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 $-3/4$.

1 Introduction

We consider the nonlinear difference equation

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

(1.1)

where $\{r_n\}_{n=0}^{\infty }$, $\{c_n\}_{n=0}^{\infty }$ are real-valued sequence and $r_n>0$ for $n > a \in \mathbb {N} \cup \{0\}$, $\mathbb {N}$ is the set of natural numbers. Here the forward difference operator $\varDelta$ is defined as $\varDelta x_n = x_{n+1} - x_n$ and $\phi _p$ is a real-valued function defined by

$$\begin{aligned} \phi _p(z)=|z|^{p-2}z \end{aligned}$$

for $z \in \mathbb {R}$ with $p>1$ (a fixed real number). Equation (1.1) is often called half-linear difference equation (for example, see [5,6,7, 9]). In the special case when $p = 2$, Eq. (1.1) becomes the linear difference equation

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

(1.2)

The form of Eq. (1.2) is called Sturm–Liouvile form difference equation.

Now, we define that all solutions of Eq. (1.1) are oscillatory or not. Oscillatory properties of Eq. (1.1) are defined using the concept of the generalized zero. A nontrivial solution $\{x_n\}$ of Eq. (1.1) is said to contains a $generalized\ zero$ in an interval $(N, N+1]$ if $x_n \ne 0$ and $x_n x_{n+1} \le 0$. Equation (1.1) is said to be disconjugate on an interval [0, N] if every solution of Eq. (1.1) has at most one generalized zero on $(0, N+1]$, and the solution $\tilde{x}_n$ given by the initial condition $\tilde{x}_0=0$, $\tilde{x}_1 \ne 0$ has no generalized zero on $(0, N+1]$. Consequently, Eq. (1.1) is said to be nonoscillatory if there exist $n \in N$ such that any nontrivial solution $\{x_n\}$ of Eq. (1.1) is disconjugate on $[N, \infty )$. Hence the solution $\{x_n\}$ is nonoscillatory if it is either eventually positive or eventually negative. Otherwise, it said to be oscillatory.

As known well, Sturm’s comparison theorem and separation theorem holds for Eq. (1.1) (see [4, pp. 386–390]). From Sturm’s separation theorem it follows that if one nontrivial solution of Eq. (1.1) is nonoscillatory (respectively, oscillatory), then all its nontrivial solutions are nonoscillatory (respectively, oscillatory). Hence, oscillatory solutions and nonoscillatory solutions do not coexist in Eq. (1.1). The basic results concerning nonoscillatory and oscillatory properties of Eqs. (1.1) and (1.2) can see the papers [1, 4,5,6,7, 9, 12, 13] and the references cited therein.

Equation (1.1) can be understood as the discrete counterpart of

$$\begin{aligned} (r(t)\phi _p(x'))'+c(t)\phi _p(x)=0, \end{aligned}$$

(1.3)

where r and c are continuous function on $[a, \infty )$ with $r(t)>0$ for $t\ge a$. As a special case of Eq. (1.3) for $p=2$, we have the Sturm–Liouville linear differential equation

$$\begin{aligned} (r(t)x')' + c(t)x = 0. \end{aligned}$$

(1.4)

Many researchers have been studied in the nonoscillation theorems for Eqs. (1.3) or (1.4). For example, the reader is referred to [2, 3, 6, 8, 10, 15, 16]. In particular, the famous nonosillation criteria, the so-called Hille–Wintner type ones, say that all nontrivial solutions of Eq. (1.4) are nonoscillatory provided

$$\begin{aligned} \limsup _{t\rightarrow \infty }\left( \int _{a}^{t}\frac{1}{r(s)}ds\right) \left( \int _{t}^{\infty }c(s)ds\right) <\frac{1}{4} \end{aligned}$$

and

$$\begin{aligned} \liminf _{t\rightarrow \infty }\left( \int _{a}^{t}\frac{1}{r(s)}ds\right) \left( \int _{t}^{\infty }c(s)ds\right) >-\frac{3}{4} \end{aligned}$$

where we suppose $\int _{a}^{\infty }r^{-1}(t)dt=\infty$ and $\int _{a}^{\infty }c(t)dt$ converges. Note that similar sufficient conditions for nonoscillation of Eq. (1.4) can be formulated also in the case when $\int _{a}^{\infty }r^{-1}(t)dt < \infty$ (see, [4]).

There are the upper and lower bounds in Hille–Wintner’s nonoscillation criteria. However, it is known by Moore [11], Wray [17] and Wu and Sugie [18] that the existence of the lower limit value $-3/4$ is not important. For example, Moore gave generalized Hille–Wintner’s nonoscillation criteria as follows.

Theorem A    Suppose that $\int _{a}^{\infty }r^{-1}(t)dt = \infty$ and $\int _{a}^{\infty }c(t)dt$ converges. if there exists a constant $k>0$ such that

$$\begin{aligned} -\sqrt{k}-k \le \left( 1+\int _{a}^{t}\frac{1}{r(s)}ds\right) \left( \int _{t}^{\infty }c(s)ds\right) \le \sqrt{k} - k \le \frac{1}{4} \end{aligned}$$

(1.5)

then all nontrivial solutions of Eq. (1.4) are nonoscillatory.

Note that similar sufficient conditions for nonoscillation of Eq. (1.4) can be formulated also in the case when $\int _{a}^{\infty }r^{-1}(t)dt < \infty$ (see, [11]). Here, we choose $k=1/2$. Then the upper and lower limit value of (1.5) becomes $(\sqrt{2}-1)/2 \thickapprox 0.207\cdots$ and $-(\sqrt{2}+1)/2 \thickapprox -1.207\ldots$. It is clear that

$$\begin{aligned} -\frac{\sqrt{2}+1}{2} \thickapprox -1.207 \cdots < -\frac{3}{4}. \end{aligned}$$

Hence, Moore’s result can extend a lower limit value $-3/4$. The results in the case of half-linear type is known for [2, p. 156].

In general, by the nature of solutions of differential equations, the nature of the solution of the difference equation are often characterized. For example, Došlý [4] extened the Hille–Wintner’s nonoscillation criteria of Eqs. (1.4)–(1.2). It is that if

$$\begin{aligned} \sum ^{\infty }c_{n} = \lim _{n\rightarrow \infty }\sum ^{n}c_{j} \end{aligned}$$

is convergent and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n}^{-1}}{\displaystyle {\sum ^{n-1}r_{j}^{-1}}}=0, \end{aligned}$$

then all nontrivial solutions of Eq. (1.2) are nonoscillatory provided

$$\begin{aligned} -\frac{3}{4}<\liminf _{n\rightarrow \infty }\sum ^{n-1}r_{j}^{-1} \sum _{j=n}^{\infty }c_{j} \quad \text{ and }\quad \limsup _{n\rightarrow \infty }\sum ^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j} <\frac{1}{4}. \end{aligned}$$

Furthermore, Došlý and Řehák [5] applied classical Hille–Wintner type nonoscillation criteria for half-linear difference equation. Equation (1.1) can be divided into two case:

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n}^{1-p^{*}}}{\displaystyle {\sum ^{n-1}r_{j}^{1-p^{*}}}}=0 \end{aligned}$$

(1.6)

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n}^{1-p^{*}}}{\displaystyle {\sum \nolimits _{j=n}^{\infty }r_{j}^{1-p^{*}}}}=0, \end{aligned}$$

(1.7)

where the conjugate exponent of p by $p^*$; that is, p and $p^*$ satisfy $1/p+1/p^{*}=1$.

Remark 1.1

By L’Hospital’s rule, conditions (1.6) and (1.7) can be replaced by a simpler condition and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n+1}}{r_{n}}=1, \end{aligned}$$

if this limit exists.

Under the supposition of (1.6) or (1.7), Došlý and Řehák [5] gave the following results.

Theorem B   Suppose that (1.6) and $\sum ^{\infty }c_{n} = \lim _{n\rightarrow \infty }\sum ^{n}c_{j}<\infty$. If

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{p-1}\sum _{j=n}^{\infty }c_{j} <\frac{1}{p}\left( \frac{p-1}{p}\right) ^{p-1} \end{aligned}$$

(1.8)

and

$$\begin{aligned} \liminf _{n\rightarrow \infty }\left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{p-1}\sum _{j=n}^{\infty }c_{j} >-\frac{2p-1}{p}\left( \frac{p-1}{p}\right) ^{p-1}, \end{aligned}$$

(1.9)

then all nontrivial solutions of Eq. (1.1) are nonoscillatory.

Theorem C   Suppose that (1.7) and $\sum ^{\infty }r_{n}^{1-p^{*}}=\lim _{n\rightarrow \infty }\sum ^{n}r_{n}^{1-p^{*}}<\infty$. If

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left( \sum _{j=n}^{\infty }r_{j}^{1-p^*}\right) ^{p-1}\sum ^{n-1}c_{j} <\frac{1}{p}\left( \frac{p-1}{p}\right) ^{p-1} \end{aligned}$$

(1.10)

and

$$\begin{aligned} \liminf _{n\rightarrow \infty }\left( \sum _{j=n}^{\infty }r_{j}^{1-p^*}\right) ^{p-1}\sum ^{n-1}c_{j}>-\frac{2p-1}{p}\left( \frac{p-1}{p}\right) ^{p-1}, \end{aligned}$$

(1.11)

then all nontrivial solutions of Eq. (1.1) are nonoscillatory.

Of course, in the case that $p=2$, the upper limit value of conditions (1.8) and (1.10), and lower limit value of conditions (1.9) and (1.11) respectively becomes 1/4 and $-3/4$. It is known that all nontrivial solutions of discrete Euler equation

$$\begin{aligned} \varDelta ^2 x_n + \frac{\lambda }{n^2} x_{n+1}=0 \end{aligned}$$

are nonoscillatory if and only if $\lambda \le 1/4$ (see [19]). Hence, it is important that the upper limit value is 1/4 according to the comparison with discrete Euler equation. Here, important questions arise:

• (Q1) Will the lower limit value for Theorem B and Theorem C really exist?

• (Q2) Is there examples that can not applied to Theorem B and C even if the linear difference equation (1.2)?

The purpose of this paper is to answer the above questions. We obtained the following new nonoscillation theorems for Eq. (1.1) in reference to Moore’s nonoscillation criteria.

Theorem 1.1

Suppose that (1.6) and $\sum ^{\infty }c_{n}=\lim _{n\rightarrow \infty }\sum ^{n}c_{j}<\infty$. If there exists a constant $k>0$ such that

$$\begin{aligned} -k^{\frac{1}{p^*}}-k<\liminf _{n\rightarrow \infty }A_{n}\le \limsup _{n\rightarrow \infty }A_{n}<k^{\frac{1}{p^*}}-k\le \frac{1}{p}\left( \frac{p-1}{p}\right) ^{p-1}, \end{aligned}$$

(1.12)

then all nontrivial solutions of Eq. (1.1) are nonoscillatory, where

$$\begin{aligned} A_{n}=\left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{p-1}\sum _{j=n}^{\infty }c_{j} \mathrm{.} \end{aligned}$$

(1.13)

Theorem 1.2

Suppose that (1.7) and $\sum ^{\infty }r_{n}^{1-p^{*}}=\lim _{n\rightarrow \infty }\sum ^{n}r_{n}^{1-p^{*}}<\infty$. If there exists a constant $k>0$ such that

$$\begin{aligned} -k^{\frac{1}{p^*}}-k< \liminf _{n\rightarrow \infty }B_{n} \le \limsup _{n\rightarrow \infty }B_{n}<k^{\frac{1}{p^*}}-k \le \frac{1}{p}\left( \frac{p-1}{p}\right) ^{p-1} , \end{aligned}$$

then all nontrivial solutions of Eq. (1.1) are nonoscillatory, where

$$\begin{aligned} B_{n}=\left( \sum _{j=n}^{\infty }r_{j}^{1-p^*}\right) ^{p-1}\sum ^{n-1}c_{j}\mathrm{.} \end{aligned}$$

Remark 1.2

Theorems 1.1 and 1.2 can be understood as the discrete counterpart of Theorem A. In the case that $k=\Big (\frac{p-1}{p}\Big )^{p}$, by using $p/p^{*}=p-1$, we see that

$$\begin{aligned} k^{\frac{1}{p^*}}-k = \left( \frac{p-1}{p}\right) ^{p-1} \left( 1-\frac{p-1}{p}\right) = \frac{1}{p}\left( \frac{p-1}{p}\right) ^{p-1} \end{aligned}$$

and

$$\begin{aligned} -k^{\frac{1}{p^*}}-k = -\left( \frac{p-1}{p}\right) ^{p-1} \left( 1+\frac{p-1}{p}\right) = -\frac{2p-1}{p}\left( \frac{p-1}{p}\right) ^{p-1}. \end{aligned}$$

Hence, the conditions of Theorems 1.1 and 1.2 becomes Theorems B and C. To prove Theorems 1.1 and 1.2, we need Riccati’s technique (see Sect. 2 for the details).

In this paper, we will give an example of $p=2$ which all nontrivial solutions of linear difference equation (1.2) are nonoscillatory even if

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

is less than the lower limit value $-3/4$ which shown by Došlý and Řehák [5] (see Sect. 3 for the details).

2 Proof of nonoscillation theorems

To prove Theorems 1.1 and 1.2, we need the following lemma.

Lemma 2.1

All nontrivial solutions of Eq. (1.1) are nonoscillatory if and only if there exists a sequence $w_{n}$ with $r_n+w_n > 0$ for $n \in [N, \infty )$, satisfying Riccati-type difference inequality

$$\begin{aligned} R[w_{n}]:=\varDelta w_{n}+c_{n} + w_{n}\Bigg (1-\frac{r_{n}}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}\Bigg )\le 0. \end{aligned}$$

(2.1)

Lemma 2.1 is called Riccati’s technique (for example, see [5,6,7, 14]). By using Lemma 2.1, we will prove Theorems 1.1 and 1.2.

Proof of Theorem 1.1

We have only to show that the Riccati difference inequality $R[w_{n}]\le 0$ has a solution $w_{n}$ with $w_{n}+r_{n}>0$ for n sufficiently large. Let

$$\begin{aligned} w_{n}=\sum _{j=n}^{\infty }c_{j}+k\left( \sum ^{n-1}r_{j}^{1-p^{*}}\right) ^{1-p} \end{aligned}$$

(2.2)

for $k>0$. Since $p^*(p-1)=p$ and (1.13), we have

$$\begin{aligned} |w_{n}|^{p^{*}}=|A_{n}+k|^{p^{*}}\left( \sum ^{n-1}r_{j}^{1-p^{*}}\right) ^{-p}. \end{aligned}$$

By the Lagrange Mean Value Theorem, we see that

$$\begin{aligned} \varDelta \left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{1-p} =(1-p){r_n^{1-p^*}\xi _{n}^{-p}}, \end{aligned}$$

where $\displaystyle {\sum ^{n-1}r_j^{1-p^{*}} \le \xi _{n} \le \sum ^{n}r_j^{1-p^{*}}}$. Hence, the equality

$$\begin{aligned} \varDelta w_{n}= & {} \sum _{j=n+1}^{\infty }c_{j}-\sum _{j=n}^{\infty }c_{j}+k\varDelta \left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{1-p}\\= & {} -c_{n}+k(1-p){r_n^{1-p^*}\xi _{n}^{-p}} \end{aligned}$$

holds. Namely, we get

$$\begin{aligned} \varDelta w_{n}+c_{n}+k(p-1){r_n^{1-p^*}\xi _{n}^{-p}}=0. \end{aligned}$$

(2.3)

Since (2.2), we have

$$\begin{aligned} \frac{|w_{n}|}{r_{n}}= & {} \frac{\left| \displaystyle {\sum _{j=n}^{\infty }c_{j}+k\left( \sum ^{n-1}r_{j}^{1-p^*} \right) ^{1-p}}\right| }{r_{n}}\\= & {} \frac{\displaystyle {\left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{1-p}} \left| \left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{p-1}\sum _{j=n}^{\infty }c_{j}+k\right| }{r_{n}}\\= & {} \left( \frac{r_{n}^{1-p^{*}}}{\displaystyle {\sum ^{n-1}r_{j}^{1-p{^*}}}} \right) ^{p-1}| A_{n}+k|. \end{aligned}$$

Then, by condition (1.6), we have $|w_{n}|/r_{n}\rightarrow 0$ as $n\rightarrow \infty$. Therefore, we have

$$\begin{aligned} w_{n}+r_{n}=r_{n}\left( 1+\frac{w_{n}}{r_{n}}\right) >0 \end{aligned}$$

and

$$\begin{aligned} \phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})>0 \end{aligned}$$

holds for $n \ge N_0$, $N_0$ sufficiently large. Taking into account that $\phi _{p}$ is a differentiable function and $(\phi _{p}(z))' \ge 0$ for $z \in \mathbb {R}$, then we see that

$$\begin{aligned} w_{n}\Bigg (1-\frac{r_{n}}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}\Bigg )= & {} w_{n}\frac{\Big [\phi _{p}\big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\big )-r_{n}\Big ]}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}\\ \\= & {} w_{n}\frac{\Big [\phi _{p}\big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\big )-\phi _{p}\big (\phi _{p^{*}}(r_{n})\big )\Big ]}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}\\ \\= & {} \frac{(p-1)|\eta _{n}|^{p-2}\phi _{p^{*}}(w_{n})}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}w_{n}\\ \\= & {} \frac{(p-1)|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\phi _{p}\big (\phi _{p^*}(r_{n})+\phi _{p^*}(w_{n})\big )}, \end{aligned}$$

where $\eta _{n}$ is between $\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})$ and $\phi _{p^{*}}(r_{n})$, we have

$$\begin{aligned} 0<\phi _{p^{*}}(r_{n})-\phi _{p^{*}}(|w_{n}|)\le \eta _{n}\le \phi _{p^{*}}(r_{n})+\phi _{p^{*}}(|w_{n}|). \end{aligned}$$

In order to clarify (2.1), by using (2.3), we only need to prove the following inequality

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\phi _{p}\big (\phi _{p^*}(r_{n}) +\phi _{p^*}(w_{n})\big )}\le k{r_n^{1-p^*}\xi _{n}^{-p}} \end{aligned}$$

(2.4)

is established. In fact, if $p \ge 2$ then

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\Big )^{p-1}}= & {} \frac{|\eta _{n}|^{p-2}|A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p}\right) ^{-p}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^*}(w_{n})\Big )^{p-1}} \\\le & {} \frac{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(|w_{n}|)\Big )^{p-2}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^*}(w_{n})\Big )^{p-1}} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p} \\= & {} \frac{\Big (\phi _{p^{*}}(r_{n})\Big )^{p-2}\Big (1+\phi _{p^{*}}\Big (\frac{|w_{n}|}{r_{n}}\Big )\Big )^{p-2}}{\Big (\phi _{p^{*}}(r_{n})\Big )^{p-1}\Big (1+\phi _{p^{*}}\Big (\frac{w_{n}}{r_{n}}\Big )\Big )^{p-1}} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p}. \end{aligned}$$

Thus we have

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n}) +\phi _{p^{*}}(w_{n})\Big )^{p-1}} \le \frac{\Big (1+\phi _{p^{*}}\Big (\frac{|w_{n}|}{r_{n}}\Big )\Big )^{p-2}}{\Big (1+\phi _{p^{*}}\Big (\frac{w_{n}}{r_{n}}\Big )\Big )^{p-1}} r_{n}^{1-p^*} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p} . \end{aligned}$$

(2.5)

Similarly, when $1< p < 2$, we see that

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n}) +\phi _{p^{*}}(w_{n})\Big )^{p-1}} \le \frac{\Big (1-\phi _{p^{*}}\Big (\frac{|w_{n}|}{r_{n}}\Big )\Big )^{p-2}}{\Big (1+\phi _{p^{*}}\Big (\frac{w_{n}}{r_{n}}\Big )\Big )^{p-1}} r_{n}^{1-p^*} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p} . \end{aligned}$$

(2.6)

Since (1.12), there exists a sufficiently small $\varepsilon > 0$ and $N_1\ge N_0$ such that

$$\begin{aligned} |A_{n}+k|\le (1-\varepsilon )^{\frac{1}{p^*}}k^{\frac{1}{p^*}} \end{aligned}$$

for $n \ge N_1$. Namely, we have

$$\begin{aligned} |A_{n}+k|^{p^*} \le (1-\varepsilon )k \end{aligned}$$

(2.7)

for $n \ge N_1$. Multiplying (2.7) by $r_{n}^{1-p^*}\left( \displaystyle {\sum ^{n}r_{j}^{1-p^*}}\right) ^{-p}$, then we get

$$\begin{aligned} |A_{n}+k|^{p^*}r_{n}^{1-p^*}\left( \sum ^{n}r_{j}^{1-p^*}\right) ^{-p}\le (1-\varepsilon )kr_{n}^{1-p^*}\left( \sum ^{n}r_{j}^{1-p^*}\right) ^{-p} \end{aligned}$$

for $n \ge N_1$. Taking into account that $\displaystyle {\left( \sum ^{n}r_{j}^{1-p^*}\right) ^{-p}\le \xi _{n}^{-p}}$, we see that

$$\begin{aligned} |A_{n}+k|^{p^*}r_{n}^{1-p^*}\left( \sum ^{n}r_{j}^{1-p^*}\right) ^{-p}\le (1-\varepsilon )kr_{n}^{1-p^*}\xi _{n}^{-p} \end{aligned}$$

for $n \ge N_1$. Hence, it follows that

$$\begin{aligned} r_{n}^{1-p^*}|A_{n}+k|^{p^*}\left( \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p}= & {} r_{n}^{1-p^*}|A_{n}+k|^{p^*}\left( \sum ^{n}r_{j}^{1-p^*}\right) ^{-p} \left( \frac{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}}{\displaystyle \sum ^{n}r_{j}^{1-p^*}}\right) ^{-p}\\\le & {} (1-\varepsilon )kr_{n}^{1-p^*}\xi _{n}^{-p} \left( \frac{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}}{\displaystyle \sum ^{n}r_{j}^{1-p^*}}\right) ^{-p} \end{aligned}$$

for $n \ge N_1$. In the case that (2.5), for the small positive number $\varepsilon$, there exists a $N_2 \ge N_0$ such that

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\Big )^{p-1}}<(1+\varepsilon ) r_{n}^{1-p^*} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p} \end{aligned}$$

for $n \ge N_2$. In the case that (2.6), there exists a $N_3 \ge N_0$ such that

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\Big )^{p-1}}<(1+\varepsilon ) r_{n}^{1-p^*} |A_{n}+k|^{p^*}\left( \displaystyle \sum ^{n-1}r_{j}^{1-p^*}\right) ^{-p} \end{aligned}$$

for $n \ge N_3$ and the same $\varepsilon >0$. Then, for arbitrary $p>1$, we have

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n})+\phi _{p^{*}}(w_{n})\Big )^{p-1}}<(1-\varepsilon ^2)kr_{n}^{1-p^*}\xi _{n}^{-p} \left( \frac{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}}{\displaystyle \sum ^{n}r_{j}^{1-p^*}}\right) ^{-p} \end{aligned}$$

for $n \ge \max \{N_1,N_2,N_3\}$. By condition (1.6), there exists a $N_4 \ge N_0$ such that

$$\begin{aligned} \left( \frac{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}}{\displaystyle \sum ^{n}r_{j}^{1-p^*}}\right) ^{-p}= \left( \frac{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}}{\displaystyle \sum ^{n-1}r_{j}^{1-p^*}+r_n^{1-p^*}}\right) ^{-p}<(1+\varepsilon ^2) \end{aligned}$$

for $n\ge N_4$ and the same $\varepsilon >0$. Thus we see that

$$\begin{aligned} \frac{|\eta _{n}|^{p-2}|w_{n}|^{p^{*}}}{\Big (\phi _{p^{*}}(r_{n}) +\phi _{p^{*}}(w_{n})\Big )^{p-1}}<(1-\varepsilon ^4)kr_{n}^{1-p^*}\xi _{n}^{-p} <kr_{n}^{1-p^*}\xi _{n}^{-p} \end{aligned}$$

for $n \ge N \ge \max \{N_1,N_2,N_3,N_4\}$. For this reason, inequality (2.4) holds. Therefore, we obtained $R[w_{n}] \le 0$ for $n\in [N,\infty )$. From Lemma 2.1, this completes the proof of Theorem 1.1. $\square$

Proof of Theorem 1.2

One can show in the same way as in the proof of Theorem 1.1 that the function

$$\begin{aligned} w_{n}=-\sum ^{n-1}c_{j}-k\left( \sum _{j=n}^{\infty }r_{j}^{1-p^*}\right) ^{1-p}, \quad k >0 \end{aligned}$$

satisfies the equality (2.3). Moreover, from (2.3) and the same way as in the proof of Theorem 1.1, we obtain the inequality (2.4). $\square$

3 Linear difference equations

In this section, we present two examples of which all nontrivial solutions of the linear difference equation (1.2) are nonoscillatory even if

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

is less than the lower limit value $-3/4$.

From Theorem 1.1, we have the following corollary.

Corollary 3.1

Suppose that (1.6) and $\sum ^{\infty }c_{n}=\lim _{n\rightarrow \infty }\sum ^{n}c_{j}<\infty$. If there exists a constant $k>0$ such that

$$\begin{aligned} -\sqrt{k}-k<\liminf _{n\rightarrow \infty }A_{n}\le \limsup _{n\rightarrow \infty }A_{n}<\sqrt{k}-k\le \frac{1}{4}, \end{aligned}$$

(3.1)

then all nontrivial solutions of Eq. (1.2) are nonoscillatory, where

$$\begin{aligned} A_{n}=\sum ^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j}\mathrm{.} \end{aligned}$$

To illustrate Corollary 3.1, we give a concrete example.

Example 3.1

Consider the difference equation

$$\begin{aligned} \varDelta ^{2} x_{n}+c_nx_{n+1}=0, \end{aligned}$$

(3.2)

where

$$\begin{aligned} c_n=\left\{ \begin{array}{l} \displaystyle \frac{\frac{1}{2}-\frac{9}{20}}{n(n+1)},\quad \text{ if }\quad n=2^m,\, m\in Z^+,\\ \\ \displaystyle \frac{-\frac{1}{2}-\frac{9}{20}}{n(n+1)},\quad \text{ if }\quad n\ne 2^m,\, m\in Z^+. \end{array} \right. \end{aligned}$$

Then all nontrivial solutions of Eq. (3.2) are nonoscillatory.

To illustrate Corollary 3.1, from $r_{n}=1$ and $c_{n}$, we can check that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n+1}}{r_{n}}=\lim _{n\rightarrow \infty }\frac{1}{1}=1 \end{aligned}$$

and

$$\begin{aligned} \sum _{j=1}^{\infty }|c_{j}|<\sum _{j=1}^{\infty }\frac{19}{20}\frac{1}{j(j+1)}=\frac{19}{20}\sum _{j=1}^{\infty }\left( \frac{1}{j}-\frac{1}{j+1}\right) =\frac{19}{20}<\infty . \end{aligned}$$

Hence, condition (1.6) and $\sum _{j=1}^{\infty }c_{n} < \infty$ are sutisfied. By a straightforward calculation, it follows that

$$\begin{aligned} \sum _{j=1}^{n-1}r_{j}^{-1}=\sum _{j=1}^{n-1}1=n-1 \end{aligned}$$

and

$$\begin{aligned} \sum _{j=n}^{\infty }c_{j}\le & {} \sum _{j=n,j\ne 2^{m} }^{\infty }\left( -\frac{1}{2}-\frac{9}{20}\right) \frac{1}{j(j+1)}+\sum _{m=\big [\log _2^n\big ]}^{\infty }\left( \frac{1}{2} -\frac{9}{20}\right) \frac{1}{2^{m}(2^{m}+1)}\\ \\< & {} \sum _{j=n}^{\infty }\left( -\frac{1}{2}-\frac{9}{20}\right) \frac{1}{j(j+1)}+2\sum _{m=\big [\log _2^n\big ]}^{\infty }\left( \frac{1}{2} -\frac{9}{20}\right) \frac{1}{2^m(2^m+1)}\\ \\< & {} -\frac{19}{20n}+\frac{2}{20}\sum _{m=\big [\log _2^n\big ]}^{\infty }\frac{1}{4^m}\\ \\= & {} -\frac{19}{20n}+\frac{2}{20}\frac{1}{4^{\big [\log _2^n\big ]}}\sum _{j=0}^{\infty }\frac{1}{4^j}\\ \\< & {} -\frac{19}{20n}+\frac{2}{20}\frac{1}{4^{\log _2^n-1}}\sum _{j=0}^{\infty }\frac{1}{4^j}\\ \\= & {} -\frac{19}{20n}+\frac{16}{30n^2}, \end{aligned}$$

where $[\cdot ]$ is greatest integer function. Thus we obtain

$$\begin{aligned} A_{n}=\sum _{j=1}^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j}<(n-1)\left( -\frac{19}{20n}+\frac{16}{30n^2}\right) . \end{aligned}$$

On the orther hand, since

$$\begin{aligned} \sum _{j=n}^{\infty }c_{j}> \sum _{j=n}^{\infty }\frac{-\frac{1}{2}-\frac{1}{20}}{j(j+1)}=\frac{-19}{20}\sum _{j=n}^{\infty }\left( \frac{1}{j}-\frac{1}{j+1}\right) =-\frac{19}{20n}, \end{aligned}$$

we see that

$$\begin{aligned} A_{n}=\sum _{j=1}^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j}>(n-1) \left( -\frac{19}{20n}\right) . \end{aligned}$$

Therefore, since

$$\begin{aligned} \liminf _{n\rightarrow \infty }A_{n}=\limsup _{n\rightarrow \infty }A_{n} =-\frac{19}{20}<-\frac{3}{4}, \end{aligned}$$

we cannot apply Theorem B directly to Eq. (3.2). However, form Corollary 3.1, if we set $k=81/100$, then

$$\begin{aligned} -\frac{171}{100}<\liminf _{n\rightarrow \infty }A_{n}\le \limsup _{n\rightarrow \infty }A_{n}<-\frac{3}{4}<\frac{9}{100}<\frac{1}{4}. \end{aligned}$$

Thus, condition (3.1) holds. Then all nontrivial solutions of Eq. (3.2) are nonoscillatory.

Fig. 1

This figure shows that Eq. (3.2) has a solution which is nonoscillatory

From Theorem 1.2, we have the following corollary.

Corollary 3.2

Suppose that (1.7) and $\sum ^{\infty }r_{n}^{-1}=\lim _{n\rightarrow \infty }\sum ^{n}r_{n}^{-1}<\infty$. If there exists a constant $k>0$ such that

$$\begin{aligned} -\sqrt{k}-k<\liminf _{n\rightarrow \infty }B_{n}\le \limsup _{n\rightarrow \infty }B_{n}<\sqrt{k}-k\le \frac{1}{4}, \end{aligned}$$

(3.3)

then all nontrivial solutions of Eq. (1.2) are nonoscillatory, where

$$\begin{aligned} B_{n}=\sum _{j=n}^{\infty }r_{j}^{-1}\sum ^{n-1}c_{j}. \end{aligned}$$

To illustrate Corollary 3.2, we give a concrete example.

Example 3.2

Consider the difference equation

$$\begin{aligned} \varDelta \Big (n(n+1)\varDelta x_{n}\Big )+\bigg (\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{1}{2}\bigg )x_{n+1}=0 \end{aligned}$$

(3.4)

for $n \in \mathbb {N}$. Then all nontrivial solutions of Eq. (3.4) are nonoscillatory.

Comparing equation (3.4) with equation (1.2), we see that

$$\begin{aligned} r_n=n(n+1) \quad \text {and} \quad c_n=\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{1}{2}. \end{aligned}$$

From $r_n$, it is easy check that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{r_{n+1}}{r_{n}}&=\lim _{n\rightarrow \infty }\frac{(n+2)(n+1)}{(n+1)n}\\&=\lim _{n\rightarrow \infty }\bigg (1+\frac{2}{n}\bigg )\\&=1 \end{aligned}$$

and

$$\begin{aligned} \sum _{j=1}^{\infty }r_{j}^{-1}&=\sum _{j=1}^{\infty }\bigg (\frac{1}{j}-\frac{1}{j+1}\bigg )\\&=1<\infty . \end{aligned}$$

Hence, condition (1.7) and $\sum _{j=1}^{\infty }r_{n}^{-1}<\infty$ are sutisfied. By a straightforward calculation, it follows that

$$\begin{aligned} \sum _{j=n}^{\infty }r_{j}^{-1}&=\sum _{j=n}^{\infty }\frac{1}{(j+1)j}\\&=\sum _{j=n}^{\infty }\bigg (\frac{1}{j}-\frac{1}{j+1}\bigg )\\&=\frac{1}{n}, \\ \sum _{j=1}^{n-1}c_{j}&=\sum _{j=1}^{n-1}\frac{1}{2}(\sin \ln j+\cos \ln j)-\sum _{j=1}^{n-1}\frac{1}{2}\\&=\sum _{j=1}^{n}\frac{1}{2}(\sin \ln j+\cos \ln j)-\frac{1}{2}(\sin \ln n+\cos \ln n)-\sum _{j=1}^{n-1}\frac{1}{2}\\ \\&=\frac{n}{2}\sum _{j=1}^{n}\frac{1}{n}\left[ \sin \ln \left( \frac{j}{n}n\right) +\cos \ln \left( \frac{j}{n}n\right) \right] -\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{n-1}{2} \\ \\&=\frac{n}{2}\sum _{j=1}^{n}\frac{1}{n}\left[ \sin \left( \ln \frac{j}{n}+\ln n\right) +\cos \left( \ln \frac{j}{n}+\ln n\right) \right] \\ \\&\quad -\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{n-1}{2} \\ \\&=\frac{n}{2}\sum _{j=1}^{n}\left[ \frac{1}{n}\cos \ln n\left( \sin \ln \frac{j}{n}+\cos \ln \frac{j}{n}\right) +\frac{1}{n}\sin \ln n\left( \cos \ln \frac{j}{n}-\sin \ln \frac{j}{n}\right) \right] \\ \\&\quad -\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{n-1}{2}. \end{aligned}$$

Hence, we see that

$$\begin{aligned} \lim _{n\rightarrow \infty }B_{n}= & {} \lim _{n\rightarrow \infty }\sum _{j=n}^{\infty }r_{j}^{-1}\sum ^{n-1}c_{j}\\ \\= & {} \lim _{n\rightarrow \infty }\frac{1}{n}\Bigg \{\frac{n}{2}\sum _{j=1}^{n}\bigg [\frac{1}{n}\cos \ln n\bigg (\sin \ln \frac{j}{n}+\cos \ln \frac{j}{n}\bigg )\\&\quad +\frac{1}{n}\sin \ln n\bigg (\cos \ln \frac{j}{n}-\sin \ln \frac{j}{n}\bigg )\bigg ]-\frac{1}{2}(\sin \ln n+\cos \ln n)-\frac{n-1}{2}\Bigg \}\\ \\= & {} \lim _{n\rightarrow \infty }\Bigg \{\frac{1}{2}\sum _{j=1}^{n}\bigg [\frac{1}{n}\cos \ln n\left( \sin \ln \frac{j}{n}+\cos \ln \frac{j}{n}\right) \\ \\&\quad +\frac{1}{n}\sin \ln n\left( \cos \ln \frac{j}{n}-\sin \ln \frac{j}{n}\right) \bigg ]-\frac{1}{2n}(\sin \ln n+\cos \ln n)-\frac{n-1}{2n}\Bigg \}\\ \\= & {} \lim _{n\rightarrow \infty }\Bigg \{\frac{1}{2}\cos \ln n\sum _{j=1}^{n}\frac{1}{n}\left( \sin \ln \frac{j}{n}+\cos \ln \frac{j}{n}\right) \\ \\&\quad +\frac{1}{2}\sin \ln n\sum _{j=1}^{n}\frac{1}{n}\left( \cos \ln \frac{j}{n}-\sin \ln \frac{j}{n}\right) -\frac{1}{2n}(\sin \ln n+\cos \ln n)-\frac{n-1}{2n}\Bigg \}. \end{aligned}$$

Taking into account that

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{j=1}^{n}\frac{1}{n}\left( \sin \ln \frac{j}{n}+\cos \ln \frac{j}{n}\right)= & {} \int _{0}^{1}(\sin \ln x+\cos \ln x)dx\\= & {} \lim _{\varepsilon \rightarrow 0^+}x\sin \ln x\Big |_{\varepsilon }^{1}=0 \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{j=1}^{n}\frac{1}{n}\left( \cos \ln \frac{j}{n}-\sin \ln \frac{j}{n}\right)= & {} \int _{0}^{1}(\cos \ln x-\sin \ln x)dx\\= & {} \lim _{\varepsilon \rightarrow 0^+}x\cos \ln x\Big |_{\varepsilon }^{1}=1, \end{aligned}$$

we can check that

$$\begin{aligned} \liminf _{n\rightarrow \infty }B_{n}=-1<-\frac{3}{4} \end{aligned}$$

and

$$\begin{aligned} \limsup _{n\rightarrow \infty }B_{n}=0. \end{aligned}$$

Then we cannot apply Theorem C directly to Eq. (3.4). However, form Corollary 3.2, if we set $k=81/100$, then

$$\begin{aligned} -\frac{171}{100}<\liminf _{n\rightarrow \infty }B_{n}\le \limsup _{n\rightarrow \infty }B_{n}=0<\frac{9}{100}<\frac{1}{4}. \end{aligned}$$

Thus, condition (3.3) holds. Then all nontrivial solutions of Eq. (3.4) are nonoscillatory.

Fig. 2

This figure shows that Eq. (3.4) has a solution which is nonoscillatory