Oscillation Test for Linear Delay Differential Equation with Nonmonotone Argument

In this article, we analyze the first order linear delay differential equation x 0 (t) + p(t)x (τ(t)) = 0, t ≥ t0, where p, τ ∈ C ([t0, ∞),R + ) and τ(t) ≤ t, limt→∞ τ(t) = ∞. Under the assumption that τ(t) is not necessarily monotone, we obtain new sufficient criterion for the oscillatory solutions of this equation. We also give an example illustrating the result.

PDF

___

[1] Akca, H., Chatzarakis, G.E., Stavroulakis, I.P., An oscillation criterion for delay differential equations with several non-monotone arguments, Applied Mathematics Letters, 59(2016), 101–108.
[2] Chatzarakis, G.E., Peics, H., Differential equations with several non-monotone arguments: An oscillation result, Applied Mathematics Letters, 68(2017), 20–26.
[3] Chao, J., On the oscillation of linear differential equations with deviating arguments, Math. in Practice and Theory, 1(1991), 32-40.
[4] Elbert, A., Stavroulakis, I.P., Oscillations of first order differential equations with deviating arguments, University of Ioannina, T.R. No 172 1990, Recent trends in differential equations, 163-178, World Sci. Ser. Appl. Anal., 1, World Sci. Publishing Co., 1992.
[5] Erbe, L.H., Zhang, B.G., Oscillation of first order linear differential equations with deviating arguments, Differ. Integral Equ., 1(1988), 305–314.
[6] Erbe, L.H., Kong, Q., Zhang, B.G., Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
[7] Fukagai, N., Kusano, T., Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl., 136(1984), 95-117.
[8] Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
[9] Jaros, J., Stavroulakis, I.P., Oscillation tests for delay equations, Rocky Mountain J. Math., 29(1999), 139–145.
[10] Kon, M., Sficas, Y.G., Stavroulakis, I.P., Oscillation criteria for delay equations, Proc. Amer. Math. Soc., 128(2000), 2989–2997.
[11] Koplatadze, R.G., Chanturija, T.A., Oscillating and monotone solutions of first-order differential equations with deviating arguments, (Russian), Differentsial’nye Uravneniya, 8(1982), 1463-1465.
[12] Koplatadze, R., Kvinikadze, G., On the oscillation of solutions of first order delay diferential inequalities and equations, Georgian Mathematical Journal, 1(6)(1994), 675–685.
[13] Kwong, M.K., Oscillation of first-order delay equations, J. Math. Anal. Appl., 156(1991), 274–286.
[14] Ladde, G.S., Lakshmikantham, V., Zhang, B.G., Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987.
[15] Philos, Ch.G., Sficas, Y.G., An oscillation criterion for first order linear delay differential equations, Canad. Math. Bull. 41(1998), 207-213.
[16] Yu, J.S., Wang, Z.C., Some further results on oscillation of neutral differential equations, Bull. Aust. Math. Soc., 46(1992), 149–157.
[17] Yu, J.S.,Wang, Z.C., Zhang, B.G., Qian, X.Z., Oscillations of differential equations with deviating arguments, PanAmerican Math. J., 2(1992), 59–78.