Local theory of complex functional differential equations

Abstract

We consider the equation ( ∗ ) f ′ ( z ) = F ( z , f ( z ) , f ( g ( z ) ) ) ( ^\ast )f’(z) = F(z,f(z),f(g(z))) where F ( z , u , w ) F(z,u,w) and g ( z ) g(z) are given analytic functions and f ( z ) f(z) is an unknown function. The question of local existence of a solution of ( ∗ ) ( ^\ast ) about a point z 0 {z_0} is natural only if g ( z 0 ) = z 0 g({z_0}) = {z_0} . We classify fixed points z 0 {z_0} of g as attractive if | g ′ ( z 0 ) | > 1 | {g’({z_0})} | > 1 , indifferent if | g ′ ( z 0 ) | = 1 | {g’({z_0})} | = 1 , and repulsive if | g ′ ( z 0 ) | > 1 | {g’({z_0})} | > 1 . In the attractive case ( ∗ ) ( ^\ast ) has a unique analytic solution satisfying an initial condition f ( z 0 ) = w 0 f({z_0}) = {w_0} . This solution depends continuously on w 0 {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.