Open Access

Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay

Advances in Difference Equations20112011:963463

DOI: 10.1155/2011/963463

Received: 5 December 2010

Accepted: 30 January 2011

Published: 24 February 2011

Abstract

We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq1_HTML.gif . An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski's measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodifferential equation.

1. Introduction

The fractional differential equations are valuable tools in the modeling of many phenomena in various fields of science and engineering; so, they attracted many researchers (cf., e.g., [16] and references therein). On the other hand, the integrodifferential equations arise in various applications such as viscoelasticity, heat equations, and many other physical phenomena (cf., e.g., [710] and references therein). Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (cf., e.g., [7, 1015] and references therein).

Neutral functional differential equations arise in many areas of applied mathematics and for this reason, the study of this type of equations has received great attention in the last few years (cf., e.g., [12, 1416] and references therein). In [12, 16], Hernández and Henríquez studied neutral functional differential equations with infinite delay. In the following, we will extend such results to fractional-order functional differential equations of neutral type with infinite delay. To the authors' knowledge, few papers can be found in the literature for the solvability of the fractional-order functional integrodifferential equations of neutral type with infinite delay.

In the present paper, we will consider the following fractional integrodifferential equation of neutral type with infinite delay in Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq2_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq5_HTML.gif is a phase space that will be defined later (see Definition 2.5). https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq6_HTML.gif is a generator of an analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq7_HTML.gif of uniformly bounded linear operators on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq8_HTML.gif . Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq9_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq10_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq13_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq14_HTML.gif ), and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq15_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq16_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq18_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq20_HTML.gif . The fractional derivative is understood here in the Caputo sense.

The aim of our paper is to study the solvability of (1.1) and present the existence of mild solution of (1.1) based on Kuratowski's measures of noncompactness. Moreover, an example is presented to show an application of the abstract results.

2. Preliminaries

Throughout this paper, we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq21_HTML.gif and denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq22_HTML.gif a real Banach space, by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq23_HTML.gif the Banach space of all linear and bounded operators on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq24_HTML.gif , and by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq25_HTML.gif the Banach space of all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq26_HTML.gif -valued continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq27_HTML.gif with the uniform norm topology.

Let us recall the definition of Kuratowski's measure of noncompactness.

Definition 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq28_HTML.gif be a bounded subset of a seminormed linear space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq29_HTML.gif . Kuratowski's measure of noncompactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq30_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ2_HTML.gif
(2.1)

This measure of noncompactness satisfies some important properties.

Lemma 2.2 (see [17]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq32_HTML.gif be bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq33_HTML.gif . Then,

  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq34_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq35_HTML.gif ,

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq36_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq37_HTML.gif denotes the closure of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq38_HTML.gif ,

     
  3. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq39_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq40_HTML.gif is precompact,

     
  4. (4)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq41_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq42_HTML.gif ,

     
  5. (5)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq43_HTML.gif ,

     
  6. (6)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq44_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq45_HTML.gif ,

     
  7. (7)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq46_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq47_HTML.gif ,

     
  8. (8)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq49_HTML.gif is the closed convex hull of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq50_HTML.gif .

     
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq51_HTML.gif , we define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ3_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq52_HTML.gif .

The following lemmas will be needed.

Lemma 2.3 (see [17]).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq53_HTML.gif is a bounded, equicontinuous set, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ4_HTML.gif
(2.3)

Lemma 2.4 (see [18]).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq54_HTML.gif and there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq55_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq56_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq57_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq58_HTML.gif is integrable and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ5_HTML.gif
(2.4)

The following definition about the phase space is due to Hale and Kato [11].

Definition 2.5.

A linear space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq59_HTML.gif consisting of functions from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq60_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq61_HTML.gif with semi-norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq62_HTML.gif is called an admissible phase space if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq63_HTML.gif has the following properties.

  1. (1)
    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq64_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq66_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq68_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq69_HTML.gif and
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ6_HTML.gif
    (2.5)

    where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq70_HTML.gif is a constant.

     
  2. (2)
    There exist a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq71_HTML.gif and a locally bounded function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq72_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq73_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ7_HTML.gif
    (2.6)

    for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq75_HTML.gif as in (1).

     
  3. (3)

    The space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq76_HTML.gif is complete.

     

Remark 2.6.

(2.5) in (1) is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq77_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq78_HTML.gif .

The following result will be used later.

Lemma 2.7 (see [19, 20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq79_HTML.gif be a bounded, closed, and convex subset of a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq80_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq81_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq82_HTML.gif be a continuous mapping of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq83_HTML.gif into itself. If the implication
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ8_HTML.gif
(2.7)

holds for every subset https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq84_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq85_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq86_HTML.gif has a fixed point.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq87_HTML.gif be a set defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ9_HTML.gif
(2.8)

Motivated by [4, 5, 21], we give the following definition of mild solution of (1.1).

Definition 2.8.

A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq88_HTML.gif satisfying the equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ10_HTML.gif
(2.9)
is called a mild solution of (1.1), where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ11_HTML.gif
(2.10)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq89_HTML.gif is a probability density function defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq90_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ12_HTML.gif
(2.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ13_HTML.gif
(2.12)

Remark 2.9.

According to [22], direct calculation gives that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ14_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq91_HTML.gif .

We list the following basic assumptions of this paper.

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq92_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq93_HTML.gif is measurable, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq95_HTML.gif is continuous for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq96_HTML.gif , and there exist two positive functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq97_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ15_HTML.gif
(2.14)
(H2) For any bounded sets https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq99_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq100_HTML.gif , there exists an integrable positive function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq101_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ16_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq103_HTML.gif .

(H3) There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq104_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ17_HTML.gif
(2.16)

(H4) For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq106_HTML.gif is measurable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq108_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq109_HTML.gif . The map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq110_HTML.gif is continuous from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq111_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq112_HTML.gif , here, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq113_HTML.gif .

(H5) There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq114_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ18_HTML.gif
(2.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq116_HTML.gif .

3. Main Result

In this section, we will apply Lemma 2.7 to show the existence of mild solution of (1.1). To this end, we consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq117_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ19_HTML.gif
(3.1)

It follows from (H1), (H3), and (H4) that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq118_HTML.gif is well defined.

It will be shown that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq119_HTML.gif has a fixed point, and this fixed point is then a mild solution of (1.1).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq120_HTML.gif be the function defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ20_HTML.gif
(3.2)

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq122_HTML.gif .

It is clear to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq123_HTML.gif satisfies (2.9) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq124_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq125_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq126_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ21_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq127_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq128_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ22_HTML.gif
(3.4)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq129_HTML.gif is a Banach space. Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ23_HTML.gif
(3.5)
Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq130_HTML.gif , from(2.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ24_HTML.gif
(3.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq131_HTML.gif .

In order to apply Lemma 2.7 to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq132_HTML.gif has a fixed point, we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq133_HTML.gif be an operator defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq134_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq135_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ25_HTML.gif
(3.7)

Clearly, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq136_HTML.gif has a fixed point is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq137_HTML.gif has one. So, it turns out to prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq138_HTML.gif has a fixed point.

Now, we present and prove our main result.

Theorem 3.1.

Assume that (H1)–(H5) are satisfied, then there exists a mild solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq139_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq140_HTML.gif .

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq142_HTML.gif , from (3.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ26_HTML.gif
(3.8)
In view of (H3),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ27_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq143_HTML.gif .

Next, we show that there exists some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq144_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq145_HTML.gif . If this is not true, then for each positive number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq146_HTML.gif , there exist a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq147_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq148_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq149_HTML.gif . However, on the other hand, we have from (3.8), (3.9), and (H4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ28_HTML.gif
(3.10)
Dividing both sides of (3.10) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq150_HTML.gif , and taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq151_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ29_HTML.gif
(3.11)

This contradicts (2.17). Hence, for some positive number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq152_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq153_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq155_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq156_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq157_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq158_HTML.gif satisfies (H1), for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq159_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ30_HTML.gif
(3.12)
In view of (3.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ31_HTML.gif
(3.13)
Noting that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ32_HTML.gif
(3.14)
we have by the Lebesgue Dominated Convergence Theorem that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ33_HTML.gif
(3.15)
Therefore, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ34_HTML.gif
(3.16)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq160_HTML.gif is continuous.

Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ35_HTML.gif
(3.17)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq162_HTML.gif , then we can see
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ36_HTML.gif
(3.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ37_HTML.gif
(3.19)

It follows the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq163_HTML.gif in the uniform operator topology for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq164_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq165_HTML.gif tends to 0, as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq166_HTML.gif . The continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq167_HTML.gif ensures that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq168_HTML.gif tends to 0, as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq169_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq170_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ38_HTML.gif
(3.20)

Clearly, the first term on the right-hand side of (3.20) tends to 0 as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq171_HTML.gif . The second term on the right-hand side of (3.20) tends to 0 as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq172_HTML.gif as a consequence of the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq173_HTML.gif in the uniform operator topology for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq174_HTML.gif .

In view of the assumption of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq175_HTML.gif and (3.8), we see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ39_HTML.gif
(3.21)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq176_HTML.gif is equicontinuous.

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq177_HTML.gif be an arbitrary subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq178_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq179_HTML.gif .

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq180_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ40_HTML.gif
(3.22)
Noting that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq181_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ41_HTML.gif
(3.23)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ42_HTML.gif
(3.24)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq182_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq183_HTML.gif .

Moreover, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq184_HTML.gif and bounded set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq185_HTML.gif , we can take a sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq186_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq187_HTML.gif (see [23], P125). Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq188_HTML.gif , noting that the choice of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq189_HTML.gif , and from Lemmas 2.2–2.4 and (H2), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ43_HTML.gif
(3.25)
It follows from Lemma 2.2 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ44_HTML.gif
(3.26)
since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq190_HTML.gif is arbitrary, we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ45_HTML.gif
(3.27)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq191_HTML.gif . Applying now Lemma 2.7, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq192_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq193_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq194_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq195_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq196_HTML.gif is a fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq197_HTML.gif which is a mild solution of (1.1).

4. Application

In this section, we consider the following integrodifferential model:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ46_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq201_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq202_HTML.gif are continuous functions, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq203_HTML.gif .

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq204_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq205_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ47_HTML.gif
(4.2)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq206_HTML.gif generates a compact, analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq207_HTML.gif of uniformly bounded, linear operators, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq208_HTML.gif .

Let the phase space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq209_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq210_HTML.gif , the space of bounded uniformly continuous functions endowed with the following norm:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ48_HTML.gif
(4.3)

then we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq211_HTML.gif in (2.6).

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq214_HTML.gif , we set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ49_HTML.gif
(4.4)

Then (4.1) can be reformulated as the abstract (1.1).

Moreover, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq215_HTML.gif , we can see
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ50_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq216_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq217_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq219_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ51_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq220_HTML.gif .

Suppose further that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq221_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ52_HTML.gif
(4.7)

then (4.1) has a mild solution by Theorem 3.1.

For example, if we put
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ53_HTML.gif
(4.8)
then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq223_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_IEq224_HTML.gif . Thus, we see
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F963463/MediaObjects/13662_2010_Article_83_Equ54_HTML.gif
(4.9)

Declarations

Acknowledgments

The authors are grateful to the referees for their valuable suggestions. F. Li is supported by the NSF of Yunnan Province (2009ZC054M). J. Zhang is supported by Tianyuan Fund of Mathematics in China (11026100).

Authors’ Affiliations

(1)
School of Mathematics, Yunnan Normal University
(2)
Department of Mathematics, Central China Normal University

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© Fang Li and Jun Zhang. 2011

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