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Path--Averaged Contractions: A New Generalization of the Banach Contraction Principle

Nicola Fabiano

TL;DR

This paper introduces PA-contractions, a fixed-point framework defined by a path-averaged contraction over orbits of pairs. It proves that continuous PA-contractions on complete metric spaces have a unique fixed point and that Picard iterations converge to it, while showing that this condition strictly generalizes Banach contractions. The authors demonstrate independence from F-, Kannan, Chatterjea, and Ćirić contractions through counterexamples and provide a comparison table to clarify non-implications. The work highlights the asymptotic nature of PA-contractions and suggests avenues for future research, including multivalued mappings and common fixed points.

Abstract

We introduce a novel class of self-mappings on metric spaces, called \textbf{PA-contractions} (Path-Averaged Contractions), defined by an averaging condition over iterated distances. We prove that every continuous PA-contraction on a complete metric space has a unique fixed point, and the Picard iterates converge to it. This condition strictly generalizes the classical Banach contraction principle. We provide examples showing that PA-contractions are independent of F-contractions, Kannan, Chatterjea, and Ćirić contractions. A comparison table highlights the distinctions. The PA-condition captures long-term contractive behavior even when pointwise contraction fails.

Path--Averaged Contractions: A New Generalization of the Banach Contraction Principle

TL;DR

This paper introduces PA-contractions, a fixed-point framework defined by a path-averaged contraction over orbits of pairs. It proves that continuous PA-contractions on complete metric spaces have a unique fixed point and that Picard iterations converge to it, while showing that this condition strictly generalizes Banach contractions. The authors demonstrate independence from F-, Kannan, Chatterjea, and Ćirić contractions through counterexamples and provide a comparison table to clarify non-implications. The work highlights the asymptotic nature of PA-contractions and suggests avenues for future research, including multivalued mappings and common fixed points.

Abstract

We introduce a novel class of self-mappings on metric spaces, called \textbf{PA-contractions} (Path-Averaged Contractions), defined by an averaging condition over iterated distances. We prove that every continuous PA-contraction on a complete metric space has a unique fixed point, and the Picard iterates converge to it. This condition strictly generalizes the classical Banach contraction principle. We provide examples showing that PA-contractions are independent of F-contractions, Kannan, Chatterjea, and Ćirić contractions. A comparison table highlights the distinctions. The PA-condition captures long-term contractive behavior even when pointwise contraction fails.

Paper Structure

This paper contains 8 sections, 2 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. PA-Contractions
  3. Fixed Point Theorem
  4. Generalization of Banach Contractions
  5. Comparison with F-Contractions
  6. Relation with Other Classical Contractions
  7. Comparison Table
  8. Conclusion

Key Result

Theorem 3

Let $(X,d)$ be a complete metric space, and let $T: X \to X$ be a continuous PA-contraction. Then $T$ has a unique fixed point $x^* \in X$, and for any $x_0 \in X$, the Picard sequence $x_n = T^n x_0$ converges to $x^*$.

Theorems & Definitions (15)

  • Definition 1: PA-Contraction
  • Remark 2
  • Theorem 3: Fixed Point Theorem for PA-Contractions
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Example 6: PA-contraction not Banach
  • Definition 7: F-Contraction Wardowski2012
  • Example 8: F-contraction not PA-contraction
  • ...and 5 more