Sufficient oscillation conditions involving lim sup and lim inf for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.
[1] E. Braverman, G.E. Chatzarakis, I.P. Stavroulakis: Iterative oscillation tests for differential equations with several non-monotone arguments. Adv. Difference Equ. 87 (2016).
[2] E. Braverman, B. Karpuz: On oscillation of differential and difference equations with non-monotone delays. Appl. Math. Comput. 218 (7) (2011) 3880–3887.
[3] G.E. Chatzarakis: Differential equations with non-monotone arguments: Iterative Oscillation results. J. Math. Comput. Sci. 6 (5) (2016) 953–964.
[4] G.E. Chatzarakis: On oscillation of differential equations with non-monotone deviating arguments. Mediterr. J. Math. 14 (2) (2017) 82.
[5] G.E. Chatzarakis, I. Jadlovská: Improved iterative oscillation tests for firs-order deviating differential equations. Opuscula Math. 38 (3) (2018) 327–356.
[6] G.E. Chatzarakis, T. Li: Oscillation criteria for delay and advanced differential equations with non-monotone arguments. Complexity 2018 (2018) 1–18. Article ID 8237634.
[7] G.E. Chatzarakis, Ö. Öcalan: Oscillations of differential equations with several non-monotone advanced arguments. Dynamical Systems 30 (3) (2015) 310–323.
[8] L.H. Erbe, Qingkai Kong, B.G. Zhang: Oscillation Theory for Functional Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, 190. Marcel Dekker, Inc., New York (1995).
[9] L.H. Erbe, B.G. Zhang: Oscillation of first order linear differential equations with deviating arguments. Differential Integral Equations 1 (3) (1988) 305–314.
[10] N. Fukagai, T. Kusano: Oscillation theory of first order functional-differential equations with deviating arguments. Ann. Mat. Pura Appl. 136 (1) (1984) 95–117.
[11] J. Jaroš, I.P. Stavroulakis: Oscillation tests for delay equations. Rocky Mountain J. Math. 29 (1) (1999) 197–207.
[12] C. Jian: On the oscillation of linear differential equations with deviating arguments. Math. in Practice and Theory 1 (1) (1991) 32–40.
[13] M. Kon, Y.G. Sficas, I.P. Stavroulakis: Oscillation criteria for delay equations. Proc. Amer. Math. Soc. 128 (10) (2000) 2989–2998.
[14] R.G. Koplatadze, T.A. Chanturija: Oscillating and monotone solutions of first-order differential equations with deviating argument. Differentsial’nye Uravneniya 18 (8) (1982) 1463–1465. (in Russian)
[15] R.G. Koplatadze, G. Kvinikadze: On the oscillation of solutions of first order delay differential inequalities and equations. Georgian Math. J. 1 (6) (1994) 675–685.
[16] M.K. Kwong: Oscillation of first-order delay equations. J. Math. Anal. Appl. 156 (1) (1991) 274–286.
[17] G. Ladas, V. Lakshmikantham, L.S. Papadakis: Oscillations of higher-order retarded differential equations generated by the retarded arguments. In: Klaus Schmitt: Delay and functional differential equations and their applications. Academic Press (1972) 219–231.
[18] G.S. Ladde: Oscillations caused by retarded perturbations of first order linear ordinary differential equations. Atti Acad. Naz. Lincei Rendiconti 63 (5) (1977) 351–359.
[19] G.S. Ladde, V. Lakshmikantham, B.G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Monographs and Textbooks in Pure and Applied Mathematics, 110, Marcel Dekker, Inc., New York (1987).
[20] X. Li, D. Zhu: Oscillation and nonoscillation of advanced differential equations with variable coefficients. J. Math. Anal. Appl. 269 (2) (2002) 462–488.
[21] 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.
[22] A.D. Myškis: Linear homogeneous differential equations of first order with deviating arguments. Uspekhi Mat. Nauk 5 (36) (1950) 160–162. (in Russian)
[23] J.S. Yu, Z.C. Wang, B.G. Zhang, X.Z. Qian: Oscillations of differential equations with deviating arguments. Panamer. Math. J. 2 (2) (1992) 59–78.
[24] B.G. Zhang: Oscillation of solutions of the first-order advanced type differential equations. Science Exploration 2 (1982) 79–82.
[25] D. Zhou: On some problems on oscillation of functional differential equations of first order. J. Shandong University 25 (1990) 434–442.