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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Existence and uniqueness of solutions for delay boundary value problems with p-Laplacian on infinite intervals

Yuming Wei1* and Patricia JY Wong2

Author affiliations

1 School of Mathematical Science, Guangxi Normal University, Guilin, 541004, China

2 School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

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Citation and License

Boundary Value Problems 2013, 2013:141  doi:10.1186/1687-2770-2013-141


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/141


Received:2 March 2013
Accepted:3 May 2013
Published:31 May 2013

© 2013 Wei and Wong; 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 is concerned with the existence and uniqueness of solutions for boundary value problems with p-Laplacian delay differential equations on the half-line. The existence of solutions is derived from the Schauder fixed point theorem, whereas the uniqueness of solution is established by the Banach contraction principle. As an application, an example is given to demonstrate the main results.

MSC: 34K10, 34B18, 34B40.

Keywords:
delay differential equation; boundary value problem; infinite interval; Schauder fixed point theorem; Banach contraction principle

1 Introduction

Boundary value problems on infinite intervals have many applications in physical problems. Such problems arise, for example, in the study of linear elasticity, fluid flows and foundation engineering (see [1,2] and the references cited therein). Boundary value problems on infinite intervals involving second-order delay differential equations are of specific interest in these applications. An interesting survey on infinite interval problems, including real world examples, history and various methods of proving solvability, can be found in the recent monograph by Agarwal et al.[3] and Agarwal and O’Regan [4]. Among the many articles dealing with boundary value problems of second-order delay differential equations, we refer the reader to [5] and the references cited therein.

Boundary value problems of second-order delay differential equations on infinite intervals are closely related to the problem of existence of global solutions with prescribed asymptotic behavior. Recently, there is a growing interest in the solutions of such boundary value problems; see, for example, [6-9]. For the basic theory of delay differential equations, the reader is referred to the books by Diekmann et al.[10] as well as by Hale and Verduyn Lunel [11]. For boundary value problems, we mention the monographs by Azbelev et al.[12] and Azbelev and Rakhmatullina [13].

To the best of our knowledge, few authors have considered the existence of solutions on infinite intervals for delay differential equations. As far as we know, only in [6,8,9] the existence and uniqueness of solutions on infinite intervals for second-order delay boundary value problems are discussed. However, no work has been done on delay boundary value problems withp-Laplacian on infinite intervals. Motivated by the work mentioned above, this paper aims to fill in the gap, and we shall tackle the existence and uniqueness of solutions to a boundary value problem of delay differential equation withp-Laplacian on infinite interval, which has been rarely discussed until now. The results we obtain improve and generalize the results mentioned in the references.

Throughout this paper, for any intervals J and X of ℝ, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M1">View MathML</a> the set of all continuous functions defined on J with values in X. Let r be a nonnegative real number. If t is a point in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> and x is a continuous real-valued function defined at least on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M3">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M4">View MathML</a> will be used for the function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M5">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M6">View MathML</a>

We notice that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M5">View MathML</a> is a Banach space equipped with the usual sup-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M8">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M9">View MathML</a>

In this paper we consider the delay differential equation with p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M10">View MathML</a>

(1.1)

subject to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M11">View MathML</a>

(1.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M12">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M15">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M16">View MathML</a>, and f is a real-valued function defined on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M17">View MathML</a>, which satisfies the following continuity condition: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M18">View MathML</a> is continuous with respect to t in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> for each given function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M20">View MathML</a> which is continuously differentiable on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>.

We are interested in global solutions of the p-Laplacian delay boundary value problem (1.1)-(1.3). By a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> of (1.1)-(1.3), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M23">View MathML</a> which is continuously differentiable on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> such that (1.1) is satisfied for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25">View MathML</a> and the conditions (1.2) and (1.3) are also fulfilled.

The main results of this paper are stated in Section 2. In Theorem 2.1, sufficient conditions are established in order that (1.1)-(1.3) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>, whereas Theorem 2.2 provides sufficient conditions for (1.1)-(1.3) to have a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>. The proof of Theorem 2.1 and Theorem 2.2 is presented in Section 3, where we employ the Schauder fixed point theorem and the well known Banach contraction principle. In Section 4, we include an example to illustrate our main results.

2 Main results

A useful integral representation of the boundary value problem (1.1)-(1.3) is given by the following lemma. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M14">View MathML</a> has the inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M30">View MathML</a>.

Lemma 2.1Letxbe a function in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31">View MathML</a>that is continuously differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>. Thenxis a solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>of the boundary value problem (1.1)-(1.3) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M34">View MathML</a>

(2.1)

Proof Let x be a function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31">View MathML</a> that is continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>. Assume that x is given by (2.1). Then (1.2) is fulfilled. Moreover, we immediately obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M37">View MathML</a>

(2.2)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M38">View MathML</a>, i.e., (1.3) holds. Furthermore, from (2.2) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M39">View MathML</a>

(2.3)

and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M40">View MathML</a>

(2.4)

which means that x satisfies (1.1). Thus, x is a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> of the boundary value problem (1.1)-(1.3).

Conversely, suppose that x is a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> of the boundary value problem (1.1)-(1.3). In view of (1.2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44">View MathML</a>. Furthermore, from (1.1) it follows that x satisfies (2.4). Taking into account the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M46">View MathML</a>, consequently we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M47">View MathML</a>

(2.5)

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M48">View MathML</a>

(2.6)

By integrating (2.6) and taking into account the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M49">View MathML</a>, we obtain for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M51">View MathML</a>

We have thus proved that x has the expression (2.1). The proof of the lemma is complete. □

The first main result of this paper is the following theorem which provides sufficient conditions for (1.1)-(1.3) to have at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>.

Theorem 2.1Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M53">View MathML</a>

(2.7)

whereFis a nonnegative real-valued function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M54">View MathML</a>which satisfies the continuity condition:

(C) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M55">View MathML</a>is continuous with respect totin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>for each given functionxin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31">View MathML</a>which is continuously differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>.

We also assume that

(A) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M59">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M60">View MathML</a>is increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M61">View MathML</a>in the sense that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M62">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M63">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M64">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M65">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M66">View MathML</a>) and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M67">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M68">View MathML</a>. Moreover, there exists a real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69">View MathML</a>so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M70">View MathML</a>

(2.8)

where the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M71">View MathML</a>depends onξ, cand is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M72">View MathML</a>

(2.9)

Then the boundary value problem (1.1)-(1.3) has at least one solutionxon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M74">View MathML</a>

(2.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M75">View MathML</a>

(2.11)

The second main result is the following theorem that establishes conditions under which the boundary value problem (1.1)-(1.3) has a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>.

Theorem 2.2Let all the conditions of Theorem 2.1 be satisfied, i.e., (2.7), (C) and (A) hold. Moreover, suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M77">View MathML</a>

(2.12)

whereKis a nonnegative continuous real-valued function on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M79">View MathML</a>

(2.13)

Then the boundary value problem (1.1)-(1.3) has a unique solutionxon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>satisfying (2.10) and (2.11).

3 Proof of main results

To prove Theorem 2.1, we shall use the fixed point technique by applying the Schauder fixed point theorem [14], whereas Theorem 2.2 is established by the Banach contraction principle [15]. We state our main tools below.

Schauder fixed point theorem[14]

LetEbe a Banach space and Ω be any nonempty convex and closed subset ofE. IfMis a continuous mapping of Ω into itself andMΩ is relatively compact, then the mappingMhas at least one fixed point, i.e., there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M81">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83">View MathML</a> be the Banach space of all bounded continuous real-valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>, endowed with the sup-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M8">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M86">View MathML</a>

We need the following compactness criterion for a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83">View MathML</a>, which is a consequence of the well-known Arzela-Ascoli theorem. This compactness criterion is an adaptation of a lemma due to Avramescu [16]. In order to formulate this criterion, we note that a set U of real-valued functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> is said to be equiconvergent at ∞ if all the functions in U are convergent in ℝ at the point ∞ and, in addition, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M89">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M90">View MathML</a> such that, for any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M91">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M92">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M93">View MathML</a>.

Compactness criterion[16]

LetUbe an equicontinuous and uniformly bounded subset of the Banach space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83">View MathML</a>. IfUis equiconvergent at ∞, it is also relatively compact.

Banach contraction principle[15]

LetEbe a Banach space and Ω be any nonempty closed subset ofE. IfMis a contraction of Ω into itself, then the mappingMhas a unique fixed point, i.e., there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M95">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82">View MathML</a>.

Throughout this section, let E denote the set of all functions x in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31">View MathML</a>, which are continuously differentiable on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> and have bounded continuous derivatives on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>. The set E is a Banach space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M101">View MathML</a>

We shall first establish a lemma which will be needed to prove the main results.

Lemma 3.1Suppose that (2.7) holds, whereFis a nonnegative real-valued function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M102">View MathML</a>, which satisfies the continuity condition (C). Assume that (A) is satisfied. Let Ω be the subset of the Banach spaceEdefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M103">View MathML</a>

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M104">View MathML</a>, define a mappingMon Ω by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M105">View MathML</a>

(3.1)

ThenMmaps Ω intoE. Moreover, MΩ is relatively compact and the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106">View MathML</a>is continuous.

Proof Since the condition (A) is satisfied, from (2.8) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M107">View MathML</a>

(3.2)

First, we shall show that M is a mapping from Ω into E, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M108">View MathML</a>. Let x be an arbitrary function in Ω. By the definition of Ω, the function x satisfies (1.2) and (2.11). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M109">View MathML</a>, it follows from (1.2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M110">View MathML</a>. By taking into account this fact and using (2.11), we can easily obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M111">View MathML</a>

(3.3)

By virtue of (1.2), (3.3) and (2.9), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M112">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M113">View MathML</a>, which ensures that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M114">View MathML</a>

(3.4)

In view of (3.4), (2.11) and the condition (A), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M115">View MathML</a>

On the other hand, (2.7) guarantees that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M116">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M117">View MathML</a>

(3.5)

From (3.2) and (3.5) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M118">View MathML</a>

(3.6)

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M119">View MathML</a>

(3.7)

Furthermore, we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M120">View MathML</a>

(3.8)

Since (3.8) holds for any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121">View MathML</a>, we immediately see that the formula (3.1) makes sense for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122">View MathML</a>, and this formula defines a mapping M from Ω into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M31">View MathML</a>.

Next, using (3.5) and (2.8), from (3.1) we obtain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M125">View MathML</a>

(3.9)

Inequality (3.9) means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M126">View MathML</a> is bounded on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> and so Mx belongs to E. We have thus proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M108">View MathML</a>.

Now, we shall prove that MΩ is relatively compact. We observe that, for any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M130">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44">View MathML</a>. By taking into account this fact as well as the definition of the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a>, we can easily conclude that it suffices to show that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M133">View MathML</a>

is relatively compact in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83">View MathML</a>. Using (3.5), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M135">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M137">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M138">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M139">View MathML</a>

In view of (3.2), this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M140">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M141">View MathML</a>, and we can easily verify that U is equicontinuous. Moreover, each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122">View MathML</a> satisfies (3.9), where c is independent of x. This guarantees that U is uniformly bounded. Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M144">View MathML</a>

and hence, noting (3.5), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M145">View MathML</a>

(3.10)

Now (3.10) together with (3.2) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M146">View MathML</a>

By using (3.2) and (3.10) again, we immediately see that U is equiconvergent at ∞. It now follows from the given compactness criterion that the set U is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M83">View MathML</a>.

Finally, we shall prove that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106">View MathML</a> is continuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M149">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M150">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M151">View MathML</a>. It is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M152">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M154">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M155">View MathML</a>. On the other hand, using (3.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M156">View MathML</a>

Thus, by taking into account the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M157">View MathML</a>

we can apply the Lebesgue dominated convergence theorem to obtain, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M159">View MathML</a>

This, together with the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M160">View MathML</a>

guarantees the pointwise convergence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M161">View MathML</a>

It remains to show that this convergence is also convergence in the sense of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M163">View MathML</a>

(3.11)

For this purpose, we consider an arbitrary subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M164">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M165">View MathML</a>. Since MΩ is relatively compact, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M166">View MathML</a> of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M164">View MathML</a> and a function u in E so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M168">View MathML</a>. As the convergence in the sense of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a> implies the pointwise convergence to the same limit function, we must have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M170">View MathML</a>, therefore (3.11) holds. Consequently, M is continuous. The proof is complete. □

Proof of Theorem 2.1 We shall apply the Schauder fixed point theorem. Let Ω be the subset of the Banach space E defined as in Lemma 3.1. Clearly, Ω is a nonempty convex and closed subset of E. By Lemma 3.1, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M106">View MathML</a> is continuous and MΩ is relatively compact. We shall show that M maps Ω into itself, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M172">View MathML</a>. Let us consider an arbitrary function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122">View MathML</a>. Following the argument in the proof of Lemma 3.1, we see that x satisfies (3.10), which together with (2.8) provides

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M174">View MathML</a>

(3.12)

Now, (3.12) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M130">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44">View MathML</a> imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M177">View MathML</a>. We have thus proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M178">View MathML</a>.

By the Schauder fixed point theorem, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M122">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M82">View MathML</a>. Hence, x has the expression (3.1), which coincides with (2.1). It follows from Lemma 2.1 that x is a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> of the boundary value problem (1.1)-(1.3). Also, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121">View MathML</a>, clearly x satisfies (2.11). Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M183">View MathML</a>, it follows from (2.11) that x also fulfills (2.10). This completes the proof of Theorem 2.1. □

Proof of Theorem 2.2 We shall employ the Banach contraction principle. Let Ω be the subset of the Banach space E defined in Lemma 3.1. Clearly, Ω is a nonempty closed subset of E. Following the argument in the proof of Theorem 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M184">View MathML</a>.

Now, we shall prove that the mapping M is a contraction. For this purpose, let us consider two arbitrary functions x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M185">View MathML</a> in Ω. From (3.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M186">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44">View MathML</a>, and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M188">View MathML</a>

(3.13)

Furthermore, by using (2.12), from (3.1) we obtain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M189">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M190">View MathML</a>

This gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M191">View MathML</a>

By the definition of the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a>, the last inequality and (3.13) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M193">View MathML</a>

(3.14)

Next, from the definition of Ω, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M194">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M44">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M196">View MathML</a>

(3.15)

Moreover, in view of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M197">View MathML</a>, we get, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M25">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M199">View MathML</a>

(3.16)

But, by the definition of the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M100">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M201">View MathML</a>

(3.17)

Thus, using (3.17) in (3.16) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M202">View MathML</a>

(3.18)

Combining (3.15) and (3.18), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M203">View MathML</a>

(3.19)

where the function μ is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M204">View MathML</a>

We can rewrite (3.19) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M205">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M206">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M207">View MathML</a>

(3.20)

But, since μ is nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M208">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M209">View MathML</a>

So, from (3.20) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M210">View MathML</a>

(3.21)

Now, using (3.17) and (3.21) in (3.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M211">View MathML</a>

where the last inequality is due to (2.13). Hence, we have shown that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M184">View MathML</a> is a contraction.

Finally, by the Banach contraction principle, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M213">View MathML</a> has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M121">View MathML</a> having the expression (3.1), which coincides with (2.1). It follows from Lemma 2.1 that x is the unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> of the boundary value problem (1.1)-(1.3). Furthermore, as in the proof of Theorem 2.1, we conclude that this unique solution x of the boundary value problem (1.1)-(1.3) satisfies (2.10) and (2.11). The proof of Theorem 2.2 is now complete. □

4 Application

Let us consider the delay boundary value problem with the p-Laplacian operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M216">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M218">View MathML</a>, and h is a continuous real-valued function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M219">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M220">View MathML</a> is a nonnegative continuous real-value function on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M222">View MathML</a>.

If the boundary value problem (1.1)-(1.3) is to be equivalent to the boundary value problem (4.1), we must define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M223">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M224">View MathML</a>. Hence, by applying Theorem 2.1 to the boundary value problem (4.1), we can be led to the following result.

Corollary 4.1Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M225">View MathML</a>

whereHis a nonnegative continuous real-valued function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M226">View MathML</a>. Suppose that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M189">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M228">View MathML</a>is increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M229">View MathML</a>in the sense that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M230">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M231">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M232">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M68">View MathML</a>.

Moreover, let there exist a real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69">View MathML</a>so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M235">View MathML</a>

where the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M236">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M237">View MathML</a>depends onξ, cand is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M238">View MathML</a>

Then the boundary value problem (4.1) has at least one solutionxsuch that (2.10) and (2.11) hold.

Note that an interesting particular case is the one where the delay <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M220">View MathML</a> is a nonnegative real constant.

Example Consider the second-order nonlinear delay differential equation of Emden-Fowler type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M240">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M242">View MathML</a> are continuous real-valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M2">View MathML</a>, and γ, β are positive real numbers. An application of Corollary 4.1 to the boundary value problem (4.2) leads us to conclude that if there exists a real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M69">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M245">View MathML</a>

(4.3)

then the boundary value problem (4.2) has at least one solution x satisfying (2.10) and (2.11).

For illustration purpose, suppose in (4.2) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M248">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M249">View MathML</a>

In this case, (4.3) is reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M250">View MathML</a>

It is not difficult to verify that this inequality is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M251">View MathML</a>

which is satisfied if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M252">View MathML</a>. By taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M253">View MathML</a>, we can conclude that the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M254">View MathML</a>

has at least one solution x satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/141/mathml/M255">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YW carried out the most part of the paper. PJYW gave the idea and revised it. All authors read and approved the final manuscript.

Acknowledgements

Project is supported by the National Natural Science Foundation of China (11061006) and the Bagui Scholars program of Guangxi.

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