Systems of nonlinear algebraic equations with unique solution

We consider nonlinear algebraic systems of the form F(x)=Ax+p,x∈ℝ+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(x)= Ax+p, x\in \mathbb {R}^{n}_{+}$\end{document}, where A is a positive matrix and p a non-negative vector. They are involved quite naturally in many applications. For such systems we prove that a positive solution x∗ exists and is unique. Moreover, we prove that x∗ is an attraction point for three Newton-type iterations. A numerical experiment, concerning the computing times for such iterations, is presented. Previously known results, involving existence and uniqueness of solution for particular functions F and matrices A, are extended and generalized.