  The nonlinear eigen-problem $ Ax+F(x)=\lambda x$ is studied where $A$ is an
$n\times n$ irreducible Stieltjes matrix. Under certain conditions, this
problem has a unique positive solution. We show that, starting from a multiple
of the positive eigenvector of $A$, the Newton-like iteration for this problem
converges monotonically. Numerical results illustrate the effectiveness of this
Newton-like method.
