Difference equations with a special type of impulses is studied. This type of impulses are called non-instantaneous and they start their action abruptly at initially given points and then continue to act on given finite intervals. Since the studied type of equation could not be solved recursively it requires to be proved and applied approximate methods. One of the approximate methods providing a constructive approach to find solutions for the nonlinear problem via linear iterates is the monotone iterative technique. It uses the method of upper and lower solutions which generates the existence of solutions in a closed sector. In this paper the approximate solution of the studied problem is obtained as a limit of successive approximations which are solutions of linear difference equation with constant coefficients and their explicit formula is given. Each term of the sequences is a lower/upper solution of the studied nonlinear problem.