Ordinary differential equations in the complex domain via Topological TransversaIity

In this paper we examine via Topological Transversality ordinary differential equations in the complex domain. Although initial value problems in the complex domain have been studied widely, very little is known on intervals (domains) of existence of a solution. The results of this paper enable us to read off immediately from the differential equation an interval of existence, although it may not be maximal. Furthermore, we introduce boundary value problems in the complex domain and obtain existence theorems for such problems. Our analysis is based on the notions of an essential map and on a priori bounds on solutions.