Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument - Stevic, S. (Український математичний журнал, 2004)
We consider the difference equation with continuous argument x(t+2)−2λx(t+1)+λ²x(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 limt→∞(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 limt→∞(x(t+1)−x(t))=0 t → ∞ if and only if α ∈ R {1}.