Positive definite solution of non-linear matrix equations through fixed point technique

The goal of this study is to solve a non-linear matrix equation of the form $\mathcal{X}=\mathcal{Q}+\sum _{i=1}^m{\mathcal{B}}_i^{\ast }\mathcal{G}(\mathcal{X}){\mathcal{B}}_i$, where $\mathcal{Q}$ is a Hermitian positive definite matrix, ${\mathcal{B}}_i^{\ast }$ stands for the conjugate transpose of an $n\times n$ matrix ${\mathcal{B}}_i$ and $\mathcal{G}$ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $\mathcal{G}(O)=O$. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $\mathcal{F}\mathcal{G}$-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.