Local theory of complex functional differential equations

Oberg, Robert J.

doi:10.1090/S0002-9947-1971-0282026-0

Local theory of complex functional differential equations
HTML articles powered by AMS MathViewer

by Robert J. Oberg PDF

Trans. Amer. Math. Soc. 161 (1971), 269-281 Request permission

Abstract:

We consider the equation $( ^\ast )f’(z) = F(z,f(z),f(g(z)))$ where $F(z,u,w)$ and $g(z)$ are given analytic functions and $f(z)$ is an unknown function. The question of local existence of a solution of $( ^\ast )$ about a point ${z_0}$ is natural only if $g({z_0}) = {z_0}$. We classify fixed points ${z_0}$ of g as attractive if $| {g’({z_0})} | < 1$, indifferent if $| {g’({z_0})} | = 1$, and repulsive if $| {g’({z_0})} | > 1$. In the attractive case $( ^\ast )$ has a unique analytic solution satisfying an initial condition $f({z_0}) = {w_0}$. This solution depends continuously on ${w_0}$ and on the functions F and g. For “most” indifferent fixed points the initial-value problem has a unique solution. Around a repulsive fixed point a solution in general does not exist, though in exceptional cases there may exist a singular solution which disappears if the equation is subjected to a suitable small perturbation.

References

Über die Iteration rationaler Funktionen

33

Sur une équation différentielle fonctionelle linéaire

48

On the theory of linear functional differential equations

30

Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
A. F. Leont′ev, Differential-difference equations, Mat. Sbornik N.S. 24(66) (1949), 347–374 (Russian). MR 0031181
Lewis-Bayard Robinson, Une pseudo-fonction et l’équation d’Izumi, Bull. Soc. Math. France 64 (1936), 66–70 (French). MR 1505047, DOI 10.24033/bsmf.1242
L. B. Robinson, A functional equation with negative exponent, Rev. Ci. (Lima) 48 (1946), 101–107. MR 21222

On the equation of Izumi having a singular solution holomorphic except at the origin and a lacunary general solution

43

Lewis Bayard Robinson, Introduction to a study of a type of functional differential and functional integral equations, Math. Mag. 23 (1950), 183–188. MR 38548, DOI 10.2307/3029826
Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 7044, DOI 10.2307/1968952