International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 793023 | https://doi.org/10.1155/2011/793023

Jaydev Dabas, Archana Chauhan, Mukesh Kumar, "Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay", International Journal of Differential Equations, vol. 2011, Article ID 793023, 20 pages, 2011. https://doi.org/10.1155/2011/793023

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

Academic Editor: D. D. Ganji
Received25 May 2011
Revised22 Jul 2011
Accepted06 Aug 2011
Published15 Oct 2011

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 form๐ท๐›ผ๐‘ก๐‘ฅ๎€ท(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๐‘“๐‘ก,๐‘ฅ๐‘ก๎€ธ[],๐ต๐‘ฅ(๐‘ก),๐‘กโˆˆ๐ฝ=0,๐‘‡,๐‘กโ‰ ๐‘ก๐‘˜,๎€ท๐‘กฮ”๐‘ฅ๐‘˜๎€ธ=๐ผ๐‘˜๎€ท๐‘ฅ๎€ท๐‘กโˆ’๐‘˜๎€ธ๎€ธ,๐‘˜=1,2,โ€ฆ,๐‘š,๐‘ฅ(๐‘ก)=๐œ™(๐‘ก),๐œ™(๐‘ก)โˆˆ๐”…โ„Ž,(1.1) where ๐‘‡>0,0<๐›ผ<1,๐ดโˆถ๐ท(๐ด)โŠ‚๐‘‹โ†’๐‘‹ is the infinitesimal generator of an ๐›ผ-resolvent family ๐‘†๐›ผ(๐‘ก)๐‘กโ‰ฅ0, the solution operator ๐‘‡๐›ผ(๐‘ก)๐‘กโ‰ฅ0 is defined on a complex Banach space ๐‘‹, ๐ท๐›ผ is the Caputo fractional derivative, ๐‘“โˆถ๐ฝร—๐”…โ„Žร—๐‘‹โ†’๐‘‹ is a given function, and ๐”…โ„Ž is a phase space defined in Section 2. Here, 0=๐‘ก0<๐‘ก1<โ‹ฏ<๐‘ก๐‘š<๐‘ก๐‘š+1=๐‘‡, ๐ผ๐‘˜โˆˆ๐ถ(๐‘‹,๐‘‹), (๐‘˜=1,2,โ€ฆ,๐‘š), are bounded functions, ฮ”๐‘ฅ(๐‘ก๐‘˜)=๐‘ฅ(๐‘ก+๐‘˜)โˆ’๐‘ฅ(๐‘กโˆ’๐‘˜),๐‘ฅ(๐‘ก+๐‘˜)=limโ„Žโ†’0๐‘ฅ(๐‘ก๐‘˜+โ„Ž) and ๐‘ฅ(๐‘กโˆ’๐‘˜)=limโ„Žโ†’0๐‘ฅ(๐‘ก๐‘˜โˆ’โ„Ž) represent the right and left limits of ๐‘ฅ(๐‘ก) at ๐‘ก=๐‘ก๐‘˜, respectively.

We assume that ๐‘ฅ๐‘กโˆถ(โˆ’โˆž,0]โ†’๐‘‹, ๐‘ฅ๐‘ก(๐‘ )=๐‘ฅ(๐‘ก+๐‘ ), ๐‘ โ‰ค0, belongs to an abstract phase space ๐”…โ„Ž. The term ๐ต๐‘ฅ(๐‘ก) is given by โˆซ๐ต๐‘ฅ(๐‘ก)=๐‘ก0๐พ(๐‘ก,๐‘ )๐‘ฅ(๐‘ )๐‘‘๐‘ , where ๐พโˆˆ๐ถ(๐ท,โ„+) is the set of all positive continuous functions on ๐ท={(๐‘ก,๐‘ )โˆˆโ„2โˆถ0โ‰ค๐‘ โ‰ค๐‘กโ‰ค๐‘‡}.

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 [1โ€“3], 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 [2โ€“7]. 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, 3โ€“5, 7โ€“12] 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 Nโ€™Guรฉrรฉkata [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 ๐ด generates an ๐›ผ-resolvent family (๐‘†๐›ผ(๐‘ก))๐‘กโ‰ฅ0 on a complex Banach space ๐‘‹ by means of classical fixed-point methods.

In [5], Benchohra et al. have considered the following nonlinear functional differential equation with infinite delay๐ท๐‘ž๐‘ฅ๎€ท(๐‘ก)=๐‘“๐‘ก,๐‘ฅ๐‘ก๎€ธ[]]],๐‘กโˆˆ0,๐‘‡,0<๐‘ž<1,๐‘ฅ(๐‘ก)=๐œ™(๐‘ก),๐‘กโˆˆโˆ’โˆž,0,(1.2) where ๐ท๐‘ž is Riemann-Liouville fractional derivative, ๐œ™โˆˆ๐”…โ„Ž, with ๐œ™(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 ๐‘‹ be a complex Banach space with its norm denoted as โ€–โ‹…โ€–๐‘‹, and ๐ฟ(๐‘‹) represents the Banach space of all bounded linear operators from ๐‘‹ into ๐‘‹, and the corresponding norm is denoted by โ€–โ‹…โ€–๐ฟ(๐‘‹). Let ๐ถ(๐ฝ,๐‘‹) denote the space of all continuous functions from ๐ฝ into ๐‘‹ with supremum norm denoted by โ€–โ‹…โ€–๐ถ(๐ฝ,๐‘‹). In addition, ๐ต๐‘Ÿ(๐‘ฅ,๐‘‹) represents the closed ball in ๐‘‹ with the center at ๐‘ฅ and the radius ๐‘Ÿ.

To describe a fractional-order functional differential equation with the infinite delay, we need to discuss the abstract phase space ๐”…โ„Ž in a convenient way (for details see [3]). Let โ„Žโˆถ(โˆ’โˆž,0]โ†’(0,โˆž) be a continuous function with โˆซ๐‘™=0โˆ’โˆžโ„Ž(๐‘ก)๐‘‘๐‘ก<โˆž. For any ๐‘Ž>0, we define

๐”…={๐œ“โˆถ[โˆ’๐‘Ž,0]โ†’๐‘‹ such that ๐œ“(๐‘ก) is bounded and measurable} and equip the space ๐”… with the norm โ€–๐œ“โ€–[โˆ’๐‘Ž,0]=sup[]๐‘ โˆˆโˆ’๐‘Ž,0||||๐œ“(๐‘ ),โˆ€๐œ“โˆˆ๐”….(2.1) Let us define by ๐”…โ„Ž=๎‚ป]๐œ“โˆถ(โˆ’โˆž,0โŸถ๐‘‹,suchthatforany๐‘>0,๐œ“โˆฃ[โˆ’๐‘,0]๎€œโˆˆ๐”…with๐œ“(0)=0and0โˆ’โˆžโ„Ž(๐‘ )โ€–๐œ“โ€–[๐‘ ,0]๎‚ผ.๐‘‘๐‘ <โˆž(2.2)

If ๐”…โ„Ž is endowed with the norm โ€–๐œ“โ€–๐”…โ„Ž=๎€œ0โˆ’โˆžโ„Ž(๐‘ )โ€–๐œ“โ€–[๐‘ ,0]๐‘‘๐‘ ,โˆ€๐œ“โˆˆ๐”…โ„Ž,(2.3) then it is known that (๐”…โ„Ž,โ€–โ‹…โ€–๐”…โ„Ž) is a Banach space.

Now, we consider the space ๐”…๎…žโ„Ž=๎€ฝ]๐‘ฅโˆถ(โˆ’โˆž,๐‘‡โŸถ๐‘‹suchthat๐‘ฅโˆฃ๐ฝ๐‘˜๎€ท๐ฝโˆˆ๐ถ๐‘˜๎€ธ๐‘ฅ๎€ท๐‘ก,๐‘‹andthereexist+๐‘˜๎€ธ๎€ท๐‘กand๐‘ฅโˆ’๐‘˜๎€ธ๎€ท๐‘กwith๐‘ฅ๐‘˜๎€ธ๎€ท๐‘ก=๐‘ฅโˆ’๐‘˜๎€ธ,๐‘ฅ0=๐œ™โˆˆ๐”…โ„Ž๎€พ,,๐‘˜=1,โ€ฆ,๐‘š(2.4) where ๐‘ฅโˆฃ๐ฝ๐‘˜ is the restriction of ๐‘ฅ to ๐ฝ๐‘˜=(๐‘ก๐‘˜,๐‘ก๐‘˜+1],๐‘˜=0,1,2,โ€ฆ,๐‘š. The function โ€–โ‹…โ€–๐”…โ€ฒโ„Ž to be a seminorm in ๐”…๎…žโ„Ž, it is defined by โ€–๐‘ฅโ€–๐”…โ€ฒโ„Ž๎€ฝ||๐‘ฅ||[]๎€พ=sup(๐‘ )โˆถ๐‘ โˆˆ0,๐‘‡+โ€–๐œ™โ€–๐”…โ„Ž,๐‘ฅโˆˆ๐”…๎…žโ„Ž.(2.5) If ๐‘ฅโˆถ]โˆ’โˆž,๐‘‡]โ†’๐‘‹,๐‘‡>0, is such that ๐‘ฅ0โˆˆ๐”…โ„Ž, then for all ๐‘กโˆˆ๐ฝ, the following conditions hold:(1)๐‘ฅ๐‘กโˆˆ๐”…โ„Ž, (2)โ€–๐‘ฅ๐‘กโ€–๐”…โ„Žโ‰ค๐ถ1(๐‘ก)sup0<๐‘ <๐‘กโ€–๐‘ฅ(๐‘ )โ€–+๐ถ2(๐‘ก)โ€–๐‘ฅ0โ€–๐”…โ„Ž, (3)โ€–๐‘ฅ(๐‘ก)โ€–โ‰ค๐ปโ€–๐‘ฅ๐‘กโ€–๐”…โ„Ž, where ๐ป>0 is a constant and ๐ถ1:[0,โˆž)โ†’[0,โˆž) is continuous, ๐ถ2:[0,โˆž)โ†’[0,โˆž) is locally bounded, and ๐ถ1,๐ถ2 are independent of ๐‘ฅ(โ‹…). For more details, see [6].

A two parameter function of the Mittag-Lefller type is defined by the series expansion๐ธ๐›ผ,๐›ฝ(๐‘ง)=โˆž๎“๐‘˜=0๐‘ง๐‘˜=1ฮ“(๐›ผ๐‘˜+๐›ฝ)๎€œ2๐œ‹๐‘–๐ถ๐œ‡๐›ผโˆ’๐›ฝ๐‘’๐œ‡๐œ‡๐›ผโˆ’๐‘ง๐‘‘๐œ‡,๐›ผ,๐›ฝ>0,๐‘งโˆˆโ„‚,(2.6) where ๐ถ is a contour which starts and ends at โˆ’โˆž and encircles the disc |๐œ‡|โ‰ค|๐‘ง|1/2 counter clockwise. For short, ๐ธ๐›ผ(๐‘ง)=๐ธ๐›ผ,1(๐‘ง). It is an entire function which provides a simple generalization of the exponent function: ๐ธ1(๐‘ง)=๐‘’๐‘ง and the cosine function: ๐ธ2(๐‘ง2)=cosh(๐‘ง),๐ธ2(โˆ’๐‘ง2)=cos(๐‘ง), 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 integral๎€œโˆž0๐‘’โˆ’๐œ†๐‘ก๐‘ก๐›ฝโˆ’1๐ธ๐›ผ,๐›ฝ(๐œ”๐‘ก๐›ผ๐œ†)๐‘‘๐‘ก=๐›ผโˆ’๐›ฝ๐œ†๐›ผโˆ’๐œ”,Re๐œ†>๐œ”1/๐›ผ,๐œ”>0,(2.7) see [12] for more details.

Definition 2.1. A closed and linear operator ๐ด is said to be sectorial if there are constants ๐œ”โˆˆ๐‘…,๐œƒโˆˆ[๐œ‹/2,๐œ‹],๐‘€>0, such that the following two conditions are satisfied: (๎“1)๐œŒ(๐ด)โŠ‚(๐œƒ,๐œ”)=๎€ฝ||||๎€พ,โ€–๐œ†โˆˆ๐ถโˆถ๐œ†โ‰ ๐œ”,arg(๐œ†โˆ’๐œ”)<๐œƒ(2)๐‘…(๐œ†,๐ด)โ€–๐ฟ(๐‘‹)โ‰ค๐‘€||||๎“๐œ†โˆ’๐œ”,๐œ†โˆˆ(๐œƒ,๐œ”).(2.8) Sectorial operators are well studied in the literature. For details see [13].

Definition 2.2 (see Definitionโ€‰โ€‰2.3 in [10]). Let ๐ด be a closed and linear operator with the domain ๐ท(๐ด) defined in a Banach space ๐‘‹. Let ๐œŒ(๐ด) be the resolvent set of ๐ด. We say that ๐ด is the generator of an ๐›ผ-resolvent family if there exist ๐œ”โ‰ฅ0 and a strongly continuous function ๐‘†๐›ผโˆถ๐‘…+โ†’๐ฟ(๐‘‹) such that {๐œ†๐›ผโˆถRe๐œ†>๐œ”}โŠ‚๐œŒ(๐ด) and (๐œ†๐›ผ๐ผโˆ’๐ด)โˆ’1๎€œ๐‘ฅ=โˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)๐‘ฅ๐‘‘๐‘ก,Re๐œ†>๐œ”,๐‘ฅโˆˆ๐‘‹,(2.9) in this case, ๐‘†๐›ผ(๐‘ก) is called the ๐›ผ-resolvent family generated by ๐ด.

Definition 2.3 (see Definitionโ€‰โ€‰2.1 in [4]). Let ๐ด be a closed linear operator with the domain ๐ท(๐ด) defined in a Banach space ๐‘‹ and ๐›ผ>0. We say that ๐ด is the generator of a solution operator if there exist ๐œ”โ‰ฅ0 and a strongly continuous function ๐‘†๐›ผโˆถ๐‘…+โ†’๐ฟ(๐‘‹) such that {๐œ†๐›ผโˆถRe๐œ†>๐œ”}โŠ‚๐œŒ(๐ด) and ๐œ†๐›ผโˆ’1(๐œ†๐›ผ๐ผโˆ’๐ด)โˆ’1๎€œ๐‘ฅ=โˆž0๐‘’โˆ’๐œ†๐‘ก๐‘†๐›ผ(๐‘ก)๐‘ฅ๐‘‘๐‘ก,Re๐œ†>๐œ”,๐‘ฅโˆˆ๐‘‹,(2.10) in this case, ๐‘†๐›ผ(๐‘ก) is called the solution operator generated by ๐ด.

The concept of the solution operator is closely related to the concept of a resolvent family (see [14] Chapter 1). For more details on ๐›ผ-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 ๐›ผ>0, of a function ๐‘“โˆถโ„+โ†’โ„ and ๐‘“โˆˆ๐ฟ1(โ„+,๐‘‹), is defined by ๐ผ0๐‘“(๐‘ก)=๐‘“(๐‘ก),๐ผ๐›ผ1๐‘“(๐‘ก)=๎€œฮ“(๐›ผ)๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๐‘“(๐‘ )๐‘‘๐‘ ,๐›ผ>0,๐‘ก>0,(2.11) where ฮ“(โ‹…) is the Euler gamma function. The Laplace transform of a function ๐‘“โˆˆ๐ฟ1(โ„+,๐‘‹) is defined by ๎๎€œ๐‘“(๐œ†)=โˆž0๐‘’โˆ’๐œ†๐‘ก๐‘“(๐‘ก)๐‘‘๐‘ก,Re(๐œ†)>๐œ”,(2.12) provided the integral is absolutely convergent for Re(๐œ†)>๐œ”.

Definition 2.5. Caputoโ€™s derivative of order ๐›ผ for a function ๐‘“โˆถ[0,โˆž)โ†’โ„ is defined as ๐ท๐›ผ๐‘ก1๐‘“(๐‘ก)=๎€œฮ“(๐‘›โˆ’๐›ผ)๐‘ก0(๐‘กโˆ’๐‘ )๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘›)(๐‘ )๐‘‘๐‘ =๐ผ๐‘›โˆ’๐›ผ๐‘“(๐‘›)(๐‘ก),(2.13) for ๐‘›โˆ’1โ‰ค๐›ผ<๐‘›,๐‘›โˆˆ๐‘. If 0<๐›ผโ‰ค1, then ๐ท๐›ผ๐‘ก1๐‘“(๐‘ก)=๎€œฮ“(1โˆ’๐›ผ)๐‘ก0(๐‘กโˆ’๐‘ )โˆ’๐›ผ๐‘“(1)(๐‘ )๐‘‘๐‘ .(2.14) Obviously, Caputoโ€™s derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order ๐›ผ>0 is given as ๐ฟ๎€ฝ๐ท๐›ผ๐‘ก๎€พ๐‘“(๐‘ก);๐œ†=๐œ†๐›ผ๎๐‘“(๐œ†)โˆ’๐‘›โˆ’1๎“๐‘˜=0๐œ†๐›ผโˆ’๐‘˜โˆ’1๐‘“(๐‘˜)(0);๐‘›โˆ’1โ‰ค๐›ผ<๐‘›.(2.15)

Lemma 2.6. If ๐‘“ satisfies the uniform Holder condition with the exponent ๐›ฝโˆˆ(0,1] and ๐ด is a sectorial operator, then the unique solution of the Cauchy problem ๐ท๐›ผ๐‘ก๐‘ฅ๎€ท(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๐‘“๐‘ก,๐‘ฅ๐‘ก๎€ธ,๐ต๐‘ฅ(๐‘ก),๐‘ก>๐‘ก0,๐‘ก0โˆˆ๐‘…,0<๐›ผ<1,๐‘ฅ(๐‘ก)=๐œ™(๐‘ก),๐‘กโ‰ค๐‘ก0,(2.16) is given by ๐‘ฅ(๐‘ก)=๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก0๐‘ฅ๎€ท๐‘ก๎€ธ๎€ท+0+๎€œ๎€ธ๎€ธ๐‘ก๐‘ก0๐‘†๐›ผ(๎€ท๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,(2.17) where ๐‘‡๐›ผ(๐‘ก)=๐ธ๐›ผ,1(๐ด๐‘ก๐›ผ1)=๎€œ๎๐ต2๐œ‹๐‘–๐‘Ÿ๐‘’๐œ†๐‘ก๐œ†๐›ผโˆ’1๐œ†๐›ผ๐‘†โˆ’๐ด๐‘‘๐œ†,๐›ผ(๐‘ก)=๐‘ก๐›ผโˆ’1๐ธ๐›ผ,๐›ผ(๐ด๐‘ก๐›ผ1)=๎€œ๎๐ต2๐œ‹๐‘–๐‘Ÿ๐‘’๐œ†๐‘ก1๐œ†๐›ผโˆ’๐ด๐‘‘๐œ†,(2.18)๎๐ต๐‘Ÿ denotes the Bromwich path. ๐‘†๐›ผ(๐‘ก) is called the ๐›ผ-resolvent family, and ๐‘‡๐›ผ(๐‘ก) is the solution operator, generated by ๐ด.

Proof. Let ๐‘กโˆ’๐‘ก0=๐‘ข, then we get ๐ท๐›ผ๐‘ข๐‘ฅ๎€ท๐‘ข+๐‘ก0๎€ธ๎€ท=๐ด๐‘ฅ๐‘ข+๐‘ก0๎€ธ๎€ท+๐‘“๐‘ข+๐‘ก0,๐‘ฅ๐‘ข+๐‘ก0๎€ท,๐ต๐‘ฅ๐‘ข+๐‘ก0๎€ธ๎€ธ,๐‘ข>0.(2.19) Taking the Laplace transform of (2.19), we have ๐œ†๐›ผ๐ฟ๎€ฝ๐‘ฅ๎€ท๐‘ข+๐‘ก0๎€ธ๎€พโˆ’๐œ†๐›ผโˆ’1๐‘ฅ๎€ท๐‘ก+0๎€ธ๎€ฝ๐‘ฅ๎€ท=๐ด๐ฟ๐‘ข+๐‘ก0๎€ฝ๐‘“๎€ท๎€ธ๎€พ+๐ฟ๐‘ข+๐‘ก0,๐‘ฅ๐‘ข+๐‘ก0๎€ท,๐ต๐‘ฅ๐‘ข+๐‘ก0.๎€ธ๎€ธ๎€พ(2.20) Since (๐œ†๐›ผ๐ผโˆ’๐ด)โˆ’1 exists, that is, ๐œ†๐›ผโˆˆ๐œŒ(๐ด), from (2.20), we obtain ๐ฟ๎€ฝ๐‘ฅ๎€ท๐‘ข+๐‘ก0๎€ธ๎€พ=๐œ†๐›ผโˆ’1(๐œ†๐›ผ๐ผโˆ’๐ด)โˆ’1๐‘ฅ๎€ท๐‘ก+0๎€ธ+(๐œ†๐›ผ๐ผโˆ’๐ด)โˆ’1๐ฟ๎€ฝ๐‘“๎€ท๐‘ข+๐‘ก0,๐‘ฅ๐‘ข+๐‘ก0๎€ท,๐ต๐‘ฅ๐‘ข+๐‘ก0.๎€ธ๎€ธ๎€พ(2.21) By the inverse Laplace transform of (2.21), we get ๐‘ฅ๎€ท๐‘ข+๐‘ก0๎€ธ=๐ธ๐›ผ,1(๐ด๐‘ข๐›ผ๎€ท๐‘ก)๐‘ฅ+0๎€ธ+๎€œ๐‘ข0(๐‘ขโˆ’๐‘ )๐›ผโˆ’1๐ธ๐›ผ,๐›ผ(๐ด(๐‘ขโˆ’๐‘ )๐›ผ๎€ท)๐‘“๐‘ +๐‘ก0,๐‘ฅ๐‘ +๐‘ก0๎€ท,๐ต๐‘ฅ๐‘ +๐‘ก0๎€ธ๎€ธ๐‘‘๐‘ .(2.22) Set ๐‘ข+๐‘ก0=๐‘ก, in (2.22), we have ๐‘ฅ(๐‘ก)=๐ธ๐›ผ,1๎€ท๐ด๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐›ผ๎€ธ๐‘ฅ๎€ท๐‘ก+0๎€ธ+โˆซ๐‘กโˆ’๐‘ก00๎€ท๐‘กโˆ’๐‘ก0๎€ธโˆ’๐‘ ๐›ผโˆ’1๐ธ๐›ผ,๐›ผ๎€ท๐ด๎€ท๐‘กโˆ’๐‘ก0๎€ธโˆ’๐‘ ๐›ผ๎€ธ๐‘“๎€ท๐‘ +๐‘ก0,๐‘ฅ๐‘ +๐‘ก0๎€ท,๐ต๐‘ฅ๐‘ +๐‘ก0๎€ธ๎€ธ๐‘‘๐‘ .(2.23) On simplification, we obtain ๐‘ฅ(๐‘ก)=๐ธ๐›ผ,1๎€ท๐ด๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐›ผ๎€ธ๐‘ฅ๎€ท๐‘ก+0๎€ธ+๎€œ๐‘ก๐‘ก0(๐‘กโˆ’๐œƒ)๐›ผโˆ’1๐ธ๐›ผ,๐›ผ(๐ด(๐‘กโˆ’๐œƒ)๐›ผ๎€ท)๐‘“๐œƒ,๐‘ฅ๐œƒ๎€ธ,๐ต๐‘ฅ(๐œƒ)๐‘‘๐œƒ.(2.24) Set ๐‘‡๐›ผ(๐‘ก)=๐ธ๐›ผ,1(๐ด๐‘ก๐›ผ) and ๐‘†๐›ผ(๐‘ก)=๐‘ก๐›ผโˆ’1๐ธ๐›ผ,๐›ผ(๐ด๐‘ก๐›ผ) in (2.24). We have ๐‘ฅ(๐‘ก)=๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘ฅ๎€ท๐‘ก+0๎€ธ+๎€œ๐‘ก๐‘ก0๐‘†๐›ผ(๎€ท๐‘กโˆ’๐œƒ)๐‘“๐œƒ,๐‘ฅ๐œƒ๎€ธ,๐ต๐‘ฅ(๐œƒ)๐‘‘๐œƒ.(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 ๐‘ฅโˆถ(โˆ’โˆž,๐‘‡]โ†’๐‘‹ is called a mild solution of (1.1) if the following holds: ๐‘ฅ0=๐œ™โˆˆ๐”…โ„Ž on (โˆ’โˆž,0] with ๐œ™(0)=0;ฮ”๐‘ฅโˆฃ๐‘ก=๐‘ก๐‘˜=๐ผ๐‘˜(๐‘ฅ(๐‘กโˆ’๐‘˜)),๐‘˜=1,โ€ฆ,๐‘š, the restriction of ๐‘ฅ(โ‹…) to the interval [0,๐‘‡)โงต{๐‘ก1,โ€ฆ,๐‘ก๐‘š} is continuous and satisfies the following integral equation: โŽงโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽฉ],๎€œ๐‘ฅ(๐‘ก)=๐œ™(๐‘ก),๐‘กโˆˆ(โˆ’โˆž,0๐‘ก0๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€บ,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ0,๐‘ก1๎€ป,๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก1๐‘ฅ๎€ท๐‘ก๎€ธ๎€ทโˆ’1๎€ธ+๐ผ1๎€ท๐‘ฅ๎€ท๐‘กโˆ’1+๎€œ๎€ธ๎€ธ๎€ธ๐‘ก๐‘ก1๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€ท๐‘ก,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก๐‘š๐‘ฅ๎€ท๐‘ก๎€ธ๎€ทโˆ’๐‘š๎€ธ+๐ผ๐‘š๎€ท๐‘ฅ๎€ท๐‘กโˆ’๐‘š+๎€œ๎€ธ๎€ธ๎€ธ๐‘ก๐‘ก๐‘š๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€ท๐‘ก,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐‘š๎€ป.,๐‘‡(2.26)

Now, we introduce the following assumptions: (H1)there exist ๐œ‡1,๐œ‡2>0 such that โ€–๐‘“(๐‘ก,๐œ‘,๐‘ฅ)โˆ’๐‘“(๐‘ก,๐œ“,๐‘ฆ)โ€–๐‘‹โ‰ค๐œ‡1โ€–๐œ‘โˆ’๐œ“โ€–๐”…โ„Ž+๐œ‡2โ€–๐‘ฅโˆ’๐‘ฆโ€–๐‘‹,๐‘กโˆˆ๐ผ,(๐œ‘,๐œ“)โˆˆ๐”…2โ„Ž,๐‘ฅ,๐‘ฆโˆˆ๐‘‹.(2.27)(H2)for each ๐‘˜=1,โ€ฆ,๐‘š, there exists ๐œŒ๐‘˜>0 such that โ€–โ€–๐ผ๐‘˜(๐‘ฅ)โˆ’๐ผ๐‘˜โ€–โ€–(๐‘ฆ)๐‘‹โ‰ค๐œŒ๐‘˜โ€–๐‘ฅโˆ’๐‘ฆโ€–๐‘‹,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹.(2.28)(H3)max1โ‰ค๐‘–โ‰ค๐‘š๎ƒฏ๎‚‹๐‘€๐‘‡๎€ท1+๐œŒ๐‘–๎€ธ+๎‚‹๐‘€๐‘†๐‘‡๐›ผ๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๎ƒฐ<1,(2.29) where ๐ถโˆ—1=sup0<๐œ<๐‘‡๐ถ1(๐œ) and ๐ตโˆ—=sup๐‘กโˆˆ[0,๐‘ก]โˆซ๐‘ก0๐พ(๐‘ก,๐‘ )๐‘‘๐‘ <โˆž and ๎‚‹๐‘€๐‘‡=sup0โ‰ค๐‘กโ‰ค๐‘‡โ€–โ€–๐‘‡๐›ผ(โ€–โ€–๐‘ก)๐ฟ(๐‘‹),๎‚‹๐‘€๐‘†=sup0โ‰ค๐‘กโ‰ค๐‘‡๐ถ๐‘’๐œ”๐‘ก๎€ท1+๐‘ก1โˆ’๐›ผ๎€ธ.(2.30)

If ๐›ผโˆˆ(0,1) and ๐ดโˆˆ๐ด๐›ผ(๐œƒ0,๐œ”0), then for any ๐‘ฅโˆˆ๐‘‹ and ๐‘ก>0, we have โ€–๐‘‡๐›ผ(๐‘ก)โ€–๐ฟ(๐‘‹)โ‰ค๐‘€๐‘’๐œ”๐‘ก and โ€–๐‘†๐›ผ(๐‘ก)โ€–๐ฟ(๐‘‹)โ‰ค๐ถ๐‘’๐œ”๐‘ก(1+๐‘ก๐›ผโˆ’1),๐‘ก>0,๐œ”>๐œ”0. Hence, we have โ€–๐‘‡๐›ผ(๐‘ก)โ€–๐ฟ(๐‘‹)โ‰ค๎‚‹๐‘€๐‘‡,โ€–๐‘†๐›ผ(๐‘ก)โ€–๐ฟ(๐‘‹)โ‰ค๐‘ก๐›ผโˆ’1๎‚‹๐‘€๐‘†. 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)โ€“(H3) are satisfied. If ๐ดโˆˆ๐”ธ๐›ผ(๐œƒ0,๐œ”0), then the system (1.1) has a unique mild solution.

Proof. Consider the operator ๐‘โˆถ๐”…๎…žโ„Žโ†’๐”…๎…žโ„Ž defined by โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉ],๎€œ(๐‘๐‘ฅ)(๐‘ก)=๐œ™(๐‘ก),๐‘กโˆˆ(โˆ’โˆž,0๐‘ก0๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€บ,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ0,๐‘ก1๎€ป,๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก1๐‘ฅ๎€ท๐‘ก๎€ธ๎€ทโˆ’1๎€ธ+๐ผ1๎€ท๐‘ฅ๎€ท๐‘กโˆ’1+๎€œ๎€ธ๎€ธ๎€ธ๐‘ก๐‘ก1๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€ท๐‘ก,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก๐‘š๐‘ฅ๎€ท๐‘ก๎€ธ๎€ทโˆ’๐‘š๎€ธ+๐ผ๐‘š๎€ท๐‘ฅ๎€ท๐‘กโˆ’๐‘š+๎€œ๎€ธ๎€ธ๎€ธ๐‘ก๐‘ก๐‘š๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฅ๐‘ ๎€ธ๎€ท๐‘ก,๐ต๐‘ฅ(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐‘š๎€ป.,๐‘‡(3.1) Let ๐‘ฆ(โ‹…)โˆถ(โˆ’โˆž,๐‘‡]โ†’๐‘‹ be the function defined by ๎‚ป]๐‘ฆ(๐‘ก)=๐œ™(๐‘ก),๐‘กโˆˆ(โˆ’โˆž,00,๐‘กโˆˆ๐ฝ,(3.2) then ๐‘ฆ0=๐œ™. For each ๐‘งโˆˆ๐ถ(๐ฝ,โ„) with ๐‘ง(0)=0, we denote by ๐‘ง the function defined by ๎‚ป];๐‘ง(๐‘ก)=0,๐‘กโˆˆ(โˆ’โˆž,0๐‘ง(๐‘ก),๐‘กโˆˆ๐ฝ.(3.3) If ๐‘ฅ(โ‹…) satisfies (2.26), then we can decompose ๐‘ฅ(โ‹…) as ๐‘ฅ(๐‘ก)=๐‘ฆ(๐‘ก)+๐‘ง(๐‘ก) for ๐‘กโˆˆ๐ฝ, which implies ๐‘ฅ๐‘ก=๐‘ฆ๐‘ก+๐‘ง๐‘ก for ๐‘กโˆˆ๐ฝ, and the function ๐‘ง(โ‹…) satisfies โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉ๎€œ๐‘ง(๐‘ก)=๐‘ก0๐‘†๐›ผ(๎€ท๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€บ๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ0,๐‘ก1๎€ป,๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก1๐‘ฆ๎€ท๐‘ก๎€ธ๎€บโˆ’1๎€ธ+๐‘ง๎€ท๐‘กโˆ’1๎€ธ+๐ผ1๐‘ฆ๎€ท๐‘ก๎€ท๎€ทโˆ’1+๎€ธ๎€ธ๐‘ง๎€ท๐‘กโˆ’1+๎€œ๎€ธ๎€ธ๎€ป๐‘ก๐‘ก1๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ก๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก๐‘š๐‘ฆ๎€ท๐‘ก๎€ธ๎€บโˆ’๐‘š๎€ธ+๐‘ง๎€ท๐‘กโˆ’๐‘š๎€ธ+๐ผ๐‘š๐‘ฆ๎€ท๐‘ก๎€ท๎€ทโˆ’๐‘š+๎€ธ๎€ธ๐‘ง๎€ท๐‘กโˆ’๐‘š+๎€œ๎€ธ๎€ธ๎€ป๐‘ก๐‘ก๐‘š๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ก๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ๐‘š๎€ป.,๐‘‡(3.4) Set ๐”…โ„Ž๎…ž๎…ž={๐‘งโˆˆ๐”…๎…žโ„Ž such that ๐‘ง0=0} and let โ€–โ‹…โ€–๐”…โ„Žโ€ฒโ€ฒ be the seminorm in ๐”…โ„Ž๎…ž๎…ž defined by โ€–๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒ=sup๐‘กโˆˆ๐ฝโ€–๐‘ง(๐‘ก)โ€–๐‘‹+โ€–โ€–๐‘ง0โ€–โ€–๐”…โ„Ž=sup๐‘กโˆˆ๐ฝโ€–๐‘ง(๐‘ก)โ€–๐‘‹,๐‘งโˆˆ๐”…โ„Ž๎…ž๎…ž,(3.5) thus (๐”…โ„Ž๎…ž๎…ž,โ€–โ‹…โ€–๐”…โ„Žโ€ฒโ€ฒ) is a Banach space. We define the operator ๐‘ƒโˆถ๐”…โ„Ž๎…ž๎…žโ†’๐”…โ„Ž๎…ž๎…ž by โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉ๎€œ(๐‘ƒ๐‘ง)(๐‘ก)=๐‘ก0๐‘†๐›ผ(๎€ท๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€บ๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ0,๐‘ก1๎€ป,๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก1๐‘ง๎€ท๐‘ก๎€ธ๎€บโˆ’1๎€ธ+๐ผ1๎€ท๐‘ง๎€ท๐‘กโˆ’1+๎€œ๎€ธ๎€ธ๎€ป๐‘ก๐‘ก1๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ก๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก๐‘š๐‘ง๎€ท๐‘ก๎€ธ๎€บโˆ’๐‘š๎€ธ+๐ผ๐‘š๎€ท๐‘ง๎€ท๐‘กโˆ’๐‘š+๎€œ๎€ธ๎€ธ๎€ป๐‘ก๐‘ก๐‘š๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ก๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ๐‘š๎€ป.,๐‘‡(3.6) It is clear that the operator ๐‘ has a unique fixed-point if and only if ๐‘ƒ has a unique fixed-point. To prove that ๐‘ƒ has a unique fixed-point, let ๐‘ง,๐‘งโˆ—โˆˆ๐”…โ„Ž๎…ž๎…ž, then for all ๐‘กโˆˆ[0,๐‘ก1]. We have โ€–๐‘ƒ(๐‘ง)(๐‘ก)โˆ’๐‘ƒ(๐‘งโˆ—)(๐‘ก)โ€–๐‘‹โ‰ค๎€œ๐‘ก0โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘ง(๐‘ )))โˆ’๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘งโˆ—๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ง๐‘ โˆ’๐‘งโˆ—๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–๐ต๎€ท๐‘ฆ(๐‘ )+๎€ธ๎€ท๐‘ง(๐‘ )โˆ’๐ต๐‘ฆ(๐‘ )+๐‘งโˆ—๎€ธโ€–โ€–๐‘‹๎‚„โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ.(3.7) For ๐‘กโˆˆ(๐‘ก1,๐‘ก2], we have โ€–๐‘ƒ(๐‘ง)(๐‘ก)โˆ’๐‘ƒ(๐‘งโˆ—)(๐‘ก)โ€–๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘กโˆ’๐‘ก1)โ€–โ€–๐ฟ(๐‘‹)๎€บโ€–โ€–๐‘ง(๐‘กโˆ’1)โˆ’๐‘งโˆ—(๐‘กโˆ’1)โ€–โ€–๐‘‹+โ€–โ€–๐ผ1(๐‘ง(๐‘กโˆ’1))โˆ’๐ผ1(๐‘งโˆ—(๐‘กโˆ’1โ€–โ€–))๐‘‹๎€ป+๎€œ๐‘ก๐‘ก1โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘ง(๐‘ )))โˆ’๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘งโˆ—๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‡๎€บโ€–โ€–๐‘ง(๐‘กโˆ’1)โˆ’๐‘งโˆ—(๐‘กโˆ’1)โ€–โ€–๐‘‹+๐œŒ1โ€–โ€–๐‘ง(๐‘กโˆ’1)โˆ’๐‘งโˆ—(๐‘กโˆ’1)โ€–โ€–๐‘‹๎€ป+๎‚‹๐‘€๐‘†๎€œ๐‘ก๐‘ก1(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ง๐‘ โˆ’๐‘งโˆ—๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–๐ต๎€ท๐‘ฆ(๐‘ )+๐‘ง๎€ธ๎€ท๐‘ฆ(๐‘ )โˆ’๐ต(๐‘ )+๐‘งโˆ—๎€ธโ€–โ€–๐‘‹๎‚„โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘‡๎€ท1+๐œŒ1๎€ธโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ+๎‚‹๐‘€๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ.(3.8) Similarly, when ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1],๐‘–=2,โ€ฆ,๐‘š, we get โ€–๐‘ƒ(๐‘ง)(๐‘ก)โˆ’๐‘ƒ(๐‘งโˆ—)(๐‘ก)โ€–๐‘‹โ‰ค๎‚‹๐‘€๐‘‡๎€ท1+๐œŒ๐‘–๎€ธโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ+๎‚‹๐‘€๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ.(3.9) Thus, for all ๐‘กโˆˆ[0,๐‘‡], we have โ€–๐‘ƒ(๐‘ง)โˆ’๐‘ƒ(๐‘งโˆ—)โ€–๐”…โ„Žโ€ฒโ€ฒโ‰คmax1โ‰ค๐‘–โ‰ค๐‘š๎ƒฏ๎‚‹๐‘€๐‘‡๎€ท1+๐œŒ๐‘–๎€ธ+๎‚‹๐‘€๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผ๎ƒฐโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ.(3.10) Hence, ๐‘ƒ is a contraction map, and therefore it has an unique fixed-point ๐‘งโˆˆ๐”…โ„Ž๎…ž๎…ž, which is a mild solution of (1.1) on (โˆ’โˆž,๐‘‡]. This completes the proof of the theorem.

The second result is established using the following Krasnoselkiiโ€™s fixed-point theorem.

Theorem 3.2. Let ๐ต be a closed-convex and nonempty subset of a Banach space ๐‘‹. Let ๐‘ƒ and ๐‘„ be two operators such that (๐‘–)๐‘ƒ๐‘ฅ+๐‘„๐‘ฆโˆˆ๐ต whenever ๐‘ฅ,๐‘ฆโˆˆ๐ต, (๐‘–๐‘–)๐‘ƒ is compact and continuous; (๐‘–๐‘–๐‘–)๐‘„ is a contraction mapping, then there exists ๐‘งโˆˆ๐ต such that ๐‘ง=๐‘ƒ๐‘ง+๐‘„๐‘ง.

Now, we make the following assumptions: (H4)๐‘“โˆถ๐ฝร—๐”…โ„Žร—๐‘‹โ†’๐‘‹ is continuous, and there exist two continuous functions ๐œ‡1,๐œ‡2โˆถ๐ฝโ†’(0,โˆž) such that โ€–๐‘“(๐‘ก,๐œ“,๐‘ฅ)โ€–๐‘‹โ‰ค๐œ‡1(๐‘ก)โ€–๐œ“โ€–๐”…โ„Ž+๐œ‡2(๐‘ก)โ€–๐‘ฅโ€–๐‘‹,(๐‘ก,๐œ“,๐‘ฅ)โˆˆ๐ฝร—๐”…ร—๐‘‹.(3.11)(H5)the function ๐ผ๐‘˜โˆถ๐‘‹โ†’๐‘‹ is continuous, and there exists ฮฉ>0 such that ฮฉ=max1โ‰ค๐‘˜โ‰ค๐‘š,๐‘ฅโˆˆ๐ต๐‘Ÿ๎€ฝโ€–โ€–๐ผ๐‘˜โ€–โ€–(๐‘ฅ)๐‘‹๎€พ.(3.12)

Before going further, we need the following lemma.

Lemma 3.3 (see Lemmaโ€‰โ€‰3.2 in [7]). Let ๐ถโˆ—1=sup0<๐œ<๐‘‡๐ถ1(๐œ),๐ถโˆ—2=sup0<๐œ<๐‘‡๐ถ2(๐œ),๐œ‡โˆ—1=sup0<๐œ<๐‘‡๐œ‡1(๐œ),๐œ‡โˆ—2=sup0<๐œ<๐‘‡๐œ‡2(๐œ)(3.13) then for any ๐‘ โˆˆ๐ฝ, ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘ง๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ ))๐‘‹โ‰ค๐œ‡โˆ—1๎‚ธ๐ถโˆ—2โ€–๐œ™โ€–๐”…โ„Ž+๐ถโˆ—1sup0โ‰ค๐œโ‰ค๐‘ โ€–โ€–๐‘ง(๐œ)๐‘‹๎‚น+๐œ‡โˆ—2๎€œ๐‘ 0๐พ(๐‘ ,๐œ)โ€–๐‘ง(๐œ)โ€–๐‘‹๐‘‘๐œ.(3.14) If โ€–๐‘งโ€–๐‘‹<๐‘Ÿ,๐‘Ÿ>0, then ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘ง๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ ))๐‘‹โ‰ค๐œ‡โˆ—1๎€บ๐ถโˆ—2โ€–๐œ™โ€–๐”…โ„Ž+๐ถโˆ—1๐‘Ÿ๎€ป+๐œ‡โˆ—2๐‘Ÿ๐ตโˆ—=๐œ†.(3.15)

Theorem 3.4. Suppose that the assumptions (H1), (H4), (H5) are satisfied with ๎ƒฌ๎‚‹๐‘€๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผ๎ƒญ<1,(3.16) then the impulsive problem (1.1) has at least one mild solution on (โˆ’โˆž,๐‘‡].

Proof. Choose ๎‚‹๐‘€๐‘Ÿโ‰ฅ[๐‘‡๎‚‹๐‘€(๐‘Ÿ+ฮฉ)+(๐‘†๐‘‡๐›ผ๐œ†/๐›ผ)] and consider ๐ต๐‘Ÿ={๐‘งโˆˆ๐”…โ„Ž๎…ž๎…žโˆถโ€–๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๐‘Ÿ}, then ๐ต๐‘Ÿ is a bounded, closed-convex subset in ๐”…โ„Ž๎…ž๎…ž.
Let ฮ“1โˆถ๐ต๐‘Ÿโ†’๐ต๐‘Ÿ and ฮ“2โˆถ๐ต๐‘Ÿโ†’๐ต๐‘Ÿ be defined as ๎€ทฮ“1๐‘ง๎€ธโŽงโŽชโŽจโŽชโŽฉ๎€บ(๐‘ก)=0,๐‘กโˆˆ0,๐‘ก1๎€ป,๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก1๐‘ง๎€ท๐‘ก๎€ธ๎€บโˆ’1๎€ธ+๐ผ1๎€ท๐‘ง๎€ท๐‘กโˆ’1๎€ท๐‘ก๎€ธ๎€ธ๎€ป,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๐‘‡๐›ผ๎€ท๐‘กโˆ’๐‘ก๐‘š๐‘ง๎€ท๐‘ก๎€ธ๎€บโˆ’๐‘š๎€ธ+๐ผ๐‘š๎€ท๐‘ง๎€ท๐‘กโˆ’๐‘š๎€ท๐‘ก๎€ธ๎€ธ๎€ป,๐‘กโˆˆ๐‘š๎€ป,๎€ทฮ“,๐‘‡(3.17)2๐‘ง๎€ธโŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๎€œ(๐‘ก)=๐‘ก0๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€บ๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ0,๐‘ก1๎€ป,๎€œ๐‘ก๐‘ก1๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ก๐‘ง(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ1,๐‘ก2๎€ป,โ‹ฎ๎€œ๐‘ก๐‘ก๐‘š๐‘†๐›ผ๎€ท(๐‘กโˆ’๐‘ )๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท๐‘ฆ,๐ต(๐‘ )+๐‘ง๎€ท๐‘ก(๐‘ )๎€ธ๎€ธ๐‘‘๐‘ ,๐‘กโˆˆ๐‘š๎€ป.,๐‘‡(3.18)
Step 1. Let ๐‘ง,๐‘งโˆ—โˆˆ๐ต๐‘Ÿ, then show that ฮ“1๐‘ง+ฮ“2๐‘งโˆ—โˆˆ๐ต๐‘Ÿ, for ๐‘กโˆˆ[0,๐‘ก1], we have โ€–โ€–(ฮ“1๐‘ง)(๐‘ก)+(ฮ“2๐‘งโˆ—โ€–โ€–)(๐‘ก)๐‘‹โ‰ค๎€œ๐‘ก0โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘งโˆ—๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘งโˆ—๐‘ โ€–โ€–๐”…โ„Žโ€ฒโ€ฒ+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ ))๐‘‹๎‚„๐‘‘๐‘ ,(3.19) and by using Lemma 3.3, we conclude that โ€–โ€–(ฮ“1๐‘ง)+(ฮ“2๐‘งโˆ—)โ€–โ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๎‚‹๐‘€๐‘†๐œ†๐‘‡๐›ผ๐›ผ<๐‘Ÿ.(3.20) Similarly, when ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], ๐‘–=1,โ€ฆ,๐‘š, we have the estimate โ€–โ€–(ฮ“1๐‘ง)(๐‘ก)+(ฮ“2๐‘งโˆ—โ€–โ€–)(๐‘ก)๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘กโˆ’๐‘ก๐‘–)[๐‘ง(๐‘กโˆ’๐‘–)+๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))]๐‘‹+๎€œ๐‘ก๐‘ก๐‘–โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘งโˆ—๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘‡๎‚€โ€–๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒ+โ€–โ€–๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎‚+๎€œ๐‘ก๐‘ก๐‘–โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)๎‚ƒ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘งโˆ—๐‘ โ€–โ€–๐”…โ„Žโ€ฒโ€ฒ+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ ))๐‘‹๎‚„โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘‡๎‚‹๐‘€(๐‘Ÿ+ฮฉ)+๐‘†๐‘‡๐›ผ๐œ†๐›ผ<๐‘Ÿ,(3.21) which implies that โ€–ฮ“1๐‘ง+ฮ“2๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๐‘Ÿ.Step 2. We will show that the mapping (ฮ“1๐‘ง)(๐‘ก) is continuous on ๐ต๐‘Ÿ. For this purpose, let {๐‘ง๐‘›}โˆž๐‘›=1 be a sequence in ๐ต๐‘Ÿ with lim๐‘ง๐‘›โ†’๐‘งโˆˆ๐ต๐‘Ÿ, then for ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], ๐‘–=0,1,โ€ฆ,๐‘š, we have โ€–โ€–(ฮ“1๐‘ง๐‘›)(๐‘ก)โˆ’(ฮ“1โ€–โ€–๐‘ง)(๐‘ก)๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘กโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹)๎€บโ€–โ€–๐‘ง๐‘›(๐‘กโˆ’๐‘–)โˆ’๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–(๐‘ง๐‘›(๐‘กโˆ’๐‘–))โˆ’๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎€ป.(3.22) Since the functions ๐ผ๐‘–,๐‘–=1,2,โ€ฆ,๐‘š are continuous, hence lim๐‘›โ†’โˆžฮ“1๐‘ง๐‘›=ฮ“1๐‘ง in ๐ต๐‘Ÿ which implies that the mapping ฮ“1 is continuous on ๐ต๐‘Ÿ.Step 3. Uniform boundedness of the map (ฮ“1๐‘ง)(๐‘ก) is an implication of the following inequality: for ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], ๐‘–=0,1,โ€ฆ,๐‘š, we have โ€–โ€–(ฮ“1โ€–โ€–๐‘ง)(๐‘ก)๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘กโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹)๎€บโ€–โ€–๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎€ปโ‰ค๎‚‹๐‘€๐‘‡(๐‘Ÿ+ฮฉ).(3.23)Step 4. To show that the map (3.17) is equicontinuous, we proceed as follows. Let ๐‘ข,๐‘ฃโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], ๐‘ก๐‘–โ‰ค๐‘ข<๐‘ฃโ‰ค๐‘ก๐‘–+1, ๐‘–=0,1,โ€ฆ,๐‘š, ๐‘งโˆˆ๐ต๐‘Ÿ, then we obtain โ€–โ€–(ฮ“1๐‘ง)(๐‘ฃ)โˆ’(ฮ“1โ€–โ€–๐‘ง)(๐‘ข)๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘ฃโˆ’๐‘ก๐‘–)โˆ’๐‘‡๐›ผ(๐‘ขโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹)โ€–โ€–๐‘ง(๐‘กโˆ’๐‘–)+๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹โ€–โ€–๐‘‡โ‰ค(๐‘Ÿ+ฮฉ)๐›ผ(๐‘ฃโˆ’๐‘ก๐‘–)โˆ’๐‘‡๐›ผ(๐‘ขโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹).(3.24) Since ๐‘‡๐›ผ is strongly continuous, the continuity of the function ๐‘กโ†ฆโ€–๐‘‡(๐‘ก)โ€– allows us to conclude that lim๐‘ขโ†’๐‘ฃโ€–๐‘‡๐›ผ(๐‘ฃโˆ’๐‘ก๐‘–)โˆ’๐‘‡๐›ผ(๐‘ขโˆ’๐‘ก๐‘–)โ€–๐ฟ(๐‘‹)=0, which implies that ฮ“1(๐ต๐‘Ÿ) is equicontinuous. Finally, combining Step 1 to Step 4 together with Ascoliโ€™s theorem, we conclude that the operator ฮ“1 is compact.
Now, it only remains to show that the map ฮ“2 is a contraction mapping. Let ๐‘ง,๐‘งโˆ—โˆˆ๐ต๐‘Ÿ and ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], ๐‘–=0,1,โ€ฆ,๐‘š, then we have โ€–โ€–(ฮ“2๐‘ง)(๐‘ก)โˆ’(ฮ“2๐‘งโˆ—โ€–โ€–)(๐‘ก)๐‘‹โ‰ค๎€œ๐‘ก๐‘ก๐‘–โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“๎€ท๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๎€ท๐‘ง(๐‘ )๎€ธ๎€ธโˆ’๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘งโˆ—๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๐‘งโˆ—โ€–โ€–(๐‘ )๎€ธ๎€ธ๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๎€œ๐‘ก๐‘ก๐‘–(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ง๐‘ โˆ’๐‘งโˆ—๐‘ โ€–โ€–๐”…โ„Žโ€ฒโ€ฒ+๐œ‡2โ€–โ€–๐ต๎€ท๐‘ฆ(๐‘ )+๎€ธ๎€ท๐‘ง(๐‘ )โˆ’๐ต๐‘ฆ(๐‘ )+๐‘งโˆ—๎€ธโ€–โ€–(๐‘ )๐‘‹๎‚„โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๐›ผ๎€ท๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—๎€ธ๐‘‡๐›ผโ€–๐‘งโˆ’๐‘งโˆ—โ€–๐”…โ„Žโ€ฒโ€ฒ,(3.25) since (๎‚‹๐‘€๐‘†/๐›ผ)(๐œ‡1๐ถโˆ—1+๐œ‡2๐ตโˆ—)๐‘‡๐›ผ<1, which implies that ฮ“2 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 (โˆ’โˆž,๐‘‡]. This completes the proof of the theorem.

Our last result is based on the following Schaeferโ€™s fixed-point theorem.

Theorem 3.5. Let ๐‘ƒ be a continuous and compact mapping on a Banach space ๐‘‹ into itself, such that the set {๐‘ฅโˆˆ๐‘‹โˆถ๐‘ฅ=๐œˆ๐‘ƒ๐‘ฅ๐‘“๐‘œ๐‘Ÿ๐‘ ๐‘œ๐‘š๐‘’0โ‰ค๐œˆโ‰ค1} is bounded, then ๐‘ƒ has a fixed-point.

Lemma 3.6 (see [5]). Let ๐‘ฃโˆถ[0,๐‘‡]โ†’[0,โˆž) be a real function, ๐‘ค(โ‹…) is nonnegative and locally integrable function on [0,๐‘‡], and there are constants ๐‘Ž>0 and 0<๐›ผ<1 such that ๎€œ๐‘ฃ(๐‘ก)โ‰ค๐‘ค(๐‘ก)+๐‘Ž๐‘ก0๐‘ฃ(๐‘ )(๐‘กโˆ’๐‘ )๐›ผ๐‘‘๐‘ .(3.26) Then there exists a constant ๐พ(๐›ผ) such that ๎€œ๐‘ฃ(๐‘ก)โ‰ค๐‘ค(๐‘ก)+๐‘Ž๐พ(๐›ผ)๐‘ก0๐‘ค(๐‘ )(๐‘กโˆ’๐‘ )๐›ผ[].๐‘‘๐‘ ,forevery๐‘กโˆˆ0,๐‘‡(3.27)

Theorem 3.7. Assume that the assumptions (H4)-(H5) are satisfied, and if ๐ดโˆˆ๐ด๐›ผ(๐œƒ0,๐œ”0) and ๎‚‹๐‘€๐‘‡<1, then the impulsive problem (1.1) has at least one mild solution on (โˆ’โˆž,๐‘‡].

Proof. We define the operator ๐‘ƒโˆถ๐”…โ„Ž๎…ž๎…žโ†’๐”…โ„Ž๎…ž๎…ž as in Theoremโ€‰โ€‰3.3. Note that ๐‘ƒ is well defined in ๐”…โ„Ž๎…ž๎…ž.We complete the proof in the following steps.Step 1. For the continuity of the map ๐‘ƒ, let {๐‘ง๐‘›} be a sequence in ๐”…โ„Ž๎…ž๎…ž such that ๐‘ง๐‘›โ†’๐‘ง in ๐”…โ„Ž๎…ž๎…ž. Since the function ๐‘“ is continuous on ๐ฝร—๐”…โ„Žร—๐‘‹, This implies that ๐‘“๎€ท๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘›๐‘ ๎€ท๐‘ฆ,๐ต(๐‘ )+๐‘ง๐‘›๎€ท(๐‘ )๎€ธ๎€ธโŸถ๐‘“๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ๎€ท,๐ต๐‘ฆ(๐‘ )+๐‘ง(๐‘ )๎€ธ๎€ธas๐‘›โŸถโˆž.(3.28) Now, for every ๐‘กโˆˆ[0,๐‘ก1], we get โ€–๐‘ƒ๐‘ง๐‘›(๐‘ก)โˆ’๐‘ƒ๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎€œ๐‘ก0โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘›๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘ง๐‘›(๐‘ )))โˆ’๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๐‘‡๐›ผ๐›ผ๐œ€,(3.29) where ๐œ€>0,๐œ€โ†’0 as ๐‘›โ†’โˆž. Moreover, we have โ€–๐‘ƒ๐‘ง๐‘›(๐‘ก)โˆ’๐‘ƒ๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎‚‹๐‘€๐‘‡๎€บโ€–โ€–๐‘ง๐‘›(๐‘กโˆ’๐‘–)โˆ’๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–(๐‘ง๐‘›(๐‘กโˆ’๐‘–))โˆ’๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎€ป+๎€œ๐‘ก๐‘ก๐‘–โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘›๐‘ ,๐ต(๐‘ฆ(๐‘ )+๐‘ง๐‘›(๐‘ )))โˆ’๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘‡๎€บโ€–โ€–๐‘ง๐‘›(๐‘กโˆ’๐‘–)โˆ’๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–(๐‘ง๐‘›(๐‘กโˆ’๐‘–))โˆ’๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎€ป+๎‚‹๐‘€๐‘†๐‘‡๐›ผ๐›ผ๐œ€,(3.30) where ๐œ€>0,๐œ€โ†’0 as ๐‘›โ†’โˆž, for all ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1],๐‘–=1,โ€ฆ,๐‘š. The impulsive functions ๐ผ๐‘˜,๐‘˜=1,โ€ฆ,๐‘š are continuous, then we get lim๐‘›โ†’โˆžโ€–P๐‘ง๐‘›โˆ’๐‘ƒ๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒ=0.(3.31) This implies that ๐‘ƒ is continuous.Step 2. ๐‘ƒ maps bounded sets into bounded sets in ๐”…โ„Ž๎…ž๎…ž. To prove that for any ๐‘Ÿ>0, there exists a ๐›พ>0 such that for each ๐‘งโˆˆ๐ต๐‘Ÿ={๐‘งโˆˆ๐”…โ„Ž๎…ž๎…žโˆถโ€–๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๐‘Ÿ}, then we have โ€–๐‘ƒ๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๐›พ, then for any ๐‘งโˆˆ๐ต๐‘Ÿ,๐‘กโˆˆ[0,๐‘ก1], we have โ€–๐‘ƒ๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎€œ๐‘ก0โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘†๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘ง๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ ))๐‘‹๎‚„๐‘‘๐‘ .(3.32) Using Lemma 3.3, we obtain โ€–๐‘ƒ๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎‚‹๐‘€๐‘†(๐‘‡๐›ผ/๐›ผ)๐œ†. Similarly, we have โ€–๐‘ƒ๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎‚‹๐‘€๐‘‡๎‚‹๐‘€(๐‘Ÿ+ฮฉ)+๐‘†๐‘‡๐›ผ๐›ผ๎€ท๐‘ก๐œ†,๐‘กโˆˆ๐‘–,๐‘ก๐‘–+1๎€ป,๐‘–=1,โ€ฆ,๐‘š.(3.33) This implies that โ€–๐‘ƒ๐‘งโ€–๐”…โ„Žโ€ฒโ€ฒโ‰ค๎‚‹๐‘€๐‘‡๎‚‹๐‘€(๐‘Ÿ+ฮฉ)+๐‘†๐‘‡๐›ผ๐›ผ[].๐œ†=๐›พ,๐‘กโˆˆ0,๐‘‡(3.34)Step 3. We will prove that ๐‘ƒ(๐ต๐‘Ÿ) is equicontinuous. Let ๐‘ข,๐‘ฃโˆˆ[0,๐‘ก1], with ๐‘ข<๐‘ฃ, we have โ€–๐‘ƒ๐‘ง(๐‘ฃ)โˆ’๐‘ƒ๐‘ง(๐‘ข)โ€–๐‘‹โ‰ค๎€œ๐‘ข0โ€–โ€–๐‘†๐›ผ(๐‘ฃโˆ’๐‘ )โˆ’๐‘†๐›ผโ€–โ€–(๐‘ขโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹+๎€œ๐‘‘๐‘ ๐‘ฃ๐‘ขโ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘ฃโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๐‘‘๐‘ โ‰ค๐‘„1+๐‘„2,(3.35) where ๐‘„1=๎€œ๐‘ข0โ€–โ€–๐‘†๐›ผ(๐‘ฃโˆ’๐‘ )โˆ’๐‘†๐›ผโ€–โ€–(๐‘ขโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๎€œ๐‘‘๐‘ โ‰ค๐œ†๐‘ข0โ€–โ€–๐‘†๐›ผ(๐‘ฃโˆ’๐‘ )โˆ’๐‘†๐›ผโ€–โ€–(๐‘ขโˆ’๐‘ )๐ฟ(๐‘‹)๐‘‘๐‘ .(3.36) Since โ€–๐‘†๐›ผ(๐‘ฃโˆ’๐‘ )โˆ’๐‘†๐›ผ(๐‘ขโˆ’๐‘ )โ€–๐ฟ(๐‘‹)๎‚‹๐‘€โ‰ค2๐‘†(๐‘ก1โˆ’๐‘ )๐›ผโˆ’1โˆˆ๐ฟ1(๐ผ,โ„+) for ๐‘ โˆˆ[0,๐‘ก1] and ๐‘†๐›ผ(๐‘ฃโˆ’๐‘ )โˆ’๐‘†๐›ผ(๐‘ขโˆ’๐‘ )โ†’0 as ๐‘ขโ†’๐‘ฃ, ๐‘†๐›ผ is strongly continuous. This implies that lim๐‘ขโ†’๐‘ฃ๐‘„1=0, ๐‘„2=๎€œ๐‘ฃ๐‘ขโ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘ฃโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๎‚‹๐‘€๐‘‘๐‘ โ‰ค๐œ†๐‘†(๐‘ฃโˆ’๐‘ข)๐›ผ๐›ผ.(3.37) Hence, lim๐‘ขโ†’๐‘ฃ๐‘„2=0. Similarly, for ๐‘ข,๐‘ฃโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1], with ๐‘ข<๐‘ฃ,๐‘–=1,โ€ฆ,๐‘š, we have โ€–๐‘ƒ๐‘ง(๐‘ฃ)โˆ’๐‘ƒ๐‘ง(๐‘ข)โ€–๐‘‹โ‰คโ€–โ€–๐‘‡๐›ผ(๐‘ฃโˆ’๐‘ก๐‘–)โˆ’๐‘‡๐›ผ(๐‘ขโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹)๎€บโ€–โ€–๐‘ง๎€ท๐‘กโˆ’๐‘–๎€ธโ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–๎€ท๐‘ง๎€ท๐‘กโˆ’๐‘–โ€–โ€–๎€ธ๎€ธ๐‘‹๎€ป+๐‘„1+๐‘„2.(3.38) Since ๐‘‡๐›ผ is also strongly continuous, so ๐‘‡๐›ผ(๐‘ฃโˆ’๐‘ก๐‘–)โˆ’๐‘‡๐›ผ(๐‘ขโˆ’๐‘ก๐‘–)โ†’0 as ๐‘ขโ†’๐‘ฃ. Thus, from the above inequalities, we have lim๐‘ขโ†’๐‘ฃโ€–๐‘ƒ๐‘ง(๐‘ฃ)โˆ’๐‘ƒ๐‘ง(๐‘ข)โ€–๐‘‹=0. So, ๐‘ƒ(๐ต๐‘Ÿ) is equicontinuous. Finally, combining Step 1 to Step 3 with Ascoliโ€™s theorem, we conclude that the operator ๐‘ƒ is compact.Step 4. We show that the set ๎€ฝ๐ธ=๐‘งโˆˆ๐”…โ„Ž๎…ž๎…ž๎€พsuchthat๐‘ง=๐œˆ๐‘ƒ๐‘งforsome0<๐œˆ<1(3.39) is bounded. Let ๐‘งโˆˆ๐ธ, then ๐‘ง(๐‘ก)=๐œˆ๐‘ƒ๐‘ง(๐‘ก) for some 0<๐œˆ<1. Then for each ๐‘กโˆˆ[0,๐‘ก1], we have โ€–๐‘ง(๐‘ก)โ€–๐‘‹๎€œโ‰ค๐œˆ๐‘ก0โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๎‚‹๐‘€๐‘‘๐‘ โ‰ค๐œˆ๐‘†๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๐‘‘๐‘ ,(3.40) for ๐‘กโˆˆ(๐‘ก๐‘–,๐‘ก๐‘–+1],๐‘–=1,โ€ฆ,๐‘š, we get โ€–๐‘ง(๐‘ก)โ€–๐‘‹๎‚ธโ€–โ€–๐‘‡โ‰ค๐œˆ๐›ผ(๐‘กโˆ’๐‘ก๐‘–)โ€–โ€–๐ฟ(๐‘‹)๎€ทโ€–โ€–๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+โ€–โ€–๐ผ๐‘–(๐‘ง(๐‘กโˆ’๐‘–โ€–โ€–))๐‘‹๎€ธ+๎€œ๐‘ก๐‘ก๐‘–โ€–โ€–๐‘†๐›ผโ€–โ€–(๐‘กโˆ’๐‘ )๐ฟ(๐‘‹)ร—โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๎€ปโ‰ค๎‚ธ๎‚‹๐‘€๐‘‘๐‘ ๐‘‡โ€–โ€–๐‘ง(๐‘กโˆ’๐‘–)โ€–โ€–๐‘‹+๎‚‹๐‘€๐‘‡๎‚‹๐‘€ฮฉ+๐‘†๎€œ๐‘ก๐‘ก๐‘–(๐‘กโˆ’๐‘ )๐›ผโˆ’1โ€–โ€–๐‘“(๐‘ ,๐‘ฆ๐‘ +๐‘ง๐‘ ,๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ )))๐‘‹๎‚น.๐‘‘๐‘ (3.41) then for all ๐‘กโˆˆ[0,๐‘‡], we have โ€–๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๎‚‹๐‘€๐‘‡ฮฉ๎‚‹๐‘€1โˆ’๐‘‡+๎‚‹๐‘€๐‘†๎‚‹๐‘€1โˆ’๐‘‡๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚ƒ๐œ‡1โ€–โ€–๐‘ฆ(๐‘ )๐‘ +๐‘ง๐‘ โ€–โ€–๐”…โ„Ž+๐œ‡2โ€–โ€–(๐‘ )๐ต(๐‘ฆ(๐‘ )+โ€–โ€–๐‘ง(๐‘ ))๐‘‹๎‚„โ‰ค๎‚‹๐‘€๐‘‘๐‘ ๐‘‡ฮฉ๎‚‹๐‘€1โˆ’๐‘‡+๎‚‹๐‘€๐‘†๐œ‡โˆ—1๐ถโˆ—2โ€–๐œ™โ€–๐”…โ„Ž๐‘‡๐›ผ๐›ผ๎‚€๎‚‹๐‘€1โˆ’๐‘‡๎‚+๎‚‹๐‘€๐‘†๎‚‹๐‘€1โˆ’๐‘‡๎€ท๐œ‡โˆ—1๐ถโˆ—1+๐œ‡โˆ—2๐ตโˆ—๎€ธ๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1sup0โ‰ค๐œโ‰ค๐‘ โ€–๐‘ง(๐œ)โ€–๐‘‹๐‘‘๐‘ โ‰ค๐œ”1+๐œ”2๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1sup0โ‰ค๐œโ‰ค๐‘ โ€–๐‘ง(๐œ)โ€–๐‘‹๐‘‘๐‘ ,(3.42) where ๐œ”1=๎‚‹๐‘€๐‘‡๎‚‹๐‘€ฮฉ/(1โˆ’๐‘‡๎‚‹๐‘€)+๐‘†๐œ‡โˆ—1๐ถโˆ—2โ€–๐œ™โ€–๐”…โ„Ž๐‘‡๐›ผ๎‚‹๐‘€/๐›ผ(1โˆ’๐‘‡) and ๐œ”2๎‚‹๐‘€=(๐‘†๎‚‹๐‘€/1โˆ’๐‘‡)(๐œ‡โˆ—1๐ถโˆ—1+๐œ‡โˆ—2๐ตโˆ—). Let ๐œโˆ—โˆˆ[0,๐‘ ] be such that sup0โ‰ค๐œโ‰ค๐‘ โ€–๐‘ง(๐œ)โ€–๐‘‹=โ€–๐‘ง(๐œโˆ—)โ€–๐‘‹,0โ‰ค๐‘ โ‰ค๐‘ก. If ๐œโˆ—โˆˆ[0,๐‘ก], then (3.43) can be written as โ€–๐‘ง(๐‘ก)โ€–๐‘‹โ‰ค๐œ”1+๐œ”2๎€œ๐‘ก0(๐‘กโˆ’๐‘ )๐›ผโˆ’1โ€–๐‘ง(๐‘ )โ€–๐‘‹๐‘‘๐‘ .(3.43) Using Lemma 3.6, there exists a constant ๐พ(๐›ผ), and (3.43) becomes (โ€–๐‘ง๐‘ก)โ€–๐‘‹โ‰ค๐œ”1๎‚ธ๎€œ1+๐‘ก0๐พ(๐›ผ)(๐‘กโˆ’๐‘ )๐›ผโˆ’1๎‚น๐‘‘๐‘ โ‰ค๐œ”1๎‚ธ1+๐พ(๐›ผ)๐‘‡๐›ผ๐›ผ๎‚น.(3.44) As a consequence of Schaeferโ€™s fixed-point theorem, we deduce that ๐‘ƒ has a fixed-point on (โˆ’โˆž,๐‘‡]. 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 form๐ท๐‘ž๐‘ก๐œ•๐‘ข(๐‘ก,๐‘ฅ)=2๐œ•๐‘ฅ2๎€œ๐‘ข(๐‘ก,๐‘ฅ)+๐‘กโˆ’โˆž+๎€œ๐ป(๐‘ก,๐‘ฅ,๐‘ โˆ’๐‘ก)๐‘„(๐‘ข(๐‘ ,๐‘ฅ))๐‘‘๐‘ ๐‘ก0๐‘˜(๐‘ ,๐‘ก)๐‘’โˆ’๐‘ข(๐‘ ,๐‘ฅ)[][]๐‘‘๐‘ ,๐‘ฅโˆˆ0,๐œ‹,๐‘กโˆˆ0,๐‘,๐‘กโ‰ ๐‘ก๐‘˜,๐‘ข][],๎€ท๐‘ก๐‘ข(๐‘ก,0)=0=๐‘ข(๐‘ก,๐œ‹),๐‘กโ‰ฅ0,(๐‘ก,๐‘ฅ)=๐œ™(๐‘ก,๐‘ฅ),๐‘กโˆˆ(โˆ’โˆž,0,๐‘ฅโˆˆ0,๐œ‹ฮ”๐‘ข๐‘–๎€ธ๎€œ(๐‘ฅ)=๐‘ก๐‘–โˆ’โˆž๐‘ž๐‘–๎€ท๐‘ก๐‘–๎€ธ[],โˆ’๐‘ ๐‘ข(๐‘ ,๐‘ฅ)๐‘‘๐‘ ,๐‘ฅโˆˆ0,๐œ‹(4.1) where ๐ท๐‘ž๐‘ก is Caputoโ€™s fractional derivative of order 0<๐‘ž<1, 0<๐‘ก1<๐‘ก2<โ‹ฏ<๐‘ก๐‘›<๐‘ are prefixed numbers, and ๐œ™โˆˆ๐”…โ„Ž. Let ๐‘‹=๐ฟ2[0,๐œ‹], and define the operator ๐ดโˆถ๐ท(๐ด)โŠ‚๐‘‹โ†’๐‘‹ by ๐ด๐‘ค=๐‘ค๎…ž๎…ž with the domain ๐ท(๐ด)โˆถ={๐‘คโˆˆ๐‘‹โˆถ๐‘ค, ๐‘คโ€ฒ are absolutely continuous, ๐‘ค๎…ž๎…žโˆˆ๐‘‹,๐‘ค(0)=0=๐‘ค(๐œ‹)}, then๐ด๐‘ค=โˆž๎“๐‘›=1๐‘›2๎€ท๐‘ค,๐‘ค๐‘›๎€ธ๐‘ค๐‘›,๐‘คโˆˆ๐ท(๐ด),(4.2) where ๐‘ค๐‘›โˆš(๐‘ฅ)=2/๐œ‹sin(๐‘›๐‘ฅ),๐‘›โˆˆโ„• is the orthogonal set of eigenvectors of ๐ด