Abstract
The paper deals with the oscillation and asymptotic behaviour of a third order nonlinear differential equation with piecewise constant arguments. We give some conditions that guarantee the oscillation and asymptotic behaviour of the solutions of this equation.
Similar content being viewed by others
References
Kim, W.J.: Oscillatory Properties of Linear Third-Order Differential Equations. Proc. Am. Math. Soc. 26, 286–293 (1970)
Tryhuk, V.: An Oscillation Criterion for Third Order Linear Differential Equations. Arch. Math. (Brno) 11(2), 99–104 (1975)
Cecchi, M.: Oscillation Criteria for a Class of Third Order Linear Differential Equations. Boll. Un. Mat. Ital. C 2(1), 297–306 (1983)
Parhi, N., Das, P.: Oscillation and Nonoscillation of Nonhomogeneous Third Order Differential Equations. Czechoslov. Math. J. 44(3), 443–459 (1994)
Parhi, N., Das, P.: On the Oscillation of a Class of Linear Homogeneous Third Order Differential Equations. Arch. Math. (Brno) 34(4), 435–443 (1998)
Dahiya, R.S.: Oscillation of the Third Order Differential Equations with and without Delay. Differential Equations and Nonlinear Mechanics (Orlando, FL, 1999), Math. Appl., vol. 528, pp. 75–87. Kluwer Academic Publisher, Dordrecht (2001)
Adamets, L., Lomtatidze, A.: Oscillation Conditions for a Third-Order Linear Equation. Differ. Uravn. 37(6), 723–729 (2001) (Russian; translation in Differ. Equations 37(6), 755–762)
Han, Z., Li, T., Sun, S., Zhang, C.: An Oscillation Criteria for Third Order Neutral Delay Differential Equations. J. Appl. Anal. 16(2), 295–303 (2010)
Das, P., Pati, J.K.: Necessary and Sufficient Conditions for the Oscillation a Third-Order Differential Equation. Electron. J. Differ. Equations 2010(174), 10 (2010)
Džurina, J., Baculíková, B.: Property (B) and Oscillation of Third-Order Differential Equations with Mixed Arguments. J. Appl. Anal. 19(1), 55–68 (2013)
Bartušek, M., Došlá, Z.: Oscillation of Third Order Differential Equation with Damping Term. Czechoslov. Math. J. 65(2), 301–316 (2015)
Shoukaku, Y.: Oscillation Criteria for Third Order Differential Equations with Functional Arguments. Math. Slovaca 65(5), 1035–1048 (2015)
Ezeilo, J.O.C.: On the Existence of Periodic Solutions of a Certain Third-Order Differential Equation. Proc. Camb. Philos. Soc. 56, 381–389 (1960)
Tabueva, V.A.: Conditions for the Existence of a Periodic Solution of a Third-Order Differential Equation. Prikl. Mat. Meh. 25, 961–962 (1961) (Russian; translated as J. Appl. Math. Mech. 25, 1445–1448)
Baculíková, B., Elabbasy, E., Saker, S., Džurina, J.: Oscillation Criteria for Third-Order Nonlinear Differential Equations. Math. Slovaca 58(2), 201–220 (2008)
Aktaş, M.F., Tiryaki, A., Zafer, A.: Oscillation Criteria for Third-Order Nonlinear Functional Differential Equations. Appl. Math. Lett. 23(7), 756–762 (2010)
Hovhannisyan, G.: On Oscillations of Solutions of Third-Order Dynamic Equation. Abstract and Applied Analysis, vol. 2012. Hindawi Publishing Corporation, Cairo (2012)
Graef, J.R., Saker, H.S.: Oscillation Theory of Third-Order Nonlinear Functional Differential Equations. Hiroshima Math. J. 43(1), 49–72 (2013)
Grace, S.R., Agarwal, R.P., Pavani, R., Thandapani, E.: On the Oscillation of Certain Third Order Nonlinear Functional Differential Equations. Appl. Math. Comput. 202(1), 102–112 (2008)
Baculíková, B.: Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments. Abstract and Applied Analysis, vol. 2011. Hindawi Publishing Corporation, Cairo (2011)
Agarwal, R.P., Baculíková, B., Džurina, J., Li, T.: Oscillation of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments. Acta Mathe. Hung. 134(1–2), 54–67 (2012)
Džurina, J., Baculíková, B.: Oscillation of Third-Order Differential Equations with Mixed Arguments. Differential and Difference Equations with Applications, Springer Proc. Math. Stat., vol. 47, pp. 375–385. Springer, New York (2013)
Shoukaku, Y.: Oscillation Criteria for Third Order Differential Equations with Functional Arguments. Math. Slovaca 65(5), 1035–1048 (2015)
Cooke, K.L., Wiener, J.: Retarded Differential Equations with Piecewise Constant Delays. J. Math. Anal. Appl. 99, 265–297 (1984)
Shah, S.M., Wiener, J.: Advanced Differential Equations with Piecewise Constant Argument Deviations. Int. J. Math. Math. Sci. 6, 243–270 (1983)
Busenberg, S., Cooke, K.L.: Models of Vertically Transmitted Diseases with Sequential-Continuous Dynamics. Nonlinear Phenom. Math. Sci., 179–197 (1982)
Zheng Rong, L., Yuan Hong, Y.: Oscillation and Nonoscillation for a Class of Nonlinear Differential Equations (Chinese). Beijing Daxue Xuebao Ziran Kexue Ban 32(1), 34–39 (1996)
Shao, Y., Liang, H.: Oscillation Criteria of Certain Even Order Nonlinear Differential Equation with Piecewise Constant Argument. Adv. Differ. Equations Control Process. 11(2), 71 (2013)
Wang, G.: Existence Theorem of Periodic Solutions for a Delay Nonlinear Differential Equation with Piecewise Constant Arguments. J. Math. Anal. Appl. 298(1), 298–307 (2004)
Wang, G., Yang, B., Debnath, L.: Periodic Positive Solutions for Adelay Nonlinear Differential Equation with Piecewise Constant Arguments. Appl. Math. Lett. 17(12), 1323–1329 (2004)
Wang, G., Cheng, S.S.: Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument. Int. J. Math. Math. Sci. (2009). doi:10.1155/2009/950797
Alwan, M.S., Liu, X., Xie, W.: Comparison Principle and Stability of Differential Equations with Piecewise Constant Arguments. J. Frankl. Inst. 350(2), 211–230 (2013)
Papaschinopoulos, G., Schinas, J.: Some Results Concerning Second and Third Order Neutral Delay Differential Equations with Piecewise Constant Argument. Czechoslov. Math. J. 44(3), 501–512 (1994)
Liang, H., Wang, G.: Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument. J. Appl, Math (2012)
Shao, Y., Liang, H.: Oscillatory and Asymptotic Behavior for Third Order Differential Equations with Piecewise Constant Argument. Far East J. Dyn. Syst. 21(1), 45 (2013)
Kneser, A.: Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Mathe. Ann. 42(3), 409–435 (1893)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bereketoglu, H., Lafci, M. & Oztepe, G.S. On the Oscillation of a Third Order Nonlinear Differential Equation with Piecewise Constant Arguments. Mediterr. J. Math. 14, 123 (2017). https://doi.org/10.1007/s00009-017-0923-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-0923-9