We study the Hermitian positive definite solutions of the matrix equation X +A*X^-qA = I with q > 0. Some properties of the solutions and the basic fixed point iterations for the equation are also discussed in some detail. Some of results in [Linear Algebra Appl., 279 (1998), 303-316], [Linear Algebra Appl., 326 (2001),27-44] and [Linear Algebra Appl. 372 (2003), 295-304] are extended.
