Asymptotic behavior of third order delay difference equations with a negative middle term

Abstract

In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term

$$ \Delta \bigl(a_{n}\Delta (\Delta w_{n})^{\alpha } \bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }-q_{n}h(w_{n-l})=0,\quad n\geq n_{0}, $$

are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.

1 Introduction

In this paper, we are concerned with the asymptotic behavior of solutions of third order delay difference equations with a negative middle term of the form

$$ \Delta \bigl(a_{n}\Delta (\Delta w_{n})^{\alpha } \bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }-q_{n}h(w_{n-l})=0,\quad n\geq n_{0}, $$

(1.1)

where $n_{0}$ is a nonnegative integer and α is a quotient of odd positive integers. Throughout this paper, we assume without further mention that: $\{a_{n}\}$ is a positive real sequence, $\{p_{n}\}$ is a nonnegative real sequence, and $\{q_{n}\}$ is a positive real sequence for all $n\geq n_{0}$, l is a positive integer, h is a continuous, nondecreasing real-valued function such that $\eta h(\eta )>0$ for $\eta \neq 0$, and $h(\eta \xi )\geq h(\eta )h(\xi )$ for $\eta \xi >0$.

By a solution of equation (1.1) we mean a nontrivial real sequence $\{w_{n}\}$ that is defined for all $n\geq n_{0}-l$ and satisfies equation (1.1) for all $n\geq n_{0}$. A nontrivial solution $\{w_{n}\}$ of equation (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative, and oscillatory otherwise. A difference equation is called nonoscillatory (oscillatory) if all its solutions are nonoscillatory (oscillatory). Following the terms used in [6], we define

$$\begin{aligned}& L_{0}w_{n}=w_{n},\qquad L_{1}w_{n}=( \Delta w_{n})^{\alpha },\qquad L_{2}w_{n}=a_{n} \Delta (L_{1}w_{n}) \quad \mbox{and} \\& L_{3}w_{n}= \Delta (L_{2}w_{n}),\quad \forall n\geq n_{0}. \end{aligned}$$

Using these notations, one can write equation (1.1) as

$$ L_{3}w_{n}-p_{n}L_{1}w_{n}-q_{n}h(w_{n-l})=0. $$

We introduce the following class of nonoscillatory (without loss of generality we say positive) solutions which give the sign structure of possible nonoscillatory solutions to equation (1.1):

$$\begin{aligned}& w_{n}\in W_{1}\quad \Leftrightarrow\quad w_{n}>0,\qquad L_{1}w_{n}>0,\qquad L_{2}w_{n}< 0,\qquad L_{3}w_{n}>0; \\& w_{n}\in W_{3} \quad \Leftrightarrow \quad w_{n}>0,\qquad L_{1}w_{n}>0,\qquad L_{2}w_{n}>0,\qquad L_{3}w_{n}>0, \end{aligned}$$

for all $n\geq n_{1}\geq n_{0}$. In Lemma 2.3, we will prove that if

$$ \sum_{n=n_{1}}^{\infty }P_{n}^{\frac{1}{\alpha }}= \infty , \quad \mbox{where } P_{n} =\prod_{s=n_{1}}^{n-1} \Biggl( 1- \frac{1}{a_{s}}\sum_{t=s}^{\infty }p_{t} \Biggr), $$

(1.2)

then the set W of all positive solutions of equation (1.1) has the following decomposition:

$$ W=W_{1}\cup W_{3}. $$

According to the well-known results in [1, 2], the oscillation criteria are often accomplished by introducing the concepts having property $(A)$ and/or $(B)$. Equation (1.1) is said to have property $(B)$ if $W=W_{3}$.

In recent years the asymptotic behavior of nonoscillatory solutions and the oscillatory behavior of solutions to different classes of third order difference equations have been the interest of many researchers, see for example [3–8, 10, 12–19] and the references cited therein. Recently, in the papers [6, 16], the authors used the comparison method and the summation averaging technique to establish some sufficient conditions for oscillation of all solutions of the third order delay difference equation

$$ \Delta \bigl(a_{n}\Delta \bigl(b_{n}(\Delta w_{n})^{\alpha }\bigr)\bigr)+p_{n}(\Delta w_{n+1})^{ \alpha }+q_{n}h(w_{n-l})=0, $$

(1.3)

where $\{p_{n}\}$ and $\{q_{n}\}$ are positive real sequences, and the auxiliary equation of second order

$$ \Delta (a_{n}\Delta z_{n})+\frac{p_{n}}{b_{n+1}}z_{n+1}=0 $$

is nonoscillatory. In [11] the authors used the oscillation of a third order difference equation of the form

$$ \Delta \bigl( a_{n}\Delta ( d_{n}\Delta w_{n} ) ^{ \gamma } \bigr) +q_{n}h(d_{n-\tau }w_{n-\tau +1})=0 $$

(1.4)

to obtain oscillation conditions for solutions of second order neutral delay difference equations.

Following this trend, in this paper, we study the asymptotic behavior of solutions of equation (1.1). Our approach depends on the application of a technique imposing one restrictive condition on the coefficients of the corresponding auxiliary equation. We show that any nonoscillatory solution $\{w_{n}\}$ of equation (1.1) satisfies $w_{n}\Delta w_{n}>0$. Further, we obtain new sufficient conditions for all solutions of equation (1.1) to have property $(B)$. Two examples are provided to illustrate the main results.

2 Main results

For the sake of simplicity, we define the following:

$$\begin{aligned}& \bar{P}_{n}=\sum_{s=n}^{\infty } \frac{1}{a_{s}}\sum_{t=s}^{\infty }q_{t},\qquad R_{1}(n,n_{1}) =\sum_{s=n_{1}}^{n-1} \frac{1}{a_{s}}, \qquad Q_{n} = p_{n}\bar{P}_{n}+q_{n}, \\& \bar{Q}_{n} =\frac{1}{a_{n}}\sum _{s=n}^{\infty }Q_{s},\qquad B(n,n_{1})= \sum_{s=n_{1}}^{n-1}R_{1}^{\frac{1}{\alpha }}(s,n_{1}) \end{aligned}$$

for $s\geq n\geq n_{1}$, where $n_{1}\geq n_{0}$. Throughout we assume that $R_{1}(n,n_{1})\rightarrow \infty$ as $n\rightarrow \infty$. To make sense of the definitions $P_{n}$ and $\bar{P_{n}}$, we also assume that

$$ \sum_{n=n_{0}}^{\infty }p_{n}< \infty ,\quad \text{and}\quad \sum_{n=n_{0}}^{ \infty }q_{n}< \infty . $$

In the sequel, and without loss of generality, we can deal only with the positive solutions of equation (1.1), since the proof for the opposite case is similar. From our technique, which will be described later, we will see that the properties of solutions to equation (1.1) are closely related to nonoscillatory solutions of an auxiliary second order difference equation

$$ \Delta (a_{n}\Delta z_{n})-p_{n}z_{n+1}=0. $$

(2.1)

First, we prove the following lemmas which will be used in the proofs of the main results.

Lemma 2.1

Let $\{z_{n}\}$ be a positive solution of (2.1) for all $n\geq n_{0}$. Then (1.1) can be written in the form

$$ \Delta \biggl(a_{n}z_{n}z_{n+1}\Delta \biggl( \frac{1}{z_{n}}(\Delta w_{n})^{ \alpha } \biggr) \biggr)-q_{n}z_{n+1}h(w_{n-l})=0 $$

(2.2)

for all $n\geq n_{0}$.

Proof

It is easy to see that

$$\begin{aligned}& \frac{1}{z_{n+1}}\Delta \biggl(a_{n}z_{n}z_{n+1} \Delta \biggl(\frac{1}{z_{n}}( \Delta w_{n})^{\alpha } \biggr)\biggr) \\& \quad = \frac{1}{z_{n+1}}\Delta \bigl(a_{n}z_{n+1} \Delta \bigl((\Delta w_{n})^{\alpha }\bigr)-a_{n}( \Delta w_{n+1})^{\alpha } \Delta z_{n}\bigr) \\& \quad = \Delta \bigl(a_{n}\Delta \bigl((\Delta w_{n})^{\alpha } \bigr)\bigr)+\frac{1}{z_{n+1}}a_{n+1}\Delta \bigl(( \Delta w_{n+1})^{\alpha }\bigr)\Delta z_{n+1} \\& \qquad {}-\frac{1}{z_{n+1}}a_{n+1}\Delta \bigl((\Delta w_{n+1})^{\alpha }\bigr)\Delta z_{n+1}-\frac{1}{z_{n+1}}(\Delta w_{n+1})^{\alpha }\Delta (a_{n}\Delta z_{n}) \\& \quad = \Delta \bigl(a_{n}\Delta \bigl((\Delta w_{n})^{\alpha } \bigr)\bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }, \end{aligned}$$

where we have used (2.1). Using the above equality in equation (1.1) and rearranging, we obtain equation (2.2). This completes the proof. □

We recall that equation (2.1) (see Theorem 6.3.4 of [1]) always has a couple of nonoscillatory solutions $\{z_{n}\}$ such that either

$$ z_{n}\Delta z_{n}>0 $$

(2.3)

or

$$ z_{n}\Delta z_{n}< 0 $$

(2.4)

for all $n\geq n_{0}$.

To find the structure of positive nonoscillatory solutions of equation (1.1), the following property of a nonoscillatory solution $\{z_{n}\}$ satisfying (2.4) plays a crucial role.

Lemma 2.2

If (1.2) holds, then (2.1) has a positive solution $\{z_{n}\}$ satisfying

$$ \sum_{n=n_{1}}^{\infty }\frac{1}{a_{n}z_{n}z_{n+1}}=\sum_{n=n_{1}}^{\infty }z_{n}^{ \frac{1}{\alpha }}= \infty . $$

(2.5)

Proof

Let $\{z_{n}\}$ be a positive solution of equation (2.1) such that (2.4) holds for all $n\geq n_{1}\geq n_{0}$. It is clear from the fact that $\Delta z_{n}<0$, there is a constant $M>0$ such that $z_{n}\leq M$. Hence

$$ \sum_{n=n_{1}}^{\infty }\frac{1}{a_{n}z_{n}z_{n+1}}= \infty . $$

On the other hand, since

$$ \Delta (a_{n}\Delta z_{n})=p_{n}z_{n+1} \geq 0, $$

then $a_{n}\Delta z_{n}$ is increasing and there exists a constant $c\leq 0$ such that $\lim_{n\rightarrow \infty }a_{n}\Delta z_{n}=c$. We claim that $c=0$, if not, then

$$ z_{n}\leq z_{n_{1}}+c\sum_{s=n_{1}}^{n-1} \frac{1}{a_{s}}\rightarrow - \infty , \quad \text{as } n\rightarrow \infty , $$

a contradiction. Hence $c=0$. Summing (2.1) from n to ∞, we have

$$ -a_{n}\Delta z_{n}=\sum_{s=n}^{\infty }p_{s}z_{s+1} \leq z_{n}\sum_{s=n}^{ \infty }p_{s} $$

or

$$ \frac{z_{n+1}}{z_{n}}\geq 1-\frac{1}{a_{n}}\sum _{s=n}^{\infty }p_{s}. $$

(2.6)

Then from (2.6) we obtain

$$ z_{n}\geq z_{n_{1}}\prod_{s=n_{1}}^{n-1} \Biggl( 1-\frac{1}{a_{s}}\sum_{t=s}^{\infty }p_{t} \Biggr) , $$

which yields

$$ z_{n}^{\frac{1}{\alpha }}\geq z_{n_{1}}^{\frac{1}{\alpha }}P_{n}^{ \frac{1}{\alpha }}. $$

(2.7)

Now summing (2.7) from $n_{1}$ to $n-1$ and then combining with (1.2) implies that the second summation in (2.5) is divergent. This completes the proof. □

Lemma 2.3

Assume that condition (1.2) holds. If $\{w_{n}\}$ is a positive solution of (1.1) for all $n\geq n_{0}$, then there is an integer $n_{1}$ such that either $w_{n}\in W_{1}$ or $w_{n}\in W_{3}$ for all $n\geq n_{1}\geq n_{0}$.

Proof

Assume that $\{w_{n}\}$ is a positive solution of equation (1.1) for all $n\geq n_{0}$. By Lemma 2.1, we may write (1.1) in an equivalent form (2.2). From Lemma 2.2, there is a positive sequence $\{z_{n}\}$ of (2.1) which satisfies (2.5), and so we see that

$$ \Delta \biggl( a_{n}z_{n}z_{n+1}\Delta \biggl( \frac{1}{z_{n}}( \Delta w_{n})^{\alpha } \biggr) \biggr) >0. $$

Then, by discrete Kneser’s theorem [1], we have

$$ w_{n}>0, \qquad (\Delta w_{n})^{\alpha }>0,\qquad \Delta \biggl(\frac{1}{z_{n}}( \Delta w_{n})^{\alpha }\biggr)< 0, $$

or

$$ w_{n}>0, \qquad (\Delta w_{n})^{\alpha }>0, \qquad \Delta \biggl(\frac{1}{z_{n}}( \Delta w_{n})^{\alpha }\biggr)>0, $$

for all $n\geq n_{1}\geq n_{0}$. Note that in both cases we have $\Delta w_{n}>0$, and by virtue of (1.1) we see that $L_{3}w_{n}>0$. The rest sign properties of $L_{i}w_{n}$, $i=1,2$, immediately follow from discrete Kneser’s theorem. The proof is now complete. □

Next, we state and prove some useful estimates which will play an important role in the proofs of our main results.

Lemma 2.4

Let $w_{n}\in W_{1}$ be a positive solution of (1.1) for all $n\geq n_{1}\geq n_{0}$. Then

$$ \frac{w_{n}}{(n-n_{1})} \textit{ is nonincreasing}, $$

(2.8)

and there is an integer $n_{2}>n_{1}$ such that

$$ L_{1}w_{n}\geq \bar{P}_{n}h(w_{n-l})\quad \textit{for all }n>n_{2}. $$

(2.9)

Proof

Let $w_{n}\in W_{1}$ be a positive solution of equation (1.1) for $n\geq n_{1}$. From the monotonicity of $L_{1}w_{n}$, we have

$$ w_{n}\geq w_{n}-w_{n_{1}}=\sum _{s=n_{1}}^{n-1}(L_{1}w_{s})^{1/ \alpha } \geq (n-n_{1})L_{1}^{1/\alpha }w_{n}. $$

(2.10)

Therefore

$$ \Delta \biggl( \frac{w_{n}}{n-n_{1}} \biggr) = \frac{(n-n_{1})L_{1}^{1/\alpha } w_{n}-w_{n}}{(n-n_{1})(n+1-n_{1})} \leq 0, $$

and so $w_{n}/(n-n_{1})$ is nonincreasing. Next, summing (1.1) from n to ∞, we obtain

$$ -L_{2}w_{n}\geq \sum_{s=n}^{\infty }p_{s}L_{1}w_{s}+ \sum_{s=n}^{ \infty }q_{s}h(w_{s-l}) \geq h(w_{n-l})\sum_{s=n}^{\infty }q_{s}. $$

Again summing, we obtain

$$ L_{1}w_{n}\geq \sum_{s=n}^{\infty } \frac{h(w_{s-l})}{a_{s}}\sum_{t=s}^{\infty }q_{t} \geq \bar{P_{n}}h(w_{n-l}). $$

This completes the proof. □

Lemma 2.5

Let $w_{n}\in W_{3}$ be a positive solution of (1.1) for all $n\geq n_{1}\geq n_{0}$. If

$$ \sum_{n=n_{1}}^{\infty } \bigl[ p_{n}R_{1}(s,n_{1})+q_{n}h \bigl(B(n-l,n_{1})\bigr) \bigr] =\infty , $$

(2.11)

and there is an integer $n_{2}>n_{1}$ such that

$$ \frac{w_{n}}{B(n,n_{1})} \textit{ is nondecreasing}\quad \textit{for all } n \geq n_{2}. $$

(2.12)

Proof

Let $w_{n}\in W_{3}$ be a positive solution of (1.1) for all $n\geq n_{1}$. Since $L_{2}w_{n}$ is increasing, there is a constant $M>0$ such that $L_{2}w_{n}\geq M$ for all $n\geq n_{1}$. Clearly,

$$ L_{1}w_{n}\geq MR_{1}(n,n_{1}) \quad \text{and}\quad w_{n}\geq M^{1/\alpha }B(n,n_{1})\quad \text{for } n\geq n_{1}. $$

We claim that condition (2.11) implies $\lim_{n\rightarrow \infty }L_{2}w_{n}=\infty$. Using the above estimates into (1.1), we obtain

$$ L_{3}w_{n}\geq Mp_{n}R_{1}(n,n_{1})+h \bigl(M^{1/\alpha }\bigr)q_{n}h\bigl(B(n-l,n_{1}) \bigr). $$

(2.13)

By summing (2.13) from $n_{1}$ to ∞, we see that the claim holds. Therefore, for any $n\geq n_{2}\geq n_{1}$, we have

$$\begin{aligned} L_{1}w_{n} =&L_{1}w_{n_{2}}+ \sum_{s=n_{2}}^{n-1} \frac{L_{2}w_{s}}{a_{s}}\leq L_{1}w_{n_{2}}+R_{1}(n,n_{2})L_{2}w_{n} \\ =&L_{1}w_{n_{2}}-R_{1}(n_{2},n_{1})L_{2}w_{n}+R_{1}(n,n_{1})L_{2}w_{n} \\ \leq &R_{1}(n,n_{1})L_{2}w_{n}, \end{aligned}$$

which yields

$$ \Delta \biggl( \frac{L_{1}w_{n}}{R_{1}(n,n_{1})} \biggr) = \frac{R_{1}(n,n_{1})L_{2}w_{n}-L_{1}w_{n}}{a_{n}R_{1}(n,n_{1})R_{1}(n+1,n_{1})}\geq 0, $$

and hence $L_{1}w_{n}/R_{1}(n,n_{2})$ is nondecreasing for all $n\geq n_{2}$. Again for any $n\geq n_{3}\geq n_{2}$, we have

$$\begin{aligned} w_{n} =&w_{n_{3}}+\sum_{s=n_{3}}^{n-1} \biggl( \frac{R_{1}(s,n_{1})L_{1}w_{s}}{R_{1}(s,n_{1})} \biggr) ^{\frac{1}{\alpha }}\leq w_{n_{3}}+B(n,n_{3}) \biggl( \frac{L_{1}w_{n}}{R_{1}(n,n_{1})} \biggr) ^{\frac{1}{\alpha }} \\ \leq &w_{n_{3}}-B(n_{3},n_{1}) \biggl( \frac{L_{1}w_{n}}{R_{1}(n,n_{1})} \biggr) ^{\frac{1}{\alpha }}+B(n,n_{1}) \biggl( \frac{L_{1}w_{n}}{R_{1}(n,n_{1})} \biggr) ^{\frac{1}{\alpha }}. \end{aligned}$$

It follows from discrete L’Hospital rule [1] that

$$ \lim_{n\rightarrow \infty }\frac{L_{1}w_{n}}{R_{1}(n,n_{1})}=\lim _{n\rightarrow \infty }L_{2}w_{n}=\infty , $$

and so we have

$$ w_{n}\leq B(n,n_{1}) \biggl( \frac{L_{1}w_{n}}{R_{1}(n,n_{1})} \biggr) ^{ \frac{1}{\alpha }},\quad n\geq n_{3}. $$

Then

$$ \Delta \biggl( \frac{w_{n}}{B(n,n_{1})} \biggr) = \frac{B(n,n_{1})(L_{1}w_{n})^{1/\alpha }-R_{1}^{1/\alpha }(n,n_{1})w_{n}}{B(n,n_{1})B(n+1,n_{1})}\geq 0. $$

Thus $w_{n}/B(n,n_{1})$ is nondecreasing for all $n\geq n_{3}$. The proof is complete. □

We conclude this section with the following remark.

Remark 2.6

It is easy to see that from Lemma 2.5, if (1.1) has property $(B)$, then any positive solution of (1.1) satisfies

$$ \lim_{n\rightarrow \infty }\frac{w_{n}}{B(n,n_{1})}=\infty , $$

which gives us information about the rate of convergence of possible positive solutions.

In the following, we present some sufficient conditions which ensure that equation (1.1) has property $(B)$.

Theorem 2.7

Let condition (1.2) hold for all $n\geq n_{1}$. If the first order delay difference equation

$$ \Delta x_{n}+\bar{Q_{n}}h(n-l-n_{1})h \bigl(x_{n-l}^{1/\alpha }\bigr)=0,\quad n\geq n_{1}, $$

(2.14)

is oscillatory, then (1.1) has property (B).

Proof

Let $\{w_{n}\}$ be a positive solution of (1.1) for $n\geq n_{0}$. From Lemma 2.3 there exists an integer $n_{1}\geq n_{0}$ such that either $w_{n}\in W_{1}$ or $w_{n}\in W_{3}$ for all $n\geq n_{1}$. If $w_{n}\in W_{1}$, then by equation (1.1) and (2.9), we have

$$ L_{3}w_{n}\geq (p_{n} \bar{P_{n}}+q_{n})h(w_{n-l})=Q_{n}h(w_{n-l}). $$

(2.15)

Summing (2.15) from n to ∞, we find

$$ -L_{2}w_{n}\geq \sum_{s=n}^{\infty }Q_{s}h(w_{s-l}) \geq \Biggl(\sum_{s=n}^{ \infty }Q_{s} \Biggr)h(w_{n-l}). $$

(2.16)

Using (2.10) in the above inequality, we obtain

$$ -\Delta (L_{1}w_{n})\geq \frac{1}{a_{n}}\Biggl( \sum_{s=n}^{\infty }Q_{s} \Biggr)h(n-l-n_{1})h\bigl(L_{1}^{1/ \alpha }w_{n-l} \bigr). $$

Letting $x_{n}=L_{1}w_{n}$, we see that the difference inequality

$$ \Delta x_{n}+\bar{Q_{n}}h(n-l-n_{1})h \bigl(x_{n-l}^{1/\alpha }\bigr)\leq 0 $$

has a positive solution. By Lemma 2.7 of [19], we see that equation (2.14) also has a positive solution, which is a contradiction. Therefore $w_{n}\in W_{3}$, which implies that equation (1.1) has property $(B)$. This completes the proof. □

Applying some known criteria for oscillation of first order delay difference equation (2.14), one can easily obtain criteria for equation (1.1) to have property $(B)$. The following one is given in [9].

Corollary 2.8

Assume that $h(u)=u^{\alpha }$ and condition (1.2) hold. If

$$ \lim_{n\rightarrow \infty }\inf \sum_{s=n-l}^{n-1} \bar{Q}_{s}(s-l-n_{1})^{1/\alpha }> \biggl( \frac{l}{l+1} \biggr) ^{l+1}, $$

(2.17)

then (1.1) has property $(B)$.

Finally, we present another result for equation (1.1) to have property $(B)$ which is applicable even to the ordinary equation.

Theorem 2.9

Let condition (1.2) hold for all $n\geq n_{1}$. Assume that

$$ \sum_{n=n_{1}}^{\infty } \Biggl( \sum _{s=n}^{\infty } \frac{1}{a_{s}}\sum _{t=s}^{\infty }Q_{t} \Biggr) ^{\frac{1}{\alpha }}=\infty , $$

(2.18)

and the function h satisfies

$$ \lim_{u=\pm \infty }\frac{u}{h^{1/\alpha }(u)}=M< \infty . $$

(2.19)

If

$$ \lim_{n\rightarrow \infty }\sup \Biggl\{ h^{1/\alpha } \biggl( \frac{1}{n-n_{1}} \biggr) \sum_{s=n_{1}}^{n-1} \Biggl( h(s-l-n_{1})\sum_{t=s}^{ \infty } \bar{Q_{t}} \Biggr) ^{\frac{1}{\alpha }} \Biggr\} >M, $$

(2.20)

then (1.1) has property $(B)$.

Proof

Let $\{w_{n}\}$ be a positive solution of (1.1) for all $n\geq n_{0}$. Then from Lemma 2.3 there exists an integer $n_{1}\geq n_{0}$ such that either $w_{n}\in W_{1}$ or $w_{n}\in W_{3}$ for all $n\geq n_{1}$. If $w_{n}\in W_{1}$, then as in the proof of Theorem 2.6 we obtain (2.16), and by summing it from n to ∞, we find that

$$ L_{1}w_{n}\geq \sum_{s=n}^{\infty }h(w_{s-l}) \sum_{t=s}^{\infty }Q_{t} \geq \Biggl( \sum_{s=n}^{\infty } \bar{Q_{s}} \Biggr) h(w_{n-l}). $$

(2.21)

Now, summing (2.21) from $n_{1}$ to $n-1$, one can easily see that

$$ w_{n}\geq \sum_{s=n_{1}}^{n-1} \Biggl( h(w_{s-l})\sum_{t=s}^{\infty } \bar{Q_{t}} \Biggr) ^{\frac{1}{\alpha }}. $$

(2.22)

Using the monotonicity property (2.8) in (2.22), we have

$$\begin{aligned} w_{n} \geq &h^{1/\alpha } \biggl( \frac{w_{n-l}}{(n-l-n_{1})} \biggr) \sum_{s=n_{1}}^{n-1} \Biggl( h(s-l-n_{1})\sum_{t=s}^{\infty } \bar{Q_{t}} \Biggr) ^{1/\alpha } \\ \geq &h^{1/\alpha } \biggl( \frac{w_{n}}{(n-n_{1})} \biggr) \sum _{s=n_{1}}^{n-1} \Biggl( h(s-l-n_{1})\sum _{t=s}^{\infty }\bar{Q_{t}} \Biggr) ^{1/\alpha }. \end{aligned}$$

Applying hypothesis $(H_{3})$ assumed on the function h and then dividing both sides of the last inequality by $h^{1/\alpha }(w_{n})$, we see that

$$ \frac{w_{n}}{h^{1/\alpha }(w_{n})}\geq h^{1/\alpha } \biggl( \frac{1}{(n-n_{1})} \biggr) \sum_{s=n_{1}}^{n-1} \Biggl( h(s-l-n_{1})\sum_{t=s}^{ \infty } \bar{Q_{t}} \Biggr) ^{1/\alpha }. $$

(2.23)

It follows from (2.18) that $\lim_{n\rightarrow \infty }w_{n}=\infty$. Taking the limit supremum on both sides of (2.23), we are led to a contradiction with (2.20). Thus $w_{n}\in W_{3}$, which means that equation (1.1) has property $(B)$. This completes the proof. □

We conclude this section with the following remark.

Remark 2.10

If all conditions of Theorem 2.7 (Theorem 2.9) are satisfied, then one can conclude that all bounded solutions of equation (1.1) are oscillatory.

3 Applications

In the following, we present two examples to illustrate the main results.

Example 3.1

Consider the third order delay difference equation

$$ \Delta ^{3}w_{n}- \frac{1}{(n+1)(n+2)}\Delta w_{n+1}- \frac{1}{n(n+1)}w_{n-1}=0,\quad n\geq 1. $$

(3.1)

Here

$$ p_{n}=\frac{1}{(n+1)(n+2)}, \qquad q_{n}= \frac{1}{n(n+1)},\qquad \alpha =1,\quad \text{and}\quad l=1. $$

A simple calculation shows that

$$\begin{aligned}& P_{n} = \frac{1}{n},\qquad \bar{P}_{n}=\sum _{s=n}^{\infty }\frac{1}{s}> \frac{1}{n}, \\& Q_{n} > \frac{1}{n(n+1)(n+2)}+\frac{1}{n(n+1)},\qquad \bar{Q}_{n}> \frac{1}{2n(n+1)}+\frac{1}{n}, \end{aligned}$$

and

$$ \lim_{n\rightarrow \infty }\inf \sum_{s=n-1}^{n-1} \bar{Q}_{s}(s-2)>1> \frac{1}{4}. $$

Hence all conditions of Corollary 2.8 are satisfied, and therefore (3.1) has property $(B)$.

Example 3.2

Consider the third order delay difference equation

$$ \Delta ^{2}\bigl((\Delta w_{n})^{3} \bigr)- \frac{(\Delta w_{n+1})^{3}}{(n+1)(n+2)}-\frac{w_{n-1}^{3}}{n(n+1)(n+2)}=0,\quad n\geq 1. $$

(3.2)

Here

$$ p_{n}=\frac{1}{(n+1)(n+2)},\qquad q_{n}= \frac{2}{n(n+1)(n+2)},\qquad \alpha =3, \quad \text{and}\quad l=1. $$

A simple calculation shows that

$$ P_{n}=\frac{1}{n},\qquad \bar{P}_{n}= \frac{1}{n},\qquad Q_{n}= \frac{3}{n(n+1)(n+2)},\qquad \bar{Q}_{n}=\frac{3}{2} \biggl( \frac{1}{n(n+1)} \biggr) ,\qquad M=1. $$

Since

$$\begin{aligned} \sum_{n=1}^{\infty } \Biggl( \sum _{s=n}^{\infty }\sum_{t=s}^{ \infty }Q_{t} \Biggr) ^{\frac{1}{3}} =&\sum_{n=1}^{\infty } \Biggl( \sum_{s=n}^{\infty }\sum _{t=s}^{\infty }\frac{3}{t(t+1)(t+2)} \Biggr) ^{\frac{1}{3}} \\ =& \biggl( \frac{3}{2} \biggr) ^{\frac{1}{3}}\sum _{n=1}^{\infty } \biggl( \frac{1}{n} \biggr) ^{\frac{1}{3}}=\infty \end{aligned}$$

and

$$\begin{aligned}& \lim_{n\rightarrow \infty }\sup \Biggl\{ \frac{1}{n-1}\sum_{s=1}^{n-1}(s-2) \Biggl( \sum _{t=s}^{\infty }\frac{3}{2} \biggl( \frac{1}{t(t+1)} \biggr) \Biggr) ^{\frac{1}{3}} \Biggr\} \\& \quad = \lim_{n\rightarrow \infty }\sup \Biggl\{ \frac{1}{(n-1)}\sum_{s=1}^{n-1} \biggl( \frac{3}{2} \biggr) ^{\frac{1}{3}} \frac{(s-2)}{s^{1/3}} \Biggr\} =\infty >1, \end{aligned}$$

we see that all conditions of Theorem 2.9 are satisfied, and hence (3.2) has property $(B)$.