The Theory and Implementation of the Sehgal Fixed Theorem in an Expanded G-metric Space using the Banach Space's Contraction Principle

Abstract

The present paper shows that classic-type results in G-metric theory can be deduced using the Banach contraction and some known fixed-point theorems directly over extended G-metric spaces. Motivated by this basis, we prove a fixed-point theorem of the Sehgal–Gusman type for operators with pointwise contractive conditions at their iterates. Furthermore, we present a broader generalization of the Banach contraction principle that encompasses a broader range of mappings and conditions that have been previously examined. This generalization broadens the scope of fixed-point theory in G-metric spaces, resulting in the development of new and more adaptable analytical instruments for the study of metric fixed points. Our proposed results will be useful to enrich the theoretical structure of functional analysis and topology, as well as to provide more applications in dynamical systems and optimization problems.