# Exponential and *l*_{p}-Stability in Volterra Equations

## Abstract

This chapter is devoted primarily to the exponential and *l*_{p}-stability of Volterra difference equations. Lyapunov functionals are the main tools in the analysis. It is pointed out that in the case of exponential stability, Lyapunov functionals are hard to extend to vector Volterra difference equations or to Volterra difference equations with infinite delay . In addition, we use nonstandard discretization scheme due to Mickens [122] and apply them to continuous Volterra integro-differential equations. We will show that under the discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms, and *z*-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved.

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