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

Abstract

Let X be a complex Banach space, αj, j = 1, ...,k, be real numbers, with ∑j = 1k αj = 1 and let (xn)n ∈ N be a sequence in X such thatunder(lim, n → ∞) {norm of matrix}{norm of matrix} fenced(xn + k - underover(∑, j = 1, k) αj xn + k - j) = 0 .It is given a sufficient and necessary condition such that the boundedness of (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.