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On the rational recursive sequence \( x_{n+1}=\dfrac {\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-k}} {\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-k}} \). (English) Zbl 1224.39015

Summary: The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\frac {\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-k}} {\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-k}}, \quad n=0,1,2,\dots, \] where the coefficients \(\alpha _{i},\beta _{i}\in (0,\infty )\) for \( i=0,1,2,\) and \(l\), \(k\) are positive integers. The initial conditions \( x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_{0} \) are arbitrary positive real numbers such that \(l<k\). Some numerical experiments are presented.

MSC:

39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A23 Periodic solutions of difference equations
39A30 Stability theory for difference equations
65Q10 Numerical methods for difference equations
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