Some new results for bounded and monotone properties of solutions of second-order quasi-linear forced difference equations. (English) Zbl 1066.39010
The second order quasi-linear forced difference equation
\[
\Delta(p_{n-1}(\Delta y_{n-1})^\alpha) = q_ny_n^\beta + r_n\;,\;n\geq n_0\in {\mathbb N}
\]
is considered with \(\Delta\) the forward difference operator, \(\{p_n\}^\infty_{n_0-1}\) being a positive sequence, \(\{q_n\}^\infty_{n_0}\) a nonnegative but not identically zero sequence, \(\{r_n\}^\infty_{n_0}\) a real sequence and \(\alpha\), \(\beta\) quotients of odd positive integers. First, some errors met in the references are pointed out and counter-examples produced. Further new results are given on positivity, unboundedness of all solutions and on oscillatory behavior or definite sign of \(y_n\Delta y_n\) for some \(n\in {\mathbb N}\). Several examples are discussed.
Reviewer: Vladimir Răsvan (Craiova)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
Keywords:
second-order quasi-linear forced difference equation; oscillation; monotone bounded solution; counter-examples; positive solution; unbounded solutionReferences:
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