# American Institute of Mathematical Sciences

## Oscillation criteria for second-order quasi-linear neutral functional differential equation

 1 Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt 2 Department of Mathematics, Faculty of Science, Hadhramout University, Seiyun, Yemen

* Corresponding author: Osama Moaaz

Received  October 2018 Revised  December 2018 Published  November 2019

In this work, new sufficient conditions for oscillation of solution of second order neutral delay differential equation are established. One objective of our paper is to further simplify and complement some results which were published lately in the literature. In order to support our results, we introduce illustrating examples.

Citation: Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020136
##### References:
 [1] R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9401-1.  Google Scholar [2] R. Arul and V. S. Shobha, Oscillation of second order quasilinear differential equations with several neutral terms, J. Progressive Research in Math., (JPRM), 7 (2016), 975-981.   Google Scholar [3] O. Bazighifan, E. M. Elabbasy and O. Moaaz, Oscillation of higher-order differential equations with distributed delay, J. Ineq. Appl., (2019), 9 pp. doi: 10.1186/s13660-019-2003-0.  Google Scholar [4] O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, Hans-Otto Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [5] E. M. Elabbasy, T. S. Hassan and O. Moaaz, Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Opuscula Mathematica, 32 (2012), 719-730.  doi: 10.7494/OpMath.2012.32.4.719.  Google Scholar [6] L. H. Erbe, Q. K. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 190. Marcel Dekker, Inc., New York, 1995.  Google Scholar [7] J. K. Hale, Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag New York, New York-Heidelberg, 1971.  Google Scholar [8] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Eequations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, 110. Marcel Dekker, Inc., New York, 1987.  Google Scholar [9] H. D. Liu, F. W. Meng and P. C. Liu, Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation, Appl. Math. Comput., 219 (2012), 2739-2748.  doi: 10.1016/j.amc.2012.08.106.  Google Scholar [10] J. W. Luo, Oscillation criteria for second-order quasi-linear neutral difference equations, Comput. Math. Appl., 43 (2002), 1549-1557.  doi: 10.1016/S0898-1221(02)00118-9.  Google Scholar [11] O. Moaaz, E. M. Elabbasy and O. Bazighifan, On the asymptotic behavior of fourth-order functional differential equations, Adv. Difference Equ., (2017), 13 pp. doi: 10.1186/s13662-017-1312-1.  Google Scholar [12] O. Moaaz, E. M. Elabbasy and E. Shaaban, Oscillation criteria for a class of third order damped differential equations, Arab J. Math. Sci., 24 (2018), 16-30.  doi: 10.1016/j.ajmsc.2017.07.001.  Google Scholar [13] M. V. Ruzhansky, Y. Je Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Trends in Mathematics, Birkhäuser/Springer, Singapore, 2017.  Google Scholar [14] S. Saker, Oscillation theory of delay differential and difference equations, VDM Verlag Dr. Muller, Saarbrucken, (2010). Google Scholar [15] H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar [16] I. P. Stavroulakis, Nonlinear delay differential inequalities, Nonlinear Anal., 6 (1982), 389-396.  doi: 10.1016/0362-546X(82)90024-4.  Google Scholar [17] S. R. Sun, T. X. Li, Z. L. Han and C. Zhang, On oscillation of second-order nonlinear neutral functional differential equations, Bull. Malays. Math. Sci. Soc. (2), 36 (2013), 541-554.   Google Scholar [18] X. H. Tang, Oscillation for first order superlinear delay differential equations, J. London Math. Soc. (2), 65 (2002), 115-122.  doi: 10.1112/S0024610701002678.  Google Scholar [19] X. L. Wang and F. W. Meng, Oscillation criteria of second-order quasi-linear neutral delay differential equations, Math. Comput. Model., 46 (2007), 415-421.  doi: 10.1016/j.mcm.2006.11.014.  Google Scholar [20] Y. Z. Wu, Y. H. Yu, J. M. Zhang and J. S. Xiao, Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type, J. Inequal. Appl., 2016 (2016), 11 pp. doi: 10.1186/s13660-016-1268-9.  Google Scholar [21] H. Yaldız and P. Agarwal, $S$-convex functions on discrete time domains, Analysis, 37 (2017), 179-184.  doi: 10.1515/anly-2017-0015.  Google Scholar

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##### References:
 [1] R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9401-1.  Google Scholar [2] R. Arul and V. S. Shobha, Oscillation of second order quasilinear differential equations with several neutral terms, J. Progressive Research in Math., (JPRM), 7 (2016), 975-981.   Google Scholar [3] O. Bazighifan, E. M. Elabbasy and O. Moaaz, Oscillation of higher-order differential equations with distributed delay, J. Ineq. Appl., (2019), 9 pp. doi: 10.1186/s13660-019-2003-0.  Google Scholar [4] O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, Hans-Otto Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [5] E. M. Elabbasy, T. S. Hassan and O. Moaaz, Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Opuscula Mathematica, 32 (2012), 719-730.  doi: 10.7494/OpMath.2012.32.4.719.  Google Scholar [6] L. H. Erbe, Q. K. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 190. Marcel Dekker, Inc., New York, 1995.  Google Scholar [7] J. K. Hale, Functional Differential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag New York, New York-Heidelberg, 1971.  Google Scholar [8] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Eequations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, 110. Marcel Dekker, Inc., New York, 1987.  Google Scholar [9] H. D. Liu, F. W. Meng and P. C. Liu, Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation, Appl. Math. Comput., 219 (2012), 2739-2748.  doi: 10.1016/j.amc.2012.08.106.  Google Scholar [10] J. W. Luo, Oscillation criteria for second-order quasi-linear neutral difference equations, Comput. Math. Appl., 43 (2002), 1549-1557.  doi: 10.1016/S0898-1221(02)00118-9.  Google Scholar [11] O. Moaaz, E. M. Elabbasy and O. Bazighifan, On the asymptotic behavior of fourth-order functional differential equations, Adv. Difference Equ., (2017), 13 pp. doi: 10.1186/s13662-017-1312-1.  Google Scholar [12] O. Moaaz, E. M. Elabbasy and E. Shaaban, Oscillation criteria for a class of third order damped differential equations, Arab J. Math. Sci., 24 (2018), 16-30.  doi: 10.1016/j.ajmsc.2017.07.001.  Google Scholar [13] M. V. Ruzhansky, Y. Je Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Trends in Mathematics, Birkhäuser/Springer, Singapore, 2017.  Google Scholar [14] S. Saker, Oscillation theory of delay differential and difference equations, VDM Verlag Dr. Muller, Saarbrucken, (2010). Google Scholar [15] H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar [16] I. P. Stavroulakis, Nonlinear delay differential inequalities, Nonlinear Anal., 6 (1982), 389-396.  doi: 10.1016/0362-546X(82)90024-4.  Google Scholar [17] S. R. Sun, T. X. Li, Z. L. Han and C. Zhang, On oscillation of second-order nonlinear neutral functional differential equations, Bull. Malays. Math. Sci. Soc. (2), 36 (2013), 541-554.   Google Scholar [18] X. H. Tang, Oscillation for first order superlinear delay differential equations, J. London Math. Soc. (2), 65 (2002), 115-122.  doi: 10.1112/S0024610701002678.  Google Scholar [19] X. L. Wang and F. W. Meng, Oscillation criteria of second-order quasi-linear neutral delay differential equations, Math. Comput. Model., 46 (2007), 415-421.  doi: 10.1016/j.mcm.2006.11.014.  Google Scholar [20] Y. Z. Wu, Y. H. Yu, J. M. Zhang and J. S. Xiao, Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type, J. Inequal. Appl., 2016 (2016), 11 pp. doi: 10.1186/s13660-016-1268-9.  Google Scholar [21] H. Yaldız and P. Agarwal, $S$-convex functions on discrete time domains, Analysis, 37 (2017), 179-184.  doi: 10.1515/anly-2017-0015.  Google Scholar
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