Asymptotic properties of iterates of certain positive linear operators

Abstract

In this paper we prove Korovkin type theorem for iterates of general positive linear operators T : C [0, 1] -> C [0, 1] and derive quantitative estimates in terms of modulus of smoothness. In particular, we show that under some natural conditions the iterates T-m : C [0, 1] -> C [0, 1] converges strongly to a fixed point of the original operator T. The results can be applied to several well-known operators; we present here the q-MKZ operators, the q-Stancu operators, the genuine q-Bernstein-Durrmeyer operators and the Cesaro operators. (C) 2012 Elsevier Ltd. All rights reserved.