UNIQUE POSITIVE DEFINITE SOLUTION OF NON-LINEAR MATRIX EQUATION ON RELATIONAL METRIC SPACES.

In this study, we consider a non-linear matrix equation of the form X = Q + Σi=1m Σj=1t Ai Fj (X) Ai where Q is a Hermitian positive definite matrix, Ai* stands for the conjugate transpose of an n × n matrix Ai and Fj are order-preserving continuous mappings from the set of all Hermitian matrices to the set of all positive definite matrices such that F(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-implicit type 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 non-linear matrix equations. [ABSTRACT FROM AUTHOR]