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

Wang Fanglei, An, An Yukun

Description

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. 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.

Sujets

• Singular
• Coné
• Fixed point theorem

Informations

• Publié par: biomed
• Publié le: 01 janvier 2013
• Nombre de lectures: 6
• Langue: English

Extrait

Wang and AnBoundary Value Problems2013,2013:7 http://www.boundaryvalueproblems.com/content/2013/1/7

R E S E A R C H

Open Access

Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter 1* 2 Fanglei Wang and Yukun An

* Correspondence: [e-mail] 1 College of Science, Hohai University, Nanjing, 210098, P.R. China Full list of author information is available at the end of the article

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 [–]. In [], 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 [] use the continuation type existence principle to investigate the following singular periodic problem:

  p–      u u+h(u)u=g(u) +c(t).

More recently, using a weak force condition, Wang [] has built some existence results for the following periodic boundary value problem:   u tt –u xx +c  u t +a  (t,x)u+a  (t,x)v=f  (t,x,u,v) +χ  (t,x),  v tt –v xx +c  v t +a  (t,x)u+a  (t,x)v=f  (t,x,u,v) +χ  (t,x).

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 tele- graph equation or regular telegraph system, see, for example, the papers [–] and the references therein. In these references, the nonlinearities are nonnegative. On the other hand, the authors [] study the semipositone telegraph system   u tt –u xx +c  u t +a  (t,x)u=b  (t,x)f(t,x,u,v),  v tt –v xx +c  v t +a  (t,x)v=b  (t,x)g(t,x,u,v),

©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.