On global attractivity of a higher order difference equation with asymptotic constant coefficients

Abdulaziz Almaslokh and Chuanxi Qian

Address:
Department of Mathematics, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, USA

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Abstract: Consider the following higher order difference equation \begin{equation*} x_{n+1}= a_n x_n+ b_n f( x_n) + c_nf(x_{n-k}), \ n=0, 1, \dots , \end{equation*} where $ f\colon [0, \infty ) \rightarrow [0, \infty ) $ is a continuous function with $f(x)>0$ for $x>0$, $\lbrace a_n\rbrace $ is a sequence in $(0,1)$, $\lbrace b_n\rbrace $ and $\lbrace c_n\rbrace $ are sequences in $[0,1)$ with $a_n+b_n+c_n=1$ and $a_n, b_n$ and $c_n$ are convergent, and $k$ is a positive integer. Our aim in this paper is to study the global attractivity of positive solutions of this equation and its applications.

AMSclassification: primary 39A10; secondary 92D25.

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

DOI: 10.5817/AM2026-1-1