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

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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

  1. \(\mathcal {G}\) 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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    Authors and Affiliations

    1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Khlong Luang, Pathum Thani, 12121, Thailand

      Kanokwan Sawangsup & Wutiphol Sintunavarat

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    1. Kanokwan Sawangsup
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    2. Wutiphol Sintunavarat
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    All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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    Correspondence to Wutiphol Sintunavarat.

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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

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