Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations. (English) Zbl 1350.39008
Summary: Non-autonomous, higher order difference equations of type
\[
x_{n+1}=\sum\limits_{i=0}^ka_ix_{n-i}+g_n\biggl(\sum\limits_{i=0}^kb_ix_{n-i}\biggr)
\]
with real variables and parameters have appeared frequently in the literature. We extend some recent results on semiconjugate factorization and reduction of order to cases where characteristic polynomials of the linear expressions \(\sum_{i=0}^ka_iu_i\) and \(\sum_{i=0}^kb_iu_i\) have complex roots. This extension yields new results on boundedness and existence of periodic solutions for equations of order 3 or greater.
MSC:
| 39A23 | Periodic solutions of difference equations |
| 39A10 | Additive difference equations |
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 39A21 | Oscillation theory for difference equations |
Keywords:
reduction of order; boundedness; periodic solutions; oscillations; non-autonomous; higher order; semiconjugate factorizationReferences:
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