Title: Oscillation of Nonlinear Delay Differential Equation with Non-Monotone Arguments
Author(s): Özkan Öcalan, Nurten Kilic, Sermin Sahin, Umut Mutlu Ozkan
Pages: 147-154
Cite as:
Özkan Öcalan, Nurten Kilic, Sermin Sahin, Umut Mutlu Ozkan, Oscillation of Nonlinear Delay Differential Equation with Non-Monotone Arguments, Int. J. Anal. Appl., 14 (2) (2017), 147-154.

Abstract


Consider the first-order nonlinear retarded differential equation

$$

x^{\prime }(t)+p(t)f\left( x\left( \tau (t)\right) \right) =0, t\geq t_{0}

$$

where $p(t)$ and $\tau (t)$ are function of positive real numbers such that $%\tau (t)\leq t$ for$\ t\geq t_{0},\ $and$\ \lim_{t\rightarrow \infty }\tau(t)=\infty $. Under the assumption that the retarded argument is non-monotone, new oscillation results are given. An example illustrating the result is also given.

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References


  1. O. Arino, I. Gy˝ ori and A. Jawhari, Oscillation criteria in delay equations, J. Differential Equations 53 (1984), 115-123. Google Scholar

  2. L. Berezansky and E. Braverman, On some constants for oscillation and stability of delay equations, Proc. Amer. Math. Soc. 139 (11) (2011), 4017-4026. Google Scholar

  3. E. Braverman, B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput. 218 (2011) 3880-3887. Google Scholar

  4. George E. Chatzarakis and Ozkan Ocalan, Oscillations of differential equations with non-monotone retarded arguments, LMS J. Comput. Math., 19 (1) (2016) 98–104. Google Scholar

  5. A. Elbert and I. P. Stavroulakis, Oscillations of first order differential equations with deviating arguments, Univ 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). Google Scholar

  6. A. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510. Google Scholar

  7. L. H. Erbe, Qingkai Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. Google Scholar

  8. L. H. Erbe and B. G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential Integral Equations, 1 (1988), 305-314. Google Scholar

  9. N. Fukagai and T. Kusano, Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl.,136 (1984), 95-117. Google Scholar

  10. K.Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992. Google Scholar

  11. M. K. Grammatikopoulos, R. G. Koplatadze and I. P. Stavroulakis, On the oscillation of solutions of first order differential equations with retarded arguments, Georgian Math. J., 10 (2003), 63-76. Google Scholar

  12. I. Gy˝ ori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. Google Scholar

  13. B. R. Hunt and J. A. Yorke, When all solutions of $x^{prime}=dsum q_{i}(t)x(t-T_{i}(t))$ oscillate, J. Differential Equations 53 (1984), 139-145. Google Scholar

  14. R. G. Koplatadze and T. A. Chanturija, Oscillating and monotone solutions of first-order differential equations with deviating arguments, (Russian), Differentsial’nye Uravneniya, 8 (1982), 1463-1465. Google Scholar

  15. G. Ladas, Sharp conditions for oscillations caused by delay, Applicable Anal., 9 (1979), 93-98. Google Scholar

  16. G. Ladas, V. Laskhmikantham and J.S. Papadakis, Oscillations of higher-order retarded differential equations generated by retarded arguments, Delay and Functional Differential Equations and Their Applications, Academic Press, New York, 1972, 219–231. Google Scholar

  17. G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987. Google Scholar

  18. A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk, 5 (1950), 160-162 (Russian). Google Scholar

  19. X.H. Tang, Oscillation of first order delay differential equations with distributed delay, J. Math. Anal. Appl. 289 (2004), 367-378. Google Scholar