Moore-type nonoscillation criteria for half-linear difference equations

Abstract

This paper deals with the half-liner difference equation

$$\begin{aligned} \varDelta (r_n\phi _p(\varDelta x_n))+c_n\phi _p(x_{n+1})=0, \end{aligned}$$

where $r_n$, $c_n$ are real-valued sequences, $r_n>0$ for $n \in \mathbb {N} \cup \{0\}$, and $\phi _p(z)=|z|^{p-2}z$ with $p>1$ and $\mathbb {N}$ is the set of natural numbers. The purpose of this paper is to use the function transformation and Riccati technology to establish a half-linear difference equations nonoscillation theorem. Our results generalize earlier nonoscillation result of Došlý and Řehák (Comput Math Appl 42:453–464, 2001). Furthermore, in the case of $p=2$, we can present two examples to determine that the solution of the linear difference equation are nonoscillatory even if

$$\begin{aligned} \sum ^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j} \quad \text {or} \quad \sum _{j=n}^{\infty }r_{j}^{-1}\sum ^{n-1}c_{j} \end{aligned}$$

is less than the lower bound $-3/4$.