In this section, we obtain sufficient conditions for the oscillation of all solutions of Eq. (1.1). Due to the assumptions and the form of our equation, we need only to give proofs for the case of eventually positive solution since the proofs for eventually negative solutions would be similar.

For convenience, for any real positive sequence \(\{ \mu_{n} \}\) which is decreasing to zero, we set

$$\begin{aligned} &B_{n} = \bigl( 1 - p_{n}\mu_{n}^{\alpha - 1} \bigr), \\ &Q_{n} = q_{n} B_{n - \ell}^{\gamma}, \\ &R_{n} = \sum_{s = n_{1}}^{n - 1} a_{s}^{ - 1/\beta}, \\ &\overline{R}_{n} = R_{n} + \frac{1}{\beta} \sum _{s = n_{1}}^{n - 1} Q_{s}R_{s + 1}R_{s - l}^{\beta} \mu_{s - \ell}^{\gamma - \beta} \end{aligned}$$

and

$$C_{n} = \frac{R_{n - \ell}}{R_{n}} $$

for \(n\ge n_{1}\), where \(n_{1} \in \mathbb{N} ( n_{0} )\) is large enough.

### Lemma 2.1

*Let*
\(\{ x_{n} \}\)
*be a positive solution of Eq*. (1.1) *for all*
\(n \in \mathbb{N} ( n_{0} )\). *Then there exists a*
\(n_{1} \in \mathbb{N} ( n_{0} )\)
*such that for all n*\(\ge n_{1}\)

$$ z_{n} > 0, \qquad \Delta z_{n} > 0,\qquad \Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) \le 0. $$

(2.1)

### Proof

The proof of the lemma can be found in [3] and hence details are omitted. □

### Lemma 2.2

*Let*
\(\{ x_{n} \}\)
*be a positive solution of Eq*. (1.1) *for all*
\(n \in \mathbb{N} ( n_{0} )\)
*and suppose Eq*. (2.1) *holds*. *Then there exists a*
\(n_{1} \in \mathbb{N} ( n_{0} )\)
*such that*

$$ z_{n} \ge R_{n}a_{n}^{1/\beta} \Delta z_{n},\quad n \ge n_{1}, $$

(2.2)

*and*

$$ \biggl\{ \frac{z_{n}}{R_{n}} \biggr\} \textit{ is decreasing for }n\ge n_{1}. $$

(2.3)

### Proof

From (2.1), we see that \(a_{n}^{1/\beta} \Delta z_{n}\) is decreasing and therefore

$$z_{n} \ge \sum_{s = n_{1}}^{n - 1} \frac{a_{s + 1}^{1/\beta} \Delta z_{s + 1}}{a_{s}^{1/\beta}} \ge R_{n}a_{n}^{1/\beta} \Delta z_{n}. $$

Further, from the last in equality, we have

$$\Delta \biggl( \frac{z_{n}}{R_{n}} \biggr) \le 0,\quad t \ge t_{1}, $$

and so \(\frac{z_{n}}{R_{n}}\) is decreasing for all \(n \ge n_{1}\). This proof is now complete. □

### Lemma 2.3

*Assume that*, *for large*
*n*, \(( p_{n}, p_{n + 1},\ldots, p_{n + k - 1} ) \ne 0\). *Then*

$$\Delta x_{n} + p_{n}x_{n - \ell}^{\alpha} = 0,\quad n = 0, 1, 2,\ldots $$

*has an eventually positive solution if and only if the corresponding inequality*

$$\Delta x_{n} + p_{n}x_{n - \ell}^{\alpha} \le 0,\quad n = 0, 1, 2,\ldots $$

*has an eventually positive solution*.

### Proof

The proof of the lemma can be found in [21] and hence details are omitted. □

### Lemma 2.4

*If*
\(0 < \alpha < 1, \ell\)
*is a positive integer and*
\(\{ p_{n}\}\)
*is a positive real sequence with*
\(\sum_{n = n_{0}}^{\infty} p_{n} = \infty\), *then every solution of eqution*
\(\Delta x_{n} + p_{n}x_{n - \ell}^{\alpha} = 0\), *is oscillatory*.

### Lemma 2.5

*If*
\(\alpha > 1\). *If there exists a*
\(\lambda> \frac{1}{l} \ln \alpha\)
*such that*
\(\lim_{n \to \infty} \inf [ p_{n}\exp ( - e^{\lambda n} ) ] > 0\), *then every solution of eqution*
\(\Delta x_{n} + p_{n}x_{n - \ell}^{\alpha} = 0\)
*is oscillatory*.

The proof of the Lemmas 2.4 and 2.5 can be found in [16] and hence details are omitted.

Next we state and prove some new oscillation results for Eq. (1.1).

### Theorem 2.1

*Let*
\(\gamma \ge \beta\)
*be holds*. *Assume that there exists a positive real sequence*
\(\{ \mu_{n} \}\)
*tending to zero such that*
\(B_{n} > 0\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If the first order delay difference equation*

$$ \Delta w_{n} + Q_{n}\overline{R}_{n} w_{n - \ell}^{\gamma /\beta} = 0 $$

(2.4)

*is oscillatory*, *then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

Let \(\{ x_{n} \}\) be a positive solution of Eq. (1.1) for all \(n \in \mathbb{N} ( n_{0} )\). Then there exists a \(n_{1} \in \mathbb{N} ( n_{0} )\) such that \(x_{n} > 0,x_{n - k} > 0\) and \(x_{n - \ell} > 0\) for all \(n \ge n_{1}\). By Lemma 2.1, the sequence \(\{ z_{n} \}\) satisfies conditions (2.1) for all \(n \ge n_{1}\). From the definition of \(z_{n}\) we have

$$ x_{n} = z_{n} - p_{n}x_{n - k}^{\alpha} \ge z_{n} - p_{n}z_{n - \ell}^{\alpha}. $$

(2.5)

Since \(\{ z_{n} \}\) is increasing and \(\{ \mu_{n} \}\) is positive, decreasing and tending to zero, we have \(z_{n} \ge \mu_{n}\) for all \(n \ge n_{1}\). Using this and \(0 < \alpha \le 1\) in (2.5), one obtains

$$x_{n} \ge B_{n}z_{n}, $$

which together with Eq. (1.1)

$$ \Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) \le - Q_{n} z_{n - \ell}^{\gamma}. $$

(2.6)

Now a simple computation shows that

$$ \Delta \bigl( z_{n} - R_{n}a_{n}^{1/\beta} \Delta z_{n} \bigr) = - R_{n + 1}\Delta \bigl( a_{n}^{1/\beta} \Delta z_{n} \bigr). $$

(2.7)

By the discrete mean value theorem [1, Theorem 1.7.2], we have

$$\begin{aligned} \Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) &= \bigl(a_{n + 1}^{\frac{1}{\beta}} \Delta z_{n + 1} \bigr)^{\beta} - \bigl(a_{n}^{\frac{1}{\beta}} \Delta z_{n}\bigr)^{\beta} \\ &\ge \beta\frac{a_{n}(\Delta z_{n})^{\beta}}{a_{n + 1}^{1/\beta} \Delta z_{n + 1}}\Delta \bigl(a_{n}^{\frac{1}{\beta}} \Delta z_{n}\bigr), \end{aligned}$$

(2.8)

where we have used \(a_{n}^{\frac{1}{\beta}} \Delta z_{n}\) is positive and decreasing. Now from (2.6), (2.7) and (2.8), one obtains

$$\begin{aligned} z_{n} &\ge R_{n}a_{n}^{\frac{1}{\beta}} \Delta z_{n} + \frac{1}{\beta} \sum_{s = n_{1}}^{n - 1} \frac{R_{s + 1}Q_{s}z_{s - l}^{\gamma} a_{s + 1}^{\frac{1}{\beta}} \Delta z_{s + 1}}{a_{s}(\Delta z_{s})^{\beta}} \\ &\ge a_{n}^{\frac{1}{\beta}} \Delta z_{n} \Biggl( R_{n} + \frac{1}{\beta} \sum_{s = n_{1}}^{n - 1} \frac{R_{s + 1}Q_{s}z_{s - l}^{\gamma}}{ a_{s}(\Delta z_{s})^{\beta}} \Biggr), \end{aligned}$$

(2.9)

where we have used \(a_{n}^{\frac{1}{\beta}} \Delta z_{n}\) is positive and decreasing. From Lemma 2.2 we have

$$ \frac{z_{n - l}}{R_{n - l}} \ge \frac{z_{n}}{R_{n}} \ge a_{n}^{\frac{1}{\beta}} \Delta z_{n},\quad n \ge n_{1}. $$

(2.10)

Substituting (2.10) in (2.9), we obtain

$$ z_{n} \ge a_{n}^{\frac{1}{\beta}} \Delta z_{n} \Biggl(R_{n} + \frac{1}{\beta} \sum_{s = n_{1}}^{n - 1} R_{s + 1}R_{s - l}^{\beta} Q_{s}z_{s - l}^{\gamma} \Biggr). $$

(2.11)

Since \(\gamma \ge \beta\), we have \(z_{n}^{\gamma - \beta} \ge \mu_{n}^{\gamma - \beta} \) for all \(n \ge n_{1}\), and using this in (2.11), one obtains

$$z_{n - l}^{\gamma} \ge \bar{R}_{n - l}\bigl(a_{n - l}^{1 / \beta} \Delta z_{n - l}\bigr)^{\gamma},\quad n \ge n_{1}. $$

Using (2.11) in (2.6), and in view of (2.1), one can see that \(w_{n} = a_{n} ( \Delta z_{n} )^{\beta} \) is a positive solution of the first order delay difference inequality

$$ \Delta w_{n} + Q_{n}\overline{R}_{n - \ell}^{\gamma} w_{n - \ell}^{\gamma /\beta} \le 0. $$

(2.12)

But by Lemma 2.3, the associated difference equation

$$\Delta w_{n} + Q_{n}\overline{R}_{n - \ell}^{\gamma} w_{n - \ell}^{\gamma /\beta} = 0 $$

also has a positive solution, which is a contradiction. Hence we complete the proof. □

### Corollary 2.2

*Let all conditions of Theorem*
2.1
*hold with*
\(\gamma = \beta\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If*

$$ \mathop{\lim} _{n \to \infty} \inf\sum_{s = n - \ell}^{n - 1} Q_{s}\overline{R}_{s - \ell}^{\gamma} > \biggl( \frac{\ell}{\ell + 1} \biggr)^{\ell + 1} $$

(2.13)

*then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

The proof follows from Theorem 2.1 and Theorem 7.6.1 of [6]. □

### Corollary 2.3

*Let all conditions of Theorem*
2.1
*hold with*
\(\gamma > \beta\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If*
\(\ell > k\)
*and there exists a*
\(\lambda > \frac{1}{\ell - k}\ln \frac{\gamma}{\beta} \)
*such that*

$$ \lim_{n \to \infty} \inf \bigl[ Q_{n} \overline{R}_{{n} - {l}}^{\gamma} \exp \bigl( - e^{\lambda n} \bigr) \bigr] > 0,\quad n \ge n_{1}. $$

(2.14)

*Then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

The proof follows from Theorem 2.1 and Lemma 2.5. □

### Theorem 2.4

*Let*
\(\gamma < \beta\)
*be holds*. *Assume that there exists a positive decreasing real sequence*
\(\{ \mu_{n} \}\)
*tending to zero such that*
\(B_{n} > 0\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If for all*
\(N \ge n_{0}\),

$$ \sum_{n = N}^{\infty} Q_{n} \Biggl( R_{n - \ell} + \frac{M^{\gamma - \beta}}{ \beta} \sum_{s = n_{1}}^{n - \ell - 1} Q_{s}R_{s + 1}R_{s - \ell}^{\gamma} \Biggr)^{\gamma} = \infty $$

(2.15)

*for any constant*
\(M > 0\), *then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

Assume that Eq. (1.1) has a positive solution such that there exists a \(n_{1} \in \mathbb{N} ( n_{0} )\) with \(x_{n} > 0,x_{n - k} > 0\) and \(x_{n - \ell} > 0\) for all \(n \ge n_{1}\). Proceeding as in the proof of Theorem 2.1 we have

$$ z_{n} \ge a_{n}^{\frac{1}{\beta}} \Delta z_{n} \Biggl( R_{n} + \frac{1}{\beta} \sum_{s = n_{1}}^{n - 1} Q_{s} R_{s + 1} R_{s - l}^{\beta} z_{s - \ell}^{\gamma - \beta} \Biggr). $$

(2.16)

Since \(z_{n}/R_{n}\) is decreasing, there exists a constant \(M > 0\) such that \(z_{n}/R_{n} \le M\) for all \(n \ge n_{1}\), and from \(\gamma < \beta\), we have \(z_{n - \ell}^{\gamma - \beta} \ge M^{\gamma - \beta} R_{n - \ell}^{\gamma - \beta} \) for all \(n \ge n_{1}\). Using this inequality in (2.16), we obtain

$$z_{n - \ell}^{\gamma} \ge \bigl( a_{n - \ell} ( \Delta z_{n - \ell} )^{\beta} \bigr)^{\gamma /\beta} \Biggl( R_{n - \ell} + \frac{M^{\gamma - \beta}}{\beta} \sum_{s = n_{1}}^{n - \ell - 1} Q_{s}R_{s + 1}R_{s - \ell}^{\gamma} \Biggr)^{\gamma},\quad n \ge n_{1}. $$

Using the last inequality in (2.6) and set \(w_{n} = a_{n} ( \Delta z_{n} )^{\beta} > 0\), we have

$$\Delta w_{n} + Q_{n} \Biggl( R_{n - \ell} + \frac{M^{\gamma - \beta}}{\beta} \sum_{s = n_{1}}^{n - \ell - 1} Q_{s}R_{s + 1} R_{s - \ell}^{\gamma} \Biggr)^{\gamma} w_{n - \ell}^{\gamma /\beta} \le 0. $$

But by Lemma 2.3, the associated difference equation

$$ \Delta w_{n} + Q_{n} \Biggl( R_{n - \ell} + \frac{M^{\gamma - \beta}}{\beta} \sum_{s = n_{1}}^{n - \ell - 1} Q_{s} R_{s + 1}R_{s - \ell}^{\gamma} \Biggr)^{\gamma} w_{n - \ell}^{\gamma /\beta} = 0 $$

(2.17)

also has a positive solution. But Lemma 2.4 and condition (2.15) imply that Eq. (2.17) is oscillatory. This contradiction completes the proof. □

In the following by employing the Riccati substitution technique, we obtain new oscillation criteria for Eq. (1.1).

### Theorem 2.5

*Let*
\(\gamma \ge \beta\)
*hold*. *Assume that there exists a positive decreasing real sequence*
\(\{ \mu_{n} \}\)
*tending to zero*, *such that*
\(B_{n} > 0\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If there exists a positive*, *nondecreasing a real sequence*
\(\{ \rho_{n} \}\)
*such that*

$$ \mathop{\lim} _{n \to \infty} \operatorname{sup} \sum_{s = N}^{n} \biggl( \rho_{s}Q_{s}C_{s}^{\gamma} \mu_{s}^{\gamma - \beta} - \frac{a_{s} ( \Delta \rho_{s} )^{1 + \beta}}{ ( \beta + 1 )^{\beta + 1} \rho_{s}^{\beta}} \biggr) = \infty, $$

(2.18)

*for sufficiently large*
\(N > n_{1}\), *then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

Let \(\{ x_{n} \}\) be a positive solution of Eq. (1.1) for all \(n \in \mathbb{N} ( n_{0} )\). Then there exists a \(n_{1} \in \mathbb{N} ( n_{0} )\) such that \(x_{n} > 0,x_{n - k} > 0\) and \(x_{n - \ell} > 0\) for all \(n \ge n_{1}\). Then, by Lemma 2.1, \(z_{n}\) satisfies conditions (2.1) for all \(n \ge n_{1}\). Define the Riccati transformation by

$$ w_{n} = \rho_{n}a_{n} \biggl( \frac{\Delta z_{n}}{z_{n}} \biggr)^{\beta},\quad n \ge n_{1}. $$

(2.19)

Then \(w_{n} > 0\), for all \(n \ge n_{1}\), and

$$ \Delta w_{n} = \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} + \rho_{n}\frac{\Delta ( a_{n} ( \Delta z_{n} )^{\beta} )}{z_{n}^{\beta}} - \frac{\rho_{n}}{\rho_{n + 1}}w_{n + 1} \frac{\Delta z_{n}^{\beta}}{z_{n}^{\beta}},\quad n \ge n_{1}. $$

(2.20)

By the discrete mean value theorem, we have

$$ \Delta z_{n}^{\beta} = z_{n + 1}^{\beta} - z_{n}^{\beta} = \beta\frac{z_{n}^{\beta} \Delta z_{n}}{z_{n + 1}}, $$

(2.21)

where we have used \(z_{n}\) is positive and increasing. Using (2.21) in (2.20), we obtain

$$\begin{aligned} \Delta w_{n} &\le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \beta \frac{\rho_{n}}{\rho_{n + 1}}w_{n + 1}\frac{\Delta z_{n}}{z_{n + 1}} - \rho_{n}Q_{n} \frac{z_{n - \ell}^{\gamma}}{z_{n}^{\beta}} \\ &\le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \beta \frac{\rho_{n}}{\rho_{n + 1}} \frac{w_{n + 1}}{a_{n}^{1 / \beta}} \frac{a_{n}^{1 / \beta} \Delta z_{n}}{z_{n + 1}} - \rho_{n}Q_{n} \frac{z_{n - \ell}^{\gamma}}{z_{n}^{\beta}} \\ &\le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \beta \frac{\rho_{n}}{\rho_{n + 1}a_{n}^{1 / \beta}} w_{n + 1}\frac{a_{n + 1}^{1 / \beta} \Delta z_{n + 1}}{z_{n + 1}} - \rho_{n}Q_{n} \frac{z_{n - \ell}^{\gamma}}{z_{n}^{\beta}}, \end{aligned}$$

where we have used \(a_{n}^{1 / \beta} \Delta z_{n}\) is positive and decreasing. Using (2.19) in the last inequality, we obtain

$$ \Delta w_{n} \le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \beta \frac{\rho_{n}}{\rho_{n + 1}^{1 + 1 / \beta} a_{n}^{1 / \beta}} w_{n + 1}^{1 + 1 / \beta} - \rho_{n}Q_{n} \frac{z_{n - \ell}^{\gamma}}{ z_{n}^{\beta}}. $$

(2.22)

From (2.3) we have

$$\frac{z_{n - \ell}}{R_{n - \ell}} \ge \frac{z_{n}}{R_{n}} $$

or

$$z_{n - \ell} \ge \frac{R_{n - \ell}}{R_{n}}z_{n} $$

and using this in (2.21) yields

$$ \Delta w_{n} \le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \frac{\beta \rho_{n}}{\rho_{n + 1}^{1 + \frac{1}{\beta}} a_{n}^{\frac{1}{\beta}}} w_{n + 1}^{1 + \frac{1}{\beta}} - \rho_{n}Q_{n}C_{n}^{\gamma} \mu_{n}^{\gamma - \beta}, $$

(2.23)

where we have used \(\gamma \ge \beta\) and \(z_{n} \ge \mu_{n}\), for all \(n \ge n_{1}\). Letting \(A = \frac{\Delta \rho_{n}}{\rho_{n + 1}}\) and \(B = \frac{\beta \rho_{n}}{\rho_{n + 1}^{1 + \frac{1}{\beta}} a_{n}^{\frac{1}{\beta}}} \) and using the inequality given in Lemma 2.6 of [15], it follows from (2.23) that

$$ \Delta w_{n} \le - \rho_{n}Q_{n}C_{n}^{\gamma} \mu_{n}^{\gamma - \beta} + \frac{a_{n} ( \Delta \rho_{n} )^{\beta + 1}}{ ( \beta + 1 )^{\beta + 1}\rho_{n}^{\beta}}. $$

(2.24)

Let \(N \ge n_{1}\) be sufficiently large and summing (2.24) from *N* to *n*, we obtain

$$\sum_{s = N}^{n} \biggl[ \rho_{s}Q_{s}C_{s}^{\gamma} \mu_{s}^{\gamma - \beta} - \frac{a_{s} ( \Delta \rho_{s} )^{\beta + 1}}{ ( \beta + 1 )^{\beta + 1}\rho_{s}^{\beta}} \biggr] \le w_{N}, $$

which contradicts (2.18). This completes the proof. □

### Theorem 2.6

*Let*
\(\gamma < \beta\)
*be holds*. *Assume that there exists a positive*, *nondecreasing real sequence*
\(\{ \mu_{n} \}\)
*tending to zero*, *such that*
\(B_{n} > 0\)
*for all*
\(n \in \mathbb{N} ( n_{0} )\). *If there exists a positive*, *nondecreasing real sequence*
\(\{ \rho_{n} \}\)
*such that*, *for some sufficiently large*
\(N \ge n_{1}\),

$$\mathop{\lim} _{n \to \infty} \operatorname{sup}\sum_{s = N}^{n} \biggl( \rho_{s}Q_{s}C_{s}^{\gamma} \mu_{s}^{\gamma - \beta} - \frac{M^{\beta - \gamma} a_{s} ( \Delta \rho_{s} )^{1 + \beta}}{ ( \beta + 1 )^{\beta + 1}\rho_{s}^{\beta}} \biggr) = \infty $$

*for any constant*
\(M > 0\), *then every solution of Eq*. (1.1) *is oscillatory*.

### Proof

The proof is similar to that of Theorem 2.5 except the inequality (2.23) is replaced by

$$\Delta w_{n} \le \frac{\Delta \rho_{n}}{\rho_{n + 1}}w_{n + 1} - \frac{\beta \rho_{n}}{\rho_{n + 1}^{1 + \frac{1}{\beta}} a_{n}^{\frac{1}{\beta}}} w_{n + 1}^{1 + \frac{1}{\beta}} - M^{\gamma - \beta} \rho_{n}Q_{n}C_{n}^{\gamma} R_{n}^{\gamma - \beta}, $$

where we have used \(\frac{z_{n}}{R_{n}} \le M\), for all \(n \ge n_{1}\) and \(\gamma < \beta\), and hence the details are omitted. This completes the proof. □