Introduction

Abstract

When dealing with the system of ODE in a complex domain one $$ \frac{{d{y^j}}}{{dt}} = {f^j}\left( {x,{y^1}, \ldots ,{y^p}} \right),j = 1, \ldots ,p $$ ((1.1.1)) assumes that f j are holomorphic (i.e. single-valued analytic) in some domain G and asks for analytic solution y l(x),..., y p(x). The local existence theorem (for brevity, we include here uniqueness and analytic dependence on initial data and parameters (if there are any)) looks like the corresponding theorem in a real domain. The most well-known proof of the latter is obtained by rewriting the system as a system of integral equations which is solved using iterations. A careful analysis of this proof reveals that it works in a complex domain as well. (We work in a small disk on the x-plane containing the initial value x 0 of the independent variable. Integration is performed along linear segments connecting x 0 to the “current” x. These integrals are estimated literally in the same way as in the real case). At some points it even becomes simpler. If a sequence of analytic functions converges uniformly in a domain on the x-plane, then its limit is an analytic function. In fact, one does not even need to assume a priori that the convergence is uniform; but in our case the proof of the convergence is based on the estimates which imply the uniform convergence in a sufficiently small disk. (For brevity, we say “function” instead of “a system of p functions”; one can also have in mind “a vector function”). So (1.1.1) has an analytic solution. Now, as regards to the uniqueness, it is easy to see (differentiating (1.1.1)) that two solutions having the same initial data must have the same derivatives of all orders. Being analytic, they must coincide. As regards to the dependence on initial data and parameters, we have an uniformly convergent sequence of functions holomorphic with respect to x, these data and parameters, so the limit is also holomorphic with respect to them (we again refer to the analyticity of the limit functions, only this time we deal with a function of several complex variables). Compare this easy argument with the situation in the real domain. In the latter case the un iform convergence does not imply any smoothness of the limit function. As regards to its dependence in x, smoothness follows immediately from the integral equation, so this is also easy, but as regards to the dependence on the initial data and parameters, one needs some extra considerations. (It is true that one can avoid them by using a simplest version of the implicit function theorem in Banach spaces, but it is not so popular).