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Statistical convergence in measure for double sequences of fuzzy-valued functions

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Abstract

The aim of this paper is to define two sorts of convergence in measure, that is, outer and inner statistical convergence, for double sequences of fuzzy-valued measurable functions and demonstrate that both kinds of convergence are equivalent in a finite measurable set. We also define the notion of statistical convergence in measure for double sequences of fuzzy-valued measurable functions and establish several interesting results. In addition, we prove the statistical version of Egorov’s theorem for double sequences of fuzzy-valued functions defined on a finite measure space.

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References

  • Altın Y, Et M, Tripathy BC (2007) On pointwise statistical convergence sequences of fuzzy mappings. J Fuzzy Math 15(2):425–433

    MathSciNet  MATH  Google Scholar 

  • Altinok H, Altin Y, Işık M (2012) Statistical convergence and strong \(p\)-Cesàro summability of order \(\beta \) in sequences of fuzzy numbers. Iran J Fuzzy Syst 9(2):65–75

    MATH  Google Scholar 

  • Altinok H, Et M (2019) Statistical convergence of order \((\beta,\gamma )\) for sequences of fuzzy numbers. Soft Comput 23:6017–6022

    Article  MATH  Google Scholar 

  • Anastassiou GA, Duman O (2008) Statistical fuzzy approximation by fuzzy positive linear operators. Comput Math Appl 55(3):573–580

    Article  MathSciNet  MATH  Google Scholar 

  • Balcerzak M, Dems K, Komisarski A (2007) Statistical convergence and ideal convergence for sequences of functions. J Math Anal Appl 328:715–729

    Article  MathSciNet  MATH  Google Scholar 

  • Belen C, Mohiuddine SA (2013) Generalized weighted statistical convergence and application. Appl Math Comput 219:9821–9826

    MathSciNet  MATH  Google Scholar 

  • Çanak I, Totur U, Önder Z (2017) A Tauberian theorem for \((C,1,1)\) summable double sequences of fuzzy numbers. Iran J Fuzzy Syst 14(1):61–75

    MathSciNet  MATH  Google Scholar 

  • Çınar M, Karakaş M, Et M (2013) On pointwise and uniform statistical convergence of order \(\alpha \) for sequences of functions. Fixed Point Theory Appl 2013:33

    Article  MathSciNet  MATH  Google Scholar 

  • Et M, Erol Y (2018) On deferred statistical convergence of order \(\beta \) of sequences of fuzzy numbers. J Intell Fuzzy Syst 35(3):3747–3755

    Article  Google Scholar 

  • Fast H (1951) Sur la convergence statistique. Coll Math 2:241–244

    Article  MathSciNet  MATH  Google Scholar 

  • Fridy JA (1985) On statistical convergence. Analysis 5(4):301–313

    Article  MathSciNet  MATH  Google Scholar 

  • Gökhan A, Güngör M, Et M (2007) Statistical convergence of double sequences of real valued functions. Int Math Forum 2(8):365–374

    Article  MathSciNet  MATH  Google Scholar 

  • Gong Z, Zhang L, Zhu X (2015) The statistical convergence for sequences of fuzzy-number-valued functions. Inf Sci 295:182–195

    Article  MathSciNet  MATH  Google Scholar 

  • Ilkhan M, Kara EE (2018) A new type of statistical Cauchy sequence and its relation to Bourbaki completeness. Cogent Math Stat 5(1):1–9

    Article  MathSciNet  MATH  Google Scholar 

  • Kadak U, Mohiuddine SA (2018) Generalized statistically almost convergence based on the difference operator which includes the \((p, q)\)-gamma function and related approximation theorems. Results Math 73(1)

  • Khan MS, Lohani QMD, Mursaleen M (2017) A novel intuitionistic fuzzy similarity measure based on double sequence by using modulus function with application in pattern recognition. Cogent Math 4

  • Kim YK, Ghil BM (1997) Integrals of fuzzy-number-valued functions. Fuzzy Sets Syst 86:213–222

    Article  MathSciNet  MATH  Google Scholar 

  • Kostyrko P, Šalát T, Wilczyński W (2000/2001) \({\cal{I}}\)-convergence. Real Anal Exchange 26:669–689

  • Mohiuddine SA, Alamri BAS (2019) Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev R Acad Cienc Exactas Fs Nat Ser A Math RACSAM 113(3):1955–1973

    Article  MathSciNet  MATH  Google Scholar 

  • Mohiuddine SA, Alotaibi A, Mursaleen M (2012) Statistical convergence of double sequences in locally solid Riesz spaces. Abstr Appl Anal 2012:9 Article ID 719729

    MathSciNet  MATH  Google Scholar 

  • Mohiuddine SA, Asiri A, Hazarika B (2019) Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int J Gen Syst 48(5):492–506

    Article  MathSciNet  Google Scholar 

  • Mohiuddine SA, Hazarika B, Alotaibi A (2017) On statistical convergence of double sequences of fuzzy valued functions. J Intell Fuzzy Syst 32:4331–4342

    Article  MATH  Google Scholar 

  • Mursaleen M, Edely OHH (2003) Statistical convergence of double sequences. J Math Anal Appl 288:223–231

    Article  MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Mohiuddine SA (2009) Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 41(5):2414–2421

    Article  MathSciNet  MATH  Google Scholar 

  • Negoita CV, Ralescu D (1975) Applications of fuzzy sets to systems analysis. Wiley, New York

    Book  MATH  Google Scholar 

  • Nuray F, Savaş E (1995) Statiatical convergence of sequences of fuzzy numbers. Math Slovaca 45(3):269–273

    MathSciNet  MATH  Google Scholar 

  • Önder Z, Çanak I, Totur U (2017) Tauberian theorems for statistically \((C,1,1)\) summable double sequences of fuzzy numbers. Open Math 15(1):157–178

    Article  MathSciNet  MATH  Google Scholar 

  • Pringsheim A (1900) Zur Ttheorie der zweifach unendlichen Zahlenfolgen. Math Ann 53:289–321

    Article  MathSciNet  MATH  Google Scholar 

  • S̆alát T (1980) On statistically convergent sequences of real numbers. Math Slovaca 30:139–150

    MathSciNet  Google Scholar 

  • Savaş E (1996) A note on double sequences of fuzzy numbers. Turkish J Math 20(2):175–178

    MathSciNet  MATH  Google Scholar 

  • Savaş E, Mursaleen M (2004) On statistically convergent double sequences of fuzzy numbers. Inf Sci 162(3–4):183–192

    Article  MathSciNet  MATH  Google Scholar 

  • Schoenberg IJ (1959) The integrability of certain functions and related summability methods. Am Math Monthly 66:361–375

    Article  MathSciNet  MATH  Google Scholar 

  • Talo Ö, Bal C (2016) On statistical summability \((N, P)\) of sequences of fuzzy numbers. Filomat 30(3):873–884

    Article  MathSciNet  MATH  Google Scholar 

  • Talo Ö, Bayazit F (2017) Tauberian theorems for statistically convergent double sequences of fuzzy numbers. J Intell Fuzzy Syst 32(3):2617–2624

    Article  MATH  Google Scholar 

  • Tripathy B, Tripathy BC (2005) On I-convergence of double sequences. Soochow J Math 31:549–560

    MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MATH  Google Scholar 

  • Zhang B (2001) On measurability of fuzzy-number-valued functions. Fuzzy Sets Syst 120:505–509

    Article  MathSciNet  Google Scholar 

  • Zygmund A (1935) Trigonometrical Series, vol. 5 of Monografýas de Matemáticas. Warszawa-Lwow

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Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant No. (RG-18-130-37). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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Authors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, 781014, India

    Bipan Hazarika

  2. Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh, 791112, India

    Bipan Hazarika

  3. Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Abdullah Alotaibi & S. A. Mohiuddine

  4. Department of General Required Courses, Mathematics, Faculty of Applied Studies, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    S. A. Mohiuddine

Authors
  1. Bipan Hazarika
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  2. Abdullah Alotaibi
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  3. S. A. Mohiuddine
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Correspondence to Bipan Hazarika.

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The authors contributed equally in writing this article and declare that they have no conflict of interest.

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Communicated by A. Di Nola.

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Hazarika, B., Alotaibi, A. & Mohiuddine, S.A. Statistical convergence in measure for double sequences of fuzzy-valued functions. Soft Comput 24, 6613–6622 (2020). https://doi.org/10.1007/s00500-020-04805-y

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