Oscillation Test for Linear Delay Differential Equation with Nonmonotone Argument

Abstract

In this article, we analyze the first order linear delay differential equation \begin{equation*} x^{\prime }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\geq t_{0}, \end{equation*} where $p,$ $\tau \in C\left( [t_{0},\infty ),\mathbb{R}^{+}\right) $ and $% \tau (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. Under the assumption that $\tau (t)$ is not necessarily monotone, we obtain new sufficient criterion for the oscillatory solutions of this equation. We also give an example illustrating the result.

Keywords

• Delay equation
• nonmonotone argument
• oscillatory solution
• nonoscillatory solution

References

1. [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. [2] Chatzarakis, G.E., Peics, H., Differential equations with several non-monotone arguments: An oscillation result, Applied Mathematics Letters, 68(2017), 20–26.
3. [3] Chao, J., On the oscillation of linear differential equations with deviating arguments, Math. in Practice and Theory, 1(1991), 32-40.
4. [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. [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. [6] Erbe, L.H., Kong, Q., Zhang, B.G., Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
7. [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. [8] Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.

9. [9] Jaros, J., Stavroulakis, I.P., Oscillation tests for delay equations, Rocky Mountain J. Math., 29(1999), 139–145.
10. [10] Kon, M., Sficas, Y.G., Stavroulakis, I.P., Oscillation criteria for delay equations, Proc. Amer. Math. Soc., 128(2000), 2989–2997.
11. [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. [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. [13] Kwong, M.K., Oscillation of first-order delay equations, J. Math. Anal. Appl., 156(1991), 274–286.
14. [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. [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. [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. [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.