International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 2, Pages 283-297
A stability theory for perturbed differential equations
Department of Mathematics, Suffolk Community College, Selden 11784, New York, USA
Received 9 August 1978
Copyright © 1979 Sheldon P. Gordon. 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.
The problem of determining the behavior of the solutions of a perturbed
differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is
and the associated perturbed differential equation is
The approach used is to examine the difference between the respective solutions and of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the difference in the case where the initial values and are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.