Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay

M. Sathish Kumar; V. Ganesan; M. Sathish Kumar; V. Ganesan

doi:10.3934/math.2020250

AIMS Mathematics

2020, Volume 5, Issue 4: 3851-3874. doi: 10.3934/math.2020250

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Research article

Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay

M. Sathish Kumar ^{1
,
,},
V. Ganesan ²

1.
Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal-637 018, Tamil Nadu, India
2.
PG and Research Department of Mathematics, Aringar Anna Government Arts College, Namakkal-637 002, Tamil Nadu, India

Received: 17 October 2019 Accepted: 17 April 2020 Published: 22 April 2020
MSC : 34C10, 34C15, 34K11

By refining the standard Riccati substitution technique, integral averaging technique and comparison principle, we obtain new oscillation and asymptotic behavior for a class of third-order neutral differential equations with discrete and distributed delay. These criteria dealing with some cases have not been covered by the existing results in the literature. We present many sufficient conditions and related examples in order to illustrate the main results.
- oscillation,
- third order,
- delay,
- neutral,
- differential equation
Citation: M. Sathish Kumar, V. Ganesan. Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay[J]. AIMS Mathematics, 2020, 5(4): 3851-3874. doi: 10.3934/math.2020250

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Abstract

By refining the standard Riccati substitution technique, integral averaging technique and comparison principle, we obtain new oscillation and asymptotic behavior for a class of third-order neutral differential equations with discrete and distributed delay. These criteria dealing with some cases have not been covered by the existing results in the literature. We present many sufficient conditions and related examples in order to illustrate the main results.

References

[1]	A. N. Stanzhitskii, A. P. Krenevich, I. G. Novak, Asymptotic equivalence of linear stochastic itô systems and oscillation of solutions of linear second-order equations, Diff. Equat., 47 (2011), 799-813. doi: 10.1134/S001226611106005X
[2]	B. Baculíková, J. Džurina, Oscillation of third-order neutral differential equations, Math. Comput. Model., 52 (2010), 215-226. doi: 10.1016/j.mcm.2010.02.011
[3]	B. Wang, Q. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica, 94 (2018), 72-80. doi: 10.1016/j.automatica.2018.04.016
[4]	C. Zhang, B. Baculíková, J. Džurina, et al. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments, Math. Slovaca, 66 (2016), 615-626. doi: 10.1515/ms-2015-0165
[5]	C. Jiang, T. Li, Oscillation criteria for third-order nonlinear neutral differential equations with distributed deviating arguments, J. Nonlinear Sci. Appl., 9 (2016), 6170-6182. doi: 10.22436/jnsa.009.12.22
[6]	C. Jiang, Y. Jiang, T. Li, Asymptotic behavior of third-order differential equations with nonpositive neutral coefficients and distributed deviating arguments, Adv. Differ. Equ., 105 (2016), 1-14.
[7]	Ch. G. Philos, Oscillation theorems for linear difierential equations of second order, Arch. Math., 53 (1989), 482-492. doi: 10.1007/BF01324723
[8]	D. D. Bainov, D. P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, Philadelphia and New York, 1991.
[9]	F. Brauer, C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, 2 Eds., Springer, New York, 2012.
[10]	G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker Inc., New York, 1987.
[11]	G. E. Chatzarakis, T. Li, Oscillation criteria for delay and advanced differential equations with nonmonotone arguments, Complexity, 2018 (2018), 1-18. doi: 10.1155/2018/8237634
[12]	H. Wang, G. Chen, Y. Jiang, et al. Asymptotic behavior of third-order neutral differential equations with distributed deviating arguments, J. Math. Comput. Sci., 17 (2017), 194-199. doi: 10.22436/jmcs.017.02.01
[13]	H. C. Wang, Q. X. Zhu, Oscillation in solutions of stochastic delay differential equations with richard's nonlinearity, Dynam. Cont. Dis. Ser. A, 15 (2008), 493-502.
[14]	J. A. D. Appleby, C. Kelly, Oscillation and non-oscillation in solutions of nonlinear stochastic delay differential equations, Electronic communications in probability, 9 (2004), 106-118.
[15]	J. A. D. Appleby, C. Kelly, Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay, Funct. Differ. Eq., 11 (2004), 235-265.
[16]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
[17]	M. Bani-Yaghoub, Analysis and applications of delay differential equations in biology and medicine, 2017, arXiv:1701.04173.
[18]	M. Sathish Kumar, S. Janaki, V. Ganesan, Some new oscillatory behavior of certain third-order nonlinear neutral differential equations of mixed type, Int. J. Appl. Comput. Math., 4 (2018), 78. doi: 10.1007/s40819-018-0508-8
[19]	Q. Zhu, H. Wang, Output feedback stabilization of stochastic feed forward systems with unknown control coefficients and unknown output function, Automatica, 87 (2018), 166-175. doi: 10.1016/j.automatica.2017.10.004
[20]	Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Syst. Control Lett., 118 (2018), 62-68. doi: 10.1016/j.sysconle.2018.05.015
[21]	Q. Zhu, Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142. doi: 10.1016/j.jmaa.2014.02.016
[22]	R. P. Agarwal, S. R. Grace, D. O' Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht, Boston, London, 2000.
[23]	S. Pinelas, E. Thandapani, S. Padmavathi, Oscillation criteria for even-order nonlinear neutral differential equations of mixed type, Bull. Math. Anal. Appl., 6 (2014), 9-22.
[24]	T. Candan, Oscillation criteria and asymptotic properties of solutions of third-order nonlinear neutral differential equations, Math. Method. Appl. Sci., 38 (2015), 1379-1392. doi: 10.1002/mma.3153
[25]	T. Li, Comparison theorems for second-order neutral differential equations of mixed type, Electron. J. Differ. Eq., 2010 (2010), 1-7.
[26]	T. Li, B. Baculíková, J. Džurina, Oscillation results for second-order neutral differential equations of mixed type, Tatra Mt. Math. Publ., 48 (2011), 101-116.
[27]	T. Li, B. Baculíková, J. Džurina, Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments, Bound. Value Probl., 2014 (2014), 1-15. doi: 10.1186/1687-2770-2014-1
[28]	T. Li, Y. V. Rogovchenko, Asymptotic behavior of higher-order quasilinear neutral differential equations, Abstr. Appl. Anal., 2014 (2014), 1-11.
[29]	V. Ganesan, M. Sathish Kumar, Oscillation of certain third order nonlinear differential equation with neutral terms, Bangmod Int. J. Math. Comput. Sci., 3 (2017), 53-60.
[30]	V. Ganesan, M. Sathish Kumar, On the oscillation of a third order nonlinear differential equations with neutral type, Ural Math. J., 3 (2017), 122-129. doi: 10.15826/umj.2017.2.013
[31]	V. Ganesan, M. Sathish Kumar, S. Janaki, et al. Nonlinear oscillation of certain third-order neutral differential equation with distributed delay, J. Mahani Math. Res. Cent., 7 (2018), 1-12.
[32]	Y. Jiang, C. Jiang, T. Li, Oscillatory behavior of third-order nonlinear neutral delay differential equations, Adv. Differ. Equ., 2016 (2016), 1-12. doi: 10.1186/s13662-015-0739-5
[33]	Y. Qi, J. Yu, Oscillation of second order nonlinear mixed neutral differential equations with distributed deviating arguments, Bull. Malays. Math. Sci. Soc., 38 (2015), 543-560. doi: 10.1007/s40840-014-0035-7
[34]	Z. Han, T. Li, C. Zhang, et al. Oscillatory behavior of solutions of certain third-order mixed neutral functional differential equations, Bull. Malays. Math. Sci. Soc., 35 (2012), 611-620.

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