Existence of concave symmetric positive solutions for a three-point boundary value problems

Ummahan Akcan,Nuket A Hamal

doi:10.1186/1687-1847-2014-313

Abstract

In this paper, we investigate the existence of triple concave symmetric positive solutions for the nonlinear boundary value problems with integral boundary conditions. The proof is based upon the Avery and Peterson fixed point theorem. An example which supports our theoretical result is also indicated.

Highlights

Consider the boundary value problem (BVP)u (x) + f (x, u(x), u (x)) =, x ∈ (, ), u( ) = u( ) = α η u(s) ds, ( . )where f : (, ) × [, ∞) × R → [, ∞) is continuous and α, η ∈ (, )
Many authors have focused on the existence of symmetric positive solutions for ordinary differential equation boundary value problems; for example, see [ – ] and the references therein
Multi-point boundary value problems included the most recent works [, – ] and boundary value problems with integral boundary conditions for ordinary differential equations have been studied by many authors; one may refer to [, – ]

Summary

Introduction

Where f : ( , ) × [ , ∞) × R → [ , ∞) is continuous and α, η ∈ ( , ). By a symmetric positive solution of BVP ). Recently, many authors have focused on the existence of symmetric positive solutions for ordinary differential equation boundary value problems; for example, see [ – ] and the references therein. Motivated by the works mentioned above, we aim to investigate existence results for concave symmetric positive solutions of BVP In Section , by applying the Avery and Peterson fixed point theorem, we obtain concave symmetric positive solutions for BVP Suppose T : P → P is completely continuous and there exist positive numbers a, b, c with a < b such that (S ): u ∈ P(α, θ , γ , b, c, d) | α(u) > b = ∅ and α(Tu) > b for u ∈ P(α, θ , γ , b, c, d);.

We now show that symmetric about

Then from

We can see from that α