The upper and lower solution method for nonlinear third-order three-point boundary value problem

Jian-Ping Sun,Ya-Hong Zhao,Qiu-Yan Ren

doi:10.14232/ejqtde.2010.1.26

Abstract

This paper is concerned with the following nonlinear third-order three-point boundary value problem � u ′′′ (t) + f (t, u(t) , u ′ (t)) = 0, t ∈ [0,1] , u(0) = u ′ (0) = 0, u ′ (1) = αu ′ (η) , where 0 < η < 1 and 0 ≤ α < 1. A new maximum principle is established and some existence

Highlights

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [6].Recently, third-order boundary value problems (BVPs for short) have received much attention
Such that m1 = m (t1) = min m (t) < 0. It follows from Taylor′s formula that there exists ξ ∈ (t1, t0)
2m1 > m′′ (ξ) ≥ λ1 m (s) ds + λ2m (ξ) ≥ λ1ξm1 + λ2m1, which implies that λ1 + λ2 > 2

Summary

Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [6]. In 2008, Guo, Sun and Zhao [7] established some existence results for at least one positive solution to the third-order three-point BVP u′′′(t) + a (t) f (u (t)) = 0, t ∈ (0, 1) , u (0) = u′ (0) = 0, u′ (1) = αu′ (η). Their main tool was the well-known Guo-Krasnoselskii fixed point theorem. In order to obtain our main results, we need the following fixed point theorem [1]. If T : [a, b] → E is an increasing compact mapping and a ≤ T a, T b ≤ b, T has a fixed point in [a, b]

Preliminaries

Main results