×

Analytic solutions of a second-order nonautonomous iterative functional differential equation. (English) Zbl 1083.34060

The second-order iterative functional-differential equation \[ x''(t)=f\left[\sum_{j=0}^{m}c_jx^{j}(z)\right]+G(z)\tag{1} \] is considered, where \(f\) and \(G\) are given analytic functions on the complex domain \(| z | <\sigma, x^{j}(z)=x(x^{j-1})(z).\) By the Schröder transformation \(x(z)=y(\alpha y^{-1}(z))\), equation (1) is reduced to an auxiliary equation in which the iterates of the unknown function are not involved and the existence of its local analytic solutions are proved for various cases with respect to \(\alpha.\) By this way, theorems on the existence of analytic solutions are proved for equation (1).

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bélair, J., Population models with state-dependent delays, (), 165-176 · Zbl 0749.92014
[2] Bélair, J.; Mackey, C.M., Consumer memory and price fluctuations on commodity markets: an integrodifferential model, J. dynamics differential equations, 1, 299-325, (1989) · Zbl 0682.34050
[3] Bessis, D.; Marmi, S.; Turchetti, G., On the singularities of divergent majorant series arising from normal form theory, Rend. mat. ser. VII, 9, 645-659, (1989) · Zbl 0723.34037
[4] Carleson, L.; Gamelin, T.W., Complex dynamics, (1993), Springer-Verlag New York · Zbl 0782.30022
[5] Driver, D., Existence theory for a delay-differential system, Contrib. differential equations, 1, 317-336, (1963)
[6] Driver, D., A two-body problem of classical electrodynamics: the one-dimensional case, Ann. phys., 21, 122-142, (1963) · Zbl 0108.40705
[7] Driver, D., A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, (), 474-484
[8] Driver, D.; Norris, J.M., Note on uniqueness for a one-dimensional two body problem of classical electrodynamics, Ann. phys., 42, 347-351, (1967)
[9] Kuczma, M., Functional equations in a single variable, Monografie mat., vol. 46, (1968), Polish Scientific Publishers Warszawa · Zbl 0196.16403
[10] Mackey, C.M.; Milton, J., Feedback delays and the origin of blood cell dynamics, Comm. theoret. biol., 1, 299-327, (1990)
[11] Eder, E., The functional differential equation \(x^\prime(t) = x(x(t))\), J. differential equations, 54, 390-400, (1984) · Zbl 0497.34050
[12] Wang, K., On the equation \(x^\prime(t) = f(x(x(t)))\), Funkcial. ekvac., 33, 405-425, (1990) · Zbl 0714.34026
[13] Si, J.G.; Cheng, S.S., Smooth solutions of a nonhomogeneous iterative functional differential equation, Proc. roy. soc. Edinburgh sect. A, 128, 821-831, (1998) · Zbl 0912.34057
[14] Cheng, S.S.; Si, J.G.; Wang, X.P., An existence theorem for iterative functional differential equations, Acta math. hungar., 94, 1-17, (2002) · Zbl 0997.34056
[15] Petahov, V.R., On a boundary value problem, (), 252-255
[16] Si, J.G.; Wang, X.P., Analytic solutions of a second-order iterative functional differential equation, J. comput. appl. math., 126, 277-285, (2000) · Zbl 0983.34056
[17] Si, J.G.; Wang, X.P., Analytic solutions of a second-order functional differential equation with state dependent delay, Results math., 39, 345-352, (2001) · Zbl 1017.34074
[18] Si, J.G.; Wang, X.P., Analytic solutions of a second-order iterative functional differential equation, Comput. math. appl., 43, 81-90, (2002) · Zbl 1008.34059
[19] Siegel, C.L., Vorlesungen über himmelsmechanik, (1956), Springer-Verlag Berlin · Zbl 0098.23601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.