Abstract
We consider the concept of weighted mean summability method of improper Riemann integrals of fuzzy-number-valued functions with the help of fuzzy Riemann–Stieltjes integrals introduced by Ren and Wu (Int J Theor Phys 52:2134–2151, 2013). The convergence of the improper Rieman integral implies its summability by weighted mean method, but inverse requirement is not true in general and is realized by means of supplementary conditions known as Tauberian conditions.
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Altın Y, Mursaleen M, Altınok H (2010) Statistical summability \((C,1)\) for sequences of fuzzy real numbers and a Tauberian theorem. J Intell Fuzzy Syst 21:379–384
Anastassiou GA, Gal SG (2001) On a fuzzy trigonometric approximation theorem of Weierstrass-type. J Fuzzy Math 9(3):701–708
Anastassiou GA (2002) Rate of convergence of fuzzy neural network operators, univariate case. J Fuzzy Math 10(3):755–780
Anastassiou GA (2004) Univariate fuzzy-random neural network approximation operators. Comput Math Appl 48:263–283
Çanak İ (2014) On the Riesz mean of sequences of fuzzy real numbers. J Intell Fuzzy Syst 26(6):2685–2688
Çanak İ (2016) On Tauberian theorems for Cesàro summability of sequences of fuzzy numbers. J Intell Fuzzy Syst 30(5):2657–2662
Diamond P, Kloeden P (1990) Metric spaces of fuzzy sets. Fuzzy Sets Syst 35:241–249
Dubois D, Prade H (1980) Fuzzy sets and systems. Academic Press, New York
Dubois D, Prade H (1987) Fuzzy numbers: an overview. In: Bezdek J (ed) Analysis of fuzzy information, vol f. CRS, Boca Raton, pp 3–39
Gal SG (2000) Approximation theory in fuzzy setting. In: Anastassiou GA (ed) Handbook of analytic-computational methods in applied mathematics. Chapman and Hall, London, pp 617–666
Hardy GH (1949) Divergent series. Clarendon press, Oxford
Ming M (1993) On embedding problem of fuzzy number space, Part 4. Fuzzy Sets Syst 58:185–193
Móricz F (2004) Necessary and sufficient Tauberian conditions in case of weighted mean summable integrals over \( \mathbb{R} _{+}\). Math Inequalities Appl 7:87–93
Nanda S (1989) On sequences of fuzzy numbers. Fuzzy Sets Syst 33(1):123–126
Önder Z, Sezer SA, Çanak İ (2015) A Tauberian theorem for the weighted mean method of summability of sequences of fuzzy numbers. J Intell Fuzzy Syst 28(3):1403–1409
Ren X, Wu C (2013) The fuzzy Riemann-Stieltjes integral. Int J Theor Phys 52:2134–2151
Sezer SA, Çanak İ (2017) Power series methods of summability for series of fuzzy numbers and related Tauberian Theorems. Soft Comput 21:1057–1064
Talo Ö, Çakan C (2012) On the Cesàro convergence of sequences of fuzzy numbers. Appl Math Lett 25(4):676–681
Talo Ö, Başar F (2013) On the slowly decreasing sequences of fuzzy numbers. Abstr Appl Anal Art. doi:10.1155/2013/891986 (ID891986)
Talo Ö, Bal C (2016) On statistical summability \(\left( \overline{N}, p\right) \) of sequences of fuzzy numbers. Filomat 30(3):873–884
Tripathy BC, Baruah A (2010) Nörlund and Riesz mean of sequences of fuzzy real numbers. Appl Math Lett 23:651–655
Wu C, Ma M (1991) The basis of fuzzy analysis. Defense Industry Press, Beijing (in Chinese)
Wu HC (1998) The fuzzy Riemann Stieltjes integral. Int J Uncertain, Fuzziness Knowl Based Syst 6(1):51–67
Wu C, Gong Z (2001) On Henstock integral of fuzzy-number-valued functions (I). Fuzzy Sets Syst 120:523–532
Yavuz E, Çoşkun H (2016) On the logarithmic summability method for sequences of fuzzy numbers, Soft Comput. doi:10.1007/s00500-016-2156-4
Yavuz E, Talo Ö, Çoşkun H (2016) Cesàro summability of integrals of fuzzy-number-valued functions. arXiv:1604.05338v1
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Belen, C. Tauberian theorems for weighted mean summability method of improper Riemann integrals of fuzzy-number-valued functions. Soft Comput 22, 3951–3957 (2018). https://doi.org/10.1007/s00500-017-2606-7
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DOI: https://doi.org/10.1007/s00500-017-2606-7