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Solvability of boundary value problem with p-Laplacian at resonance

Weihua Jiang

Author Affiliations

College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei, 050018, P.R. China

Boundary Value Problems 2014, 2014:36  doi:10.1186/1687-2770-2014-36

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/36


Received:29 October 2013
Accepted:24 January 2014
Published:7 February 2014

© 2014 Jiang; 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

By generalizing the extension of the continuous theorem of Ge and Ren and constructing suitable Banach spaces and operators, we investigate the existence of solutions for p-Laplacian boundary value problems at resonance. An example is given to illustrate our results.

MSC: 34B15.

Keywords:
continuous theorem; resonance; p-Laplacian; boundary value problem

1 Introduction

In this paper, we will study the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M1">View MathML</a>

(1.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M2">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M7">View MathML</a>.

A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Mawhin’s continuous theorem [1] is an effective tool to solve this kind of problems when the differential operator is linear, see [2-10] and references cited therein. But it does not work for nonlinear cases such as boundary value problems with a p-Laplacian, which attracted the attention of mathematicians in recent years [11-15]. Ge and Ren extended Mawhin’s continuous theorem [15] and many authors used their results to solve boundary value problems with a p-Laplacian, see [16,17]. In this new theorem, two projectors P and Q must be constructed. But it is difficult to give the projector Q in many boundary value problems with a p-Laplacian. In this paper, we generalize the extension of the continuous theorem and show that the p-Laplacian problem is solvable when Q is not a projector. And we will use this new theorem to discuss problems (1.1) and (1.2), respectively.

In this paper, we will always suppose that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M8">View MathML</a> are nonnegative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M10">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M11">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M12">View MathML</a>.

2 Preliminaries

Definition 2.1[15]

Let X and Y be two Banach spaces with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M14">View MathML</a>, respectively. A continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M15">View MathML</a> is said to be quasi-linear if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M16">View MathML</a> is a closed subset of Y,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M17">View MathML</a> is linearly homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M19">View MathML</a>,

where domM denote the domain of the operator M.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M21">View MathML</a> be the complement space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M22">View MathML</a> in X, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M23">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M24">View MathML</a> be a projector and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a> an open and bounded set with the origin <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M26">View MathML</a>.

Definition 2.2 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28">View MathML</a> is a continuous and bounded operator. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M29">View MathML</a> by N. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M30">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31">View MathML</a> is said to be M-quasi-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a> if there exists a vector subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M33">View MathML</a> of Y satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M34">View MathML</a> and two operators Q, R with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36">View MathML</a>, being continuous, bounded, and satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38">View MathML</a> continuous and compact such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28">View MathML</a>,

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40">View MathML</a>,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42">View MathML</a>,

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M43">View MathML</a> is the zero operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M44">View MathML</a>,

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M45">View MathML</a>.

Theorem 2.1LetXandYbe two Banach spaces with the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M14">View MathML</a>, respectively, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a>be an open and bounded nonempty set. Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M49">View MathML</a>

is a quasi-linear operator and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M28">View MathML</a>isM-quasi-compact. In addition, if the following conditions hold:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M54">View MathML</a>,

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M55">View MathML</a>,

then the abstract equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M56">View MathML</a>has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M57">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59">View MathML</a>is a homeomorphism with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M60">View MathML</a>.

Proof The proof is similar to the one of Lemma 2.1 and Theorem 2.1 in [15]. □

We can easily get the following inequalities.

Lemma 2.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M61">View MathML</a>, we have

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63">View MathML</a>.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M65">View MathML</a>.

In the following, we will always suppose that q satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M66">View MathML</a>.

3 The existence of a solution for problem (1.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M67">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M69">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M70">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M71">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M73">View MathML</a> are Banach spaces.

Define operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M75">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M76">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, c satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M79">View MathML</a>

(3.1)

Lemma 3.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, there is only one constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M81">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M83">View MathML</a>and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M84">View MathML</a>is continuous.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M86">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M87">View MathML</a> is continuous and strictly decreasing in ℝ. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M89">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M91">View MathML</a>. Thus, there exists a unique constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M92">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M93">View MathML</a>, i.e. there is only one constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M81">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M96">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M97">View MathML</a>, assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M99">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M102">View MathML</a> being strictly increasing, we obtain, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M103">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M104">View MathML</a>

This is a contradiction. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M105">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M106">View MathML</a>

This is a contradiction, too. So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M107">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M108">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M84">View MathML</a> is continuous. The proof is completed. □

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M110">View MathML</a> is a solution if and only if it satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58">View MathML</a>. For convenience, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M113">View MathML</a>.

Lemma 3.2Mis a quasi-linear operator.

Proof It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M114">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M116">View MathML</a>, then c satisfies (3.1). On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M77">View MathML</a>, take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M119">View MathML</a>

By a simple calculation, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M116">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M122">View MathML</a>

By the continuity of T, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M123">View MathML</a> is closed. So, M is quasi-linear. The proof is completed. □

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>.

Proof The proof is simple. Therefore, we omit it. □

Take a projector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M129">View MathML</a> and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M130">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M131">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M132">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M34">View MathML</a>.

By the continuity and boundedness of T, we can easily see that Q is continuous and bounded in Y. It follows from Lemma 3.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M136">View MathML</a>.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M137">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M138">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M139">View MathML</a>. By (H2) and the Arzela-Asscoli theorem, we can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38">View MathML</a> is continuous and compact, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a> is an open bounded set.

Lemma 3.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a>is an open bounded set. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31">View MathML</a>isM-quasi-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>.

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M148">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M150">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M152">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M153">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M155">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M156">View MathML</a>

i.e. Definition 2.2(c) holds. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M158">View MathML</a>

So, Definition 2.2(d) holds. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31">View MathML</a> is M-quasi-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>. The proof is completed. □

Theorem 3.1Assume that the following conditions hold.

(H3) There exists a nonnegative constantKsuch that one of (1) and (2) holds:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M164">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M168">View MathML</a>.

(H4) There exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M169">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M170">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M171">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M173">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M174">View MathML</a>.

Then boundary value problem (1.1) has at least one solution.

In order to prove Theorem 3.1, we show two lemmas.

Lemma 3.5Suppose (H3) and (H4) hold. Then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M175">View MathML</a>

is bounded inX.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M176">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M177">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M178">View MathML</a>. By (H3), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M180">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M182">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M183">View MathML</a>

(3.2)

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M184">View MathML</a>, (H4), and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M185">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63">View MathML</a>, by Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M187">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M188">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M190">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M191">View MathML</a>, by Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M192">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M193">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M195">View MathML</a>.

These, together with (3.2), mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M196">View MathML</a> is bounded in X. □

Lemma 3.6Assume (H3) holds. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M197">View MathML</a>

is bounded inX, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M199">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M201">View MathML</a>. By (H3), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M202">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M203">View MathML</a> is bounded. The proof is completed. □

Proof of Theorem 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M204">View MathML</a>, where r is large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M206">View MathML</a>.

By Lemmas 3.5 and 3.6, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M213">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59">View MathML</a> is a homeomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M216">View MathML</a>

Define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M217">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M219">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M220">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M222">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M223">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M224">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M225">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M226">View MathML</a>, we now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M227">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M228">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M229">View MathML</a>

(3.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M230">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M231">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M232">View MathML</a>. So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M233">View MathML</a>. A contradiction with the definition of ρ. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212">View MathML</a>.

By the homotopy of degree, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M237">View MathML</a>

By Theorem 2.1, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111">View MathML</a> has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>. The proof is completed. □

Example Let us consider the following boundary value problem at resonance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M240">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M241">View MathML</a>.

Corresponding to problem (1.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M247">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M248">View MathML</a>. By a simple calculation, we find that the conditions (H1)-(H4) hold. By Theorem 3.1, we obtain the result that problem (3.4) has at least one solution.

4 The existence of a solution for problem (1.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M67">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M251">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M252">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M71">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M73">View MathML</a> are Banach spaces.

Define operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M75">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M258">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M263">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M264">View MathML</a>

(4.1)

Lemma 4.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, there is only one constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M266">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M267">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M268">View MathML</a>. And<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M269">View MathML</a>are continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270">View MathML</a>.

The proof is similar to Lemma 3.1.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M110">View MathML</a> is a solution if and only if it satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M111">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58">View MathML</a>. For convenience, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M274">View MathML</a>.

Lemma 4.2Mis a quasi-linear operator.

Proof It is easy to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M275">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M277">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M263">View MathML</a> satisfy (4.1). On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M260">View MathML</a>, take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M283">View MathML</a>

By simple calculation, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M277">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M286">View MathML</a>

By the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M123">View MathML</a> is closed. So, M is quasi-linear. The proof is completed. □

Take a projector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M129">View MathML</a> and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M130">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M292">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M293">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M295">View MathML</a>.

By the continuity and boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270">View MathML</a>, we can easily see that Q is continuous and bounded in Y. It follows from Lemma 3.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M298">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M299">View MathML</a>.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M137">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M301">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M139">View MathML</a>. By (H2) and the Arzela-Asscoli theorem, we can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M38">View MathML</a> is continuous and compact, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a> is an open bounded set.

Lemma 4.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M25">View MathML</a>is an open bounded set. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31">View MathML</a>isM-quasi-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>.

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M148">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M313">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M152">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M153">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M40">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M155">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M319">View MathML</a>

i.e. Definition 2.2(c) holds. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M149">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M321">View MathML</a>

Thus, Definition 2.2(d) holds. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M31">View MathML</a> is M-quasi-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>. The proof is completed. □

Theorem 4.1Assume that the following conditions hold:

(H5) There exists a nonnegative constantLsuch that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M162">View MathML</a>then either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M326">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M327">View MathML</a>

(H6) There exist nonnegative constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M329">View MathML</a>such that one of (1) and (2) holds:

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M330">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M331">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M332">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M333">View MathML</a>

(H7) There exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M334">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M335">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M171">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M173">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M174">View MathML</a>.

Then boundary value problem (1.2) has at least one solution.

In order to prove Theorem 4.1, we show two lemmas.

Lemma 4.4Suppose (H5)-(H7) hold. Then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M340">View MathML</a>

is bounded inX.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M176">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M177">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M270">View MathML</a>. By (H5) and (H6), there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M345">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M346">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M347">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M349">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M350">View MathML</a>

(4.2)

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M184">View MathML</a>, (H7), and (4.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M352">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M63">View MathML</a>, by Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M354">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M355">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M356">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M357">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M191">View MathML</a>, by Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M359">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M360">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M362">View MathML</a>.

These, together with (4.2), mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M196">View MathML</a> is bounded in X. □

Lemma 4.5Assume (H6) holds. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M364">View MathML</a>

is bounded inX, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M58">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M199">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M368">View MathML</a>. By (H6), we see that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M369">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M371">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M203">View MathML</a> is bounded. The proof is completed. □

Proof of Theorem 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M204">View MathML</a>, where r is large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M374">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M375">View MathML</a>.

By Lemmas 4.4 and 4.5, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M213">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M59">View MathML</a> is a homeomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M385">View MathML</a>

Take the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M386">View MathML</a> is the same as the one in Proof of Theorem 3.1.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M388">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M389">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M391">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M223">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M224">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M394">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M226">View MathML</a>, we now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M228">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M398">View MathML</a>

(4.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M399">View MathML</a>, we have either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M400">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M401">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M400">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M403">View MathML</a>. So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M404">View MathML</a>. This is a contradiction with the definition of ρ. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M405">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M401">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M407">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M408">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M409">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M410">View MathML</a>. This is a contradiction with the definition of ρ, too. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M212">View MathML</a>.

By the homotopy of degree, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M414">View MathML</a>

By Theorem 2.1, we find that (1.2) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/36/mathml/M32">View MathML</a>. The proof is completed. □

Competing interests

The author declares that she has no competing interests.

Author’s contributions

All results belong to WJ.

Acknowledgements

This work is supported by the National Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108).

The author is grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript.

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