Open Access Research

Positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces

Yan Sun

Author Affiliations

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

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China

Boundary Value Problems 2012, 2012:86  doi:10.1186/1687-2770-2012-86


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/86


Received:5 June 2012
Accepted:23 July 2012
Published:6 August 2012

© 2012 Sun; 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 work, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro differential equation in a real Banach space. Some new existence results of positive solutions are established by applying fixed-point index theory together with comparison theorem. Some discussions and an example are given to demonstrate the applications of our main results.

MSC: 34B15, 34B25, 45J05.

Keywords:
measure of non-comparison; positive solution; boundary value problem; impulsive integro-differential equation

1 Introduction

In this paper, we study the existence of positive solutions to second-order singular nonlinear impulsive integro-differential equation of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M13">View MathML</a>, and P is a positive cone in E. θ is a zero element of E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M15">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M16">View MathML</a>

(1.2)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M19">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M21">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M23">View MathML</a> denote the jump of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M25">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M27">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M31">View MathML</a> represent the right-hand limit and left-hand limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M25">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26">View MathML</a>, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M35">View MathML</a> and may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M36">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M37">View MathML</a>.

Boundary value problems for impulsive differential equations arise from many nonlinear problems in sciences, such as physics, population dynamics, biotechnology, and economics etc. (see [1,2,4-14,16-18]). As it is well known that impulsive differential equations contain jumps and/or impulses which are main characteristic feature in computational biology. Over the past 15 years, a significant advance has been achieved in theory of impulsive differential equations. However, the corresponding theory of impulsive integro-differential equations in Banach spaces does not develop rapidly. Recently, Guo [5-8] established the existence of a solution, multiple solutions and extremal solutions for nonlinear impulsive integro-differential equations with nonsingular argument in Banach spaces. The main tools of Guo [5-8] are the Schauder fixed-point theorem, fixed-point index theory, upper and lower solutions together with the monotone iterative technique, respectively. The conditions of the Kuratowski measure of non-compactness in Guo [5-8] play an important role in the proof of the results. But all kinds of compactness type conditions is difficult to verify in abstract spaces. As a result, it is an interesting and important problem to remove or weak compactness type conditions.

Inspired and motivated greatly by the above works, the aim of the paper is to consider the existence of positive solutions for the boundary value problem (1.1) under simpler conditions. The main results of problem (1.1) are obtained by making use of fixed-point index theory and fixed-point theorem. More specifically, in the proof of these theorems, we construct a special cone for strict set contraction operator. Our main results in essence improve and generalize the corresponding results of Guo [5-8]. Moreover, our method is different from those in Guo [5-8].

The rest of the paper is organized as follows: In Section 2, we present some known results and introduce conditions to be used in the next section. The main theorem formulated and proved in Section 3. Finally, in Section 4, some discussions and an example for singular nonlinear integro-differential equations are presented to demonstrate the application of the main results.

2 Preliminaries and lemmas

In this section, we shall state some necessary definitions and preliminaries results.

Definition 2.1 Let E be a real Banach space. A nonempty closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M38">View MathML</a> is called a cone if it satisfies the following two conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M40">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M41">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M43">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M44">View MathML</a>.

A cone is said solid if it contains interior points, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M45">View MathML</a>. A cone P is called to be generating if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M46">View MathML</a>, i.e., every element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M47">View MathML</a> can be represented in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M48">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M49">View MathML</a>. A cone P in E induces a partial ordering in E given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M50">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M51">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M53">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M54">View MathML</a>; if cone P is solid and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M55">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M56">View MathML</a>.

Definition 2.2 A cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M38">View MathML</a> is said to be normal if there exists a positive constant N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M61">View MathML</a>.

Definition 2.3 Let E be a metric space and S be a bounded subset of E. The measure of non-compactness <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M62">View MathML</a> of S is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M63">View MathML</a>

Definition 2.4 An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M64">View MathML</a> is said to be completely continuous if it is continuous and compact. B is called a k-set-contraction (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M65">View MathML</a>) if it is continuous, bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M66">View MathML</a> for any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M67">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M62">View MathML</a> denotes the measure of noncompactness of S.

A k-set-contraction is called a strict-set contraction if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M69">View MathML</a>. An operator B is said to be condensing if it is continuous, bounded, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M70">View MathML</a> for any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M71">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M72">View MathML</a>.

Obviously, if B is a strict-set contraction, then B is a condensing mapping, and if operator B is completely continuous, then B is a strict-set contraction.

It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M73">View MathML</a> is a solution of the problem (1.1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M74">View MathML</a> is a solution of the following nonlinear integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M75">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M76">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M77">View MathML</a>. In what follows, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M79">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M80">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M12">View MathML</a>. By making use of (2.1), we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M82">View MathML</a> has the following properties.

Proposition 2.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M84">View MathML</a>.

Proposition 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M86">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M88">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M89">View MathML</a>

(2.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M90">View MathML</a>. It is easy to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M91">View MathML</a> is a Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M92">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M93">View MathML</a> is a cone in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M94">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M96">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M98">View MathML</a>. Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M100">View MathML</a> is a cone in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M102">View MathML</a>, by making use of the mean value theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M103">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M104">View MathML</a>), obviously we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M105">View MathML</a> exists and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M106">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M107">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M108">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M111">View MathML</a>.

A map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M112">View MathML</a> is called a nonnegative solution of problem (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M114">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M24">View MathML</a> satisfies problem (1.1). An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M117">View MathML</a> is called a positive solution of problem (1.1) if y is a nonnegative solution of problem (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M118">View MathML</a>.

For convenience and simplicity in the following discussion, we denote

where ν denote 0 or ∞.

To establish the existence of multiple positive solutions in E of problem (1.1), let us list the following assumptions, which will stand throughout the paper:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M121">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M122">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M124">View MathML</a> are Lebesgue integrable functionals on J (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M125">View MathML</a>) and satisfying

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M128">View MathML</a> and there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M131">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132">View MathML</a>) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M133">View MathML</a>

(2.4)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M134">View MathML</a>

(H3) for any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M138">View MathML</a> together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M139">View MathML</a> are relatively compact sets,

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M140">View MathML</a>,

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M141">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M142">View MathML</a>

(2.5)

We shall reduce problem (1.1) to an integral equation in E. To this end, we first consider operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M143">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M144">View MathML</a>

(2.6)

Lemma 2.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M145">View MathML</a>is a solution of problem (1.1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M146">View MathML</a>is a solution of the following impulsive integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M147">View MathML</a>

(2.7)

i.e., yis a fixed point of operatorAdefined by (2.6) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M148">View MathML</a>.

Proof First suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M149">View MathML</a> is a solution of problem (1.1). It is easy to see by the integration of problem (1.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M150">View MathML</a>

(2.8)

Integrate again, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M151">View MathML</a>

(2.9)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M37">View MathML</a> in (2.8) and (2.9), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M153">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M154">View MathML</a>

(2.11)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M155">View MathML</a>

(2.12)

We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M156">View MathML</a>

(2.13)

Substituting (2.12) and (2.13) into (2.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M157">View MathML</a>

Conversely, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M102">View MathML</a> is a solution of the integral equation (2.7). Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M159">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M160">View MathML</a>). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M26">View MathML</a>, direct differentiation of the integral equation (2.7) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M162">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M163">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M165">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M166">View MathML</a>). It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M168">View MathML</a>. The proof is complete. □

Thanks to (2.1), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M169">View MathML</a>

In the following, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M170">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M171">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>).

Lemma 2.2 ([12])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M176">View MathML</a>be a bounded set. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M177">View MathML</a>is equi-continuous on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M178">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132">View MathML</a>). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M180">View MathML</a>

where ϒ and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M181">View MathML</a>denote the Kuratowski measures of noncompactness of bounded sets inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M97">View MathML</a>, respectively.

Lemma 2.3 ([15])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M183">View MathML</a>be bounded equicontinuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M184">View MathML</a>is continuous onJand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M185">View MathML</a>

Lemma 2.4 ([15])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M186">View MathML</a>is relatively compact if and only if each element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M187">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M188">View MathML</a>are uniformly bounded and equicontinuous on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M178">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132">View MathML</a>).

Lemma 2.5 ([15])

LetEbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M191">View MathML</a>ifHis countable and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M192">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M195">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M196">View MathML</a>is integrable onJ, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M197">View MathML</a>

Lemma 2.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M198">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M199">View MathML</a>, from Proposition 2.1 and (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M200">View MathML</a>

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M201">View MathML</a>, by (2.6) and Proposition 2.2, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M202">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M198">View MathML</a>. □

Lemma 2.7Suppose that (H1) and (H3) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204">View MathML</a>is completely continuous.

Proof Firstly, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204">View MathML</a> is continuous. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M208">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M209">View MathML</a>). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M212">View MathML</a>, then

(2.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M214">View MathML</a>

(2.15)

Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>, from the Lebesgue dominated convergence theorem together with (2.14) and (2.15), we know that

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M204">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M218">View MathML</a> be any bounded set, then there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M219">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M220">View MathML</a>. Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M222">View MathML</a>, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M223">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M224">View MathML</a>

(2.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M132">View MathML</a>. Let

Integrating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M229">View MathML</a> from 0 to 1 and exchanging integral sequence, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M230">View MathML</a>

(2.17)

Thus, by (H1) and (2.17), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M231">View MathML</a>. Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M232">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M47">View MathML</a>, from (2.16), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M234">View MathML</a>

(2.18)

From (2.17), (2.18), and the absolutely continuity of integral function, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M235">View MathML</a> is equicontinuous.

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M238">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M235">View MathML</a> is uniformly bounded. By virtue of Lemma 2.3 and (H3), we know that

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M241">View MathML</a>. Therefore, A is compact. To sum up, the conclusion of Lemma 2.7 follows. □

The main tools of the paper are the following well-known fixed-point index theorems (see [2-4]).

Lemma 2.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M242">View MathML</a>be a completely continuous mapping and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M243">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244">View MathML</a>. Thus, we have the following conclusions:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M245">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M247">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M248">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M244">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M250">View MathML</a>.

3 Main results

In this section, we establish the existence of positive solutions for problem (1.1) by making use of Lemma 2.8.

Theorem 3.1Suppose that (H1)-(H4) hold. Then problem (1.1) has at least one positive solution.

Proof From (H4), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M251">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M252">View MathML</a> and also there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M253">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M255">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M256">View MathML</a>

(3.1)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M257">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M258">View MathML</a>, by virtue of (3.1), we know that

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M260">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M261">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M262">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M263">View MathML</a> is a bounded open subsets in E, and so for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M264">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>, we obtain

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M267">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M268">View MathML</a>

(3.3)

From (3.2) and (3.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M269">View MathML</a>

Therefore, A has at least one fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M270">View MathML</a>. Consequently, problem (1.1) has at least one positive solution. □

Theorem 3.2Suppose that (H1)∼(H3) and (H5) are satisfied. Then problem (1.1) has at least one positive solution.

Proof From (H5), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M271">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M272">View MathML</a> and also there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M273">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M255">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M276">View MathML</a>

(3.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M277">View MathML</a>. By virtue of (3.4), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M278">View MathML</a>

(3.5)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M279">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M280">View MathML</a>, by virtue of (3.5), we know that

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M260">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M283">View MathML</a>

(3.6)

By the same method as the selection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M284">View MathML</a> in Theorem 3.1, we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M285">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M286">View MathML</a>

(3.7)

According to (3.6) and (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M287">View MathML</a>

Therefore, A has at least one fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M288">View MathML</a>. Consequently, problem (1.1) has at least one positive solution. The proof is complete. □

4 Concerned results and applications

In this section, we deal with a special case of the problem (1.1). The method is just similar to what we have done in Section 3, so we omit the proof of some main results of the section. Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M289">View MathML</a> is treated in the following theorem. Under the case, the problem (1.1) reduces to the following boundary value problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M290">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M292">View MathML</a>.

Theorem 4.1Assume that (H2) holds, and the following conditions are satisfied:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M121">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M295">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M124">View MathML</a>are Lebesgue integrable functionals onJ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M298">View MathML</a>) and satisfying

(C2) for any bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M135">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M298">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M302">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M138">View MathML</a>together with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M304">View MathML</a>are relatively compact sets.

(C3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M305">View MathML</a>, wheremis defined by (2.4). Then the problem (4.1) has at least one positive solution.

Theorem 4.2Assume that (H2) and (C1)∼(C2) hold, and the following condition is satisfied:

(C4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M306">View MathML</a>, wheremis defined by (2.4). Then the problem (4.1) has at least one positive solution.

To illustrate how our main results can be used in practice, we present an example.

Example 4.1 Consider the following boundary value problem for scalar second-order impulsive integro-differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M307">View MathML</a>

(4.2)

5 Conclusion

The problem (4.2) has at least one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M308">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M310">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M311">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M312">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M313">View MathML</a>. By simple computation, we know that

Then conditions (H1)∼(H4) are satisfied. Therefore, by Theorem 3.1, the problem (4.2) has at least one positive solution.

Remark 5.1 In [12], by requiring that f satisfies some noncompact measure conditions and P is a normal cone, Guo established the existence of positive solutions for initial value problem. In the paper, we impose some weaker condition on f, we obtain the positive solution of the problem (1.1).

Remark 5.2 For the special case when the problem (1.1) has no singularities and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/86/mathml/M315">View MathML</a>, our results still hold. Obviously, our theorems generalize and improve the results in [9-12].

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author is very grateful to Professor Lishan Liu and Professor R. P. Agarwal for their making many valuable comments. The author would like to express her thanks to the editor of the journal and the anonymous referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. The author was supported financially by the Foundation of Shanghai Municipal Education Commission (Grant Nos. DYL201105).

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