On new existence of a unique common solution to a pair of non-linear matrix equations

• Published in: arXiv.org 2020-06
• Main Authors: Garai, Hiranmoy, Dey, Lakshmi Kanta, Sintunavarat, Wutiphol, Som, Sumit, Raha, Sayandeepa
• Format: Article
• Language: English
• Subjects: Iterative methods; Mathematical analysis; Matrix methods; Metric space; Nonlinear equations
• ISSN: 2331-8422

The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 + \displaystyle \sum_{i=1}^{m} {A_i}^*G(X)A_i \end{eqnarray*} where \(Q_1,Q_2\in P(n)\), \(A_i\in M(n)\) and \(F,G:P(n)\to P(n)\) are certain functions and \(r,s>1\). In order to achieve our target, we take the help of elegant properties of Thompson metric on the set of all \(n \times n\) Hermitian positive definite matrices. To proceed this, we first derive a common fixed point result for a pair of mappings utilizing a certain class of control functions in a metric space. Then, we obtain some sufficient conditions to assure a unique positive definite common solution to the said equations. Finally, to validate our results, we provide a couple of numerical examples with diagrammatic representations of the convergence behaviour of iterative sequences.