Stability for a class of difference equations

We consider the following non-autonomous and nonlinear difference equations with unbounded delays: { x i + 1 = q x i − ∑ j = − ∞ i a i , j f i − j ( x j ) , i = 0 , 1 , 2 , … , x j = ϕ j , − ∞ < j ≤ 0 , where 0 < q < 1 and f j ( x ) ( 0 ≤ j < + ∞ ) are suitable functions. We establish sufficient conditions for the zero solution of the above equation to be globally asymptotically stable. These conditions improve the well known stability conditions for linear and nonlinear difference equations.