A numerical algorithm for approximation of the solutions of nonlinear matrix equations with the relation-theoretic contractions

Abstract

The purpose of this paper is to introduce the numerical algorithm for approximation the solutions of nonlinear matrix equations with the relation-theoretic contractions. First, we introduce the new contraction namely $(\psi,\phi,\Re)$-contraction and prove the fixed point theorem for relation-theoretic $(\psi,\phi,\Re)$-contractions in a metric space endowed with a $T$-orbital transitivity. We also give an example to show the benefit of our theorems. As applications, we apply our main results together with the Thompson metric to solve the nonlinear matrix equations
\begin{equation*}
X^{r}=Q+\sum_{i=1}^{m}A_{i}^{\ast}\mathcal{G}_{i}(X)A_{i},
\end{equation*}
where $r\geq1$, $Q$ is an $n \times n$ positive definite matrix, $A_{i}$ is an $n\times n$ non-singular matrix with its conjugate transpose $A_i^*$, and $\mathcal{G}_{i}$ is a continuous order preserving mapping from a set of all $n \times n$ positive definite matrices into itself for all $i=1,2,3,\ldots, m$. Finally, we furnish a numerical example to support results of our applications.