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

  • S. Stevic

Abstract

We consider the difference equation with continuous argument $$x(t + 2) - 2\lambda x(t + 1) + \lambda ^2 x(t) = f(t,x(t)),$$ where λ > 0, t ∈ [0, ∞), and f: [0, ∞) × RR. 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 $$\mathop {\lim }\limits_{t \to \infty } (x(t + 2) - (1 - \alpha )x(t + 1) - \alpha x(t)) = 0$$ for some α ∈ R. Then it always follows from the boundedness of x(t) that $$\mathop {\lim }\limits_{t \to \infty } (x(t + 1) - x(t)) = 0$$ t → ∞ if and only if α ∈ R {1}.
Published
25.08.2004
How to Cite
StevicS. “Asymptotic Behavior of Solutions of a Nonlinear Difference Equation With Continuous Argument”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, no. 8, Aug. 2004, pp. 1095–1100, https://umj.imath.kiev.ua/index.php/umj/article/view/3824.
Section
Research articles