Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

Ye Xue,Guowei Zhang

doi:10.14232/ejqtde.2016.1.97

Abstract

Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

Highlights

For the first time Liu [7] considered the existence of positive solutions to the following secondorder three-point boundary value problems x (t) + λh(t) f (x(t)) = 0, t ∈ [0, 1], (1.1)x(0) = 0, x(1) = δx(η), where λ is a positive parameter, η ∈ (0, 1), f ∈ C([0, ∞), [0, ∞)) is nondecreasing, δ ∈ (0, 1) and h(t) is continuous and especially changes sign on [0, 1] which is different from the nonnegative assumption in most of these studies.Karaca [4] studied the problems with more general boundary conditions x (t) + h(t) f (x(t)) = 0, t ∈ [0, 1], αx(0) = βx (0), x(1) = δx(η), (1.2)Y
We investigate the boundary-value problem x (t) + h(t) f (x(t)) = 0, t ∈ [0, 1], x(0) = βx (0), x(1) = x(η), where β ≥ 0, η ∈ (0, 1), f ∈ C([0, ∞), [0, ∞)) is nondecreasing, and importantly h changes sign on [0, 1]
We investigate the existence of positive solutions to the three-point boundary-value problem x (t) + h(t) f (x(t)) = 0, t ∈ [0, 1], (1.4)

Summary

Introduction

For the first time Liu [7] considered the existence of positive solutions to the following secondorder three-point boundary value problems x (t) + λh(t) f (x(t)) = 0, t ∈ [0, 1],. The authors of [4, 7] showed the existence of at least one positive solution by applying the fixed-point theorem in a cone. We investigate the existence of positive solutions to the three-point boundary-value problem x (t) + h(t) f (x(t)) = 0, t ∈ [0, 1], (1.4). The existence of positive solutions is obtained via a special cone (see (2.5)) in terms of superlinear or sublinear behavior of f by the Guo–Krasnosel’skiı fixed-point theorem in a cone. Other relevant research can be seen in [1, 2, 5, 8, 10]

Preliminaries

Main results