Solution of a class of nonlinear matrix equations

In this paper we solve nonlinear matrix equations of the formXδ=Q+∑i=1p(Ai⁎Fi(X)Ai)ri andXδ=Q+∑i=1p(Ai⁎Fi(X)Ai)ri+∑j=1q(Bj⁎Gj(X)Bj)qj, where δ∈(−∞,−1]∪[1,∞), ri,qj∈[−1,1], Q∈P(n), the collection of all n×n Hermitian positive definite matrices and Ai,Bj's are n×n matrices, also Fi,Gj's are monotone mappings from P(n) into P(n). Examples are given to illustrate that the equations can not be solved by previously known theorems.