Random fixed point equations and inverse problems by collage theorem

In this paper we are interested in the direct and inverse problems for the following class of random fixed point equations $T(w,x(w))=x(w)$ where $T:\Omega\times X\to X$ is a given operator, $\Omega$ is a probability space and $X$ is a complete metric space. The inverse problem is solved by recourse to the collage theorem for contractive maps. We then consider two applications: (i) random integral equations and (ii) random iterated function systems with greyscale maps (RIFSM), for which noise is added to the classical IFSM.