Studies on λ-statistical convergence of sequences in generalized probabilistic metric spaces
A study on convergence of sequences in probabilistic G-metric spaces
Abstract
In this paper, we introduce the notion of λ-statistical convergence of sequences in probabilistic G-metric spaces. We study some basic properties of λ-statistical convergence of sequences. Also introducing the notion of λ-statistically Cauchy sequences, we study its relationship with λ-statistical convergence in probabilistic G-metric spaces. Further, introducing the notion of λ-statistically pre-Cauchy sequences in probabilistic G-metric spaces, we examine the connections among all these new concepts and establish some basic facts.
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References
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Abstract and applied analysis, 2014.
and Applications 2021 (2021), no. 1, 1–11.
[2] H¨useyin C¸ akallı, New kinds of continuities, Computers & Mathematics with Applications 61 (2011), no. 4, 960–965.
[3] J Connor, J Fridy, and J Kline, Statistically pre-Cauchy sequences, Analysis 14 (1994), no. 4, 311–318.
[4] Henry Fast, Sur la convergence statistique, Colloquium mathematicae, 1951, pp. 241–244.
[5] John A Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–314.
[6] Argha Ghosh and Samiran Das, Some further studies on strong Iλ-statistical convergence in probabilistic metric spaces,
Analysis 42 (2022), no. 1, 11–22.
[7] Argha Ghosh and Samiran Das, Strongly λ-statistically and strongly Vall´ee-Poussin pre-Cauchy sequences in probabilistic metric spaces,
Tamkang Journal of Mathematics 53 (2022), no. 4, 373–384.
[8] P Hu and FENG Gu, Some fixed point theorems of λ-contractive mappings in Menger PSM-spaces, Journal of Nonlinear
Functional Analysis 2020 (2020), no. 2020, 33.
[9] Prasanta Malik and Samiran Das, Further results on strong λ-statistical convergence of sequences in probabilistic
metric spaces, Boletim da Sociedade Paranaense de Matem´atica 41 (2023), 1–12.
[10] Karl Menger, Statistical metrics, Proc. Nat. Acad. Sci. 28 (1942), 535–537.
[11] Mohammad Mursaleen, λ-statistical convergence, Mathematica slovaca 50 (2000), no. 1, 111–115.
[12] Mohammad Mursaleen and A Alotaibi, Statistical summability and approximation by de la Vall´ee-Poussin mean,
Applied Mathematics Letters 24 (2011), no. 3, 320–324.
[13] Zead Mustafa, Hamed Obiedat, and Fadi Awawdeh, Some fixed point theorem for mapping on complete G-metric
spaces, Fixed Point Theory and Algorithms for Sciences and Engineering (2008).
[14] Zead Mustafa, Wasfi Shatanawi, Malik Bataineh, and Andrei Volodin, Existence of fixed point results in G-metric
spaces, International Journal of Mathematics and Mathematical Sciences 2009 (2009), 10.
[15] Zead Mustafa and Brailey Sims, A new approach to generalized metric spaces, Journal of Nonlinear and convex Analysis
7 (2006), no. 2, 289.
[16]- Zead Mustafa and Brailey Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory and Algorithms
for Sciences and Engineering (2009).
[17] Isaac J Schoenberg, The integrability of certain functions and related summability methods, The American mathematical
monthly 66 (1959), no. 5, 361–775.
[18] Berthold Schweizer and Abe Sklar, Probabilistic metric spaces, Courier Corporation, 2011.
[19] Berthold Schweizer, Abe Sklar, et al., Statistical metric spaces, Pacific J. Math 10 (1960), no. 1, 313–334.
[20] Berthold Schweizer, Abe Sklar, and Edward Thorp, The metrization of statistical metric spaces. (1960).
[21] Celaleddin S¸en¸cimen and Serpil Pehlivan, Strong statistical convergence in probabilistic metric spaces, Stochastic analysis
and applications 26 (2008), no. 3, 651–664.
[22] Robert Tardiff, Topologies for probabilistic metric spaces, Pacific Journal of Mathematics 65 (1976), no. 1, 233–251.
[23] Edward Thorp, Generalized topologies for statistical metric spaces, Mathematical Sciences Directorate, Office of Scientific
Research, US Air Force, 1960.
[24] Caili Zhou, Shenghua Wang, Ljubomir ´Ciri´c, and Saud M Alsulami, Generalized probabilistic metric spaces and fixed
point theorems, Fixed Point Theory and Applications 2014 (2014), no. 1, 1–15.
[25] Chuanxi Zhu, Wenqing Xu, and Zhaoqi Wu, Some fixed point theorems in generalized probabilistic metric spaces,
Abstract and applied analysis, 2014.
Published
2025-02-14
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