| Title:
|
Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients (English) |
| Author:
|
Tripathy, Arun K. |
| Author:
|
Santra, Shyam S. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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146 |
| Issue:
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2 |
| Year:
|
2021 |
| Pages:
|
185-197 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $$ (r(t)(z'(t))^\gamma )' +\sum _{i=1}^m q_i(t)x^{\alpha _i}(\sigma _i(t))=0, \quad t\geq t_0, $$ where $z(t)=x(t)+p(t)x(\tau (t))$. Under the assumption $\int ^{\infty }(r(\eta ))^{-1/\gamma } {\rm d}\eta =\infty $, we consider two cases when $\gamma >\alpha _i$ and $\gamma <\alpha _i$. Our main tool is Lebesgue's dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem. (English) |
| Keyword:
|
oscillation |
| Keyword:
|
non-oscillation |
| Keyword:
|
neutral |
| Keyword:
|
delay |
| Keyword:
|
Lebesgue's dominated convergence theorem |
| MSC:
|
34C10 |
| MSC:
|
34K11 |
| DOI:
|
10.21136/MB.2020.0063-19 |
| . |
| Date available:
|
2021-05-20T13:54:17Z |
| Last updated:
|
2021-06-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148931 |
| . |
| Reference:
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[1] Agarwal, R. P., Bohner, M., Li, T., Zhang, C.: Oscillation of second-order differential equations with a sublinear neutral term.Carpathian J. Math. 30 (2014), 1-6. Zbl 1324.34078, MR 3244084, 10.37193/CJM.2014.01.01 |
| Reference:
|
[2] Agarwal, R. P., Bohner, M., Li, T., Zhang, C.: Oscillation of second-order Emden-Fowler neutral delay differential equations.Ann. Mat. Pura Appl. (4) 193 (2014), 1861-1875. Zbl 1308.34083, MR 3275266, 10.1007/s10231-013-0361-7 |
| Reference:
|
[3] Agarwal, R. P., Bohner, M., Li, T., Zhang, C.: Even-order half-linear advanced differential equations: Improved criteria in oscillatory and asymptotic properties.Appl. Math. Comput. 266 (2015), 481-490. Zbl 1410.34191, MR 3377575, 10.1016/j.amc.2015.05.008 |
| Reference:
|
[4] Agarwal, R. P., Zhang, C., Li, T.: Some remarks on oscillation of second order neutral differential equations.Appl. Math. Comput. 274 (2016), 178-181. Zbl 1410.34192, MR 3433126, 10.1016/j.amc.2015.10.089 |
| Reference:
|
[5] Baculíková, B., Džurina, J.: Oscillation theorems for second order neutral differential equations.Comput. Math. Appl. 61 (2011), 94-99. Zbl 1207.34081, MR 2739438, 10.1016/j.camwa.2010.10.035 |
| Reference:
|
[6] Baculíková, B., Džurina, J.: Oscillation theorems for second-order nonlinear neutral differential equations.Comput. Math. Appl. 62 (2011), 4472-4478. Zbl 1236.34092, MR 2855589, 10.1016/j.camwa.2011.10.024 |
| Reference:
|
[7] Baculíková, B., Li, T., Džurina, J.: Oscillation theorems for second order neutral differential equations.Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Article ID 74, 13 pages. Zbl 1340.34238, MR 2838502, 10.14232/ejqtde.2011.1.74 |
| Reference:
|
[8] Bohner, M., Grace, S. R., Jadlovská, I.: Oscillation criteria for second-order neutral delay differential equations.Electron. J. Qual. Theory Differ. Equ. 2017 (2017), Article ID 60, 12 pages. Zbl 1413.34213, MR 3690229, 10.14232/ejqtde.2017.1.60 |
| Reference:
|
[9] Brands, J. J. A. M.: Oscillation theorems for second-order functional differential equations.J. Math. Anal. Appl. 63 (1978), 54-64. Zbl 0384.34049, MR 0470416, 10.1016/0022-247X(78)90104-X |
| Reference:
|
[10] Chatzarakis, G. E., Džurina, J., Jadlovská, I.: New oscillation criteria for second-order half-linear advanced differential equations.Appl. Math. Comput. 347 (2019), 404-416. Zbl 1428.34088, MR 3880830, 10.1016/j.amc.2018.10.091 |
| Reference:
|
[11] Chatzarakis, G. E., Grace, S. R., Jadlovská, I., Li, T., Tunç, E.: Oscillation criteria for third-order Emden-Fowler differential equations with unbounded neutral coefficients.Complexity 2019 (2019), Article ID 5691758, 7 pages. Zbl 1429.34071, 10.1155/2019/5691758 |
| Reference:
|
[12] Chatzarakis, G. E., Jadlovská, I.: Improved oscillation results for second-order half-linear delay differential equations.Hacet. J. Math. Stat. 48 (2019), 170-179. MR 3976169, 10.15672/hjms.2017.522 |
| Reference:
|
[13] Džurina, J.: Oscillation theorems for second order advanced neutral differential equations.Tatra Mt. Math. Publ. 48 (2011), 61-71. Zbl 1265.34232, MR 2841106, 10.2478/v10127-011-0006-4 |
| Reference:
|
[14] Džurina, J., Grace, S. R., Jadlovská, I., Li, T.: Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term.Math. Nachr. 293 (2020), 910-922. Zbl 07206438, MR 4100546, 10.1002/mana.201800196 |
| Reference:
|
[15] Grace, S. R., Džurina, J., Jadlovská, I., Li, T.: An improved approach for studying oscillation of second-order neutral delay differential equations.J. Inequal. Appl. 2018 (2018), Article ID 193, 11 pages. MR 3833834, 10.1186/s13660-018-1767-y |
| Reference:
|
[16] Hale, J.: Theory of Functional Differential Equations.Applied Mathematical Sciences 3. Springer, New York (1977). Zbl 0352.34001, MR 0508721, 10.1007/978-1-4612-9892-2_3 |
| Reference:
|
[17] Karpuz, B., Santra, S. S.: Oscillation theorems for second-order nonlinear delay differential equations of neutral type.Hacet. J. Math. Stat 48 (2019), 633-643. MR 3974570, 10.15672/HJMS.2017.542 |
| Reference:
|
[18] Li, H., Zhao, Y., Han, Z.: New oscillation criterion for Emden-Fowler type nonlinear neutral delay differential equations.J. Appl. Math. Comput. 60 (2019), 191-200. Zbl 1422.34194, MR 3969079, 10.1007/s12190-018-1208-6 |
| Reference:
|
[19] Li, Q., Wang, R., Chen, F., Li, T.: Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients.Adv. Difference Equ. 2015 (2015), Article ID 35, 7 pages. Zbl 1350.34053, MR 3304960, 10.1186/s13662-015-0377-y |
| Reference:
|
[20] Li, T., Rogovchenko, Y. V.: Oscillation theorems for second-order nonlinear neutral delay differential equations.Abstr. Appl. Anal. 2014 (2014), Article ID 594190, 5 pages. Zbl 07022675, MR 3226209, 10.1155/2014/594190 |
| Reference:
|
[21] Li, T., Rogovchenko, Y. V.: Oscillation of second-order neutral differential equations.Math. Nachr. 288 (2015), 1150-1162. Zbl 1342.34090, MR 3367905, 10.1002/mana.201300029 |
| Reference:
|
[22] Li, T., Rogovchenko, Y. V.: Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations.Monatsh. Math. 184 (2017), 489-500. Zbl 1386.34118, MR 3709308, 10.1007/s00605-017-1039-9 |
| Reference:
|
[23] Pinelas, S., Santra, S. S.: Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays.J. Fixed Point Theory Appl. 20 (2018), Article ID 27, 13 pages. Zbl 1387.34095, MR 3761383, 10.1007/s11784-018-0506-9 |
| Reference:
|
[24] Qian, Y., Xu, R.: Some new oscillation criteria for higher-order quasi-linear neutral delay differential equations.Differ. Equ. Appl. 3 (2011), 323-335. Zbl 1235.34184, MR 2856412, 10.7153/dea-03-20 |
| Reference:
|
[25] Santra, S. S.: Existence of positive solution and new oscillation criteria for nonlinear first-order neutral delay differential equations.Differ. Equ. Appl. 8 (2016), 33-51. Zbl 1337.34071, MR 3462235, 10.7153/dea-08-03 |
| Reference:
|
[26] Santra, S. S.: Oscillation analysis for nonlinear neutral differential equations of second order with several delays.Mathematica 59 (2017), 111-123. Zbl 07101431, MR 3937284 |
| Reference:
|
[27] Santra, S. S.: Oscillation analysis for nonlinear neutral differential equations of second order with several delays and forcing term.Mathematica 61 (2019), 63-78. Zbl 07101459, MR 3938805, 10.24193/mathcluj.2019.1.06 |
| Reference:
|
[28] Tripathy, A. K., Panda, B., Sethi, A. K.: On oscillatory nonlinear second order neutral delay differential equations.Differ. Equ. Appl. 8 (2016), 247-258. Zbl 1341.34072, MR 3488819, 10.7153/dea-08-12 |
| Reference:
|
[29] Wong, J. S. W.: Necessary and sufficient conditions for oscillation of second order neutral differential equations.J. Math. Anal. Appl. 252 (2000), 342-352. Zbl 0976.34057, MR 1797859, 10.1006/jmaa.2000.7063 |
| Reference:
|
[30] Zhang, C., Agarwal, R. P., Bohner, M., Li, T.: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators.Bull. Malays. Math. Sci. Soc. (2) 38 (2015), 761-778. Zbl 1318.34124, MR 3323739, 10.1007/s40840-014-0048-2 |
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