FUZZY KOROVKIN THEORY AND INEQUALITIES

Abstract

Here we study the fuzzy positive linear operators acting on fuzzy continuous functions. We prove the fuzzy Riesz representation theorem, the fuzzy Shisha–Mond type inequalities and fuzzy Korovkin type theorems regarding the fuzzy convergence of fuzzy positive linear operators to the fuzzy unit in various cases. Special attention is paid to the study of fuzzy weak convergence of finite positive measures to the unit Dirac measure. All convergences are with rates and are given via fuzzy inequalities involving the fuzzy modulus of continuity of the engaged fuzzy valued function. The assumptions for the Korovkin theorems are minimal and of natural realization, fulfilled by almost all example – fuzzy positive linear operators. The surprising fact is that the real Korovkin test functions assumptions carry over here in the fuzzy setting and they are the only enough to impose the conclusions of fuzzy Korovkin theorems. We give a lot of examples and applications to our theory, namely: to fuzzy Bernstein operators, to fuzzy Shepard operators, to fuzzy Szasz–Mirakjan and fuzzy Baskakov-type operators and to fuzzy convolution type operators.

We work in general, basically over real normed vector space domains that are compact and convex or just convex. On the way to prove the main theorems we establish a lot of other interesting and important side results This chapter relies on [24].