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On the Borel summability method for convergence of triple sequences of Bernstein–Stancu operators of fuzzy numbers

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Abstract

We define Borel rough summable of triple sequences and discuss some fundamental results related to Borel rough summable of triple Bernstein–Stancu operators based on (p, q)-integers. Further, we study rough-weighted generalized statistical convergence method for some approximation properties for (p, q)-analogue of Bernstein–Stancu operators for triple sequence of fuzzy numbers. With the help of an example we illustrate our approximation results.

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References

  • Alotaibi A, Mursaleen M, Mohiuddine SA (2015) Korovkin type approximation theorems for \(\sigma \)-convergence of double sequences. J Nonlinear Convex Anal 16(1):183–192

    MathSciNet  Google Scholar 

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

    Google Scholar 

  • Anastassiou GA (2005) On Basic Fuzzy Korovkin Theory. Studia Universitatis Babeş Bolyai-Mathematica 50:3–10

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Aytar S (2008a) Rough statistical convergence. Numer Funct Anal Optim 29(3–4):291–303

    MathSciNet  Google Scholar 

  • Aytar S (2008b) The rough limit set and the core of a real sequence. Numer Funct Anal Optim 29(3–4):283–290

    MathSciNet  Google Scholar 

  • Aytar S, Pehlivan S (2008) Statistical convergence of sequences of fuzzy numbers and sequences of \(\lambda \)-cuts. Int J Gen Syst 37(2):231–237

    MathSciNet  Google Scholar 

  • Belen C, Mursaleen M, Yildirim M (2012) Statistical \(A\)-summability of double sequences and a Korovkin type approximation theorem. Bull Korean Math Soc 49(4):851–861

    MathSciNet  Google Scholar 

  • Çanak I (2015) Tauberian theorems for the product of Borel and Hölder summability methods. Analele Ştiintifice ale Universitatii “Al I Cuza” din Iaşi. https://doi.org/10.2478/aicu-2014-0039

  • Çanak I, Totur Ü (2010) Some Tauberian theorems for Borel summability methods. Appl Math Lett 23:302–305

    MathSciNet  Google Scholar 

  • Diamond P, Kloeden P (1990) Metric spaces of fuzzy sets. Fuzzy Sets Syst 35:241–249

    MathSciNet  Google Scholar 

  • Dündar E, Cakan C (2014) Rough \(I\)-convergence. Demonstratio Math 47(3):638–651

    MathSciNet  Google Scholar 

  • Dutta H, Gogoi J (2019) Weighted \(\lambda \)-statistical convergence connecting a statistical summability of sequences of fuzzy numbers and Korovkin-type approximation theorems. Soft Comput 23:12883–12895

    Google Scholar 

  • Edely OHH, Mohiuddine SA, Noman AK (2010) Korovkin type approximation theorems obtained through generalized statistical convergence. Appl Math Lett 23:1382–1387

    MathSciNet  Google Scholar 

  • Esi A, Araci S, Acikgoz M (2016) Statistical convergence of Bernstein operators. Appl Math Inf Sci 10(6):2083–2086

    MathSciNet  Google Scholar 

  • Esi A, Araci S, Esi A (2017a) \(\lambda \)-statistical convergence of Bernstein polynomial sequences. Adv Appl Math Sci 16(3):113–119

    Google Scholar 

  • Esi A, Subramanian N, Esi A (2017b) On triple sequence space of Bernstein operator of rough \(I\)-convergence pre-Cauchy sequences. Proyecciones 36(4):567–587

    MathSciNet  Google Scholar 

  • Gadjiev AD, Orhan C (2002) Some approximation theorems via statistical convergence. Rocky Mt J Math 32:129–138

    MathSciNet  Google Scholar 

  • Hazarika B, Subramanian N, Esi A (2018) On rough weighted ideal convergence of triple sequence of Bernstein polynomials. Proc Jangjeon Math Soc 21(3):497–506

    MathSciNet  Google Scholar 

  • Hazarika B, Alotaibi A, Mohiuddine SA (2020) Statistical convergence in measure for double sequences of fuzzy-valued functions. Soft Comput 24(9):6613–6622

    Google Scholar 

  • Kadak U (2016a) On weighted statistical convergence based on \((p, q)\)-integers and related approximation theorems for functions of two variables. J Math Anal Appl 443:752–764

    MathSciNet  Google Scholar 

  • Kadak U (2016b) On weighted statistical convergence based on \((p, q)\)-integers and related approximation theorems for functions of two variables. J Math Anal Appl 443(2):752–764

    MathSciNet  Google Scholar 

  • Kadak U (2017) Weighted statistical convergence based on generalized difference operator involving \((p, q)\)-Gamma function and its applications to approximation theorems. J Math Anal Appl 448(2):1633–1650

    MathSciNet  Google Scholar 

  • Karaisa A, Kadak U (2017) On \(\alpha \beta \)-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem. Filomat 31(12):3749–3760

    MathSciNet  Google Scholar 

  • Khan K, Lobiyal DK (2017) Bézier curves based on Lupaş-analogue of Bernstein functions in CAGD. J Comput Appl Math 317:458–477

    MathSciNet  Google Scholar 

  • Khan A, Sharma V (2018) Statistical approximation by \((p, q)\)-analogue of Bernstein–Stancu operators. Azerb J Math 8(2):100–121

    MathSciNet  Google Scholar 

  • 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(1):1–12

    MathSciNet  Google Scholar 

  • Karakaya V, Chishti TA (2009) Weighted statistical convergence. Iran J Sci Technol Trans A Sci 33:219–223

    MathSciNet  Google Scholar 

  • Korovkin PP (1960) Linear operators and approximation theory. Hindustan Publishing Corporation, Delhi

    Google Scholar 

  • Mohiuddine SA, Hazarika B, Alghamdi MA (2019a) Ideal relatively uniform convergence with Korovkin and Voronoskaya type approximation theorems. Filomat 33(14):4549–4560

    MathSciNet  Google Scholar 

  • Mohiuddine SA, Asiri A, Hazarika B (2019b) 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

    MathSciNet  Google Scholar 

  • Mursaleen M, Alotaibi A (2013) Korovkin type approximation theorem for statistical \(A\)-summability of double sequences. J Comput Anal Appl 15(6):1036–1045

    MathSciNet  Google Scholar 

  • Mursaleen M, Khan F, Khan A (2015a) Approximation properties for modified \(q\)-Bernstein–Kantorovich operators. Numer Funct Anal Optim 36(9):1178–1197

    MathSciNet  Google Scholar 

  • Mursaleen M, Ansari KJ, Khan A (2015b) on \((p,q)\)-analogue of Bernstein Operators. Appl Math Comput 266:874–882 [Erratum: Appl Math Comput 266(2015) 874–882]

  • Mursaleen M, Nasiruzzaman M, Khan A, Ansari KJ (2016a) Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p, q)\)-integers. Filomat 30(3):639–648

    MathSciNet  Google Scholar 

  • Mursaleen M, Sarsenbi Abdizhahan M, Khan T (2016b) On \((p, q)\)-analogue of two parametric Stancu–Beta operators. J Inequal Appl 2016:190

  • Mursaleen M, Nasiruzzaman M, Ansari KJ, Alotaibi A (2017) Generalized \((p, q)\)-Bleimann–Butzer–Hahn operators and some approximation results. J Inequal Appl 2017:310

  • Nanda S (1989) On sequence of fuzzy numbers. Fuzzy Sets Syst 33:123–126

  • Phu HX (2001) Rough convergence in normed linear spaces. Numer Funct Anal Optim 22:199–222

  • Phu HX (2002) Rough continuity of linear operators. Numer Funct Anal Optim 23:139–146

    MathSciNet  Google Scholar 

  • Phu HX (2003) Rough convergence in infinite dimensional normed spaces. Numer Funct Anal Optim 4:285–301

    MathSciNet  Google Scholar 

  • Sahiner A, Tripathy BC (2008) Some \(I\)-related properties of triple sequences. Selcuk J Appl Math 9(2):9–18

    MathSciNet  Google Scholar 

  • Sahiner A, Gürdal M, Duden FK (2007) Triple sequences and their statistical convergence. Selcuk J Appl Mah 8(2):49–55

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Talo Ö, Başar F (2009) Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations. Comput Math Appl 58(4):717–733

    MathSciNet  Google Scholar 

  • Yavuz E, Çoşkun H (2006) On the Borel summability method of sequences of fuzzy numbers. J Intell Fuzzy Syst 30:2111–2117

    Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, SASTRA University, Thanjavur, 613 401, India

    A. Indumathi & N. Subramanian

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

    Bipan Hazarika

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  1. A. Indumathi
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  2. N. Subramanian
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  3. Bipan Hazarika
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Correspondence to Bipan Hazarika.

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Indumathi, A., Subramanian, N. & Hazarika, B. On the Borel summability method for convergence of triple sequences of Bernstein–Stancu operators of fuzzy numbers. Soft Comput 25, 683–697 (2021). https://doi.org/10.1007/s00500-020-05178-y

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