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Solutions and nonnegative solutions for a weighted variable exponent impulsive integro-differential system with multi-point and integral mixed boundary value problems

Rong Dong and Qihu Zhang*

Author Affiliations

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

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Boundary Value Problems 2013, 2013:161  doi:10.1186/1687-2770-2013-161


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/161


Received:19 March 2013
Accepted:18 June 2013
Published:5 July 2013

© 2013 Dong and Zhang; 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 investigates the existence of solutions for a weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian impulsive integro-differential system with multi-point and integral mixed boundary value problems via Leray-Schauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.

MSC: 34B37.

Keywords:
weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian; impulsive integro-differential system; Leray-Schauder’s degree

1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian integro-differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M4">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, with the following impulsive boundary value conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M9">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M10">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M11">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M14">View MathML</a> is called the weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M17">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>); <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M20">View MathML</a> is nonnegative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M21">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M23">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M24">View MathML</a>; T and S are linear operators defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M28">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M30">View MathML</a>, we say the problem is nonresonant, but if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>, we say the problem is resonant.

Throughout the paper, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M33">View MathML</a> means functions which are uniformly convergent to 0 (as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M34">View MathML</a>); for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M36">View MathML</a> will denote the jth component of v; the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M39">View MathML</a> will denote the absolute value and the Euclidean norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M47">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M48">View MathML</a> the interior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M50">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M51">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M52">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M55">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M56">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M57">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M58">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M59">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M60">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M61">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M62">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M63">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M64">View MathML</a>

In the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M66">View MathML</a> will be simply denoted by PC and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>, respectively. We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M68">View MathML</a>

The study of differential equations and variational problems with nonstandard <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-growth conditions has attracted more and more interest in recent years (see [1-4]). The applied background of these kinds of problems includes nonlinear elasticity theory [4], electro-rheological fluids [1,3], and image processing [2]. Many results have been obtained on these kinds of problems; see, for example, [5-15]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [16-19]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M70">View MathML</a> (a constant), (1)-(4) becomes the well-known p-Laplacian problem. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a> is a general function, one can see easily <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M72">View MathML</a> in general, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M73">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74">View MathML</a> represents a non-homogeneity and possesses more nonlinearity, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74">View MathML</a> is more complicated than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76">View MathML</a>. For example:

(a) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M77">View MathML</a> is a bounded domain, the Rayleigh quotient

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M78">View MathML</a>

is zero in general, and only under some special conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M79">View MathML</a> (see [9]), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M80">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M81">View MathML</a>) is an interval, the results show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M82">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M83">View MathML</a> is monotone. But the property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M84">View MathML</a> is very important in the study of p-Laplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.

(b) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M70">View MathML</a> (a constant) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M87">View MathML</a>, then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems (see [21]), but it is invalid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74">View MathML</a>. It is another difference between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M74">View MathML</a>.

In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [22-29]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M76">View MathML</a>, results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [30-33]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M92">View MathML</a>-Laplacian impulsive differential equation with multi-point boundary value conditions, when the problem is nonresonant. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [35-38].

In this paper, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a> is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [34] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M95">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M96">View MathML</a>, the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, p-Laplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between p-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M97">View MathML</a> in general and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M2">View MathML</a>-Laplacian is non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)-(4), we usually need the nonlinear term that satisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99">View MathML</a> growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M100">View MathML</a> growth condition.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M101">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M102">View MathML</a> is assumed to be Caratheodory, by which we mean:

(i) For almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M104">View MathML</a> is continuous;

(ii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M105">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M106">View MathML</a> is measurable on J;

(iii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M107">View MathML</a>, there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M108">View MathML</a> such that, for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M105">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M114">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M115">View MathML</a>

We say a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M116">View MathML</a> is a solution of (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M118">View MathML</a> absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M50">View MathML</a>, which satisfies (1) a.e. on J.

In this paper, we always use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M121">View MathML</a> to denote positive constants, if it cannot lead to confusion. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M122">View MathML</a>

We say f satisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99">View MathML</a> growth condition if f satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M124">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M126">View MathML</a>.

We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:

Case (i): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M128">View MathML</a>;

Case (ii): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>;

Case (iii): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M132">View MathML</a>.

This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)-(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>. In Section 4, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>. Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M138">View MathML</a>.

2 Preliminary

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M139">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M140">View MathML</a>. Obviously, φ has the following properties.

Lemma 2.1 (see [34])

φis a continuous function and satisfies:

(i) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M142">View MathML</a>is strictly monotone, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M143">View MathML</a>

(ii) There exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M145">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M146">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M147">View MathML</a>

It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M142">View MathML</a> is a homeomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a> for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M152">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M153">View MathML</a> is continuous and sends bounded sets to bounded sets.

In this section, we will do some preparation and give the operator equation which has the same solutions of (1)-(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M154">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M155">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M156">View MathML</a>.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M158">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M159">View MathML</a>.

We will discuss it in three cases, respectively.

2.1 Case (i)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M128">View MathML</a>. If u is a solution of (6) with (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M162">View MathML</a>

(7)

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M163">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164">View MathML</a> is dependent on a, b and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M166">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M167">View MathML</a>

By solving for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M168">View MathML</a> in (7) and integrating, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M169">View MathML</a>

which together with boundary value condition (4) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M171">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M172">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M173">View MathML</a>

then W is a Banach space.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M175">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M176">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M177">View MathML</a>

Throughout the paper, we denote

Lemma 2.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M181">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>. Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M185">View MathML</a>has the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M187">View MathML</a>

(8)

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M188">View MathML</a>.

(ii) The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M189">View MathML</a>, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M191">View MathML</a>

where the notation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192">View MathML</a>means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M193">View MathML</a>

Proof (i) From Lemma 2.1, it is immediate that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M194">View MathML</a>

and hence, if (8) has a solution, then it is unique.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M195">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M196">View MathML</a>, it is easy to see that there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M197">View MathML</a> such that the absolute value of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198">View MathML</a>th component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164">View MathML</a> satisfies

Thus the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198">View MathML</a>th component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M203">View MathML</a> keeps sign on J, namely, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103">View MathML</a>, we have

Obviously, we have

then it is easy to see that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M198">View MathML</a>th component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M208">View MathML</a> keeps the same sign of . Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M210">View MathML</a>

Let us consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M211">View MathML</a>

(9)

According to the preceding discussion, all the solutions of (9) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M212">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M213">View MathML</a>

it means the existence of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M214">View MathML</a>.

In this way, we define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M215">View MathML</a>, which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M216">View MathML</a>.

(ii) By the proof of (i), we also obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217">View MathML</a> sends bounded sets to bounded sets, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M218">View MathML</a>

It only remains to prove the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M220">View MathML</a> be a convergent sequence in W and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M221">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M34">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M223">View MathML</a> is a bounded sequence, it contains a convergent subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M224">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M225">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M226">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M227">View MathML</a>, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M226">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M229">View MathML</a>, which together with (i) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M230">View MathML</a>, it means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M217">View MathML</a> is continuous. This completes the proof. □

Now we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M232">View MathML</a> the Nemytskii operator associated to f defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M233">View MathML</a>

(10)

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M234">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M235">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237">View MathML</a>.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M238">View MathML</a> is continuous and sends bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> to bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a>, and hence it is compact continuous.

If u is a solution of (6) with (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M241">View MathML</a>

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M243">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M244">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M245">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M246">View MathML</a>

Lemma 2.3 (i) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M247">View MathML</a>is continuous and sends equi-integrable sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

(ii) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M250">View MathML</a>is continuous and sends bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

Proof (i) It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M255">View MathML</a>. Since and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M257">View MathML</a>

it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M258">View MathML</a> is a continuous operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

Let now U be an equi-integrable set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M262">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M263">View MathML</a>

We want to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M264">View MathML</a> is a compact set.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M265">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M266">View MathML</a>, then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M267">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M268">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M269">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M270">View MathML</a>

Hence the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M271">View MathML</a> is uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M271">View MathML</a> (which we rename the same) which is convergent in PC. According to the bounded continuity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M273">View MathML</a>, we can choose a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M274">View MathML</a> (which we still denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M275">View MathML</a>) which is convergent in PC, then is convergent in PC.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M277">View MathML</a>

it follows from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M278">View MathML</a> and the integrability of in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M281">View MathML</a> is convergent in PC. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M265">View MathML</a> is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

(ii) It is easy to see from (i) and Lemma 2.2.

This completes the proof. □

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M284">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M285">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M286">View MathML</a> is compact continuous.

Lemma 2.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>. Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M295">View MathML</a>

(12)

Proof Suppose that u is a solution of (1)-(4). By integrating (1) from 0 to t, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M296">View MathML</a>

(13)

It follows from (13) and (4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M297">View MathML</a>

(14)

Combining the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164">View MathML</a>, we can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M299">View MathML</a>

Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M300">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M301">View MathML</a>

By the condition of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M164">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M303">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M304">View MathML</a>

(16)

It follows from (15) and (16) that (4) is satisfied.

From (12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M305">View MathML</a>

(17)

It follows from (17) that (3) is satisfied.

Hence u is a solution of (1)-(4). This completes the proof. □

2.2 Case (ii)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M307">View MathML</a>. If u is a solution of (6) with (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M308">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M309">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M310">View MathML</a> is dependent on a, b and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165">View MathML</a>. Boundary value condition (4) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M312">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M314">View MathML</a>

Throughout the paper, we denote .

Lemma 2.5The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M316">View MathML</a>has the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M318">View MathML</a>has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M319">View MathML</a>.

(ii) The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M320">View MathML</a>, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M322">View MathML</a>

where the notation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192">View MathML</a>means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M324">View MathML</a>

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M325">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M326">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237">View MathML</a>.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M329">View MathML</a> is continuous and sends bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> to bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a>, and hence it is compact continuous.

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M333">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M334">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M335">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M336">View MathML</a>

Similar to the proof of Lemma 2.3, we have the following.

Lemma 2.6 (i) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M337">View MathML</a>is continuous and sends equi-integrable sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

(ii) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M340">View MathML</a>is continuous and sends bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

Let us define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M343">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M344">View MathML</a>

It is easy to see that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M345">View MathML</a>is compact continuous.

Lemma 2.7Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>, thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M348">View MathML</a>

Proof Similar to the proof of Lemma 2.4, we omit it here. □

2.3 Case (iii)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M132">View MathML</a>. If u is a solution of (6) with (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M351">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M352">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M353">View MathML</a> is dependent on a, b and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M165">View MathML</a>.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M355">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M356">View MathML</a>

(18)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M357">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M358">View MathML</a>

(19)

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M360">View MathML</a>

From (18) and (19), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M361">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M362">View MathML</a> can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M363">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M176">View MathML</a>. Moreover, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M365">View MathML</a>

Lemma 2.8Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M367">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>.

Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M375">View MathML</a>has the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M174">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M377">View MathML</a>has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M378">View MathML</a>.

(ii) The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M379">View MathML</a>, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M190">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M381">View MathML</a>

where the notation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M192">View MathML</a>means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M383">View MathML</a>

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M384">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M385">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M237">View MathML</a>.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M388">View MathML</a> is continuous and sends bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> to bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M37">View MathML</a>, and hence it is compact continuous.

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M242">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M392">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M393">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M394">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M395">View MathML</a>

Similar to the proof of Lemma 2.3, we have

Lemma 2.9 (i) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M396">View MathML</a>is continuous and sends equi-integrable sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M248">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

(ii) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M399">View MathML</a>is continuous and sends bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>to relatively compact sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>.

Let us define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M402">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M403">View MathML</a>

It is easy to see that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M404">View MathML</a>is compact continuous.

Lemma 2.10Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>.

Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M416">View MathML</a>

Proof Similar to the proof of Lemma 2.4, we omit it here. □

3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>.

When f satisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a> growth condition, we have the following theorem.

Theorem 3.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99">View MathML</a>growth condition; and operatorsAandBsatisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M429">View MathML</a>

(20)

then problem (1)-(4) has at least a solution.

Proof First we consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M430">View MathML</a>

Denote

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M432">View MathML</a> is defined in (10).

Obviously, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M433">View MathML</a>) has the same solution as the following operator equation when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M434">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M435">View MathML</a>

(21)

It is easy to see that the operator is compact continuous for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>. It follows from Lemma 2.2 and Lemma 2.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M438">View MathML</a> is compact continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>.

We claim that all the solutions of (21) are uniformly bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>. In fact, if it is false, we can find a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M443">View MathML</a> for (21) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M444">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M445">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M446">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M447">View MathML</a> .

From Lemma 2.2, we have

Thus

(22)

From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M433">View MathML</a>), we have

It follows from (11) and Lemma 2.2 that

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M453">View MathML</a>. If the above inequality holds then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M454">View MathML</a>

(23)

It follows from (14), (20) and (22) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M455">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M457">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M458">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M459">View MathML</a>

(24)

It follows from (23) and (24) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M460">View MathML</a> is uniformly bounded.

Thus, we can choose a large enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M461">View MathML</a> such that all the solutions of (21) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M462">View MathML</a>. Therefore the Leray-Schauder degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M463">View MathML</a> is well defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M465">View MathML</a>

It is easy to see that u is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M466">View MathML</a> if and only if u is a solution of the following usual differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M467">View MathML</a>

Obviously, system (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M468">View MathML</a>) possesses a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M469">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M470">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M471">View MathML</a>

which implies that (1)-(4) has at least one solution. This completes the proof. □

Theorem 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and operatorsAand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481">View MathML</a>satisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M482">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>.

Then problem (1) with (2), (4) and (5) has at least a solution.

Proof Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M486">View MathML</a>.

From Theorem 3.1, it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M487">View MathML</a>

(25)

(a) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M488">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M489">View MathML</a> is a large enough positive constant. From the definition of D, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M490">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M491">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M492">View MathML</a>. Thus (25) is valid.

(b) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M493">View MathML</a>, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M494">View MathML</a>

There are two cases: Case (i): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M495">View MathML</a>; Case (ii): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M496">View MathML</a>.

Case (i): Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M498">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M499">View MathML</a>

Thus (25) is valid.

Case (ii): Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M491">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M492">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M502">View MathML</a>

Thus (25) is valid.

Thus problem (1) with (2), (4) and (5) has at least a solution. This completes the proof. □

Let us consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M503">View MathML</a>

(26)

where ε is a parameter, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M504">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M505">View MathML</a> are Caratheodory. We have the following theorem.

Theorem 3.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M515">View MathML</a>

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M516">View MathML</a>

We consider the existence of solutions of the following equation with (2)-(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M517">View MathML</a>

(27)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M518">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M519">View MathML</a> is defined in (10).

We know that (27) with (2)-(4) has the same solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M520">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M521">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M522">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M523">View MathML</a>. As in the proof of Theorem 3.1, we know that all the solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M524">View MathML</a> are uniformly bounded, then there exists a large enough <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M461">View MathML</a> such that all the solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M526">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M462">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M528">View MathML</a> is compact continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M531">View MathML</a>

(28)

Since f and h are Caratheodory, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M532">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M533">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M534">View MathML</a>. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M535">View MathML</a>

Thus, when ε is small enough, from (28), we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M536">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M520">View MathML</a> has no solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M538">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>, when ε is small enough. It means that the Leray-Schauder degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M540">View MathML</a> is well defined for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M437">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M542">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M522">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M544">View MathML</a>, from the proof of Theorem 3.1, we can see that the right-hand side is nonzero. Thus (26) with (2)-(4) has at least one solution when ε is small enough. This completes the proof. □

Theorem 3.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M291">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M292">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M554">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

4 Existence of solutions in Case (ii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>.

When f satisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a> growth condition, we have the following.

Theorem 4.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99">View MathML</a>growth condition; and operatorsAandBsatisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M564">View MathML</a>

then problem (1)-(4) has at least a solution.

Proof Similar to the proof of Theorem 3.1, we omit it here. □

Theorem 4.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and operatorsAand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481">View MathML</a>satisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M569">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M570">View MathML</a>

then problem (1) with (2), (4) and (5) has at least a solution.

Proof Similar to the proof of Theorem 3.2, we omit it here. □

Theorem 4.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M574">View MathML</a>

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.3, we omit it here. □

Theorem 4.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M130">View MathML</a>; fsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M578">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

5 Existence of solutions in Case (iii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M582">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M583">View MathML</a>.

When f satisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a> growth condition, we have the following theorem.

Theorem 5.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>;

whenfsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M99">View MathML</a>growth condition; and operatorsAandBsatisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M597">View MathML</a>

then problem (1)-(4) has at least a solution.

Proof Similar to the proof of Theorem 3.1, we omit it here. □

Theorem 5.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>;

whenfsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and operatorsAand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M481">View MathML</a>satisfy the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M611">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M612">View MathML</a>

then problem (1) with (2), (4) and (5) has at least a solution.

Proof Similar to the proof of Theorem 3.2, we omit it here. □

Theorem 5.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>;

whenfsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M625">View MathML</a>

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.3, we omit it here. □

Theorem 5.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M366">View MathML</a>, g, hsatisfy one of the following:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M368">View MathML</a>;

(20) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M179">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M184">View MathML</a>;

whenfsatisfies the sub-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M419">View MathML</a>growth condition; and we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M638">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M483">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M484">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

In the following, we will consider the existence of nonnegative solutions. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M642">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M643">View MathML</a> means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M644">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>.

Theorem 5.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M367">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M649">View MathML</a>. We also assume:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M650">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M651">View MathML</a>;

(20) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M653">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>;

(30) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M657">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>;

(40) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>.

Then every solution of (1)-(4) is nonnegative.

Proof Let u be a solution of (1)-(4). From Lemma 2.10, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M660">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M661">View MathML</a>. If it is false, then there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M663">View MathML</a>.

It follows from (10) and (20) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M664">View MathML</a>

(29)

Thus (29) and condition (30) hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M665">View MathML</a>

(30)

Similar to the proof before Lemma 2.8, from the boundary value conditions, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M666">View MathML</a>

(31)

From (29) and (30), we get a contradiction to (31). Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M661">View MathML</a>.

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M668">View MathML</a>

(32)

If it is false, then there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M670">View MathML</a>

It follows from (10) and (20) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M671">View MathML</a>

(33)

Thus (33) and condition (30) hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M672">View MathML</a>

(34)

From (33), (34), we get a contradiction to (31). Thus (32) is valid.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M673">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M674">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M675">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M676">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M677">View MathML</a> is decreasing, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M678">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M679">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M680">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M682">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M683">View MathML</a>

It follows from condition (30) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M684">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M685">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M684">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M687">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M688">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>.

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M662">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M691">View MathML</a>

(35)

from (4) and (35), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M692">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M693">View MathML</a>.

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M694">View MathML</a>

(36)

from (4), (36) and condition (40), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M695">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M696">View MathML</a>.

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M697">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M698">View MathML</a>. The proof is completed. □

Corollary 5.6Under the conditions of Theorem 5.1, we also assume:

(10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M650">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M651">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M701">View MathML</a>;

(20) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M653">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705">View MathML</a>;

(30) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M657">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705">View MathML</a>;

(40) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M183">View MathML</a>;

(50) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M713">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M714">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M715">View MathML</a>.

Then (1)-(4) has a nonnegative solution.

Proof Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M716">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M717">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M718">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M719">View MathML</a> satisfies the Caratheodory condition, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M720">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M721">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M8">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M723">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M724">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M725">View MathML</a> are continuous and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M726">View MathML</a>

It is not hard to check that

(20)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M727">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M103">View MathML</a> uniformly, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M729">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M126">View MathML</a>;

(30)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M731">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>;

(40)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M733">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M654">View MathML</a>.

Let us consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M735">View MathML</a>

(37)

It follows from Theorem 5.1 and Theorem 5.5 that (37) has a nonnegative solution u. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M705">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M737">View MathML</a>, and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M738">View MathML</a>

Thus u is a nonnegative solution of (1)-(4). This completes the proof. □

Note (i) Similarly, we can get the existence of nonnegative solutions of (26) with (2)-(4).

(ii) Similarly, under the conditions of Case (ii), we can discuss the existence of nonnegative solutions.

6 Examples

Example 6.1 Consider the existence of solutions of (1)-(4) under the following assumptions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M739">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M740">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M741">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M742">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M743">View MathML</a>.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M744">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M745">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M746">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M747">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/161/mathml/M19">View MathML</a>); then the conditions of Theorem 3.1 are satisfied, then (1)-(4) has a 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).

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