DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

VARIATION-OF-PARAMETERS FORMULAE AND LIPSCHITZ
STABILITY CRITERIA FOR NONLINEAR MATRIX
DIFFERENTIAL EQUATIONS WITH INITIAL
TIME DIFFERENCE
COSKUN YAKAR
This paper investigates the relationship between an unperturbed matrix differential equation
and a perturbed matrix differential system which both have different initial positions
and an initial time difference. Variation-of-parameter techniques are employed to obtain
integral formulae and to establish Lipschitz stability criteria for nonlinear matrix differential
systems and make use of the variational system associated with the unperturbed
differential system.
Copyright © 2006 Coskun Yakar. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The method of variation-of-parameters formulae (VPF) has been a very useful technique
in the qualitative theory of system of differential equations and nonlinear matrix differential
equations since it is a practical tool in the investigation of the properties of solutions.
Recently in [1–3], the study of nonlinear matrix initial value problems with an initial time
difference (ITD) has been initiated and the corresponding theory of differential inequalities
has been investigated. Below, we will derive VPF showing the relationship between
unperturbed matrix differential systems with different initial conditions and unperturbed
and perturbed systems with different initial conditions.
The qualitative behavior of matrix differential equations has been explored extensively
and the investigation of initial value problems with a perturbation in the space variable
is well known when the perturbation is restricted to the space variable with the initial
time unchanged [1–4, 6, 7]. Recently, several investigations have been initiated to explore
the qualitative behavior of matrix differential systems that have a different initial position
and a different initial time. We call this type of stability analysis ITD stability analysis.
In Section 3, variation-of-parameter formulae were used to investigate the relationship
between (1) unperturbed matrix equations with different initial conditions and (2) unperturbed
and perturbed matrix equations with ITD. ITD Lipschitz stability criteria for
matrix differential systems are established by employing the variational system associated
with the unperturbed matrix differential system.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1201–1216