Two results in metric fixed point theory

TL;DR

This paper proves two fixed point theorems for contractive mappings in metric and Banach spaces, one for mappings into the space and another using the continuation method with Leray-Schauder boundary conditions.

Contribution

It introduces two new fixed point theorems for contractive mappings, expanding applicability to mappings into spaces and boundary condition cases.

Findings

Established fixed point theorems for mappings into complete metric spaces.

Applied continuation method with Leray-Schauder boundary condition in Banach spaces.

Extended fixed point theory to new classes of mappings.

Abstract

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space $X$ into $X$, and the second with an application of the continuation method to the case where they satisfy the Leray-Schauder boundary condition in Banach spaces.