Stability of nonlinear Volterra equations

Abstract

Abstract. Using a novel approach, we present some new explicit criteria for global exponential stability of the zero solution of general nonlinear time-varying Volterra difference equations. Furthermore, an explicit stability bound for equations subject to nonlinear time-varying perturbations is given. Finally, the obtained results are used to study uniform attraction of equilibrium of discrete-time bidirectional associative memory (BAM) neural networks. Some illustrative examples are given.Key words: global exponential stability, Volterra equations. global exponential stability of nonlinear time-varying Volterra difference equations in the literature. In particular, several abstract criteria for exponential stability of some nonlinear Volterra difference equations can be found in [5, 15,22]. Stability of nonlinear Volterra equationsIn this paper, we present a new approach to global exponential stability of nonlinear time-varying Volterra difference equations. Our approach is based on the celebrated Perron-Frobenius theorem and the comparison principle. Consequently, we get some new explicit criteria for global exponential stability of the zero solution of general nonlinear time-varying Volterra difference equations. Furthermore, we derive an explicit stability bound for equations subject to nonlinear time-varying perturbations. Finally, the obtained results are applied to study uniform attraction of equilibrium of discrete-time bidirectional associative memory neural networks. Some illustrative examples are given. PreliminariesIn this section, we present notation and some preliminary results which will be of use in what follows. Let TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 DOI: 10.1515/bpasts-2016 Stability of nonlinear Volterra equations Abstract. Using a novel approach, we present some new explicit criteria for global exponential stability of the zero solution of general nonlinear time-varying Volterra difference equations. Furthermore, an explicit stability bound for equations subject to nonlinear time-varying perturbations is given. Finally, the obtained results are used to study uniform attraction of equilibrium of discrete-time bidirectional associative memory (BAM) neural networks. Some illustrative examples are given. BULLETIN OF THE POLISH ACADEMY OF SCIENCES