Positive definite solution of two kinds of nonlinear matrix equations

Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations X=Σi=1m Ai* XδiAi and X=Σi=1m (Ai* XAi)δi, (0<|δi|<1) always have a unique positive definite solution.Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as δ=max{|δi|, i=1,2, ..., m} decreases. Perturbation bounds for the unique positive definite solution are derived in the end.