Abstract
AbstractThe nonlinear matrix equation \( X^{r} + \sum\nolimits_{i = 1}^{m} {A_{i}^{*} } X^{{\delta_{i} }} A_{i} = Q \) is studied. A necessary condition for the existence of positive definite solutions of this equation is derived. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed.KeywordsNonlinear matrix equationPositive definiteExtremal positive solutionIterative methods
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