Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument

Abstract

We consider the difference equation with continuous argument x(t + 2) - 2λ x(t + 1) + λ 2 x(t) = f(t,x(t)), where λ > 0, t [0, ∞), and f: [0, ∞) × R → R. Conditions for the existence and uniqueness of continuous asymptotically periodic solutions of this equation are given. We also prove the following result: Let x(t) be a real continuous function such that lim (x(t + 2) - (1 -α)x(t + 1) - αx(t)) = 0 for some α R. Then it always follows from the boundedness of x(t) that lim(x(t + 1) - x(t)) = 0 t → ∞ if and only if α R {1}.