Systems of nonlinear algebraic equations with unique solution

Abstract

We consider nonlinear algebraic systems of the form F ( x ) = Ax + p , x ? ? + n $F(x)= Ax+p, x\in \mathbb {R}^{n}_{+}$ , 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.