Global attractivity of a higher order nonlinear difference equation with decreasing terms - Xiao Wang; Chuanxi Qian (Electronic Journal of Qualitative Theory of Differential Equations)
In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation \begin{equation*} x(n+1)= ax(n)+ bf( x(n)) + cf(x(n-k)), \qquad n=0, 1, \dots \end{equation*} where $a, b $ and $c$ are constants with $0<a<1, 0\leq b<1, 0\leq c <1$ and $a+b+c=1$, $f\in C[[0, \infty), [0, \infty)] $ with $f(x)>0$ for $x>0$, and $k$ is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.