| Title:
|
On transformations of functional-differential equations (English) |
| Author:
|
Čermák, Jan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
29 |
| Issue:
|
2 |
| Year:
|
1993 |
| Pages:
|
227-234 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper contains applications of Schrőder’s equation to differential equations with a deviating argument. There are derived conditions under which a considered equation with a deviating argument intersecting the identity $y=x$ can be transformed into an equation with a deviation of the form $\tau (x)=\lambda x$. Moreover, if the investigated equation is linear and homogeneous, we introduce a special form for such an equation. This special form may serve as a canonical form suitable for the investigation of oscillatory and asymptotic properties of the considered equation. (English) |
| Keyword:
|
Functional-differential equation |
| Keyword:
|
singular case |
| Keyword:
|
transformation |
| Keyword:
|
canonical form |
| MSC:
|
34K05 |
| MSC:
|
34K99 |
| idZBL:
|
Zbl 0802.34079 |
| idMR:
|
MR1263124 |
| . |
| Date available:
|
2008-06-06T21:25:04Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107485 |
| . |
| Reference:
|
[1] Eĺsgolc, L. E.: Vvedenije v teoriju differencialnych uravnenij s otklonjajuščimsa argumentom.Nauka, Moscow, 1964. (Russian) MR 0170049 |
| Reference:
|
[2] Heard, M. L.: Asymptotic behavior of solutions of the functional differential equation $x^{\prime }(t)=ax(t)+bx(t^{\alpha }), \alpha >1$.J.Math.Anal.Appl. 44 (1973), 745–757. Zbl 0289.34115, MR 0333405 |
| Reference:
|
[3] Kato, T., McLeod, J. B.: The functional-differential equation $y^{\prime }(x)=ay(\lambda x)+by(x)$.Bull. Amer. Math. Soc. 77 (1971), 891–937. MR 0283338 |
| Reference:
|
[4] Kuczma, M.: Functional Equations in a Single Variable.Polish Scient.Publ., Warszawa, 1968. Zbl 0196.16403, MR 0228862 |
| Reference:
|
[5] Lade, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments.Marcel Dekker, New York, 1983. |
| Reference:
|
[6] Lim, E.-B.: Asymptotic behavior of solutions of the functional differential equation $x^{\prime }(t)=Ax(\lambda t)+Bx(t), \lambda >0$.J.Math.Anal.Appl. 55 (1976), 794–806. Zbl 0336.34060, MR 0447749 |
| Reference:
|
[7] Neuman, F.: On transformations of differential equations and systems with deviating argument.Czechoslovak Math.J. 31(106) (1981), 87-90. Zbl 0463.34051, MR 0604115 |
| Reference:
|
[8] Neuman, F.: Transformation and canonical forms of functional-differential equation.Proc. Roy.Soc.Edinburgh 115A (1990), 349-357. MR 1069527 |
| Reference:
|
[9] Pandofi, L.: Some observations on the asymptotic behaviors of the solutions of the equation $x^{\prime }(t)=A(t)x(\lambda t)+B(t)x(t), \lambda >0$.J.Math.Anal.Appl. 67 (1979), 483–489. MR 0528702 |
| Reference:
|
[10] Szekeres, G.: Regular iteration of real and complex functions.Acta Math. 100 (1958), 203–258. Zbl 0145.07903, MR 0107016 |
| Reference:
|
[11] Tryhuk, V.: The most general transformation of homogeneous retarded linear differential equations of the $n$-th order.Math.Slovaka 33 (1983), 15–21. Zbl 0514.34058, MR 0689272 |
| . |
