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The qualitative behavior of solutions of a nonlinear difference equation. (English) Zbl 1128.39005

Summary: This paper is concerned with the qualitative behavior of solutions to the difference equation

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

where the initial conditions ${x}_{-k},\cdots ,{x}_{-1},{x}_{0}$ are non-negative, $k\in \left\{1,2,3,\cdots \right\}$, and the parameters $p$, $q$ are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem introduced by M. R. S. Kulenovic and G. Ladas [Dynamics of second order rational difference equations, Chapman & Hall/CRC, Boca Raton, FL (2002; Zbl 0981.39011)]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations 65Q05 Numerical methods for functional equations (MSC2000)