Asymptotics and oscillation of higher-order functional dynamic equations with Laplacian and deviating arguments

Résumé

In this paper, we deal with the asymptotics and oscillation of the solutions of higher-order nonlinear dynamic equations with Laplacian and mixed nonlinearities of the form $$\begin{aligned}& \bigl\{ r_{n-1}(t) \phi_{\alpha_{n-1}} \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)\phi _{\alpha_{1}} \bigl[x^{\Delta}(t) \bigr] \bigr)^{\Delta}\cdots \bigr)^{\Delta} \bigr)^{\Delta } \bigr] \bigr\} ^{\Delta} \\& \quad {}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi _{\gamma _{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0 \end{aligned}$$ on an above-unbounded time scale. By using a generalized Riccati transformation and integral averaging technique we study asymptotic behavior and derive some new oscillation criteria for the cases without any restrictions on $g(t)$ and $\sigma(t)$ and when n is even and odd. Our results obtained here extend and improve the results of Chen and Qu (J. Appl. Math. Comput. 44(1-2):357-377, 2014) and Zhang et al. (Appl. Math. Comput. 275:324-334, 2016).

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