1 Introduction

In this paper, we shall study the oscillatory behavior of the solutions of nonlinear second-order delay difference equations of the form

$$\begin{aligned} \varDelta (r\left( t \right) ({\varDelta x\left( t \right) )}^{\alpha })+q\left( t \right) x^{\beta }\left( t-m+1 \right) =0. \end{aligned}$$
(1.1)

We shall assume that

  1. (i)

    \(\{q(t)\}\) and \(\{r(t)\}\) are positive real sequences,

  2. (ii)

    \({\upalpha }\, \mathrm {and}\, {\upbeta }\) are ratios of positive odd integers,

  3. (iii)

    \(m {\ge } 1\) is a positive integer.

Moreover, it is assumed that

$$\begin{aligned} R (t,{\, }\mathrm {t}_{\mathrm {0}})= \sum \limits _{\mathrm {s=\, }\mathrm {t}_{\mathrm {0}}}^{\mathrm {t-1}} {\mathrm {r}^{\mathrm {-}\frac{\mathrm {1}}{{\upalpha }}}\mathrm {(s)}} \rightarrow \infty \text{ as } \text{ t } \rightarrow \infty . \end{aligned}$$
(1.2)

Recall that a solution of (1.1) is a nontrivial real-valued sequence \(\{x(t)\}\) satisfying (1.1) for \(t \ge \, t_{0}-m+1\).

Solutions vanishing identically in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. An equation itself is said to be oscillatory if all its solutions are oscillatory.

The problem of investigating oscillation criteria for various types of difference equations has been a very active research area over the past several decades. A large number of papers and monographs have been devoted to this problem; for a few examples, see [1,2,3,4,5,6,7,8,9,10,11, 13,14,15] and the references contained therein.

The main goal of this paper was to provide some new oscillation criteria for Eq. (1.1) via comparison with a second-order linear difference equation or a first-order linear delay difference equation whose oscillatory behavior is discussed intensively in the literature. We will demonstrate the usefulness of our main results via some applications to neutral difference equations and some examples.

2 Comparison Theorems

To obtain our result, we need the following two lemmas:

Lemma 2.1

Let \(\{q (t)\}\) be a sequence of positive real numbers; m is a positive real number and f: R\(\rightarrow \) R is a continuous nondecreasing function, and \(x\, f(x) > 0\) for \(x\ne 0.\) If the first-order delay differential inequality

$$\begin{aligned} \varDelta y(t)+q(t)f(y(t-m+1))\le 0 \end{aligned}$$

has an eventually positive solution, so does the delay equation

$$\begin{aligned} \varDelta y(t)+q(t)f(y(t-m+1))=0. \end{aligned}$$

This Lemma is an extension of the discrete analogue of known results. See Lemma 6.2.2 in [2] and also in [11]. The proof is immediate.

Lemma 2.2

Let \(\{x(t)\}\) be an eventually increasing solution of Eq. (1.1). Then \(x^{\beta -\alpha }(t)\ge \varphi (t)\), where \(\varphi \left( t \right) \) is  given  by

$$\begin{aligned} \varphi \left( t \right) \, = \left\{ {\begin{array}{l} 1\quad if\, \alpha =\beta \, \\ a\quad if\, \alpha <\beta \, \\ \mathrm {b}R^{\beta -\alpha }\left( t,\, t_{1} \right) \, if\, \alpha >\beta , \end{array}} \right. \end{aligned}$$
(2.1)

where  a and b are positive constants and all large \(t\ge t_{1}\ge t_{0.}\)

Proof

Since \(\{x(t)\}\) is a positive increasing solution of Eq. (1.1), there exists a constant \(c > 0\) such that x(t) \(\ge c\, for\, all\, t\ge t_{1\, }for\, some\, t_{1}\ge t_{0}\). Now, one can easily find that

$$\begin{aligned} x(t)\ge \sum \limits _{s=t_{1}}^{t-1} {\, r^{-1/\alpha }(s)\left( r^{1/\alpha }(s)\varDelta x(s) \right) } \ge R(t,t_{1})\left( r^{1/\alpha }(t)\varDelta x(t) \right) . \end{aligned}$$
(2.2)

Since \(r(t)\left( \varDelta x(t) \right) ^{\alpha }\) is positive and non-increasing on\(\, [t_{1},\infty )\), there exists a constant \(C > 0\) such that

$$\begin{aligned} r(t)\left( \varDelta x(t) \right) ^{\alpha } < C \quad \mathrm{for} \quad t\ge t_{1}. \end{aligned}$$

Summing this inequality from \(t_{1}\) to t-1, we have

$$\begin{aligned} x\left( t \right) \le CR\left( t,t_{1} \right) \mathrm {\, for\, t\, \ge \, }t_{1} \end{aligned}$$

  and  for  some  constant  \(C>\, 0\)  and so,

$$\begin{aligned} x^{\beta -\alpha }\left( t \right) \ge \varphi \left( t \right) = \left\{ \, {\begin{array}{l} 1\quad if\, \alpha =\beta \, \\ a\quad if\, \alpha <\beta \, \\ \mathrm {b}R^{\beta -\alpha }(t,\mathrm {t}_{\mathrm {1}})\quad if\, \alpha >\beta , \end{array}} \right. \end{aligned}$$

where \( a = \quad c^{\beta -\alpha }\, and\, b=C^{\beta -\alpha }\). This proves the Lemma.

For t\(\, \ge t_{1}\ge t_{0}\) , we let

$$\begin{aligned} Q_{1}\left( t,t_{1} \right) =\frac{1}{\alpha }q\left( t \right) \, \left( \frac{R^{\alpha }\left( t-m+1,\mathrm {t}_{\mathrm {1}} \right) }{R\left( t+1,\mathrm {t}_{\mathrm {1}} \right) } \right) \varphi \left( t-m+1 \right) \end{aligned}$$

and

$$\begin{aligned} Q_{2}\left( t,t_{1} \right) =\frac{1}{\alpha }\frac{R^{\alpha }\left( t-m+1,\mathrm {t}_{\mathrm {1}} \right) }{r(t-m+1)}q(t)\varphi \left( t-m+1 \right) . \end{aligned}$$

Now, we present our first oscillation result for Eq. (1.1) via comparison with second-order linear difference equation.

Theorem 2.1

Let \({\alpha \ge 1}\),the conditions (i)–(iii) and (1.2) hold. If the second-order linear difference equation

$$\begin{aligned} \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +Q_{1}\left( t,t_{1} \right) x\left( t+1 \right) =0 \end{aligned}$$
(2.3)

is oscillatory for all large \(t\ge t_{1,\, }\, \) then Eq. (1.1) is oscillatory.

Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq. (1.1), say \(x(t) > 0\), and \(x(t - m + 1)) > 0\) for \(t\ge t_{1}\mathrm {\, for\, some\, }\mathrm {t}_{1}\ge t_{0}.\) The proof if x(t) is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper. Then, it follows from Eq. (1.1) that

$$\begin{aligned} \varDelta (r\left( t \right) ({\varDelta x\left( t \right) )}^{\alpha })=-q\left( t \right) \, x^{\beta -\alpha }\left( t-m+1 \right) x^{\alpha }\left( t-m+1 \right) <0 , \end{aligned}$$
(2.4)

It is easy to see that there exists a \(\mathrm {t}_{2}\ge t_{1}\) such that

$$\begin{aligned} x(t)> 0, \varDelta x\left( t \right) >\, 0\quad \mathrm{and}\quad \varDelta (r\left( t \right) \left( {\varDelta x\left( t \right) )}^{\alpha } \right) \, < 0,\, \mathrm{for}\,\, t\, \ge \mathrm {t}_{2}. \end{aligned}$$

By using (2.1) in (2.4) we have

$$\begin{aligned} \varDelta (r\left( t \right) ({\varDelta x\left( t \right) )}^{\alpha })\le -q\left( t \right) \, \varphi (t-m+1)x^{\alpha }\left( t-m+1 \right) \end{aligned}$$

Inequality (2.4) can be written in the following form:

$$\begin{aligned} \, \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) ^{\alpha }+q\left( t \right) \, \varphi \left( t-m+1 \right) x^{\alpha }\left( t-m+1 \right) \le 0. \end{aligned}$$

\(\varDelta -\)derivative yields

$$\begin{aligned} \varDelta (r\left( t \right) ({\varDelta x\left( t \right) )}^{\alpha })= & {} \varDelta \left( r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right) \right) ^{\alpha }\\ {}\ge & {} \alpha \left( r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right) \right) ^{\alpha -1}\varDelta \left( r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right) \right) , \end{aligned}$$

or,

$$\begin{aligned} \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) {+ }\frac{1}{\alpha }\left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) ^{1-\alpha }q\left( t \right) \, \varphi \left( t{-}m{+}1 \right) x^{\alpha }\left( t{-}m{+}1 \right) \le 0. \nonumber \\ \end{aligned}$$
(2.5)

Using (2.2), there exists\(\, t_{3} \ge t_{2}\) such that

$$\begin{aligned} x (t - m +1) \ge R\left( t-m+1,t_{1} \right) \, r^{\frac{1}{\alpha }}\left( t-m+1 \right) \varDelta x(t-m+1)\quad \mathrm{for}\,\, t \ge t_{3},\nonumber \\ \end{aligned}$$
(2.6)

using the fact that \(r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right) \)is a nonincreasing sequence, we see that

$$\begin{aligned} r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right) \le r^{\frac{1}{\alpha }}\left( t-m+1 \right) \varDelta x(t-m+1)\quad \mathrm{for}\,\, t \ge t_{3}, \end{aligned}$$
(2.7)

and using (2.6) in (2.7) we get

$$\begin{aligned} \, r^{\frac{1}{\alpha }}\left( t \right) \varDelta x\left( t \right)\le & {} r^{\frac{1}{\alpha }}\left( t-m+1 \right) \varDelta x\, \left( t-m+1 \right) \\ \,\le & {} R^{-1}\left( t-m+1,t_{1} \right) x(t-m+1). \end{aligned}$$

Substituting this inequality in (2.5) and using the fact that \({\upalpha }\ge \mathrm {1,\, we\, see\, that}\)

$$\begin{aligned}&\varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +\frac{1}{\alpha }\left( R^{-1}\left( t-m+1,t_{1} \right) x(t-m+1) \right) ^{1-\alpha } \\&\quad \quad \times q\left( t \right) \, \varphi \left( t-m+1 \right) x^{\alpha }\left( t-m+1 \right) \le 0, \end{aligned}$$

or,

$$\begin{aligned}&\varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +\frac{1}{\alpha }R^{\alpha -1}\left( t-m+1,t_{1} \right) \\&\quad \times q\left( t \right) \, \varphi \left( t-m+1 \right) x\left( t-m+1 \right) \le 0. \end{aligned}$$

From (2.2) It follows that

$$\begin{aligned} \varDelta \left( \frac{\mathrm {x(t)}}{R(t,\mathrm {t}_{\mathrm {1}})} \right) \mathrm {=}\frac{R(t,\mathrm {t}_{\mathrm {1}})\varDelta x\mathrm {(t)-}\frac{\mathrm {x(t)}}{\mathrm {r}^{\mathrm {1}/\upalpha } \mathrm {(t)}}}{{R(t+1,\mathrm {t}_{\mathrm {1}})R(t,}\mathrm {t}_{\mathrm {1}})}\mathrm {\le 0}, \end{aligned}$$

i.e. \(\frac{x(t)}{R(t,\mathrm {t}_{\mathrm {1}})}\) is eventually nonincreasing for \(t\ge t_{2}.\) Thus, we have

$$\begin{aligned} \frac{x(t-m+1))}{R(t-m+1,\mathrm {t}_{\mathrm {1}})}{\, }\ge \quad \frac{x(t+1))}{R(t+1,\mathrm {t}_{\mathrm {1}})}, \end{aligned}$$

and so,

$$\begin{aligned}&\varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +\frac{1}{\alpha }R^{\alpha -1}\left( t-m+1,t_{1} \right) \\&\quad \times q\left( t \right) \, \left( \frac{R\left( t-m+1,\mathrm {t}_{\mathrm {1}} \right) }{R\left( t+1,\mathrm {t}_{\mathrm {1}} \right) } \right) \varphi \left( t-m+1 \right) x\left( t+1 \right) \le 0, \end{aligned}$$

or,

$$\begin{aligned} \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) + \, Q_{1}\left( t,t_{1} \right) x\left( t+1 \right) \le 0. \end{aligned}$$

But by Lemma 1 of [14], the corresponding Eq. (2.3) has a positive solution. We derive a contradiction which completes the proof.

By Applying Theorem 3.5 in [8] to Eq. (2.3), we have the following oscillation result;

Corollary 2.1

Let \({\upalpha \ge 1,the}\), conditions (i)–(iii) and (1.2) hold. If there exists a nondecreasing positive sequence \(\{\pi (t)\}\) such that for any \(t\ge t_{0}\)

$$\begin{aligned} \mathop {\text{ lim } \text{ sup }}\limits _{t \rightarrow \infty } \sum \limits _{t_{0}}^{t-1} {\left[ \, \pi \left( s \right) \, Q_{1}\left( t,t_{1} \right) \, -\frac{r^{\frac{1}{\alpha }}(s)}{4}\left( \frac{\varDelta \pi (s)}{\pi (s)} \right) ^{2}\, \right] =\infty } ,\, \end{aligned}$$
(2.8)

then Eq. (1.1) is oscillatory.

Next, we present our second oscillation result for Eq. (1.1) via comparison with first-order delay difference equation.

Theorem 2.2

Let \(0<{\upalpha }\le 1\), the conditions (i)–(iii) and (1.2) hold. If the first-order linear delay difference equation

$$\begin{aligned} \varDelta w(t)+ Q_{2}\left( t,t_{1} \right) w\left( t-m+1 \right) =0, \end{aligned}$$
(2.9)

is oscillatory for all large \(t\ge t_{1\, }\, \), then Eq. (1.1) is oscillatory.

Proof

Let x(t) be a nonoscillatory solution of Eq. (1.1), say \(x(t) > 0\) and x (\(\tau (t)) > 0\) for \(t\ge t_{1}\mathrm {\, for\, some\, }\mathrm {t}_{1}\ge t_{0}.\) The proof if x(t) is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper. Proceeding as in the proof of Theorem 2.1, we obtain the inequalities (2.5)–(2.7). Using (2.6) into (2.5), we have

$$\begin{aligned}&\, \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) + \quad \frac{1}{\alpha }\left( r^{\frac{1}{\alpha }}(t)\varDelta x(t) \right) ^{1-\alpha }q(t)\varphi \left( t-m+1 \right) \\ {}&\quad \quad \times \left( R\left( t-m+1,t_{1} \right) \, r^{\frac{1}{\alpha }}(t-m+1)\varDelta x(t-m+1) \right) ^{\alpha }\le 0 , \quad \mathrm{for}\,\, t \ge t_{3}, \end{aligned}$$

or,

$$\begin{aligned}&\varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +\frac{1}{\alpha }\left( r^{\frac{1}{\alpha }}(t-m+1)\varDelta x(t-m+1) \right) ^{1-\alpha } \\&\quad \times {\, }\frac{\left( R\left( t-m+1,t_{1} \right) \right) ^{\alpha }}{r(t-m+1)}q(t)\varphi \left( t-m+1 \right) \\&\quad \times \left( r^{\frac{1}{\alpha }}(t-m+1)\varDelta x(t-m+1) \right) ^{\alpha }\le 0, \end{aligned}$$

or,

$$\begin{aligned} \varDelta w(t)+ Q_{2}\left( t,t_{1} \right) {\, }w(t-m+1)\le 0, \end{aligned}$$
(2.10)

where, \(w(t) = r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) >0.\) Summing inequality (2.10) from \(t \ge t_{3} \)to u and letting u \(\rightarrow \quad \infty \), we obtain

$$\begin{aligned} w(t) \ge \sum \limits _{s=t}^\infty {Q_{2}\left( t,t_{1} \right) {\, }w(s-m+1)} \quad \mathrm{for}\,\, t \ge t_{3}. \end{aligned}$$

The function w(t) is strictly decreasing on [\(t_{3}\), \(\infty )\). It follows from Lemma 2.1 that the corresponding difference Eq. (2.9) also has a positive solution. We arrive at a contradiction which completes the proof.

By summing Eq. (2.9) from t - m\(+\)1 to t -1, we have the following result:

Corollary 2.2

Let \(0<{\upalpha }\le 1\),the conditions (i)–(iii), and (1.2) hold. If for all large \(t\ge t_{1\, }\)

$$\begin{aligned} \mathop {\text {lim inf}}\limits _{t \rightarrow \infty } \sum \limits _{s=t-m+1}^{t-1} {{\, }Q_{2}\left( t,t_{1} \right) } =\infty ,\, \end{aligned}$$
(2.11)

then Eq. (1.1) is oscillatory.

Example 2.1

Consider the second order difference equation

$$\begin{aligned} \varDelta \left( \varDelta x(t \right) ^{3})+\, q\left( t \right) x^{\beta }\left( t-m+1 \right) =0, \end{aligned}$$
(2.12)

Here,\(\, \alpha =3,\) and \({\upbeta >0\, }\)and \(q\left( t \right) \, \)is a positive of real sequence, r\((t) =\)1, R(t,\(\, t_{0}) = t-t_{0}\).

If the second-order linear difference equation

$$\begin{aligned} \varDelta ^{2}y(t)+\frac{C}{\alpha }{\, \left( \frac{t-m+1-t_{1}}{t{+}1{-}\mathrm {t}_{\mathrm {1}}} \right) \left( t{-}m{+}1{-}t_{0} \right) }^{2}q\left( t \right) \varphi \left( t{-}m{+}1 \right) y\left( t{+}1 \right) {=}\,0 \end{aligned}$$

is oscillatory for all large \(t\ge t_{1\, }and\, any\, constant\, C\in (0\, ,1]\) .All condition of Theorem 2.1 are satisfied and hence we see that Eq. (2.12) is oscillatory.

3 Applications

In this section we apply our previous results to neutral second-order difference equations of the form

$$\begin{aligned} \varDelta (r\left( t \right) ({\varDelta y\left( t \right) )}^{\alpha })+q\left( t \right) x^{\beta }\left( t-m+1 \right) =0, \end{aligned}$$
(3.1)

where \((I) y(t) = x(t) + \mathrm {p}_{\mathrm {1}}\left( \mathrm {t} \right) \mathrm {x}^{{\upgamma }}\left( \mathrm {t-}k_{1} \right) \mathrm {+}\mathrm {p}_{\mathrm {2}}\left( \mathrm {t} \right) \mathrm {x}^{{\updelta }}\left( \mathrm {t-}k_{2} \right) \) or, (II) y(t) = x(t) \(+\) p(t) \(\mathrm {x}^{{\upgamma }}\left( \mathrm {t-k} \right) ,\)

\({\upgamma \, \text {and}\, \updelta \, \text {are}\, \text {ratios}\, \text {of}\, \text {positive}\, \text {odd}\, \text {integers}\, \text {with}\, 0<\, \upgamma \le 1\, \text {and}\, \updelta \ge 1}\) , k,\({\, }k_{1}\) and \(k_{2\, }\)are positive integers and \(\{p(t)\}, \{\mathrm {p}_{\mathrm {1}}\left( \mathrm {t} \right) \}\) and \(\{\mathrm {p}_{\mathrm {2}}\left( \mathrm {t} \right) \}\) are positive sequences of real numbers.

To obtain our results we need the following lemma:

Lemma 3.1

[12]. If X and Y are nonnegative, then

$$\begin{aligned} X^{\lambda }-\left( 1-\lambda \right) Y^{\lambda }-\lambda XY^{\lambda -1}\le \lambda {\, <\, 1,} \end{aligned}$$
(3.2)

where equality holds if and only if X \(=\) Y.

Now, we present our oscillation result for Eq. (3.1) with (I),

i.e., second-order equation with sublinear and superlinear neutral terms.

Theorem 3.1

Let the conditions (i)–(iii), and (1.2) hold and let

$$\begin{aligned} \lim _{t\rightarrow \infty }{R^{{\updelta -1}}\left( t,t_{0} \right) p_{2}\left( t \right) =0} \quad and\quad \lim _{t\rightarrow \infty }p_{1}\left( t \right) =\, 0\,. \end{aligned}$$
(3.3)

Equation (3.1) is oscillatory if one of the following conditions holds for all large:

$$\begin{aligned} t\ge t_{1\, }and\, any\, constant\, C\in (0\, ,1]: \end{aligned}$$
  1. (I)

    The second-order linear difference equation

    $$\begin{aligned} \varDelta \left( r^{\frac{1}{\alpha }}(t)\varDelta x\left( t \right) \right) +C \, Q_{1}\left( t,t_{1} \right) x\left( t+1 \right) =0, \end{aligned}$$
    (3.4)

    with \({\upalpha \ge 1,}\) is oscillatory.

  2. (II)

    Let \({\upalpha \ge 1\, and\, assume\, that\, }\)there exists a nondecreasing positive sequence \(\{\pi (t)\}\) such that for any \(t\ge t_{1}\ge t_{0}\)

    $$\begin{aligned} \mathop {\text{ lim } \text{ sup }}\limits _{t \rightarrow \infty } \sum \limits _{t_{0}}^{t-1} {\left[ \, C\pi \left( s \right) \, Q_{1}\left( t,t_{1} \right) \, -\frac{r^{\frac{1}{\alpha }}\left( s \right) }{4}\left( \frac{\varDelta \pi \left( s \right) }{\pi \left( s \right) } \right) ^{2}\, \right] =\infty } ,\, \end{aligned}$$
    (3.5)
  3. (III)

    The first-order linear delay difference equation

    $$\begin{aligned} \varDelta \, w(t)+ C Q_{2}\left( t,t_{1} \right) \, w\left( t-m+1 \right) =0 \end{aligned}$$
    (3.6)

    with \({\upalpha >0}\) is oscillatory.

  4. (IV)

    Let \(0<{\upalpha }\le 1\, \)and condition (2.11).

Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq. (1.1), say \(x(t)> 0, x (\mathrm {t-}k_{1})> 0, x (\mathrm {t-}k_{2}) > 0\), and \(x(t-m+1) > 0\) and \(y(t) > 0\) for \(t\ge t_{1}\mathrm {\, for\, some\, }\mathrm {t}_{1}\ge t_{0}.\) The proof if x(t) is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper. Then, it follows from (3.1) that

$$\begin{aligned} \, \varDelta \left( r(t)\left( \varDelta y(t) \right) ^{\alpha } \right) \le -\, q\left( t \right) x^{\beta }\left( t-m+1 \right) <0 , \end{aligned}$$
(3.7)

Since x(t) \(\le \) y(t), it follows from the definition of y(t) that

$$\begin{aligned} x\left( t \right)= & {} y\left( t \right) -\, p_{1}\left( t \right) x^{\gamma }\left( \mathrm {t-}k_{1} \right) -p_{2}\left( t \right) x^{\delta }\left( \mathrm {t-}k_{2} \right) \nonumber \\&\ge {\, }y\left( t \right) \mathrm {-}\, p_{1}\left( t \right) y^{\gamma }\left( \mathrm {t-}k_{1} \right) -p_{2}\left( t \right) y^{\delta }\left( \mathrm {t-}k_{2} \right) \nonumber \\&\ge y\left( t \right) \mathrm {-}\, p_{1}\left( t \right) y^{\gamma }\left( t \right) -p_{2}\left( t \right) y^{\delta }\left( t \right) {\, } \nonumber \\&\ge y\left( t \right) \, \mathrm {-}{\, p}_{2}\left( t \right) \frac{y\left( t \right) }{y^{1-\delta }\left( t \right) }{-\, }p_{1}\left( t \right) \left[ y^{\gamma }\left( t \right) -y\left( t \right) \right] -p_{1}(t)y(t){\, }\, \end{aligned}$$
(3.8)

By applying (3.2) with

$$\begin{aligned} \lambda =\gamma \, ,X=y\quad \mathrm{and}\quad Y= (\frac{1}{\gamma })^{\frac{1}{\gamma -1}}, \end{aligned}$$

we obtain

$$\begin{aligned} y^{\gamma }\left( t \right) -y\left( t \right) \le \left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\quad \mathrm{for}\,\, t\ge t_{1}. \end{aligned}$$
(3.9)

Substituting (3.9) into (3.8) we find

$$\begin{aligned} x(t)&\ge y(t)-p_{2}(t)\frac{y(t)}{y^{1-\delta }(t)}-p_{1}(t)y(t){\, -}{p}_{1} (t)\left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\, \nonumber \\&{ \ge }y(t)-p_{2}(t)\frac{y(t)}{y^{1-\delta }(t)}-p_{1}(t)y(t)) \nonumber \\&{-\, }\mathrm {p}_{1}(t)\left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\, \nonumber \\&{ \ge }y(t)\left[ 1-p_{2}\left( t\right) \frac{1}{y^{1-\delta }\left( t \right) }-p_{1}\left( t \right) -\frac{{p}_{1}(t)\left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\, }{y(t)} \right] {\, } \end{aligned}$$
(3.10)

Since \(y(t) > 0\) and \(\varDelta y(t) > 0\) on [\(\mathrm {t}_{2}\), \(\infty )\), there exists a constant \(c_{1} > 0\) such that

$$\begin{aligned} y(t) \ge c_{1}\, \text {for t} \ge \mathrm {t}_{2}. \end{aligned}$$
(3.11)

Since \(r(t)\left( \varDelta y(t) \right) ^{\alpha }\) is positive and non-increasing on\(\, [t_{1},\infty )\), there exist a constant \(C > 0\) and a \(\mathrm {t}_{3}\ge t_{2}\)such that

$$\begin{aligned} r(t)\left( \varDelta y(t) \right) ^{\alpha } < C\quad \mathrm{for} \,\, t\ge t_{3}. \end{aligned}$$
(3.12)

Summing the inequality (3.12) from \(t_{3}\) to t-1 we have

$$\begin{aligned} y\left( t \right) \le CR\left( t,t_{3} \right) \mathrm {\, for\, t\, \ge \, }t_{4}{,\, } \end{aligned}$$
(3.13)

for some \(t_{4}{\ge \, }t_{3}\)  and  for  some  constant  \(C>\, 0\).

Using (3.11) and (3.13) in (3.10) gives

$$\begin{aligned} x(t)\ge & {} y(t)\left[ 1-p_{2}\left( t \right) \left( CR\left( t,t_{3} \right) \right) ^{\delta -1}{\, }-p_{1}\left( t \right) (1+\frac{\left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\, }{c_{1}}) \right] \nonumber \\ x(t)\,\ge & {} y(t)\left[ 1-\mathrm {B}\left[ p_{2}\left( t \right) \left( R\left( t,t_{3} \right) \right) ^{\delta -1}+p_{1}\left( t \right) \right] \right] \mathrm {\, for\, t\, \ge \, }t_{4}{\, } \end{aligned}$$
(3.14)

where B \(=\) max \(\{\mathrm {1}+\frac{\left( 1-\gamma \right) \gamma ^{\frac{\gamma }{1-\gamma }}\, }{c_{1}},\, C^{\delta -1}\}\).

Now, in view of (3.3), for any \(\rho \in \) (0, 1) there exists \(t_{\rho }\ge t_{4}\) such that

$$\begin{aligned} x(t) \ge \rho y(t)\quad \mathrm{for}\, t\, \ge t_{\rho }. \end{aligned}$$
(3.15)

Fix \(\rho \in \) (0, 1) and choose \(t_{\rho }\)by (3.15). Since \(\lim _{t\rightarrow \infty }{\tau (t)} = \infty \), we can choose \(t_{5} \ge t_{\rho }\) such that \(\tau \) (t) \(\ge t_{\rho }\) for all t \(\ge t_{5}\). Thus, from (3.15) we have

$$\begin{aligned} x(\tau (t)) \ge \rho y(\tau (t))\quad \mathrm{for}\, t\, \ge \, t_{5}. \end{aligned}$$
(3.16)

Using this inequality in Eq. (3.1) we find

$$\begin{aligned} \varDelta \left( r(t)\left( \varDelta y(t) \right) ^{\alpha } \right) +q(t)\rho ^{\beta }y^{\beta }(t-m+1)\le 0. \end{aligned}$$
(3.17)

The rest of the proof is similar to that of Theorem 2.1 and hence is omitted.

Example 3.1

Consider the second-order neutral difference equation

$$\begin{aligned} \varDelta \left( \left( \varDelta \left[ \mathrm {x}\left( \mathrm {t} \right) \mathrm {+\, }\frac{1}{t}\mathrm {x}^{\frac{\mathrm {1}}{\mathrm {3}}}\left( \mathrm {t-}k_{1} \right) \mathrm {+}\frac{1}{t^{3}}\mathrm {x}^{\mathrm {3}}\left( \mathrm {t-}k_{2} \right) \right] \right) ^{3} \right) +\, q\left( t \right) x^{\beta }\left( t-m+1 \right) =0,\nonumber \\ \end{aligned}$$
(3.18)

and the second-order difference equation with a sublinear neutral term of the form

$$\begin{aligned} \varDelta \left( \left( \varDelta \left[ \mathrm {\, x}\left( \mathrm {t} \right) \mathrm {+\, }\frac{1}{t}\mathrm {x}^{\frac{\mathrm {1}}{\mathrm {3}}}\left( \mathrm {t-}k_{1} \right) \right] \right) ^{3} \right) +\, q\left( t \right) x^{\beta }\left( t-m+1 \right) =0 . \end{aligned}$$
(3.19)

Here,\(\, \upalpha =3, {\upgamma =}\frac{\mathrm {1}}{\mathrm {3}}{,\, \updelta =3}\) and \({\upbeta >0}\) , k,\({\, }k_{1}\, \)and \(k_{2\, }\)are positive integers and p(t)\(=\)1/t\(={\mathrm {\, p}}_{\mathrm {1}}\left( \mathrm {t} \right) \) ,

\(\mathrm {p}_{\mathrm {2}}\left( \mathrm {t} \right) = \frac{1}{t^{3}}\, ,\, q\left( t \right) \, \)are positive sequences of real numbers. \(r(t) =1, R(t,\, t_{0}) = t-1-t_{0}\).

If the second-order linear difference equation

$$\begin{aligned} \varDelta ^{2}y(t){+}\frac{C}{\alpha }{\, \left( \frac{t-m+1-t_{1}}{t+1-\mathrm {t}_{\mathrm {1}}} \right) \left( t{-}m{+}1{-}t_{0} \right) ^{2}}q\left( t \right) \, \varphi \left( t-m+1 \right) y\left( t{+}1 \right) =0, \end{aligned}$$

is oscillatory for all large \(t\ge t_{1\, }and\, any\, constant\, C\in (0\, ,1]\) .Applying Theorems 2.1 and 3.2 we see that both Eqs. (3.18) and (3.19) are oscillatory.

Next, we present the following oscillation result for Eq. (3.1) with (II),

i.e., second-order equation with a sublinear neutral term.

Theorem 3.2

Let the conditions (i)–(iii), and (1.2) hold and let \({\mathrm {lim}}_{t\rightarrow \infty }p_{1}\left( t \right) =\, 0\, .\)

Then the conclusions of Theorem 3.1 hold.

Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq. (1.1), say \(x(t)> 0, x (\mathrm {t-k}))> 0 \text {and} x(t-m+1) > 0\) and \(y(t) > 0\) for \(t\ge t_{1}\mathrm {\, for\, some\, }\mathrm {t}_{1}\ge t_{0}.\) The proof if x(t) is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper. Then, it follows from (3.1) that

$$\begin{aligned} \, \varDelta \left( r(t)\left( \varDelta y(t) \right) ^{\alpha } \right) \le -\, q\left( t \right) x^{\beta }\left( t-m+1 \right) <0 , \end{aligned}$$
(3.20)

Since x(t) \(\le \) y(t), it follows from the definition of y(t) with y(t) is a nondecreasing sequence that

$$\begin{aligned} x\left( t \right) = y\left( t \right) -\, p\left( t \right) x^{\gamma }\left( \mathrm {t-k} \right) \\ x\left( t \right) \ge y\left( t \right) -\, p\left( t \right) y^{\gamma }(t) \\ x\left( t \right) \ge y\left( t \right) \left( \mathrm {1-}\frac{p\left( t \right) }{y^{1-\gamma }\left( t \right) } \right) {.} \end{aligned}$$

Using (3.11), there exists a constant b \(\in (0\, ,1]\) such that

$$\begin{aligned} x(t) \ge b\, y(t) \quad \mathrm{for}\,\, \text{ t } \ge t_{1}. \end{aligned}$$

The rest of the proof is similar to that of Theorem 2.1 and hence is omitted.

Example 3.1. Consider the second-order neutral difference equation

$$\begin{aligned} \varDelta \left( \left( \varDelta \left[ \mathrm {\, x}\left( \mathrm {t} \right) \mathrm {+\, }\frac{1}{t}\mathrm {x}^{\frac{\mathrm {1}}{\mathrm {3}}}\left( \mathrm {t-}k_{1} \right) \mathrm {+}\frac{1}{t^{3}}\mathrm {x}^{\mathrm {3}}\left( \mathrm {t-}k_{2} \right) \right] \right) ^{1/3} \right) +\, q\left( t \right) x^{\beta }\left( t-m+1 \right) =0, \nonumber \\ \end{aligned}$$
(3.21)

and the second-order difference equation with a sublinear neutral term of the form

$$\begin{aligned} \varDelta \left( \left( \varDelta \left[ \mathrm {\, x}\left( \mathrm {t} \right) \mathrm {+\, }\frac{1}{t}\mathrm {x}^{\frac{\mathrm {1}}{\mathrm {3}}}\left( \mathrm {t-}k_{1} \right) \right] \right) ^{1/3} \right) +\, q\left( t \right) x^{\beta }\left( t-m+1 \right) =0 . \end{aligned}$$
(3.22)

Here,\(\, \alpha =1/3, {\upgamma =}\frac{\mathrm {1}}{\mathrm {3}}{,\, \updelta =3}\) and \({\upbeta >0}\) , k,\({\, }k_{1}\, \)and \(k_{2\, }\)are positive integers and p(t)\(=\)1/t\(={\mathrm {\, p}}_{\mathrm {1}}\left( \mathrm {t} \right) \),

\(\mathrm {p}_{\mathrm {2}}\left( \mathrm {t} \right) = \frac{1}{t^{3}}\, ,\, q\left( t \right) \, \)are positive of real numbers. r(t) \(=\)1, R(t,\(\, t_{0}) =\) t-1-\(t_{0}\).

If the first-order linear delay difference equation

$$\begin{aligned} \varDelta w(t)+\frac{C}{\alpha }{\, \left( t-m+1-t_{1} \right) }^{\frac{1}{3}}q\left( t \right) \, \varphi \left( t-m+1 \right) w\left( t-m\, +1 \right) =0, \end{aligned}$$

is oscillatory for all large \(t\ge t_{2\, }and\, any\, constant\, C\in (0\, ,1]\) , then Eqs. (3.21) and (3.22) are oscillatory.

4 General Remarks

The results of this paper are presented in a new form and of high degree of generality.

Our main task here is to reduce the oscillation of half-linear delay difference equations and/or nonlinear delay difference equations to the oscillation of linear or first-order difference equations whose oscillatory behavior is known and literature is filled with all types of criteria.