[Paper Review] Using an implicit function to prove common fixed point theorems

Mohammad Imdad, Rqeeb Gubran+1 more|arXiv (Cornell University)|May 18, 2016

Fixed Point Theorems Analysis17 references6 citations

TL;DR

This paper introduces a novel implicit function framework to unify and generalize common fixed point theorems in ordered metric spaces, extending Ćirić's quasi-contraction result to nonlinear, order-theoretic settings. It proves the existence and uniqueness of common fixed points under a generalized contraction condition, with convergence of T-S sequences to the fixed point, and provides a sharpened version of Berinde and Vetro's result in metric spaces.

ABSTRACT

In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike several other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to Ćirić (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105).

Motivation & Objective

• To unify and generalize a broad class of known and unknown contraction conditions in fixed point theory using a single implicit function.
• To extend Ćirić's classical quasi-contraction fixed point theorem to an order-theoretic, nonlinear setting via an implicit function approach.
• To correct and refine an incorrect order-theoretic result from Berinde and Vetro (2012) by providing a sharpened, valid version in metric spaces.
• To establish conditions under which a pair of self-mappings has a unique common fixed point, with convergence of T-S sequences.
• To generalize and extend existing results in the literature by introducing new regularity and compatibility conditions (e.g., I-regularity, O-compatibility).

Proposed method

• Define a general implicit function $ F \in \mathfrak{F} $ that subsumes multiple contraction types, including quasi-contractions and comparison function-based contractions.
• Introduce new concepts such as $ S $-increasing/decreasing mappings, $ T $-$ S $-sequences, and $ (T,S) $-directed sets to handle order-theoretic structures.
• Use $ \overline{\rm O} $-compatibility, $ \underline{\rm O} $-compatibility, and weak compatibility to ensure convergence and uniqueness of fixed points.
• Apply $ I $-regularity and $ D $-regularity conditions to guarantee the limit of $ T $-$ S $-sequences lies in the appropriate set.
• Construct a framework where the implicit inequality $ F(d(Tx,Ty), d(Sx,Sy), \dots) \leq 0 $ implies the existence of a coincidence point.
• Prove convergence of $ T $-$ S $-sequences to the unique common fixed point under weak compatibility and directedness conditions.

Experimental results

Research questions

• RQ1 Can a single implicit function framework unify and generalize a wide range of known contraction conditions in common fixed point theory?
• RQ2 Can Ćirić’s quasi-contraction fixed point theorem be extended to an order-theoretic, nonlinear setting using an implicit function approach?
• RQ3 What are the minimal conditions (e.g., regularity, compatibility, directedness) required to ensure the existence and uniqueness of a common fixed point in ordered metric spaces?
• RQ4 How can Berinde and Vetro’s (2012) result on coincidence points be corrected and sharpened in the context of metric spaces?
• RQ5 Under what conditions does the $ T $-$ S $-sequence converge to the unique common fixed point?

Key findings

• The proposed implicit function framework generalizes numerous known contraction conditions, including quasi-contractions and comparison function-based contractions.
• A nonlinear order-theoretic version of Ćirić’s fixed point theorem is established, extending it to ordered metric spaces with $ S $-monotone mappings.
• The paper corrects an error in Berinde and Vetro’s (2012) order-theoretic result and provides a valid, sharpened version of Theorem 1 in metric spaces.
• Under weak compatibility and $ (T,S) $-directedness (or total order), the pair $ (T,S) $ has a unique common fixed point.
• For any initial point $ x_0 $, the $ T $-$ S $-sequence $ \{Tx_n\} $ converges to the unique common fixed point.
• A counterexample shows that the linear form of the contraction (with $ k \geq 1/2 $) fails, proving that the nonlinear form with a comparison function is essential and cannot be replaced by a linear one.