Perturbed Operators on Banach Spaces

TL;DR

This paper establishes conditions under which weakly coercive perturbed operators on Banach spaces are diffeomorphisms, using Fredholm theory, inverse mapping, and fixed point theorems, with specific corollaries and applications.

Contribution

It provides new sufficient conditions for perturbed operators to be C^1-diffeomorphisms on Banach spaces, including special cases like linear and contractive perturbations.

Findings

Perturbed operators are diffeomorphisms under certain conditions.

Corollaries for linear, contractive, and Hilbert space cases.

Applications demonstrate the theoretical results.

Abstract

Let X be a Banach Space over K=R or C, and let f:=F+C be a weakly coercive operator from X onto X, where F is a C^1-operator, and C a C^1 compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C^1-diffeomorphism. Three corollaries are given. The first one, when F is a linear homeomorphism. The second one, when F is a k-contractive perturbation of the identity. The third one, when X is a Hilbert space and F a particular linear operator. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems, and the Banach fixed point theorem. As an application two examples are given