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 ω1(f, 1/ √ n )a nd that they preserve the quasi-concavity of f .S ince B (M) n (f ) generates in a simple way a fuzzy number of the same support [a, b ]w ithf , 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 ,w e 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.