On Nonseparated Three-Point Boundary Value Problems for Linear Functional Differential Equations

A Rontó,M Rontó

doi:10.1155/2011/326052

Abstract

For a system of linear functional differential equations, we consider a three-point problem with nonseparated boundary conditions determined by singular matrices. We show that, to investigate such a problem, it is often useful to reduce it to a parametric family of two-point boundary value problems for a suitably perturbed differential system. The auxiliary parametrised two-point problems are then studied by a method based upon a special kind of successive approximations constructed explicitly, whereas the values of the parameters that correspond to solutions of the original problem are found from certain numerical determining equations. We prove the uniform convergence of the approximations and establish some properties of the limit and determining functions.

Highlights

The aim of this paper is to show how a suitable parametrisation can help when dealing with nonseparated three-point boundary conditions determined by singular matrices
We consider the system of n linear functional differential equations 1.9 subjected to the nonseparated inhomogeneous three-point boundary conditions of form 1.10
The presence of unknown parameters leads one to a certain system of determining equations, from which one finds those numerical values of the parameters that correspond to the solutions of the given three-point boundary value problem

Summary

Introduction

The aim of this paper is to show how a suitable parametrisation can help when dealing with nonseparated three-point boundary conditions determined by singular matrices. We construct a suitable numerical-analytic scheme allowing one to approach a three-point boundary value problem through a certain iteration procedure. Methods of the so-called numerical-analytic type, in a sense, combine, advantages of the mentioned approaches and are usually based upon certain iteration processes constructed explicitly Such an approach belongs to the few of them that offer constructive possibilities both for the investigation of the existence of a solution and its approximate construction. The aim of this paper is to extend the techniques used in 46 for the system of n linear functional differential equations of the form x t P0 t x t P1 txβtft , t ∈ 0, T , 1.7 subjected to the inhomogeneous three-point Cauchy-Nicoletti boundary conditions x1 0 x10, . C a, b , Rn is the Banach space of the continuous functions a, b → Rn with the standard uniform norm; L1 a, b , Rn is the usual Banach space of the vector functions a, b → Rn with Lebesgue integrable components; L Rn is the algebra of all the square matrices of dimension n with real elements; r Q is the maximal, in modulus, eigenvalue of a matrix Q ∈ L Rn ; 1k is the unit matrix of dimension k; 0i,j is the zero matrix of dimension i × j; 0i 0i,i

Problem Setting and Freezing Technique

Auxiliary Estimates

Successive Approximations

Some Properties of the Limit Function

On the Numerical-Analytic Algorithm of Solving the Problem