Oscillation of deviating differential equations. (English) Zbl 1499.34344

Chatzarakis, George E.

Math. Bohem. 145, No. 4, 435-448 (2020).

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:

34K11	Oscillation theory of functional-differential equations
34K06	Linear functional-differential equations

Keywords:

differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution; Gronwall inequality

Software:

Matlab

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