| Title | HIGHER ORDER FUZZY KOROVKIN THEORY VIA INEQUALITIES |
| Publication Type | Journal Article |
| Year of Publication | 2006 |
| Authors | Anastassiou, GA |
| Secondary Title | Communications in Applied Analysis |
| Volume | 10 |
| Issue | 3 |
| Start Page | 361 |
| Pagination | 395 |
| Date Published | 08/2006 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 26E50, 28E10, 41A17, 41A25, 41A36, 47S40 |
| Abstract | Here is studied with rates the fuzzy uniform and L p , p ≥ 1, convergence of a sequence of fuzzy positive linear operators to the fuzzy unit operator acting on spaces of fuzzy differentiable functions. This is done quantitatively via fuzzy Korovkin type inequalities involving the fuzzy modulus of continuity of a fuzzy derivative of the engaged function. From there we deduce general fuzzy Korovkin type theorems with high rate of convergence. The surprising fact is that basic real positive linear operator simple assumptions enforce here the fuzzy convergences. At the end we give applications. Our results are univariate and multivariate. The assumptions are minimal and natural fulfilled by almost all example—fuzzy positive linear operators. |
| URL | http://www.acadsol.eu/en/articles/10/3/7.pdf |
| Short Title | Higher Order Fuzzy Theory |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York and London, 1985.
[2] George A. Anastassiou, Moments in Probability and Approximation Theory, Pitman, Longman, 287, UK, 1993. [3] George A. Anastassiou, Fuzzy approximation by fuzzy convolution type operators, Computers and Mathematics with Applications, 48 (2004), 1369-1386. [4] George A. Anastassiou, On basic Fuzzy Korovkin theory, Studia Univ. Babes - Bolyai, Mathematica, 2005, Romania, To Appear. [5] George A. Anastassiou, Fuzzy Korovkin theorems and inequalities, 2005, To Appear. [6] G.A. Anastassiou and S. Gal, On a fuzzy trigonometric approximation theorem of Weierstrass-type, Journal of Fuzzy Mathematics, 9 (2001), no. 3, 701-708. [7] G.A. Anastassiou and S. Gal, On Fuzzy trigonometric Korovkin theory, J. of Nonlinear Functional Analysis and Applic., Korea (2006), To Appear. [8] R. Cristescu, Ordered structures in ordered vector spaces, Ed. St. Enciclop., Bucharest, 1983, In Romanian. [9] S. Gal, Approximation theory in Fuzzy setting, Chapter 13, pp. 617-666, In: Handbook of Analyticomputational Methods in Applied Mathematics (Ed. G.A. Anastassiou), Chapman and Hall/CRC, Boca Raton, 2000. [10] R. Goetschel Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43. [11] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[12] L.V. Kantorovich, On the moment problem for a finite interval, Dokl. Akad. Nauk SSSR, 14 (1937), 531-537, In Russian. [13] J.H.B. Kemperman, The general moment problem, a geometric approach, The Annals of Mathematical Statistics, 39 (1968), no. 1, 93-122. [14] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., Delhi, India, 1960. [15] O. Shisha and B. Mond, The degree of convergence of sequences of linear positive operators, Nat. Acad. of Sci. U.S., 60 (1968), 1196-1200. [16] Congxin Wu and Ming Ma, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and Systems, 44 (1991), 33-38. [17] Congxin Wu and Zengtai Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems, 120 (2001), 523-532. |