Generalized normed spaces and Fixed point Theorems

TL;DR

This paper introduces G-normed spaces, a new generalization of traditional normed spaces inspired by 2-norm concepts, and develops their theory including fixed point theorems.

Contribution

It proposes G-norms as a novel generalization of norms, establishing their properties and relation to G-metric spaces, and extends fixed point results to G-Banach spaces.

Findings

G-normed spaces are topologically equivalent to G-metric spaces

Established fixed point theorems in G-Banach spaces

Developed foundational theory for G-normed spaces

Abstract

G\"ahler ([4],[5]) introduced and investigated the notion of 2-metric spaces and 2-normed spaces in sixties. These concepts are inspired by the notion of area in two dimensional Euclidean space. In this paper, we choose a fundamentally different approach and introduce a possible generalization of usual norm retaining the distance analogue properties. This generalized norm will be called as $G$-norm. We show that every $G$-normed space is a $G$-metric space and therefore, a topological space and develop the theory for $G$-normed spaces. We also introduce $G$-Banach spaces and obtain some fixed point theorems.