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Global behavior of a higher order nonlinear difference equation. (English) Zbl 1066.39008

The authors consider the difference equation

${x}_{n+1}=f\left({x}_{n},{x}_{n-k}\right)$

for $k>1$ fixed. The function $f\left(u,v\right)$ is continuous with respect to both arguments on $\left(0,\infty \right)$, decreasing in $u$ and increasing in $v$, with $f\left(\overline{x},v\right)/v$ non-increasing in $v$, where $\overline{x}$ is the unique positive equilibrium of the equation. Two kinds of global solutions are considered: oscillatory solutions and semi-cycles. Several results on existence of various global solutions are given, incorporating previous results concerning global behavior of equations that are special cases of the above.

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