  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.
