MMN-2250

# Nonoscillation of even order Euler type half-linear difference equations

*Vojtech Ruzicka*;

## Abstract

We establish nonoscillation criteria for the even order half-linear difference equation of Euler type
\begin{equation*}
\sum_{l=0}^n (-1)^{n-l} \beta_{n-l} \Delta^{n-l}\left(k^{(\alpha-lp)}\Phi\left(\Delta^{n-l}x_{k+l}\right)\right) = 0, \quad \beta_n := 1,
\end{equation*}
where $\Phi(t) := |t|^{p-1} \mathop{\mathrm{sgn}} t$, $p \in (1,\infty)$, $n \in \mathbb{N}$, $k^{(\beta)}$ denotes the falling factorial power (for $\beta \in \mathbb{R}$) and $\alpha, \beta_0, \beta_1, \hdots, \beta_{n-1}$ are real constants. For the two-term equation
\begin{equation*}
(-1)^n \Delta^n\left(k^{(\alpha)}\Phi\left(\Delta^n x_k\right)\right) + \beta_0 k^{(\alpha-np)} \Phi(x_{k+n}) = 0
\end{equation*}
we establish the constant $\gamma_{n,p,\alpha}$ such that the two-term equation is nonoscillatory if $\beta_0 > -\gamma_{n,p,\alpha}$. The criteria are derived using the variational technique and they are further extended via the theory of regularly varying sequences.

Vol. 19 (2018), No. 2, pp. 1137-1161