On the structure of the solution set of nonlinear equations

Abstract

1. LET E, F be Banach spaces, UC E be open and bounded and f: ir-, F be a continuous map such that f(x) f 0 for every x E aU. We say that f is 0-epi (zero-epi) if for every continuous and compact map h: ir ---, F such that h(x) = 0 for every x E aU, the nonlinear operator equation f(x) = h(x) (1.1) has a solution x E U. The notion of 0-epi maps was introduced in [6] where the main properties of 0-epi maps were also established and they were used to obtain existence results for nonlinear operator equations of the form