# zbMATH — the first resource for mathematics

Improved iterative oscillation tests for first-order deviating differential equations. (English) Zbl 1405.34056
This paper is devoted to study the oscillation of first order differential equations with non-monotone arguments of the delay type $x'+p(t)x(\tau(t))=0,$ as well as the advanced type $x'-q(t)x(\sigma(t))=0.$ Improved oscillation criteria for these equations of lim sup and lim inf types are obtained. The authors provide two examples to show the significance and the improvement of these new results over the known ones.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K06 Linear functional-differential equations
Full Text:
##### References:
 [1] E. Braverman, G.E. Chatzarakis, I.P. Stavroulakis, Iterative oscillation tests for dif- ferential equations with several non-monotone arguments, Adv. Difference Equ. 2016, 1–18. · Zbl 1348.34121 [2] E. Braverman, B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput. 218 (2011), 3880–3887. · Zbl 1256.39013 [3] G.E. Chatzarakis, Differential equations with non-monotone arguments: Iterative Oscil- lation results, J. Math. Comput. Sci. 6 (2016) 5, 953–964. [4] G.E. Chatzarakis, On oscillation of differential equations with non-monotone deviating arguments, Mediterr. J. Math. (2017), 14:82. · Zbl 1369.34088 [5] G.E. Chatzarakis, T. Li, Oscillation criteria for delay and advanced differential equations with non-monotone arguments, Complexity (2018), in press. [6] G.E. Chatzarakis, Ö. Öcalan, Oscillations of differential equations with several non-monotone advanced arguments, Dynamical Systems: An International Journal (2015), 1–14. Improved iterative oscillation tests for first-order deviating differential equations355 [7] L.H. Erbe, B.G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential Integral Equations 1 (1988), 305–314. · Zbl 0723.34055 [8] L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equa- tions, Marcel Dekker, New York, 1995. [9] N. Fukagai, T. Kusano, Oscillation theory of first order functional-differential equations with deviating arguments, Ann. Mat. Pura Appl. 136 (1984), 95–117. · Zbl 0552.34062 [10] J. Jaroš, I.P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 45 (2000), 2989–2997. · Zbl 0951.34045 [11] C. Jian, On the oscillation of linear differential equations with deviating arguments, Math. in Practice and Theory 1 (1991), 32–40. [12] M. Kon, Y.G. Sficas, I.P. Stavroulakis, Oscillation criteria for delay equations, Proc. Amer. Math. Soc. 128 (1994), 675–685. [13] R.G. Koplatadze, T.A. Chanturija, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsial’nye Uravneniya 18 (1982), 1463–1465, 1472 [in Russian]. · Zbl 0496.34044 [14] R.G. Koplatadze, G. Kvinikadze, On the oscillation of solutions of first-order delay differential inequalities and equations, Georgian Math. J. 3 (1994), 675–685. · Zbl 0810.34068 [15] Y. Kuang, Delay differential equations with application in population dynamics, Academic Press, Boston, 1993. · Zbl 0777.34002 [16] T. Kusano, On even-order functional-differential equations with advanced and retarded arguments, J. Differential Equations 45 (1982), 75–84. · Zbl 0512.34059 [17] M.K. Kwong, Oscillation of first-order delay equations, J. Math. Anal. Appl. 156 (1991), 274–286. · Zbl 0727.34064 [18] G. Ladas, I.P. Stavroulakis, Oscillations caused by several retarded and advanced argu- ments, J. Differential Equations 44 (1982), 134–152. · Zbl 0452.34058 [19] G. Ladas, V. Lakshmikantham, L.S. Papadakis, Oscillations of higher-order retarded dif- ferential equations generated by the retarded arguments, Delay and Functional Differential Equations and their Applications, Academic Press, New York, 1972, 219–231. [20] G.S. Ladde, Oscillations caused by retarded perturbations of first order linear ordinary differential equations, Atti Acad. Naz. Lincei Rendiconti 63 (1978), 351–359. · Zbl 0402.34058 [21] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equa- tions with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987. · Zbl 0622.34071 [22] X. Li, D. Zhu, Oscillation and nonoscillation of advanced differential equations with variable coefficients, J. Math. Anal. Appl. 269 (2002), 462–488. · Zbl 1013.34067 [23] H.A. El-Morshedy, E.R. Attia, New oscillation criterion for delay differential equations with non-monotone arguments, Appl. Math. Lett. 54 (2016), 54–59. · Zbl 1331.34132 [24] A.D. Myshkis, Lineare Differentialgleichungen mit nacheilendem Argument, Deutscher Verlag. Wiss. Berlin, 1955, Translation of the 1951 Russian edition. 356George E. Chatzarakis and Irena Jadlovská [25] A.D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk 5 (1950), 160–162 [in Russian]. · Zbl 0041.42108 [26] I.P. Stavroulakis, Oscillation criteria for delay and difference equations with non-monotone arguments, Appl. Math. Comput. 226 (2014), 661–672. · Zbl 1354.34120 [27] J.S. Yu, Z.C. Wang, B.G. Zhang, X.Z. Qian, Oscillations of differential equations with deviating arguments, Panamer. Math. J. 2 (1992) 2, 59–78. · Zbl 0845.34082 [28] B.G. Zhang, Oscillation of solutions of the first-order advanced type differential equations, Science Exploration 2 (1982), 79–82. [29] D. Zhou, On some problems on oscillation of functional differential equations of first order, J. Shandong University 25 (1990), 434–442. · Zbl 0726.34060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.