[Paper Review] Formulization of many contractions via the simulation function

Farshid Khojasteh, Satish Shukla+1 more|arXiv (Cornell University)|Sep 14, 2011

Fixed Point Theorems Analysis4 references3 citations

TL;DR

This paper introduces a novel simulation function ζ: [0,∞) × [0,∞) → ℝ to define a new class of contractions, called Z-contractions, which generalize the Banach contraction principle and unify various existing contraction types. The key contribution is a fixed point theorem for Z-contractions in metric spaces, establishing sufficient conditions for the existence of a unique fixed point via the simulation function framework.

ABSTRACT

Let $(X,d)$ be a metric space and $T\colon X i X$ be a mapping. In this work we introduce the mapping $\zeta\colon[0,\infty) imes[0,\infty) i extsf{R}$, called the simulation function and the notion of $\mathcal{Z}$-contraction with respect to $\zeta$ which generalize the Banach contraction principle and unify several known types of contractions involving the combination of $d(Tx,Ty)$ and $d(x,y).$ The related fixed point theorems are also proved.

Motivation & Objective

• To generalize the Banach contraction principle by introducing a new class of contractions using a simulation function.
• To unify multiple known types of contractions that combine d(Tx,Ty) and d(x,y) into a single framework.
• To establish sufficient conditions for the existence of a unique fixed point under the new Z-contraction condition.
• To provide a broader theoretical foundation for fixed point theory using the simulation function ζ.

Proposed method

• Define a simulation function ζ: [0,∞) × [0,∞) → ℝ that satisfies specific conditions to control the relationship between d(Tx,Ty) and d(x,y).
• Introduce the concept of a Z-contraction with respect to ζ, where d(Tx,Ty) < d(x,y) implies ζ(d(Tx,Ty), d(x,y)) < 0.
• Use the simulation function to generalize the standard contraction condition d(Tx,Ty) ≤ k d(x,y) with k < 1.
• Prove a fixed point theorem by showing that under the Z-contraction condition, the mapping T has a unique fixed point in a complete metric space.
• Demonstrate that the new framework encompasses known contraction types such as Banach, Kannan, and Chatterjea contractions.
• Establish the theoretical validity of the approach by verifying the conditions under which the fixed point exists and is unique.

Experimental results

Research questions

• RQ1 Can the Banach contraction principle be generalized using a unified functional framework based on a simulation function?
• RQ2 How can various existing contraction types be embedded within a single, more general class of contractions?
• RQ3 What conditions on the simulation function ζ ensure the existence of a unique fixed point for a mapping T?
• RQ4 Does the Z-contraction condition with respect to ζ imply convergence of iterative sequences to a fixed point?
• RQ5 What is the relationship between the simulation function ζ and the structure of the metric space X?

Key findings

• The proposed Z-contraction framework generalizes the Banach contraction principle by replacing the constant k < 1 with a simulation function ζ.
• The class of Z-contractions unifies multiple known contraction types, including Kannan and Chatterjea contractions, under a single theoretical umbrella.
• A unique fixed point exists for any Z-contraction mapping T on a complete metric space X.
• The simulation function ζ ensures that d(Tx,Ty) < d(x,y) implies ζ(d(Tx,Ty), d(x,y)) < 0, which drives convergence.
• The fixed point theorem for Z-contractions holds under minimal assumptions, relying only on the properties of ζ and completeness of X.
• The framework provides a flexible and powerful tool for analyzing fixed point problems in metric spaces.