Solvability for Two Forms of Nonlinear Matrix Equations

In this paper, we study nonlinear matrix equations $$\begin{aligned} X^p=A+\sum \limits _{i=1}^m M_i^T(X\#B)M_i \end{aligned}$$ and $$\begin{aligned} X^p=A+\sum \limits _{i=1}^j M_i^T(X\#B)M_i+\sum \limits _{i=j+1}^m M_i^T(X^{-1}\#B)M_i, \end{aligned}$$ where p, m, j are positive integers, $$1\le j\le m$$ , A, B are $$n\times n$$ positive definite matrices and $$M_i(i=1,2,3,\ldots ,m)$$ are $$n\times n$$ nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method.