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The purpose of this study is to provide necessary and sufficient conditions for exponential asymptotic stability in the large and uniform asymptotic stability of perturbations of linear systems with unbounded delays. A strong relationship is established between the two types of asymptotic stability. It is found that if the exponential estimate of the solution of a system tends to zero as t → ∞ the system is said to be uniformly asymptotically stable. But if the solution of a system approaches the origin faster than any exponential function, then the system is said to be exponentially asymptotically stable. Utilizing the exponential estimate of the solution, stability criteria for the linear part of our system of interest is derived. With enough smoothness conditions on the perturbation function, and appeal made to Lyapunov's stability results and some Gronwall-type inequalities the required stability results are established for the linear perturbation.
Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 529-536