Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

A Boichuk,D Khusainov,J Diblík,M Růžičková

doi:10.1155/2011/631412

Abstract

Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems ofnordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity:ż(t)=Az(t-τ)+g(t)+εZ(z(hi(t),t,ε), t∈[a,b], assuming that these solutions satisfy the initial and boundary conditionsz(s):=ψ(s) if s∉[a,b], lz(⋅)=α∈Rm. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to anexplicitandanalyticalform of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functionall) does not coincide with the number of unknowns in the differential system with a single delay.

Highlights

We derive some auxiliary results concerning the theory of differential equations with delay
Lz t : zt − A t Sh0 z t φ t, t ∈ a, b, 1.5 where φ is an n-dimensional column-vector defined by the formula φ t : g t A t ψh[0] t ∈ Lp a, b
Transformations 1.3, 1.4 make it possible to add the initial function ψ s, s < a to nonhomogeneity generating an additive and homogeneous operation not depending on ψ and without the classical assumption regarding the continuous connection of solution z t with the initial function ψ t at the point t a

Summary

Introduction

We derive some auxiliary results concerning the theory of differential equations with delay. A solution of differential system 1.5 is defined as an n-dimensional column vectorfunction z ∈ Dp a, b , absolutely continuous on a, b , with a derivative z ∈ Lp a, b satisfying 1.5 almost everywhere on a, b Such approach makes it possible to apply well-developed methods of linear functional analysis to 1.5 with a linear and bounded operator L. It is well-known see: 1, 2 that a nonhomogeneous operator equation 1.5 with delayed argument is solvable in the space Dp a, b for an arbitrary right-hand side φ ∈ Lp a, b and has an n-dimensional family of solutions dim ker L n in the form b ztXtcKt, s φ s ds ∀c ∈ Ên , 1.7 a. The problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayed matrix exponential defined below

A Delayed Matrix Exponential

Fredholm Boundary-Value Problem

Perturbed Weakly Nonlinear Boundary Value Problems

Example