Bounded solutions of nonlinear discrete equations

The goal of this paper is to give sufficient conditions for the existence of bounded solutions of a system of discrete equations u(k+1)=F(k,u(k))u(k+1)=F(k,u(k)), k=a,a+1,a+2,…k=a,a+1,a+2,… with a∈N, N={0,1,…}N={0,1,…} and F:N(a)×Rn→RnF:N(a)×Rn→Rn. A principle giving sufficient conditions for the existence of at least one initial problem generating solution with its graph staying in a previously prescribed domain for every k∈{a,a+1,a+2,…}k∈{a,a+1,a+2,…} is formulated. Illustrative examples are considered as well.