[Paper Review] Fixed points of multiplicative contraction mappings on multiplicative metric spaces

Muttalip Özavşar, Adem C. Çevikel|arXiv (Cornell University)|May 23, 2012

Fixed Point Theorems Analysis21 citations

TL;DR

This paper introduces multiplicative contraction mappings on multiplicative metric spaces and establishes fixed point theorems for such mappings on complete multiplicative metric spaces. It proves that the set of positive real numbers ℝ₊ is complete under the multiplicative metric, and demonstrates that under specific contraction conditions involving a constant λ ∈ [0, 1/2), a mapping has a unique fixed point, with iterative sequences converging to it.

ABSTRACT

In this paper, we first discussed multiplicative metric mapping by giving some topological properties of the relevant multiplicative metric space. As an interesting result of our discussions, we observed that the set of positive real numbers $\\mathbb{R}_+$ is a complete multiplicative metric space with respect to the multiplicative absolute value function. Furthermore, we introduced concept of multiplicative contraction mapping and proved some fixed point theorems of such mappings on complete multiplicative metric spaces

Motivation & Objective

• To investigate the topological structure of multiplicative metric spaces and establish foundational properties such as completeness.
• To define and analyze multiplicative contraction mappings, generalizing classical contraction principles to multiplicative settings.
• To prove fixed point theorems for such mappings on complete multiplicative metric spaces, ensuring existence and uniqueness of fixed points.
• To demonstrate the applicability of the results through concrete examples, including mappings on ℝ₊ and subsets of ℝ².
• To extend classical fixed point theory from standard metric spaces to multiplicative metric spaces using multiplicative triangle inequality and reverse triangle inequality.

Proposed method

• Defining a multiplicative metric d:X×X→ℝ satisfying d(x,y)>1 for x≠y, d(x,y)=1 iff x=y, symmetry, and multiplicative triangle inequality d(x,z)≤d(x,y)·d(y,z).
• Introducing the multiplicative absolute value function |a|⁎=a if a≥1, 1/a if a<1, to construct multiplicative metrics on ℝ₊ⁿ and ℂⁿ.
• Establishing the multiplicative reverse triangle inequality: |d(x,z)/d(y,z)|⁎ ≤ d(x,y) for all x,y,z∈X.
• Defining multiplicative open and closed balls, and proving that open balls are multiplicative open sets, forming a topology on X.
• Proposing multiplicative contraction mappings satisfying d(fx,fy)≤(d(fx,y)·d(fy,x))⁵ for λ∈[0,1/2), and using iterative sequences to prove convergence.
• Using the completeness of the space to show that Cauchy sequences (xn) converge to a limit z, and proving d(fz,z)=1 implies fz=z via limit arguments.

Experimental results

Research questions

• RQ1 Is the set of positive real numbers ℝ₊ complete under the multiplicative metric d(x,y)=|x/y|⁎?
• RQ2 Can fixed point theorems for multiplicative contraction mappings be established in complete multiplicative metric spaces?
• RQ3 What conditions ensure the existence and uniqueness of a fixed point for a mapping f:X→X on a complete multiplicative metric space?
• RQ4 How do iterative sequences (fⁿx) behave in such spaces, and do they converge to the fixed point?
• RQ5 Can classical fixed point results be generalized to multiplicative metric spaces using multiplicative triangle and reverse triangle inequalities?

Key findings

• The set of positive real numbers ℝ₊ is a complete multiplicative metric space under the multiplicative absolute value function |a|⁎.
• The multiplicative metric d(x,y)=|x/y|⁎ on ℝ₊ satisfies all required axioms, including the multiplicative triangle inequality.
• For a mapping f satisfying d(fx,fy)≤(d(fx,y)·d(fy,x))⁵ with λ∈[0,1/2), a unique fixed point exists in any complete multiplicative metric space.
• Iterative sequences (fⁿx) converge to the fixed point in complete multiplicative metric spaces under the stated contraction condition.
• The mapping f(x)=e^{x−1−x³/10} on X=[0.1,1] satisfies the multiplicative contraction condition with λ=0.997 and has a unique fixed point at approximately 0.7411.
• The example on X⊂ℝ² with d((a,b),(c,d))=(|a/c|⁎·|b/d|⁎)^{1/3} confirms the existence of a unique fixed point at (1,1) under the contraction condition with λ=1/2.