Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/16641
Title: Solvability of second-order nonlinear three-point boundary value problems
Authors: Kwong, MK
Wong, JSW
Keywords: Second-order ordinary differential equation
Sign changing nonlinearities
Three-point boundary value problem
Issue Date: 2010
Publisher: Pergamon Press
Source: Nonlinear analysis : theory, methods and applications, 2010, v. 73, no. 8, p. 2343-2352 How to cite?
Journal: Nonlinear analysis : theory, methods and applications 
Abstract: We are interested in the existence of nontrivial solutions to the three-point boundary value problem (BVP): u″(t)+f(t,u(t))=0,0<t<1 u′(0)=0,u(1)=αu(η)+βu′(η), where 0<η<1, f(t,u)∈C([0,1]×ℝ,ℝ) and α, β are real constants. Fixed-point theorems and degree theory are frequently used to study such problems. Recently, the authors demonstrated that, in many situations, the shooting method proves to be an effective approach, often leading to better results with shorter proofs. Here we present another such example. Assume that f(t,0)≢0 and that there exist nonnegative functions k, h∈L 1(0,1) such that |f(t,w)|≤k(t)|w|+h(t) for all (t,w)∈[0,1]×ℝ. Sun and Liu [13] (for the special case β=0), and Sun [14] (for the special case α=0) showed that, if the L1 norm
k
1 is sufficiently small, then there exists a nontrivial solution to the BVP (*). In this paper, their results are improved using the shooting method.
URI: http://hdl.handle.net/10397/16641
ISSN: 0362-546X
DOI: 10.1016/j.na.2010.04.062
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