Effectualness of the fixed point results on the nonlinear matrix equations $$\mathcal {X}=\mathcal {L}_1+\sum _{i=1}^{m}\mathcal {M}_i^*\mathbb {S}(\mathcal {X})\mathcal {M}_i$$ and $$\mathcal {X}=\mathcal {L}_2+\sum _{i=1}^{m}\mathcal {M}_i^*\mathbb {T}(\mathcal {X})\mathcal {M}_i$$

Abstract

We shall give a notion to obtain some adequate conditions for the existence and uniqueness of a positive definite common solution to a pair of non-linear matrix equations. In pursuit of this, our interest lies in presenting some invigorating results containing altering distance functions and control functions in metric spaces. Using these results, we employ some firm conditions for the existence and uniqueness of a positive definite common solution to the pair of non-linear matrix equations. We also figure out a systematic applicable area of our findings. Eventually, we give precise examples to assert one of the prominent results with a numerical approximation of convergence of iterated sequence using a diagram.