Eventual Stability for Lipschitz Functional Differential Systems

In this paper it is established that for Lipschitz junctional differential systems, the eventual uniform asymptotic stability of the origin is preserved under absolutely diminishing perturbations. Iii two recent papers, A. Strauss and J. A. Yorke have investigated "eventual" stability properties for systems of ordinary differential equations. 1,2 In particular, they have shown that for Lipschitz systems, diminishing perturbations preserve eventual uniform asymptotic stability.1 It is the purpose of this paper to extend a somewhat weaker form of this result to functional differential systems. Namely, it will be shown that for Lipschitz functional differential systems, the eventual uniform asymptotic stability of the origin is preserved under absolutely diminishing perturbations. The following notation will be used in this paper: En is the space of n-vectors, and for x in En, | x | denotes any vector norm. For a given number r > 0, C denotes the linear space of continuous functions mapping the interval [--r, 0] into En, and for ...