Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 1, Pages 49-67
doi:10.1155/S1048953394000055

Delay differential systems with discontinuous initial data and existence and uniqueness theorems for systems with impulse and delay

S. V. Krishna1 and A. V. Anokhin2

1Andhra University, Department of Mathematics, Visakhapatnam 530 003, India
2Chelyabinsk State University, Department of Mathematics, Chelyabinsk 454138, Russia

Received 1 August 1993; Revised 1 February 1994

Copyright © 1994 S. V. Krishna and A. V. Anokhin. 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.

Abstract

The main purpose of this paper is to discuss some qualitative aspects of differential equations with delays and impulses. Such systems are encountered in modeling the dynamics of prices and cultured populations. However, any such discussion has to be based on some existence and uniqueness results for delay equations with discontinuous initial data. This is the content of the first part of the paper. For an impulsive system, we observe a phenomenon of existence of infinite number of solutions subject to impulses arbitrarily close to a fixed time. Conditions, when such solutions exist and when they do not, are discussed.