A fixed point theorem for monotone maps and its applications to nonlinear matrix equations (Q2337062)

Summary: By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation \(X = Q + A^\ast f(X) A\), where \(f\) is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation \(X = k Q + A^\ast(\hat{X} - C)^q A\) and prove that the equation has a unique positive definite solution when \(\hat{Q} \geq C\) and \(k > 1\) and \(0 < q < 1\). For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.