# Some extensions of Banach'\ CP in $G$-metric spaces

## TL;DR

This paper extends the Banach contraction principle within $G$-metric spaces by introducing variable powers of mappings and relaxing continuity requirements, broadening the applicability of fixed point results.

## Contribution

It generalizes existing fixed point theorems in $G$-metric spaces by allowing contractive conditions to depend on points and subsets, without requiring continuity.

## Key findings

- Established fixed point results without continuity assumptions.
- Extended contraction conditions to powers depending on points.
- Generalized known theorems in $G$-metric spaces.

## Abstract

We present different extensions of the Banach contraction principle in the $G$-metric space setting. More precisely, we consider mappings for which the contractive condition is satisfied by a power of the mapping and for which the power depends on the specified point in the space. We first state the result in the continuous case and later, show that the continuity is indeed not necessary. Imitating some techniques obtained in the metric case, we prove that under certain conditions, it is enough for the contractive condition to be verified on a proper subset of the space under consideration. These results generalize well known comparable results.