| Title:
|
Asymptotic properties of one differential equation with unbounded delay (English) |
| Author:
|
Svoboda, Zdeněk |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
239-248 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the asymptotic behavior of the solutions of a differential equation with unbounded delay. The results presented are based on the first Lyapunov method, which is often used to construct solutions of ordinary differential equations in the form of power series. This technique cannot be applied to delayed equations and hence we express the solution as an asymptotic expansion. The existence of a solution is proved by the retract method. (English) |
| Keyword:
|
asymptotic expansion |
| Keyword:
|
retract method |
| MSC:
|
34A25 |
| MSC:
|
34E05 |
| MSC:
|
34K25 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1265.34274 |
| idMR:
|
MR2978269 |
| DOI:
|
10.21136/MB.2012.142869 |
| . |
| Date available:
|
2012-06-08T10:16:14Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142869 |
| . |
| Reference:
|
[1] Cezari, L.: Asymptotic Behaviour and Stability Problems in Ordinary Differenal Equations.Springer (1959). |
| Reference:
|
[2] Diblík, J.: Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative.English. Russian original Sib. Math. J. 23 654-662 (1983); translation from Sib. Mat. Zh. 23 80-91 (1982). Zbl 0521.34003, MR 0673540, 10.1007/BF00971283 |
| Reference:
|
[3] Diblík, J., Svoboda, Z.: An existence criterion of positive solutions of $p$-type retarded functional differential equations.J. Comput. Appl. Math. 147 (2002), 315-331. Zbl 1019.34072, MR 1933599, 10.1016/S0377-0427(02)00439-9 |
| Reference:
|
[4] Diblík, J., Koksch, N.: Existence of global solutions of delayed differential equations via retract approach.Nonlinear Anal., Theory Methods Appl. 64 (2006), 1153-1170. Zbl 1101.34046, MR 2200483, 10.1016/j.na.2005.06.030 |
| Reference:
|
[5] Lakshmikantham, V., Wen, L., Zhang, B.: Theory of Differential Equations with Unbounded Delay.Kluwer Academic Publishers, Dordrecht (1994). Zbl 0823.34069, MR 1319339 |
| Reference:
|
[6] Svoboda, Z.: Asymptotic behaviour of solutions of a delayed differential equation.Demonstr. Math. (1995), 28 9-18. Zbl 0832.34086, MR 1330074 |
| Reference:
|
[7] Svoboda, Z.: Asymptotic integration of solutions of a delayed differential equation.Sborník VA Brno ada B 1 (1998), 7-18. |
| Reference:
|
[8] Šmarda, Z.: The existence and asymptotic behaviour of solutions of certain class of the integro-differential equations.Arch. Math., Brno 26 (1990), 7-18. Zbl 0728.45005, MR 1188069 |
| . |
