Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos

Let X be a complex Banach space, αj, j = 1, … ,k, be real numbers, with and let be a sequence in X such that It is given a sufficient and necessary condition such that the boundedness of always implies limn→∞ xn+1 − xn = 0. We also present a sufficient condition which guarantees that every slowly varying solution of the difference equation xn+1 = f(xn, … ,xn−k) is convergent, if f is a real function.