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Existence of solutions and nonnegative solutions for a class of p ( t ) -Laplacian differential systems with multipoint and integral boundary value conditions

Guizhen Zhi1, Yunrui Guo2, Yan Wang1 and Qihu Zhang1*

Author Affiliations

1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China

2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan, 453003, China

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Boundary Value Problems 2013, 2013:106  doi:10.1186/1687-2770-2013-106


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/106


Received:12 November 2011
Accepted:12 April 2013
Published:26 April 2013

© 2013 Zhi et al.; licensee Springer

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

Abstract

This paper explores the existence of solutions for a class of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian differential systems with multipoint and integral boundary value conditions via Leray-Schauder’s degree. Moreover, the existence of nonnegative solutions is discussed.

MSC: 34B10.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian; Leray-Schauder degree; fixed point

1 Introduction

In this paper, we consider the existence of solutions for the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>); <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M8">View MathML</a> is called <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M11">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M14">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M16">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M17">View MathML</a>, they are both nonnegative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M19">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M20">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M22">View MathML</a> are nonnegative constants; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24">View MathML</a> are positive parameters.

The study of differential equations and variational problems with variable exponent growth conditions has attracted more and more attention in recent years. Many results have been obtained on these problems, for example, [1-16]. We refer to [3,12,16] for the applied background of these problems. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M25">View MathML</a> (a constant), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26">View MathML</a> becomes the well-known p-Laplacian. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a> is a general function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26">View MathML</a> represents a non-homogeneity and possesses more nonlinearity, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M26">View MathML</a> is more complicated than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M30">View MathML</a> (see [7]).

In recent years, because of the wide mathematical and physical background (see [17-19]), the existence of positive solutions for the p-Laplacian equation group has received extensive attention. Especially, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M31">View MathML</a>, the existence of positive solutions for the equation group boundary value problems has been obtained (see [20-25]). On the integral boundary value problems, we refer to [26-30]. But as for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian equation group, there are few papers dealing with the existence of solutions, especially the existence of solutions for the systems with multipoint and integral boundary value problems. Therefore, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a> is a general function, this paper mainly investigates the existence of solutions for a class of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian differential systems with multipoint and integral boundary value conditions. Moreover, we discuss the existence of nonnegative solutions.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M36">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M37">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>) is assumed to be Carathéodory, by which we mean:

(i) For almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M40">View MathML</a> is continuous;

(ii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M41">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M42">View MathML</a> is measurable on J;

(iii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M43">View MathML</a>, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M44">View MathML</a> such that, for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M46">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M50">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M51">View MathML</a>

Throughout the paper, we denote

The inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M55">View MathML</a> will denote the absolute value and the Euclidean norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M56">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M35">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M59">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M60">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M61">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M64">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M65">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M66">View MathML</a>. Spaces C, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67">View MathML</a> and W will be equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M70">View MathML</a>, respectively. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M73">View MathML</a> are Banach spaces. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M74">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M75">View MathML</a>.

We say a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M76">View MathML</a> is a solution of (P) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M65">View MathML</a> satisfies the differential equation in (P) a.e. on J and the boundary value conditions.

In this paper, we always use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M78">View MathML</a> to denote positive constants if this does not lead to confusion. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M79">View MathML</a>

We say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>) satisfies a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a> growth condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M84">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M85">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M86">View MathML</a>. We say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies a general growth condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> does not satisfy a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a> growth condition.

We will discuss the existence of solutions for (P) in the following two cases:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a> growth condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies a general growth condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>.

This paper is organized as follows. In Section 2, we do some preparation. In Section 3, we discuss the existence of solutions of (P). Finally, in Section 4, we discuss the existence of nonnegative solutions for (P).

2 Preliminary

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M95">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M98">View MathML</a> has the following properties.

Lemma 2.1 (see [5])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M98">View MathML</a>is a continuous function and satisfies the following:

(i) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101">View MathML</a>is strictly monotone, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M102">View MathML</a>

(ii) There exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M104">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M105">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M106">View MathML</a>

It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101">View MathML</a> is a homeomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M56">View MathML</a> for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M100">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M112">View MathML</a> the inverse operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M101">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M114">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M112">View MathML</a> is continuous and sends bounded sets into bounded sets.

Let us now consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M116">View MathML</a>

(1)

with the boundary value condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M117">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M118">View MathML</a>. If u is a solution of (1) with (2), by integrating (1) from 0 to t, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M119">View MathML</a>

(3)

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M120">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121">View MathML</a> is dependent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M122">View MathML</a>. Define operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M123">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M124">View MathML</a>

From (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M125">View MathML</a>

(4)

By integrating (4) from 0 to t, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M126">View MathML</a>

From (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M128">View MathML</a>

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M129">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M130">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M131">View MathML</a>

(5)

It is easy to obtain the following lemma.

Lemma 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M132">View MathML</a>is continuous and sends bounded sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a>to bounded sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M135">View MathML</a>

(6)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136">View MathML</a> is a compact continuous mapping.

Let us now consider another problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M137">View MathML</a>

(7)

with the boundary value condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M138">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M139">View MathML</a>. Similar to the discussion of the solutions of (1) with (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M140">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M141">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M143">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M146">View MathML</a>

(9)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M148">View MathML</a>

(10)

From (9) and (10), we have

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M151">View MathML</a>

Lemma 2.3The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M152">View MathML</a>has the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150">View MathML</a>, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M154">View MathML</a>

(11)

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M155">View MathML</a>.

(ii) The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M156">View MathML</a>, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M157">View MathML</a>

Proof (i) It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M158">View MathML</a>

From Lemma 2.1, it is immediate that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M159">View MathML</a>

and hence, if (11) has a solution, then it is unique.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M160">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M161">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M150">View MathML</a>, it is easy to see that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M163">View MathML</a> such that the ith component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M164">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M165">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M166">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M167">View MathML</a> keeps sign on J and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M168">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M169">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M170">View MathML</a>

Thus the ith component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M171">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M172">View MathML</a> is nonzero and keeps sign, and then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M173">View MathML</a>

Let us consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M174">View MathML</a>

(12)

It is easy to see that all the solutions of (12) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M175">View MathML</a>. So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M176">View MathML</a>

which implies the existence of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M177">View MathML</a>.

In this way, we define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M178">View MathML</a>, which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M179">View MathML</a>

(ii) By the proof of (i), we also obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180">View MathML</a> sends bounded sets to bounded sets, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M181">View MathML</a>

It only remains to prove the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M183">View MathML</a> be a convergent sequence in C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M184">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M185">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M186">View MathML</a> is a bounded sequence, then it contains a convergent subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M187">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M188">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M189">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M190">View MathML</a>, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M189">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M192">View MathML</a>. From (i), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M193">View MathML</a>, it means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M180">View MathML</a> is continuous. The proof is completed. □

Now, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M195">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M196">View MathML</a>

(13)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197">View MathML</a> is continuous and sends bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a> into bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M53">View MathML</a>, and hence it is a compact continuous mapping.

If u is a solution of (1) with (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M200">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M201">View MathML</a>

If u is a solution of (7) with (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M202">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M203">View MathML</a>

We denote

Lemma 2.4The operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M205">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>) are continuous and send equi-integrable sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M208">View MathML</a>.

Proof We only prove that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M209">View MathML</a> is continuous and sends equi-integrable sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a> to relatively compact sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67">View MathML</a>, the rest is similar.

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M212">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M129">View MathML</a>. Since

it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M209">View MathML</a> is a continuous operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67">View MathML</a>.

Let now U be an equi-integrable set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M133">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M219">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M220">View MathML</a>

We want to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M221">View MathML</a> is a compact set.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M222">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M223">View MathML</a>, then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M224">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M225">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M226">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M227">View MathML</a>

Hence the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228">View MathML</a> is uniformly bounded and equicontinuous. By the Ascoli-Arzela theorem, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228">View MathML</a> (which we still denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M228">View MathML</a>) convergent in C. According to the bounded continuous of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121">View MathML</a>, we can choose a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M232">View MathML</a> (which we still denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M232">View MathML</a>) which is convergent in C, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M234">View MathML</a> is convergent in C.

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M235">View MathML</a> and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M236">View MathML</a>, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M237">View MathML</a> is convergent in C. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M222">View MathML</a> is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67">View MathML</a>. This completes the proof. □

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M240">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M241">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M243">View MathML</a> are both compact continuous.

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M244">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>) the Nemytskii operator associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M247">View MathML</a>

Lemma 2.5<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a>is a solution of (P) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a>is a solution of the following abstract equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M250">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution to (P), according to the proof before Lemma 2.5, it is easy to obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution to (S).

Conversely, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution to (S), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M254">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M255">View MathML</a>

It follows from (S) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M256">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M257">View MathML</a>

By the condition of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M259">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M260">View MathML</a>

From (S), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M261">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M262">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M263">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M264">View MathML</a>

From (S), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M265">View MathML</a>

By the condition of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M267">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M268">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M269">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M270">View MathML</a>

Moreover, from (S), it is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M271">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M272">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution of (P).

This completes the proof. □

3 Existence of solutions

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for (P), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a> growth condition or a general growth condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>).

We denote (S) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M277">View MathML</a>

where

Theorem 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a>satisfies a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a>growth condition, then the problem (P) has at least one solution for any fixed parameter<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M281">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M38">View MathML</a>).

Proof Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M283">View MathML</a>

where

According to Lemma 2.5, we know that (P) has the same solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M285">View MathML</a>

(14)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M286">View MathML</a>.

It is easy to see that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M243">View MathML</a> are compact continuous. According to Lemma 2.2, Lemma 2.3 and Lemma 2.4, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M290">View MathML</a> are compact continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M291">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M67">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M293">View MathML</a> is compact continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M294">View MathML</a> to W.

We claim that all the solutions of (14) are uniformly bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295">View MathML</a>. In fact, if it is false, we can find a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M296">View MathML</a> for (14) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M297">View MathML</a> as n<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M298">View MathML</a>.

From Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M299">View MathML</a>

which together with the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M300">View MathML</a> growth condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301">View MathML</a> implies that

(15)

From (14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M303">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M304">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M305">View MathML</a>. From the above inequality we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M306">View MathML</a>

(16)

It follows from (14) and (15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M307">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M309">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M310">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M311">View MathML</a>

(17)

It follows from (16) and (17) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M312">View MathML</a>.

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M313">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M314">View MathML</a>.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M315">View MathML</a> is bounded.

Thus, we can choose a large enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M316">View MathML</a> such that all the solutions of (14) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M317">View MathML</a>. Therefore, the Leray-Schauder degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M318">View MathML</a> is well defined for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M320">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M321">View MathML</a>

(18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M322">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M323">View MathML</a> are defined in (5) and (13), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a> is the unique solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325">View MathML</a>.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution of the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M329">View MathML</a>

(19)

Obviously, (19) possesses a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M331">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M332">View MathML</a>

Therefore (P) has at least one solution. This completes the proof. □

In the following, we investigate the existence of solutions for (P) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M80">View MathML</a> satisfies a general growth condition.

Denote

Assume the following.

(A1) Let a positive constant ε be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a> is defined in (18), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197">View MathML</a> are defined in (5) and (13), respectively.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M343">View MathML</a> is an open bounded domain in W. We have the following theorem.

Theorem 3.2Assume that (A1) is satisfied. If positive parameters<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24">View MathML</a>are small enough, then the problem (P) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M346">View MathML</a>.

Proof Similarly, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M347">View MathML</a>. By Lemma 2.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M349">View MathML</a>

with (2) and (8) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a solution of the following abstract equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M351">View MathML</a>

(20)

From the proof of Theorem 3.1, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M293">View MathML</a> is compact continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M294">View MathML</a> to W. According to Leray-Schauder’s degree theory, we only need to prove that

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M354">View MathML</a> has no solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M355">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M357">View MathML</a>,

then we can conclude that the system (P) has a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M358">View MathML</a>.

(1) If there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M295">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M360">View MathML</a> is a solution of (20), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> and λ satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M362">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M363">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M360">View MathML</a>, there exists an i such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M365">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M366">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M365">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M368">View MathML</a>) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M369">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M370">View MathML</a>. On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M371">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M372">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M373">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M375">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M376">View MathML</a>, holding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M377">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M378">View MathML</a> is continuous, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M379">View MathML</a> is small enough, from (A1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M380">View MathML</a>

It is a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M381">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M383">View MathML</a>) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M384">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M385">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M386">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M387">View MathML</a>, and we can find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M388">View MathML</a>

(21)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M375">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301">View MathML</a> is Carathéodory, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M391">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M392">View MathML</a>

From Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136">View MathML</a> is continuous, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M394">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M379">View MathML</a> is small enough, from (A1) and (21), we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M396">View MathML</a>

It is a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M397">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M398">View MathML</a>. Similar to the proof of (i), we get a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M399">View MathML</a>.

Summarizing this argument, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M354">View MathML</a> has no solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M355">View MathML</a> when positive parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M24">View MathML</a> are small enough.

(2) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a> is defined in (18)) is the unique solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M325">View MathML</a>, and (A1) holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335">View MathML</a>, we can see that the Leray-Schauder degree

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M409">View MathML</a>

This completes the proof. □

As applications of Theorem 3.2, we have the following.

Corollary 3.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M410">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M412">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M413">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>are small enough, then the problem (P) possesses at least one solution.

Proof It is easy to have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M417">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M418">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M343">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M420">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M413">View MathML</a>, then there exists a small enough ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M422">View MathML</a>

From Lemma 2.2 and the small enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M424">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M425">View MathML</a> is valid.

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339">View MathML</a>.

Obviously, it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M427">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M428">View MathML</a> and the small enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337">View MathML</a>.

Thus, the conditions of (A1) are satisfied, then the problem (P) possesses at least one solution. □

Corollary 3.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M410">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M435">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M436">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M281">View MathML</a>are small enough, then the problem (P) possesses at least one solution.

Proof From Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M440">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M441">View MathML</a> is dependent on the small enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M442">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>, then it follows from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M121">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M445">View MathML</a> is small enough, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M446">View MathML</a>

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M339">View MathML</a>.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M425">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M449">View MathML</a> and the small enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>, it is easy to have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M336">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M337">View MathML</a>.

Thus, the conditions of (A1) are satisfied, then the problem (P) possesses at least one solution. □

We denote

Assume the following.

(A2) Let positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M457">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M458">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M459">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M460">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M462">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M324">View MathML</a> is defined in (18), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M197">View MathML</a> are defined in (5) and (13), respectively.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466">View MathML</a> is an open bounded domain in W. We have the following.

Corollary 3.5Assume that

whereϵ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M468">View MathML</a>are positive constants; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M469">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M470">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414">View MathML</a>. Then the problem (P) possesses at least one solution.

Proof Similar to the proof of Theorem 3.2, we only need to prove that (A2) is satisfied, then we can conclude that the problem (P) possesses at least one solution.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M473">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466">View MathML</a>, it is easy to have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M475">View MathML</a>

where we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M476">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M470">View MathML</a>, then there exists a big enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M456">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M479">View MathML</a>

From Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M480">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461">View MathML</a> is valid.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M482">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M466">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M484">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M485">View MathML</a>, then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M457">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M487">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M488">View MathML</a>), which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M489">View MathML</a>

From Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M490">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M462">View MathML</a> is valid.

Obviously, it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M493">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M459">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M460">View MathML</a>.

Thus, the conditions of (A2) are satisfied, then the problem (P) possesses at least one solution. □

Corollary 3.6Assume that

whereϵ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M468">View MathML</a>are positive constants; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M499">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M500">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M501">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M502">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M414">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M415">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M416">View MathML</a>are small enough, then the problem (P) possesses at least one solution.

Proof Similar to the proof of Corollary 3.5, we conclude that (A2) is satisfied. Then the problem (P) possesses at least one solution. □

4 Existence of nonnegative solutions

In the following, we deal with the existence of nonnegative solutions of (P). For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M506">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M507">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M508">View MathML</a>) means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M509">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M510">View MathML</a>) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M512">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M513">View MathML</a> means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M514">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M515">View MathML</a> means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M516">View MathML</a>.

Theorem 4.1We assume that

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M517">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M519">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518">View MathML</a>;

(30) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M521">View MathML</a>;

(40) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M522">View MathML</a>.

Then every solution of (P) is nonnegative.

Proof (i) We shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M523">View MathML</a> is nonnegative.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M524">View MathML</a> is a solution of (P), from Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M525">View MathML</a>

which together with (5), (10) and (40) implies that

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M527">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a>. Holding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M529">View MathML</a> is decreasing, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M530">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M226">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M532">View MathML</a>.

According to the boundary value condition (2) and condition (30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M533">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M534">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M523">View MathML</a> is nonnegative.

(ii) We shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M536">View MathML</a> is nonnegative.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M524">View MathML</a> is a solution of (P), From Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M538">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539">View MathML</a>. If it is false, then there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M541">View MathML</a>, which together with condition (20) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M542">View MathML</a>

(22)

Similar to the proof of Lemma 2.3, the boundary value condition (8) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M543">View MathML</a>

(23)

From (22) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M541">View MathML</a>, we get a contradiction to (23).

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539">View MathML</a>.

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M546">View MathML</a>

(24)

If it is false, then there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M548">View MathML</a>

which together with condition (20) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M549">View MathML</a>

(25)

From (25) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539">View MathML</a>, we get a contradiction to (23). Thus (24) is valid.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M551">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M552">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M553">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M554">View MathML</a> is decreasing, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M555">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M556">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M557">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M308">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M559">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M560">View MathML</a>

We can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M561">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M562">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M561">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M564">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M565">View MathML</a>

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M540">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M567">View MathML</a>

which together with (8) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M568">View MathML</a>

(26)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M539">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M570">View MathML</a>

(27)

It follows from (26) and (27) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M571">View MathML</a>

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M572">View MathML</a>

(28)

from (8) and (28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M573">View MathML</a>

(29)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M574">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M575">View MathML</a>

(30)

Combining (29) and (30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M576">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M536">View MathML</a> is nonnegative.

Combining (i) and (ii), we find that every solution of (P) is nonnegative. □

Corollary 4.2We assume that

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M517">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M580">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M519">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M518">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M580">View MathML</a>;

(30) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M521">View MathML</a>;

(40) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M522">View MathML</a>.

Then we have

(a) Under the conditions of Theorem 3.1, (P) has at least one nonnegative solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a>;

(b) Under the conditions of Theorem 3.2, (P) has at least one nonnegative solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a>.

Proof (a) Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M588">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M589">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M590">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M7">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M592">View MathML</a> satisfies the Carathéodory condition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M593">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M594">View MathML</a>.

We assume the following.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M595">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M39">View MathML</a> uniformly, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M597">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M86">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M599">View MathML</a> satisfies a sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M82">View MathML</a> growth condition.

Let us consider the existence of solutions of the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M601">View MathML</a>

(31)

According to Theorem 3.1, (31) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a>. From Theorem 4.1, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is nonnegative. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M248">View MathML</a> is a nonnegative solution of (P).

(b) It is similar to the proof of (a).

This completes the proof. □

5 Examples

Example 5.1 Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M605">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M606">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M607">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M608">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M610">View MathML</a> are Caratheodory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M611">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M612">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M613">View MathML</a>, then the conditions of Theorem 3.1 are satisfied, then (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M614">View MathML</a>) has a solution.

Example 5.2 Consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M615">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M616">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M606">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M607">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M619">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M301">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M610">View MathML</a> are Caratheodory, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M612">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M613">View MathML</a>, the conditions of Corollary 4.2 are satisfied, then (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M625">View MathML</a>) has a nonnegative solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

Partly supported by the National Science Foundation of China (10701066 & 10971087).

References

  1. Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal.. 156, 121–140 (2001). Publisher Full Text OpenURL

  2. Antontsev, S, Shmarev, S: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat.. 53, 355–399 (2009)

  3. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math.. 66(4), 1383–1406 (2006). Publisher Full Text OpenURL

  4. Diening, L: Maximal function on generalized Lebesgue spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M626">View MathML</a>. Math. Inequal. Appl.. 7(2), 245–253 (2004)

  5. Fan, XL, Zhang, QH: Existence of solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M628">View MathML</a>-Laplacian Dirichlet problem. Nonlinear Anal.. 52, 1843–1852 (2003). Publisher Full Text OpenURL

  6. Fan, XL, Wu, HQ, Wang, FZ: Hartman-type results for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian systems. Nonlinear Anal.. 52, 585–594 (2003). Publisher Full Text OpenURL

  7. Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M632">View MathML</a>-Laplacian Dirichlet problem. J. Math. Anal. Appl.. 302, 306–317 (2005). Publisher Full Text OpenURL

  8. Fu, YQ: The principle of concentration compactness in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M634">View MathML</a> spaces and its application. Nonlinear Anal.. 71(5-6), 1876–1892 (2009). Publisher Full Text OpenURL

  9. El Hamidi, A: Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. Appl.. 300, 30–42 (2004). Publisher Full Text OpenURL

  10. Hudzik, H: On generalized Orlicz-Sobolev space. Funct. Approx. Comment. Math.. 4, 37–51 (1976)

  11. Kováčik, O, Rákosník, J: On spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M636">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M637">View MathML</a>. Czechoslov. Math. J.. 41, 592–618 (1991)

  12. Růžička, M: Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000)

  13. Zhang, QH: Existence of positive solutions for elliptic systems with nonstandard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M640">View MathML</a>-growth conditions via sub-supersolution method. Nonlinear Anal.. 67(4), 1055–1067 (2007). Publisher Full Text OpenURL

  14. Zhang, QH, Qiu, ZM, Liu, XP: Existence of solutions for a class of weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M1">View MathML</a>-Laplacian system multi-point boundary value problems. J. Inequal. Appl.. 2008, Article ID 791762 (2008)

  15. Zhang, QH: Boundary blow-up solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/106/mathml/M644">View MathML</a>-Laplacian equations with exponential nonlinearities. J. Inequal. Appl.. 2008, Article ID 279306 (2008)

  16. Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv.. 29, 33–36 (1987). Publisher Full Text OpenURL

  17. Bandle, CV, Kwong, MK: Semilinear elliptic problems in annular domains. Z. Angew. Math. Phys.. 40, 245–257 (1989). Publisher Full Text OpenURL

  18. Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1980)

  19. Wang, HY: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ.. 109, 1–7 (1994). Publisher Full Text OpenURL

  20. Hu, L, Wang, LL: Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations. J. Math. Anal. Appl.. 335, 1052–1060 (2007). Publisher Full Text OpenURL

  21. Liu, LS, Liu, BM, Wu, YH: Positive solutions of singular boundary value problems for nonlinear differential systems. Appl. Math. Comput.. 186, 1163–1172 (2007). Publisher Full Text OpenURL

  22. Liu, WW, Liu, LS, Wu, YH: Positive solutions of a singular boundary value problem for systems of second-order differential equations. Appl. Math. Comput.. 208, 511–519 (2009). Publisher Full Text OpenURL

  23. Wei, ZL, Zhang, MC: Positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems. Appl. Math. Comput.. 197, 135–148 (2008). Publisher Full Text OpenURL

  24. Yang, ZL: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal.. 62, 1251–1265 (2005). Publisher Full Text OpenURL

  25. Zhou, YM, Xu, Y: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations. J. Math. Anal. Appl.. 320, 578–590 (2006). Publisher Full Text OpenURL

  26. Feng, MQ, Ji, DH, Ge, WG: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math.. 222, 351–363 (2008). Publisher Full Text OpenURL

  27. Feng, MQ: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett.. 24, 1419–1427 (2011). Publisher Full Text OpenURL

  28. Feng, MQ, Zhang, XM, Ge, WG: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl.. 2011, Article ID 720702 (2011)

  29. Ma, HL: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal.. 68, 645–651 (2008). Publisher Full Text OpenURL

  30. Zhang, XM, Ge, WG: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput.. 219, 3553–3564 (2012). Publisher Full Text OpenURL