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Dynamics of a higher-order rational difference equation. (English) Zbl 1106.39005

The authors consider the dynamics of the difference equation

${y}_{n+1}=\frac{p+q{y}_{n}}{1+{y}_{n-k}},\phantom{\rule{1.em}{0ex}}n=0,1,2,\cdots$

where the initial conditions ${y}_{-k},\cdots ,{y}_{-1},{y}_{0}$ are non-negative, $k\in ℕ$, and the parameters $p$ and $q$ are non-negative. They study characteristics such as the global stability, the boundedness of positive solutions and the character of semicycles of above difference equation. Firstly, they show that every solution of the difference equation is bounded from above and from below by positive constants. Next, the results about the character of semicycles are presented. Finally, they discuss the global asymptotic stability and show that, when $k$ is even, the positive unique equilibrium is globally asymptotically stable if and only if $q<1$, and when $k$ is odd, the positive unique equilibrium is globally asymptotically stable for all values of the parameters $p$ and $q$.

The results obtained solve an open problem from the monograph by M. R. S. Kulenovic and G. Ladas [Dynamics of second order rational difference equations with open problems and conjectures. Boca Raton, FL: Chapman & Hall/CRC. (2002; Zbl 0981.39011)].

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations