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Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions

Min Jia1*, Xinguang Zhang2* and Xuemai Gu1

Author Affiliations

1 Communication Research Center, Harbin Institute of Technology, Harbin, 150080, China

2 School of Mathematical and Informational Sciences, Yantai University, Yantai, 264005, China

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Boundary Value Problems 2012, 2012:70  doi:10.1186/1687-2770-2012-70


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/70


Received:7 April 2012
Accepted:15 June 2012
Published:3 July 2012

© 2012 Jia et al.; licensee Springer

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

Abstract

This paper investigates the existence and uniqueness of nontrivial solutions to a class of fractional nonlocal multi-point boundary value problems of higher order fractional differential equation, this kind of problems arise from viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system. Some sufficient conditions for the existence and uniqueness of nontrivial solutions are established under certain suitable growth conditions, our proof is based on Leray-Schauder nonlinear alternative and Schauder fixed point theorem.

MSC: 34B15, 34B25.

Keywords:
fractional differential equation; nontrivial solution; Green function; Leray-Schauder nonlinear alternative

1 Introduction

The purpose of this paper is to establish the existence and uniqueness of nontrivial solutions to the following higher fractional differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M5">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M13">View MathML</a> is the standard Riemann-Liouville derivative, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M14">View MathML</a> is continuous.

Differential equations of fractional order occur more frequently in different research areas such as engineering, physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system, etc. [1-6].

For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas et al. [7], Miller and Ross [8], Podlubny [9], the papers [10-24] and the references therein.

Recently, Salem [10] has investigated the existence of Pseudo solutions for the nonlinear m-point boundary value problem of a fractional type. In particular, he considered the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M15">View MathML</a>

(1.2)

where x takes values in a reflexive Banach space E<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M18">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M19">View MathML</a> denotes the kth Pseudo-derivative of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20">View MathML</a> denotes the Pseudo fractional differential operator of order α. By means of the fixed point theorem attributed to O’Regan, a criterion was established for the existence of at least one Pseudo solution for the problem (1.2).

More recently, Zhang [11] has considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of unknown functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M21">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20">View MathML</a> is the standard Riemann-Liouville fractional derivative of order αq may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M23">View MathML</a> and f may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M24">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M25">View MathML</a>. By using the fixed point theorem of a mixed monotone operator, a unique existence result of positive solution to the problem (1.3) was established. And then, Goodrich [12] was concerned with a partial extension of the problem (1.3) by extending boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M26">View MathML</a>

(1.4)

The author derived the Green’s function for the problem (1.4) and showed that it satisfies certain properties. Then, by using cone theoretic techniques, a general existence theorem for (1.4) was obtained when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M27">View MathML</a> satisfies some growth conditions.

In recent work [13], Rehman and Khan have investigated the multi-point boundary value problems for fractional differential equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M28">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M29">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M30">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M31">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M32">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M33">View MathML</a>. By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for BVP (1.5) provided that the nonlinear function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M34">View MathML</a> is continuous and satisfies certain growth conditions. However, Rehman and Khan only considered the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M29">View MathML</a> and the case of the nonlinear term f was not considered comprehensively.

Notice that the results dealing with the existence and uniqueness of solution for multi-point boundary value problems of fractional order differential equations are relatively scarce when the nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. Thus, the aim of this paper is to establish the existence and uniqueness of nontrivial solutions for the higher nonlocal fractional differential equations (1.1) where nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. In our study, the proof is based on the reduced order method as in [11] and the main tool is the Leray-Schauder nonlinear alternative and the Schauder fixed point theorem.

2 Basic definitions and preliminaries

Definition 2.1 A function x is said to be a solution of BVP (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M36">View MathML</a> and satisfies BVP (1.1). In addition, x is said to be a nontrivial solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M37">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M38">View MathML</a> and x is solution of BVP (1.1).

For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later. These and other related results and their proofs can be found, for example, in [6-9].

Definition 2.2 (see [8])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M40">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M41">View MathML</a> then the αth Riemann-Liouville fractional integral is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M42">View MathML</a>

whenever the right-hand side is defined. Similarly, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M44">View MathML</a>, we define the αth Riemann-Liouville fractional derivative to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M45">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M46">View MathML</a> is the unique positive integer satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M48">View MathML</a>.

Remark 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M49">View MathML</a> with order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M51">View MathML</a>

Lemma 2.1 (see [7])

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M53">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M54">View MathML</a>

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M56">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M57">View MathML</a>

Lemma 2.2 (see [8])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M58">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M59">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M60">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M63">View MathML</a>). Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M64">View MathML</a>stands for the standard Riemann-Liouville fractional integral of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M20">View MathML</a>denotes the Riemann-Liouville fractional derivative as Definition 2.1.

Lemma 2.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M69">View MathML</a>, then the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M70">View MathML</a>

(2.1)

has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M71">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M72">View MathML</a>is the Green function of BVP (2.1), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M73">View MathML</a>

(2.2)

(2.3)

Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M75">View MathML</a>

(2.4)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M76">View MathML</a> and (2.4), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M77">View MathML</a>. Consequently, a general solution of (2.3) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M78">View MathML</a>

(2.5)

By (2.5) and Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M79">View MathML</a>

(2.6)

So, from (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M80">View MathML</a>

(2.7)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M81">View MathML</a>, combining with (2.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M82">View MathML</a>

So, substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M83">View MathML</a> into (2.5), the unique solution of the problem (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M84">View MathML</a>

The proof is completed. □

Lemma 2.4<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M85">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M86">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M87">View MathML</a>

(2.8)

Proof Obviously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M86">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M90">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M91">View MathML</a>

This completes the proof. □

Now let us consider the following modified problem of BVP (1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M92">View MathML</a>

(2.9)

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M94">View MathML</a>. Then (2.9) can be transformed into (1.1). Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M95">View MathML</a>is a solution of the problem (2.9), then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M96">View MathML</a>is a solution of the problem (1.1).

Proof Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93">View MathML</a> into (1.1), by Lemmas 2.1 and 2.2, we can obtain that

(2.10)

and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M99">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M100">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M101">View MathML</a>. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M103">View MathML</a>, (2.9) is transformed into (1.1).

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M95">View MathML</a> be a solution for the problem (2.9). Then, from Lemma 2.1, (2.9) and (2.10), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M105">View MathML</a>

Notice

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M106">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M107">View MathML</a>. Thus from (2.10), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M108">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M109">View MathML</a>

Moreover, it follows from the monotonicity and property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M110">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M111">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M93">View MathML</a> is a solution of the problem (1.1). □

Now let us define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M113">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M114">View MathML</a>

(2.11)

Clearly, the fixed point of the operator T is a solution of BVP (2.9); and consequently is also a solution of BVP (1.1) from Lemma 2.5.

Lemma 2.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M113">View MathML</a>is a completely continuous operator.

Proof Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M116">View MathML</a> is continuous, by using the Ascoli-Arzela theorem and standard arguments, the result can easily be shown. □

Lemma 2.7 (see [25])

LetXbe a real Banach space, Ω be a bounded open subset ofX, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M118">View MathML</a>is a completely continuous operator. Then, either there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M120">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M121">View MathML</a>, or there exists a fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M122">View MathML</a>.

3 Main results

For the convenience of expression in rest of the paper, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M123">View MathML</a>.

Theorem 3.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M124">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>. Moreover, there exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M126">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M127">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M128">View MathML</a>

(3.2)

whereMis defined by (2.8). Then BVP (1.1) has at least one nontrivial solution.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M129">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M131">View MathML</a>

By condition (3.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M132">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M133">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M134">View MathML</a>

On the other hand, from (3.2), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M135">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M136">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M137">View MathML</a>.

Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M138">View MathML</a>, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M140">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M141">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M142">View MathML</a>

(3.3)

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M143">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M144">View MathML</a>

thus we have, by hypothesis (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M145">View MathML</a>

Consequently, from (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M146">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M147">View MathML</a>

This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M120">View MathML</a>. By Lemma 2.7, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M149">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M124">View MathML</a>; so then, by Lemma 2.5, BVP (1.1) has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M151">View MathML</a>. This completes the proof. □

Theorem 3.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M152">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>. Moreover, there exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M126">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M155">View MathML</a>

(3.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M156">View MathML</a>are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.

Proof By Lemma 2.6, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M157">View MathML</a> is a completely continuous operator.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M158">View MathML</a>

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M159">View MathML</a>

and define a ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M160">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M161">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M162">View MathML</a>

On the other hand, it follows from (3.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M163">View MathML</a>

(3.5)

In view of (3.5), we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M164">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M165">View MathML</a>. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M166">View MathML</a>. Hence the Schauder fixed point theorem implies the existence of a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M167">View MathML</a> for BVP (2.9). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M168">View MathML</a>, then by Lemma 2.5, BVP (1.1) has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M151">View MathML</a>. This completes the proof. □

Theorem 3.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M170">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>. Moreover, there exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M172">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M173">View MathML</a>

(3.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M174">View MathML</a>are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.

Proof The proof is similar to that of Theorem 3.2, so it is omitted. □

Remark 3.1 In [13], the authors studied the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M175">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M176">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M30">View MathML</a>, but the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M178">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62">View MathML</a> was not considered. Here we extend the results of [13] and fill the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M178">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M181">View MathML</a>.

Theorem 3.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M170">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>. Moreover, there exist nonnegative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M184">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M185">View MathML</a>

(3.7)

and (3.2) holds. Then BVP (1.1) has a unique nontrivial solution.

Proof In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M186">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M187">View MathML</a>

From Theorem 3.1, we know BVP (1.1) has a nontrivial solution.

But in this case, we prefer to concentrate on the uniqueness of a nontrivial solution for BVP (1.1). Let T be given in (2.11), we shall show that T is a contraction. In fact, by (3.7), a similar method to Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M188">View MathML</a>

And then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M189">View MathML</a>

Then (3.2) implies that T is indeed a contraction. Finally, we use the Banach fixed point theorem to deduce the existence of a unique nontrivial solution to BVP (1.1). □

Corollary 3.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M190">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>, and (3.1) holds. Then BVP (1.1) has at least one nontrivial solution if one of the following conditions holds

(1) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M193">View MathML</a>

(3.8)

(2) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M194">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M195">View MathML</a>

(3.9)

(3) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M194">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M197">View MathML</a>

(3.10)

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M198">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M62">View MathML</a>) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M200">View MathML</a>

(3.11)

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M201">View MathML</a>

From the proof of Theorem 3.1, we only need to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M202">View MathML</a>

(1) If (3.8) holds, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M203">View MathML</a>, and by using H<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M204">View MathML</a>lder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M205">View MathML</a>

(2) In this case, it follows from (3.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M206">View MathML</a>

(3) In this case, it follows from (3.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M207">View MathML</a>

(4) If (3.11) is satisfied, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M208">View MathML</a>

This completes the proof of Corollary 3.1. □

Corollary 3.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M209">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M125">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M211">View MathML</a>

(3.12)

Then BVP (1.1) has at least one nontrivial solution.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M212">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M213">View MathML</a>

by (3.12), there exists a large enough constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M214">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M216">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M217">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M218">View MathML</a>

Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M220">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M221">View MathML</a>

we prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M222">View MathML</a>

In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M223">View MathML</a>

Then it follows from Theorem 3.1 that BVP (1.1) has at least one nontrivial solution. □

4 Examples

Example 4.1 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M224">View MathML</a>

(4.1)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M228">View MathML</a>, and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M229">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M230">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M231">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M232">View MathML</a>

Thus the condition (3.2) in Theorem 3.1 is satisfied, and from Theorem 3.1, BVP (4.1) has a nontrivial solution. □

Example 4.2 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M233">View MathML</a>

(4.2)

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M234">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/70/mathml/M235">View MathML</a>

Thus Theorem 3.4 guarantees a nontrivial solution for BVP (4.2). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between all authors. Each of the authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Acknowledgement

The authors thank the referee for helpful comments and suggestions which led to an improvement of the paper. The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2010091), the National Science Foundation for Post-doctoral Scientists of China (Grant No. 2012M510956).

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