2019
Том 71
№ 4

# Stevic S.

Articles: 2
Article (English)

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

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1095–1100

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}.

Brief Communications (English)

### A Note on the Recursive Sequence $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 570-574

We present some comments on the behavior of solutions of the difference equation $x_{n + 1} = p_kx_n + p_{k − 1}x_{n − 1} +...+ p_1x_{n − k + 1}$, $n = −1, 0, 1,…,$ where $p_i ≥ 0, i = 1,..., k, k ∈ N$, and $x_{−k},..., x_{−1} ∈ R$.