BOUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

Abstract

In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed non- linear system : Lyapunov's second method, the use of integral inequalities , and the method of variation of constants formula. The method incorporating in- tegral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov's method. The notion of h-stability (hS) was introduced by Pinto (15,16) with the in- tention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some pertur- bations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Using this notion, Choi and Ryu (3,5) investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al. (8) studied the boundedness of solutions for nonlinear perturbed systems. In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.