BEBERAPA SIFAT BARISAN CAUCHY−I DAN I−LOCALIZED PADA RUANG METRIK−G

SOME PROPERTIES OF CAUCHY−I AND I−LOCALIZED ON G−METRIC SPACES

Ruang metrik-G biasanya disimbolkan dengan (X, G) yang merupakan generalisasi dari ruang metrik. Salah satu sifat yang banyak diteliti adalah mengenai kekonvergenan di ruang metrik−G. Dalam ruang metrik−G, suatu barisan (xn) di ruang metrik−G dikatakan konvergen−G ke x jika limx→∞(x, xn, xm) = 0 yang berarti untuk setiap ε ∈ R, ε > 0, terdapat K ∈ N sedemikian hingga untuk setiap m, n ∈ N dengan m, n ≥ K berlaku G(x, xn, xm) < ε. Banyak matematikawan yang membahas konsep kekonvergenan salah satunya gagasan kekonvergenan secara statistik yang pertama kali dipublikasikan oleh Fast (1951). Penelitian ini bertujuan mengkaji tentang kekonvergenan−I yang merupakan generalisasi kekonvergenan statistik. Kemudian akan diberikan butki keterhubungan antara barisan Cauchy−I yang merupakan perluasan konsep kekonvergenan−I dengan kekonvergenan pada ruang metrik−G. Tidak hanya itu, pada penelitian ini juga dikaji sifat-sifat barisan Cauchy−I dan Cauchy−I ∗ , dan barisan I−localized di ruang metrik−G

The G−metric space is typically denoted as (X, G), which is a generalization of metric spaces. One of the extensively studied properties in G−metric spaces is convergence. In a G−metric space, a sequence (xn) is said to G−converge to x if limn→∞ G(x, xn, xm) = 0, which means that for every ε ∈ R, ε > 0, there exists K ∈ N such that for all m, n ∈ N with m, n ≥ K, we have G(x, xn, xm) < ε. 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 I−convergence, which is a generalization of statistical convergence. Furthermore, the study establishes the connection between I−Cauchy sequences, an extension of I−convergence, and convergence in G−metric spaces. Additionally, the properties of I−Cauchy and I ∗−Cauchy sequences, as well as I−localized sequences in G−metric spaces, are examined. The objective of this research is to enhance the understanding of the properties of I−Cauchy sequences within the context of G−metric spaces. The new findings from this study can contribute to the development of the theory of G−metric spaces and expand our understanding of I−convergence in sequences and I−Cauchy sequences. These results may also have potential applications in various fields involving mathematical analysis and modeling