Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 412052, 12 pages
http://dx.doi.org/10.1155/2012/412052
Research Article

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

Woula Themistoclakis and Antonia Vecchio

Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sede di Napoli, National Research Council of Italy (CNR), Via P. Castellino, 111, 80131 Napoli, Italy

Received 15 November 2011; Accepted 13 March 2012

Academic Editor: Rigoberto Medina

Copyright © 2012 Woula Themistoclakis and Antonia Vecchio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ( 𝑥 𝑟 ) 𝐴 ) 𝑥 𝑟 + 1 = 𝑝 , 𝑟 = 0 , 1 , … . The global convergence is proved by considering a related fixed-point problem.