Global stability for nonlinear difference equations with variable coefficients

Note that for f(x) = e − 1, (1.1) becomes a logistic equation with piecewise constant arguments. Definition 1.1 The zero solution of (1.1) is said to be uniformly stable, if for any ε > 0 and nonnegative integer n0, there is a δ = δ(ε) > 0 such that max{|x(n0 − j)| | j = −k,−k + 1, · · · , 0} < δ, implies that the solution {x(n)}n=0 of (1.1) satisfies |x(n)| < ε, n = n0, n0 + 1, · · · . Definition 1.2 The zero solution of (1.1) is called globally attractive, if every solution of (1.1) tends to zero as n → ∞. Definition 1.3 The zero solution of (1.1) is called globally asymptotically stable, if it is uniformly stable and globally attractive. For the case of f(x) = e − 1 and aj(n) = raj/( ∑m j=0 aj), 0 ≤ j ≤ m, by Gopalsamy et al.[2], r ≤ log 2/(m + 1) is a sufficient condition of global asymptotic stability for the zero solution of (1.1). This condition is improved by So and Yu [8] as follows. Theorem A (see So and Yu [8]). Assume that f(x) = ex−1 and aj(n) = r(n)aj/( ∑m j=0 aj) (aj, 0 ≤ j ≤ m are constants) in (1.1) with (1.3) and n ∑ k=n−m m ∑