On the convergence of sequences in $\mathbb{R}^+$ through weighted geometric means via multiplicative calculus and application to intuitionistic fuzzy numbers

TL;DR

This paper investigates the convergence of sequences in positive real numbers using weighted geometric means and multiplicative calculus, extending these concepts to intuitionistic fuzzy numbers with new convergence criteria and illustrative examples.

Contribution

It introduces new convergence concepts based on multiplicative calculus and applies them to intuitionistic fuzzy numbers, providing necessary and sufficient conditions for convergence.

Findings

Established conditions for convergence via weighted geometric means.

Developed multiplicative analogues of classical oscillation and two-sided conditions.

Applied convergence criteria to sequences of intuitionistic fuzzy numbers.

Abstract

We define weighted geometric mean method of convergence for sequences in $\mathbb{R}^+$ by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in $\mathbb{R}^+$ follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of $^{\oplus}$convergence, $^{\otimes}$convergence, $(\bar{N},p)-^{\oplus}$convergence, $(\bar{G},p)-^{\otimes}$convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence.