Bulletin of the Australian Mathematical Society

Research Article

POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

YANG-WEN ZHANGa1 and HONG-XU LIa2 c1

a1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China (email: zhangyan_569088080@qq.com)

a2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China (email: hoxuli@sohu.com)

Abstract

In this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,):

\[ \begin {cases} -(p(t)x'(t))'+q(t)x(t)=\lambda g(t)f(x(t)), \quad 0\leq t\leq 1,\\ x'(0)=x'(1)=0. \end {cases} \]

By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

(Received October 09 2011)

2010 Mathematics subject classification

  • primary 34B18; secondary 47H10

Keywords and phrases

  • dependence of parameter;
  • Neumann boundary value problem;
  • positive solution;
  • fixed point theorem

Correspondence:

c1 For correspondence; e-mail: hoxuli@sohu.com

Footnotes

This work is supported by the NNSF of China (Grant No. 11071042).