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Algebraic Systems with Positive Coefficients and Positive Solutions

Author

Listed:
  • Ana Maria Acu

    (Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No. 5-7, 550012 Sibiu, Romania
    These authors contributed equally to this work.)

  • Ioan Raşa

    (Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania
    These authors contributed equally to this work.)

  • Ancuţa Emilia Şteopoaie

    (Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Ana Maria Acu & Ioan Raşa & Ancuţa Emilia Şteopoaie, 2022. "Algebraic Systems with Positive Coefficients and Positive Solutions," Mathematics, MDPI, vol. 10(8), pages 1-10, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1327-:d:795372
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    References listed on IDEAS

    as
    1. Fuad Aleskerov & Hasan Ersel & Dmitri Piontkovski, 2011. "Linear Algebra for Economists," Springer Texts in Business and Economics, Springer, number 978-3-642-20570-5, September.
    2. Yongqiang Du & Guang Zhang & Wenying Feng, 2016. "Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-7, January.
    3. Moghadam, Amin Abrishami & Soheili, Ali R. & Bagherzadeh, Amir Saboor, 2022. "Numerical solution of fourth-order BVPs by using Lidstone-collocation method," Applied Mathematics and Computation, Elsevier, vol. 425(C).
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