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Some Differential Equations Related to Iteration Theory

Published online by Cambridge University Press:  20 November 2018

J´nos Aczél  and
Detlef Gronau
J´nos Aczél
Affiliation:
University of Waterloo, Waterloo, Ontario
Detlef Gronau
Affiliation:
Universität Graz, Graz, Austria
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In connection with the translation equation

(T)

three differential equations arise together with a differential initial condition. They are satisfied by the differentiable solutions of (T) and of the initial condition

(I)

These equations are attributed in [9] to E. Jabotinsky who seems to have been the first who treated these equations in connection with the theory of analytic iteration (see [6], cf. [7, 8], but see also [1, 2, 3]).

Gronau (see [9]) asked whether, conversely, it is true that all solutions of each of these “Jabotinsky differential equations”, possibly with some further initial conditions added, are also solutions of the translation equation. In this paper we give counter examples but also partial positive answers to these questions. First we show how one obtains the Jabotinsky equations from (T) and (I).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 695 - 717
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Aczél, J., Einige aus Funktionalgleichungen zweier Verànderlichen ableitbare Dijferentialgleichungen, Acta Sci. Math. (Szeged) 13 (1949), 179–189.Google Scholar
2. Aczél, J., Losung der Vektor-Funktionalgleichung der homogenen und inhomogenen ndimensionalen einparametrigen “Translation”, der erzeugenden Funktion von Kettenreaktionen und des stationàren und nichtstationàren B ew e gun gs integrals, Acta Math. Acad. Sci. Hung. 6 (1955), 131–141.Google Scholar
3. Aczél, J., Lectures on functional equations and their applications (Academic Press, New York-London, 1966).Google Scholar
4. Cartan, H., Differential calculus (Hermann, Paris/Houghton, Mifflin, Boston, 1971).Google Scholar
5. Courant, R. and John, F., Introduction to calculus and analysis (John Wiley & Sons, New York, 1974).Google Scholar
6. Jabotinsky, E., Analytic iteration, Trans. Amer. Math. Soc. 108 (1963), 457–477.Google Scholar
7. Reich, L., On a differential equation arising in iteration theory in rings of formal power series in one variable, Lecture Notes in Math. 1163 (Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986)., 135–148.Google Scholar
8. Reich, L., Holomorphe Losungen der Differentialgleichung von E. Jabotinsky, Ôsterr. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 195 (1986), 157–166.Google Scholar
9. Targonski, Gy., New directions and open problems in iteration theory, Ber. Math.-Stat. Sek. Forschungsz-Graz. 229 (1984).Google Scholar