  We study the iterations of a class of curvature image operators
$\Lambda_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016)
2133--2165). The fixed points of these operators are the solutions of the $L_p$
Minkowski problems with the positive continuous prescribed data $\varphi$. One
of our results states that if $p\in (-n,1)$ and $\varphi$ is even, or if $p\in
(-n,-n+1]$, then the iterations of these operators applied to suitable convex
bodies sequentially converge in the Hausdorff distance to fixed points.
