This paper studies the solvability, existence of unique solution, closed-form
solution and numerical solution of matrix equation X=Af(X)B+C with f(X)=XT,f(X)=Xˉ and f(X)=XH, where X is the
unknown. It is proven that the solvability of these equations is equivalent to
the solvability of some auxiliary standard Stein equations in the form of
W=AWB+C where the dimensions of the coefficient
matrices A,B and C are the same as those of
the original equation. Closed-form solutions of equation X=Af(X)B+C can then
be obtained by utilizing standard results on the standard Stein equation. On
the other hand, some generalized Stein iterations and accelerated Stein
iterations are proposed to obtain numerical solutions of equation equation
X=Af(X)B+C. Necessary and sufficient conditions are established to guarantee
the convergence of the iterations