2017
Том 69
№ 6

All Issues

Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

Agarwal R. P., Manojlović J. V.

Full text (.pdf)


Abstract

We consider a class of fourth-order nonlinear difference equations of the form $$ \Delta^2(p_n(\Delta^2y_n)^{\alpha})+q_n y^{\beta}_{n+3}=0, \quad n\in {\mathbb N} $$ where $\alpha, \beta$ are the ratios of odd positive integers, and $\{p_n\}, \{q_n\}$ are positive real sequences defined for all $n\in {\mathbb N} $. We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of the convergence or divergence conditions of the sums $$ \sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$

English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 1, pp 6-28.

Citation Example: Agarwal R. P., Manojlović J. V. Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations // Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 8–27.

Full text