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Common Fixed Point of the Commutative F-contraction Self-mappings with uniquely bounded sequence

Djamel Deghoul, Zoheir Chebel, Abdellatif Boureghda, Salah Benyoucef

TL;DR

This work addresses the problem of guaranteeing a unique common fixed point for two commuting self-maps $f$ and $g$ on a complete metric space under an $F$-contraction relationship, by replacing Jungck's range-inclusion condition with a bounded Picard sequence criterion. The authors develop a Banach-contraction–type framework, proving that if $f$ is continuous and there exists $x_0$ with $igl\{f^n(x_0)\bigr\}$ bounded, then $f$ and $g$ have a unique common fixed point $l$, with the iterates $(f\circ g)^n(x_0)$ converging to $l$. Key contributions include establishing the contractive inequality $d(g^{n}(x), g^{n}(y)) \le k^{n} d(f^{n}(x), f^{n}(y))$, demonstrating convergence and uniqueness, and extending the results to corollaries, triples of mappings, and compact spaces. This extends fixed-point theory beyond inclusion-based Jungck results and offers a Banach-like analogue with potential applications in analysis and numerical computation.

Abstract

We establish the existence of a common fixed point for mappings that satisfy and extend the F-contraction condition. To support our findings, we present pertinent definitions and properties associated with F-contraction mappings. Additionally, we establish an analogue to the Banach contraction theorem. Our results contribute to the broader understanding of this field by extending and generalizing existing findings in the literature.

Common Fixed Point of the Commutative F-contraction Self-mappings with uniquely bounded sequence

TL;DR

This work addresses the problem of guaranteeing a unique common fixed point for two commuting self-maps and on a complete metric space under an -contraction relationship, by replacing Jungck's range-inclusion condition with a bounded Picard sequence criterion. The authors develop a Banach-contraction–type framework, proving that if is continuous and there exists with bounded, then and have a unique common fixed point , with the iterates converging to . Key contributions include establishing the contractive inequality , demonstrating convergence and uniqueness, and extending the results to corollaries, triples of mappings, and compact spaces. This extends fixed-point theory beyond inclusion-based Jungck results and offers a Banach-like analogue with potential applications in analysis and numerical computation.

Abstract

We establish the existence of a common fixed point for mappings that satisfy and extend the F-contraction condition. To support our findings, we present pertinent definitions and properties associated with F-contraction mappings. Additionally, we establish an analogue to the Banach contraction theorem. Our results contribute to the broader understanding of this field by extending and generalizing existing findings in the literature.
Paper Structure (4 sections, 19 theorems, 24 equations)

This paper contains 4 sections, 19 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Main results and theorems
  4. Conclusions

Key Result

Theorem 2.1

chebel Consider a continuous contraction mapping $g - k - f$ in the complete metric space $X$. Additionally, assume that the mappings $f$ and $g$ commute, and there exists an element $x_{0} \in X$ such that the Picard sequence $\{ f_{n} (x_{0})\}_{n\geq 0}$ converges to $t_{0} \in X$. In this case,

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 23 more