Open Access

Existence results for abstract semilinear evolution differential inclusions with nonlocal conditions

Boundary Value Problems20132013:153

DOI: 10.1186/1687-2770-2013-153

Received: 12 April 2013

Accepted: 10 June 2013

Published: 1 July 2013

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
{ x ( t ) A x ( t ) + F ( t , x ( t ) ) , t I = [ 0 , 1 ] , x ( 0 ) = g ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ1_HTML.gif
(1.1)

where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e., C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq1_HTML.gif-semigroup) T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif in a Banach space X, and F : [ 0 , 1 ] × X P c ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq3_HTML.gif, g : C ( [ 0 , 1 ] ; X ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq4_HTML.gif 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 [27]. 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 T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif are compact. In [912] 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 F ( t , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq5_HTML.gif is l.s.c. for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif. 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 ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq7_HTML.gif 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 T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif 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 T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif: g is either compact or Lipschitz continuous and T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is an equicontinuous semigroup. So, our work extends and improves many main results such as those in [812, 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 ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq8_HTML.gif be a real Banach space. Let P c ( X ) = { A X : nonempty, closed, convex } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq9_HTML.gif. A multivalued map G : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq10_HTML.gif is convex (closed) valued if G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq11_HTML.gif is convex (closed) for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq12_HTML.gif. We say that G is bounded on bounded sets if G ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq13_HTML.gif 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 x 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq14_HTML.gif the set G ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq15_HTML.gif is a nonempty, closed subset of X, and if for each open set N of X containing G ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq15_HTML.gif, there exists an open neighborhood M of x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq16_HTML.gif such that G ( M ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq17_HTML.gif. Also, G is said to be completely continuous if G ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq13_HTML.gif is relatively compact for every bounded subset B X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq18_HTML.gif. 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., x n x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq19_HTML.gif, y n y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq20_HTML.gif, y n G ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq21_HTML.gif imply y 0 G ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq22_HTML.gif). Moreover, the following conclusions hold. Let D X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq23_HTML.gif and G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq11_HTML.gif be closed for all x D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq24_HTML.gif, if G is u.s.c. and D is closed, then graph ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq25_HTML.gif is closed. If G ( D ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq26_HTML.gif is compact and D is closed, then G is u.s.c. if and only if graph ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq25_HTML.gif is closed. Finally, we say that G has a fixed point if there exists x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq12_HTML.gif such that x G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq27_HTML.gif.

We denote by C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif the space of X-valued continuous functions on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq29_HTML.gif with the norm x = sup { x ( t ) ; t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq30_HTML.gif, and by L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq31_HTML.gif the space of X-valued Bochner functions on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq32_HTML.gif with the norm x = 0 1 x ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq33_HTML.gif.

A C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq1_HTML.gif-semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is said to be compact if T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is compact for any t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq34_HTML.gif. If the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is compact, then t T ( t ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq35_HTML.gif is equicontinuous at all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq34_HTML.gif with respect to x in all bounded subsets of X; i.e., the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is equicontinuous. If A is the generator of an analytic semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif or a differentiable semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif, then T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is an equicontinuous C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq1_HTML.gif-semigroup (see [21]). In this paper, we suppose that A generates an equicontinuous semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif.

Definition 2.1 A function x C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq36_HTML.gif is a mild solution of (1.1) if
  1. (1)

    x ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq37_HTML.gif,

     
  2. (2)

    x ( 0 ) = g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq38_HTML.gif, where v S F , x = { v L 1 ( I , X ) : v ( t ) F ( t , x ( t ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq39_HTML.gif.

     

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

Lemma 2.2 [22]

If W C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq40_HTML.gif is bounded, then α ( W ( t ) ) α ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq41_HTML.gif for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif, where W ( t ) = { x ( t ) ; x W } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq42_HTML.gif. Furthermore, if W is equicontinuous on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq32_HTML.gif, then α ( W ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq43_HTML.gif is continuous on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq32_HTML.gif, and α ( W ) = sup { α ( W ( t ) ) ; t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq44_HTML.gif.

Lemma 2.3 [22]

If { W n } n = 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq45_HTML.gif is a decreasing sequence of bounded closed nonempty subsets of X and lim n + α ( W n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq46_HTML.gif, then n = 1 + W n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq47_HTML.gif is nonempty and compact in X.

Lemma 2.4 [23]

If { u n } n = 1 L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq48_HTML.gif is uniformly integrable, then α ( { u n ( t ) } n = 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq49_HTML.gif is measurable and
α ( { 0 t u n ( s ) d s } n = 1 ) 2 0 t α ( { u n ( s ) } n = 1 ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ2_HTML.gif
(2.1)

Lemma 2.5 [24]

If the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif is equicontinuous and η L 1 ( 0 , 1 ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq50_HTML.gif, then the set { t 0 t T ( t s ) x ( s ) d s ; x L 1 ( 0 , 1 ; + ) , x ( s ) η ( s ) , for a.e. s [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq51_HTML.gif is equicontinuous on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq32_HTML.gif.

Lemma 2.6 [25]

If W is bounded, then for each ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq52_HTML.gif, there is a sequence { u n } n = 1 W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq53_HTML.gif such that
α ( W ) 2 α ( { u n } n = 1 ) + ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ3_HTML.gif
(2.2)
A countable set { f n } n = 1 L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq54_HTML.gif is said to be semicompact if
  1. (a)

    it is integrably bounded: f n ( t ) ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq55_HTML.gif for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif and every n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq56_HTML.gif, where ω ( ) L 1 ( 0 , 1 ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq57_HTML.gif;

     
  2. (b)

    the set { f n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq58_HTML.gif is relatively compact for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

     

Lemma 2.7 [26]

Every semicompact set is weakly compact in the space L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq31_HTML.gif.

Lemma 2.8 [16, 26]

If { f n } n = 1 L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq54_HTML.gif is semicompact, then { 0 t T ( t s ) f n ( s ) d s } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq59_HTML.gif is relatively compact in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif and, moreover, if f n f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq60_HTML.gif, then
0 t T ( t s ) f n ( s ) d s 0 t T ( t s ) f 0 ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equa_HTML.gif

as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq61_HTML.gif.

The map F : W X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq62_HTML.gif is said to be α contraction if there exists a positive constant k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq63_HTML.gif such that
α ( F ( Q ) ) k α ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equb_HTML.gif

for any bounded closed subset Q W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq64_HTML.gif.

Lemma 2.9 [2730] (Fixed point theorem)

If W X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq65_HTML.gif is a nonempty, bounded, closed, convex and compact subset, the map F : W 2 W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq66_HTML.gif is upper semicontinuous with F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq67_HTML.gif a nonempty, closed, convex subset of W for each x W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq68_HTML.gif, then F has at least one fixed point in W.

Lemma 2.10 [26] (Fixed point theorem)

If W X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq65_HTML.gif is nonempty, bounded, closed and convex, the map F : W 2 W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq66_HTML.gif is a closed α contraction map with F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq67_HTML.gif a nonempty, convex and compact subset of W for each x W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq68_HTML.gif, then F has at least one fixed point in W.

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. (1)

    The C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq1_HTML.gif semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif generated by A is equicontinuous. We denote N = sup { T ( t ) ; t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq69_HTML.gif.

     
  2. (2)

    g : C ( [ 0 , 1 ] ; X ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq70_HTML.gif is continuous and compact, there exist positive constants c and d such that g ( x ) c x + d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq71_HTML.gif, x C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq72_HTML.gif.

     
  3. (3)

    The multivalued operator F : [ 0 , 1 ] × X P c ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq73_HTML.gif satisfies the hypotheses: the set S F , x = { v L 1 ( I , X ) : v ( t ) F ( t , x ( t ) ) ;  for a.e.  t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq74_HTML.gif is nonempty.

     

t F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq75_HTML.gif is measurable for every x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq12_HTML.gif;

x F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq76_HTML.gif is u.s.c. for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif;
  1. (4)
    There exists L L 1 ( 0 , 1 ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq77_HTML.gif such that for any bounded D X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq23_HTML.gif,
    α ( F ( t , D ) ) L ( t ) α ( D ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equc_HTML.gif
     
for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq78_HTML.gif.
  1. (5)
    There exist a function m L 1 ( 0 , 1 ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq79_HTML.gif and a nondecreasing continuous function Ω : + + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq80_HTML.gif such that
    F ( t , x ) m ( t ) Ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equd_HTML.gif
     

for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq12_HTML.gif, and a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

Remark 3.1 If dim X < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq81_HTML.gif, then S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq82_HTML.gif for each x C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq83_HTML.gif (see Lasota and Opial [31]). If dim X = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq84_HTML.gif and x C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq36_HTML.gif, then the set S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq85_HTML.gif is nonempty if and only if the function Y : [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq86_HTML.gif defined by Y ( t ) = inf { v : v F ( t , x ( t ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq87_HTML.gif belongs to L 1 ( 0 , 1 ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq88_HTML.gif (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 { x n } n = 1 C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq89_HTML.gif and { v n } n = 1 L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq90_HTML.gif, where v n S F , x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq91_HTML.gif, such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq92_HTML.gif, v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq93_HTML.gif, then v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq94_HTML.gif.

Now we give the existence results under the above hypotheses.

Theorem 3.3 If (1)-(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constant R with
0 1 m ( s ) d s < N ( c R + d ) R 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ4_HTML.gif
(3.1)
Proof Define the operator Γ : C ( [ 0 , 1 ] ; X ) C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq95_HTML.gif by
( Γ x ) ( t ) = { y ( t ) C ( [ 0 , 1 ] ; X ) : y ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s ; v S F , x } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Eque_HTML.gif
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. (1)

    We contract a bounded, convex, closed and compact set W C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq96_HTML.gif such that Γ maps W into itself.

     
In view of (3.1), we know there exists a constant η > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq97_HTML.gif such that
0 1 m ( s ) d s < T 0 + η R 1 N Ω ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ5_HTML.gif
(3.2)

where T 0 = N ( c R + d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq98_HTML.gif.

Then there exists a positive integer K such that
T 0 + η T 0 + K η 1 N Ω ( s ) d s < 0 1 m ( s ) d s T 0 + η T 0 + ( K + 1 ) η 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ6_HTML.gif
(3.3)
Hence, we get 0 = t 0 < t 1 < t 2 < < t K 1 < t K = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq99_HTML.gif such that
0 t 1 m ( s ) d s = T 0 + η T 0 + 2 η 1 N Ω ( s ) d s , t 1 t 2 m ( s ) d s = T 0 + 2 η T 0 + 3 η 1 N Ω ( s ) d s , , t K 2 t K 1 m ( s ) d s = T 0 + ( K 1 ) η T 0 + K η 1 N Ω ( s ) d s , t K 1 1 m ( s ) d s T 0 + K η T 0 + ( K + 1 ) η 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equf_HTML.gif

We denote W 0 = { x C ( [ 0 , 1 ] ; X ) , x ( t ) i = sup { x ( t ) : t [ t i 1 , t i ] } T 0 + i η , i = 1 , 2 , , K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq100_HTML.gif, then W 0 C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq101_HTML.gif is nonempty, bounded, closed and convex.

For any x W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq102_HTML.gif, we have
( Γ x ) ( t ) = { y ( t ) : y ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s ; v ( t ) S F , x } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equg_HTML.gif
Therefore
y ( t ) T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s N ( c x + d ) + N 0 t m ( s ) Ω ( x ( s ) ) d s N ( c ( T 0 + K η ) + d ) + N 0 t m ( s ) Ω ( x ( s ) ) d s N ( c R + d ) + N 0 t m ( s ) Ω ( x ( s ) ) d s T 0 + N 0 t m ( s ) Ω ( x ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equh_HTML.gif
and
y i = sup { y ( t ) : t [ t i 1 , t i ] } sup { T 0 + N 0 t m ( s ) Ω ( x ( s ) ) d s : t [ t i 1 , t i ] } T 0 + N 0 t i m ( s ) Ω ( x ( s ) ) d s T 0 + N [ 0 t 1 m ( s ) Ω ( x ( s ) ) d s + t 1 t 2 m ( s ) Ω ( x ( s ) ) d s + + t i 1 t i m ( s ) Ω ( x ( s ) ) d s ] T 0 + N [ 0 t 1 m ( s ) d s Ω ( T 0 + η ) + t 1 t 2 m ( s ) d s Ω ( T 0 + 2 η ) + + t i 1 t i m ( s ) d s Ω ( T 0 + i η ) ] T 0 + N [ T 0 + η T 0 + 2 η 1 N Ω ( s ) d s Ω ( T 0 + η ) + T 0 + 2 η T 0 + 3 η 1 N Ω ( s ) d s Ω ( T 0 + 2 η ) + + T 0 + i η T 0 + ( i + 1 ) η 1 N Ω ( s ) d s Ω ( T 0 + i η ) ] T 0 + i η , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equi_HTML.gif

which implies Γ : W 0 2 W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq103_HTML.gif is a bounded operator.

Define W 1 = conv ¯ Γ ( W 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq104_HTML.gif, where conv ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq105_HTML.gif means the closure of the convex hull in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq106_HTML.gif. Then W 1 C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq107_HTML.gif is nonempty bounded closed convex on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq29_HTML.gif with W 1 W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq108_HTML.gif. Let W n + 1 = conv ¯ Γ ( W n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq109_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq56_HTML.gif. Similarly to the above discussions, we know that W n + 1 W n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq110_HTML.gif for n = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq111_HTML.gif as W 1 W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq108_HTML.gif and W 1 , W 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq112_HTML.gif are both nonempty, closed, bounded and convex. Thus { W n } n = 1 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq113_HTML.gif is a decreasing sequence consisting of subsets of C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq114_HTML.gif. Moreover, set
W = n = 1 + W n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equj_HTML.gif

then W is a convex, closed and bounded subset of C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq114_HTML.gif and Γ ( W ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq115_HTML.gif.

Now, we claim that W is nonempty and compact in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq114_HTML.gif. To do so, from Lemma 2.6, we know for arbitrary given ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq116_HTML.gif, there exist sequences { v n } n = 1 + S F , W n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq117_HTML.gif such that
α ( W n + 1 ( t ) ) = α ( ( Γ W n ) ( t ) ) 2 α ( 0 t T ( t s ) v n ( s ) n = 1 d s ) + ε 4 0 t α ( T ( t s ) v n ( s ) n = 1 ) d s + ε 4 N 0 t α ( v n ( s ) n = 1 ) d s + ε 4 N 0 t α ( F ( s , W n ( s ) ) d s + ε 4 N 0 t L ( s ) α ( W n ( s ) ) d s + ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equk_HTML.gif
Since this is true for arbitrary ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq116_HTML.gif, we have
α ( W n + 1 ( t ) ) 4 N 0 t L ( s ) α ( W n ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equl_HTML.gif
Because W n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq118_HTML.gif is decreasing for n, we can define
μ ( t ) = lim n + α ( ( W n ) ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equm_HTML.gif
Let n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq119_HTML.gif, we have
μ ( t ) 4 N 0 t L ( s ) μ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equn_HTML.gif
It implies that μ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq120_HTML.gif for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq121_HTML.gif. By Lemma 2.2, we know that lim n + α ( W n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq122_HTML.gif. Using Lemma 2.3, we obtain W = n = 1 + W n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq123_HTML.gif is nonempty and compact in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq106_HTML.gif.
  1. (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 graph ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq124_HTML.gif is closed. Let { x n } n = 1 W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq125_HTML.gif with x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq92_HTML.gif in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif, and y n Γ x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq126_HTML.gif with y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq127_HTML.gif in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif. Moreover, let { v n } n = 1 L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq90_HTML.gif be a sequence such that v n S F , x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq91_HTML.gif for any n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq56_HTML.gif, and
y n ( t ) = T ( t ) g ( x n ) + 0 t T ( t s ) v n ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equo_HTML.gif

As x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq92_HTML.gif in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif, we know that { x n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq128_HTML.gif is a bounded set of C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq28_HTML.gif, we denote R x = sup { x n : n = 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq129_HTML.gif.

From hypothesis (5), we obtain
v n ( t ) F ( t , x n ( t ) m ( t ) Ω ( x n ) m ( t ) Ω ( R x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equp_HTML.gif

Then we have the set { v n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq130_HTML.gif is integrably bounded for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

From hypothesis (4), we know
α ( { v n ( t ) } n = 1 ) α ( F ( t , { x n ( t ) } n = 1 ) ) L ( t ) α ( { x n ( t ) } n = 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equq_HTML.gif

for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif. Then the set { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is relatively compact for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

So, the set { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is semicompact. By applying Lemma 2.7, it yields that { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is weakly compact in L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq31_HTML.gif. We get that there exists v L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq132_HTML.gif such that v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq93_HTML.gif. Therefore, we infer that
0 t T ( t s ) v n ( s ) d s 0 t T ( t s ) v ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equr_HTML.gif
Further, we have
y n ( t ) T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equs_HTML.gif
and hence
y ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equt_HTML.gif
By Lemma 3.2, it implies that v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq94_HTML.gif, i.e., y Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq133_HTML.gif. Therefore graph ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq124_HTML.gif is closed. And hence Γ has closed values on W.
  1. (4)

    Γ is u.s.c. on W.

     

Since Γ W ¯ W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq134_HTML.gif is compact, W is closed and graph ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq124_HTML.gif 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 x Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq135_HTML.gif, 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 [812] the authors discuss the nonlocal initial value problem (1.1) when T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif 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 L ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq136_HTML.gif, and we only require T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq2_HTML.gif to be an equicontinuous semigroup. So, Theorem 3.3 generalizes and improves the related results in [812, 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 R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq137_HTML.gif such that
N ( c R + d ) + N 0 1 m ( s ) d s Ω ( R ) R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ7_HTML.gif
(3.4)
Proof In view of (3.4), we get
0 1 m ( s ) d s R N ( c R + d ) N Ω ( R ) < N ( c R + d ) R 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equu_HTML.gif

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

Remark 3.6 If N = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq138_HTML.gif, c = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq139_HTML.gif, d = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq140_HTML.gif, Ω ( x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq141_HTML.gif and 0 1 m ( s ) d s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq142_HTML.gif. We cannot obtain a constant R such that
1 3 R + R R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equv_HTML.gif
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 R = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq143_HTML.gif such that
0 1 m ( s ) d s = 1 < ln 3 = N ( c R + d ) R 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equw_HTML.gif

So, Theorem 3.3 is better than Theorem 3.5.

Theorem 3.7 [12]

If (1)-(5) are satisfied and g ( x ) d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq144_HTML.gif, then there is at least one mild solution for (1.1) provided that
0 1 m ( s ) d s < d + 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ8_HTML.gif
(3.5)
Proof In view of (3.5), we get there exists a constant R such that
0 1 m ( s ) d s < 0 R + d R 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equx_HTML.gif

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:
  1. (6)

    There exists a constant c + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq145_HTML.gif such that g ( u ) g ( v ) c u v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq146_HTML.gif for all u , v C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq147_HTML.gif. Therefore, g ( x ) c x + d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq71_HTML.gif, where d = g ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq148_HTML.gif.

     
Theorem 3.8 If (1) and (3)-(6) are satisfied and
N c + 4 N 0 1 L ( s ) d s < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equy_HTML.gif
then there is at least one mild solution for (1.1) provided that there exists a constant R satisfying
0 1 m ( s ) d s < T 0 + 1 N Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ9_HTML.gif
(3.6)

Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know Γ : W 0 2 W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq103_HTML.gif is a bounded map with convex values and is closed on W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq149_HTML.gif.

Now, we prove the values of Γ are compact in C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq114_HTML.gif.

Let x C ( [ 0 , 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq150_HTML.gif and y n Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq151_HTML.gif. To prove that Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq152_HTML.gif is compact, we have to show that y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq153_HTML.gif has a subsequence converging to a point y Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq154_HTML.gif. We have v n S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq155_HTML.gif such that
y n ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v n ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equz_HTML.gif
From hypothesis (5), we obtain
v n ( t ) F ( t , x ( t ) ) m ( t ) Ω ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equaa_HTML.gif

Then we have the set { v n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq130_HTML.gif is integrably bounded for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

From hypothesis (4), we know
α ( { v n ( t ) } n = 1 ) α ( F ( t , x ( t ) ) ) L ( t ) α ( x ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equab_HTML.gif

for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif. Then the set { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is relatively compact for a.e. t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq6_HTML.gif.

So, the set { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is semicompact. By applying Lemma 2.7, it yields that { v n ( t ) } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq131_HTML.gif is weakly compact in L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq31_HTML.gif. We get that there exists v L 1 ( 0 , 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq132_HTML.gif such that v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq93_HTML.gif. Therefore, we infer that
0 t T ( t s ) v n ( s ) d s 0 t T ( t s ) v ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equac_HTML.gif
and
lim n + y n ( t ) = T ( t ) g ( x ) + 0 t T ( t s ) v ( s ) d s = y ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equad_HTML.gif

By Lemma 3.2, it implies that v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq94_HTML.gif, i.e., y Γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq133_HTML.gif. Therefore Γ has compact values.

Next, we prove Γ is an α contraction map. For any B W 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq156_HTML.gif, we have
α ( ( Γ B ) ( t ) ) = α ( T ( t ) g ( B ) + 0 t T ( t s ) S F , B d s ) N c α ( B ) + α ( 0 t T ( t s ) S F , B d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equae_HTML.gif
From Lemma 2.6, we know for arbitrary given ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq116_HTML.gif, there exist sequences { v n } n = 1 + S F , B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq157_HTML.gif such that
α ( 0 t T ( t s ) S F , B d s ) = 2 α ( 0 t T ( t s ) v n ( s ) n = 1 d s ) + ε 4 0 t α ( T ( t s ) v n ( s ) n = 1 ) d s + ε 4 N 0 t α ( v n ( s ) n = 1 ) d s + ε 4 N 0 t α ( F ( s , B ( s ) ) ) d s + ε 4 N 0 t L ( s ) α ( B ( s ) ) d s + ε 4 N α ( B ) 0 1 L ( s ) d s + ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equaf_HTML.gif
Since this is true for arbitrary ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq116_HTML.gif, we have
α ( 0 t T ( t s ) S F , B d s ) 4 N α ( B ) 0 1 L ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equag_HTML.gif
Therefore, we obtain
α ( Γ B ) ( N c + 4 N 0 1 L ( s ) d s ) α ( B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equah_HTML.gif

Noting N c + 4 N 0 t L ( s ) d s < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq158_HTML.gif, 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:
{ u ( ζ , t ) t + Σ | α | 2 m a α ( ζ ) D α u ( ζ , t ) f ( t , u ( ζ , t ) ) , ( ζ , t ) Ω × [ 0 , 1 ] , u ( ζ , t ) = 0 , ( ζ , t ) Ω × [ 0 , 1 ] , u ( ζ , 0 ) = Ω 0 1 k ( t , ζ , η , u ( η , t ) ) d t d η , ζ Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ10_HTML.gif
(4.1)

where Ω is a bounded domain in n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq159_HTML.gif with a smooth boundary Ω, a α ( ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq160_HTML.gif is a smooth real function on Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq161_HTML.gif.

We suppose that
  1. (a)

    The differential operator Σ | α | 2 m a α ( ζ ) D α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq162_HTML.gif is strongly elliptic [21].

     
  2. (b)

    The function k : [ 0 , 1 ] × Ω × Ω × https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq163_HTML.gif satisfies the following conditions:

     

(b1) k ( t , ζ , η , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq164_HTML.gif is a continuous function about r for a.e. ( t , ζ , η ) [ 0 , 1 ] × Ω × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq165_HTML.gif.

(b2) k ( t , ζ , η , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq164_HTML.gif is measurable about ( t , ζ , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq166_HTML.gif for each fixed r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq167_HTML.gif.

(b3) For any R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq137_HTML.gif, there is β R L 1 ( [ 0 , 1 ] × Ω × Ω × ; + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq168_HTML.gif such that
| k ( t , ζ , η , r ) k ( t , ζ , η , r ) | β R ( t , ζ , ζ , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equai_HTML.gif
for all ( t , ζ , η , r ) , ( t , ζ , η , r ) ( [ 0 , 1 ] × Ω × Ω × ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq169_HTML.gif with | r | R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq170_HTML.gif, and
lim Δ ζ 0 Ω 0 1 β R ( t , ζ , Δ ζ , η ) d t d η = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equaj_HTML.gif

uniformly for ζ Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq171_HTML.gif.

(b4) There exist a ( ) L ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq172_HTML.gif and d ( ) L 2 ( [ 0 , 1 ] × Ω × Ω , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq173_HTML.gif such that
| k ( t , ζ , η , r ) | a ( t ) r + d ( t , ζ , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equak_HTML.gif

for all ( t , ζ , η , r ) ( [ 0 , 1 ] × Ω × Ω × ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq174_HTML.gif.

Let D ( A ) = H 2 m H 0 m ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq175_HTML.gif and Au ( ζ ) = Σ | α | 2 m a α ( ζ ) D α u ( ζ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq176_HTML.gif, then A generates an analytic semigroup on X = L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq177_HTML.gif ([21]). We suppose
g ( u ) ( ζ ) = Ω 0 1 k ( t , ζ , η , u ( η , t ) ) d t d η . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equal_HTML.gif

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

Then equation (4.1) can be regarded as the following nonlocal semilinear evolution equation:
{ u ( t ) Au ( t ) + f ( t , u ( t ) ) , t I = [ 0 , 1 ] , u ( 0 ) = g ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_Equ11_HTML.gif
(4.2)

By using Theorem 3.3, the problem (4.1) has at least one mild solution u C ( [ 0 , 1 ] ; L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-153/MediaObjects/13661_2013_Article_410_IEq178_HTML.gif provided that hypotheses (3)-(5) and (3.1) hold.

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Nanjing Institute of Technology
(2)
Department of Mathematics, Yangzhou University

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