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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter

Fanglei Wang1* and Yukun An2

Author Affiliations

1 College of Science, Hohai University, Nanjing, 210098, P.R. China

2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

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Boundary Value Problems 2013, 2013:7  doi:10.1186/1687-2770-2013-7


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/7


Received:26 July 2012
Accepted:29 December 2012
Published:16 January 2013

© 2013 Wang and An; 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

In this paper, we study the multiplicity of positive doubly periodic solutions for a singular semipositone telegraph equation. The proof is based on a well-known fixed point theorem in a cone.

MSC: 34B15, 34B18.

Keywords:
semipositone telegraph equation; doubly periodic solution; singular; cone; fixed point theorem

1 Introduction

Recently, the existence and multiplicity of positive periodic solutions for a scalar singular equation or singular systems have been studied by using some fixed point theorems; see [1-9]. In [10], the authors show that the method of lower and upper solutions is also one of common techniques to study the singular problem. In addition, the authors [11] use the continuation type existence principle to investigate the following singular periodic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M1">View MathML</a>

More recently, using a weak force condition, Wang [12] has built some existence results for the following periodic boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M2">View MathML</a>

The proof is based on Schauder’s fixed point theorem. For other results concerning the existence and multiplicity of positive doubly periodic solutions for a single regular telegraph equation or regular telegraph system, see, for example, the papers [13-17] and the references therein. In these references, the nonlinearities are nonnegative.

On the other hand, the authors [18] study the semipositone telegraph system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M3">View MathML</a>

where the nonlinearities f, g may change sign. In addition, there are many authors who have studied the semipositone equations; see [19,20].

Inspired by the above references, we are concerned with the multiplicity of positive doubly periodic solutions for a general singular semipositone telegraph equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M4">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M5">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M6">View MathML</a> is a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8">View MathML</a> may change sign and is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M9">View MathML</a>, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M10">View MathML</a>

The main method used here is the following fixed-point theorem of a cone mapping.

Lemma 1.1[21]

LetEbe a Banach space, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M11">View MathML</a>be a cone inE. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M13">View MathML</a>are open subsets ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M15">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M16">View MathML</a>be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M18">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M20">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M18">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M20">View MathML</a>.

ThenThas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M25">View MathML</a>.

The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, we give the main result.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26">View MathML</a> be the torus defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M27">View MathML</a>

Doubly 2π-periodic functions will be identified to be functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26">View MathML</a>. We use the notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M29">View MathML</a>

to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M30">View MathML</a> denotes the space of distributions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26">View MathML</a>.

By a doubly periodic solution of Eq. (1) we mean that a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M32">View MathML</a> satisfies Eq. (1) in the distribution sense, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M33">View MathML</a>

First, we consider the linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M34">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M37">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38">View MathML</a> be the differential operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M39">View MathML</a>

acting on functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26">View MathML</a>. Following the discussion in [14], we know that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38">View MathML</a> has the resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M43">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M44">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M45">View MathML</a> is the unique solution of Eq. (2), and the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M43">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M47">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M48">View MathML</a>) or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M49">View MathML</a> is compact. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M50">View MathML</a> is a completely continuous operator.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M51">View MathML</a>, the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M52">View MathML</a> of the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M38">View MathML</a> is explicitly expressed; see Lemma 5.2 in [14]. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M52">View MathML</a>, we have

For convenience, we assume the following condition holds throughout this paper:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M57">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M26">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M59">View MathML</a>.

Finally, if −ξ is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M60">View MathML</a> in Eq. (2), the author [13] has proved the following unique existence and positive estimate result.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M37">View MathML</a>. Then Eq. (2) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M63">View MathML</a>is a linear bounded operator with the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M64">View MathML</a>is a completely continuous operator;

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M65">View MathML</a>, a.e<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M67">View MathML</a>has the positive estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M68">View MathML</a>

(3)

3 Main result

Theorem 3.1Assume (H1) holds. In addition, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8">View MathML</a>satisfies

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M70">View MathML</a>, uniformly<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M71">View MathML</a>,

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M72">View MathML</a>is continuous,

(H4) there exists a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M73">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M74">View MathML</a>

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M75">View MathML</a>, where the limit function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M76">View MathML</a>,

then Eq. (1) has at least two positive doubly periodic solutions for sufficiently smallλ.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M49">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M78">View MathML</a>. Define a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M79">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M81">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M83">View MathML</a>. By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M73">View MathML</a>is a nonnegative function, the linear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M85">View MathML</a>

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M86">View MathML</a>. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M86">View MathML</a>satisfies the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M88">View MathML</a>

Lemma 3.3If the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M89">View MathML</a>

has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M90">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M91">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M92">View MathML</a>is a positive doubly periodic solution of Eq. (1).

Proof of Theorem 3.1 Step 1. Define the operator T as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M93">View MathML</a>

We obtain the conclusion that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M94">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M95">View MathML</a> is completely continuous.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M96">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M97">View MathML</a>, and T is defined. On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M96">View MathML</a>, the complete continuity is obvious by Lemma 2.1. And we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M99">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M100">View MathML</a>.

Now we prove that the operator T has one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M91">View MathML</a> for all sufficiently small λ.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M75">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M104">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M105">View MathML</a>

Furthermore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M106">View MathML</a>. It follows that

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M108">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M110">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M111">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M113">View MathML</a>, we can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M114">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M115">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M116">View MathML</a>

By the Fatou lemma, one has

Hence, there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M119">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M120">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M121">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M122">View MathML</a>. On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M113">View MathML</a>, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M124">View MathML</a>

From above, we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M125">View MathML</a>

Therefore, by Lemma 1.1, the operator T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M126">View MathML</a> and

So, Eq. (1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M128">View MathML</a>.

Step 2. By conditions (H2) and (H3), it is clear to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M129">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M130">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M131">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M132">View MathML</a>. Then define the operator A as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M133">View MathML</a>

It is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M134">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M135">View MathML</a> is completely continuous.

And for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M136">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M137">View MathML</a>

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M138">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M139">View MathML</a>

Thus, from the above inequality, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M141">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M142">View MathML</a>, then there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M143">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M144">View MathML</a>

where μ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M145">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M146">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M147">View MathML</a>

By Lemma 2.1, it is clear to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M148">View MathML</a>

Therefore, by Lemma 1.1, A has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M150">View MathML</a>, which is another positive periodic solution of Eq. (1).

Finally, from Step 1 and Step 2, Eq. (1) has two positive doubly periodic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M152">View MathML</a> for sufficiently small λ. □

Example

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M153">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/7/mathml/M8">View MathML</a> satisfies the conditions (H1)-(H5).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is concerned with a singular semipositone telegraph equation with a parameter and represents a somewhat interesting contribution in the investigation of the existence and multiplicity of doubly periodic solutions of the telegraph equation. All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this paper.

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