The Investigation on Two Kinds of Nonlinear Matrix Equations

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In this paper, we consider two kinds of nonlinear matrix equations $$X+ \sum _{i=1}^{m}B_{i}^*X^{t_{i}}B_{i}=I\;(0<t_{i}<1)$$ and $$X^{s}-\sum _{i=1}^{m}A_{i}^*X^{p_{i}}A_{i}=I\;(p_{i}>1,\;s\ge 1)$$ . By means of the integral representation of matrix functions, properties of Kronecker product and the monotonic p-concave operator fixed point theorem, we derive necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equations. We also obtain some properties of the Hermitian positive definite solutions, the bounds of the determinant’s sum for $$A_{i}^{*}A_{i}$$ and the spectral radius of $$A_{i}$$ .