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On a general system of difference equations defined by homogeneous functions

  • Nouressadat Touafek EMAIL logo
Published/Copyright: June 8, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 3

Abstract

The aim of this paper is to study the following second order system of difference equations

xn+1=f(yn,yn−1),yn+1=g(xn,xn−1)

where n ∈ ℕ0, the initial values x−1, x0, y−1 and y0 are positive real numbers, the functions f, g : (0, +∞)2 → (0, +∞) are continuous and homogeneous of degree zero. In this study, we establish results on local stability of the unique equilibrium point and to deal with the global attractivity, and so the global stability, some general convergence theorems are provided. Necessary and sufficient conditions on existence of prime period two solutions of our system are given. Also, a result on oscillatory solutions is proved. As applications of the obtained results, concrete models of systems of difference equations defined by homogeneous functions of degree zero are investigated. Our system generalize some existing works in the literature and our results can be applied to study new models of systems of difference equations. For interested readers, we left in the conclusion as open problems two more general systems of higher order defined by homogenous functions of degree zero.

MSC 2010: 39A05; 39A21; 39A23; 39A30
Keywords: Homogeneous difference equations; global and local stability; periodicity; oscillation

This work was supported by DGRSDT (MESRS, DZ).

Acknowledgement

The author would like to thank the two referees for their comments, remarks and suggestions which significantly improved the presentation of the paper.

  1. (Communicated by Michal Fečkan)

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Received: 2020-06-01
Accepted: 2020-07-11
Published Online: 2021-06-08
Published in Print: 2021-06-25

© 2021 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2021-0014
Keywords for this article
Homogeneous difference equations; global and local stability; periodicity; oscillation

Articles in the same Issue

  1. Regular papers
  2. Properties of implication in effect algebras
  3. On a nonlinear relation for computing the overpartition function
  4. Six-cycle systems
  5. Representation of bifinite domains by BF-closure spaces
  6. L-fuzzy cosets in universal algebras
  7. Density of sets with missing differences and applications
  8. Degree of independence of numbers
  9. A generalization of a result on the sum of element orders of a finite group
  10. A New fuzzy McShane integrability
  11. Hankel determinants of second and third order for the class 𝓢 of univalent functions
  12. Herglotz's theorem for Jacobi-Dunkl positive definite sequences
  13. Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind
  14. Existence on solutions of a class of casual differential equations on a time scale
  15. On a general system of difference equations defined by homogeneous functions
  16. Jordan amenability of banach algebras
  17. A new notion of orthogonality involving area and length
  18. Reeb flow invariant ∗-Ricci operators on trans-Sasakian three-manifolds
  19. A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function
  20. On first countable quasitopological homotopy groups
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