Global stability for nonlinear difference equations with variable coefficients

Consider the following nonlinear difference equation with variable coefficients: ⎧⎪ ⎪⎨⎪ ⎪⎩ x(n+1) = x(n) − m j=0 aj (n)fj x(n− j) , n= 0, 1, 2, . . . , x(j) = xj , −m j 0, where aj (n) 0, 0 j m, m j=0 aj (n) > 0 and ∞n=0 m j=0 aj (n)=+∞.We assume that there exists a strictly monotone increasing function f (x) on (−∞,+∞) such that f (0) = 0, 0 < fj (x) f (x) 1, x = 0, 0 j m, and limx→−∞f (x) is finite if f (x) = x. In this paper, we establish sufficient conditions for the zero solution of the above equation to be globally asymptotically stable. Applying these conditions to some special cases, we improve the “3/2 criteria” type stability conditions for linear and nonlinear difference equations. © 2006 Elsevier Inc. All rights reserved.