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A global convergence result for a higher order difference equation. (English) Zbl 1180.39003

Summary: Let \(f(z_1,\dots,z_k)\in C(I^k,I)\) be a given function, where \(I\) is (bounded or unbounded) subinterval of \(\mathbb R\), and \(k\in\mathbb N\). Assume that \(f(y_1,y_2,\dots,y_k)\geq f(y_2,\dots,y_k,y_1)\) if \(y_1\geq \max\{y_2,\dots,y_k\}\), \(f(y_1,y_2,\dots,y_k)\leq f(y_2,\dots,y_k,y_1)\) if \(y_1\leq \min\{y_2,\dots,y_k\}\), and \(f\) is non-decreasing in the last variable \(z_k\). We then prove that every bounded solution of an autonomous difference equation of order \(k\), namely, \(x_n= f(x_{n-1},\dots,x_{n-k})\), \(n=0,1,2,\dots\), with initial values \(x_{-k},\dots,x_{-1}\in I\), is convergent, and every unbounded solution tends either to \(+\infty\) or to \(-\infty\).

MSC:

39A10 Additive difference equations
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