Some Properties ???? −Cauchy and ???? −localized Sequence on ???? −metric Space

The metric space is typically denoted as (????, ????), which is a generalization of metric spaces. One of the extensively studied properties in ???? −metric spaces is convergence. In a ???? −metric space, a sequenc (????????) is said to ???? −converge to ???? if ???????????? ????(????, ????????, ????????) = ????, which means that for every ???? ∈ ℝ, ???? > ????, there exists ???? ∈ ℕ such that for all ????, ???? ∈ ℕ with ????, ???? ≥ ????, we have ????(????, ????????, ????????) < ????. Many mathematicians have discussed the concept of convergence, including the notion of statistical convergence first introduced by Fast (1951). This research aims to investigate the concept of ???? −convergence, which is a generalization of statistical convergence. Furthermore, the study establishes the connection between ???? −Cauchy sequences, an extension of ???? −convergence, and convergence in ???? −metric spaces. Additionally, the properties of ???? −Cauchy and ???? ∗ −Cauchy sequences, as well as ???? −localized sequences in ???? −metric spaces, are examined. The objective of this research is to enhance the understanding of the properties of ???? −Cauchy sequences within the context of G−metric spaces. The new findings from this study can contribute to the development of the theory of ???? −metric spaces and expand our understanding of ???? −convergence in sequences and ???? −Cauchy sequences. These results may also have potential applications in various fields involving mathematical analysis and modeling