Local theory of complex functional differential equations
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- 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
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 161 (1971), 269-281
- MSC: Primary 34.75
- DOI: https://doi.org/10.1090/S0002-9947-1971-0282026-0
- MathSciNet review: 0282026