟 and 𝑡(𝑡𝑖,𝑡𝑖+1], 𝑖=0,1,,𝑚, then we have (Γ2𝑧)(𝑡)(Γ2𝑧)(𝑡)𝑋𝑡𝑡𝑖𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑋𝑀𝑑𝑠𝑆𝑡𝑡𝑖(𝑡𝑠)𝛼1𝜇1𝑧𝑠𝑧𝑠𝔅+𝜇2𝐵𝑦(𝑠)+𝑧(𝑠)𝐵𝑦(𝑠)+𝑧(𝑠)𝑋𝑀𝑑𝑠𝑆𝛼𝜇1𝐶1+𝜇2𝐵𝑇𝛼𝑧𝑧𝔅,(3.25) since (𝑀𝑆/𝛼)(𝜇1𝐶1+𝜇2𝐵)𝑇𝛼<1, which implies that Γ2 is a contraction mapping. Hence, by the Krasnoselkii fixed-point theorem, we can conclude that the problem (1.1) has at least one solution on (,𝑇]. This completes the proof of the theorem.

Our last result is based on the following Schaefer’s fixed-point theorem.

Theorem 3.5. Let 𝑃 be a continuous and compact mapping on a Banach space 𝑋 into itself, such that the set {𝑥𝑋𝑥=𝜈𝑃𝑥𝑓𝑜𝑟𝑠𝑜𝑚𝑒0𝜈1} is bounded, then 𝑃 has a fixed-point.

Lemma 3.6 (see [5]). Let 𝑣[0,𝑇][0,) be a real function, 𝑤() is nonnegative and locally integrable function on [0,𝑇], and there are constants 𝑎>0 and 0<𝛼<1 such that 𝑣(𝑡)𝑤(𝑡)+𝑎𝑡0𝑣(𝑠)(𝑡𝑠)𝛼𝑑𝑠.(3.26) Then there exists a constant 𝐾(𝛼) such that 𝑣(𝑡)𝑤(𝑡)+𝑎𝐾(𝛼)𝑡0𝑤(𝑠)(𝑡𝑠)𝛼[].𝑑𝑠,forevery𝑡0,𝑇(3.27)

Theorem 3.7. Assume that the assumptions (H4)-(H5) are satisfied, and if 𝐴𝐴𝛼(𝜃0,𝜔0) and 𝑀𝑇<1, then the impulsive problem (1.1) has at least one mild solution on (,𝑇].

Proof. We define the operator 𝑃𝔅𝔅 as in Theorem  3.3. Note that 𝑃 is well defined in 𝔅.We complete the proof in the following steps.Step 1. For the continuity of the map 𝑃, let {𝑧𝑛} be a sequence in 𝔅 such that 𝑧𝑛𝑧 in 𝔅. Since the function 𝑓 is continuous on 𝐽×𝔅×𝑋, This implies that 𝑓𝑠,𝑦𝑠+𝑧𝑛𝑠𝑦,𝐵(𝑠)+𝑧𝑛(𝑠)𝑓𝑠,𝑦𝑠+𝑧𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)as𝑛.(3.28) Now, for every 𝑡[0,𝑡1], we get 𝑃𝑧𝑛(𝑡)𝑃𝑧(𝑡)𝑋𝑡0𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑛𝑠,𝐵(𝑦(𝑠)+𝑧𝑛(𝑠)))𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑆𝑇𝛼𝛼𝜀,(3.29) where 𝜀>0,𝜀0 as 𝑛. Moreover, we have 𝑃𝑧𝑛(𝑡)𝑃𝑧(𝑡)𝑋𝑀𝑇𝑧𝑛(𝑡𝑖)𝑧(𝑡𝑖)𝑋+𝐼𝑖(𝑧𝑛(𝑡𝑖))𝐼𝑖(𝑧(𝑡𝑖))𝑋+𝑡𝑡𝑖𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑛𝑠,𝐵(𝑦(𝑠)+𝑧𝑛(𝑠)))𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑇𝑧𝑛(𝑡𝑖)𝑧(𝑡𝑖)𝑋+𝐼𝑖(𝑧𝑛(𝑡𝑖))𝐼𝑖(𝑧(𝑡𝑖))𝑋+𝑀𝑆𝑇𝛼𝛼𝜀,(3.30) where 𝜀>0,𝜀0 as 𝑛, for all 𝑡(𝑡𝑖,𝑡𝑖+1],𝑖=1,,𝑚. The impulsive functions 𝐼𝑘,𝑘=1,,𝑚 are continuous, then we get lim𝑛P𝑧𝑛𝑃𝑧𝔅=0.(3.31) This implies that 𝑃 is continuous.Step 2. 𝑃 maps bounded sets into bounded sets in 𝔅. To prove that for any 𝑟>0, there exists a 𝛾>0 such that for each 𝑧𝐵𝑟={𝑧𝔅𝑧𝔅𝑟}, then we have 𝑃𝑧𝔅𝛾, then for any 𝑧𝐵𝑟,𝑡[0,𝑡1], we have 𝑃𝑧(𝑡)𝑋𝑡0𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑆𝑡0(𝑡𝑠)𝛼1𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝑑𝑠.(3.32) Using Lemma 3.3, we obtain 𝑃𝑧(𝑡)𝑋𝑀𝑆(𝑇𝛼/𝛼)𝜆. Similarly, we have 𝑃𝑧(𝑡)𝑋𝑀𝑇𝑀(𝑟+Ω)+𝑆𝑇𝛼𝛼𝑡𝜆,𝑡𝑖,𝑡𝑖+1,𝑖=1,,𝑚.(3.33) This implies that 𝑃𝑧𝔅𝑀𝑇𝑀(𝑟+Ω)+𝑆𝑇𝛼𝛼[].𝜆=𝛾,𝑡0,𝑇(3.34)Step 3. We will prove that 𝑃(𝐵𝑟) is equicontinuous. Let 𝑢,𝑣[0,𝑡1], with 𝑢<𝑣, we have 𝑃𝑧(𝑣)𝑃𝑧(𝑢)𝑋𝑢0𝑆𝛼(𝑣𝑠)𝑆𝛼(𝑢𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋+𝑑𝑠𝑣𝑢𝑆𝛼(𝑣𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑑𝑠𝑄1+𝑄2,(3.35) where 𝑄1=𝑢0𝑆𝛼(𝑣𝑠)𝑆𝛼(𝑢𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑑𝑠𝜆𝑢0𝑆𝛼(𝑣𝑠)𝑆𝛼(𝑢𝑠)𝐿(𝑋)𝑑𝑠.(3.36) Since 𝑆𝛼(𝑣𝑠)𝑆𝛼(𝑢𝑠)𝐿(𝑋)𝑀2𝑆(𝑡1𝑠)𝛼1𝐿1(𝐼,+) for 𝑠[0,𝑡1] and 𝑆𝛼(𝑣𝑠)𝑆𝛼(𝑢𝑠)0 as 𝑢𝑣, 𝑆𝛼 is strongly continuous. This implies that lim𝑢𝑣𝑄1=0, 𝑄2=𝑣𝑢𝑆𝛼(𝑣𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝜆𝑆(𝑣𝑢)𝛼𝛼.(3.37) Hence, lim𝑢𝑣𝑄2=0. Similarly, for 𝑢,𝑣(𝑡𝑖,𝑡𝑖+1], with 𝑢<𝑣,𝑖=1,,𝑚, we have 𝑃𝑧(𝑣)𝑃𝑧(𝑢)𝑋𝑇𝛼(𝑣𝑡𝑖)𝑇𝛼(𝑢𝑡𝑖)𝐿(𝑋)𝑧𝑡𝑖𝑋+𝐼𝑖𝑧𝑡𝑖𝑋+𝑄1+𝑄2.(3.38) Since 𝑇𝛼 is also strongly continuous, so 𝑇𝛼(𝑣𝑡𝑖)𝑇𝛼(𝑢𝑡𝑖)0 as 𝑢𝑣. Thus, from the above inequalities, we have lim𝑢𝑣𝑃𝑧(𝑣)𝑃𝑧(𝑢)𝑋=0. So, 𝑃(𝐵𝑟) is equicontinuous. Finally, combining Step 1 to Step 3 with Ascoli’s theorem, we conclude that the operator 𝑃 is compact.Step 4. We show that the set 𝐸=𝑧𝔅suchthat𝑧=𝜈𝑃𝑧forsome0<𝜈<1(3.39) is bounded. Let 𝑧𝐸, then 𝑧(𝑡)=𝜈𝑃𝑧(𝑡) for some 0<𝜈<1. Then for each 𝑡[0,𝑡1], we have 𝑧(𝑡)𝑋𝜈𝑡0𝑆𝛼(𝑡𝑠)𝐿(𝑋)𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝜈𝑆𝑡0(𝑡𝑠)𝛼1𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑑𝑠,(3.40) for 𝑡(𝑡𝑖,𝑡𝑖+1],𝑖=1,,𝑚, we get 𝑧(𝑡)𝑋𝑇𝜈𝛼(𝑡𝑡𝑖)𝐿(𝑋)𝑧(𝑡𝑖)𝑋+𝐼𝑖(𝑧(𝑡𝑖))𝑋+𝑡𝑡𝑖𝑆𝛼(𝑡𝑠)𝐿(𝑋)×𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋𝑀𝑑𝑠𝑇𝑧(𝑡𝑖)𝑋+𝑀𝑇𝑀Ω+𝑆𝑡𝑡𝑖(𝑡𝑠)𝛼1𝑓(𝑠,𝑦𝑠+𝑧𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))𝑋.𝑑𝑠(3.41) then for all 𝑡[0,𝑇], we have 𝑧(𝑡)𝑋𝑀𝑇Ω𝑀1𝑇+𝑀𝑆𝑀1𝑇𝑡0(𝑡𝑠)𝛼1𝜇1𝑦(𝑠)𝑠+𝑧𝑠𝔅+𝜇2(𝑠)𝐵(𝑦(𝑠)+𝑧(𝑠))𝑋𝑀𝑑𝑠𝑇Ω𝑀1𝑇+𝑀𝑆𝜇1𝐶2𝜙𝔅𝑇𝛼𝛼𝑀1𝑇+𝑀𝑆𝑀1𝑇𝜇1𝐶1+𝜇2𝐵𝑡0(𝑡𝑠)𝛼1sup0𝜏𝑠𝑧(𝜏)𝑋𝑑𝑠𝜔1+𝜔2𝑡0(𝑡𝑠)𝛼1sup0𝜏𝑠𝑧(𝜏)𝑋𝑑𝑠,(3.42) where 𝜔1=𝑀𝑇𝑀Ω/(1𝑇𝑀)+𝑆𝜇1𝐶2𝜙𝔅𝑇𝛼𝑀/𝛼(1𝑇) and 𝜔2𝑀=(𝑆𝑀/1𝑇)(𝜇1𝐶1+𝜇2𝐵). Let 𝜏[0,𝑠] be such that sup0𝜏𝑠𝑧(𝜏)𝑋=𝑧(𝜏)𝑋,0𝑠𝑡. If 𝜏[0,𝑡], then (3.43) can be written as 𝑧(𝑡)𝑋𝜔1+𝜔2𝑡0(𝑡𝑠)𝛼1𝑧(𝑠)𝑋𝑑𝑠.(3.43) Using Lemma 3.6, there exists a constant 𝐾(𝛼), and (3.43) becomes (𝑧𝑡)𝑋𝜔11+𝑡0𝐾(𝛼)(𝑡𝑠)𝛼1𝑑𝑠𝜔11+𝐾(𝛼)𝑇𝛼𝛼.(3.44) As a consequence of Schaefer’s fixed-point theorem, we deduce that 𝑃 has a fixed-point on (,𝑇]. This completes the proof of the theorem.

4. Applications

To illustrate the application of the theory, we consider the following partial integro-differential equation with fractional derivative of the form𝐷𝑞𝑡𝜕𝑢(𝑡,𝑥)=2𝜕𝑥2𝑢(𝑡,𝑥)+𝑡+𝐻(𝑡,𝑥,𝑠𝑡)𝑄(𝑢(𝑠,𝑥))𝑑𝑠𝑡0𝑘(𝑠,𝑡)𝑒𝑢(𝑠,𝑥)[][]𝑑𝑠,𝑥0,𝜋,𝑡0,𝑏,𝑡𝑡𝑘,𝑢][],𝑡𝑢(𝑡,0)=0=𝑢(𝑡,𝜋),𝑡0,(𝑡,𝑥)=𝜙(𝑡,𝑥),𝑡(,0,𝑥0,𝜋Δ𝑢𝑖(𝑥)=𝑡𝑖𝑞𝑖𝑡𝑖[],𝑠𝑢(𝑠,𝑥)𝑑𝑠,𝑥0,𝜋(4.1) where 𝐷𝑞𝑡 is Caputo’s fractional derivative of order 0<𝑞<1, 0<𝑡1<𝑡2<<𝑡𝑛<𝑏 are prefixed numbers, and 𝜙𝔅. Let 𝑋=𝐿2[0,𝜋], and define the operator 𝐴𝐷(𝐴)𝑋𝑋 by 𝐴𝑤=𝑤 with the domain 𝐷(𝐴)={𝑤𝑋𝑤, 𝑤 are absolutely continuous, 𝑤𝑋,𝑤(0)=0=𝑤(𝜋)}, then𝐴𝑤=𝑛=1𝑛2𝑤,𝑤𝑛𝑤𝑛,𝑤𝐷(𝐴),(4.2) where 𝑤𝑛(𝑥)=2/𝜋sin(𝑛𝑥),𝑛 is the orthogonal set of eigenvectors of 𝐴. It is well known that 𝐴 is the infinitesimal generator of an analytic semigroup (𝑇(𝑡))𝑡0 in 𝑋 and is given by𝑇(𝑡)𝑤=𝑛=1𝑒𝑛2𝑡𝑤,𝑤𝑛𝑤𝑛,𝑤𝑋,andevery𝑡>0.(4.3) From these expressions, it follows that (𝑇(𝑡))𝑡0 is a uniformly bounded compact semigroup, so that 𝑅(𝜆,𝐴)=(𝜆𝐴)1 is a compact operator for all 𝜆𝜌(𝐴),thatis,𝐴𝔸𝛼(𝜃0,𝜔0). Let (𝑠)=𝑒2𝑠,𝑠<0, then 𝑙=0(𝑠)𝑑𝑠=1/2, and define𝜙=0(𝑠)sup[]𝜃𝑠,0𝜙(𝜃)𝐿2𝑑𝑠.(4.4) Hence, for (𝑡,𝜙)[0,𝑏]×𝔅, where 𝜙(𝜃)(𝑥)=𝜙(𝜃,𝑥),(𝜃,𝑥)(,0]×[0,𝜋]. Set 𝑢(𝑡)(𝑥)=𝑢(𝑡,𝑥),𝑓(𝑡,𝜙,𝐵𝑢(𝑡))(𝑥)=0𝐻(𝑡,𝑥,𝜃)𝑄(𝜙(𝜃)(𝑥))𝑑𝜃+𝐵𝑢(𝑡)(𝑥),(4.5) where 𝐵𝑢(𝑡)(𝑥)=𝑡0𝑘(𝑠,𝑡)𝑒𝑢(𝑠,𝑥)𝑑𝑠. Then with these settings, the above equation (4.1) can be written in the abstract form of the equations (1.1). The functions 𝐻, 𝑘, and 𝑄 are satisfying some conditions, and 𝑞𝑖 are continuous and 𝑑𝑖=0(𝑠)𝑞2𝑖(𝑠)𝑑𝑠< for 𝑖=1,2,,𝑛. Suppose further that (1)the function 𝐻(𝑡,𝑥,𝜃) is continuous in [0,𝑏]×[0,𝜋]×(,0] and 𝐻(𝑡,𝑥,𝜃)0, 0𝐻(𝑡,𝑥,𝜃)𝑑𝜃=𝑝1(𝑡,𝑥)<,(2)the function 𝑄() is continuous and for each (𝜃,𝑦)(,0]×[0,𝜋], 0𝑄(𝑢(𝜃)(𝑥))(0𝑒2𝑠𝑢(𝑠,)𝐿2𝑑𝑠).

Now, we can see that 𝑓(𝑡,𝜙,𝐵𝑢(𝑡))𝐿2=𝜋00(𝑡,𝑥,𝜃)𝑄(𝜙(𝜃)(𝑥))𝑑𝜃+𝐵𝑢(𝑡)(𝑥)𝑑𝑥21/2𝜋00(𝑡,𝑥,𝜃)0𝑒2𝑠𝜙(𝑠)()𝐿2𝑑𝑠𝑑𝜃2𝑑𝑥1/2+𝜋0𝑡0𝑘(𝑠,𝑡)𝑒𝑢(𝑠,𝑥)𝑑𝑠2𝑑𝑥1/2𝜋00(𝑡,𝑥,𝜃)0𝑒2𝑠sup𝑠[𝜃,0]𝜙(𝑠)𝐿2𝑑𝑠𝑑𝜃2𝑑𝑥1/2+𝜋0𝑡0𝑘(𝑠,𝑡)𝑒𝑢(𝑠,𝑥)𝑑𝑠2𝑑𝑥1/2𝜋00(𝑡,𝑥,𝜃)𝑑𝜃2𝑑𝑥1/2𝜙𝔅+𝐵𝑢(𝑡)𝐿2𝜋0(𝑝(𝑡,𝑥))2𝑑𝑥1/2𝜙𝔅+𝐵𝑢(𝑡)𝐿2𝑝(𝑡)𝜙𝔅+𝐵𝑢(𝑡)𝐿2.(4.6)

If we take 𝜇1(𝑡)=𝑝(𝑡) and 𝜇2(𝑡)=1, hence 𝑓 satisfies (H4), and similarly we can show that 𝐼𝑘 satisfy (H5). All conditions of Theorem 3.7 are now fulfilled, so we deduce that the system (4.1) has a mild solution on (,𝑇].

Acknowledgment

The authors express their deep gratitude to the referees for their valuable suggestions and comments for improvement of the paper.

References

  1. X.-B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis, Theory, Methods & Applications, vol. 74, no. 5, pp. 2003–2011, 2011. View at Publisher · View at Google Scholar
  2. J. Y. Park, K. Balachandran, and N. Annapoorani, “Existence results for impulsive neutral functional integrodifferential equations with infinite delay,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3152–3162, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. X. Zhang, X. Huang, and Z. Liu, “The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 775–781, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. R. P. Agarwal, B. De Andrade, and G. Siracusa, “On fractional integro-difierential equations with state-dependent delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1143–1149, 2011. View at Publisher · View at Google Scholar
  5. M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978. View at Google Scholar · View at Zentralblatt MATH
  7. G. M. Mophou and G. M. N'Guérékata, “Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 61–69, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Shateri and D. D. Ganji, “Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique,” International Journal of Differential Equations, vol. 2010, Article ID 954674, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, “Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 117–124, 2010. View at Publisher · View at Google Scholar
  10. D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 11, pp. 3692–3705, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar
  12. I. Podlubny, Fractional Differential Equations, Acadmic press, New York, NY, USA, 1993.
  13. M. Haase, The Functional Calculus for