On Zeros of Homotopic Mappings, Fixed Point Problems, and Inverse Functions

TL;DR

This paper investigates conditions under which smooth mappings between Banach spaces have zeros, with applications to fixed-point problems and the Implicit Function Theorem, advancing understanding of inverse function existence.

Contribution

It introduces new conditions for the existence of zeros in homotopic mappings between Banach spaces, linking fixed-point problems and inverse function theory.

Findings

Established criteria for zeros of smooth Banach space mappings

Connected fixed-point problems with inverse function conditions

Extended Implicit Function Theorem applications

Abstract

This article examines a family of smooth mappings between Banach spaces and establishes conditions for the existence of their zeros. Applications to fixed-point problems and the Implicit Function Theorem are also discussed.