On two classes of mixed-type Lyapunov equations

The Hermitian positive definite solutions of the mixed-type Lyapunov equations X=AXB∗+BXA∗+QX=AXB∗+BXA∗+Q and AX+XA∗+BXB∗+Q=0AX+XA∗+BXB∗+Q=0 are studied in this paper. Based on the Bhaskar and Lakshmikantham’s fixed point theorem, new sufficient conditions for the existence of Hermitian positive definite solutions are derived. Iterative methods are proposed to compute the Hermitian positive definite solutions. Numerical examples are used to illustrate the convergence of the new methods.