Open Access Research

Solvability of impulsive partial neutral second-order functional integro-differential equations with infinite delay

Shengli Xie

Author Affiliations

Department of Mathematics and Physics, Anhui University of Architecture, Zipeng Road, Hefei, Anhui, 230601, P.R. China

Boundary Value Problems 2013, 2013:203  doi:10.1186/1687-2770-2013-203


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/203


Received:5 April 2013
Accepted:5 August 2013
Published:9 September 2013

© 2013 Xie; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral second-order functional integro-differential equations with infinite delay in Banach spaces. The compactness condition of the impulsive term, some restrictive conditions on a priori estimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As applications, some examples are provided to illustrate the obtained results.

MSC: 34K30, 34K40, 35R10, 47D09.

Keywords:
impulsive partial neutral functional integro-differential equations; mild solutions; fixed point; Banach spaces

1 Introduction

Consider the following impulsive partial neutral second-order functional integro-differential systems with infinite delay in a Banach space X:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M1">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M2">View MathML</a>

(2)

where A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M3">View MathML</a>, on X. In both cases, the history <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M6">View MathML</a> belongs to some abstract phase space ß defined axiomatically; g, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) are appropriate functions; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M11">View MathML</a> are fixed numbers and the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M12">View MathML</a> represents the jump of the function x at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M13">View MathML</a>, which is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M15">View MathML</a>.

The study of impulsive functional differential equations is linked to their utility in simulating processes and phenomena subject to short-time perturbations during their evolution. The perturbations are performed discretely and their duration is negligible in comparison with the total duration of the processes and phenomena. Now impulsive partial neutral functional differential equations have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. With regard to this matter, we refer the reader to [1-12] and references therein. However, in order to obtain the existence of solutions in these study papers, the compactness condition on the associated family of operators and the impulsive term, some similar restrictive conditions on a priori estimation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M16">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M17">View MathML</a>

(4)

are used. In [13-16], authors used a strict set contraction mapping fixed point theorem without the compactness assumption on the associated family of operators to obtain the existence results of system (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M18">View MathML</a> is not an integral operator and the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M19">View MathML</a>

(5)

improved and generalized some results in [1,7]. However, the compactness condition of the impulsive terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20">View MathML</a>, some similar restrictive conditions on a priori estimation (3), (4) and the restrictive condition on measure of noncompactness estimation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M21">View MathML</a>

(6)

are used in [13-16]. So far we have not seen the existence results of system (2).

In this paper, using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions of impulsive partial neutral second-order functional integro-differential systems (1) and (2). The compactness condition of impulsive terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20">View MathML</a>, some restrictive conditions on a priori estimation and measure of noncompactness estimation (3), (4) and (6) have been deleted. Our conditions are simple and our results essentially improve and extend some corresponding results in papers [1,2,13,14]. As applications, some examples are provided to illustrate the obtained results.

2 Preliminaries

In this paper, X is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M23">View MathML</a> and A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M24">View MathML</a>, on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M25">View MathML</a> is the sine function associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M3">View MathML</a>, which is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>. We designate by N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M30">View MathML</a> certain constants such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M32">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M33">View MathML</a>. We refer the reader to [17] for the necessary concepts about cosine functions. Next, we only mention a few results and notations needed to establish our results. As usual we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M34">View MathML</a> the domain of A endowed with the graph norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M36">View MathML</a>. Moreover, the notation E stands for the space formed by the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M28">View MathML</a>, for which the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M38">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M39">View MathML</a>. It was proved by Kisyński [18] that the space E endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M40">View MathML</a>

is a Banach space. The operator-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M41">View MathML</a> is a strongly continuous group of linear operators on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M42">View MathML</a> generated by the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M43">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M44">View MathML</a>. It follows from this that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M45">View MathML</a> is a bounded linear operator and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M46">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M47">View MathML</a>) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M48">View MathML</a>. Furthermore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M49">View MathML</a> is a locally integrable function, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M50">View MathML</a> defines an E-valued continuous function. This is a consequence of the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M51">View MathML</a>

defines an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M52">View MathML</a>-valued continuous function. Next, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M53">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M54">View MathML</a> stands for the Banach space of bounded linear operators from E into X, and we abbreviate this notation to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M55">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M56">View MathML</a>.

To describe appropriately our system (1), we say that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M57">View MathML</a> is a normalized piecewise continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M58">View MathML</a> if u is piecewise continuous and left continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M59">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M60">View MathML</a> the space formed by the normalized piecewise continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M58">View MathML</a> into X. In particular, we introduce the space PC formed by all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M62">View MathML</a> such that u is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M65">View MathML</a> exists for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M66">View MathML</a>. It is clear that PC endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M67">View MathML</a> is a Banach space.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M68">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M69">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M70">View MathML</a>. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M73">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M74">View MathML</a>. From Lemma 1.1 in [1], we know that a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M71">View MathML</a> is relatively compact if and only if each set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M74">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M77">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>).

For system (2), we give the precise meaning of the derivative in (2). We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M68">View MathML</a> is piecewise smooth if x is continuously differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M81">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M81">View MathML</a>, there are the right derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M84">View MathML</a> and the left derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M85">View MathML</a>. Furthermore, we denote the space by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M86">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M87">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M88">View MathML</a> is a Banach space.

In this work we employ an axiomatic definition of the phase space ß introduced by Hale and Kato [19] which appropriated to treat retarded impulsive differential equations. For other abstract phase spaces, we can refer to [20,21].

Definition 2.1[19]

The phase space ß is a linear space of functions mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M89">View MathML</a> into X endowed with a seminorm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M90">View MathML</a>. We assume that ß satisfies the following axioms.

(A) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M92">View MathML</a>) is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M94">View MathML</a>, then for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M95">View MathML</a> the following conditions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M96">View MathML</a> is in ß,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M97">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M98">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M99">View MathML</a> is a constant; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M100">View MathML</a>, K is continuous, M is locally bounded and H, K, M are independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M101">View MathML</a>.

(B) The space ß is complete.

In this paper we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M102">View MathML</a> the Kuratowski measure of noncompactness of X, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M103">View MathML</a> the Kuratowski measure of noncompactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M104">View MathML</a> and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M105">View MathML</a> the Kuratowski measure of noncompactness of PC.

The following lemma is easy to get.

Lemma 2.2If the cosine function family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M107">View MathML</a>, is equicontinuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M108">View MathML</a>, then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M109">View MathML</a>

is equicontinuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110">View MathML</a>.

Lemma 2.3[16,22]

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M111">View MathML</a>is bounded, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M112">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M114">View MathML</a>.

(2) IfWis piecewise equicontinuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M115">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M116">View MathML</a>is piecewise continuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M118">View MathML</a>.

(3) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M111">View MathML</a>is bounded and piecewise equicontinuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M120">View MathML</a>is piecewise continuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M121">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M122">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M123">View MathML</a>.

(4) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M124">View MathML</a>is bounded and the elements of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M125">View MathML</a>are equicontinuous on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M126">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M128">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M129">View MathML</a>denotes the Kuratowski measure of noncompactness in the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M130">View MathML</a>.

Lemma 2.4[23]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M131">View MathML</a>be an integrable function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M132">View MathML</a>. Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M133">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M87">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M135">View MathML</a>is integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M136">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M110">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M138">View MathML</a>

Lemma 2.5[24]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M139">View MathML</a>. If there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M140">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M141">View MathML</a>) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M142">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M143">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M144">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M145">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M146">View MathML</a>

Lemma 2.6[25] (Mónch)

LetXbe a Banach space, Ω be a bounded open subset inXand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M147">View MathML</a>. Assume that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M148">View MathML</a>is continuous and satisfies the following conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M151">View MathML</a>,

(2) Dis relatively compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M152">View MathML</a>for any countable set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M153">View MathML</a>.

ThenFhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M154">View MathML</a>.

3 Main results

Firstly, we discuss the existence of mild solutions for the impulsive second-order system (1).

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M155">View MathML</a> is said to be a mild solution of system (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M157">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M158">View MathML</a>

(7)

For system (1), we make the following hypotheses.

(H1) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M159">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) satisfy the following conditions:

(1) For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M162">View MathML</a> are strongly measurable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M163">View MathML</a> are continuous for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>;

(2) There are integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M165">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M167">View MathML</a>

(3) For any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M168">View MathML</a>, there are integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M169">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M171">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M172">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>).

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M176">View MathML</a>) satisfies the following conditions:

(1) For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M178">View MathML</a> are strongly measurable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M179">View MathML</a> are continuous for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M180">View MathML</a>;

(2) There are continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M181">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M183">View MathML</a>

(3) For any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M168">View MathML</a>, there are continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M185">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M187">View MathML</a>

(H3) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M188">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) are continuous and there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M193">View MathML</a>

Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M194">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M195">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M196">View MathML</a>

Theorem 3.2Suppose that the cosine function family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, is equicontinuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M200">View MathML</a>satisfy the condition (H1), (H2) and (H3) are satisfied. Then the impulsive second-order system (1) has at least one mild solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a> be the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M202">View MathML</a> endowed with the supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M203">View MathML</a>. The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M204">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M205">View MathML</a>

(8)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M206">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M209">View MathML</a>. Thus F is well defined with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (H1), (H2) and (H3), we can show that F is continuous (see [5]). It is easy to see that if x is a fixed point of F, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M211">View MathML</a> is a mild solution of system (1).

Firstly, we show that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M212">View MathML</a>

is bounded. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M213">View MathML</a>, then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M215">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216">View MathML</a>, notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M218">View MathML</a> is continuous nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M219">View MathML</a>. We have, by (8) and (H1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M220">View MathML</a>

(9)

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M221">View MathML</a>

(10)

By well-known Gronwall’s lemma and (10), there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M222">View MathML</a> independent of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. It follows from this and the condition (H3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M227">View MathML</a>

Nextly, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M228">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M229">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M230">View MathML</a>. Similar to (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M231">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M232">View MathML</a>. Equation (11) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M233">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M234">View MathML</a>. Using Gronwall’s lemma once again and (12), there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M235">View MathML</a> independent of v and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M238">View MathML</a>. Thence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M240">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241">View MathML</a>.

It is similar to the proof above, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M242">View MathML</a> independent of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M245">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M247">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M250">View MathML</a> is bounded.

Lastly, we verify that all the conditions of Lemma 2.6 are satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M252">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M253">View MathML</a> is a bounded open set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M147">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M149">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M258">View MathML</a>.

Nextly, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M259">View MathML</a> be a countable set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M260">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M261">View MathML</a>

(13)

It follows from (H1)-(H3) and Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M262">View MathML</a> is equicontinuous on every interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>), which together with (13) implies that V is equicontinuous on every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216">View MathML</a>, by the property of noncompactness measure, (H1)(3), (H2)(3) and Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M268">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M269">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M272">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M273">View MathML</a>

From this and Gronwall’s lemma, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. Therefore V is a relative compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M277">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M278">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M279">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M281">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M283">View MathML</a>, similar to (14), it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M284">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M285">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M287">View MathML</a>. Equation (15) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M288">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M290">View MathML</a> and V is a relative compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M291">View MathML</a>.

Similarly, we can show that V is a relative compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M292">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246">View MathML</a>), so V is a relative compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. Lemma 2.6 concludes that F has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M295">View MathML</a>. Let x be a fixed point of F on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M297">View MathML</a> is a mild solution of system (1). □

Nextly, we discuss the existence of mild solutions for the impulsive system (2).

Definition 3.3 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M155">View MathML</a> is said to be a mild solution of system (2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M299">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M301">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M302">View MathML</a>

(16)

Differentiate (16) to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M303">View MathML</a>

(17)

Let functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M304">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M306">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M307">View MathML</a>

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M308">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M310">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M311">View MathML</a> be the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M312">View MathML</a> endowed with the supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M313">View MathML</a>.

We make the following hypotheses for convenience.

(Hf) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M314">View MathML</a> satisfies the following conditions:

(1) For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M315">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M316">View MathML</a> is strongly measurable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M317">View MathML</a> is continuous for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>;

(2) There is an integrable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M319">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M320">View MathML</a>

(3) For any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M321">View MathML</a>, there is an integrable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M322">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M323">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M325">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M327">View MathML</a>.

(Hg) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M328">View MathML</a> satisfies the following conditions:

(1) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M329">View MathML</a> is continuous, there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M330">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M331">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M333">View MathML</a>

(2) For every bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M334">View MathML</a>, the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M335">View MathML</a> is uniformly equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M337">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M338">View MathML</a>;

(3) For any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>.

(HI) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M343">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M344">View MathML</a>) are continuous and there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M346">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M347">View MathML</a>

Theorem 3.4Let the conditions (Hf), (Hg) and (HI) be satisfied, the cosine function family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, be equicontinuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M350">View MathML</a>. Then system (2) has at least one mild solution.

Proof Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M351">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M352">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M164">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M355">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M356">View MathML</a>

(18)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M357">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M358">View MathML</a>

(19)

The product space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M359">View MathML</a> is endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M360">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M362">View MathML</a> are well defined and with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (Hf), (Hg) and (HI), we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M364">View MathML</a> is continuous. It is easy to see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M365">View MathML</a> is a fixed point of Γ, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M211">View MathML</a> is a mild solution of system (2).

Firstly, we show that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M367">View MathML</a>

is bounded. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M213">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M371">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216">View MathML</a>, it follows from (18), (19) and (Hf)(2), (Hg)(1), (HI) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M373">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M374">View MathML</a>

(21)

Equations (20) and (21) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M375">View MathML</a>

(22)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M376">View MathML</a>, by Gronwall’s lemma and (22), there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M380">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. It follows from this and the condition (HI) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M385">View MathML</a>

Nextly, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M228">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M387">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M388">View MathML</a>. Similar to (20) and (21), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M389">View MathML</a>

(23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M390">View MathML</a>

(24)

We have, by (23) and (24),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M391">View MathML</a>

(25)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M392">View MathML</a>

Using Gronwall’s lemma once again and (25), there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M235">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M394">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M238">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241">View MathML</a>.

It is similar to the proof above, there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M245">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M247">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M403">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M250">View MathML</a> is bounded.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M406">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M253">View MathML</a> is a bounded open set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M408">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M251">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M410">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M411">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M223">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M259">View MathML</a> is a countable set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M414">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M415">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M416">View MathML</a>

(26)

It follows from (18), (19) and (Hg)(2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M417">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) are equicontinuous on every interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>), which together with (26) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M421">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) are equicontinuous on every interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M265">View MathML</a>.

In the following, we verify that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M424">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M425">View MathML</a> is relatively compact in PC. Without loss of generality, we do not distinguish <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M426">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M73">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M426">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) is the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M126">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M216">View MathML</a>, by the condition (Hf)(3), (Hg)(3) and Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M433">View MathML</a>

(27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M434">View MathML</a>

(28)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M435">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) are continuous nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M219">View MathML</a>, (27) and (28) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M438">View MathML</a>

(29)

By Gronwall’s lemma and (29), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M439">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M226">View MathML</a>. Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M277">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M445">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M446">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) is continuous, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M448">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M283">View MathML</a>, it is similar to (27) and (28), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M450">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M451">View MathML</a>

(31)

Equations (30) and (31) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M452">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M439">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M241">View MathML</a>. Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) are relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M291">View MathML</a>.

Similarly, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) are relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M292">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M246">View MathML</a>). So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M430">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M422">View MathML</a>) are relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. In view of Lemma 2.6, we conclude that Γ has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M466">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M365">View MathML</a> be a fixed point of Γ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M201">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M469">View MathML</a> is a mild solution of system (2). □

Theorem 3.5Let the conditions (Hf), (Hg)(2) and (HI) be satisfied, the cosine function family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, be equicontinuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M350">View MathML</a>. Furthermore, suppose that the following condition is satisfied:

(Hg)(1′) The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M329">View MathML</a>is continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M474">View MathML</a>satisfies the Lipschitz condition, that is, there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M330">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M476">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332">View MathML</a>. Then system (2) has at least one mild solution.

Proof We have, by the condition (Hg)(1′),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M478">View MathML</a>

The rest of the proof is similar to the proof of Theorem 3.4, we omit it. □

Remark 3.6 The similar restrictive conditions (3), (4) and (6) were used in [1,2,13,14] even if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M332">View MathML</a>.

4 Examples

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M480">View MathML</a> and let A be the operator given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M481">View MathML</a> with the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M482">View MathML</a>

It is well known that A is the infinitesimal generator of a strongly continuous cosine family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, on X. Moreover, A has discrete spectrum, the eigenvalues are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M485">View MathML</a>, with corresponding normalized eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M486">View MathML</a>, and the following properties hold:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M487">View MathML</a> is an orthonormal basis of X.

(b) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M488">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M489">View MathML</a>. Moreover, it follows from this expression that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M490">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M25">View MathML</a> is compact for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M492">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M493">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M494">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>. Additionally, we observe that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M496">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M497">View MathML</a>, are not compact.

(c) If Φ denotes the group of translations on X defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M498">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M499">View MathML</a> is the extension of x with period 2π, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M500">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M501">View MathML</a>, where B is the infinitesimal generator of the group Φ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M502">View MathML</a>; see [1] for details. In particular, we observe that the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M503">View MathML</a> is compact.

In the next application, ß should be the phase space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M504">View MathML</a> in [1], where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M505">View MathML</a> is a positive Lebesgue integrable function. We can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M506">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M507">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M508">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M509">View MathML</a>.

Example 4.1 Consider the partial neutral functional integro-differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M510">View MathML</a>

(32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M511">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M512">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M513">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M514">View MathML</a> and

(i) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M515">View MathML</a> is continuous and there is a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M516">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M517">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M518">View MathML</a>.

(ii) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M519">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M520">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521">View MathML</a>.

(iii) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M522">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M523">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521">View MathML</a>.

(iv) The function φ defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M525">View MathML</a> belongs to ß.

Assuming that the conditions (i)-(iv) are satisfied, then system (32) can be modeled as the abstract impulsive Cauchy problem (1) by defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M526">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M527">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M528">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M532">View MathML</a> are bounded linear operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M533">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M534">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M536">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M537">View MathML</a>. If the cosine function family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, is equicontinuous, all the conditions of Theorem 3.2 are satisfied (see [1] for details), so system (32) has at least one mild solution. However, if we select <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M540">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M541">View MathML</a> in (iii), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M542">View MathML</a>. But

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M543">View MathML</a>

the restrictive conditions (3), (4) and (6) do not hold. Thus, our results are different from the corresponding known results.

Example 4.2 Consider the partial neutral functional integro-differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M544">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M545">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M546">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M514">View MathML</a> and

(v) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M548">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M549">View MathML</a> are measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M550">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M551">View MathML</a>

(34)

(vi) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M552">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M553">View MathML</a>.

Assuming that the conditions (ii)-(vi) are satisfied, system (33) can be modeled as the abstract Cauchy problem (2) by defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M554">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M555">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M20">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M10">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M474">View MathML</a> are bounded linear operators and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M560">View MathML</a>

Moreover, for every bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M334">View MathML</a>, it follows from (34) and the proof in [26] that the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M335">View MathML</a> is uniformly equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M263">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M72">View MathML</a>. If the cosine function family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/203/mathml/M29">View MathML</a>, is equicontinuous, all the conditions of Theorem 3.4 are satisfied, so system (33) has at least one mild solution.

Remark 4.3 From the results of this paper, we know that the compactness condition of the impulsive term, the restrictive conditions on a priori estimation and noncompactness measure estimation can be deleted for the existence results of abstract impulsive functional Volterra integro-differential equations Cauchy problems.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the referees for their careful reading and helpful suggestions that have led to the present improved version of the original paper. The work was supported by the Natural Science Foundation of Anhui Province (11040606M01), Anhui Educational Committee (KJ2011A061, KJ2011B052), China.

References

  1. Hernández, EM, Rabelloa, M, Henríquez, HR: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl.. 331, 1135–1158 (2007). Publisher Full Text OpenURL

  2. Chang, YK, Arjunam, MM, Kavitha, V: Existence results for a second order impulsive functional differential equation with sates-dependent delay. Diff. Eqs. & Appl.. 1(3), 325–339 (2009)

  3. Balachandran, K, Annapoorani, N: Existence results for impulsive neutral evolution integro-differential equations with infinite delay. Nonlinear Analysis: Hybrid Systems.. 3, 674–684 (2009). Publisher Full Text OpenURL

  4. Cuevas, C, Hernández, EM, Rabelo, M: The existence of solutions for impulsive neutral functional differential equations. Comput. Math. Appl.. 58, 744–757 (2009). Publisher Full Text OpenURL

  5. Park, JY, Balachandran, K, Annapoorani, N: Existence results for impulsive neutral functional integrodifferential equations with infinite delay. Nonlinear Anal.. 71, 3152–3162 (2009). Publisher Full Text OpenURL

  6. Rabelo, MN, Henrique, M, Siracusa, G: Existence of integro-differential solutions for a class of abstract partial impulsive differential equations. J. Inequalities and Appl.. 135, 1–19 (2011)

  7. Changa, YK, Angurajb, A, Arjunan, MM: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Analysis: Hybrid Systems.. 2, 209–218 (2008). Publisher Full Text OpenURL

  8. Hernández, EM, Henríquez, HR, McKibben, MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal.. 70, 2736–2751 (2009). Publisher Full Text OpenURL

  9. Chalishajar, DN: Controllability of impulsive partial neutral functional differential equation with Infinite Delay. Int. J. Math. Anal.. 5(8), 369–380 (2011). PubMed Abstract OpenURL

  10. Hernández, EM, Henríquez, HR, McKibben, MA: Existence of solutions for second order partial neutral functional differential equations. Integr. equ. oper. theory.. 62, 191–217 (2008). Publisher Full Text OpenURL

  11. Hernández, EM, McKibben, MA: Some comments on: Existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations. Comput. Math. Appl.. 50, 655–669 (2005). Publisher Full Text OpenURL

  12. Hernandez, E, Balachandran, K, Annapoorani, N: Existence results for a damped second order abstract functional differential equation with impulses. Mathematical and Computer Modelling.. 50, 1583–1594 (2009). Publisher Full Text OpenURL

  13. Runping, Y: Impulsive partial neutral functional differential equation of second-order with infinite delay. Inter. J. Nonlinear Sci.. 12(2), 245–256 (2011). PubMed Abstract OpenURL

  14. Vijayakumar, V, Sivasankarmn, S, Arjunan, MM: Existence of solutions for second-order impulsive neutral functional integro-differential equations with infinite delay. Nonlinear Stud.. 19(2), 327–343 (2012)

  15. Vijayakumar, V, Sivasankaran, S, Arjunan, MM: Existence of solutions for double perturbed impulsive neutral functional evolution equations. J. KSIAM.. 15(4), 253–265 (2011)

  16. Runping, Y: Existence of solutions for impulsive partial neutral functional differential equation with infinite delay. Nonlinear Anal.. 73, 155–162 (2010). Publisher Full Text OpenURL

  17. Fattorini, HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland, Amsterdam (1985)

  18. Kisy’nski, J: On cosine operator functions and one parameter group of operators. Studia Math.. 49, 93–105 (1972)

  19. Hale, JK, Kato, J: Phase space for retarded equations with infinite delay. Funkcial Ekvac. 21, 11–41 (1978)

  20. Chalishajar, DN: Controllability of Second Order Impulsive Neutral Functional Differential Inclusions with Infinite Delay. J Optim Theory Appl.. 154(2), 672–684 (2012). Publisher Full Text OpenURL

  21. Chalishajar, DN, Acharya, FS: Controllability of second order semi linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces. Bull. Korean Math. Soc.. 48(4), 813–838 (2011). Publisher Full Text OpenURL

  22. Dajun, G, Lakshmikantham, V, Liu, X: Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht (1996)

  23. Henríquez, HR, Vásquez, CH: Differentiability of solutions of second-order functional differential equations with unbounded delay. J. Math. Anal. Appl.. 280(2), 284–312 (2003). Publisher Full Text OpenURL

  24. Heine, HP: On the behavior of measure of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal.. 7, 1351–1371 (1983). Publisher Full Text OpenURL

  25. Mönch, H: Boundary value problems for nonlinear ordinary equations of second order in Banach spaces. Nonlinear Anal.. 4, 985–999 (1980). Publisher Full Text OpenURL

  26. Hernández, E: Existence results for partial neutral functional integro-differential equations with unbounded delay. J. Math. Anal. Appl.. 292, 194–210 (2004). PubMed Abstract | Publisher Full Text OpenURL