Nonoscillation of even order Euler type half-linear difference equations

摘要

We establish nonoscillation criteria for the even order half-linear difference equation of Euler typebegin{equation*}sum_{l=0}^n (-1)^{n-l} beta_{n-l} Delta^{n-l}left(k^{(alpha-lp)}Phileft(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 equationbegin{equation*}(-1)^n Delta^nleft(k^{(alpha)}Phileft(Delta^n x_kright)right) + beta_0 k^{(alpha-np)} Phi(x_{k+n}) = 0end{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.