Oscillation results for second order half-linear functional dynamic equations with unbounded neutral coefficients on time scales

Year 2020, Volume: 69 Issue: 1, 668 - 683, 30.06.2020

Orhan Özdemir

https://doi.org/10.31801/cfsuasmas.549184

Cited By: 4

Abstract

This study aims to present some new sufficient conditions for the oscillatory behavior of solutions to a class of second order half-linear functional dynamic equations with mixed neutral term i.e., the neutral term contains both retarded and advanced arguments. The results obtained are applicable in the case where the studied equation has unbounded neutral coefficients and they are new even for the linear case. Illustrative examples are also provided.

Keywords

Oscillation, second order, neutral dynamic equations, time scales

References

• Agarwal, R. P., Grace, S. R. and O'Regan, D., Oscillation criteria for certain nth order differential equations with deviating arguments, J. Math. Appl. Anal.,262 (2001), 601--622.
• Agarwal, R. P., Grace, S. R. and O'Regan, D., The oscillation of certain higher-order functional dfferential equations, Math. Comput. Model., 37 (2003), 705--728.
• Agarwal, R. P., O'Regan, D. and Saker, S. H., Philos-type oscillation criteria for second order half linear dynamic equations, Rocky Mountain J. Math., 37 (2007), 1085--1104.
• Agarwal, R. P., Bohner, M., Li, T. and Zhang, C., A Philos-type theorem for third-order nonlinear retarded dynamic equations, Appl. Math. Comput., 249 (2014),527--531.
• Agarwal, R. P., Bohner, M., Li, T. and Zhang, C., Oscillation criteria for second-order dynamic equations on time scales, Appl. Math. Lett., 31 (2014), 34--40.
• Agwa, H. A., Khodier, A. M. M. and Arafa, H. M., Oscillation of second-order nonlinear neutral dynamic equations with mixed arguments on time scales, Journal of Basic and Applied Research International, 17 (2016), 49--66.
• Baculikova, B. and Dzurina, J., Oscillation theorems for second-order nonlinear neutral differential equations, Comput. Math. Appl., 62(12) (2011), 4472--4478.
• Bohner, M. and Peterson, A., Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, 2001.
• Bohner, M. and Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
• Candan, T. and Dahiya, R. S., On the oscillation of certain mixed neutral equations, Appl. Math. Lett., 21 (2008), 222--226.
• Chatzarakis, G. E. and Miliaras, G. N., Covergence of the solutions for a neutral difference equation with negative coefficients, Tatra Mt. Math. Publ., 54 (2013),31--43.
• Grace, S. R., On the oscillations of mixed neutral equations, J. Math. Anal. Appl., 194 (1995), 377--388.
• Grace, S. R., Dzurina, J., Jadlovska, I. and Li, T., An improved approach for studying oscillation of second-order neutral delay differential equations,J. Inequal. Appl., 2018:193 (2018), 13 pages.
• Graef, J. R., Tunc, E. and Grace, S. R., Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation, Opuscula Math., 37(6) (2017), 839--852.
• Hale, J., Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
• Han, Z., Li, T., Zhang, C. and Sun, Y., Oscillation criteria for certain second-order nonlinear neutral differential equations of mixed type, Abstr. Appl. Anal., 2011 (2011), Article ID:387483, 1--9.
• Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Reprint of the 1952 edition, Cambridge University Press, Cambridge, 1988.
• Hilger, S., Analysis on measure chains A unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18--56.
• Ji, T., Tang, S. and Thandapani, E., Oscillation of second-order neutral dynamic equations with mixed arguments, Appl. Math. Inf. Sci., 8 (2014), 2225--2228.
• Karpuz, B., Manojlovic, J. V., Ocalan, O. and Shoukaku, Y., Oscillation criteria for a class of second-order neutral delay differential equations, Appl. Math. Comput., 210 (2009), 303--312.
• Karpuz, B., Ocalan, O. and Y{\i}ld{\i}z, M. K., Oscillation of a class of difference equations of second order, Math. Comput. Model., 49 (2009), 912--917.
• Kolmanovskii, V. and Myshkis, A., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
• Li, T., Comparison theorems for second-order neutral differential equations of mixed type, Electron. J. Differ. Eq., 2010 (2010), No. 167, 1--7.
• Li, T., Agarwal, R. P. and Bohner, M., Some oscillation results for second-order neutral dynamic equations, Hacet. J. Math. Stat., 41 (2012), 715--721.
• MacDonald, N., Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, 1989.
• Ozdemir, O. and Tunc, E., Asymptotic behavior and oscillation of solutions of third order neutral dynamic equations with distributed deviating arguments, Bull. Math. Anal. Appl., 10(2) (2018), 31--52.
• Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math., 53 (1989), 482--492.
• Qi, Y. and Yu, J., Oscillation of second order nonlinear mixed neutral differential equations with distributed deviating arguments, Bull. Malays. Math. Sci. Soc., 38 (2015), 543--560.
• Saker, S. H., Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math., 187 (2006), 123--141.
• Thandapani, E., Padmavathi, S. and Pinelas, S., Oscillation criteria for even-order nonlinear neutral differential equations of mixed type, Bull. Math. Anal. Appl., 6(1)(2014), 9--22.
• Tunc, E. and Graef, J. R., Oscillation results for second order neutral dynamic equatitons with distributed deviating arguments, Dynam. Syst. Appl., 23(2-3) (2014), 289--303.
• Tunc, E. and Grace, S. R., On oscillatory and asymptotic behavior of a second order nonlinear damped neutral differential equation, International Journal of Differential Equations, 2016 (2016), Article ID: 3746368, 8 pages.
• Tunc, E. and Ozdemir, O., On the asymptotic and oscillatory behavior of solutions of third-order neutral dynamic equations on time scales, Adv. Difference Equ., 2017:127 (2017), 13 pages.
• Tunc, E. and Liu, H., Oscillatory behavior for second-order damped differential equation with nonlinearities including Riemann--Stieltjes integrals, Electron. J. Differ. Equ., 2018(54) (2018), 1--12.
• Tunc, E. and Ozdemir, O., On the oscillation of second-order half-linear functional differential equations with mixed neutral term, J. Taibah Univ. Sci., 13(1) (2019), 481--489.
• Yan, J., Oscillations of second order neutral functional differential equations, Appl. Math. Comput., 83 (1997), 27--41.
• Zhang, C., Baculikova, B., Dzurina, J. and Li, T., Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments, Math. Slovaca, 66(3) (2016), 615--626.