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

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 $$\sum\limits_{n=n_0}^{\infty}\frac n{p_n^{1/\alpha}}\quad \text{and}\quad \sum\limits_{n=n_0}^{\infty}\left(\frac n{p_n}\right)^{1/\alpha}.$$