Asymptotic behavior of solutions of second order nonlinear difference equations

Abstract

In this paper, we study the asymptotic behavior of the second order difference equation
(*)   Δ(r(n)Δx(n))+f(n, x(n))=0.
we obtain some sufficient conditions which ensure that all the solutions of (*) are bounded, and also obtain some conditions which guarantee that for every solution x(n) of (*) satisfies |x(n)|=O(R(n, n₀)) as n→∞, where R(n, s)=∑\limits_k=sⁿ⁻¹\frac{1}{r(k)}.