Nonlinear equations based on jointly homogeneous mappings

In this paper, we consider a nonlinear equation based on two-variable monotone and jointly homogeneous functions on an open convex normal conex=g(h(a,x),k(b,x))x=g(h(a,x),k(b,x))and show that it has a unique positive solution and furthermore the solution map is again monotone and jointly homogeneous. We apply our results to nonlinear matrix equations involving the geometric mean of positive definite matrices, which include some known nonlinear matrix equations.