# Solutions of a class of nonlinear matrix equations

## TL;DR

This paper establishes conditions for the existence of Hermitian positive definite solutions to a class of nonlinear matrix equations and proposes iterative methods for computing these solutions.

## Contribution

It provides necessary and sufficient conditions for solutions and introduces iterative algorithms for solving the nonlinear matrix equations.

## Key findings

- Derived conditions for solution existence.
- Developed iterative solution methods.
- Discussed maximal and minimal solutions.

## Abstract

In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \geq 1$, $ A, B$ are nonsingular matrices and $Q$ is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation.