Abstract

The present paper is devoted to the initial problem of the nonlinear discrete system x(k+1)= \sum_{j=1}^{p}A_{j}(k)x(k−h_{j})+f(k,x(k−h_{1}),x(k−h_{2}),…,x(k−h_{p})), x(k)=x_{0}for k=k_{0}−h_{p},k_{0}−h_{p}+1,…,0, where k (≥k0), k_{0}. h_{j}, p (≥1)∈Z^{+}:={0,1,2,…}, 0=h_{1}<h_{2}<⋯<h_{p}, x(k)∈R^{n}, A_{j}(k)∈R^{n x n}, f(k,0,…,0)=0 for all k∈Z^{+}. In an earlier work of the first author [Constrained control problems of discrete processes, Singapore, World Scientific (1996; Zbl 0917.93002)], it was proved that the particular case of the last initial value problem with k_{0}=0 has a solution given by x(k)=P_{k}x_{0}+\sum_{s=0}^{k-1}G_{s}^{k} f(s,x(s−h_{1}),…,x(s−h_{p})), where P_{k}, G_{k}^{s+1} are some matrices formulated in terms of A_{j}(k), j=1,2,…,p. Sufficient conditions for asymptotic stability of (1) expressed in terms of A_{j}(k), G_{k}^{s+1} are obtained when the perturbation f(⋅) obeys a general Hölder-type condition. The main tool used in the paper is a new discrete Gronwall-type inequality of the form x(k)≤C+\sum_{s=0}^{k-1}\sum_{j=1}^{p}a_{j}(i)x(i−hj)^{m}, where m, C are positive numbers, a_{j}(k)≥0, z(k)≥0, z(k)≤C for k=−h_{p},…,0.