Existence of Nontrivial Solutions for Unilaterally Asymptotically Linear Three-Point Boundary Value Problems

Hongyu Li

doi:10.1155/2014/263042

Abstract

Using fixed point theorems in ordered Banach spaces with the lattice structure, we consider the existence of nontrivial solutions under the condition that the nonlinear term can change sign and study the existence of sign-changing solutions for some second order three-point boundary value problems. Our results improve and generalize on those in the literatures.

Highlights

In this paper, we shall discuss the existence of nontrivial solutions for the following boundary value problem:−u󸀠󸀠 (t) = g (t, u (t)), 0 ≤ t ≤ 1, (1)u (0) = 0, u (1) = αu (β), where g : [0, 1] × (−∞, +∞) → (−∞, +∞) is continuous, 0 < α < 1, 0 < β < 1.Many problems of different areas of physics and applied mathematics can be changed into multipoint boundary value problems for ordinary differential equations
Let P be a cone of Banach space E. x is said to be a positive fixed point of A if x ∈ (P \ {θ}) is a fixed point of A; x is said to be a negative fixed point of A if x ∈ ((−P) \ {θ}) is a fixed point of A; x is said to be a sign-changing fixed point of A if x ∉ (P ∪ (−P)) is a fixed point of A
Let E be a Banach space with a lattice structure, let P be a normal cone of E, and let A : E → E be completely continuous and quasi-additive on lattice

Summary

Introduction

We shall discuss the existence of nontrivial solutions for the following boundary value problem:. Some authors have studied the existence and multiplicity of positive solutions for nonlinear multipoint boundary value problems under the condition that the nonlinear term may be nonnegative by applying Krasnosel’skii’s fixed point theorem, theory of fixed point index, and so on (see [3,4,5,6,7,8]). Some authors considered the existence of nontrivial solutions when the nonlinear term can be negative; for example, see [9,10,11] and references therein. In [13], by using the fixed point index method, Xu and Sun have considered the existence of signchanging solutions for the following three-point boundary value problem: y󸀠󸀠 (t) + g (y (t)) = 0, 0 ≤ t ≤ 1, (2).

Preliminaries and Some Lemmas

Main Results

Applications