Asymptotic properties of iterates of certain positive linear operators

In this paper we prove Korovkin type theorem for iterates of general positive linear operators $T:C[0,1]\to C[0,1]$ and derive quantitative estimates in terms of modulus of smoothness. In particular, we show that under some natural conditions the iterates $T^m:C[0,1]\to C[0,1]$ 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.