# zbMATH — the first resource for mathematics

Oscillation of deviating differential equations. (English) Zbl 07286023
Summary: Consider the first-order linear delay (advanced) differential equation $x'(t)+p(t)x(\tau(t))=0\quad(x'(t)-q(t)x(\sigma(t))=0),\quad t\geq t_{0},$ where $$p$$ $$(q)$$ is a continuous function of nonnegative real numbers and the argument $$\tau(t)$$ $$(\sigma(t))$$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions $\limsup\limits_{t\rightarrow\infty}\int_{\tau(t)}^{t}p(s)ds>1\quad\biggl(\limsup\limits_{t\rightarrow\infty}\int_{t}^{\sigma(t)}q(s)ds>1\bigg)$ and $\liminf_{t\rightarrow\infty}\int_{\tau(t)}^{t}p(s)ds>\frac{1}{\mathrm{e}}\quad\biggl(\liminf_{t\rightarrow\infty}\int_{t}^{\sigma(t)}q(s)ds>\frac{1}{\mathrm{e}}\bigg)$ are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.
##### MSC:
 34K06 Linear functional-differential equations 34K11 Oscillation theory of functional-differential equations
Matlab
Full Text:
##### References:
 [1] Braverman, E.; Karpuz, B., On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput. 218 (2011), 3880-3887 [2] Chatzarakis, G. E., Differential equations with non-monotone arguments: Iterative oscillation results, J. Math. Comput. Sci. 6 (2016), 953-964 [3] Chatzarakis, G. E., On oscillation of differential equations with non-monotone deviating arguments, Mediterr. J. Math. 14 (2017), Paper No. 82, 17 pages [4] Chatzarakis, G. E.; Jadlovská, I., Improved iterative oscillation tests for first-order deviating differential equations, Opusc. Math. 38 (2018), 327-356 [5] Chatzarakis, G. E.; Jadlovská, I., Oscillations in differential equations caused by non-monotone arguments, (to appear) in Nonlinear Stud [6] Chatzarakis, G. E.; Li, T., Oscillation criteria for delay and advanced differential equations with nonmonotone arguments, Complexity 2018 (2018), Article ID 8237634, 18 pages [7] Chatzarakis, G. E.; Ocalan, “ O. ”, Oscillations of differential equations with several non-monotone advanced arguments, Dyn. Syst. 30 (2015), 310-323 [8] El-Morshedy, H. A.; Attia, E. R., New oscillation criterion for delay differential equations with non-monotone arguments, Appl. Math. Lett. 54 (2016), 54-59 [9] Erbe, L. H.; Kong, Q.; Zhang, B. G., Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics 190. Marcel Dekker, New York (1995) [10] Erbe, L. H.; Zhang, B. G., Oscillation for first order linear differential equations with deviating arguments, Differ. Integral Equ. 1 (1988), 305-314 [11] Fukagai, N.; Kusano, T., Oscillation theory of first order functional-differential equations with deviating arguments, Ann. Mat. Pura Appl., IV. Ser. 136 (1984), 95-117 [12] Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations. With Applications, Clarendon Press, Oxford (1991) [13] Koplatadze, R. G.; Chanturiya, T. A., Oscillating and monotone solutions of first order differential equations with deviating argument, Differ. Uravn. 18 (1982), 1463-1465 Russian [14] 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 [15] Kwong, M. K., Oscillation of first-order delay equations, J. Math. Anal. Appl. 156 (1991), 274-286 [16] Ladas, G.; Lakshmikantham, V.; Papadakis, J. S., Oscillations of higher-order retarded differential equations generated by the retarded arguments, Delay and Functional Differential Equations and Their Applications Academic Press, New York (1972), 219-231 K. Schmitt [17] Ladde, G. S., Oscillations caused by retarded perturbations of first order linear ordinary differential equations, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 63 (1977), 351-359 [18] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G., Oscillation Theory of Differential Equations with Deviating Arguments, Pure and Applied Mathematics 110. Marcel Dekker, New York (1987) [19] Li, X.; Zhu, D., Oscillation and nonoscillation of advanced differential equations with variable coefficients, J. Math. Anal. Appl. 269 (2002), 462-488 [20] Myshkis, A. D., Linear homogeneous differential equations of the first order with deviating arguments, Usp. Mat. Nauk 5 (1950), 160-162 Russian [21] Yu, J. S.; Wang, Z. C.; Zhang, B. G.; Qian, X. Z., Oscillations of differential equations with deviating arguments, Panam. Math. J. 2 (1992), 59-78 [22] Zhang, B. G., Oscillation of solutions of the first-order advanced type differential equations, Sci. Exploration 2 (1982), 79-82 [23] Zhou, D., On some problems on oscillation of functional differential equations of first order, J. Shandong Univ., Nat. Sci. Ed. 25 (1990), 434-442
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.