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 ∑j=1kαj=1 and let (xn)n∈N(xn)n∈N be a sequence in X such thatlimn→∞xn+k-∑j=1kαjxn+k-j=0.It is given a sufficient and necessary condition such that the boundedness of (xn)n∈N(xn)n∈N 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.