Opuscula Math. 43, no. 2 (2023), 131-143
https://doi.org/10.7494/OpMath.2023.43.2.131

 
Opuscula Mathematica

Global attractivity of a higher order nonlinear difference equation with unimodal terms

Abdulaziz Almaslokh
Chuanxi Qian

Abstract. In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms \[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), \quad n=0, 1, \ldots,\] where \(a\), \(b\) and \(c\) are constants with \(0\lt a\lt 1\), \(0\leq b\lt 1\), \(0\leq c \lt 1\) and \(a+b+c=1\), \(g\in C[[0, \infty), [0, \infty)]\) is decreasing, and \(k\) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.

Keywords: higher order difference equation, positive equilibrium, unimodal term, global attractivity, population model.

Mathematics Subject Classification: 39A10, 92D25.

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  • Abdulaziz Almaslokh
  • Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
  • Chuanxi Qian (corresponding author)
  • Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
  • Communicated by Stevo Stević.
  • Received: 2022-12-13.
  • Revised: 2023-01-03.
  • Accepted: 2023-01-06.
  • Published online: 2023-03-27.
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Cite this article as:
Abdulaziz Almaslokh, Chuanxi Qian, Global attractivity of a higher order nonlinear difference equation with unimodal terms, Opuscula Math. 43, no. 2 (2023), 131-143, https://doi.org/10.7494/OpMath.2023.43.2.131

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