1 Introduction

This work is concerned with studying the oscillation properties of the second-order neutral delay differential equations with damping term of the form

$$ \bigl( r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+p ( t ) \bigl( z^{ \prime } ( t ) \bigr) ^{\alpha }+q ( t ) f \bigl( x \bigl( \sigma ( t ) \bigr) \bigr) =0, \quad t\geq t_{0}, $$
(1)

where

$$ z ( t ) =x ( t ) +c ( t ) x \bigl( \tau ( t ) \bigr) . $$
(2)

Throughout this work, we assume \(\alpha \in Q_{\mathrm{odd}}^{+}:= \{ a/b: a,b\in \mathbf{Z} ^{+}\text{ are odd} \} \), r, c, τ, σ, p, \(q\in C ( [t_{0},\infty ) ) \), r is positive, c, p and q are nonnegative, \(\tau ( t ) \leq t\), \(\sigma ( t ) \leq t \), \(\sigma ^{\prime } ( t ) >0\), \(\lim_{t\rightarrow \infty }\tau ( t ) =\lim_{t \rightarrow \infty }\sigma ( t ) =\infty \), \(f\in C ( \mathbf{R} ,\mathbf{R} ) \), and it satisfies the following property:

$$ f ( x ) >kx^{\beta }\quad \text{for all }x\neq 0, $$
(3)

where \(k>0\) is a constant and \(\beta \in Q_{\mathrm{odd}}^{+}\). Furthermore, this study requires that

$$ \int _{t_{0}}^{\infty }\frac{1}{r^{1/\alpha } ( h ) }\exp \biggl( \frac{-1}{\alpha } \int _{t_{0}}^{h} \frac{p ( s ) }{r ( s ) }\,\mathrm{d}s \biggr) \,\mathrm{d}h< \infty $$
(4)

and

$$ c ( t ) < \min \biggl\{ 1, \frac{\eta ( t ) }{\eta ( \tau ( t ) ) } \biggr\} , $$

where

$$ \eta ( t ) := \int _{t}^{\infty } \biggl( \frac{1}{r ( v ) }\exp \biggl( - \int _{t_{0}}^{v} \frac{p ( s ) }{r ( s ) }\,\mathrm{d}s \biggr) \biggr) ^{1/\alpha }\,\mathrm{d}v. $$

A real-valued function \(x\in C ( [t_{x},\infty ),\mathbf{R} ) \), \(t_{x}\geq t_{0}\) is a solution of (1) if x satisfies (1) on \([t_{x},\infty )\), and it has the property that \(z ( t ) \) and \(r ( t ) ( z^{\prime } ( t ) ) ^{ \alpha }\) are continuously differentiable for \(t\in {}[ t_{x},\infty )\). A nontrivial solution x is said to be oscillatory, if it has arbitrary large zeros. Otherwise, it is said to be non-oscillatory. The set of all eventually positive solutions of (1) is denoted by ℜ, that is, if \(x\in \Re \), then there exists a \(t_{1}\geq t_{0}\) large enough such that \(x ( t ) >0\) for all \(t\geq t_{1}\). We only focus on solutions of (1), which exist on \([ t_{0},\infty ) \) and satisfy \(\sup \{ \vert x ( t ) \vert :t_{x}\leq t\}>0\) for every \(t\geq t_{x}\).

Half-linear differential equations arise in real problems; for instance, in the study of non-Newtonian fluid theory and the turbulent flow of a polytropic gas in a porous medium; see [1, 2].

In the past few years, there has been research studying the asymptotic properties and the oscillatory behavior of solutions of differential equations with different order. This research focused on developing and improving the oscillation criteria for differential equations. References [3–9] improved the oscillation criteria for noncanonical second-order equations with delay and advanced argument. For canonical second-order delay equations, Refs. [10, 11] developed the oscillation criteria. The results in [12–21] dealt with the issue of oscillation of equations of higher order. For differential equations with damping, we present the following results that are closely related to this paper.

Tunc and Kaymaz [22] studied the oscillatory behavior of equations with damping term

$$ z^{\prime \prime } ( t ) +r ( t ) z^{\prime } ( t ) +q ( t ) x \bigl( \sigma ( t ) \bigr) =0, \quad t\geq t_{0}>0, $$

under the condition

$$ \int _{t_{0}}^{\infty }\exp \biggl( - \int _{t_{0}}^{t}r ( s ) \,\mathrm{d}s \biggr) \,\mathrm{d}t=\infty . $$

They are extended to more general second-order linear and/or nonlinear neutral differential equations with damping in [23–25]. Saker et al. [26] established Kamenev-type and Philos-type theorems for oscillation of equation with damping term

$$ \bigl( a ( t ) x^{\prime } ( t ) \bigr) ^{ \prime }+p ( t ) x^{\prime } \bigl( \sigma ( t ) \bigr) +q ( t ) f \bigl( x \bigl( g ( t ) \bigr) \bigr) =0. $$

Results in [26] are extended and improved results in [27–29].

In this paper, we try to improve the oscillation criteria for solutions of (1) by creating new and more effective criteria. In a noncanonical case, we always have two cases of derivatives signs for corresponding function z, which often leads to two independent conditions to ensure oscillatory. As an extension of the results of [4], we create a new criterion for oscillation of (1), which in turn is a simplification of the previous results in [26].

Although the theoretical advantage lies in reducing the number of conditions that are sufficient to verify the oscillation of solutions of differential equations, but sometimes a single condition is less effective in practical applications. So, we also follow the usual approach to creating two independent criteria of oscillation.

Remark 1

The functional inequalities in this paper are supposed to hold eventually, that is, they are satisfied for all t large enough.

2 Preliminaries

In the following, we provide some notations which help us to easily display the results. Moreover, we present the auxiliary lemmas which help in validating the main results.

Notation 1

For the sake of brevity, we define the operators \(\gamma := ( \alpha / ( \alpha +1 ) ) ^{ \alpha +1}\),

$$ \mu ( t ) :=\exp \biggl( \int _{t_{1}}^{t} \frac{p ( s ) }{r ( s ) }\,\mathrm{d}s \biggr) , $$

and

$$ \varrho ( t ) := \textstyle\begin{cases} 1 & \mbox{if } \alpha = \beta , \\ l_{1} & \mbox{if } \alpha >\beta , \\ l_{2}\eta ^{\beta -\alpha } ( t ) & \mbox{if } \alpha < \beta ,\end{cases}$$

where \(l_{1}\), \(l_{2}\) are positive constants.

Notation 2

For the sake of brevity, we define the functions

$$ G ( t ) :=k\mu ( t ) q ( t ) \bigl( 1-c \bigl( \sigma ( t ) \bigr) \bigr) ^{\beta } $$

and

$$ Q ( t ) :=k\mu ( t ) q ( t ) \biggl( 1-c \bigl( \sigma ( t ) \bigr) \frac{\eta ( \tau ( \sigma ( t ) ) ) }{\eta ( \sigma ( t ) ) } \biggr) ^{\beta }. $$

Lemma 1

Let \(\Theta ( \theta ) =A_{1}\theta -A_{2} ( \theta -A_{3} ) ^{ ( \alpha +1 ) /\alpha }\), where \(A_{2}>0, A_{1}\) and \(A_{3}\) are constants. Then the maximum value of Θ on R at \(\theta ^{\ast }=A_{3}+ ( \alpha A_{1}/ ( ( \alpha +1 ) A_{2} ) ) ^{\alpha }\) is

$$ \max_{\theta \in R} \Theta ( \theta ) = \Theta \bigl( \theta ^{\ast } \bigr) =A_{1}A_{3}+ \frac{\gamma }{\alpha }\frac{A_{1}^{\alpha +1}}{A_{2}^{\alpha }}. $$
(5)

Lemma 2

Assume that \(x\in \Re \). Then:

  1. (i)

    The function z satisfies \(z ( t ) >0\), \(( \mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha } ) ^{\prime }\leq 0\) and one of the next cases:

    $$\begin{aligned}& ( \mathbf{D}_{1} )\quad z^{\prime } ( t ) > 0; \\& ( \mathbf{D}_{2} )\quad z^{\prime } ( t ) < 0. \end{aligned}$$
  2. (ii)

    If z satisfies (\(\mathbf{D}_{2} \)), then

    $$ z^{\beta -\alpha } ( t ) \geq \varrho ( t ) $$
    (6)

    and

    $$ \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }\leq -Q ( t ) z^{\beta } \bigl( \sigma ( t ) \bigr) . $$
    (7)

Proof

Since \(x\in \Re \), it is obvious that there exists a \(t_{1}\geq t_{0}\) such that \(x ( t )\), \(x ( \tau ( t ) ) \) and \(x ( \sigma ( t ) ) \) are positive functions. As a direct conclusion from the definition (2), we find that \(z ( t ) >0\). Next, from (1) and (3), we obtain

$$\begin{aligned} \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime } =&-\mu ( t ) q ( t ) f \bigl( x \bigl( \sigma ( t ) \bigr) \bigr) \\ \leq &-k\mu ( t ) q ( t ) x^{\beta } \bigl( \sigma ( t ) \bigr) , \end{aligned}$$
(8)

which means that \(\mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha }\) is a nonincreasing function and has a fixed sign. Since \(\mu ( t ) r ( t ) >0\), we get either \(z^{\prime } ( t ) >0\) or \(z^{\prime } ( t ) <0\).

Now, let \(z^{\prime } ( t ) <0\) for all \(t\geq t_{2}\geq t_{1}\), where \(t_{2}\) is large enough. Then there exists a constant \(K_{1}>0\) such that

$$ z^{\beta -\alpha } ( t ) \geq K_{1}^{\beta -\alpha }=l_{1} \quad \text{for }\alpha \geq \beta . $$

On the other hand, using the fact that \(( \mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha } ) ^{\prime }\leq 0\), we get

$$ \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha }\leq \mu ( t_{2} ) r ( t_{2} ) \bigl( z^{\prime } ( t_{2} ) \bigr) ^{\alpha }=-K_{2}< 0 $$
(9)

and therefore

$$ z^{\prime } ( t ) \leq \biggl( \frac{-K_{2}}{\mu ( t ) r ( t ) } \biggr) ^{1/\alpha }. $$
(10)

Integrating (10) from t to ∞ we obtain \(-z ( t ) \leq -K_{2}^{1/\alpha }\eta ( t ) \). Thus,

$$ z^{\beta -\alpha } ( t ) \geq K_{2}^{ \frac{\beta -\alpha }{\alpha }}\eta ^{\beta -\alpha } ( t ) :=l_{2}\eta ^{\beta -\alpha } ( t ) \quad \text{for }\alpha < \beta , $$

i.e., (6) holds. Moreover, we have

$$ z ( t ) \geq - \int _{t}^{\infty }z^{\prime } ( \theta ) \,\mathrm{d}\theta \geq -\eta ( t ) \bigl( \mu ( t ) r ( t ) \bigr) ^{1/\alpha }z^{ \prime } ( t ) $$
(11)

and so

$$ \frac{\mathrm{d}}{\mathrm{d}t} \biggl( \frac{z ( t ) }{\eta ( t ) } \biggr) = \frac{\eta ( t ) z^{\prime } ( t ) + ( \mu ( t ) r ( t ) ) ^{-1/\alpha }z ( t ) }{\eta ^{2} ( t ) }\geq 0. $$

Hence,

$$ x ( t ) \geq \biggl( 1-c ( t ) \frac{\eta ( \tau ( t ) ) }{\eta ( t ) } \biggr) z ( t ) , $$

which, in view of (8), gives (7). The proof of the lemma is complete. □

Lemma 3

Let \(x\in \Re \), the function z satisfy (\(\mathbf{D}_{2} \)) and

$$ \int _{t_{0}}^{t} \biggl( \frac{1}{\mu ( v ) r ( v ) }\int _{t_{1}}^{v}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha } \,\mathrm{d}v=\infty . $$
(12)

Then

$$ \lim_{t\rightarrow \infty }x ( t ) =\lim_{t\rightarrow \infty }z ( t ) =0. $$
(13)

Proof

Suppose that \(x\in \Re \) and the function z satisfies (\(\mathbf{D}_{2} \)). Since z is a positive decreasing function, we have \(\lim_{t\rightarrow \infty }z ( t ) =\epsilon \geq 0\).

Let \(\epsilon >0\). From Lemma 2, there exists a \(t_{1}\geq t_{0}\) such that

$$ \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }\leq -Q ( t ) z^{\beta } \bigl( \sigma ( t ) \bigr) \leq - \epsilon ^{\beta }Q ( t ) . $$

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

$$ -z^{\prime } ( t ) \geq \epsilon ^{\beta /\alpha } \biggl( \frac{1}{\mu ( t ) r ( t ) } \int _{t_{1}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }. $$
(14)

Integrating (14) from \(t_{1}\) to t, we get

$$ z ( t ) \leq z ( t_{1} ) -\epsilon ^{\beta / \alpha } \int _{t_{1}}^{t} \biggl( \frac{1}{\mu ( v ) r ( v ) }\int _{t_{1}}^{v}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha } \,\mathrm{d}v. $$

Taking \(\lim_{t\rightarrow \infty }\) of this inequality and using (12), we are led to a contradiction with positivity of z. Therefore, \(\epsilon =0\) and (13) hold. This completes the proof. □

Lemma 4

Let \(x\in \Re \), the function z satisfy (\(\mathbf{D}_{2} \)) and (12) hold. If, moreover there exists a constant \(\delta \in [ 0,1 ) \) such that

$$ \eta ( t ) \biggl( \varrho ( t ) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }\geq \delta , $$
(15)

then

$$ \frac{\mathrm{d}}{\mathrm{d}t} \biggl( \frac{z ( t ) }{\eta ^{\delta } ( t ) } \biggr) \leq 0. $$
(16)

Proof

Suppose that \(x\in \Re \) and the function z satisfies (\(\mathbf{D}_{2} \)). From Lemma 2, we see that (7) holds. Integrating (7) from \(t_{1}\) to t, we obtain

$$\begin{aligned} \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \leq & \mu ( t_{1} ) r ( t_{1} ) \bigl( z^{\prime } ( t_{1} ) \bigr) ^{ \alpha }- \int _{t_{1}}^{t}Q ( s ) z^{\beta } \bigl( \sigma ( s ) \bigr) \,\mathrm{d}s \\ \leq &\mu ( t_{1} ) r ( t_{1} ) \bigl( z^{ \prime } ( t_{1} ) \bigr) ^{\alpha }-z^{\beta } \bigl( \sigma ( t ) \bigr) \int _{t_{1}}^{t}Q ( s ) \,\mathrm{d}s. \end{aligned}$$
(17)

From Lemma 3, clearly (13) holds. Thus,

$$ \mu ( t_{1} ) r ( t_{1} ) \bigl( z^{\prime } ( t_{1} ) \bigr) ^{\alpha }+z^{\beta } \bigl( \sigma ( t ) \bigr) \int _{t_{0}}^{t_{1}}Q ( s ) \,\mathrm{d}s>0, $$
(18)

for \(t\geq t_{2}\), where \(t_{2}\) is large enough. Combining (17) and (18) and using the fact that \(z^{\beta -\alpha } ( t ) \geq \varrho ( t ) \), we find

$$ \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha }\leq -z^{\beta } \bigl( \sigma ( t ) \bigr) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \leq -\varrho ( t ) z^{\alpha } ( t ) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s, $$
(19)

which, in view of (15), gives

$$\begin{aligned} z^{\prime } ( t ) \leq &- \bigl( \mu ( t ) r ( t ) \bigr) ^{-1/\alpha }z ( t ) \biggl( \varrho ( t ) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha } \\ \leq &-\frac{\delta }{\eta ( t ) } \bigl( \mu ( t ) r ( t ) \bigr) ^{-1/\alpha }z ( t ) . \end{aligned}$$

Thus,

$$ \frac{\mathrm{d}}{\mathrm{d}t} \biggl( \frac{z ( t ) }{\eta ^{\delta } ( t ) } \biggr) = \frac{\eta ^{\delta -1} ( t ) ( \eta ( t ) z^{\prime } ( t ) +\delta ( \mu ( t ) r ( t ) ) ^{-1/\alpha }z ( t ) ) }{\eta ^{2\delta } ( t ) }\leq 0. $$

This completes the proof. □

3 The one criterion theorems

In the following theorems, we obtain a criterion which ensure oscillation of (1) without verifying the extra condition.

Theorem 1

If

$$ \int _{t_{1}}^{\infty } \biggl( \frac{1}{\mu ( h ) r ( h ) }\int _{t_{1}}^{h}\eta ^{\beta } \bigl( \sigma ( v ) \bigr) Q ( v ) \,\mathrm{d}v \biggr) ^{1/\alpha } \,\mathrm{d}h=\infty , $$
(20)

for any \(t_{1}\geq t_{0}\), then all solutions of (1) are oscillatory.

Proof

Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).

Let (\(\mathbf{D}_{2} \)) hold. As in the proof of Lemma 2, we arrive at (19). Integrating (19) from t to ∞, we find

$$ z ( t ) \geq K_{2}^{1/\alpha }\eta ( t ) $$
(21)

for all \(t\geq t_{2}\geq t_{1}\), where \(t_{2}\) is large enough. From (7) in Lemma 2, we have

$$ \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }\leq -K_{2}^{ \beta /\alpha }Q ( t ) \eta ^{\beta } \bigl( \sigma ( t ) \bigr) . $$

Integrating this inequality from \(t_{2}\) to t, we get

$$ z^{\prime } ( t ) \leq -K_{2}^{\beta /\alpha ^{2}} \biggl( \frac{1}{\mu ( t ) r ( t ) } \int _{t_{2}}^{t}\eta ^{ \beta } \bigl( \sigma ( v ) \bigr) Q ( v ) \,\mathrm{d}v \biggr) ^{1/\alpha }. $$

Integrating again from \(t_{2}\) to t, we obtain

$$ z ( t ) \leq z ( t_{2} ) -K_{2}^{\beta /\alpha ^{2}} \int _{t_{2}}^{t} \biggl( \frac{1}{\mu ( h ) r ( h ) }\int _{t_{2}}^{h}\eta ^{\beta } \bigl( \sigma ( v ) \bigr) Q ( v ) \,\mathrm{d}v \biggr) ^{1/\alpha } \,\mathrm{d}h. $$

Taking \(\lim_{t\rightarrow \infty }\) of this inequality and using (20), we are led to a contradiction with positivity of z.

Next, we let (\(\mathbf{D}_{1} \)) hold. Since \(z ( t ) \geq x ( t ) \), \(\tau ( t ) \leq t\) and \(z^{\prime } ( t ) >0\), we get \(x ( t ) \geq ( 1-c ( t ) ) z ( t ) \). Hence, (1) becomes

$$ \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }\leq -G ( t ) z^{\beta } \bigl( \sigma ( t ) \bigr) . $$
(22)

Since \(\eta ( \tau ( \sigma ( t ) ) ) \geq \eta ( \sigma ( t ) ) \), we obtain

$$ 1-c \bigl( \sigma ( t ) \bigr) \frac{\eta ( \tau ( \sigma ( t ) ) ) }{\eta ( \sigma ( t ) ) }\leq 1-c \bigl( \sigma ( t ) \bigr) . $$
(23)

We note that the function \(\int ^{t}\eta ^{\beta } ( \sigma ( v ) ) Q ( v ) \,\mathrm{d}v\) is unbounded due to (20) and (4). Taking into account \(\eta ^{\prime } ( t ) \leq 0 \) and (23), it is easy to see that

$$ \int ^{\infty }G ( v ) \,\mathrm{d}v\geq \int ^{\infty }Q ( v ) \,\mathrm{d}v=\infty . $$
(24)

Combining (22) and (23), we have

$$\begin{aligned} \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \leq & \mu ( t_{1} ) r ( t_{1} ) \bigl( z^{\prime } ( t_{1} ) \bigr) ^{ \alpha }- \int _{t_{1}}^{t}G ( v ) z^{\beta } \bigl( \sigma ( v ) \bigr) \,\mathrm{d}v \\ \leq &\mu ( t_{1} ) r ( t_{1} ) \bigl( z^{ \prime } ( t_{1} ) \bigr) ^{\alpha }-z^{\beta } \bigl( \sigma ( t_{2} ) \bigr) \int _{t_{1}}^{t}G ( v ) \,\mathrm{d}v, \end{aligned}$$

which, in view of (24), contradicts the positivity \(z^{\prime } ( t ) \). The proof of the theorem is complete. □

Theorem 2

If

$$ \limsup_{t\rightarrow \infty } \eta ( t ) \biggl( \varrho ( t ) \int _{t_{1}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }>1 $$
(25)

for any \(t_{1}\geq t_{0}\), then all solutions of (1) are oscillatory.

Proof

Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).

Let (\(\mathbf{D}_{2} \)) hold. From the proofs of Lemmas 2 and 4, we arrive at (11) and (17), respectively. Combining (11) and (17), we get

$$ \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha }\leq \varrho ( t ) \eta ^{ \alpha } ( t ) \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \int _{t_{1}}^{t}Q ( s ) \,\mathrm{d}s $$

and so

$$ \eta ( t ) \biggl( \varrho ( t ) \int _{t_{1}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }\leq 1, $$

which contradicts (25).

Next, we let \(z^{\prime } ( t ) >0\) for \(t\geq t_{1}\). We note that (25) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete. □

4 Theorems of two independent criteria

Theorem 3

Assume that (12) holds. If

$$ \limsup_{t\rightarrow \infty } \eta ( t ) \biggl( \varrho ( t ) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }>1, $$
(26)

then all solutions of (1) are oscillatory.

Proof

For the proof of this lemma, it suffices to use (19) [from the proof of Lemma 4] instead of (17) in the proof of Theorem 2. □

Using Lemma 4, we obtain the new results which improve the previous theorems when

$$ \limsup_{t\rightarrow \infty } \eta ( t ) \biggl( \varrho ( t ) \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }\leq 1. $$

Theorem 4

Assume that (12) and there exists a \(\delta \in [ 0,1 ) \) such that (15) holds. If

$$ \limsup_{t\rightarrow \infty } \eta ( t ) \biggl( \varrho ( t ) \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{\delta \beta } \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/\alpha }>1, $$
(27)

then all solutions of (1) are oscillatory.

Proof

Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).

Let (\(\mathbf{D}_{2} \)) hold. Using Lemma 4, we arrive at (16) and (19). From (16), we conclude that

$$ z \bigl( \sigma ( t ) \bigr) \geq \frac{\eta ^{\delta } ( \sigma ( t ) ) }{\eta ^{\delta } ( t ) }z ( t ) , $$
(28)

which with (19) gives

$$ \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha }\leq -z^{\beta } ( t ) \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{ \delta \beta } \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s. $$
(29)

As in the proof of Lemma 2, we arrive at (11). Combining (11) and (29), we obtain

$$\begin{aligned} \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \leq &-\varrho ( t ) z^{ \alpha } ( t ) \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{\delta \beta } \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \\ \leq &\varrho ( t ) \eta ^{\alpha } ( t ) \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{\delta \beta } \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s, \end{aligned}$$

then

$$ \eta ( t ) \biggl( \varrho ( t ) \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{ \delta \beta } \int _{t_{0}}^{t}Q ( s ) \,\mathrm{d}s \biggr) ^{1/ \alpha }\leq 1. $$

This contradicts (27).

Next, we let (\(\mathbf{D}_{1} \)) hold. We note that (12) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete. □

Theorem 5

Assume that (12) and there exists a \(\delta \in [ 0,1 ) \) such that (15) holds. If there exist a function \(\rho \in C^{1} ( [ t_{0},\infty ) , ( 0, \infty ) ) \) and a \(t_{1}\in [ t_{0},\infty ) \) such that

$$ \limsup_{t\rightarrow \infty } \biggl\{ \frac{\eta ^{\alpha } ( t ) }{\rho ( t ) } \int _{t_{0}}^{t} \biggl( \rho ( v ) Q ( v ) \varrho ( v ) \biggl( \frac{\eta ( \sigma ( v ) ) }{\eta ( v ) } \biggr) ^{ \delta \alpha }-\gamma \frac{\mu ( v ) r ( v ) ( \rho ^{\prime } ( v ) ) ^{\alpha +1}}{\alpha ^{\alpha +1}\rho ^{\alpha } ( v ) } \biggr) \,\mathrm{d}v \biggr\} >1, $$
(30)

then all solutions of (1) are oscillatory.

Proof

Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).

Let (\(\mathbf{D}_{2} \)) hold. From Lemma 2, we see that (6), (7) and (11) hold. We define the function

$$ w ( t ) :=\rho ( t ) \biggl( \frac{\mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha }}{z^{\alpha } ( t ) }+ \frac{1}{\eta ^{\alpha } ( t ) } \biggr) . $$
(31)

Using (11), we see that \(w ( t ) \geq 0\) for all \(t\geq t_{2}\geq t_{1}\). Differentiating (31), we get

$$\begin{aligned} w^{\prime } ( t ) =&\frac{\rho ^{\prime } ( t ) }{\rho ( t ) }w ( t ) +\rho ( t ) \frac{ ( \mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha } ) ^{\prime }}{z^{\alpha } ( t ) }-\alpha \rho ( t ) \frac{\mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha +1}}{z^{\alpha +1} ( t ) } \\ &{}+\alpha \rho ( t ) \frac{1}{ ( \mu ( t ) r ( t ) ) ^{1/\alpha }\eta ^{\alpha +1} ( t ) }. \end{aligned}$$

Combining (6), (7) and (31), we obtain

$$\begin{aligned} w^{\prime } ( t ) \leq &- \frac{\alpha }{ ( \rho ( t ) \mu ( t ) r ( t ) ) ^{1/\alpha }} \biggl( w ( t ) - \frac{\rho ( t ) }{\eta ^{\alpha } ( t ) } \biggr) ^{1+1/\alpha }-\rho ( t ) Q ( t ) \varrho ( t ) \frac{z^{\alpha } ( \sigma ( t ) ) }{z^{\alpha } ( t ) } \\ &{}+\alpha \rho ( t ) \frac{1}{ ( \mu ( t ) r ( t ) ) ^{1/\alpha }\eta ^{\alpha +1} ( t ) }+ \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }w ( t ) . \end{aligned}$$
(32)

Using inequality (5) with \(A_{1}:=\rho ^{\prime }/\rho \), \(A_{2}:=\alpha / ( \rho \mu r ) ^{-1/\alpha }\), \(A_{3}=\rho /\eta ^{\alpha }\) and \(\theta :=w\), we get

$$ \frac{\rho ^{\prime }}{\rho }w- \frac{\alpha }{ ( \rho \mu r ) ^{1/\alpha }} \biggl( w-\frac{\rho }{\eta ^{\alpha }} \biggr) ^{1+1/ \alpha }\leq \frac{\rho ^{\prime }}{\eta ^{\alpha }}+ \frac{\gamma }{\alpha ^{\alpha +1}}\mu r \frac{ ( \rho ^{\prime } ) ^{\alpha +1}}{\rho ^{\alpha }}, $$

which, in view of (32), gives

$$\begin{aligned} w^{\prime } ( t ) \leq &\frac{\rho ^{\prime } ( t ) }{\eta ^{\alpha } ( t ) }+ \frac{\gamma }{\alpha ^{\alpha +1}}\mu ( t ) r ( t ) \frac{ ( \rho ^{\prime } ( t ) ) ^{\alpha +1}}{\rho ^{\alpha } ( t ) }-\rho ( t ) Q ( t ) \varrho ( t ) \frac{z^{\alpha } ( \sigma ( t ) ) }{z^{\alpha } ( t ) } \\ &{}+\alpha \rho ( t ) \frac{1}{ ( \mu ( t ) r ( t ) ) ^{1/\alpha }\eta ^{\alpha +1} ( t ) }. \end{aligned}$$
(33)

As in the proof of Theorem 4, we arrive at (28). Thus, (33) becomes

$$ w^{\prime } ( t ) \leq -\rho ( t ) Q ( t ) \varrho ( t ) \biggl( \frac{\eta ( \sigma ( t ) ) }{\eta ( t ) } \biggr) ^{\delta \alpha }+ \frac{\gamma }{\alpha ^{\alpha +1}}\mu ( t ) r ( t ) \frac{ ( \rho ^{\prime } ( t ) ) ^{\alpha +1}}{\rho ^{\alpha } ( t ) }+ \biggl( \frac{\rho ( t ) }{\eta ^{\alpha } ( t ) } \biggr) ^{\prime }. $$

Integrating this inequality from \(t_{2}\) to t, we have

$$\begin{aligned}& \int _{t_{2}}^{t} \biggl( \rho ( v ) Q ( v ) \varrho ( v ) \biggl( \frac{\eta ( \sigma ( v ) ) }{\eta ( v ) } \biggr) ^{\delta \alpha }-\gamma \frac{\mu ( v ) r ( v ) ( \rho ^{\prime } ( v ) ) ^{\alpha +1}}{\alpha ^{\alpha +1}\rho ^{\alpha } ( v ) } \biggr) \,\mathrm{d}v \\& \quad \leq \biggl( \frac{\rho ( t ) }{\eta ^{\alpha } ( t ) }-w ( t ) \biggr) \bigg\vert _{t_{2}}^{t} \\& \quad \leq - \biggl( \rho ( t ) \frac{\mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha }}{z^{\alpha } ( t) }\biggr) \bigg\vert _{t_{2}}^{t}. \end{aligned}$$
(34)

From (11), we have

$$ - \frac{ ( \mu ( t ) r ( t ) ) ^{1/\alpha }z^{\prime } ( t ) }{z ( t ) } \leq \frac{1}{\eta ( t ) }, $$

which, in view of (34), implies

$$ \frac{\eta ^{\alpha } ( t ) }{\rho ( t ) }\int _{t_{2}}^{t} \biggl( \rho ( v ) Q ( v ) \varrho ( v ) \biggl( \frac{\eta ( \sigma ( v ) ) }{\eta ( v ) } \biggr) ^{\delta \alpha }-\gamma \frac{\mu ( v ) r ( v ) ( \rho ^{\prime } ( v ) ) ^{\alpha +1}}{\alpha ^{\alpha +1}\rho ^{\alpha } ( v ) } \biggr) \,\mathrm{d}v\leq 1. $$

Taking the lim sup on both sides of this inequality, we are led to a contradiction with (30).

The rest of proof is the same and hence we omit it. The proof of the theorem is complete. □

Example 1

Consider the differential equation

$$ \biggl( t^{\alpha +1} \biggl[ \biggl( x ( t ) +c_{0}x \biggl( \frac{t}{2} \biggr) \biggr) ^{\prime } \biggr] \biggr) ^{\prime }+t \biggl( \biggl( x ( t ) +c_{0}x \biggl( \frac{t}{2} \biggr) \biggr) ^{\prime } \biggr) +q_{0}t^{3}x ( \lambda t ) =0, $$
(35)

where \(t\geq 1\), \(q_{0}\in ( 0,\infty ) \), \(c_{0}\in ( 0,1/16 ) \) and \(\lambda \in ( 0,1 ] \). From Theorem 2, we deduce that Eq. (35) is oscillatory if

$$ \frac{1}{16}q_{0} ( 1-16c_{0} ) >1. $$
(36)

To apply Theorem 4, we first note that (12) and (15) are satisfied with

$$ \delta =\frac{1}{16}q_{0} ( 1-16c_{0} ) . $$

Therefore, (35) is oscillatory if

$$ \frac{1}{16\lambda ^{4\delta }}q_{0} ( 1-16c_{0} ) >1. $$