A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

Abstract EN

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 ⁎ 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 ⁎ ( X ^ - C ) q A and prove that the equation has a unique positive definite solution when Q ^ ≥ 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.