Algebraic Systems with Positive Coefficients and Positive Solutions

Abstract

The paper is devoted to the existence, uniqueness and nonuniqueness of positive solutions to nonlinear algebraic systems of equations with positive coefficients. Such systems appear in large numbers of applications, such as steady-state equations in continuous and discrete dynamical models, Dirichlet problems, difference equations, boundary value problems, periodic solutions and numerical solutions for differential equations. We apply Brouwer’s fixed point theorem, Krasnoselskii’s fixed point theorem and monotone iterative methods in order to extend some known results and to obtain new results. We relax some hypotheses used in the literature concerning the strict monotonicity of the involved functions. We show that, in some cases, the unique positive solution can be obtained by a monotone increasing iterative method or by a monotone decreasing iterative method. As a consequence of one of our results, we recover the existence of a non-negative solution of the Leontief system and describe a monotone iterative method to find it.

Modified positive linear operators, iterates and systems of linear equations

“…Algebraic systems with positive coefficients and positive solutions appear in several applications. Existence/uniqueness results and iterative methods to finding the solution can be found in many papers, see, e.g., [4,6,10,15,17,23] and the references therein. In this section we use an algorithm for approximating the solutions of systems like ( 2) and (11), see also [9,20].…”