Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Hubert Cremer, Über die Iteration rationaler Funktionen, Jber. Deutsch. Math.-Verein. 33 (1925), 185-210.
  • [2] P. Flamant, Sur une équation différentielle fonctionelle linéaire, Rend. Circ. Mat. Palermo 48 (1924), 135-208.
  • [3] S. Izumi, On the theory of linear functional differential equations, Tôhoku Math. J. 30 (1929), 10-18.
  • [4] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [5] A. F. Leont′ev, Differential-difference equations, Mat. Sbornik N.S. 24(66) (1949), 347–374 (Russian). MR 0031181
  • [6] Lewis-Bayard Robinson, Une pseudo-fonction et l’équation d’Izumi, Bull. Soc. Math. France 64 (1936), 66–70 (French). MR 1505047
  • [7] L. B. Robinson, A functional equation with negative exponent, Revista Ci., Lima 48 (1946), 101–107. MR 0021222
  • [8] -, On the equation of Izumi having a singular solution holomorphic except at the origin and a lacunary general solution, Tôhoku Math. J. 43 (1937), 310-313.
  • [9] Lewis Bayard Robinson, Introduction to a study of a type of functional differential and functional integral equations, Math. Mag. 23 (1950), 183–188. MR 0038548,
  • [10] Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 0007044,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34.75

Retrieve articles in all journals with MSC: 34.75

Additional Information

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

American Mathematical Society