Abstract
In this paper, we consider the nonlinear matrix equation \(X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,\) where \(A_i (i=1,2,\ldots ,m)\) are \(n\times n\) nonsingular complex matrices, Q is a \(n\times n\) Hermitian positive definite (HPD) matrix, \(p\ge 1, m\ge 1\) are positive integers, and \(\delta \in (0,1)\). We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.
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References
Al-Dubiban AM (2015) Positive definite solutions of the matrix equation \(X^{r}-\sum _{i=1}^m A_i^*X^{-\delta _i}A_i=I\), Abstr Appl Anal 8.(473965)
Bhatia R (1997) Matrix analysis. In: Graduate Texts in Mathematics, Spring, Berlin
Bose S, Hossein SM, Paul K (2017) Solution of a class of nonlinear matrix. Linear Algebra Appl 530:109–126
Cai J, Chen JL (2019) Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation \(X^s+A^*X^{-t}A=Q,\). J Southeast Univ 35:142–146
Cai J, Chen GL (2009) Some investigation on Hermitian positive definite solutions of the matrix equation \(X^s+A^*X^{-t}A=Q,\) Linear Algebra Appl 430: 2448–2456
Duan X, Liao A(2008) On the existence of Hermitian positive definite solutions of the matrix equation \(X^{s}+A^*X^{-t}A=Q,\) Linear Algebra Appl 429:673–687
Duan XF, Liao AP (2009) On Hermitian positive definite solution of the matrix equation \(X-\sum _{i=1}^m A_i^*X^rA_i=Q\). J Comput Appl Math 229(1):27–36
Duan XF, Chang HX, Duan FJ (2012a) Positive definite solution of the matrix equation \(X-\sum _{i=1}^mA_i^*X^{-1}A_i=Q\). Numer Math J Chin Univ 34(2):141–152
Engwerda JC (1993) On the existence of a positive definite solution of the matrix equation \(X+A^TX^{-1}A=I\). Linear Algebra Appl 194:91–108
Guo D, Lakshmikantham V (1988) Nonlinear problems in abstract cones. Academic Press, New York
Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, Cambridge
Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge
Hasanov VI, Ivanov IG (2005) On the matrix equation \(X-A^*X^{-n}A=I\). Appl Math Comput 168(2):1340–1356
Hossein SM, Bose S, Paul K (2018) On positive definite solution of nonlinear matrix equations. Linear Multilinear Algebra 66:881–893
Huang BH, Ma CF (2018) Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation. Numer Algorithms 79:153–178
Jia Z, Zhao M, Wang M, Ling S (2014) Solvability theory and iteration method for one Self-adjoint polynomial matrix equation. J Appl Math 7(681605)
Lim Y (2009) Solving the nonlinear matrix equation \(X-\sum _{i=1}^{m}M_iX^{\delta _{i}}M_i^*=Q\) via a contraction principle. Linear Algebra Appl 430:1380–1383
Lim Y (2011) Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means. Linear Algebra Appl 434:1884–1892
Li L, Wang QW, Shen SQ (2015) On positive definite solutions of the nonlinear matrix equations \(X\pm A^*X^qA=Q,\) Appl. Math Comput 271:556–566
Liu AJ, Chen GL (2014) On the Hermitian positive definite solutions of nonlinear matrix equation \(X^{s}+\sum _{i=1}^m A_i^*X^{-t_i}A_i=Q\). Appl Math Comput 243:950–959
Liu PP, Zhang SG, Li QC (2011) Hermite positive definite solution of a class of matrix equation. Proc Eng 15:1884–1888
Long JH, Hu XY, Zhang L (2008) On the Hermitian positive definite solution of the nonlinear matrix equation \(X+A^\ast X^{-1}A+B^\ast X^{-1}B=I\). Bull Braz Math Soc (NS) 39:371–386
Meng J, Kim HM (2016) The positive definite solution to a nonlinear matrix equation. Linear Multilinear Algebra 64:653–666
Meng J, Kim H-M (2018) The positive definite solution of the nonlinear matrix equation \(X^s-A^*X-tA=Q\). Numer Funct Anal Opt 39:398–412
Mouhadjer L, Benahmed B (2016) Fixed point theorem in ordered Banach spaces and applications to matrix equations. Positivity 20:981–998
Nieto JJ, Rodriguez-Lopez R (2005) Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 22:223–239
Peng JJ, Liao AP, Peng ZY, Chen ZC (2019) Newton’s iterative method to solve a nonlinear matrix. Linear Multilinear Algebra 67:1867–1878
Thompson AC (1963) On certain contraction mappings in a partially ordered vector space. Proc Am Math Soc 14:438–443
Yin XY, Fang L (2013) Perturbation analysis for the positive definite solution of the nonlinear matrix equation \(X-\sum _{i=1}^m A_i^*X^{-1}A_i=Q\). J Appl Math Comput 43:199–211
Zhang XD, Feng XL (2017) The Hermitian positive definite solution of the nonlinear matrix equation. Int J Nonlinear Sci Numer Simul 18:293–301
Zhou B, Cai GB, Lam J (2013) Positive definite solutions of the nonlinear matrix equation \(X+A^H{\bar{X}}^{-1}A=I,\) Appl. Math Comput 219(14):7377–7391
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This paper was supported financially by Shanxi Province Science Foundation (201901D111020) and Graduate Science and Technology Innovation Project of Shanxi (2019BY014).
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Communicated by Jinyun Yuan.
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Jin, Z., Zhai, C. Investigation of positive definite solution of nonlinear matrix equation \(X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i\). Comp. Appl. Math. 40, 74 (2021). https://doi.org/10.1007/s40314-021-01463-0
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DOI: https://doi.org/10.1007/s40314-021-01463-0