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.

中文翻译：

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两种形式的非线性矩阵方程的可解性

在本文中，我们研究非线性矩阵方程

$$ \ begin {aligned} X ^ p = A + \ sum \ limits _ {i = 1} ^ m M_i ^ T（X \ #B）M_i \ end {aligned} $$

和

$$ \ begin {aligned} X ^ p = A + \ sum \ limits _ {i = 1} ^ j M_i ^ T（X \ #B）M_i + \ sum \ limits _ {i = j + 1} ^ m M_i ^ T（X ^ {-1} \＃B）M_i，\ end {aligned} $$

其中p，  m，  j是正整数，\（1 \ le j \ le m \），A，  B是\（n \ n n \）个正定矩阵和\（M_i（i = 1,2,3， \ ldots，m）\）是\（n × n \）个非奇异实数矩阵。基于有序Banach空间中单调和混合单调算子的一些不动点定理和锥的一些性质，我们证明这些方程始终具有唯一的正定解。此外，通过采用多步平稳迭代方法，可以给出一个迭代序列以逼近唯一正定解。