On the Solution of a Class of Nonlinear Systems Governed by an 𝑀-Matrix

Abstract

We consider a weakly nonlinear system of the form (𝐼+𝜑(𝑥)𝐴)𝑥=𝑝, where 𝜑(𝑥) is a real function of the unknown vector 𝑥, and (𝐼+𝜑(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+𝜑(𝑥𝑟)𝐴)𝑥𝑟+1=𝑝, 𝑟=0,1,…. The global convergence is proved by considering a related fixed-point problem.