Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms

Grace, Said R.; Graef, John R.; Li, Tongxing; Tunç, Ercan

doi:10.2478/tmmp-2021-0023

Abstract

The authors examine the oscillation of second-order nonlinear differential equations with mixed nonlinear neutral terms. They present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated by some examples.

References

[1] AGARWAL, R.P.— BOHNER, M.— LI, T.— ZHANG, C.: Oscillation of second-order differential equations with a sublinear neutral term, Carpathian J. Math. 30 (2014), 1–6.10.37193/CJM.2014.01.01
Search in Google Scholar Back to article
[2] AGARWAL, R.P.— GRACE, S.R.—O’REGAN, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 2010.
Search in Google Scholar Back to article
[3] AGARWAL, R.P.—GRACE, S.R.—O’REGAN, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht, 2010.
Search in Google Scholar Back to article
[4] BOHNER, M.—GRACE, S.R.—JADLOVSKÁ, I.: Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 60, 1–12.
Search in Google Scholar Back to article
[5] GRACE, S. R.: Oscillatory behavior of second-order nonlinear differential equations with a nonpositive neutral term, Mediterr. J. Math. 14 (2017), Paper no. 229, 1–12.
Search in Google Scholar Back to article
[6] GRACE, S. R.—GRAEF, J. R.: Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term, Math. Model. Anal. 23 (2018), 217–226.10.3846/mma.2018.014
Search in Google Scholar Back to article
[7] GRAEF, J. R.—GRACE, S. R.—TUNÇ, E.: Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term, Opuscula Math. 39 (2019), 39–47.10.7494/OpMath.2019.39.1.39
Search in Google Scholar Back to article
[8] GRAEF, J. R.—GRACE, S.R.—TUNÇ, E.: Oscillation of even-order nonlinear differential equations with sublinear and superlinear neutral terms, Publ. Math. Debrecen (to appear).
Search in Google Scholar Back to article
[9] GRAEF, J. R.—GRAMMATIKOPOULOS, M. K.—SPIKES, P.W.: On the asymptotic behavior of solutions of a second order nonlinear neutral delay differential equation, J. Math. Anal. Appl. 156 (1991), 23–39.10.1016/0022-247X(91)90379-E
Search in Google Scholar Back to article
[10] GRAEF, J. R.—SPIKES, P. W.: On the oscillation of an nth-order nonlinear neutral delay differential equation, J. Comput. Appl. Math. 41 (1992), 35–40.10.1016/0377-0427(92)90235-P
Search in Google Scholar Back to article
[11] HALE, J. K.: Theory of Functional Differential Equations. Springer, New York, 1977.10.1007/978-1-4612-9892-2
Search in Google Scholar Back to article
[12] HARDY, G. H.—LITTLEWOOD, I.E.—POLYA, G.: Inequalities. Reprint of the 1952 edition, Cambridge University Press, Cambridge, 1988.
Search in Google Scholar Back to article
[13] KOPLATADZE, R.G.—CHANTURIYA, T. A.: Oscillating and monotone solutions of first-order differential equations with deviating argument, Differ. Uravn. 18 (1982), 1463–1465. (In Russian)
Search in Google Scholar Back to article
[14] LADAS, G.—STAVROULAKIS, I. P.: Oscillation caused by several retarded and advanced arguments, J. Differ. Equations 44 (1982), 134–152.10.1016/0022-0396(82)90029-8
Search in Google Scholar Back to article
[15] LI, T.—ROGOVCHENKO, YU. V.: Oscillation of second-order neutral differential equations, Math. Nachr. 288 (2015), 1150–1162.10.1002/mana.201300029
Search in Google Scholar Back to article
[16] LI, T.—ROGOVCHENKO, YU. V.—ZHANG, C.: Oscillation results for second-order nonlinear neutral differential equations, Adv. Differ. Equ. 2013 (2013), Paper no. 336, 13 pp.
Search in Google Scholar Back to article
[17] PHILOS, CH. G.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168–178.10.1007/BF01223686
Search in Google Scholar Back to article
[18] QIN, H.—SHANG, N.—LU, Y.: A note on oscillation criteria of second order nonlinear neutral delay differential equations, Comput. Math. Appl. 56 (2008), 2987–2992.10.1016/j.camwa.2008.09.004
Search in Google Scholar Back to article
[19] TAMILVANAN, S.—THANDAPANI, E.—DŽURINA, J.: Oscillation of second order nonlinear differential equation with sub-linear neutral term, Differ. Equ. Appl. 9 (2017), 29–35.
Search in Google Scholar Back to article
[20] XU, R.—MENG, F.: Some new oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. Math. Comput. 182 (2006), 797–803.10.1016/j.amc.2006.04.042
Search in Google Scholar Back to article
[21] WONG, J. S. W.: Necessary and sufficient conditions for oscillation of second order neutral differential equations, J. Math. Anal. Appl. 252 (2000), 342–352.10.1006/jmaa.2000.7063
Search in Google Scholar Back to article
[22] WU, H.—ERBE, L.—PETERSON, A.: Oscillation of solution to second-order half-linear delay dynamic equations on time scales, Electron. J. Differential Equations 2016 (2016), no. 71, 1–15.
Search in Google Scholar Back to article

Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms

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