On the solution of a class of nonlinear systems governed by an M -matrix

We consider a weakly nonlinear system of the form (I + phi(x)A)x = p, where phi(x) is a real function of the unknown vector x, and (I + phi(x)A) is an M-matrix. We propose to solve it by means of a sequence of linear systems defined by the iteration procedure (I + phi(x(r))A)x(r + 1) = p, r = 0, 1, ... . The global convergence is proved by considering a related fixed-point problem.