On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Abstract

Continuing our previous investigations, we give simple sufficient conditions for global stability of the zero solution of the difference equation x_n+1 = qx_n + f_n (x_n ,..., x_n-k ), n ∈ Z, where nonlinear functions f_n satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2k + 2) /3] disjoint subintervals, and, for q from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition.