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Classifications and existence of positive solutions of a higher order nonlinear difference equation. (English) Zbl 0970.39015

This paper is concerned with a class of higher-order nonlinear difference equations of the form \[ \Delta(r_n\Delta^{m-1} x_n)+f(n,x_n)=0,\;n=K,K+1, \dots,\tag{1} \] where \(K\) is a fixed integer, \(m\) is an integer greater than or equal to 2, \(\{r_n\}^\infty_{n=k}\) is a positive sequence and \(f(n,x)\) is a real-valued function defined on \(\{K,K+1, \dots\} \times\mathbb{R}\) which is continuous in the second variable \(x\) and satisfies \(f(n,x) >0\) for \(x>0\). A classification scheme for the eventually positive solutions to (1) is given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of such solutions are provided.

MSC:

39A11 Stability of difference equations (MSC2000)
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