Asymptotic properties of iterates of certain positive linear operators

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 Tm: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.