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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local theory of complex functional differential equations


Author: Robert J. Oberg
Journal: Trans. Amer. Math. Soc. 161 (1971), 269-281
MSC: Primary 34.75
MathSciNet review: 0282026
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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 $ \vert {g'({z_0})} \vert < 1$, indifferent if $ \vert {g'({z_0})} \vert = 1$, and repulsive if $ \vert {g'({z_0})} \vert > 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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0282026-0
PII: S 0002-9947(1971)0282026-0
Keywords: Functional differential equations, local existence theory, perturbation of solutions, differential equations over the complex plane, differential-difference equations
Article copyright: © Copyright 1971 American Mathematical Society