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Existence results for abstract semilinear evolution differential inclusions with nonlocal conditions

Tao Zhu1*, Chao Song1 and Gang Li2

Author Affiliations

1 Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211100, P.R. China

2 Department of Mathematics, Yangzhou University, Yangzhou, 225002, P.R. China

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Boundary Value Problems 2013, 2013:153  doi:10.1186/1687-2770-2013-153

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/153


Received:12 April 2013
Accepted:10 June 2013
Published:1 July 2013

© 2013 Zhu et al.; 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

In this paper, we use a new method to study semilinear evolution differential inclusions with nonlocal conditions in Banach spaces. We derive conditions for F and g for the existence of mild solutions. The results obtained here improve and generalize many known results.

MSC: 34A60, 34G20.

Keywords:
semilinear evolution differential inclusions; mild solutions; measure of noncompactness; upper semicontinuous

1 Introduction

In this paper, we discuss the nonlocal initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M1">View MathML</a>

(1.1)

where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2">View MathML</a>-semigroup) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> in a Banach space X, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M5">View MathML</a> are given X-valued functions.

The study of nonlocal evolution equations was initiated by Byszewski [1]. Since these represent mathematical models of various phenomena in physics, Byszewski’s work was followed by many others [2-7]. Subsequently, many authors have contributed to the study of the differential inclusions (1.1). Differential inclusions (1.1) were first considered by Aizicovici and Gao [8] when g and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> are compact. In [9-12] the semilinear evolution differential inclusions (1.1) were discussed when A generates a compact semigroup. Xue and Song [13] established the existence of mild solutions to the differential inclusions (1.1) when A generates an equicontinuous semigroup and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M7">View MathML</a> is l.s.c. for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>. In [14] the author proved the existence of mild solutions of the differential inclusions (1.1) when A generates an equicontinuous semigroup and a Banach space X which is separable and uniformly smooth. In [15] Zhu and Li studied the differential inclusions (1.1) when F admits a strongly measurable selector. In [16] the differential inclusions (1.1) were discussed when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M9">View MathML</a> is a family of linear (not necessarily bounded) operators. In [17] local and global existence results for differential inclusions with infinite delay in a Banach space were considered. Benchohra and Ntouyas [18] studied the second-order initial value problems for delay integrodifferential inclusions. In [19,20] the impulsive multivalued semilinear neutral functional differential inclusions were discussed in the case that the linear semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is compact. The purpose of this paper is to continue the study of these authors. By using a new method, we prove the existence results of mild solutions for (1.1) under the following conditions of g and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a>: g is either compact or Lipschitz continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is an equicontinuous semigroup. So, our work extends and improves many main results such as those in [8-12,14,15].

The organization of this work is as follows. In Section 2, we recall some definitions and facts about set-valued analysis and the measure of noncompactness. In Section 3, we give the existence of mild solutions of the nonlocal initial value problem (1.1). In Section 4, an example is given to show the applications of our results.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M13">View MathML</a> be a real Banach space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M14">View MathML</a>. A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M15">View MathML</a> is convex (closed) valued if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M16">View MathML</a> is convex (closed) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17">View MathML</a>. We say that G is bounded on bounded sets if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M18">View MathML</a> is bounded in X for each bounded set B of X. The map G is called upper semicontinuous (u.s.c.) on X if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M19">View MathML</a> the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M20">View MathML</a> is a nonempty, closed subset of X, and if for each open set N of X containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M20">View MathML</a>, there exists an open neighborhood M of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M23">View MathML</a>. Also, G is said to be completely continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M18">View MathML</a> is relatively compact for every bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M25">View MathML</a>. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M28">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M29">View MathML</a>). Moreover, the following conclusions hold. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M16">View MathML</a> be closed for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M32">View MathML</a>, if G is u.s.c. and D is closed, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M33">View MathML</a> is closed. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M34">View MathML</a> is compact and D is closed, then G is u.s.c. if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M33">View MathML</a> is closed. Finally, we say that G has a fixed point if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M37">View MathML</a>.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a> the space of X-valued continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M39">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M40">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41">View MathML</a> the space of X-valued Bochner functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M43">View MathML</a>.

A <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2">View MathML</a>-semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is said to be compact if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is compact for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M47">View MathML</a>. If the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M49">View MathML</a> is equicontinuous at all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M47">View MathML</a> with respect to x in all bounded subsets of X; i.e., the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is equicontinuous. If A is the generator of an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> or a differentiable semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is an equicontinuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2">View MathML</a>-semigroup (see [21]). In this paper, we suppose that A generates an equicontinuous semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M58">View MathML</a> is a mild solution of (1.1) if

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M59">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M60">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M61">View MathML</a>.

To prove the existence results in this paper, we need the following lemmas.

Lemma 2.2[22]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M62">View MathML</a>is bounded, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M63">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M65">View MathML</a>. Furthermore, ifWis equicontinuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M67">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M69">View MathML</a>.

Lemma 2.3[22]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M70">View MathML</a>is a decreasing sequence of bounded closed nonempty subsets ofXand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M71">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M72">View MathML</a>is nonempty and compact inX.

Lemma 2.4[23]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M73">View MathML</a>is uniformly integrable, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M74">View MathML</a>is measurable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M75">View MathML</a>

(2.1)

Lemma 2.5[24]

If the semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a>is equicontinuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M77">View MathML</a>, then the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M78">View MathML</a>is equicontinuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M42">View MathML</a>.

Lemma 2.6[25]

IfWis bounded, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M80">View MathML</a>, there is a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M81">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M82">View MathML</a>

(2.2)

A countable set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M83">View MathML</a> is said to be semicompact if

(a) it is integrably bounded: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M84">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M87">View MathML</a>;

(b) the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M88">View MathML</a> is relatively compact for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

Lemma 2.7[26]

Every semicompact set is weakly compact in the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41">View MathML</a>.

Lemma 2.8[16,26]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M83">View MathML</a>is semicompact, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M92">View MathML</a>is relatively compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>and, moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M95">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M96">View MathML</a>.

The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M97">View MathML</a> is said to be α contraction if there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M98">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M99">View MathML</a>

for any bounded closed subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M100">View MathML</a>.

Lemma 2.9[27-30] (Fixed point theorem)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M101">View MathML</a>is a nonempty, bounded, closed, convex and compact subset, the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M102">View MathML</a>is upper semicontinuous with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M103">View MathML</a>a nonempty, closed, convex subset ofWfor each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M104">View MathML</a>, thenFhas at least one fixed point in W.

Lemma 2.10[26] (Fixed point theorem)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M101">View MathML</a>is nonempty, bounded, closed and convex, the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M102">View MathML</a>is a closedαcontraction map with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M103">View MathML</a>a nonempty, convex and compact subset ofWfor each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M104">View MathML</a>, thenFhas at least one fixed point inW.

3 Main results

In this section, by using the measure of noncompactness and fixed point theorems, we give the existence results of the nonlocal initial value problem (1.1). Here we list the following hypotheses.

(1) The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M2">View MathML</a> semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> generated by A is equicontinuous. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M111">View MathML</a>.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M112">View MathML</a> is continuous and compact, there exist positive constants c and d such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M114">View MathML</a>.

(3) The multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M115">View MathML</a> satisfies the hypotheses: the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M120">View MathML</a> is nonempty.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M116">View MathML</a> is measurable for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M118">View MathML</a> is u.s.c. for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>;

(4) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M121">View MathML</a> such that for any bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M123">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M124">View MathML</a>.

(5) There exist a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M125">View MathML</a> and a nondecreasing continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M126">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M127">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M17">View MathML</a>, and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

Remark 3.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M130">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M131">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M132">View MathML</a> (see Lasota and Opial [31]). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M58">View MathML</a>, then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M135">View MathML</a> is nonempty if and only if the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M136">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M137">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M138">View MathML</a> (see Hu and Papageorgiou [32]).

The following lemma plays a crucial role in the proof of the main theorem.

Lemma 3.2[26]

Under assumptions (3)-(5), if we consider sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M139">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M140">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M141">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144">View MathML</a>.

Now we give the existence results under the above hypotheses.

Theorem 3.3If (1)-(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constantRwith

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M145">View MathML</a>

(3.1)

Proof Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M146">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M147">View MathML</a>

We shall show that the multivalued Γ has at least one fixed point. The fixed point is then a mild solution of the problem (1.1).

(1) We contract a bounded, convex, closed and compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M148">View MathML</a> such that Γ maps W into itself.

In view of (3.1), we know there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M149">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M150">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M151">View MathML</a>.

Then there exists a positive integer K such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M152">View MathML</a>

(3.3)

Hence, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M153">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M154">View MathML</a>

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M155">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M156">View MathML</a> is nonempty, bounded, closed and convex.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M157">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M158">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M159">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M160">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M161">View MathML</a> is a bounded operator.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M162">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M163">View MathML</a> means the closure of the convex hull in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M164">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M165">View MathML</a> is nonempty bounded closed convex on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M167">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M168">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86">View MathML</a>. Similarly to the above discussions, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M170">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M171">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M173">View MathML</a> are both nonempty, closed, bounded and convex. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M174">View MathML</a> is a decreasing sequence consisting of subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175">View MathML</a>. Moreover, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M176">View MathML</a>

then W is a convex, closed and bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M178">View MathML</a>.

Now, we claim that W is nonempty and compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175">View MathML</a>. To do so, from Lemma 2.6, we know for arbitrary given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180">View MathML</a>, there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M181">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M182">View MathML</a>

Since this is true for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M184">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M185">View MathML</a> is decreasing for n, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M186">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M187">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M188">View MathML</a>

It implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M189">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M190">View MathML</a>. By Lemma 2.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M191">View MathML</a>. Using Lemma 2.3, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M192">View MathML</a> is nonempty and compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M164">View MathML</a>.

(2) We shall show that Γ is closed on W with closed convex values. It is very easy to see that Γ has convex values.

Let us now verity that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194">View MathML</a> is closed. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M195">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M198">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M199">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M140">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M141">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M86">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M204">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M142">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M207">View MathML</a> is a bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M38">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M209">View MathML</a>.

From hypothesis (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M210">View MathML</a>

Then we have the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M211">View MathML</a> is integrably bounded for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

From hypothesis (4), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M213">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>. Then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is relatively compact for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

So, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is semicompact. By applying Lemma 2.7, it yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is weakly compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41">View MathML</a>. We get that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M220">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143">View MathML</a>. Therefore, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M222">View MathML</a>

Further, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M223">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M224">View MathML</a>

By Lemma 3.2, it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M226">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194">View MathML</a> is closed. And hence Γ has closed values on W.

(4) Γ is u.s.c. on W.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M228">View MathML</a> is compact, W is closed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M194">View MathML</a> is closed, we can come to the conclusion that Γ is u.s.c. (see [30]).

Finally, due to fixed point Lemma 2.9, Γ has at least one point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M230">View MathML</a>, and x is a mild solution to the semilinear evolution differential inclusions with the nonlocal conditions (1.1). Thus the proof is complete. □

Remark 3.4 In [8-12] the authors discuss the nonlocal initial value problem (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> is compact. In [14] the existence of mild solutions of the differential inclusions (1.1) is proved when A generates an equicontinuous semigroup and Banach space X is separable and uniformly smooth. In this paper, by using a new method, we prove the operator Γ maps compact set W into itself. We do not impose any restriction on the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M232">View MathML</a>, and we only require <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M3">View MathML</a> to be an equicontinuous semigroup. So, Theorem 3.3 generalizes and improves the related results in [8-12,14].

Theorem 3.5[15]

If (1)-(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M234">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M235">View MathML</a>

(3.4)

Proof In view of (3.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M236">View MathML</a>

From Theorem 3.3, the nonlocal initial value problem (1.1) has at least one mild solution. □

Remark 3.6 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M241">View MathML</a>. We cannot obtain a constant R such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M242">View MathML</a>

By using Theorem 3.5, we do not know whether or not equation (1.1) has a mild solution. But we know there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M243">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M244">View MathML</a>

So, Theorem 3.3 is better than Theorem 3.5.

Theorem 3.7[12]

If (1)-(5) are satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M245">View MathML</a>, then there is at least one mild solution for (1.1) provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M246">View MathML</a>

(3.5)

Proof In view of (3.5), we get there exists a constant R such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M247">View MathML</a>

By Theorem 3.3, we complete the proof of this theorem. □

Next, we give the existence result for (1.1) when g is Lipschitz continuous.

We suppose that:

(6) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M248">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M249">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M250">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M113">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M252">View MathML</a>.

Theorem 3.8If (1) and (3)-(6) are satisfied and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M253">View MathML</a>

then there is at least one mild solution for (1.1) provided that there exists a constantRsatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M254">View MathML</a>

(3.6)

Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M161">View MathML</a> is a bounded map with convex values and is closed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M256">View MathML</a>.

Now, we prove the values of Γ are compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M175">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M259">View MathML</a>. To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M260">View MathML</a> is compact, we have to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M261">View MathML</a> has a subsequence converging to a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M262">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M263">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M264">View MathML</a>

From hypothesis (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M265">View MathML</a>

Then we have the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M211">View MathML</a> is integrably bounded for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

From hypothesis (4), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M268">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>. Then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is relatively compact for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M8">View MathML</a>.

So, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is semicompact. By applying Lemma 2.7, it yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M215">View MathML</a> is weakly compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M41">View MathML</a>. We get that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M220">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M143">View MathML</a>. Therefore, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M277">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M278">View MathML</a>

By Lemma 3.2, it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M144">View MathML</a>, i.e.,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M226">View MathML</a>. Therefore Γ has compact values.

Next, we prove Γ is an α contraction map. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M281">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M282">View MathML</a>

From Lemma 2.6, we know for arbitrary given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180">View MathML</a>, there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M284">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M285">View MathML</a>

Since this is true for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M287">View MathML</a>

Therefore, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M288">View MathML</a>

Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M289">View MathML</a>, therefore Γ is an α contraction map.

Finally, due to Lemma 2.10, Γ has at least one fixed point. This completes the proof. □

4 An example

In this section, as an application of our main results, an example is presented. We consider the following partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M290">View MathML</a>

(4.1)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M291">View MathML</a> with a smooth boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M292">View MathML</a> is a smooth real function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M293">View MathML</a>.

We suppose that

(a) The differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M294">View MathML</a> is strongly elliptic [21].

(b) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M295">View MathML</a> satisfies the following conditions:

(b1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M296">View MathML</a> is a continuous function about r for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M297">View MathML</a>.

(b2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M296">View MathML</a> is measurable about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M299">View MathML</a> for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M300">View MathML</a>.

(b3) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M234">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M302">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M303">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M304">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M305">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M306">View MathML</a>

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M307">View MathML</a>.

(b4) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M309">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M310">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M311">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M312">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M313">View MathML</a>, then A generates an analytic semigroup on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M314">View MathML</a> ([21]). We suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M315">View MathML</a>

From [33], we obtain g satisfies hypothesis (2).

Then equation (4.1) can be regarded as the following nonlocal semilinear evolution equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M316">View MathML</a>

(4.2)

By using Theorem 3.3, the problem (4.1) has at least one mild solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/153/mathml/M317">View MathML</a> provided that hypotheses (3)-(5) and (3.1) hold.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The research was supported by the Scientific Research Foundation of Nanjing Institute of Technology (No: QKJA2011009).

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