Fixed point theorems for weak, partial, Bianchini and Chatterjea-Bianchini contractions in semimetric spaces with triangle functions

TL;DR

This paper extends fixed point theorems to semimetric spaces with triangle functions, covering various contraction types and revealing new results for weak and partial contractions.

Contribution

It generalizes classical fixed point theorems to semimetric spaces with triangle functions, introducing new insights for weak and partial contractions.

Findings

Extended fixed point theorems to semimetric spaces with triangle functions

Derived corollaries for metric, b-metric, ultrametric, and distance spaces

Discovered new results for weak and partial contractions

Abstract

This paper advances a line of research in fixed point theory initiated by M. Bessenyei and Z. P\'ales, building on their introduction of the triangle function concept in [J. Nonlinear Convex Anal, Vol 18 (3), 515-524 (2017)]. By applying this concept, the study revises several well-known fixed point theorems in metric spaces, extending their applicability to semimetric spaces with triangle functions. The paper focuses on general theorems involving weak, partial, Bianchini and Chatterjea-Bianchini contractions, deriving corollaries relevant to metric spaces, $b$-metric spaces, ultrametric spaces, and distance spaces with power triangle functions. Notably, several new and interesting findings emerge in the context of weak and partial contractions.