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On Hermitian positive definite solutions of a nonlinear matrix equation

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Abstract

In this paper, the Hermitian positive definite solutions of the matrix equation \(X^s +A^* X^{ - t}A = Q\), where A is an \(n \times n\) nonsingular complex matrix, Q is an \(n \times n\) Hermitian positive definite matrix and \(s, t> 0\), are discussed. Some conditions for the existence of Hermitian positive definite solutions of this equation are derived. In addition, two iterative methods to obtaining the maximum or minimum Hermitian positive definite solutions of this equation are proposed. In addition, a necessary and sufficient condition for the existence of these solutions is presented. Theoretical results are illustrated by some numerical examples.

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Acknowledgements

We would like to thank to the anonymous referees for their valuable comments and introducing some helpful references to improve this paper.

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    Mohsen Masoudi

  2. Department of Applied Mathematics & Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

    Abbas Salemi

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  1. Mohsen Masoudi
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  2. Abbas Salemi
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Correspondence to Mohsen Masoudi.

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Masoudi, M., Salemi, A. On Hermitian positive definite solutions of a nonlinear matrix equation . J. Fixed Point Theory Appl. 23, 30 (2021). https://doi.org/10.1007/s11784-021-00867-z

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