Global attractivity of a higher-order nonlinear difference equation. (English) Zbl 1201.39006

Jia, Xiu-Mei; Hu, Lin-Xia

Appl. Math. Comput. 216, No. 3, 857-861 (2010).

Authors’ abstract: The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation \[ x_{n+1} = \frac{1-x_n}{A+x_{n-k}}, \quad n=0,1,\dots , \] where \(A\in (-\infty ,-1),k\) is a positive integer and initial conditions \(x_{-k},\dots , x_0\in (-\infty ,0]\). It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

Reviewer: Miloš Čanak (Beograd)

MSC:

39A20	Multiplicative and other generalized difference equations
39A30	Stability theory for difference equations
39A22	Growth, boundedness, comparison of solutions to difference equations

Keywords:

period-two solution; invariant interval; global attractor; local asymptotic stability; rational difference equation; negative solutions

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