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The main goal of this article is to study the existence of a unique positivedefinite 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 certainfunctions and $r,s>1$. In order to achieve our target, we take the help ofelegant properties of Thompson metric on the set of all $n \times n$ Hermitianpositive definite matrices. To proceed this, we first derive a common fixedpoint result for a pair of mappings utilizing a certain class of controlfunctions in a metric space. Then, we obtain some sufficient conditions toassure a unique positive definite common solution to the said equations.Finally, to validate our results, we provide a couple of numerical exampleswith diagrammatic representations of the convergence behaviour of iterativesequences.

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