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Oscillation Criteria for Functional Nonlinear Dynamic Equations with \({\gamma}\) -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals

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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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Author notes

  1. B. A. Glalah

    Present address: Department of Mathematics, Faculty of Science, University of Hail, Hail, 2440, Kingdom of Saudi Arabia

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

    E. El-Shobaky

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    E. M. Elabbasy

  3. Department of Mathematics, Faculty of Science, University of Hail, Hail, 2440, Kingdom of Saudi Arabia

    T. S. Hassan

  4. Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City, 6th of October Branch, October, Egypt

    B. A. Glalah

Authors
  1. E. El-Shobaky
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  2. E. M. Elabbasy
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  3. T. S. Hassan
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  4. B. A. Glalah
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Corresponding author

Correspondence to T. S. Hassan.

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El-Shobaky, E., Elabbasy, E.M., Hassan, T.S. et al. Oscillation Criteria for Functional Nonlinear Dynamic Equations with \({\gamma}\) -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals. Mediterr. J. Math. 13, 981–1003 (2016). https://doi.org/10.1007/s00009-015-0553-z

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  • DOI: https://doi.org/10.1007/s00009-015-0553-z

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