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Classification of nonoscillatory solutions of higher order neutral type difference equations. (English) Zbl 0855.39014
The authors consider the difference equation $\Delta^m [y_n - p_n y_{n - k}] + \delta q_n y_{\sigma (n + m - 1)} = 0, \tag{*}$ where $$m \geq 2$$, $$\delta = \pm 1$$, $$k \in \{0, 1, \dots\}$$, $$\sigma (n) \leq n$$, $$\lim_{n \to \infty} \sigma (n) = \infty$$, $$|p_n |< \lambda < 1$$, $$q_n > 0$$. Sufficient and necessary conditions for existence of nonoscillatory solutions of (*) possessing asymptotic properties $\lim_{n \to \infty} \bigl( (y_n - p_n y_{n - k})/n^j \bigr) = \text{constant} \neq 0, \quad \text{for some} \quad j \in \{0, 1, \dots, m - 1\},$ or $\lim_{n \to \infty} \bigl( (y_n - p_n y_{n - k})/n^j \bigr) = 0 \quad \text{and} \quad \lim_{n \to \infty} \bigl( (y_n - p_n y_{n - k})/n^{j - 1} \bigr) = \pm \infty$ for some $$j \in \{1, 2, \dots, m - 1\}$$ with $$(-1)^{m - j} \delta = - 1$$ are given.
For earlier results in this direction see e.g. A. Drozdowicz and J. Popenda [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 29, No. 2, 161-168 (1990; Zbl 0717.39001)], J. Korczak and M. Migda [Demonstr. Math. 21, No. 3, 615-630 (1988; Zbl 0673.39001) and Fasc. Math. 20, 89-95 (1989; Zbl 0715.39002)], J. Popenda [Ann. Pol. Math. 44, 95-111 (1984; Zbl 0553.39002)] and Proc. Indian Acad. Sci., Math. Sci. 95, 141-153 (1986; Zbl 0628.39003)], W. F. Trench [J. Math. Anal. Appl. 179, No. 1, 135-153 (1993; Zbl 0796.39001)].

MSC:
 39A12 Discrete version of topics in analysis 39A10 Additive difference equations
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