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

Radu Precup, Andrei Stan

TL;DR

This work introduces vector $B$-metric spaces as a matrix-generalization of $b$-metric spaces, where $d:X\times X\to\mathbb{R}^n_+$ and $d(u,w)\leq B(d(u,v)+d(v,w))$. It develops a comprehensive fixed-point theory in this setting, including Perov-type contractions, error estimates, and stability results, as well as an Avramescu-type theorem for two-mapping systems. The paper also extends Ekeland's variational principle and Caristi's fixed point theorem to vector $B$-metric spaces, and derives corresponding $b$-metric variants, thereby providing a broad variational-analytic framework for vector-valued distances. These results generalize and unify many known scalar $b$-metric results and open the door to new applications in nonlinear analysis and optimization with vector-valued metrics.

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

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

TL;DR

This work introduces vector -metric spaces as a matrix-generalization of -metric spaces, where and . It develops a comprehensive fixed-point theory in this setting, including Perov-type contractions, error estimates, and stability results, as well as an Avramescu-type theorem for two-mapping systems. The paper also extends Ekeland's variational principle and Caristi's fixed point theorem to vector -metric spaces, and derives corresponding -metric variants, thereby providing a broad variational-analytic framework for vector-valued distances. These results generalize and unify many known scalar -metric results and open the door to new applications in nonlinear analysis and optimization with vector-valued metrics.

Abstract

In this paper, we extend the concept of -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 -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

Paper Structure

This paper contains 13 sections, 22 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fixed point theorems in vector $B$-metric spaces
  4. Perov type fixed point theorem
  5. Error estimates
  6. Stability results
  7. Avramescu type fixed point theorem
  8. Ekeland's principle and Caristi's fixed point theorem in vector $B$-metric spaces
  9. Classical results
  10. Ekeland variational principle in vector $B$-metric spaces
  11. New versions of the Ekeland variational principle in $b$-metric spaces
  12. Conclusion and further research
  13. Aknowledgements

Key Result

Proposition 2.1

Let $M\in \mathcal{M}_{n\times n}\left( \mathbb{R} _{+}\right)$ and let $I$ be the identity matrix of size $n.$ The following statements are equivalent:

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 29 more