Some general fixed-point theorems for nonlinear mappings connected with one Cauchy theorem

TL;DR

This paper introduces a new geometric approach to fixed-point theorems that generalizes classical results like Brouwer and Schauder, applicable to both single-valued and multi-valued nonlinear mappings across various vector spaces.

Contribution

The work develops a generalized fixed-point theorem based on geometric and convexity principles, extending classical theorems to broader classes of mappings and spaces.

Findings

Generalized fixed-point theorems for nonlinear mappings

Applicable to multi-valued and single-valued cases

Provided sufficient conditions for theorem applicability

Abstract

In this work, using a new geometrical approach we study to the existence of the fixed-point of mappings that independence of the smoothness, and also of their single-values or multi-values. This work proved the theorems that generalize in some sense the Brouwer and Schauder fixed-point theorems, and also such type results in multi-valued cases. One can reckon this approach is based on the generalization of the one theorem Cauchy and on the convexity properties of sets. As the used approach is based on the geometry of the image of the examined mappings that are independent of the topological properties of the space we could to prove the general results for almost every vector space. The general results we applied to the study of the nonlinear equations and inclusions in VTS, and also by applying these results are investigated different concrete nonlinear problems. Here provided also sufficient conditions under which the conditions of the theorems will fulfill.