Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

Jian-Ping Sun,Xiao-Yun Wang

doi:10.1155/2010/106962

Abstract

We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: , , , where and are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters is also studied.

Highlights

Boundary value problems BVPs for short consisting of fourth-order differential equation and four-point homogeneous boundary conditions have received much attention due to their striking applications
1 a nonempty closed convex set P ⊆ X is said to be a cone if mP ⊆ P for all m ≥ 0 and P ∩ −P {0}, where 0 is the zero element of X; 2 every cone P in X defines a partial ordering in X by u ≤ v ⇔ v − u ∈ P ; 3 a cone P is said to be normal if there exists M > 0 such that 0 ≤ u ≤ v implies that u ≤M v ;
4 a cone P is said to be solid if the interior P of P is nonempty

Summary

Introduction

Boundary value problems BVPs for short consisting of fourth-order differential equation and four-point homogeneous boundary conditions have received much attention due to their striking applications. Under the following assumptions: A1 α, β, γ, δ, a, b, c, and d are nonnegative constants with β > 0, δ > 0, ρ1 : αγ αδ γβ > 0, ρ2 : ad bc ac ξ2 − ξ1 > 0, −aξ[1] b > 0, and c ξ2 − 1 d > 0; A2 f t, u : 0, 1 × 0, ∞ → 0, ∞ is continuous and monotone increasing in u for every t ∈ 0, 1 ; A3 there exists 0 ≤ θ < 1 such that f t, ku ≥ kθf t, u for any t ∈ 0, 1 , k ∈ 0, 1 , u ∈ 0, ∞ , 1.8 we prove the uniqueness of positive solution for the BVP 1.5 – 1.7 and study the dependence of this solution on the parameters λi i 1, 2, 3, 4

Preliminary Lemmas

Main Result