Oscillatory Behavior for Certain Theorems and Examples of Higher order Nonlinear Delay Differential Equations

Abstract

In this paper the oscillatory behaviour of higher order nonlinear delay differential equation theorems and examples are investigated. Some new oscillatory main results of higher order nonlinear delay differential equations are given. We discuss the relation of Riccati transformation of the nonlinear delay differential equation to studying properties of the two higher order differential equations. Furthermore, an average integrating method is introduced as a asymptotic approach to study the oscillatory behavior. Some results are extended to nonlinear delay differential equations of any order. An example is also discussed, to illustrate the efficiency of the results obtained.

Keywords

• oscillatory,higher order nonlinear delay differential equations

References

1. [1] A. Skerlik, Oscillation theorems for third order nonlinear differential equations, Math. Slovaca, 4 (1992), 471–484.
2. [2] A. Tiryaki, M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54–68.
3. [3] B. Bacul´ıkov´a, J. Dˇzurina, Oscillation of third-order nonlinear differential equations, Applied Mathematics Letters 24 (2011), 466–470.
4. [4] B.Bacul´ıkov´a, J. Dˇzurina, Oscillation theorems for higher order neutral differential equations, Applied Mathematics and Computation 219 (2012), 3769–3778.
5. [5] C. Fabry, J.Mawhin, M. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull London Math soc. 18 (1986), 173-180.
6. [6] C. Hou, S. Cheng, Asymptotic Dichotomy in a Class of Fourth-Order Nonlinear Delay Differential Equations with Damping, Abstract and Applied Analysis, 20 (2009), 1–7.
7. [7] Ch. G. Philo, Oscillation theorems for linear differential equation of second order, Arch. Math., 53 (1989), 482–492.
8. [8] G. Ladas, Oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. Differential Equations, 10 (1971), 281–290.

9. [9] G.S. Ladde,V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.
10. [10] I. T. Kiguradze, B. Puza, On periodic solutions of system of differential equations with deviating arguments, Nonlinear Anal., 42 (2000), 229-242.
11. [11] J. H. Shen, The nonoscillatory solutions of delay differential equations with impulses, Appl. Math. comput. 77 (1996), 153-165.
12. [12] J. V. MANOJLOVIC, Oscillation Criteria for Second-Order Half-Linear Differential Equations, Mathematical and Computer Modelling, 30 (1999), 109-119.
13. [13] J. Y. Patricia Wong, Ravi P. Agarwal, Oscillatory Behavior of Solutions of Certain Second Order Nonlinear Differential Equations, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 198 (1996), 337-354.
14. [14] Jiaowan Luo and Lokenath Debnath, Oscillations of Second-Order Nonlinear Ordinary Differential Equations with Impulses, Journal of Mathematical Analysis and Applications, 240 (1) (1999), 105-114.
15. [15] K. Gopalsamy, B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139 (1989), 110-122.
16. [16] Balamuralitharan S, Periodic solutions of fourth-order delay differential equation, Bulletin of the Iranian Mathematical Society, Vol. 41 (2015), No. 2,307 - 314
17. [17] L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1994.
18. [18] Lijun Pan, Periodic solutions for higher order differential equations with deviating argument, Journal of Mathematical Analysis and Applications, 343 (2), (2008), 904-918.
19. [19] M. F. Aktas, A. Tiryaki, A. Zafer, Integral criteria for oscillation of third order nonlinear differential equations, Nonlinear Anal., 71 (2009), 1496-1502.
20. [20] S. Balamuralitharan, PERIODIC SOLUTIONS FOR THIRD ORDER DELAY DIFFERENTIAL EQUATION IMPULSES WITH FREDHOLM OPERATOR OF INDEX ZERO, Konuralp Journal of Mathematics, Volume 4 No. 2 pp. 158-168 (2016).
21. [21] M.F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Appl. Math. Lett. 23 (2010), 756–762.
22. [22] N. Parhi, S. K. Nayak, Nonoscillation of second-order nonhomogeneous differential equations, J. Math. Anal. Appl., 102 (1984), 62–74.
23. [23] O. Doˇsl`y, A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, Hiroshima Math. J., 36 (2006), 203–219.
24. [24] P. Hartman, On nonoscillatory linear differential equations of second order, Amer. J. Math., 74 (1952), 389–400.
25. [25] PAUL WALTMAN, An Oscillation Criterion for a Nonlinear Second Order Equation, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 10 (1965), 439-441.
26. [26] Balamuralitharan S, PERIODIC SOLUTIONS FOR THIRD AND FOURTH ORDER DELAY DIFFERENTIAL EQUATION IMPULSES WITH FREDHOLM OPERATOR OF INDEX ZERO, Differential Equations and Control Processes, N 3, page no: 17-37, 2015.
27. [27] Quanxin Zhanga, Li Gaoa, Yuanhong Yub, Oscillation criteria for third-order neutral differential equations with continuously distributed delay, Applied Mathematics Letters, 25 (2012), 1514-1519.
28. [28] R. P. Agarwal, M. F. Aktasm A. Tiryaki, On oscillation criteria for third order nonlinear delay differential equations, Arch. Math., 45 (2009), 1–18.
29. [29] D.Seethalakshmi and S.Balamuralitharan, Blow-up Global Solutions of Third Order Differential Equation, International Journal of Pure and Applied Mathematics, Volume 113, No. 11, 2017, 105 -113 .
30. [30] R.P. Agarwal, S. R. Grace, D. O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Acad. Publ,Drdrechet, 2000.
31. [31] R. P. Agarwal, S. R. Grace, P. J. Y. Wong, Oscillation theorems for certain higher order nonlinear functional differential equations, Appl. Anal. Disc. Math., 2 (2008), 1–30.
32. [32] R. P. AGARWAL, SHIOW-LING SHIEH AND CHEH-CHIH YEH, Oscillation Criteria for Second-Order Retarded Differential Equations, Mathl. Comput. Modelling, 26(4)(1997), l-11.
33. [33] S. H. Saker, Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca, 56 (2006), 433-450.
34. [34] S. Lu, W. Ge, Sufficient conditions for the existence of periodic solutions to some second order differential equation with a deviating argument, J. Math. Anal. Appl. 308 (2005), 393-419.
35. [35] S. R. GRACE, Oscillation Theorems for Nonlinear Differential Equations of Second Order, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 171 (1992), 220-241.
36. [36] S.H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, Journal of Computational and Applied Mathematics, 177 (2005), 375-387.
37. [37] S.R. Grace, R.P. Agarwal, M.F. Aktas, On the oscillation of third order functional differential equations, Indian J. Pure Appl. Math. 39 (2008), 491–507.
38. [38] V.Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Scientific Singapore, 1989.
39. [39] W. E. Mahfoud, Oscillatory and asymptotic behavior of solutions of N-th order nonlinear delay differential equations, J. Differential Equations, 24 (1977), 75–98.
40. [40] Zhimin He and Weigao Ge, Oscillations of second-order nonlinear impulsive ordinary differential equations, Journal of Computational and Applied Mathematics, 158 (2) (2003), 397-406.
41. [41] S. Balamuralitharan, Majorant Cauchy Problem of a Priori Inequality with Nonlinear fractional Differential Equations, Mathematical Sciences Letters, 6, No. 3, 299-303 (2017).