Criteria for stability of Volterra difference equations

Continuous and discrete Volterra-type difference equations arise in many applications. In particular, when studying models of population dynamics, modeling various economic or physical processes, in management theory, and medicine. The paper deals with the problem of asymptotic stability of the zero solution of a linear difference equation of Volterra type in convolutions. The definitions of stability and asymptotic stability of the zero solution of this equation are given. The article presents sufficient conditions for the asymptotic stability of linear Volterra difference equations. The corresponding theorems are proved using the z-transform method. The obtained criteria of asymptotic stability of the zero solution are restrictions on the coefficients of the original equation, that is, they represent a certain region of stability in the space of the equation parameters. The obtained criteria are compared with some known sufficient conditions for the asymptotic stability of finite-dimensional linear difference equations. The main advantage of the obtained sufficient conditions for asymptotic stability of a linear difference equation of Volterra type is the visibility of these criteria and ease of their application. In addition, this type of criteria is useful if the coefficients of the equation are not known exactly.