Open Access Research

Solvability for p-Laplacian boundary value problem at resonance on the half-line

Weihua Jiang

Author Affiliations

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

Boundary Value Problems 2013, 2013:207  doi:10.1186/1687-2770-2013-207


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/207


Received:5 April 2013
Accepted:23 August 2013
Published:11 September 2013

© 2013 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

The existence of solutions for p-Laplacian boundary value problem at resonance on the half-line is investigated. Our analysis relies on constructing the suitable Banach space, defining appropriate operators and using the extension of Mawhin’s continuation theorem. An example is given to illustrate our main result.

MSC: 70K30, 34B10, 34B15.

Keywords:
p-Laplacian; resonance; half-line; multi-point boundary value problem; continuation theorem

1 Introduction

A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Resonance problems can be expressed as an abstract equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M1">View MathML</a>, where L is a noninvertible operator. When L is linear, Mawhin’s continuation theorem [1] is an effective tool in finding solutions for these problems, see [2-10] and references cited therein. But it does not work when L is nonlinear, for instance, p-Laplacian operator. In order to solve this problem, Ge and Ren [11] proved a continuation theorem for the abstract equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M1">View MathML</a> when L is a noninvertible nonlinear operator and used it to study the existence of solutions for the boundary value problems with a p-Laplacian:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M3">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M6">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M7">View MathML</a> is nonlinear when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M8">View MathML</a>.

As far as the boundary value problems on unbounded domain are concerned, there are many excellent results, see [12-15] and references cited therein.

To the best of our knowledge, there are few papers that study the p-Laplacian boundary value problem at resonance on the half-line. In this paper, we investigate the existence of solutions for the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M9">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12">View MathML</a>.

In order to obtain our main results, we always suppose that the following conditions hold.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M16">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M18">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M19">View MathML</a>, there exists a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M20">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M21">View MathML</a>

2 Preliminaries

For convenience, we introduce some notations and a theorem. For more details, see [11].

Definition 2.1[11]

Let X and Y be two Banach spaces with the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M23">View MathML</a>, respectively. A continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M24">View MathML</a> is said to be quasi-linear if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M25">View MathML</a> is a closed subset of Y,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M26">View MathML</a> is linearly homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M28">View MathML</a>, where domM denote the domain of the operator M.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M30">View MathML</a> be the complement space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M31">View MathML</a> in X, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M32">View MathML</a>. On the other hand, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33">View MathML</a> is a subspace of Y, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M34">View MathML</a> is the complement of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33">View MathML</a> in Y, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M36">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38">View MathML</a> be two projectors and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39">View MathML</a> an open and bounded set with the origin <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M40">View MathML</a>.

Definition 2.2[11]

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42">View MathML</a> is a continuous operator. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M43">View MathML</a> by N. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M44">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45">View MathML</a> is said to be M-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46">View MathML</a> if there exist a vector subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M33">View MathML</a> of Y satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M48">View MathML</a> and an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M49">View MathML</a> being continuous and compact such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42">View MathML</a>,

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M51">View MathML</a>,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M52">View MathML</a>,

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M53">View MathML</a> is the zero operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M54">View MathML</a>,

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M55">View MathML</a>.

Theorem 2.1[11]

LetXandYbe two Banach spaces with the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M23">View MathML</a>, respectively, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39">View MathML</a>an open and bounded nonempty set. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M59">View MathML</a>

is a quasi-linear operator and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42">View MathML</a>M-compact. In addition, if the following conditions hold:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M64">View MathML</a>,

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M65">View MathML</a>,

then the abstract equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M66">View MathML</a>has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M67">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M69">View MathML</a>is a homeomorphism with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M70">View MathML</a>.

3 Main result

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M71">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M72">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M73">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M74">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M75">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M77">View MathML</a> are Banach spaces.

Define operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M79">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M80">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M81">View MathML</a>

Then the boundary value problem (1.1) is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M82">View MathML</a>.

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M83">View MathML</a>

It is clear that KerM is linearly homeomorphic to ℝ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M84">View MathML</a> is closed. So, M is a quasi-linear operator.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M87">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M89">View MathML</a>. We can easily obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M38">View MathML</a> are projectors. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M36">View MathML</a>.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M95">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M97">View MathML</a>. By (H1) and (H2), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94">View MathML</a> is continuous.

Lemma 3.1[15]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M99">View MathML</a>is compact if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M101">View MathML</a>are both equicontinuous on any compact intervals of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M102">View MathML</a>and equiconvergent at infinity.

Lemma 3.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M94">View MathML</a>is compact.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39">View MathML</a> be nonempty and bounded. There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M19">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M107">View MathML</a>. It follows from (H2) that there exists a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M20">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M109">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M42">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M114">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M115">View MathML</a> are equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M117">View MathML</a> are equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M119">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M120">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M121">View MathML</a>

(3.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M125">View MathML</a>

It follows from the absolute continuity of integral that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M126">View MathML</a> are equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M128">View MathML</a> is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M129">View MathML</a>, by (3.1), we can obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M130">View MathML</a> are equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M116">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M112">View MathML</a>, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M133">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M128">View MathML</a> is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M135">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M136">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M137">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M138">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M139">View MathML</a>

(3.2)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M140">View MathML</a>

(3.3)

there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M141">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M142">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M143">View MathML</a>

(3.4)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M144">View MathML</a>, by (3.2), (3.3) and (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M146">View MathML</a>

By Lemma 3.1, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M147">View MathML</a> is compact. The proof is completed. □

In the spaces X and Y, the origin <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M148">View MathML</a>. In the following sections, we denote the origin by 0.

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M39">View MathML</a>be nonempty, open and bounded. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45">View MathML</a>isM-compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46">View MathML</a>.

Proof By (H2), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M41">View MathML</a> is continuous. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M153">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M107">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M155">View MathML</a> is a zero operator, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M156">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M158">View MathML</a>. So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M51">View MathML</a>. It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M160">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M162">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M163">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M165">View MathML</a>, thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M166">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M167">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M168">View MathML</a>

These, together with Lemma 3.2, mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M45">View MathML</a> is M-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46">View MathML</a>. The proof is completed. □

In order to obtain our main results, we need the following additional conditions.

(H3) There exist nonnegative functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M173">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M175">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M176">View MathML</a>

(H4) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M177">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178">View MathML</a>, then one of the following inequalities holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M179">View MathML</a>

Lemma 3.4Assume that (H3) and (H4) hold. The set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M180">View MathML</a>

is bounded inX.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M181">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M182">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M183">View MathML</a>. By (H4), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M184">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M185">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M186">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M187">View MathML</a>

Considering (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M188">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M189">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M190">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M191">View MathML</a>

(3.6)

By (3.5), (3.6) and (H3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M192">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M193">View MathML</a>

This, together with (3.6), means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M194">View MathML</a> is bounded. The proof is completed. □

Lemma 3.5Assume that (H4) holds. The set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M195">View MathML</a>

is bounded inX.

Proof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M196">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M199">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M200">View MathML</a>

By (H4), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M201">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M202">View MathML</a> is bounded. The proof is completed. □

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

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M203">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M204">View MathML</a>. It follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M202">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M209">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211">View MathML</a>.

Define a homeomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M212">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M213">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M214">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178">View MathML</a>, take the homotopy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M216">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M217">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M218">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M219">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M223">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M224">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M225">View MathML</a>

A contradiction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M178">View MathML</a>, take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M230">View MathML</a>

and the contradiction follows analogously. So, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M211">View MathML</a>.

By the homotopy of degree, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M234">View MathML</a>

By Theorem 2.1, we can get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M82">View MathML</a> has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M46">View MathML</a>. The proof is completed. □

4 Example

Let us consider the following boundary value problem at resonance

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M237">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M12">View MathML</a>.

Corresponding to problem (1.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M242">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/207/mathml/M246">View MathML</a>. By simple calculation, we can get that conditions (H1)-(H4) hold. By Theorem 3.1, we obtain that problem (4.1) has at least one solution.

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 the improvement of the original manuscript.

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