On the boundedness of solutions of functional differential systems

Abstract

It is known [3] that the study of some of the properties of an ordinary differential system can be reduced to that of a single differential equation. It is natural to expect that the same should also be possible for functional differential systems. With this end in view a comparison theorem stating that a continuous function satisfying a functional differential inequality can be majorized by the maximal solution of the corresponding functional differential equation, has been proved [6]. Using this comparison theorem it has been shown [5, 7] that the stability properties of ~ functional differential system can be reduced to that of a single functional differential equation. The purpose of the present work is to show that the Second Method of Lyapunov can be extended to reduce the study of certain boundedness properties of a functional differential system to that of a single functional differential equation. The results obtained have further been extended to perturbed systems. The results are natural extensions of some of those in [3]. Finally, an example is given to exhibit the approach. 2. Preliminaries Let R ~ denote the Euclidean n-space, I the interval 0 ~ t ~ ~ and C ~ -~ Cn{[ - T, 0], Rn}, �9 ~ 0, the space of continuous functions from [-- ~,0] into R n./~1 and C 1 will be denoted by/~ and C, respectively, whereas R + and C + will denote the nonnegative real line and the nonnegative continuous functions belonging to C, respectively. For any vector x E R n the i th component will be denoted by X i and its norm by Ixl = ( i)2 i=1