Local Theory of Complex Functional Differential Equations

We consider the equation <!-- MATH $( ^\ast )f'(z) = F(z,f(z),f(g(z)))$ --> where and are given analytic functions and is an unknown function. The question of local existence of a solution of about a point is natural only if <!-- MATH $g({z_0}) = {z_0}$ --> . We classify fixed points of g as attractive if <!-- MATH $| {g'({z_0})} | < 1$ --> <img width="103" height="41" align="MIDDLE" border="0" src="images/img9.gif" alt="$ \vert {g'({z_0})} \vert < 1$">, indifferent if <!-- MATH $| {g'({z_0})} | = 1$ --> , and repulsive if <!-- MATH $| {g'({z_0})} | > 1$ --> 1$">. In the attractive case has a unique analytic solution satisfying an initial condition <!-- MATH $f({z_0}) = {w_0}$ --> . This solution depends continuously on 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.