On positive definite solutions of the nonlinear matrix equations X±A*XqA=QX±A*XqA=Q

In this paper, we investigate the nonlinear matrix equations X+A*XqA=Q(01)X−A*XqA=Q(q>1). Necessary and sufficient conditions for the (unique) existence of Hermitian positive definite solutions of these equations are derived. Effective iterative algorithms are also provided to obtain the unique solution of X+A*XqA=Q(01)X−A*XqA=Q(q>1). Numerical examples are included to illustrate the efficiency of the presented iteration algorithms.