Existence of positive solutions to periodic boundary value problems with sign-changing Green's function

Zhong Shengren, An, An Yukun

Description

This paper deals with the periodic boundary value problems where is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping. This paper deals with the periodic boundary value problems where is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping.

Sujets

• Coné
• Fixed point theorem

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 8
• Langue: English

Extrait

Zhong and AnBoundary Value Problems2011,2011:8 http://www.boundaryvalueproblems.com/content/2011/1/8

R E S E A R C HOpen Access Existence of positive solutions to periodic boundary value problems with signchanging Green’s function 1 2* Shengren Zhongand Yukun An

* Correspondence: anykna@nuaa. edu.cn 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics Nanjing, 210016, PR China Full list of author information is available at the end of the article

Abstract This paper deals with the periodic boundary value problems 2 u+ρu=f(u), 0<t<T   u(0) =u(T),u(0) =u(T), 3π where 0< ρ≤ is a constant and in which case the associated Green’s function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping. Keywords:periodic boundary value problem, positive solution, signchanging Green’s function, cone, fixed point theorem

1 Introduction The periodic boundary value problems  u+a(t)u=f(t,u), 0<t<T (1)   u(0) =u(T),u(0) =u(T), 1 wherefis a continuous orL Caratheodory type function have been extensively studied. A very popular technique to obtain the existence and multiplicity of positive solutions to the problem is Krasnosel’skii’s fixed point theorem of cone expansion/ compression type, see for example [14], and the references contained therein. In those papers, the following condition is an essential assumptions: (A) The Green functionG(t,s) associated with problem (1) is positive for all (t,s)Î [0,T] × [0,T]. Under condition (A), Torres get in [4] some existence results for (1) with jumping nonlinearities as well as (1) with a repulsive or attractive singularity, and the authors in [3] obtained the multiplicity results to (1) whenf(t,u) has a repulsive singularity nearx 2 = 0 andf(t,u) is superlinear nearx= +∞. In [2], a special case,a(t)≡mand π m∈(0, , was considered, the multiplicity results to (1) are obtained when the non linear termf(t,u) is singular atu= 0 and is superlinear atu=∞. Recently, in [5], the hypothesis (A) is weakened as (B) The Green functionG(t,s) associated with problem (1) is nonnegative for all (t,s)Î[0,T] × [0,T] but vanish at some interior points.

© 2011 Zhong 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.