Approximation of fuzzy numbers by nonlinear bernstein operators of max-product kind

Abstract

In this paper firstly we extend from [0, 1] to an arbitrary compact interval [a, b], the definition of the nonlinear Bernstein operators of max-product kind, B(M) n (f), n N, by proving that their order of uniform approximation to f is 1(f, 1/n) and that they preserve the quasi-concavity of f. Since B(M) n (f) generates in a simple way a fuzzy number of the same support [a, b] with f, it turns out that these results are very suitable in the approximation of the fuzzy numbers. Thus, besides the approximation properties, for sufficiently large n, we prove that these nonlinear operators preserve the non-degenerate segment core of the fuzzy number f and, in addition, the segment cores of B(M) n (f), n N, approximate the segment core of f with the order 1/n. © 2011. The authors-Published by Atlantis Press.