Document Type : Research Paper
Author
Department of Mathematics, Faculty of Science, Gazi University, P.O.Box 06500, Ankara, Turkey.
Abstract
In this study, we purpose to extend approximation properties of the $ (p,q)$-Bernstein-Chlodowsky operators from real function spaces to fuzzy function spaces. Firstly, we define fuzzy $ (p,q)$-Bernstein-Chlodowsky operators, and we give some auxiliary results. Later, we give a fuzzy Korovkin-type approximation theorem for these operators. Additionally, we investigate rate of convergence by using first order fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Eventually, we give an estimate for fuzzy asymptotic expansions of the fuzzy $ (p,q)$-Bernstein-Chlodowsky operators.
Keywords
[1] O. Agratini and G. Nowak, On a generalization of Bleimann, Butzer and Hahn operators based on q-integers}, Math. Comput. Model., {53} (5-6) (2011), pp. 699-706.
[2] G.A. Anastassiou, Fuzzy Mathematics: Approximation Theory}, Springer-Verlag, Berlin, 2010.
[3] G.A. Anastassiou and S.G. Gal, On a fuzzy trigonometric approximation theorem of Weierstrass-type}, J. Fuzzy Math., {9} (3) (2001), pp. 701-708.
[4] K.J. Ansari and A. Karaisa, On the approximation by Chlodowsky
type generalization of $ (p,q)$-Bernstein operators}, Int. J. Nonlinear Anal. Appl., {8} (2) (2017), pp. 181-200.
type generalization of $ (p,q)$-Bernstein operators}, Int. J. Nonlinear Anal. Appl., {8} (2) (2017), pp. 181-200.
[5] A. Aral, A generalization of Szasz-Mirakyan operators based on q-integers}, Math. Comput. Model., {47} (9-10) (2008), pp. 1052-1062.
[6] B. Bede and S.G. Gal, Best approximation and Jackson-type estimates by generalized fuzzy polynomials}, J. Concr. Appl. Math. , {2} (3) (2004), pp. 213-232.
[7] I. Buyukyazici and E. Ibikli, Inverse theorems for Bernstein-Chlodowsky type polynomials}, J. Math. Kyoto U., {46} (2006), pp. 21-29.
[8] I. Chlodowsky, Sur le developpement des fonctions definies dan un intervalle infini en series de polnomes de M.S. Bernstein}, Compositio Math., {4} (1937), pp. 380-393.
[9] L. Coroianu, G.S. Gal and B. Bede, Approximation of fuzzy numbers by max-product Bernstein operators}, Fuzzy Set Syst., {257}, (2014), pp. 41-66.
[10] B. Daraby and J. Jafari, Some properties of fuzzy real numbers}, Sahand Commun. Math. Anal., {3} (1), (2016), pp. 21-27.
[10] B. Daraby and J. Jafari, Some properties of fuzzy real numbers}, Sahand Commun. Math. Anal., {3} (1), (2016), pp. 21-27.
[11] D. Dubois and H. Prada, Fuzzy Numbers: An overview in: "Analysis of fuzzy information, vol. 1, Mathematical Locig", pp. 3-39} , CRC Press, Boca Raton, 1987.
[12] R. Gotschel and W. Voxman, Elementary fuzzy calculus}, Fuzzy Set Syst., {18} (1986), pp. 31-43.
[13] H. Karsli and V. Gupta, Some approximation properties of q-Chlodowsky operators}, Appl. Math. Comput., {195} (2008), pp. 220-229.
[14] A.D. Gadjiev, R.O. Efendiev and E. Ibikli, Generalized Bernstein-Chlodowsky polynomials}, Rocky Mt. J. Math, {28} (4) (1998), pp. 1267-1277.
[15] E.A. Gadjieva and E. Ibikli, Weighted approximation by Bernstein-Chlodowsky polynomials}, Indian J. Pure Ap. Math., {30} (1) (1999), pp. 83-87.
[16] S.G. Gal, A fuzzy variant of the Weierstrass' approximation theorem}, J. Fuzzy Math. , {1} (4) (1993), pp. 865-872.
[17] S.G. Gal, Degree of approximation of fuzzy mappings by fuzzy polynomials}, J. Fuzzy Math. , {2} (4) (1994), pp. 847-853.
[18] S.G. Gal, Approximate selections for fuzzy-set valued mappings and applications}, J. Fuzzy Math. , {3} (4) (1995), pp. 941-947.
[19] S.G. Gal, Linear continuous functionals on FN -type spaces}, J. Fuzzy Math. , {17} (3) (2009), pp. 535-553.
[20] S.G. Gal and V. Gupta, Approximation of vector-valued functions to random and fuzzy approximation}, An. Univ. Oradea Fasc. Mat., {16} (2009), pp. 233-240.
[21] M. Golmohammadi, S. Najafzadeh and M. Foroutan, Some Properties of Certain Subclass of Meromorphic Functions Associated with (p,q) -derivative}, Sahand Commun. Math. Anal., {17} (4) (2020), pp. 71-84.
[22] H.G. İnce İlarslan and E.Y. Ozkan, Approximation properties of bivariate generalization of Meyer König and Zeller type operators}, An. c{S}tiin~{I}c{t}. Univ. Al. I. Cuza Iac{s}i Mat. (N.S.), 63 (1) (2017), 181-191.
[23] N. İspir and E.Y. Ozkan, Approximation properties complex q-Balazs-Szabados operators in compact disks}, J. Inequal. Appl., 2013: {361} (2013).
[24] P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions}, Dokl. Akad. Nauk. SSSR , {90} (1953), pp. 961-964.
[25] A. Lupac{s}, A q-analogue of the Bernstein operator}, University of Cluj-Napoca, Seminar on Numerical and Statistical, No.9, 1987.
[26] V.N. Mishra, M. Mursaleen, S. Pandey and A. Alotaibi, Approximation properties of Chlodowsky variant of (p,q) Bernstein-Stancu-Schurer operators}, J. Inequal. Appl., 2017:{176} (2017).
[27] M. Mursaleen, K.J. Ansari and A. Khan, On $ (p,q)$-analogue of Bernstein operators}, Appl. Math. Comput., {266} (2015), pp. 874-882.
[28] M. Mursaleen, K.J. Ansari and A. Khan, Erratum to "On $ (p,q)$-analogue of Bernstein operators}, Appl. Math. Comput., {278} (2016), pp. 70-71.
[29] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by $ (p,q)$-analogue of Bernstein-Stancu operators}, Appl. Math. Comput., {264} (2015), pp. 392-402
[30] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results for Bernstein-Kantorovich operators based on $ (p,q)$-calculus}, U.P.B. Sci. Bull. Series A, {78} (4) (2016), pp. 129-142.
[31] M. Mursaleen, A.A.H. Al-Abied and A. Alotaibi, On $ (p,q)$-Sz azs-Mirakyan operators and their approximation properties}, J. Inequal. Appl., 2017:{196} (2017).
[32] M. Mursaleen, MD Naziruzzaman, K.J. Ansari and A. Alotaibi, Generalized $ (p,q)$-Bleimann-Butzer-Hahn operators and some approximation results}, J. Inequal. Appl., 2017:{310} (2017).
[26] V.N. Mishra, M. Mursaleen, S. Pandey and A. Alotaibi, Approximation properties of Chlodowsky variant of (p,q) Bernstein-Stancu-Schurer operators}, J. Inequal. Appl., 2017:{176} (2017).
[27] M. Mursaleen, K.J. Ansari and A. Khan, On $ (p,q)$-analogue of Bernstein operators}, Appl. Math. Comput., {266} (2015), pp. 874-882.
[28] M. Mursaleen, K.J. Ansari and A. Khan, Erratum to "On $ (p,q)$-analogue of Bernstein operators}, Appl. Math. Comput., {278} (2016), pp. 70-71.
[29] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by $ (p,q)$-analogue of Bernstein-Stancu operators}, Appl. Math. Comput., {264} (2015), pp. 392-402
[30] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results for Bernstein-Kantorovich operators based on $ (p,q)$-calculus}, U.P.B. Sci. Bull. Series A, {78} (4) (2016), pp. 129-142.
[31] M. Mursaleen, A.A.H. Al-Abied and A. Alotaibi, On $ (p,q)$-Sz azs-Mirakyan operators and their approximation properties}, J. Inequal. Appl., 2017:{196} (2017).
[32] M. Mursaleen, MD Naziruzzaman, K.J. Ansari and A. Alotaibi, Generalized $ (p,q)$-Bleimann-Butzer-Hahn operators and some approximation results}, J. Inequal. Appl., 2017:{310} (2017).
[33] E.Y. Ozkan and N. İspir, Approximation by $ (p,q)$-analogue of Balazs-Szabados operators}, Filomat, {32} (6) (2018), pp. 2257-2271.
[34] E.Y. Ozkan, Quantitative estimates for the tensor product $ (p,q)$-Balazs-Szabados operators}, Filomat, {34} (3) (2020), pp. 779-793.
[35] E.Y. Ozkan, Statistical approximation properties of q-Balazs-Szabados-Stancu operators}, Filomat, {28} (9) (2014), pp. 1943-1952.
[35] E.Y. Ozkan, Statistical approximation properties of q-Balazs-Szabados-Stancu operators}, Filomat, {28} (9) (2014), pp. 1943-1952.
[36] E.Y. Ozkan, Approximation properties of Kantorovich type q-Balazs-Szabados operators}, Demonstr. Math., {52} (1) (2019), pp. 10-19.
[37] E.Y. Ozkan, Approximation properties of bivariate complex q-Balazs-Szabados operators of tensor product kind}, J. Inequal. Appl., 2014: {20} (2014).
[38] E.Y. Ozkan, An upper estimate of complex q-Balazs-Szabados-Kantorovich operators on compact disks}, G.U.J. Sci., 29 (2016), 479-486.
[39] A.E. Piriyeva, On order of approximation of functions by generalized Bernstein-Chlodowsky polynomials}, Proc. Inst. Math. Mech., {21} (2004), pp. 157-164.
[40] L-T. Shu, G. Zhou and Q-B. Chai, On the convergence of a family of Chlodowsky type Bernstein-Stancu-Shurer operators}, J. Funct. Spaces, 2018: {6385451} (2018), 15 pp.
[41] T. Tiberiu, Meyer-K"{o}nig and Zeller operators based on q-integers}, Rev. Anal. Numer. Theory Approx., {29} (2) (2000), pp. 221-229.
[42] C. Wu and L. Danghang, A fuzzy variant Weierstrass approximation theorem}, J. Fuzzy Math., {7} (1) (1999), pp. 101-104.
[43] C. Wu and M. Ming, On embedding problem of fuzzy number space Part I}, Fuzzy Set Syst., {44} (1991), pp. 33-38.
[44] C. Wu and G. Zengtai, On Henstock integral of fuzzy-number-valued functions Part I}, Fuzzy Set Syst., {120} (3) (2001), pp. 523-532.
[45] L.A. Zareh, Fuzzy sets}, Inform. and Control, {8} (1965), pp. 338-353.
)
