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Solving the nonlinear matrix equation $X=Q+{\sum }_{i=1}^{m}{M}_{i}{X}^{{\delta }_{i}}{M}_{i}^{*}$ via a contraction principle. (English) Zbl 1162.15008

Author’s abstract: “We consider the nonlinear matrix equation $X=Q+{\sum }_{i=1}^{m}{M}_{i}{X}^{{\delta }_{i}}{M}_{i}^{*}$ where $Q$ is positive (resp. semidefinite) definite and the ${M}_{i}$’s are arbitrary (resp. nonsingular) matrices. We prove that if $\delta :=max\left\{|{\delta }_{i}|:1⩽i⩽m\right\}<1$, then the equation has a unique positive definite solution which is realized as the unique fixed point of a strict contraction with the Lipschitz constant less than or equal to $\delta$. Furthermore, we show that the solution map varying over the determining coefficient matrices is continuous.”

The reader is recommended to read the following two papers to know previous results and methods related to the equation above: M.-J. Huang, C.-Y. Huang and T.-M. Tsai [ibid. 413, No. 1, 202–211 (2006; Zbl 1092.47053)] and X. Duan, A. Liao and B. Tang [ibid. 429, No. 1, 110–121 (2008; Zbl 1148.15012)].

##### MSC:
 15A24 Matrix equations and identities