Solving a class of matrix equations via the Bhaskar–Lakshmikantham coupled fixed point theorem

We consider a class of matrix equations of the type X = Q + ∑ i = 1 m A i ∗ X A i − ∑ i = 1 m B i ∗ X B i , where Q is an n × n positive definite matrix and A i , B i are n × n arbitrary matrices. Using the recently presented coupled fixed point theorem of Bhaskar and Lakshmikantham, we prove the existence and uniqueness of positive definite solutions to such equations. Some numerical examples are also presented to show the efficiency of our approach.