Abstract :
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.
