数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (1): 29-38.doi: 10.1016/S0252-9602(11)60205-1

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TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

李宇华|梁占平   

  1. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
  • 收稿日期:2007-07-14 修回日期:2009-08-27 出版日期:2011-01-20 发布日期:2011-01-20
  • 基金资助:

    Project supported by the National Natural Science Foundation of China (11071149, 10771128) and the NSF of Shanxi Province (2006011002, 2010011001-1).

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

 LI Yu-Hua, LIANG Zhan-Ping   

  1. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
  • Received:2007-07-14 Revised:2009-08-27 Online:2011-01-20 Published:2011-01-20
  • Supported by:

    Project supported by the National Natural Science Foundation of China (11071149, 10771128) and the NSF of Shanxi Province (2006011002, 2010011001-1).

摘要:

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: −u′′(t) = λf(t, u(t)) for all t ∈(0, 1) subjecting to u(0)=0 and αu(η)=u(1), where η∈(0,1), α∈[0,1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t=0, t=1, and u=0. By the fixed point index theory and approximation method, we establish that there exists λ∗ ∈ (0,+1], such that the above problem has at least two positive solutions for any λ ∈ (0, λ) under certain conditions on the nonlinear term f.

关键词: Three-point singular boundary value problem, positive solutions, fixed point index

Abstract:

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: −u′′(t) = λf(t, u(t)) for all t ∈(0, 1) subjecting to u(0)=0 and αu(η)=u(1), where η∈(0,1), α∈[0,1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t=0, t=1, and u=0. By the fixed point index theory and approximation method, we establish that there exists λ∗ ∈ (0,+1], such that the above problem has at least two positive solutions for any λ ∈ (0, λ) under certain conditions on the nonlinear term f.

Key words: Three-point singular boundary value problem, positive solutions, fixed point index

中图分类号: 

  • 47H10