This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Existence solutions for boundary value problem of nonlinear fractional q-difference equations

Wen-Xue Zhou12* and Hai-Zhong Liu1

Author Affiliations

1 Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, P.R. China

2 School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China

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Advances in Difference Equations 2013, 2013:113  doi:10.1186/1687-1847-2013-113


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/113


Received:10 December 2012
Accepted:4 April 2013
Published:19 April 2013

© 2013 Zhou and Liu; licensee Springer

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

Abstract

In this paper, we discuss the existence of weak solutions for a nonlinear boundary value problem of fractional q-difference equations in Banach space. Our analysis relies on the Mönch’s fixed-point theorem combined with the technique of measures of weak noncompactness.

MSC: 26A33, 34B15.

Keywords:
boundary value problem; fractional q-difference equations; Caputo fractional derivative; weak solutions

1 Introduction

Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps because of its demonstrated applications in various fields of science and engineering [1]. Many researchers studied the existence of solutions to fractional boundary value problems, for example, [2-10].

The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [11,12]; basic definitions and properties of q-difference calculus can be found in [13,14].

The fractional q-difference calculus had its origin in the works by Al-Salam [15] and Agarwal [16]. More recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q-difference calculus were made, for example, q-analogues of the integral and differential fractional operators properties such as Mittage-Leffler function [17], just to mention some.

El-Shahed and Hassan [18] studied the existence of positive solutions of the q-difference boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M1">View MathML</a>

Ferreira [19] considered the existence of positive solutions to nonlinear q-difference boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M2">View MathML</a>

Ferreira [20] studied the existence of positive solutions to nonlinear q-difference boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M3">View MathML</a>

El-Shahed and Al-Askar [21] studied the existence of positive solutions to nonlinear q-difference equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M5">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M6">View MathML</a> is the fractional q-derivative of the Caputo type.

Ahmad, Alsaedi and Ntouyas [22] discussed the existence of solutions for the second-order q-difference equation with nonseparated boundary conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M11">View MathML</a> is a fixed constant, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M12">View MathML</a> is a fixed real number.

Ahmad and Nieto [23] discussed a nonlocal nonlinear boundary value problem (BVP) of third-order q-difference equations given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M14">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M16">View MathML</a> is a fixed constant, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M17">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M18">View MathML</a> is a real number.

This paper is mainly concerned with the existence results for the following fractional q-difference equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M19">View MathML</a>

(1.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M5">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M6">View MathML</a> is the fractional q-derivative of the Caputo type. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M22">View MathML</a> is a given function satisfying some assumptions that will be specified later, and E is a Banach space with norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M23">View MathML</a>.

To investigate the existence of solutions of the problem above, we use Mönch’s fixed-point theorem combined with the technique of measures of weak noncompactness, which is an important method for seeking solutions of differential equations. This technique was mainly initiated in the monograph of Banaś and Goebel [24], and subsequently developed and used in many papers; see, for example, Banaś et al.[25], Guo et al.[26], Krzyska and Kubiaczyk [27], Lakshmikantham and Leela [28], Mönch [29], O’Regan [30,31], Szufla [32,33] and the references therein. As far as we know, there are very few results devoted to weak solutions of nonlinear fractional differential equations [34-38]. Motivated by the above mentioned papers, the purpose of this paper is to establish the existence results for the boundary value problem (1.1) by virtue of the Mönch’s fixed-point theorem combined with the technique of measures of weak noncompactness.

The remainder of this article is organized as follows. In Section 2, we provide some basic definitions, preliminaries facts and various lemmas, which are needed later. In Section 3, we give main results of the problem (1.1). In the end, we also give an example for the illustration of the theories established in this paper.

2 Preliminaries and lemmas

In this section, we present some basic notations, definitions and preliminary results, which will be used throughout this paper.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M24">View MathML</a> and define [13]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M25">View MathML</a>

The q-analogue of the power <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M26">View MathML</a> is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M27">View MathML</a>

If α is not a positive integer, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M28">View MathML</a>

Note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M30">View MathML</a>. The q-gamma function is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M31">View MathML</a>

and satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M32">View MathML</a>.

The q-derivative of a function f is here defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M33">View MathML</a>

and q-derivatives of higher order by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M34">View MathML</a>

The q-integral of a function f defined in the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M35">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M36">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M37">View MathML</a> and f is defined in the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M38">View MathML</a>, its integral from a to b is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M39">View MathML</a>

Similarly, as done for derivatives, an operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M40">View MathML</a> can be defined, namely,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M41">View MathML</a>

The fundamental theorem of calculus applies to these operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M42">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M43">View MathML</a>, that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M44">View MathML</a>

and if f is continuous at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M45">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M46">View MathML</a>

Basic properties of the two operators can be found in the book mentioned in [13]. We now point out three formulas that will be used later (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M47">View MathML</a> denotes the derivative with respect to variable i) [19]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M48">View MathML</a>

Remark 2.1 We note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M49">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M50">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M51">View MathML</a>[19].

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M52">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M53">View MathML</a> denote the Banach space of real-valued Lebesgue integrable functions on the interval J, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M54">View MathML</a> denote the Banach space of real-valued essentially bounded and measurable functions defined over J with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M55">View MathML</a>.

Let E be a real reflexive Banach space with norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M56">View MathML</a> and dual <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M57">View MathML</a>, and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M58">View MathML</a> denote the space E with its weak topology. Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M59">View MathML</a> is the Banach space of continuous functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M60">View MathML</a> with the usual supremum norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M61">View MathML</a>.

Moreover, for a given set V of functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M62">View MathML</a>, let us denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M63">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M64">View MathML</a>.

Definition 2.1 A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M65">View MathML</a> is said to be weakly sequentially continuous if h takes each weakly convergent sequence in E to a weakly convergent sequence in E (i.e. for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M66">View MathML</a> in E with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M67">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M68">View MathML</a> then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M69">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M70">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M71">View MathML</a>).

Definition 2.2[39]

The function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M60">View MathML</a> is said to be Pettis integrable on J if and only if there is an element <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M73">View MathML</a> corresponding to each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M74">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M75">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M76">View MathML</a>, where the integral on the right is supposed to exist in the sense of Lebesgue. By definition, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M77">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M78">View MathML</a> be the space of all E-valued Pettis integrable functions in the interval J.

Lemma 2.1[39]

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M79">View MathML</a>is Pettis integrable and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M80">View MathML</a>is a measurable and an essentially bounded real-valued function, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M81">View MathML</a>is Pettis integrable.

Definition 2.3[40]

Let E be a Banach space, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M82">View MathML</a> the set of all bounded subsets of E, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M83">View MathML</a> the unit ball in E. The De Blasi measure of weak noncompactness is the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M84">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M85">View MathML</a>

Lemma 2.2[40]

The De Blasi measure of noncompactness satisfies the following properties:

(a) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M86">View MathML</a>;

(b) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M87">View MathML</a>is relatively weakly compact;

(c) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M88">View MathML</a>;

(d) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M89">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M90">View MathML</a>denotes the weak closure ofS;

(e) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M91">View MathML</a>;

(f) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M92">View MathML</a>;

(g) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M93">View MathML</a>;

(h) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M94">View MathML</a>.

The following result follows directly from the Hahn-Banach theorem.

Lemma 2.3LetEbe a normed space with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M95">View MathML</a>. Then there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M76">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M97">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M98">View MathML</a>.

Definition 2.4[16]

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99">View MathML</a> and f be a function defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M100">View MathML</a>. The fractional q-integral of the Riemann-Liouville type is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M101">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M102">View MathML</a>

Definition 2.5[14]

The fractional q-derivative of the Riemann-Liouville type of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99">View MathML</a> is defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M104">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M105">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M106">View MathML</a> is the smallest integer greater than or equal to α.

Definition 2.6[14]

The fractional q-derivative of the Caputo type of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M99">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M108">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M106">View MathML</a> is the smallest integer greater than or equal to α.

Lemma 2.4[14]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M110">View MathML</a>and letfbe a function defined on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M100">View MathML</a>. Then the next formulas hold:

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M112">View MathML</a>,

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M113">View MathML</a>.

Lemma 2.5[32]

LetDbe a closed convex and equicontinuous subset of a metrizable locally convex vector space<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M59">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M115">View MathML</a>. Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M116">View MathML</a>is weakly sequentially continuous. If the implication

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M117">View MathML</a>

(2.1)

holds for every subsetVofD, thenAhas a fixed point.

3 Main results

Let us start by defining what we mean by a solution of the problem (1.1).

Definition 3.1 A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M118">View MathML</a> is said to be a solution of the problem (1.1) if u satisfies the equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M119">View MathML</a> on J, and satisfy the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M120">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M121">View MathML</a>.

For the existence results on the problem (1.1), we need the following auxiliary lemmas.

Lemma 3.1[19]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M122">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M123">View MathML</a>. Then, the following equality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M124">View MathML</a>

Lemma 3.2[14]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M122">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M126">View MathML</a>. Then the following equality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M127">View MathML</a>

We derive the corresponding Green’s function for boundary value problem (1.1), which will play major role in our next analysis.

Lemma 3.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M128">View MathML</a>be a given function, then the boundary-value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M129">View MathML</a>

(3.1)

has a unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M130">View MathML</a>

(3.2)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131">View MathML</a>is defined by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M132">View MathML</a>

(3.3)

Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131">View MathML</a>is called the Green’s function of boundary value problem (3.1).

Proof By Lemma 2.4 and Lemma 3.2, we can reduce the equation of problem (3.1) to an equivalent integral equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M134">View MathML</a>

(3.4)

Applying the boundary conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M120">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M136">View MathML</a>

(3.5)

So, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M137">View MathML</a>

(3.6)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M138">View MathML</a>

(3.7)

Then, by the condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M121">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M140">View MathML</a>

(3.8)

Therefore, the unique solution of problem (3.1) is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M141">View MathML</a>

which completes the proof. □

Remark 3.1 From the expression of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131">View MathML</a>, it is obvious that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M131">View MathML</a> is continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M144">View MathML</a>. Denote by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M145">View MathML</a>

To prove the main results, we need the following assumptions:

(H1) For each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>, the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M147">View MathML</a> is weakly sequentially continuous;

(H2) For each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148">View MathML</a>, the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M149">View MathML</a> is Pettis integrable on J;

(H3) There exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M150">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M151">View MathML</a>

(H3)′ There exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M150">View MathML</a> and a continuous nondecreasing function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M153">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M154">View MathML</a>

(H4) For each bounded set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M155">View MathML</a>, and each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M157">View MathML</a>

(H5) There exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M158">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M159">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M160">View MathML</a>.

Theorem 3.1LetEbe a reflexive Banach space and assume that (H1)-(H3) are satisfied. If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M161">View MathML</a>

(3.9)

then the problem (1.1) has at least one solution onJ.

Proof Let the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M162">View MathML</a> defined by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M163">View MathML</a>

(3.10)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M164">View MathML</a> is the Green’s function defined by (3.3). It is well known the fixed points of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> are solutions of the problem (1.1).

First notice that, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M167">View MathML</a> (assumption (H2)). Since, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M168">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M169">View MathML</a> is Pettis integrable for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a> by Lemma 2.1, and so the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> is well defined.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M172">View MathML</a>, and consider the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M173">View MathML</a>

Clearly, the subset D is closed, convex and equicontinuous. We shall show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> satisfies the assumptions of Lemma 2.5. The proof will be given in three steps.

Step 1: We will show that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> maps D into itself.

Take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177">View MathML</a> and assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M178">View MathML</a>. Then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M180">View MathML</a>. Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M181">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M182">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M183">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M184">View MathML</a>, so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M185">View MathML</a>. Then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M186">View MathML</a>, such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M187">View MathML</a>. Hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M188">View MathML</a>

this means that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M189">View MathML</a>.

Step 2: We will show that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> is weakly sequentially continuous.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M191">View MathML</a> be a sequence in D and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M192">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M193">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Fix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Since f satisfies assumptions (H1), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M196">View MathML</a> converge weakly uniformly to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M197">View MathML</a>. Hence, the Lebesgue dominated convergence theorem for Pettis integrals implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M198">View MathML</a> converges weakly uniformly to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M199">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M200">View MathML</a>. Repeating this for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a> shows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M202">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M203">View MathML</a> is weakly sequentially continuous.

Step 3: The implication (2.1) holds. Now let V be a subset of D such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M204">View MathML</a>. Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M205">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M207">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177">View MathML</a>, is bounded in E. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M209">View MathML</a> is weakly relatively compact since a subset of a reflexive Banach space is weakly relatively compact if and only if it is bounded in the norm topology. Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M210">View MathML</a>

thus, V is relatively weakly compact in E. In view of Lemma 2.5, we deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> has a fixed point, which is obviously a solution of the problem (1.1). This completes the proof. □

Remark 3.2 In Theorem 3.1, we presented an existence result for weak solutions of the problem (1.1) in the case where the Banach space E is reflexive. However, in the nonreflexive case, conditions (H1)-(H3) are not sufficient for the application of Lemma 2.5; the difficulty is with condition (2.1).

Theorem 3.2LetEbe a Banach space, and assume assumptions (H1), (H2), (H3), (H4) are satisfied. If (3.9) holds, then the problem (1.1) has at least one solution onJ.

Theorem 3.3LetEbe a Banach space, and assume assumptions (H1), (H2), (H3)′, (H4), (H5) are satisfied. If (3.9) holds, then the problem (1.1) has at least one solution onJ.

Proof Assume that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M162">View MathML</a> is defined by the formula (3.10). It is well known the fixed points of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> are solutions of the problem (1.1).

First notice that, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M148">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M167">View MathML</a> (assumption (H2)). Since, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M216">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M169">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a> is Pettis integrable (Lemma 2.1), and thus, the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> makes sense.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M220">View MathML</a>, and consider the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M221">View MathML</a>

(3.11)

clearly, the subset <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222">View MathML</a> is closed, convex and equicontinuous. We shall show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> satisfies the assumptions of Lemma 2.5. The proof will be given in three steps.

Step 1: We will show that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> maps <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222">View MathML</a> into itself.

Take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M226">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177">View MathML</a> and assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M178">View MathML</a>. Then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M180">View MathML</a>. Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M231">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M182">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M183">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M234">View MathML</a>, so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M185">View MathML</a>. Then there exist <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M186">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M237">View MathML</a>

Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M238">View MathML</a>

this means that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M239">View MathML</a>.

Step 2: We will show that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> is weakly sequentially continuous.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M191">View MathML</a> be a sequence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222">View MathML</a> and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M192">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M193">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Fix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Since f satisfies assumptions (H1), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M196">View MathML</a>, converging weakly uniformly to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M197">View MathML</a>. Hence, the Lebesgue dominated convergence theorem for Pettis integral implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M198">View MathML</a> converging weakly uniformly to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M199">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M200">View MathML</a>. We do it for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a> so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M202">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M254">View MathML</a> is weakly sequentially continuous.

Step 3: The implication (2.1) holds. Now let V be a subset of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M204">View MathML</a>. Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M205">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M259">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M177">View MathML</a>, is bounded in E. Using this fact, assumption (H4), Lemma 2.2 and the properties of the measure β, we have for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M262">View MathML</a>

which gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M263">View MathML</a>

This means that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M264">View MathML</a>

By (3.9), it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M265">View MathML</a>, that is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M266">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M146">View MathML</a>, and then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M268">View MathML</a> is relatively weakly compact in E. In view of Lemma 2.5, we deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/113/mathml/M165">View MathML</a> has a fixed point which is obviously a solution of the problem (1.1). This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was supported by the National Natural Science Foundation of China (11161027, 11262009). The authors are thankful to the referees for their careful reading of the manuscript and insightful comments.

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