Existence and Uniqueness of Solutions for a Second-Order Delay Differential Equation Boundary Value Problem on the Half-Line

Yuming Wei

doi:10.1155/2008/752827

Abstract

This paper is concerned with the existence and uniqueness of solutions for the second-order nonlinear delay differential equations. By the use of the Schauder fixed point theorem, the existence of the solutions on the half-line is derived. Via the Banach contraction principle, another result concerning the existence and uniqueness of solutions on the half-line is established. The main results in this paper extend some of the existing literatures.

Highlights

Boundary value problems on unbounded interval have many applications in physical problems
Boundary value problems on unbounded interval concerning second-order delay differential equations are of specific interest in these applications
For second-order delay differential equations, boundary value problems on the halfline are closely related to the problems of existence of global solutions on the half-line with prescribed asymptotic behavior

Summary

Introduction

Boundary value problems on unbounded interval have many applications in physical problems. Such problems arise, for example, in the study of linear elasticity, fluid flows, and foundation engineering see 1 and the references therein. Boundary value problems on unbounded interval concerning second-order delay differential equations are of specific interest in these applications. For second-order delay differential equations, boundary value problems on the halfline are closely related to the problems of existence of global solutions on the half-line with prescribed asymptotic behavior. There is, in particular, a growing interest in solutions of such boundary value problems see, e.g., 3. Concerning initial value problems, we refer to the monograph by Lakshmikantham and Leela 6 , while, regarding boundary value problems, we mention the monographs by Azbelev et al 7 and Azbelev and Rakhmatullina 8

Boundary Value Problems

The Schauder theorem

Compactness criterion

The Banach contraction principle

Findings

Thus by taking into account the fact that