Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument
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, ∞) × 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 $$\mathop {\lim }\limits_{t \to \infty } (x(t + 2) - (1 - \alpha )x(t + 1) - \alpha x(t)) = 0$$ 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}.
English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 8, pp 1300-1307.
Citation Example: Stevic S. Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument // Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1095–1100.
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