On positive definite solutions of the family of matrix equations X+A*X-nA=QX+A*X-nA=Q

The nonlinear matrix equation X+A*X-nA=QX+A*X-nA=Q and properties of its positive definite solutions are studied. Sufficient conditions for existence the minimal XSXS and special XlXl positive definite solutions are derived and iterative procedures for computing these solution are discussed.