On modified $$\mathcal {Z}$$ Z -contractions and an iterative scheme for solving nonlinear matrix equations

Abstract

In this work, we introduce the notion of a $$\mathcal {Z}_\mathfrak {R}$$ -contraction mapping, where $$\mathfrak {R}$$ is an binary relation on its domain, which improves upon the idea of Khojasteh et al. (Filomat 29:1189–1194, 2015). We establish some fixed point results for $$\mathcal {Z}_\mathfrak {R}$$ -contraction mappings in complete metric spaces endowed with a transitive relation and also give two illustrative examples. Moreover, we show that N-th order fixed point theorems are derived from our main results. As an application, we apply our main result to study a class of nonlinear matrix equation. Finally, as numerical experiments, we approximate the definite solution of a nonlinear matrix equation using MATLAB.