Asymptotic stability for functional differential equations

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

“…This theory will unify, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ASYMPTOTIC STABILITY FOR NONAUTONOMOUS FDE'S 3557 simplify, complete, and improve our results [4,5,10,11] and those of other papers [6,7,9,12,15,19,25,26,27]. For example, Zhang's theorems [27] based on the inequality (1.4) or (1.5) can be obtained by the choice…”

“…First, however, let us point out another direction of the further development of the theory. It has been observed [4,5,10,21,25,26,27] that using the · 2 norm also in condition (2.1) about the negative definiteness of V makes these theorems more effective. This can be illuminated by the example of Equation (1.9) with the Lyapunov functional (2.8) whose derivative with respect to Equation (1.9) can be estimated by…”

“…Having been motivated by such examples, in [4,5] we initiated the use of Lyapunov functionals whose derivatives satisfy (1.2) or (1.3). If we want to deduce AS from these inequalities, then, obviously, we have to require…”

“…Parts B in Theorems 2.2 and 2.3 show that these conditions can be essentially weakened if the Lyapunov functional is decrescent in Krasovskiȋ's sense. The method of proof brings also assumptions for AS at the same time; so we included parts A, proved in [5] by other methods.…”

“…Considering nonautonomous functional differential equations (FDE's for short), in [4,5] we initiated the use of Lyapunov functionals whose derivatives satisfy one of the inequalities…”