MR3468732 G. INFANTE, R. KOPLATADZE and I. P. STAVROULAKIS G. INFANTE, R. KOPLATADZE and I. P. STAVROULAKIS Oscillation Criteria for Differential Equations with Several Retarded Arguments Funkcialaj Ekvacioj. Serio Internacia 58 2015 347--364 http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-3/58_347.pdf http://www.ams.org/mathscinet-getitem?mr=MR3468732 Consider the first-order linear differential equation with several retarded arguments $x'(t)+\sum\limits_{i=1}^{m}p_i(t)x(\tau_i(t))=0$, $t\geq t_0$, where the functions $p_i$, $\tau_i\in C([t_0,\infty),\mathbb{R}^+)$, for every $i=1,2,\ldots,m$, $\tau_i(t)\leq t$ for $t\geq t_0$ and $\lim_{t\rightarrow\infty}\tau_i(t)=\infty$. In this paper the state-of-the-art on the oscillation of all solutions to these equations is reviewed and new sufficient conditions for the oscillation are established, especially in the case of nonmonotone arguments. Examples illustrating the results are given. Oscillation, Retarded, Differential equations, Non-monotone arguments. Primary 34K11; Secondary 34K06. 58-347 2015 Oscillation Criteria for Differential Equations with Several Retarded Arguments G. INFANTE, R. KOPLATADZE and I. P. STAVROULAKIS G. INFANTE, R. KOPLATADZE and I. P. STAVROULAKIS
1 Arino, O.; Gyori, I.; Jawhari, A. Oscillation criteria in delay equations J. Differential Equations 53 1984 115-123 MR0747409
2 Berezansky, L.; Braverman, E. On some constants for oscillation and stability of delay equations Proc. Amer. Math. Soc. 139 2011 4017-4026 MR2823047
3 Braverman, E.; Karpuz, B. On oscillation of differential and difference equations with non-monotone delays Appl. Math. Comput. 218 2011 3880-3887 MR2851485
4 Elbert, A.; Stavroulakis, I. P. Oscillations of first order differential equations with deviating arguments Recent trends in differential equations, World Sci. Ser. Appl. Anal., 1, World Sci. Publ., River Edge, NJ 1992 pp. 163-178 MR1180110
5 Elbert, A.; Stavroulakis, I. P. Oscillation and nonoscillation criteria for delay differential equations Proc. Amer. Math. Soc. 123 1995 1503-1510 MR1242082
6 Erbe, L. H.; Kong, Qingkai; Zhang, B. G. Oscillation Theory for Functional Differential Equations Monographs and Textbooks in Pure and Applied Mathematics, 190, Marcel Dekker, Inc., New York 1995 MR1309905
7 Erbe, L. H.; Zhang, B. G. Oscillation for first order linear differential equations with deviating arguments Differential Integral Equations 1 1988 305-314 MR0929918
8 Fukagai, N.; Kusano, T. Oscillation theory of first order functional differential equations with deviating arguments Ann. Mat. Pura Appl. (4) 136 1984 95-117 MR0765918
9 Gopalsamy, K. Stability and Oscillations in Delay Differential Equations of Population Dynamics Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht 1992 MR1163190
10 Grammatikopoulos, M. K.; Koplatadze, R. G.; Stavroulakis, I. P. On the oscillation of solutions of first order differential equations with retarded arguments Georgian Math. J. 10 2003 63-76 MR1990688
11 Gyori, I.; Ladas, G. Oscillation Theory of Delay Differential Equations with Applications Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York 1991 MR1168471 12 Hale, J. K. Theory of Functional Differential Equations Second edition. Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg 1977 MR0508721
13 Hunt, B. R.; Yorke, J. A. When all solutions of $x^{\prime }=\sum q_i(t)x(t-T_i(t))$ oscillate J. Differential Equations 53 1984 139-145 MR0748236
14 Jaroš, J.; Stavroulakis, I. P. Oscillation tests for delay equations Rocky Mountain J. Math. 29 1999 139-145 MR1687662
15 Jian, C. Oscillation of linear differential equations with deviating argument Math. Practice Theory 1 1991 32-41 (in Chinese) MR1107456
16 Kon, M.; Sficas, Y. G.; Stavroulakis, I. P. Oscillation criteria for delay equations Proc. Amer. Math. Soc. 128 2000 2989-2997 MR1694869
17 Koplatadze, R. G.; Chanturija, T. A. On the oscillatory and monotonic solutions of first order differential equations with deviating arguments Differentsial'nye Uravneniya 18 1982 1463-1465 MR0671174
18 Koplatadze, R. G.; Kvinikadze, G. On the oscillation of solutions of first order delay differential inequalities and equations Georgian Math. J. 1 1994 675-685 MR1296574
19 Kwong, M. K. Oscillation of first-order delay equations J. Math. Anal. Appl. 156 1991 274-286 MR1102611
20 Ladas, G. Sharp conditions for oscillations caused by delay Applicable Anal. 9 1979 93-98 MR0539534
21 Ladas, G.; Laskhmikantham, V.; Papadakis, J. S. Oscillations of higher-order retarded differential equations generated by retarded arguments Delay and Functional Differential Equations and Their Applications, Academic Press, New York 1972 pp. 219-231 MR0387776
22 Ladas, G.; Stavroulakis, I. P. Oscillations caused by several retarded and advanced arguments J. Differential Equations 44 1982 134-152 MR0651691
23 Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G. Oscillation Theory of Differential Equations with Deviating Arguments Monographs and Textbooks in Pure and Applied Mathematics, 110. Marcel Dekker, Inc., New York 1987 MR1017244
24 Li, B. Oscillations of first order delay differential equations Proc. Amer. Math. Soc. 124 1996 3729-3737 MR1363175
25 Myshkis, A. D. Linear homogeneous differential equations of first order with deviating arguments Uspehi Matem. Nauk (N.S.) 5 1950 160-162 (Russian) MR0036423
26 Sficas, Y. G.; Stavroulakis, I. P. Oscillation criteria for first-order delay equations Bull. London Math. Soc. 35 2003 239-246 MR1952402
27 Wang, Z. C.; Stavroulakis, I. P.; Qian, X. Z. A Survey on the oscillation of solutions of first order linear differential equations with deviating arguments Appl. Math. E-Notes 2 2002 171-191 MR1979424
28 Yu, J. S.; Wang, Z. C.; Zhang, B. G.; Qian, X. Z. Oscillations of differential equations with deviating arguments Panamer. Math. J. 2 1992 59-78 MR1160129
29 Zhou, Y.; Yu, Y. H. On the oscillation of solutions of first order differential equations with deviating arguments Acta Math. Appl. Sinica (English Ser.) 15 1999 297-302 MR1714967