[Paper Review] A new approach of couple fixed point results on JS-metric spaces

Tanusri Senapati, Lakshmi Kanta Dey|arXiv (Cornell University)|Jun 20, 2016

Fixed Point Theorems Analysis4 references3 citations

TL;DR

This paper establishes new coupled fixed point theorems in JS-metric spaces, a generalized metric framework that includes standard metric, b-metric, and dislocated metric spaces. By introducing a novel $D_+$-induced metric on $X^2$ and applying a contraction condition with a constant $k \in [0,1)$, the authors prove the existence and uniqueness of coupled fixed points under mixed monotonicity and completeness assumptions, with simplified proofs distinct from prior methods.

ABSTRACT

In this article, we study coupled fixed point theorems in newly appeared JS-metric spaces. It is important to note that the class of JS-metric spaces includes standard metric space, dislocated metric space, b-metric space etc. The purpose of this paper is to present several coupled fixed point results in a more general way. Moreover, the techniques used in our proofs are indeed different from the comparable existing literature. Finally, we present a non- trivial example to validate our main result.

Motivation & Objective

• To extend coupled fixed point results from standard metric spaces to the more general JS-metric space framework.
• To generalize Bhaskar and Lakshmikantham's coupled fixed point theorems using a contraction condition involving $k \in [0,1)$.
• To establish uniqueness conditions for coupled fixed points under comparability and boundedness assumptions.
• To simplify proofs by leveraging the weaker structure of JS-metric spaces compared to traditional metric spaces.
• To provide a non-trivial example validating the main theoretical results.

Proposed method

• Define a new metric $D_+$ on $X^2$ by $D_+((x,y),(u,v)) = D(x,u) + D(y,v)$, proving $(X^2, D_+)$ is a JS-metric space.
• Use the contraction condition $D_+(T_F(X), T_F(Y)) \leq k D_+(X,Y)$ for $k \in [0,1)$ to ensure convergence of iterative sequences.
• Apply the $\limsup$-based JS-metric axiom (D3) to control limit behavior and ensure convergence to a fixed point.
• Construct iterative sequences $(x_n, y_n)$ from initial points with mixed monotonicity and show they converge to a coupled fixed point.
• Prove uniqueness using contraction mapping principles and boundedness assumptions on comparable or bounded pairs.
• Use the $D_+$-metric to analyze convergence of iterates $T_F^n(z^*)$ to both $w$ and $\tilde{z}$, implying equality via limit uniqueness.

Experimental results

Research questions

• RQ1 Can coupled fixed point theorems in standard metric spaces be generalized to JS-metric spaces, which include b-metric and dislocated metric spaces?
• RQ2 How can the contraction condition from Bhaskar and Lakshmikantham be adapted to the JS-metric setting while preserving existence and uniqueness?
• RQ3 What conditions ensure the uniqueness of a coupled fixed point in JS-metric spaces when elements are comparable or bounded?
• RQ4 Does the use of the $D_+$-metric on $X^2$ preserve the topological properties needed for convergence and fixed point existence?
• RQ5 Under what conditions does the equality $\tilde{x} = \tilde{y}$ hold for a coupled fixed point $(\tilde{x}, \tilde{y})$ in JS-metric spaces?

Key findings

• The authors prove the existence of a coupled fixed point for a mapping $F: X^2 \to X$ under a $k$-contraction condition with $k \in [0,1)$ in a complete JS-metric space with mixed monotonicity.
• The constructed sequence $(x_n, y_n)$ converges to a coupled fixed point $\tilde{z} = (\tilde{x}, \tilde{y})$, where $\tilde{x} = F(\tilde{x}, \tilde{y})$ and $\tilde{y} = F(\tilde{y}, \tilde{x})$, under the given contraction and completeness assumptions.
• Uniqueness of the coupled fixed point is established if two such points are comparable or if they share a common upper or lower bound with finite $D_+$-distance.
• If $\tilde{x}$ and \tilde{y}$ are comparable with $D(\tilde{x}, \tilde{y}) < \infty$, then $\tilde{x} = \tilde{y}$, implying the components of the fixed point are equal.
• The proof techniques are distinct from prior literature, relying on the $\limsup$-based JS-metric axiom and the $D_+$-metric structure, leading to simpler convergence arguments.
• A non-trivial example is constructed to validate the main result, demonstrating the applicability of the theory in a concrete JS-metric space.