Open Access

Nonlocal Impulsive Cauchy Problems for Evolution Equations

Advances in Difference Equations20102011:784161

DOI: 10.1155/2011/784161

Received: 17 October 2010

Accepted: 19 November 2010

Published: 25 November 2010

Abstract

Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required. An application to partial differential equations is also presented.

1. Introduction

Impulsive equations arise from many different real processes and phenomena which appeared in physics, chemical technology, population dynamics, biotechnology, medicine, and economics. They have in recent years been an object of investigations with increasing interest. For more information on this subject, see for instance, the papers (cf., e.g., [16]) and references therein.

On the other hand, Cauchy problems with nonlocal conditions are appropriate models for describing a lot of natural phenomena, which cannot be described using classical Cauchy problems. That is why in recent years they have been studied by many researchers (cf., e.g., [4, 712] and references therein).

In [4], the authors combined the two directions and studied firstly a class of nonlocal impulsive Cauchy problems for evolution equations by investigating the existence for mild (in generalized sense) solutions to the problems. In this paper, we study further the existence of solutions to the following nonlocal impulsive Cauchy problem for evolution equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq1_HTML.gif is the infinitesimal generator of an analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq3_HTML.gif is a real Banach space endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq4_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ2_HTML.gif
(1.2)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq8_HTML.gif are given continuous functions to be specified later.

By going a new way, that is, by combining operator semigroups, the techniques of approximate solutions, noncompact measures, and the fixed point theory, we obtain new existence results for problem (1.1), which generalize and improve some previous theorems since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required in the present paper.

The organization of this work is as follows. In Section 2, we recall some definitions, and facts about fractional powers of operators, mild solutions and Hausdorff measure of noncompactness. In Section 3, we give the existence results for problem (1.1) when the nonlocal item and impulsive functions are only assumed to be continuous. In Section 4, we give an example to illustrate our abstract results.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq9_HTML.gif be a real Banach space. We denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq10_HTML.gif the space of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq11_HTML.gif -valued continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq12_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ3_HTML.gif
(2.1)
and by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq13_HTML.gif the space of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq14_HTML.gif -valued Bochner integrable functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq15_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq16_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ4_HTML.gif
(2.2)
It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq17_HTML.gif is a Banach space with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ5_HTML.gif
(2.3)
In this paper, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq18_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq19_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ6_HTML.gif
(2.4)

Throughout this paper, we assume the following.

(H1) The operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq20_HTML.gif is the infinitesimal generator of a compact analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq21_HTML.gif on Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq23_HTML.gif (the resolvent set of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq24_HTML.gif ).

In the remainder of this work, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq25_HTML.gif .

Under the above conditions, it is possible to define the fractional power https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq27_HTML.gif , of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq28_HTML.gif as closed linear operators. And it is known that the following properties hold.

Theorem 2.1 (see [13, Pages 69–75]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq29_HTML.gif and assume that (H1) holds. Then,

  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq30_HTML.gif is a Banach space with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq31_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq32_HTML.gif ,

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq33_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq34_HTML.gif ,

     
  3. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq35_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq37_HTML.gif ,

     
  4. (4)

    for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq39_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq40_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq41_HTML.gif such that

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ7_HTML.gif
    (2.5)
     
  5. (5)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq42_HTML.gif is a bounded linear operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq43_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq44_HTML.gif ,

     
  6. (6)

    if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq45_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq46_HTML.gif .

     

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq47_HTML.gif that the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq48_HTML.gif endowed the graph norm from now on.

Definition 2.2.

A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq49_HTML.gif is said to be a mild solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq50_HTML.gif if the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq51_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq52_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq53_HTML.gif and the following integral equation is satisfied:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ8_HTML.gif
(2.6)
To discuss the compactness of subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq54_HTML.gif , we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq56_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ9_HTML.gif
(2.7)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq57_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq58_HTML.gif the set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ10_HTML.gif
(2.8)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq59_HTML.gif . Then it is easy to see that the following result holds.

Lemma 2.3.

A set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq60_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq61_HTML.gif if and only if the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq62_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq63_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq64_HTML.gif .

Next, we recall that the Hausdorff measure of noncompactness https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq65_HTML.gif on each bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq66_HTML.gif of Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq67_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ11_HTML.gif
(2.9)

Some basic properties of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq68_HTML.gif are given in the following Lemma.

Lemma 2.4 (see [14]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq69_HTML.gif be a real Banach space and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq70_HTML.gif be bounded. Then,

  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq71_HTML.gif is precompact if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq72_HTML.gif ;

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq73_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq75_HTML.gif mean the closure and convex hull of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq76_HTML.gif , respectively;

     
  3. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq77_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq78_HTML.gif ;

     
  4. (4)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq79_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq80_HTML.gif ;

     
  5. (5)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq81_HTML.gif ;

     
  6. (6)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq82_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq83_HTML.gif ;

     
  7. (7)

    let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq84_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq85_HTML.gif Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq86_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq87_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq88_HTML.gif being bounded.

     

We note that a continuous map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq89_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq90_HTML.gif -contraction if there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq91_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq92_HTML.gif for all bounded closed https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq93_HTML.gif .

Lemma 2.5 (see Darbo-Sadovskii's fixed point theorem in [14]).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq94_HTML.gif is bounded closed and convex, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq95_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq96_HTML.gif -contraction, then the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq97_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq98_HTML.gif .

3. Main Results

In this section, by using the techniques of approximate solutions and fixed points, we establish a result on the existence of mild solutions for the nonlocal impulsive problem (1.1) when the nonlocal item https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq99_HTML.gif and the impulsive functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq100_HTML.gif are only assumed to be continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq102_HTML.gif , respectively.

In practical applications, the values of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq103_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq104_HTML.gif near zero often do not affect https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq105_HTML.gif . For example, it is the case when
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ12_HTML.gif
(3.1)

So, to prove our main results, we introduce the following assumptions.

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq106_HTML.gif is a continuous function, and there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq107_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq108_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq109_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq110_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq111_HTML.gif . Moreover, there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq112_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq113_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq114_HTML.gif .

(H3)There exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq115_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq116_HTML.gif is a continuous function, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq117_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq118_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq120_HTML.gif . Moreover, there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq121_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ13_HTML.gif
(3.2)
for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq122_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq123_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ14_HTML.gif
(3.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq125_HTML.gif .

(H4)The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq126_HTML.gif is continuous a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq127_HTML.gif ; the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq128_HTML.gif is strongly measurable for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq129_HTML.gif . Moreover, for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq130_HTML.gif , there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq131_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq132_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq133_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq134_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ15_HTML.gif
(3.4)

(H5) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq135_HTML.gif is continuous for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq136_HTML.gif , and there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq137_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq138_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq140_HTML.gif .

We note that, by Theorem 2.1, there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq143_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ16_HTML.gif
(3.5)

For simplicity, in the following we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq144_HTML.gif and will substitute https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq145_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq146_HTML.gif below.

Theorem 3.1.

Let (H1)–(H5) hold. Then the nonlocal impulsive Cauchy problem (1.1) has at least one mild solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq147_HTML.gif , provided
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ17_HTML.gif
(3.6)
To prove the theorem, we need some lemmas. Next, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq148_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq149_HTML.gif the maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq150_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ18_HTML.gif
(3.7)
In addition, we introduce the decomposition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq151_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ19_HTML.gif
(3.8)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq153_HTML.gif .

Lemma 3.2.

Assume that all the conditions in Theorem 3.1 are satisfied. Then for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq154_HTML.gif , the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq155_HTML.gif defined by (3.7) has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq156_HTML.gif .

Proof.

To prove the existence of a fixed point for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq157_HTML.gif , we will use Darbu-Sadovskii's fixed point theorem.

Firstly, we prove that the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq158_HTML.gif is a contraction on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq159_HTML.gif . For this purpose, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq160_HTML.gif . Then for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq161_HTML.gif and by condition (H3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ20_HTML.gif
(3.9)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ21_HTML.gif
(3.10)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq162_HTML.gif is a contraction by condition (3.6).

Secondly, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq165_HTML.gif are completely continuous operators. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq166_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq167_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ22_HTML.gif
(3.11)
in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq168_HTML.gif . By the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq169_HTML.gif with respect to the second argument, we deduce that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq171_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq172_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq173_HTML.gif , and we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ23_HTML.gif
(3.12)
Then by the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq176_HTML.gif , and using the dominated convergence theorem, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ24_HTML.gif
(3.13)

in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq177_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq178_HTML.gif are continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq179_HTML.gif .

Next, for the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq180_HTML.gif we refer to the proof of [4, Theorem  3.1].

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq181_HTML.gif and any bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq182_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq183_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ25_HTML.gif
(3.14)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq184_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq185_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq186_HTML.gif by the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq187_HTML.gif . On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq188_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ26_HTML.gif
(3.15)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq189_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq190_HTML.gif , we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ27_HTML.gif
(3.16)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq191_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq192_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq193_HTML.gif is a compact operator.

Now, we prove the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq194_HTML.gif . For this purpose, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ28_HTML.gif
(3.17)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ29_HTML.gif
(3.18)
Thus according to Lemma 2.3, we only need to prove that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ30_HTML.gif
(3.19)
is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq195_HTML.gif , as the remaining cases for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq197_HTML.gif , can be dealt with in the same way; here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq198_HTML.gif is any bounded subset in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq199_HTML.gif . And, we recall that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq201_HTML.gif , which means that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ31_HTML.gif
(3.20)

Thus, by the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq202_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq203_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq204_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq205_HTML.gif .

Next, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq206_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ32_HTML.gif
(3.21)

Thus, the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq207_HTML.gif is equicontinuous due to the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq208_HTML.gif and the strong continuity of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq209_HTML.gif . By the Arzela-Ascoli theorem, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq210_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq211_HTML.gif . The same idea can be used to prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq212_HTML.gif is precompact for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq213_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq214_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq215_HTML.gif , that is, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq216_HTML.gif is compact.

Thus, for any bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq217_HTML.gif , we have by Lemma 2.4,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ33_HTML.gif
(3.22)

Hence, the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq218_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq219_HTML.gif -contraction in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq220_HTML.gif .

Now, in order to apply Lemma 2.5, it remains to prove that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq221_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq222_HTML.gif . Suppose this is not true; then for each positive integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq223_HTML.gif , there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq225_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq226_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ34_HTML.gif
(3.23)
Dividing on both sides by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq227_HTML.gif and taking the lower limit as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq228_HTML.gif , we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ35_HTML.gif
(3.24)

This is a contradiction with inequality (3.6). Therefore, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq229_HTML.gif such that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq230_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq231_HTML.gif into itself. By Darbu-Sadovskii's fixed point theorem, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq232_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq233_HTML.gif . This completes the proof.

Lemma 3.3.

Assume that all the conditions in Theorem 3.1 are satisfied. Then the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq234_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq235_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq236_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ36_HTML.gif
(3.25)

and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq237_HTML.gif is the constant in (H2).

Proof.

The proof will be given in several steps. In the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq238_HTML.gif is a number in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq239_HTML.gif .

Step 1.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq240_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq241_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq242_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq243_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ37_HTML.gif
(3.26)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq244_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq247_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq248_HTML.gif , and we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq249_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ38_HTML.gif
(3.27)
By condition (H3), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq250_HTML.gif is well defined and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq251_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ39_HTML.gif
(3.28)
On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq252_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq253_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq255_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ40_HTML.gif
(3.29)
Now, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq256_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ41_HTML.gif
(3.30)

By the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq257_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq258_HTML.gif , we get that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq259_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq260_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq262_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq263_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq264_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq265_HTML.gif .

By the same reasoning, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq266_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq267_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq268_HTML.gif , we claim that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq269_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq270_HTML.gif . In fact, (H3) implies that for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq272_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ42_HTML.gif
(3.31)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ43_HTML.gif
(3.32)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq273_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq274_HTML.gif .

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq275_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq276_HTML.gif , and so is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq277_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq278_HTML.gif .

Thus, by (3.29) and Lemma 2.4, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ44_HTML.gif
(3.33)

By (3.6), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq279_HTML.gif , which implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq280_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq281_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq282_HTML.gif .

Step 2.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq283_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq284_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq285_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ45_HTML.gif
(3.34)
and define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq286_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ46_HTML.gif
(3.35)
By (H3), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq287_HTML.gif is well defined and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq288_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ47_HTML.gif
(3.36)
So, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq289_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq290_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ48_HTML.gif
(3.37)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ49_HTML.gif
(3.38)
According to the proof of Step 1, we know that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ50_HTML.gif
(3.39)

are all precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq292_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq293_HTML.gif .

Next, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq294_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq295_HTML.gif . Firstly, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq296_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq297_HTML.gif . Thus according to Lemma 2.3, it remains to prove that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ51_HTML.gif
(3.40)
is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq298_HTML.gif . And, we recall that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq299_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq300_HTML.gif , which means that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ52_HTML.gif
(3.41)
By Step 1, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq301_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq302_HTML.gif . Without loss of generality, we may suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ53_HTML.gif
(3.42)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq303_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq304_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq305_HTML.gif . Thus, by the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq306_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq307_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ54_HTML.gif
(3.43)

as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq308_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq309_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq310_HTML.gif . And, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq311_HTML.gif , by the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq312_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq313_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq314_HTML.gif is also relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq315_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq316_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq317_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq318_HTML.gif .

Next, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq319_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ55_HTML.gif
(3.44)

Thus, the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq320_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq321_HTML.gif due to the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq322_HTML.gif and the strong continuity of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq323_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq324_HTML.gif . By the Arzela-Ascoli theorem, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq325_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq326_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq327_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq328_HTML.gif .

Thus, by Lemma 2.4, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ56_HTML.gif
(3.45)

By (3.6), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq329_HTML.gif , which implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq330_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq331_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq332_HTML.gif .

Step 3.

The same idea can be used to prove the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq333_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq334_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq335_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq336_HTML.gif . This completes the proof.

Proof of Theorem 3.1.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq337_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq338_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ57_HTML.gif
(3.46)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq339_HTML.gif comes from the condition (H2). Then, by condition (H2), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq340_HTML.gif .

By Lemma 3.3, without loss of generality, we may suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq341_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq342_HTML.gif . Thus, by the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq343_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq344_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ58_HTML.gif
(3.47)
as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq345_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ59_HTML.gif
(3.48)
is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq346_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq347_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq348_HTML.gif are both precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq349_HTML.gif . And https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq350_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq351_HTML.gif . Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ60_HTML.gif
(3.49)
Therefore, by Lemma 2.4, we know that the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq352_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq353_HTML.gif . Without loss of generality, we may suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq354_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq355_HTML.gif . On the other hand, we also have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ61_HTML.gif
(3.50)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq356_HTML.gif in both sides, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ62_HTML.gif
(3.51)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq357_HTML.gif is a mild solution of the nonlocal impulsive problem (1.1). This completes the proof.

Remark 3.4.

From Lemma 3.3 and the above proof, it is easy to see that we can also prove Theorem 3.1 by showing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq358_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq359_HTML.gif .

The following results are immediate consequences of Theorem 3.5.

Theorem 3.5.

Assume (H1), (H3)–(H5) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq360_HTML.gif , then the impulsive Cauchy problem (1.1) has at least one mild solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq361_HTML.gif , provided
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ63_HTML.gif
(3.52)

Theorem 3.6.

Assume (H1), (H2), (H4), and (H5) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq362_HTML.gif , then the nonlocal impulsive problem (1.1) has at least one mild solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq363_HTML.gif , provided https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq364_HTML.gif .

Theorem 3.7.

Assume (H1), (H4), and (H5) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq365_HTML.gif , then the impulsive problem (1.1) has at least one mild solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq366_HTML.gif , provided https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq367_HTML.gif .

Remark 3.8.

Theorems  3.5-3.6 are new even for many special cases discussed before, since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required.

4. Application

In this section, to illustrate our abstract result, we consider the following differential system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ64_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq368_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq369_HTML.gif are given real numbers for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq370_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq371_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq372_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq373_HTML.gif are functions to be specified below.

To treat the above system, we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq374_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq375_HTML.gif and we consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq376_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ65_HTML.gif
(4.2)
with domain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ66_HTML.gif
(4.3)

The operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq377_HTML.gif is the infinitesimal generator of an analytic compact semigroup https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq378_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq379_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq380_HTML.gif has a discrete spectrum, the eigenvalues are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq381_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq382_HTML.gif , with the corresponding normalized eigenvectors https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq383_HTML.gif , and the following properties are satisfied.

  1. (a)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq384_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq385_HTML.gif .

     
  2. (b)

    For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq386_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq387_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq388_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq389_HTML.gif .

     
  3. (c)

    For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq390_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq391_HTML.gif . In particular, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq392_HTML.gif .

     
  4. (d)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq393_HTML.gif is given by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq394_HTML.gif with the domain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq395_HTML.gif .

     

Assume the following.

  1. (1)
    The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq396_HTML.gif is continuously differential with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq397_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq398_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq399_HTML.gif , and there exists a real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq400_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq401_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq402_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq403_HTML.gif . Moreover,
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ67_HTML.gif
    (4.4)
     
  2. (2)

    For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq404_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq405_HTML.gif is continuous, and for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq406_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq407_HTML.gif is measurable and, there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq408_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq409_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq410_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq411_HTML.gif .

     
  3. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq412_HTML.gif is a continuous function for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq413_HTML.gif , and there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq414_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq415_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq416_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq417_HTML.gif .

     
Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq418_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq419_HTML.gif , respectively, as follows. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq420_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ68_HTML.gif
(4.5)
From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq421_HTML.gif and assumption (1), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ69_HTML.gif
(4.6)
Thus, system (4.1) can be transformed into the abstract problem (1.1), and conditions (H2), (H3), (H4), and (H5) are satisfied with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_Equ70_HTML.gif
(4.7)

If (3.6) holds (it holds when the related constants are small), then according to Theorem 3.1, the problem (4.1) has at least one mild solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F784161/MediaObjects/13662_2010_Article_71_IEq422_HTML.gif .

Declarations

Acknowledgments

The authors would like to thank the referees for helpful comments and suggestions. J. Liang acknowledges support from the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). Z. Fan acknowledges support from the NSF of China (11001034) and the Research Fund for Shanghai Postdoctoral Scientific Program (10R21413700).

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Jiao Tong University
(2)
Department of Mathematics, Changshu Institute of Technology

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Copyright

© J. Liang and Z. Fan. 2011

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