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

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}, where mathcal{Q} is a Hermitian positive definite matrix, mathcal{A}_{i}^{*} stands for the conjugate transpose of an ntimes n matrix mathcal{A}_{i}, and mathcal{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. 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.