Global asymptotic stability in a perturbed higher-order linear difference equation
In this note, we give a sufficient condition for the global asymptotic stability of the zero solution of the difference equation x(n + 1) = Sigma(i=0)(k) p(i)(n)x(n - i) + f(n,x(n),x(n - 1),...,x(n 1)), n = 0,1..., where k and 1 are nonnegative integers, the coefficients pi(n) are real numbers, and the nonlinearity f satisfies the growth condition \f(n, x(0), x(1),... x(1))\ < q max(0less than or equal toiless than or equal tol) \xi\, for n = 0, 1.... and x(i) is an element of R, 0less than or equal toiless than or equal tol, where q is a constant. The stability condition is formulated in terms of the fundamental solution of the unperturbed equation k y(n + 1) = Sigma(i=0)(k) p(i) (n)y (n - i) (C) 2003 Elsevier Science Ltd. All rights reserved.
global asymptotic stability difference equation perturbed equation