[Paper Review] Functional contractions in local Branciari metric spaces

Mihai Turinici|arXiv (Cornell University)|Aug 22, 2012

Fixed Point Theorems Analysis17 references20 citations

TL;DR

This paper establishes a fixed point theorem for functional contractions in local Branciari metric spaces, extending classical results by Banach and Rus. It proves that under a generalized contractive condition involving a strongly regressive function φ, a self-map T has a unique fixed point if the space is complete or orbitally complete, generalizing prior work in asymmetric and generalized metric spaces.

ABSTRACT

A fixed point result is given for a class of functional contractions over local Branciari metric spaces. It extends some contributions in the area due to Fora et al [Mat. Vesnik, 61 (2009), 203-208].

Motivation & Objective

• To extend fixed point theory to local Branciari metric spaces, which generalize standard and Branciari metrics by allowing a variable polyhedral inequality condition.
• To establish conditions under which a functional contraction mapping has a unique fixed point in such spaces.
• To show that the Hausdorff property and completeness assumptions can be relaxed by using orbital completeness and local structure.
• To generalize existing results from Fora et al., Das and Dey, and Leader by incorporating Matkowski-type contractions with strongly regressive functions.
• To demonstrate that the regularity condition on the space (e.g., Hausdorff) is not necessary, confirming earlier observations by Kikina and Kikina.

Proposed method

• Define a local Branciari metric as a reflexive sufficient symmetric satisfying a (2+k)-polyhedral inequality for each effectively denumerable subset M, with k depending on M.
• Introduce the concept of d-convergence and d-Cauchy sequences in this generalized metric framework.
• Use a functional contraction condition: d(Tx,Ty) ≤ φ(F(x,y)), where φ is a strongly regressive function in the class Fr(R+), and F involves distances between x, y, Tx, and Ty.
• Prove that under orbital completeness, the T-orbital sequence (T^n x) d-converges to a unique fixed point.
• Apply a contradiction argument assuming the limit z ≠ Tz, and use the polyhedral inequality with a chain of points to derive φ(ρ) ≥ ρ, contradicting the strong regressivity of φ.
• Extend the result to the case of standard completeness and show that the result reduces to known theorems when d is a standard metric or satisfies the tetrahedral inequality.

Experimental results

Research questions

• RQ1 Can fixed point results for functional contractions be extended to local Branciari metric spaces without requiring the Hausdorff property?
• RQ2 What conditions on the contraction function φ ensure the existence and uniqueness of a fixed point in such spaces?
• RQ3 Is orbital completeness sufficient to guarantee convergence of iterative sequences in local Branciari metric spaces?
• RQ4 How does the (2+k)-polyhedral inequality generalize the triangular and tetrahedral inequalities in fixed point theory?
• RQ5 To what extent can the regularity assumptions on the metric space be weakened while preserving the fixed point property?

Key findings

• A unique fixed point exists for any self-map T that satisfies a functional contraction condition with a strongly regressive function φ in a local Branciari metric space.
• The fixed point is the d-limit of the T-orbital sequence (T^n x) for any initial point x, provided the space is T-orbitally complete.
• The result generalizes the classical Banach contraction principle when d is a standard metric and extends Leader’s result in the metric case.
• The proof shows that assuming the limit z ≠ Tz leads to ρ ≤ φ(ρ), contradicting the strong regressivity of φ, hence ρ = 0.
• The framework allows removal of the upper semicontinuity condition on φ in earlier results by Fora et al., as it is not required for the convergence argument.
• The theorem includes the results of Das and Dey (2012) as a special case when φ is a Boyd-Wong type function.