@article{Shil2022, 
author = {Sourav Shil and Hemant Kumar Nashine},
title = {Positive definite solution of non-linear matrix equations through fixed point technique},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {4},
pages = {6259-6281},
keywords = {positive definite matrix, nonlinear matrix equation, fixed point, relational metric space},
url = {https://www.sciopen.com/article/10.3934/math.2022348},
doi = {10.3934/math.2022348},
abstract = {The goal of this study is to solve a non-linear matrix equation of the form        X    =      Q    +      ∑          i      =      1              m                  B              i              ∗            G    (      X    )            B              i      , where        Q   is a Hermitian positive definite matrix,              B              i              ∗       stands for the conjugate transpose of an    n  ×  n matrix              B              i       and        G   an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that        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        F    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.}
}