Convergence: Rough set with Variational inequalities

Abstract

This research paper presents a uniqueness approach towards the Rough Metric and Variational inequalities, Subsequently we fold our paper in to two different parts:

FOLD-I]

To address the shortcomings of classical set theory when confronted with real-world complexities and uniqueness of fixed points, we present fixed point results under rough metric inspiring results from probability, fuzzy sets, rough sets, and soft sets, we establish novel fixed point results for Compact Rough Metric Spaces, motivated by seminal works, Revisiting foundational principles from literature, our study contributes to uniqueness under rough fixed-point theorems. Moreover the study on Pawlak rough sets and we given some examples on rough set theory Emphasizing the significance of alternative models in addressing complex, uncertain phenomena. The established fixed-point theorems underscore the efficacy of Compact Rough Metric Spaces in providing solutions.

FOLD-II]

In the second fold iterative strategies for general variational inequalities presented, subsequently this study devoted to the fixed-point formulation and general variational inequalities. Scrutinizing convergence under specific conditions, we introduce the extragradient method and modified double projection methods as distinctive implicit methodologies. Furthermore we given fixed point results for general variational inequalities which help to find uniqueness for solution, our research provides a numerical example for practical illustration. This work contributes to advancing mathematical methodologies for problem-solving, emphasizing the practical applicability of alternative models and iterative strategies for fixed points.