On generalizations of some fixed point theorems in semimetric spaces with triangle functions

TL;DR

This paper extends several fixed point theorems to semimetric spaces with triangle functions, broadening their applicability across various mathematical and computational disciplines.

Contribution

It generalizes classical fixed point theorems to semimetric spaces with triangle functions, encompassing diverse space types like metric, ultrametric, and b-metric spaces.

Findings

Generalized Banach, Kannan, Chatterjea, and Cirić-Reich-Rus fixed point theorems.

Derived corollaries for various semimetric spaces.

Applications in optimization, modeling, and algorithm design.

Abstract

In the present paper, we prove generalizations of Banach, Kannan, Chatterjea, \'Ciri\'c-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mappings contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by M. Bessenyei and Z. P\'ales. Such an approach allows us to derive corollaries for various types of semimetric spaces including metric spaces, ultrametric spaces, b-metric spaces etc. The significance of these generalized theorems extends across multiple disciplines, including optimization, mathematical modeling, and computer science. They may serve to establish stability conditions, demonstrate the existence of optimal solutions, and improve algorithm design.