Fixed Points of Meir-Keeler and Leader Contractions with bounded orbits in b-Metric Spaces

TL;DR

This paper proves fixed-point theorems for certain contractions in b-metric spaces, overcoming previous limitations and unifying various fixed-point results under a broader framework.

Contribution

It introduces a strengthened condition for Meir-Keeler contractions that guarantees fixed points in b-metric spaces, expanding the scope of fixed-point theory.

Findings

Fixed points exist for non-expansive Leader contractions with bounded orbits.

Classical Meir-Keeler contractions may fail to have fixed points in b-metric spaces.

The results hold without the triangle inequality, only requiring unique limits.

Abstract

We establish fixed-point theorems for Meir-Keeler-type contractions in b-metric spaces. While Lu et al. demonstrated via an explicit counterexample that classical Meir-Keeler contractions may fail to admit fixed points in this setting, we prove that a natural strengthening of the conditions yields existence results. Specifically, we show that every non-expansive Leader contraction with bounded orbits in a b-metric space possesses a fixed point. To contextualize our findings, we present a hierarchical diagram illustrating that the fixed-point theory of non-expansive Leader contractions subsumes earlier results, including Meir-Keeler contractions, the primary focus of this work. Our proofs hold in arbitrary b-metric spaces, without relying on the triangle inequality, requiring instead only the assumption of unique limits. This work not only resolves the limitation exposed by Lu et al.'s counterexample but also establishes a unifying framework for future research in the literature.