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.
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Notes
\(\mathcal {G}\) is order preserving if \(A,B \in H(n)\) with \(A \preceq B\) implies that \(\mathcal {G}(A) \preceq \mathcal {G}(B)\).
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Acknowledgements
The second author would like to thank the Thailand Research Fund and Office of the Higher Education Commission under Grant no. MRG5980242 for financial support during the preparation of this manuscript.
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Sawangsup, K., Sintunavarat, W. On modified \(\mathcal {Z}\)-contractions and an iterative scheme for solving nonlinear matrix equations. J. Fixed Point Theory Appl. 20, 80 (2018). https://doi.org/10.1007/s11784-018-0563-0
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DOI: https://doi.org/10.1007/s11784-018-0563-0