| Summary: | Abstract In this paper, we study the existence of solution for the following non-linear matrix equations: X=Q+∑i=1nAi∗XAi−∑i=1nBi∗XBi,X=Q+∑i=1nAi∗ϒ(X)Ai, $$\begin{aligned}& X=Q+ \sum^{n}_{i=1} A^{*}_{i} X A_{i}- \sum ^{n}_{i=1} B^{*}_{i} X B_{i}, \\& X=Q+ \sum^{n}_{i=1} A^{*}_{i} \varUpsilon (X) A_{i}, \end{aligned}$$ where Q is a Hermitian positive definite matrix, Ai $A_{i}$, Bi $B_{i}$ are arbitrary m×m $m\times m$ matrices and ϒ:H(m)→P(m) $\varUpsilon: \mathcal{H}(m)\rightarrow \mathcal{P}(m)$ is an order preserving continuous map such that ϒ(0)=0 $\varUpsilon (0)=0$. To this aim, we establish several common fixed point theorems for two mapping satisfying a rational FR $F_{\mathcal{R}}$-contractive condition, where R $\mathcal{R}$ is a binary relation.
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