Almost Convergence of Complex Uncertain Triple Sequences
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RESEARCH ARTICLE
Almost Convergence of Complex Uncertain Triple Sequences Birojit Das1 • Binod Chandra Tripathy2 • Piyali Debnath1 • Jagannath Nath1 Baby Bhattacharya1
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Received: 4 November 2019 / Revised: 16 August 2020 / Accepted: 21 October 2020 Ó The National Academy of Sciences, India 2020
Abstract In this paper, we present the concept of complex uncertain triple sequences and study almost convergence therein. Almost convergence with respect to all five aspects in uncertain space, viz., almost convergence in mean, measure, distribution, almost surely and uniformly almost surely, are initiated and interrelationships among them are established. We have also studied almost Cauchy triple sequence of complex uncertain variables and established some results. It is known that every convergent sequence is a Cauchy sequence but the converse is not true in general. But taking complex uncertain variables in triple sequences, we find that a complex uncertain triple sequence is an almost Cauchy sequence if and only if it is almost convergent. Statement In this article we have introduced and investigated almost convergence of triple sequences of complex uncertain variables. Studies on complex uncertain sequences has been initiated in the last decade. Now it is drawing attention of researcher and this article will motivate for further investigation and application. & Binod Chandra Tripathy [email protected]; [email protected]; [email protected] Birojit Das [email protected] Piyali Debnath [email protected] Jagannath Nath [email protected] Baby Bhattacharya [email protected] 1
Department of Mathematics, National Institute of Technology Agartala, Tripura 799046, India
2
Department of Mathematics, Tripura University, Agartala, Tripura 799022, India
Keywords Complex uncertain triple sequence Complex uncertain Variable Almost convergence in mean Almost convergence in measure Almost surely Cauchy sequence Mathematics Subject Classification 60B10 60B1 60E05 60F25 40A05 40A30 40D25 40F05
1 Introduction The theory of uncertainty is first introduced by Liu [1]. After that, it has been studied in various fields of mathematics like calculus [2], risk and stability analysis [3], set theory [4], logic [5], process [6], finance [7], graph theory [8], sequence and series [1]. It became a separate branch of mathematics, and nowadays, the research on uncertainty theory became quite famous. Convergence of sequences plays a pivotal role in the study of fundamental theory of mathematics [9–11]. Liu applied the theory of uncertainty on sequences and established the properties of convergence of uncertain measure by introducing convergence in measure, in mean, in distribution and in almost surely of an uncertain sequence. You [12] extended this study to convergence in uniformly almost surely and established the interrelationship with the previous four types of convergence. Guo and Xu [13] presented a necessary and sufficient condition of convergence in mean square for uncertain
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