Fixed point results and the Ekeland variational principle in vector $B$-metric spaces

TL;DR

This paper extends fixed point theory and Ekeland's variational principle to vector-valued b-metric spaces with matrix-based triangle inequalities, broadening the scope of these concepts in generalized metric spaces.

Contribution

It introduces the concept of vector b-metric spaces with matrix constants and establishes fixed point theorems and variational principles within this new framework.

Findings

Established fixed point theorems in vector b-metric spaces

Proved a variant of Ekeland's variational principle for these spaces

Derived a version of Caristi's fixed-point theorem

Abstract

In this paper, we extend the concept of \( b \)-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the \( b \)-metric setting: fixed-point theorems, stability results, and a variant of Ekeland's variational principle. As a consequence, we also derive a variant of Caristi's fixed-point theorem