### Oscillation Results for Second Order Nonlinear Differential Equation with Delay and Advanced Arguments

Thandapani, Ethiraju;Selvarangam, Srinivasan;Vijaya, Murugesan;Rama, Renu

• 투고 : 2014.04.11
• 심사 : 2015.11.03
• 발행 : 2016.03.23
• 5 4

#### 초록

In this paper we study the oscillation criteria for the second order nonlinear differential equation with delay and advanced arguments of the form $$([x(t)+a(t)x(t-{\sigma}_1)+b(t)x(t+{\sigma}_2)]^{\alpha})^{{\prime}{\prime}}+q(t)x^{\beta}(t-{\tau}_1)+q(t)x^{\gamma}(t+{\tau}_2)=0,\;t{\geq}t_0$$ where ${\sigma}_1$, ${\sigma}_2$, ${\tau}_1$ and ${\tau}_2$ are nonnegative constants and ${\alpha}$, ${\beta}$ and ${\gamma}$ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

#### 키워드

Oscillation;Second order;Nonlinear;Differential equation;Delay and advanced argument

#### 참고문헌

1. R. P. Agarwal, S. L. Shieh, and C. C. Yeh, Oscillation criteria for second-order retarded differential equations, Math.Comput. Modelling, 26(4)(1997), 1-11.
2. B. Baculikova, Oscillation criteria for second order nonlinear differential equations, Arch. Math., 42(2)(2006), 141-149.
3. J. G. Dong, Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Comput. Math. Appl., 59(12)(2010), 3710-3717. https://doi.org/10.1016/j.camwa.2010.04.004
4. J. Dzurina, On the second order functional differential equations with advanced and retarded arguments, Nonlinear Times Digest, 1(1994), 179-187.
5. J. Dzurina and S. Kulasar, Oscillation criteria for second order neutral functional differential equations, Publ. Math. Debrecen., 59(1-2)(2001), 25-33.
6. J. Dzurina, J. Busa and E. A. Airyan, Oscillation criteria for second-order differential equations of neutral type with mixed arguments, Differ. Equ., 38(2002), 137-140. https://doi.org/10.1023/A:1014872030186
7. J. Dzurina and I. P. Stavouralakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140(2-3)(2003), 445-453. https://doi.org/10.1016/S0096-3003(02)00243-6
8. J. Dzurina and D. Hudakova, Oscillation of second order neutral delay differential equations, Mathematica Bohemica, 134 (1)(2009), 31-38.
9. L. H. Erbe and Q. Kong, Oscillation results for second order neutral differential equations, Funkcial. Ekvac., 35(3)(1992), 545-555.
10. S. R. Grace, Oscillation criteria for n-th order neutral functional differential equations, J. Math, Anal. Appl., 184(1994), 44-55. https://doi.org/10.1006/jmaa.1994.1182
11. S. R. Grace, On the oscillations of mixed neutral equations, J. Math. Anal. Appl., 194(2)(1995), 377-388. https://doi.org/10.1006/jmaa.1995.1306
12. S. R. Grace, Oscillations of mixed neurtal functional-differential equations, Appl. Math. Comput., 68(1)(1995), 1-13. https://doi.org/10.1016/0096-3003(94)00075-F
13. I. Gyori and G. Ladas, Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
14. Z. Han, T. Li, S. Sun, and W. Chen, On the oscillation of second-order neutral delay differential equations, Adv. Difference Equ., 2010, Article ID 289340, 8 pages, 2010.
15. G. S. Ladde, V. Lakshimikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, 110 , Marcel Dekker, New York, 1987.
16. L. Liu and Y. Bai, New oscillation criteria for second-order nonlinear neutral delay differential equations, J. Comput. Appl. Math., 231(2)(2009), 657-663. https://doi.org/10.1016/j.cam.2009.04.009
17. Ch. G. Philos, Oscillation theorems for linear differential equations of second order, Archiv der Mathematik, 53(5)(1989), 482-492. https://doi.org/10.1007/BF01324723
18. S. H. Saker, Oscillation of second order neutral delay differential equations of Emden-Fowler type, Acta Math. Hungar, 100(1-2)(2003), 37-62. https://doi.org/10.1023/A:1024699900047
19. S. Sun, T. Li, Z. Han, and Y. sun, Oscillation of second-order neutral functional differential equations with mixed nonlinearities, Abstract and Applied Analysis, 2011, Article ID 927690, 15 pages, 2011.
20. Shuhong Tang, Cunchen Gao, E. Thandapani, and Tongxing Li, Oscillation theorem for second order netural differential equations of mixed type, Far East J. Math. Sci., 2011.
21. E. Thandapani and R. Rama, Comparision and oscillation theorems for second oredr nonlinear netural differential equation, Serdica Math. J., 39(2013), 1-16.
22. R. Xu and F. Meng, Oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. Math.Comput.,192(1)(2007), 216-222. https://doi.org/10.1016/j.amc.2007.01.108
23. J. R. Yan, Oscillation of higher order neutral differential equations of mixed type, Isreal J. Math., 115(2000), 125-136. https://doi.org/10.1007/BF02810583