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Keywords:
difference equations; nonlinear; asymptotic behavior; nonoscillatory solutions
difference equations; nonlinear; asymptotic behavior; nonoscillatory solutions
Summary:
The authors consider the difference equation \[ \Delta ^{m} [y_{n} - p_{n} y_{n - k}] + \delta q_{n} y_{\sigma (n + m - 1)} = 0 \qquad \mathrm {(\ast )}\] where $m \ge 2$, $\delta = \pm 1$, $k \in N_0 = \lbrace 0,1, 2, \dots \rbrace $, $\Delta y_{n} = y_{n + 1} - y_{n}$, $q_{n} > 0$, and $\lbrace \sigma (n)\rbrace $ is a sequence of integers with $\sigma (n) \le n$ and $\lim _{n \rightarrow \infty } \sigma (n) = \infty $. They obtain results on the classification of the set of nonoscillatory solutions of ($\ast $) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.
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