Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.