| Title:
|
On the oscillation of third-order quasi-linear neutral functional differential equations (English) |
| Author:
|
Thandapani, E. |
| Author:
|
Li, Tongxing |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
181-199 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation
\begin{equation*} \big [a(t)\big ([x(t)+p(t)x(\delta (t))]^{\prime \prime }\big )^\alpha \big ]^{\prime }+q(t)x^\alpha (\tau (t))=0\,, E \end{equation*}
where $\alpha >0$, $0\le p(t)\le p_0<\infty $ and $\delta (t)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results. (English) |
| Keyword:
|
third-order |
| Keyword:
|
neutral functional differential equations |
| Keyword:
|
oscillation and asymptotic behavior |
| MSC:
|
34C10 |
| MSC:
|
34K11 |
| idZBL:
|
Zbl 1249.34194 |
| idMR:
|
MR2852380 |
| . |
| Date available:
|
2011-11-11T08:51:14Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141706 |
| . |
| Reference:
|
[1] Baculíková, B., Džurina, J.: Oscillation of third-order functional differential equations.Electron. J. Qual. Theory Differ. Equ. 43 (2010), 1–10. Zbl 1211.34077, MR 2678385 |
| Reference:
|
[2] Baculíková, B., Džurina, J.: Oscillation of third-order neutral differential equations.Math. Comput. Modelling 52 (2010), 215–226. Zbl 1201.34097, MR 2645933, 10.1016/j.mcm.2010.02.011 |
| Reference:
|
[3] Baculíková, B., Džurina, J.: Oscillation of third-order nonlinear differential equations.Appl. Math. Lett. 24 (2011), 466–470. Zbl 1209.34042, MR 2749728, 10.1016/j.aml.2010.10.043 |
| Reference:
|
[4] Baculíková, B., Elabbasy, E. M., Saker, S. H., Džurina, J.: Oscillation criteria for third-order nonlinear differential equations.Math. Slovaca 58 (2008), 1–20. Zbl 1174.34052, MR 2391214, 10.2478/s12175-008-0068-1 |
| Reference:
|
[5] Erbe, L.: Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations.Pacific J. Math. 64 (1976), 369–385. Zbl 0339.34030, MR 0435508, 10.2140/pjm.1976.64.369 |
| Reference:
|
[6] Grace, S. R., Agarwal, R. P., Pavani, R., Thandapani, E.: On the oscillation of certain thir d order nonlinear functional differential equations.Appl. Math. Comput. 202 (2008), 102–112. MR 2437140, 10.1016/j.amc.2008.01.025 |
| Reference:
|
[7] Han, Z., Li, T., Zhang, C., Sun, S.: An oscillation criterion for third order neutral delay differential equations.J. Appl. Anal., to appear. MR 2740504 |
| Reference:
|
[8] Hanan, M.: Oscillation criteria for third order differential equations.Pacific J. Math. 11 (1961), 919–944. MR 0145160, 10.2140/pjm.1961.11.919 |
| Reference:
|
[9] Hartman, P., Winter, A.: Linear differential and difference equations with monotone solutions.Amer. J. Math. 75 (1953), 731–743. MR 0057404, 10.2307/2372548 |
| Reference:
|
[10] Karpuz, B., Öcalan, Ö., Öztürk, S.: Comparison theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations.Glasgow Math. J. 52 (2010), 107–114. MR 2587820, 10.1017/S0017089509990188 |
| Reference:
|
[11] Philos, Ch. G.: Oscillation theorems for linear differential equations of second order.Arch. Math. 53 (1989), 482–492. Zbl 0661.34030, MR 1019162, 10.1007/BF01324723 |
| Reference:
|
[12] Saker, S. H., Džurina, J.: On the oscillation of certain class of third-order nonlinear delay differential equations.Math. Bohem. 135 (2010), 225–237. Zbl 1224.34217, MR 2683636 |
| Reference:
|
[13] Zhong, J., Ouyang, Z., Zou, S.: Oscillation criteria for a class of third-order nonlinear neutral differential equations.J. Appl. Anal. (2010), to appear. |
| . |
