Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

Jian Liu,Hanying Feng,Xingfang Feng

doi:10.1155/2014/603203

Jian Liu, Hanying Feng + Show 1 more

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https://doi.org/10.1155/2014/603203

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Journal: Journal of Applied Mathematics	Publication Date: Jan 1, 2014
Citations: 8	License type: CC BY 3.0

Affiliation: Shijiazhuang University

Abstract

This paper is concerned with the following second-order three-point boundary value problemu″t+β2ut+λqtft,ut=0,t∈0 , 1,u0=0,u(1)=δu(η), whereβ∈(0,π/2),δ>0,η∈(0,1), andλis a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values ofλby means of the fixed point index theory.

Highlights

For given positive numbers η ∈ (0, 1) and β ∈ (0, π/2), the existence, multiplicity, and nonexistence of positive solutions for the following boundary value problem (BVP for short) u󸀠󸀠 (t) + β2u (t) + λq (t) f (t, u (t)) = 0, t ∈ (0, 1) (1)u (0) = 0, u (1) = δu (η) are considered, where λ is a positive parameter, f ∈ C([0, 1]× [0 + ∞), [0 + ∞)), and q : (0, 1) → [0, +∞) may be singular at t = 0 and 1.A function u(t) ∈ C2(0, 1) is said to be a solution of BVP (1) if u satisfies BVP (1)
If u(t) > 0 for any t ∈ (0, 1), u is said to be a positive solution of BVP (1)
Nonlinear three-point boundary value problems have been studied by many authors using the fixed point index theorem, Leray-Schauder continuation theorem, nonlinear alternative of Leray-Schauder, coincidence degree theory, and fixed point theorem in cones

Summary

Introduction

For given positive numbers η ∈ (0, 1) and β ∈ (0, π/2), the existence, multiplicity, and nonexistence of positive solutions for the following boundary value problem (BVP for short) u󸀠󸀠 (t) + β2u (t) + λq (t) f (t, u (t)) = 0, t ∈ (0, 1) (1). In [8], positive solutions for the following three-point boundary value problem at resonance x󸀠󸀠 (t) = f (t, x (t)) , t ∈ (0, 1) , (2). The existence of positive solutions is studied by means of fixed point index theory. Journal of Applied Mathematics the results of existence, multiplicity, and nonexistence of positive solutions for BVP (1) were established via the fixed point index theory.

Preliminaries

Proofs of the Main Results