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

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• 作者：Taher S Hassan
• 关键词：asymptotic behavior ; oscillation ; higher order ; dynamic equations ; dynamic inequality ; time scales
• 刊名：Advances in Difference Equations
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1615KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847
• 卷排序：2017

文摘

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).Keywordsasymptotic behavioroscillationhigher orderdynamic equationsdynamic inequalitytime scalesMSC34K1134N0539A1039A1339A2139A991 IntroductionWe are concerned with the asymptotic and oscillatory behavior of the higher-order nonlinear functional dynamic equation $$\begin{aligned}& \bigl\{ r_{n-1}(t) \phi_{\alpha_{n-1}} \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)\phi _{\alpha_{1}} \bigl[x^{\Delta}(t) \bigr] \bigr)^{\Delta}\cdots \bigr)^{\Delta} \bigr)^{\Delta } \bigr] \bigr\} ^{\Delta} \\& \quad {}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi _{\gamma _{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0 \end{aligned}$$ (1.1) on an above-unbounded time scale \({\mathbb{T}}\), assuming without loss of generality that \(t_{0}\in{\mathbb{T}}\). For \(A\subset {\mathbb{T}}\) and \(B\subset{\mathbb{R}}\), we denote by \(C_{\mathrm{rd}}(A,B)\) the space of right-dense continuous functions from A to B and by \(C_{\mathrm{rd}}^{1}(A,B)\) the set of functions in \(C_{\mathrm{rd}}(A,B)\) with right-dense continuous Δ-derivatives. We refer the readers to the books by Bohner and Peterson [3, 4] for an excellent introduction of calculus of time scales. Throughout this paper, we suppose that: (i)\(n,N\in\mathbb{N}\), \(n\geq2\), and \(\phi_{\beta }(u):=\vert u\vert ^{\beta-1}u\), \(\beta>0\);