Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

We consider nonlinear difference equations of unbounded order of the form x i = b i - ? i j = 0 a i , j f i - j ( x j ) , i = 0 , 1 , 2 , ... , where f j ( x ) ( j = 0 , ... , i ) are suitable functions. We establish sufficient conditions for the boundedness and the convergence of x i as i -> + ? . Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations