The authors consider the nonlinear matrix equation
where is an unknown matrix, is the identity matrix and is a positive integer. The equations and have many applications, and iterative procedures for solving the equation have been proposed [see C.-H. Guo and P. Lancaster, Math. Comput. 68, 1589–1603 (1999; Zbl 0940.65036) and A. Ferrante and B. C. Levy, Linear Algebra Appl. 247, 359–373 (1996; Zbl 0876.15011 )]. The iterative positive definite solutions and the properties of the equations and have been discussed by I. G. Ivanov and S. M. El-Sayed [Linear Algebra Appl. 279, 303–316 (1998; Zbl 0935.65041)], and by I. G. Ivanov, V. I. Hasanov and B. V. Minchov [Linear Algebra Appl. 326, No. 1–3, 27–44 (2001; Zbl 0979.15007)].
In this paper, the authors review the existing methods for solving the equation and derive a sufficient condition for this equation to have a unique positive definite solution. Moreover, the convergence of the iterative methods proposed by S. M. El-Sayed [Comput. Math. Appl. 41, 579–588 (2001; Zbl 0984.65043)] is proved under weaker restrictions for the matrix .