A relational-theoretic approach to get solution of nonlinear matrix equations

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Abstract
In this study, we consider a nonlinear matrix equation of the form $\mathcal{X}= \mathcal{Q} + \sum_{i=1}^{m} \mathcal{A}_{i}^{*} \mathcal{G} (\mathcal{X})\mathcal{A}_{i}$ X = Q + ∑ i = 1 m A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is a Hermitian positive definite matrix, $\mathcal{A}_{i}^{*}$ A i ∗ stands for the conjugate transpose of an $n\times n$ n × n matrix $\mathcal{A}_{i}$ A i , and $\mathcal{G}$ G is an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $\mathcal{G}(O)=O$ G ( O ) = O . We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation. For this, we derive some fixed point results for Suzuki-FG contractive mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order). We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of nonlinear matrix equations through an illustration with graphical representations.

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