| Title:
|
Oscillation of deviating differential equations (English) |
| Author:
|
Chatzarakis, George E. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
435-448 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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$$ \limsup \limits _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>1\quad \biggl (\limsup \limits _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>1\bigg ) $$ and $$ \liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>\frac {1}{\rm e}\quad \biggl (\liminf _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>\frac {1}{\rm e}\bigg ) $$ 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. (English) |
| Keyword:
|
differential equation |
| Keyword:
|
non-monotone argument |
| Keyword:
|
oscillatory solution |
| Keyword:
|
nonoscillatory solution |
| Keyword:
|
Grönwall inequality |
| MSC:
|
34K06 |
| MSC:
|
34K11 |
| idZBL:
|
07286023 |
| idMR:
|
MR4221844 |
| DOI:
|
10.21136/MB.2020.0002-19 |
| . |
| Date available:
|
2020-11-18T09:58:36Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148434 |
| . |
| Reference:
|
[1] Braverman, E., Karpuz, B.: On oscillation of differential and difference equations with non-monotone delays.Appl. Math. Comput. 218 (2011), 3880-3887. Zbl 1256.39013, MR 2851485, 10.1016/j.amc.2011.09.035 |
| Reference:
|
[2] Chatzarakis, G. E.: Differential equations with non-monotone arguments: Iterative oscillation results.J. Math. Comput. Sci. 6 (2016), 953-964. |
| Reference:
|
[3] Chatzarakis, G. E.: On oscillation of differential equations with non-monotone deviating arguments.Mediterr. J. Math. 14 (2017), Paper No. 82, 17 pages. Zbl 1369.34088, MR 3620160, 10.1007/s00009-017-0883-0 |
| Reference:
|
[4] Chatzarakis, G. E., Jadlovská, I.: Improved iterative oscillation tests for first-order deviating differential equations.Opusc. Math. 38 (2018), 327-356. Zbl 1405.34056, MR 3781617, 10.7494/OpMath.2018.38.3.327 |
| Reference:
|
[5] Chatzarakis, G. E., Jadlovská, I.: Oscillations in differential equations caused by non-monotone arguments.(to appear) in Nonlinear Stud. MR 4159431 |
| Reference:
|
[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. Zbl 1407.34045, 10.1155/2018/8237634 |
| Reference:
|
[7] Chatzarakis, G. E., Ocalan, " O. ": Oscillations of differential equations with several non-monotone advanced arguments.Dyn. Syst. 30 (2015), 310-323. Zbl 1330.34107, MR 3373715, 10.1080/14689367.2015.1036007 |
| Reference:
|
[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. Zbl 1331.34132, MR 3434455, 10.1016/j.aml.2015.10.014 |
| Reference:
|
[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). Zbl 0821.34067, MR 1309905 |
| Reference:
|
[10] Erbe, L. H., Zhang, B. G.: Oscillation for first order linear differential equations with deviating arguments.Differ. Integral Equ. 1 (1988), 305-314. Zbl 0723.34055, MR 929918 |
| Reference:
|
[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. Zbl 0552.34062, MR 765918, 10.1007/BF01773379 |
| Reference:
|
[12] Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations. With Applications.Clarendon Press, Oxford (1991). Zbl 0780.34048, MR 1168471 |
| Reference:
|
[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. Zbl 0496.34044, MR 0671174 |
| Reference:
|
[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. Zbl 0810.34068, MR 1296574, 10.1007/BF02254685 |
| Reference:
|
[15] Kwong, M. K.: Oscillation of first-order delay equations.J. Math. Anal. Appl. 156 (1991), 274-286. Zbl 0727.34064, MR 1102611, 10.1016/0022-247X(91)90396-H |
| Reference:
|
[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. Zbl 0273.34052, MR 0387776, 10.1016/B978-0-12-627250-5.50013-7 |
| Reference:
|
[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. Zbl 0402.34058, MR 0548601 |
| Reference:
|
[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). Zbl 0832.34071, MR 1017244 |
| Reference:
|
[19] Li, X., Zhu, D.: Oscillation and nonoscillation of advanced differential equations with variable coefficients.J. Math. Anal. Appl. 269 (2002), 462-488. Zbl 1013.34067, MR 1907126, 10.1016/S0022-247X(02)00029-X |
| Reference:
|
[20] Myshkis, A. D.: Linear homogeneous differential equations of the first order with deviating arguments.Usp. Mat. Nauk 5 (1950), 160-162 Russian. Zbl 0041.42108, MR 0036423 |
| Reference:
|
[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. Zbl 0845.34082, MR 1160129 |
| Reference:
|
[22] Zhang, B. G.: Oscillation of solutions of the first-order advanced type differential equations.Sci. Exploration 2 (1982), 79-82. MR 713776 |
| Reference:
|
[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. Zbl 0726.34060 |
| . |
