Some results on the invertibility of nonlinear operators

The objective of this paper is to present certain conditions guaranteeing i nvertibility of a nonlinear operator between normed linear spaces. The idea is to approximate the given operator by an invertible, possibly linear ope rator, and reduce the problem to the contraction mapping principle. Several theorems of this kind are given, which appear as generalizations of some e arly results by I. W. Sandberg, and estimates for an "approximate inverse" are established. Finally, introducing certain "invertibility indices," furt her sufficient and necessary conditions for invertibility are given.