Bounded solutions of nonlinear difference equations with advanced arguments

Sufficient conditions of existence and uniqueness of α-bounded and bounded solutions to the difference equation with advancedd arguments x(n + 1) = A(n)x(n) + B(n)x(σ1(n)) + f(n, x(n), x(σ2(n)), σi(n) n + 1, I = 1,2, are given. It is proven that under certain conditions it is possible to find positive numbers R, μ, such that from every initial condition ξ satisfying ξ R, a unique bounded solution, belonging to the ball x μ, starts.