Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence

Purshottam Agrawal; Behar Baxhaku

doi:10.33993/jnaat522-1350

Authors

Purshottam Agrawal Department of Mathematics, Indian Institute of Technology Roorkee, India
Behar Baxhaku Department of Mathematics, University of Prishtina "Hasan Prishtina", Kosovo https://orcid.org/0000-0002-8990-1440

DOI:

https://doi.org/10.33993/jnaat522-1350

Keywords:

Korovkin theorem, fuzzy number, $eq-stat^{D}_{CE}$, fuzzy positive linear operators, fuzzy modulus of continuity

Abstract views: 172

Abstract

We establish a fuzzy Korovkin type approximation theorem by using $eq-stat^{D}_{CE}$(deferred Ces\'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over $[a,b]$. Further, we determine the rate of convergence via fuzzy modulus of continuity.

Downloads

Download data is not yet available.

References

R. P. Agnew, On deferred Cesaro means, Ann. Math. 33 (3) (1932), 413–421. DOI: https://doi.org/10.2307/1968524

P. N. Agrawal , A. M. Acu and R. Chauhan, Weighted A–statistical convergence and Bogel approximation by operators of exponential type, J. Math. Inequal. 16(3) (2022), 827–850. https://doi.org/dx.doi.org/10.7153/jmi-2022-16-57 DOI: https://doi.org/10.7153/jmi-2022-16-57

P. N. Agrawal , A. M. Acu and T. Garg, Approximation of Bogel continuous functions and deferred weighted A– statistical convergence by Bernstein-Kantorovich operators on a triangle, J. Math. Inequal. 15(4) (2021), 1695–1711. https://doi.org/dx.doi.org/10.7153/jmi-2021-15-116 DOI: https://doi.org/10.7153/jmi-2021-15-116

P. N. Agrawal , B. Baxhaku and R. Shukla, Characterization of deferred type statistical convergence and P-summability method for operators: Applications to q-Lagrange–Hermite operator, Math. Methods Appl. Sci., 46(4) (2023), 4449–4465. https://doi.org/10.1002/mma.8770 DOI: https://doi.org/10.1002/mma.8770

M. Aiyub, K. Saini and K. Raj, Korovkin type approximation theorem via lacunary equistatistical convergence in fuzzy spaces, J. Math. Comput. Sci. 25 (2022), 312–321. http://dx.doi.org/10.22436/jmcs.025.04.02 DOI: https://doi.org/10.22436/jmcs.025.04.02

G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Studies in Fuzziness and Soft Computing, Springer-Verlag Berlin, Heidelberg (2010). DOI: https://doi.org/10.1007/978-3-642-11220-1

G. A. Anastassiou, Fuzzy approximation by fuzzy convolution type operators, Comput. Math. Appl., 48 (2004), 1369-1386. https://doi.org/10.1016/j.camwa.2004.10.027 DOI: https://doi.org/10.1016/j.camwa.2004.10.027

G. A. Anastassiou, On basic Fuzzy Korovkin Theory, Studia Univ. Babes - Bolyai, Mathematica, Vol. L, No. 4, (2005), 3-10.

G. A. Anastassiou, K. Demirci and S. Karakus A-summability and fuzzy trigonometric Korovkin-type approximation, J. Fuzzy Math. 19(2) (2011), 453–462.

G. A. Anastassiou, and O. Duman, Statistical fuzzy approximation by fuzzy positive linear operators, Comput. Math. Appl. 55 (2008), 573–580. https://doi.org/10.1016/j.camwa.2007.05.007 DOI: https://doi.org/10.1016/j.camwa.2007.05.007

G. A. Anastassiou and S. G. Gal, On fuzzy trigonometric Korovkin theory, Nonlinear Funct. Anal. Appl., 11 (2006), 385-395.

B. Baxhaku , P. N. Agrawal and R. Shukla, Some fuzzy Korovkin type approximation theorems via power series summability method, Soft Computing, 26(21) (2022), 11373–11379. https://doi.org/0.1007/s00500-022-07429-6 DOI: https://doi.org/10.1007/s00500-022-07429-6

A. A. Das, S. K. Paikray, T. Pradhan and H. Dutta, Statistical (C, 1)(E, μ)-summability and associated fuzzy approximation theorems with statistical fuzzy rates, Soft Computing 24 (2020), 10883–10892. https://doi.org/10.1007/s00500-019-04591-2 DOI: https://doi.org/10.1007/s00500-019-04591-2

K. Demirci, F. Dirik, and S. Yildiz, Approximation via equi-statistical convergence in the sense of power series method, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116 65 (2022), https://doi.org/10.1007/s13398-021-01191-4 DOI: https://doi.org/10.1007/s13398-021-01191-4

O. Duman and C. Orhan, Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett. 18 (2005), 1339–1344. https://doi.org/10.1016/j.aml.2005.02.029 DOI: https://doi.org/10.1016/j.aml.2005.02.029

H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. https://doi.org/10.4064/cm-2-3-4-241-244 DOI: https://doi.org/10.4064/cm-2-3-4-241-244

A. R. Freedman and J. J. Sember, Densities and summability., Pacific J. Math. 95 (1981), 293–305. https://doilorg/10.2140/PJM.1981.95.293 DOI: https://doi.org/10.2140/pjm.1981.95.293

A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), 129–138. https://doi.org/10.1216/rmjm/1030539612 DOI: https://doi.org/10.1216/rmjm/1030539612

S. Gal, Approximation theory in Fuzzy setting, Handbook of Analytic-Computational Methods in Applied Mathematics, Chapman & Hall/CRC, Boca Raton, (2000). DOI: https://doi.org/10.1201/9781420036053.ch13

R. Goetschel Jr. and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Set Syst., 10 (1983), 87-99. https://doi.org/10.1016/S0165-0114(83)80107-9 DOI: https://doi.org/10.1016/S0165-0114(83)80107-9

R. Goetschel Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Set Syst., 18 (1986), 31-43. https://doi.org/10.1016/0165-0114(86)90026-6 DOI: https://doi.org/10.1016/0165-0114(86)90026-6

E. H. Ismail and C. P. May, On a family of approximation operators, J. Math. Anal. Appl. 63 (1978), 446–462. https://doi.org/0022-247X/78/0632tl446$02.00/ DOI: https://doi.org/10.1016/0022-247X(78)90090-2

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

P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi, (1960)

M. Matloka, Sequence of fuzzy numbers, BUSEFAL 28 (1986), 28–37. DOI: https://doi.org/10.1016/S0009-9260(86)80158-1

S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin’s type approximation theorem., J. Inequal. Appl. 2016, 101 (2016). https://doi.org/10.1186/s13660-016-1040-1 DOI: https://doi.org/10.1186/s13660-016-1040-1

S. Nanda, On sequence of fuzzy numbers, Fuzzy sets Syst. 33 (1989), 123–126. https://doi.org/10.1016/0165-0114(89)90222-4 DOI: https://doi.org/10.1016/0165-0114(89)90222-4

F. Nuray and E. Savas Statistical convergence of sequences of fuzzy numbers, Math. Slovaca 45 (1995), 269–273.

M. Patro, S. K. Paikray and H. Dutta, Statistical deferred Euler summability mean and associated Korovkin type approximation theorem, Science & Technology Asia 25(1) (2020), 31–37. https://doi.org/10.14456/scitechasia.2020.4

K. Raj, K. Saini and M. Mursaleen, Applications of the fractional difference operator for studying Euler statistical convergence of sequences of fuzzy real numbers and associated Korovkin-type theorems, Probl. Anal. Issues Anal. 11(29), No. 3 (2022), 91–108. https://doi.org/10.15393/j3.art.2022.11770

K. Saini, K. Raj and M. Mursaleen, Deferred Cesaro and deferred Euler equistatistical convergence and its implications to Korovkin-type approximation theorem, Int. J. General Syst. 50(5) (2021), 567–579. https://doi.org/10.1080/03081079.2021.1942867 DOI: https://doi.org/10.1080/03081079.2021.1942867

H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Mishra, Deferred weighted A-statistical convergence based upon the (p, q)-Lagrange polynomials and its applications to approximation theorems, J. Appl. Anal. 24(1) (2018), 1-16. DOI: https://doi.org/10.1515/jaa-2018-0001

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74. DOI: https://doi.org/10.4064/cm-2-2-98-108

P. V. Subrahmanyam, Cesaro summability for fuzzy real numbers. p-adic analysis, summability theory, fuzzy analysis and applications (INCOPASFA) (Chennai, 1998). J. Anal. 7 (1999), 159–168.

Wu Cong-Xin, and Ma Ming, On embedding problem of fuzzy number space: Part 1, Fuzzy Set Syst., 44 (1991), 33-38. https://doi.org/10.1016/0165-0114(91)90030-T DOI: https://doi.org/10.1016/0165-0114(91)90030-T

E. Yavuz, Fuzzy trigonometric Korovkin type approximation via power series methods of summability, U. P. B. Sci. Bull. Series A 80(3) (2018), 123-132.

E. Yavuz, Basic trigonometric Korovkin approximation for fuzzy valued functions of two variables, C. R. Acad. Bulg. Sci. 74(9) (2021), 1276–1288. https://doi.org/10.48550/arXiv.1901.06682 DOI: https://doi.org/10.7546/CRABS.2021.09.02

L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X