Oscillation Criteria for Functional Nonlinear Dynamic Equations with $${\gamma}$$ -Laplacian,Damping and Nonlinearities Given by Riemann–Stieltjes Integrals |
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| Authors: | E El-Shobaky E M Elabbasy T S Hassan B A Glalah |
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| Institution: | 1.Department of Mathematics, Faculty of Science,Ain Shams University,Cairo,Egypt;2.Department of Mathematics, Faculty of Science,Mansoura University,Mansoura,Egypt;3.Department of Mathematics, Faculty of Science,University of Hail,Hail,Kingdom of Saudi Arabia;4.Department of Basic Science, Higher Technological Institute,Tenth of Ramadan City,October,Egypt |
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| Abstract: | In this paper, we present oscillation criteria for the second-order nonlinear dynamic equation \({a(t)\phi_{\gamma} (x^{\Delta}(t))]^{\Delta} + p(t)\phi_{\gamma}(x^{\Delta^{\sigma}}(t)) + q_{0}(t) \phi_{\gamma}(x(g_{0}(t)))+\sum_{i=1}^{2}\int_{a_{i}}^{b_{i}}q_{i}(t,s)\phi_{\alpha_{i}(s)}(x(g_{i}(t,s))) \Delta \zeta_{i}(s)=0}\) on a time scale \({\mathbb{T}}\) which is unbounded above. Our results generalize and improve some known results for oscillation of second-order nonlinear dynamic equation. Some examples are given to illustrate the main results. |
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