In this section, we establish oscillation criteria for (1) and (3) by the Riccati transformation and comparison technique.
Theorem 3.1
If the equation
$$ \eta ^{\prime } ( t ) + \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}}\frac{q _{i} ( t ) \tau ^{3 ( p_{2}-1 ) } ( t ) }{r ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( \tau ( t ) ) }\eta ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } \bigl( \tau ( t ) \bigr) =0 $$
(11)
is oscillatory, then (1) is oscillatory.
Proof
Let (1) have a nonoscillatory solution in \([ t _{0},\infty ) \). Then there exists \(t _{1}\geq t _{0}\) such that \(x ( t ) >0\) and \(x ( \tau _{i} ( t ) ) >0\) for \(t \geq t _{1}\). Let
$$ \eta ( t ) :=r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}>0 \quad \text{[from Lemma 2.4]}, $$
which with (1) gives
$$ \eta ^{\prime } ( t ) +q ( t ) x^{p_{2}-1} \bigl( \tau ( t ) \bigr) =0. $$
(12)
Since x is positive and increasing, we see \(\lim_{t \rightarrow \infty }x ( t ) \neq 0\). So, using Lemma 2.1, we find
$$ x^{p_{2}-1} \bigl( \tau ( t ) \bigr) \geq \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}}\tau ^{3 ( p_{2}-1 ) } ( t ) \bigl( x^{\prime \prime \prime } \bigl( \tau ( t ) \bigr) \bigr) ^{p_{2}-1} $$
(13)
for all \(\lambda \in ( 0,1 ) \). By (12) and (13), we see that
$$ \eta ^{\prime } ( t ) + \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}}q _{i} ( t ) \tau ^{3 ( p_{2}-1 ) } ( t ) \bigl( x^{\prime \prime \prime } \bigl( \tau ( t ) \bigr) \bigr) ^{p_{2}-1}\leq 0. $$
So, η is a positive solution of the inequality
$$ \eta ^{\prime } ( t ) + \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}}\frac{q ( t ) \tau ^{3 ( p_{2}-1 ) } ( t ) }{r ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( \tau ( t ) ) }\eta ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } \bigl( \tau ( t ) \bigr) \leq 0. $$
By using [40, Theorem 1], we find that (11) also has a positive solution, which is a contradiction. The proof is complete. □
Corollary 3.2
Let \(p_{2}= p_{1}\) and (2) hold. If
$$ \underset{t \rightarrow \infty }{\lim \inf } \int _{\tau ( t ) }^{t }\ \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}} \frac{q ( s ) \tau ^{3 ( p_{2}-1 ) } ( s ) }{r ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( \tau ( s ) ) }\,\mathrm{d}s> \frac{1}{\mathrm{e}}, $$
(14)
then (1) is oscillatory.
Theorem 3.3
Let \(p_{2}\geq p_{1}\) and (6) hold for some \(\mu \in ( 0,1 ) \). If
$$ u^{\prime \prime } ( t ) +M^{p_{2}-p_{1}}\widetilde{R} ( t ) u ( t ) =0 $$
(15)
is oscillatory, then (1) is oscillatory.
Proof
Assume to the contrary that (1) has a nonoscillatory solution in \([ t _{0},\infty ) \). Without loss of generality, we only need to be concerned with positive solutions of equation (1). Then there exists \(t _{1}\geq t _{0}\) such that \(x ( t ) >0\) and \(x ( \tau _{i} ( t ) ) >0\) for \(t \geq t _{1}\). From Lemmas 2.2 and 2.4, we have that
$$ x^{\prime } ( t ) >0,\qquad x^{\prime \prime } ( t ) < 0\quad \text{and}\quad x^{\prime \prime \prime } ( t ) >0 $$
(16)
for \(t \geq t _{2}\), where \(t _{2}\) is sufficiently large. Now, integrating (1) from t to l, we have
$$ r ( l ) \bigl( x^{\prime \prime \prime } ( l ) \bigr) ^{p_{1}-1}=r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}- \int _{t }^{l}q ( s ) x^{p_{2}-1} \bigl( \tau ( s ) \bigr) \,\mathrm{d}s. $$
(17)
Using Lemma 3 in [34] with (16), we get
$$ \frac{x ( \tau ( t ) ) }{x ( t ) }\geq \lambda \frac{\tau ( t ) }{t }, $$
which with (17) gives
$$ r ( l ) \bigl( x^{\prime \prime \prime } ( l ) \bigr) ^{p_{1}-1}-r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}+\lambda ^{p_{2}-1} \int _{t }^{l}q _{i} ( s ) \biggl( \frac{\tau ( s ) }{s} \biggr) ^{p_{2}-1}x^{p_{1}-1} ( s ) \,\mathrm{d}s\leq 0. $$
It follows, by \(x^{\prime }>0\), that
$$ r ( l ) \bigl( x^{\prime \prime \prime } ( l ) \bigr) ^{p_{1}-1}-r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}+\lambda ^{p_{2}-1}x^{p_{1}-1} ( t ) \int _{t }^{l}q ( s ) \biggl( \frac{\tau ( s ) }{s} \biggr) ^{p_{2}-1}\,\mathrm{d}s\leq 0. $$
(18)
Taking \(l\rightarrow \infty \), we have
$$ -r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}+\lambda ^{p_{2}-1}x^{p_{1}-1} ( t ) \int _{t }^{\infty }q ( s ) \biggl( \frac{\tau ( s ) }{s} \biggr) ^{p_{2}-1}\,\mathrm{d}s\leq 0, $$
that is,
$$ x^{\prime \prime \prime } ( t ) \geq \frac{\lambda ^{ ( p_{2}-1 ) / ( p_{1}-1 ) }}{r ^{1/ ( p_{1}-1 ) } ( t ) }x^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( t ) \biggl( \int _{t }^{\infty }q ( s ) \biggl( \frac{\tau ( s ) }{s} \biggr) ^{p_{2}-1}\,\mathrm{d}s \biggr) ^{1/ ( p_{1}-1 ) }. $$
Integrating the above inequality from t to ∞, we obtain
$$ -x^{\prime \prime } ( t ) \geq \lambda ^{ ( p_{2}-1 ) / ( p_{1}-1 ) }x^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( t ) \int _{t }^{\infty } \biggl( \frac{1}{r ( \eta ) } \int _{\eta }^{\infty }q ( s ) \biggl( \frac{\tau _{i} ( s ) }{s} \biggr) ^{p_{2}-1} \mathrm{d}s \biggr) ^{1/ ( p_{1}-1 ) }\mathrm{\,}\mathrm{d}\eta , $$
hence
$$ x^{\prime \prime } ( t ) \leq -\widetilde{R} ( t ) x^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( t ) . $$
(19)
Letting
$$ \phi ( t ) = \frac{x^{\prime } ( t ) }{x ( t ) }, $$
then \(\phi ( t ) >0\) for \(t \geq t _{1}\) and
$$ \phi ^{\prime } ( t ) = \frac{x^{\prime \prime } ( t ) }{x ( t ) }- \biggl( \frac{x^{\prime } ( t ) }{x ( t ) } \biggr) ^{2}. $$
By using (19) and the definition of \(\phi ( t ) \), we see that
$$ \phi ^{\prime } ( t ) \leq -\widetilde{R} ( t ) \frac{x^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( t ) }{x ( t ) }-\phi ^{2} ( t ) . $$
(20)
Since \(x^{\prime } ( t ) >0\), there exists a constant \(M>0\) such that \(x ( t ) \geq M\) for all \(t \geq t _{2}\). Then (20) becomes
$$ \phi ^{\prime } ( t ) +\phi ^{2} ( t ) +M^{p_{2}-p_{1}} \widetilde{R} ( t ) \leq 0. $$
(21)
From [39], we obtain that (15) is nonoscillatory if and only if there exists \(t _{3}>\max \{ t _{1},t _{2} \} \) such that (21) holds, which is a contradiction. Theorem is proved. □
Theorem 3.4
Let \(p_{2}\geq p_{1}\), \(\tau _{i}^{\prime } ( t ) >1\) and (6) hold for some \(\mu \in ( 0,1 ) \). If
$$ \biggl( \frac{1}{\tau ^{\prime } ( t ) }u^{\prime } ( t ) \biggr) ^{\prime }+M^{ ( p_{2}-1 ) / ( p_{1}-2 ) }R ( t ) u ( t ) =0 $$
(22)
is oscillatory, then (1) is oscillatory.
Proof
From the proof of Theorem 3.3, we find that (17) holds. So, it follows from \(\tau _{i}^{\prime } ( t ) \geq 0\) and \(x^{\prime } ( t ) \geq 0\) that
$$ r ( l ) \bigl( x^{\prime \prime \prime } ( l ) \bigr) ^{p_{1}-1}-r ( t ) \bigl( x^{\prime \prime \prime } ( t ) \bigr) ^{p_{1}-1}+x^{p_{2}-1} \bigl( \tau ( t ) \bigr) \int _{t }^{l}q ( s ) \,\mathrm{d}s\leq 0. $$
(23)
Thus, (16) becomes
$$ x^{\prime \prime } ( t ) \leq -R ( t ) x^{ ( p_{2}-1 ) / ( p_{1}-1 ) } \bigl( \tau _{i} ( t ) \bigr) . $$
(24)
Letting
$$ \delta ( t ) = \frac{x^{\prime } ( t ) }{x ( \tau ( t ) ) }, $$
(25)
then \(\delta ( t ) >0\) for \(t \geq t _{1}\), and
$$\begin{aligned} \delta ^{\prime } ( t ) =& \frac{x^{\prime \prime } ( t ) }{x ( \tau ( t ) ) }- \frac{x^{\prime } ( t ) }{x^{2} ( \tau ( t ) ) }x^{\prime } \bigl( \tau ( t ) \bigr) \tau ^{\prime } ( t ) \\ \leq & \frac{x^{\prime \prime } ( t ) }{x ( \tau ( t ) ) }- \tau ^{\prime } ( t ) \biggl( \frac{x^{\prime } ( t ) }{x ( \tau ( t ) ) } \biggr) ^{2}. \end{aligned}$$
From (24) and (25), we find that
$$ \delta ^{\prime } ( t ) +M^{ ( p_{2}-1 ) / ( p_{1}-2 ) }R ( t ) +\tau ^{\prime } ( t ) \delta ^{2} ( t ) \leq 0. $$
(26)
From [39], we find that (22) is nonoscillatory if and only if there exists \(t _{3}>\max \{ t _{1},t _{2} \} \) such that (26) holds, which is a contradiction. Theorem is proved. □
Corollary 3.5
Let \(p_{2}=p_{1}\) and (6) hold. If
$$ \underset{t \rightarrow \infty }{\lim } \frac{1}{H ( t ,t _{0} ) } \int _{t _{0}}^{t } \biggl( H ( t ,s ) \widetilde{R} ( s ) -\frac{1}{4}h^{2} ( t ,s ) \biggr) \,\mathrm{d}s= \infty $$
or
$$ \underset{t \rightarrow \infty }{\lim \inf }t \int _{t }^{\infty } \widetilde{R} ( s ) \,\mathrm{d}s> \frac{1}{4}, $$
(27)
then (1) is oscillatory.
Corollary 3.6
Let \(p_{2}=p_{1}\) and (6) hold. If \(\varepsilon \in ( 0,1/4 ] \) such that
$$ t ^{2}\widetilde{R} ( s ) \geq \varepsilon $$
and
$$ \underset{t \rightarrow \infty }{\lim \sup } \biggl( t ^{\varepsilon -1} \int _{t _{0}}^{t }s^{2-\varepsilon }\widetilde{R} ( s ) \,\mathrm{d}s+t ^{1-\widetilde{\varepsilon }} \int _{t }^{\infty }s^{ \widetilde{\varepsilon }}\widetilde{R} ( s ) \,\mathrm{d}s \biggr) >1, $$
where \(\widetilde{\varepsilon }=\frac{1}{2} ( 1- \sqrt{1-4\varepsilon } ) \), then (1) is oscillatory.
Corollary 3.7
Let \(p_{1}=p_{2}\) and (4) hold. If
$$ \underset{t \rightarrow \infty }{\lim \inf }\ \int _{\tau (t )}^{t } \frac{\lambda ^{p_{2}-1}}{6^{p_{2}-1}} \frac{\vartheta _{t _{0}} ( s ) q ( s ) \tau _{i}^{3 ( p_{2}-1 ) } ( s ) }{\vartheta _{t _{0}}^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( \tau ( s ) ) r ^{ ( p_{2}-1 ) / ( p_{1}-1 ) } ( \tau ( s ) ) } \,\mathrm{d}s>\frac{1}{\mathrm{e}}, $$
then (3) is oscillatory.
Corollary 3.8
Let \(p_{1}=p_{2}\), (4), and
$$ \int _{t _{0}}^{\infty } \biggl( M^{p_{2}-p_{1}}\beta ( s ) \vartheta _{t _{0}} ( s ) q ( s ) \frac{\tau ^{3\kappa } ( s ) }{s^{3\kappa }}- \frac{2^{p_{1}-1}}{p_{1}{}^{p_{1}}} \frac{r ( s ) ( \beta ^{\prime } ( s ) ) ^{p_{1}}}{\mu ^{p_{1}-1}s^{2 ( p_{1}-1 ) }\beta ^{p_{1}-1} ( s ) } \biggr) \,\mathrm{d}s=\infty , $$
(28)
hold for some \(\mu \in ( 0,1 ) \). If
$$ \underset{t \rightarrow \infty }{\lim } \frac{1}{H ( t ,t _{0} ) } \int _{t _{0}}^{t } \biggl( H ( t ,s ) \widehat{R} ( s ) -\frac{1}{4}h^{2} ( t ,s ) \biggr) \,\mathrm{d}s=\infty $$
or
$$ \underset{t \rightarrow \infty }{\lim \inf } \int _{t }^{\infty }\widehat{R} ( s ) \,\mathrm{d}s>\frac{1}{4}, $$
then (3) is oscillatory.
Corollary 3.9
Let \(p_{1}=p_{2}\) and (28) hold. If \(\varepsilon \in ( 0,1/4 ] \) such that
$$ t ^{2}\widehat{R} ( s ) \geq \varepsilon $$
and
$$ \underset{t \rightarrow \infty }{\lim \sup } \biggl( t ^{\varepsilon -1} \int _{t _{0}}^{t }s^{2-\varepsilon }\widehat{R} ( s ) \,\mathrm{d}s+t ^{1-\widetilde{\varepsilon }} \int _{t }^{\infty }s^{\widetilde{\varepsilon }}\widehat{R} ( s ) \,\mathrm{d}s \biggr) >1, $$
where ε̃ is defined as in Corollary 3.6, then (3) is oscillatory.