Fixed Points Theorems in Hausdorff M-distance Spaces

TL;DR

This paper establishes fixed point theorems in a generalized space with a distance function valued in a partially ordered monoid, broadening applicability beyond traditional metric spaces and exploring solutions to integral equations.

Contribution

It introduces fixed point theorems in M-distance spaces, generalizing existing results without requiring linear structure in the distance range.

Findings

Fixed point theorems in M-distance spaces are proven.

Applications to solutions of Fredholm integral equations in L-spaces.

Generalization includes metric and uniform spaces.

Abstract

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including metric and uniform spaces. On the other hand, compared to the so-called cone metric spaces and $K$-metric spaces, we do not require that the distance function range has a linear structure. We also consider several applications of the obtained fixed point theorems. In particular, we consider the questions of the existence of solutions of the Fredholm integral equation in $L$-spaces.