A new generalization of Browder degree

TL;DR

This paper introduces a new, simplified generalization of Browder's degree for certain mappings, facilitating easier calculations and demonstrated through an application in liquid crystal phase transitions.

Contribution

It presents a novel generalization of Browder's degree for $(S)_+$ mappings, improving computational simplicity and applicability in nonlinear operator theory.

Findings

Degree remains unchanged for specific mappings in reflexive Banach spaces

Simplifies calculations of the Browder degree

Successfully applied to phase transition problems in liquid crystals

Abstract

A new generalization of the Browder's degree for the mappings of the type $(S)_+$ is presented. The main idea is rooted in the observation that the Browder's degree remains unchanged for the mappings of the form $A: Y\to X^*$, where $Y$ is a reflexive uniformly convex Banach space continuously embedded in the Banach space $X$. The advantage of the suggested degree lies in the simplicity it provides for the calculations of the degree associated with nonlinear operators. An application from the theory of phase transition in liquid crystals is presented for which the suggested degree has been successfully applied.