Algebraic Systems with Positive Coefficients and Positive Solutions

Ana Maria Acu, Ioan Raşa and Ancuţa Emilia Şteopoaie
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Ana Maria Acu: Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No. 5-7, 550012 Sibiu, Romania
Ioan Raşa: Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania
Ancuţa Emilia Şteopoaie: Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania

Mathematics, 2022, vol. 10, issue 8, 1-10

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.

Keywords: Brouwer’s fixed point theorem; Krasnoselskii’s fixed point theorem; nonlinear algebraic systems of equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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