Now we are ready to prove the main results of the paper. The function *Q* which appears in these criteria is a function defined by (11).

We will distinguish two cases: \sigma (t)\le \tau (t) and \tau (t)\le \sigma (t). Let us start with the first case.

**Theorem 1** *Suppose that* (7) *and*

{\int}^{\mathrm{\infty}}\frac{1}{{r}^{1/\alpha}(t)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t=\mathrm{\infty}

(14)

*hold*. *Further suppose that* \sigma (t)\le t, \sigma (t)\le \tau (t) *and there exist positive mutually conjugate numbers* *l*, {l}^{\ast} *and positive functions* \rho \in {C}^{1}([{t}_{0},\mathrm{\infty}),{\mathbb{R}}^{+}), \phi (t) *such that*

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}\rho (s)Q(s)-\frac{1}{{(\alpha +1)}^{\alpha +1}}\frac{\rho (s)r(\sigma (s))}{{({\sigma}^{\prime}(s))}^{\alpha}}\hfill \\ \phantom{\rule{1em}{0ex}}\times [{l}^{\alpha -1}{\left(\frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}\right)}^{\alpha +1}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\hfill \\ \phantom{\rule{1em}{0ex}}\times {(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\right)}^{\prime}\frac{{\tau}^{\prime}(s)}{{p}^{\alpha}(\sigma (s))\phi (s)})}_{+}^{\alpha +1}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}.\hfill \end{array}

(15)

*Then* (1) *is oscillatory*.

*Proof* Suppose, by contradiction, that all of the assumptions of the theorem hold and there exists a solution x(t) of (1) and a number {t}_{1}>{t}_{0} which satisfies

min\{x(t),x(\tau (t)),x(\sigma (t)),x\left(\tau (\sigma (t))\right)\}>0

for every t>{t}_{1}.

Condition (14) ensures that the corresponding function *z* is eventually increasing. In fact, from (1) we have

{(r(t)\mathrm{\Phi}({z}^{\prime}(t)))}^{\prime}=-q(t)\mathrm{\Phi}\left(x(\sigma (t))\right)<0

for t\in ({t}_{1},\mathrm{\infty}). Hence r(t)\mathrm{\Phi}({z}^{\prime}(t)) is decreasing and either

\mathrm{\Phi}({z}^{\prime}(t))>0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\mathrm{\Phi}({z}^{\prime}(t))<0

for large *t*.

Suppose that there exists T>{t}_{1} such that \mathrm{\Phi}({z}^{\prime}(t))<0 for t\ge T. There exists a positive constant *M* such that

r(t)\mathrm{\Phi}({z}^{\prime}(t))<-M<0

and

{z}^{\prime}(t)<-{\mathrm{\Phi}}^{-1}(M){r}^{-1/\alpha}(t)

for t\ge T. Integrating this inequality over the interval (T,t) we get

z(t)\le z(T)-{\mathrm{\Phi}}^{-1}(M){\int}_{T}^{t}{r}^{-1/\alpha}(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.

Letting t\to \mathrm{\infty} we have a negative upper bound for the function *z* and large *t*. However, the positivity of both x(t) and x(\tau (t)) implies positivity of *z*. This contradiction proves that \mathrm{\Phi}({z}^{\prime}(t))>0 and {z}^{\prime}(t)>0 eventually.

Consequently, we will work on the interval ({t}_{2},\mathrm{\infty}) where {t}_{2} is such that

min\{x(t),x(\tau (t)),x(\sigma (t)),x\left(\sigma (\tau (t))\right),{z}^{\prime}(t),{z}^{\prime}(\tau (t)),{z}^{\prime}(\sigma (t)),{z}^{\prime}\left(\tau (\sigma (t))\right)\}>0

for every t>{t}_{2}.

Define

\omega (t)=\rho (t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}}{{z}^{\alpha}(\sigma (t))}.

(16)

Clearly \omega (t)>0 and

{\omega}^{\prime}(t)={\rho}^{\prime}(t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}}{{z}^{\alpha}(\sigma (t))}+\rho (t)\frac{{(r(t){({z}^{\prime}(t))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\sigma (t))}-\alpha \rho (t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}{z}^{\prime}(\sigma (t)){\sigma}^{\prime}(t)}{{z}^{\alpha +1}(\sigma (t))}.

From \sigma (t)\le t and from the monotonicity of r(t)\mathrm{\Phi}({z}^{\prime}(t)) we have

{z}^{\prime}(\sigma (t))\ge {\left(\frac{r(t)}{r(\sigma (t))}\right)}^{1/\alpha}{z}^{\prime}(t)

and combining these computations we get

{\omega}^{\prime}(t)\le \frac{{\rho}^{\prime}(t)}{\rho (t)}\omega (t)+\rho (t)\frac{{(r(t){({z}^{\prime}(t))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\sigma (t))}-\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{\omega}^{\frac{\alpha +1}{\alpha}}(t).

(17)

Further we define

v(t)=\rho (t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}}{{z}^{\alpha}(\sigma (t))},

(18)

we use the obvious fact v(t)>0 and differentiate

\begin{array}{rcl}{v}^{\prime}(t)& =& {\rho}^{\prime}(t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}}{{z}^{\alpha}(\sigma (t))}+\rho (t)\frac{{(r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\sigma (t))}\\ -\alpha \rho (t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}{z}^{\prime}(\sigma (t)){\sigma}^{\prime}(t)}{{z}^{\alpha +1}(\sigma (t))}.\end{array}

Using the monotonicity of r(t)\mathrm{\Phi}({z}^{\prime}(t)) and \sigma (t)\le \tau (t) we have

{z}^{\prime}(\sigma (t))\ge {\left(\frac{r(\tau (t))}{r(\sigma (t))}\right)}^{1/\alpha}{z}^{\prime}(\tau (t))

and hence

{v}^{\prime}(t)\le \frac{{\rho}^{\prime}(t)}{\rho (t)}v(t)+\rho (t)\frac{{(r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\sigma (t))}-\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{v}^{\frac{\alpha +1}{\alpha}}(t).

(19)

Multiplying (17) by {l}^{\alpha -1}, (19) by {({l}^{\ast})}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}, adding the resulting inequalities and using (10), we get

\begin{array}{c}{l}^{\alpha -1}{\omega}^{\prime}(t)+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}{v}^{\prime}(t)\hfill \\ \phantom{\rule{1em}{0ex}}\le -\rho (t)Q(t)+{l}^{\alpha -1}[\frac{{\rho}^{\prime}(t)}{\rho (t)}\omega (t)-\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{\omega}^{\frac{\alpha +1}{\alpha}}(t)]\hfill \\ \phantom{\rule{2em}{0ex}}+{\left({l}^{\ast}\right)}^{\alpha -1}[\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{{\rho}^{\prime}(t)}{\rho (t)}v(t)-\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{v}^{\frac{\alpha +1}{\alpha}}(t)].\hfill \end{array}

Using the product rule for derivatives we obtain

\begin{array}{c}{l}^{\alpha -1}{\omega}^{\prime}(t)+{\left({l}^{\ast}\right)}^{\alpha -1}{(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}v(t))}^{\prime}\hfill \\ \phantom{\rule{1em}{0ex}}\le -\rho (t)Q(t)+{l}^{\alpha -1}[\frac{{\rho}^{\prime}(t)}{\rho (t)}\omega (t)-\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{\omega}^{\frac{\alpha +1}{\alpha}}(t)]\hfill \\ \phantom{\rule{2em}{0ex}}+{\left({l}^{\ast}\right)}^{\alpha -1}[(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{{\rho}^{\prime}(t)}{\rho (t)}+{\left(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\right)}^{\prime})v(t)\hfill \\ \phantom{\rule{2em}{0ex}}-\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{\alpha {\sigma}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\sigma (t))}{v}^{\frac{\alpha +1}{\alpha}}(t)]\hfill \end{array}

and Lemma 1 implies

\begin{array}{c}{l}^{\alpha -1}{\omega}^{\prime}(t)+{\left({l}^{\ast}\right)}^{\alpha -1}{(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}v(t))}^{\prime}\hfill \\ \phantom{\rule{1em}{0ex}}\le -\rho (t)Q(t)+{l}^{\alpha -1}\frac{{\alpha}^{\alpha}}{{(\alpha +1)}^{\alpha +1}}{\left(\frac{{\rho}_{+}^{\prime}(t)}{\rho (t)}\right)}^{\alpha +1}\frac{\rho (t)r(\sigma (t))}{{\alpha}^{\alpha}{({\sigma}^{\prime}(t))}^{\alpha}}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{\alpha}^{\alpha}}{{(\alpha +1)}^{\alpha +1}}{(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{{\rho}^{\prime}(t)}{\rho (t)}+{\left(\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\right)}^{\prime})}_{+}^{\alpha +1}\hfill \\ \phantom{\rule{2em}{0ex}}\times \frac{{({\tau}^{\prime}(t))}^{\alpha}}{{p}^{{\alpha}^{2}}(\sigma (t)){\phi}^{\alpha}(t)}\frac{\rho (t)r(\sigma (t))}{{\alpha}^{\alpha}{({\sigma}^{\prime}(t))}^{\alpha}}.\hfill \end{array}

Integrating from {t}_{2} to *t*

\begin{array}{c}{l}^{\alpha -1}\omega (t)-{l}^{\alpha -1}\omega ({t}_{2})+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}v(t)-{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma ({t}_{2}))\phi ({t}_{2})}{{\tau}^{\prime}({t}_{2})}v({t}_{2})\hfill \\ \phantom{\rule{1em}{0ex}}\le -{\int}_{{t}_{2}}^{t}\rho (s)Q(s)-\frac{1}{{(\alpha +1)}^{\alpha +1}}\frac{\rho (s)r(\sigma (s))}{{({\sigma}^{\prime}(s))}^{\alpha}}\hfill \\ \phantom{\rule{2em}{0ex}}\times [{l}^{\alpha -1}{\left(\frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}\right)}^{\alpha +1}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\hfill \\ \phantom{\rule{2em}{0ex}}\times {(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\right)}^{\prime}\frac{{\tau}^{\prime}(s)}{{p}^{\alpha}(\sigma (s))\phi (s)})}_{+}^{\alpha +1}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\hfill \end{array}

Multiplying by −1 and taking into account the fact that both \omega (t) and \frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}v(t) are nonnegative we get a finite upper bound for the integral from (15), which contradicts (15). □

**Remark 1** Under the conditions l={l}^{\ast}=2, \phi (t)=1 we can obtain [[3], Theorem 3.1] as a corollary of Theorem 1, since the inequality

{(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\right)}^{\prime}\frac{{\tau}^{\prime}(s)}{{p}^{\alpha}(\sigma (s))\phi (s)})}_{+}\le \frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}+{\left({\left(\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\right)}^{\prime}\frac{{\tau}^{\prime}(s)}{{p}^{\alpha}(\sigma (s))\phi (s)}\right)}_{+}

holds.

The following corollary is in fact a variant of Theorem 1 if p(t) is bounded above by a nonnegative number and {\tau}^{\prime}(t) is bounded below by a positive number. Since (15) is not simply monotone with respect to p(t) and \tau (t), we have to include the corresponding estimates in the opening part of the proof.

**Corollary 1** *Suppose that* (7), (14), \sigma (t)\le t *and* \sigma (t)\le \tau (t) *are satisfied and there exist constants* {p}_{0}\ge 0 *and* {\tau}_{0}>0 *such that* p(t)\le {p}_{0}<\mathrm{\infty} *and* {\tau}^{\prime}(t)\ge {\tau}_{0}. *If there exist positive mutually conjugate numbers* *l*, {l}^{\ast}, *and positive functions* \rho (t), \phi (t) *such that*

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}\rho (s)Q(s)-\frac{1}{{(\alpha +1)}^{\alpha +1}}\frac{\rho (s)r(\sigma (s))}{{({\sigma}^{\prime}(s))}^{\alpha}}\hfill \\ \phantom{\rule{1em}{0ex}}\times [{l}^{\alpha -1}{\left(\frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}\right)}^{\alpha +1}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}_{0}^{\alpha}\phi (s)}{{\tau}_{0}}{(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}_{0}^{\alpha}\phi (s)}{{\tau}_{0}}\right)}^{\prime}\frac{{\tau}_{0}}{{p}_{0}^{\alpha}\phi (s)})}_{+}^{\alpha +1}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty},\hfill \end{array}

(20)

*then* (1) *is oscillatory*.

*Proof* The proof is the same as the proof of Theorem 1, we just use (12) instead of (10) and in the remaining part of the proof we replace p(t) by {p}_{0} and {\tau}^{\prime}(t) by {\tau}_{0}. □

**Example 1** For the Euler type equation (2) with 0<{\lambda}_{2}\le {\lambda}_{1}<1 we have q(t)=\frac{\beta}{{t}^{\alpha +1}}, r(t)=1, \sigma (t)={\lambda}_{2}t, {\sigma}^{\prime}(t)={\lambda}_{2}, \tau (t)={\lambda}_{1}t, {\tau}^{\prime}(t)={\tau}_{0}={\lambda}_{1}, q(\tau (t))=\frac{\beta}{{\lambda}_{1}^{\alpha +1}{t}^{\alpha +1}}. Denote \phi (t)={\lambda}_{1}^{\alpha +1} and \rho (t)={t}^{\alpha}. With this setting we have q(t)=\phi (t)q(\tau (t)) and hence Q(t)=q(t). Further \rho (t)Q(t)=\frac{\beta}{t}, \frac{{\rho}^{\prime}(t)}{\rho (t)}=\frac{\alpha}{t}, \frac{{p}_{0}^{\alpha}\phi (t)}{{\tau}_{0}}={({p}_{0}{\lambda}_{1})}^{\alpha} and (20) becomes

[\beta -{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{1}{{\lambda}_{2}^{\alpha}}[{l}^{\alpha -1}+{\left({l}^{\ast}\right)}^{\alpha -1}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}]]\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}^{t}\frac{1}{s}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}

and (2) is oscillatory if

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{1}{{\lambda}_{2}^{\alpha}}[{l}^{\alpha -1}+{\left({l}^{\ast}\right)}^{\alpha -1}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}].

(21)

Note that if l={l}^{\ast}=2, then this condition becomes

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{{2}^{\alpha -1}}{{\lambda}_{2}^{\alpha}}[1+{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}],

and since for {\lambda}_{1}<1 we have {\lambda}_{1}^{\alpha}<1<\frac{1}{{\lambda}_{1}}, this oscillation constant is smaller than the oscillation constant from (3).

Further, if {p}_{0}=0 and {\lambda}_{1}=1={\lambda}_{2}, then (1) becomes (5). Condition (21) becomes

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}{l}^{\alpha -1}

and, since l>1 is arbitrary, we get

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}

which is well known to be an optimal and non-improvable oscillation constant for (5). In this sense we consider our result as reasonably sharp.

Finally, taking into account that {l}^{\ast}=\frac{l}{l-1}, condition (21) becomes

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{1}{{\lambda}_{2}^{\alpha}}[{l}^{\alpha -1}+{\left(\frac{l}{l-1}\right)}^{\alpha -1}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}].

A simple computation shows that the function

f(l)={l}^{\alpha -1}+{\left(\frac{l}{l-1}\right)}^{\alpha -1}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha},\phantom{\rule{1em}{0ex}}l>1,

(22)

satisfies

{f}^{\prime}(l)=(\alpha -1){l}^{\alpha -2}[1-\frac{1}{{(l-1)}^{\alpha}}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}]

and has a global minimum at l=1+{p}_{0}{\lambda}_{1}. Thus the choice l=1+{p}_{0}{\lambda}_{1} in (21) produces the smallest oscillation constant

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{{(1+{p}_{0}{\lambda}_{1})}^{\alpha}}{{\lambda}_{2}^{\alpha}}.

**Example 2** Baculíková *et al.* [[5], Example 2.1] considered the equation

{\left(t\right|{z}^{\prime}(t)|{z}^{\prime}(t))}^{\prime}+\frac{b}{{t}^{2}}|x(\beta t)|x(\beta t)=0

(23)

with z(t)=x(t)+{p}_{0}x(\omega t), {p}_{0}>0, b>0 and 0<\beta <1. They proved that under the condition \beta <\omega <1 (23) is oscillatory if

bln\frac{\omega}{\beta}>\frac{\omega +{p}_{0}^{2}}{2\beta e\omega}

(24)

(note that this condition is misprinted in [5]). This condition naturally produces poor oscillation constant if *β* is close to *ω*. In our notation we have \alpha =2, q(t)=b/{t}^{2}, r(t)=t, \sigma (t)=\beta t, {\sigma}^{\prime}(t)=\beta, \tau (t)=\omega t, {\tau}^{\prime}(t)={\tau}_{0}=\omega. We choose \phi (t)={\omega}^{2} and \rho (t)=t. Thus (20) takes the form

[b-\frac{l+{l}^{\ast}{p}_{0}^{2}\omega}{{3}^{3}\beta}]\underset{t\to \mathrm{\infty}}{lim}{\int}_{{t}_{0}}^{t}\frac{1}{s}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}.

Taking into account that {l}^{\ast}=\frac{l}{l-1} and that the function l+{l}^{\ast}{p}_{0}^{2}\omega has a local minimum {(1+{p}_{0}\sqrt{\omega})}^{2} at the point l=1+{p}_{0}\sqrt{\omega}, we find that (23) is oscillatory if

b>\frac{{(1+{p}_{0}\sqrt{\omega})}^{2}}{27\beta}.

(25)

This condition completes condition (24). It is possible to find constants *ω* and *β* for which (25) is better than (24), as well as constants where the opposite is true. The fact that both estimates depend heavily on the parameters is illustrated by Figure 1.

The following corollary suggests another modification of the proof of Theorem 1: we replace condition (7) by weaker condition (8) and add conditions which ensure that *x* possesses the same type of monotonicity as *z*.

**Corollary 2** *Suppose that* p(t)\equiv {p}_{0}, (8), (14), \sigma (t)\le t *and* \sigma (t)\le \tau (t) *hold*. *If* (20) *holds for some mutually conjugate numbers* *l*, {l}^{\ast} *and positive functions* \rho (t), \phi (t), *then every solution of* (1) *is either oscillatory*, *or the first derivative of this solution is oscillatory*.

*Proof* Suppose, by contradiction, that the assumptions are satisfied and *x* is an eventually positive solution of (1) such that {x}^{\prime}(t) is not oscillatory.

We proceed as in Theorem 1 with modifications mentioned in the proof of Corollary 1. To ensure that Lemma 3 can be applied even though (7) need not to hold note that from the fact that *z* is eventually increasing, p(t) constant and x(t) not oscillatory we conclude easily that *x* is also eventually increasing. □

In the following example we show an application of Corollary 2 to the equation where \sigma (\tau (t))\ne \tau (\sigma (t)).

**Example 3** Consider the equation

{\left[\mathrm{\Phi}\left({(x(t)+{p}_{0}x(t-{\lambda}_{1}))}^{\prime}\right)\right]}^{\prime}+\frac{\beta}{{t}^{\alpha +1}}\mathrm{\Phi}(x({\lambda}_{2}t))=0

with 0\le {p}_{0}, \alpha \ge 1, \beta >0, {\lambda}_{1}>0, {\lambda}_{2}\in (0,1). We have q(t)=\beta /{t}^{\alpha +1}, \sigma (t)={\lambda}_{2}t, {\sigma}^{\prime}(t)={\lambda}_{2}, \tau (t)=t-{\lambda}_{1}, {\tau}^{\prime}(t)={\tau}_{0}=1, \sigma (\tau (t))={\lambda}_{2}t-{\lambda}_{2}{\lambda}_{1}\ge {\lambda}_{2}t-{\lambda}_{1}=\tau (\sigma (t)), \sigma (t)\le \tau (t) for large *t* and q(\tau (t))\ge q(t). We choose \phi (t)=1 and \rho (t)={t}^{\alpha}. With this setting the condition (20) takes the form

[\beta -{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{1}{{\lambda}_{2}^{\alpha}}({l}^{\alpha -1}+{\left({l}^{\ast}\right)}^{\alpha -1}{p}_{0}^{\alpha})]\underset{t\to \mathrm{\infty}}{lim}ln\frac{t}{{t}_{0}}=\mathrm{\infty}.

Using this computation and using the fact that the function {l}^{\alpha -1}+{({l}^{\ast})}^{\alpha -1}{p}_{0}^{\alpha} takes global minimum on l\in (1,\mathrm{\infty}) for l=1+{p}_{0} and {l}^{\ast}=1+1/{p}_{0} we see that the condition

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{{(1+{p}_{0})}^{\alpha}}{{\lambda}_{2}^{\alpha}}

guarantees that either every solution or derivative of every solution of the equation is oscillatory.

In the following theorem we drop the condition \sigma (t)\le \tau (t) and use the opposite \sigma (t)\ge \tau (t). In this case we modify the denominator in the Riccati type substitutions (16) and (18).

**Theorem 2** *Suppose that* (7), (14), \tau (t)\le t *and* \sigma (t)\ge \tau (t) *hold*. *Further suppose that there exist positive mutually conjugate numbers* *l*, {l}^{\ast} *and positive functions* \rho \in {C}^{1}([{t}_{0},\mathrm{\infty}),{\mathbb{R}}^{+}), \phi (t) *such that*

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}\rho (s)Q(s)-\frac{1}{{(\alpha +1)}^{\alpha +1}}\frac{\rho (s)r(\tau (s))}{{({\tau}^{\prime}(s))}^{\alpha}}\hfill \\ \phantom{\rule{1em}{0ex}}\times [{l}^{\alpha -1}{\left(\frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}\right)}^{\alpha +1}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\hfill \\ \phantom{\rule{1em}{0ex}}\times {(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}^{\alpha}(\sigma (s))\phi (s)}{{\tau}^{\prime}(s)}\right)}^{\prime}\frac{{\tau}^{\prime}(s)}{{p}^{\alpha}(\sigma (s))\phi (s)})}_{+}^{\alpha +1}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}.\hfill \end{array}

*Then* (1) *is oscillatory*.

*Proof* Suppose, by contradiction, that all the conditions are satisfied and an eventually positive solution x(t) of (1) exists. As in the proof of Theorem 1, we can show that r(t)\mathrm{\Phi}({z}^{\prime}(t)) is decreasing eventually and {z}^{\prime}(t) increasing eventually. Let us work on the interval ({t}_{2},\mathrm{\infty}) where {t}_{2} is such that

min\{x(t),x(\tau (t)),x(\sigma (t)),x\left(\tau (\sigma (t))\right),{z}^{\prime}(t),{z}^{\prime}(\tau (t)),{z}^{\prime}(\sigma (t)),{z}^{\prime}\left(\tau (\sigma (t))\right)\}>0

for every t>{t}_{2}.

Define

\omega (t)=\rho (t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}}{{z}^{\alpha}(\tau (t))}.

As in the proof of Theorem 1, we have \omega (t)>0 and

{\omega}^{\prime}(t)={\rho}^{\prime}(t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}}{{z}^{\alpha}(\tau (t))}+\rho (t)\frac{{(r(t){({z}^{\prime}(t))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\tau (t))}-\alpha \rho (t)\frac{r(t){({z}^{\prime}(t))}^{\alpha}{z}^{\prime}(\tau (t)){\tau}^{\prime}(t)}{{z}^{\alpha +1}({\tau}^{\prime}(t))}.

From \tau (t)\le t and from the monotonicity of r(t)\mathrm{\Phi}({z}^{\prime}(t)) we have

{z}^{\prime}(\tau (t))\ge {\left(\frac{r(t)}{r(\tau (t))}\right)}^{1/\alpha}{z}^{\prime}(t)

and combining these computations we get

{\omega}^{\prime}(t)\le \frac{{\rho}^{\prime}(t)}{\rho (t)}\omega (t)+\rho (t)\frac{{(r(t){({z}^{\prime}(t))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\tau (t))}-\frac{\alpha {\tau}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\tau (t))}{\omega}^{\frac{\alpha +1}{\alpha}}(t).

Further we define

v(t)=\rho (t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}}{{z}^{\alpha}(\tau (t))},

differentiate

\begin{array}{rcl}{v}^{\prime}(t)& =& {\rho}^{\prime}(t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}}{{z}^{\alpha}(\tau (t))}+\rho (t)\frac{{(r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\tau (t))}\\ -\alpha \rho (t)\frac{r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha}{z}^{\prime}(\tau (t)){\tau}^{\prime}(t)}{{z}^{\alpha +1}(\tau (t))},\end{array}

and conclude

{v}^{\prime}(t)=\frac{{\rho}^{\prime}(t)}{\rho (t)}v(t)+\rho (t)\frac{{(r(\tau (t)){({z}^{\prime}(\tau (t)))}^{\alpha})}^{\prime}}{{z}^{\alpha}(\tau (t))}-\frac{\alpha {\tau}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\tau (t))}{v}^{\frac{\alpha +1}{\alpha}}(t).

Similarly as in the proof of Theorem 1 and using the fact that monotonicity of z(t) and inequality \sigma (t)\ge \tau (t) imply Q(t)\frac{z(\sigma (t))}{z(\tau (t))}\ge Q(t), we get

\begin{array}{c}{l}^{\alpha -1}{\omega}^{\prime}(t)+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}{v}^{\prime}(t)\hfill \\ \phantom{\rule{1em}{0ex}}\le -\rho (t)Q(t)+{l}^{\alpha -1}[\frac{{\rho}^{\prime}(t)}{\rho (t)}\omega (t)-\frac{\alpha {\tau}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\tau (t))}{\omega}^{\frac{\alpha +1}{\alpha}}(t)]\hfill \\ \phantom{\rule{2em}{0ex}}+{\left({l}^{\ast}\right)}^{\alpha -1}[\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{{\rho}^{\prime}(t)}{\rho (t)}v(t)-\frac{{p}^{\alpha}(\sigma (t))\phi (t)}{{\tau}^{\prime}(t)}\frac{\alpha {\tau}^{\prime}(t)}{{\rho}^{1/\alpha}(t){r}^{1/\alpha}(\tau (t))}{v}^{\frac{\alpha +1}{\alpha}}(t)].\hfill \end{array}

The remaining part of the proof is the same as in Theorem 1. □

**Remark 2** Similarly as in Remark 1, [[3], Theorem 3.3] is a corollary of Theorem 2.

**Corollary 3** *Suppose that* (7), (14), \tau (t)\le t *and* \sigma (t)\ge \tau (t) *hold*. *Furthermore*, *suppose that there exist constants* {p}_{0}\ge 0 *and* {\tau}_{0}>0 *such that* p(t)\le {p}_{0}<\mathrm{\infty} *and* {\tau}^{\prime}(t)\ge {\tau}_{0}. *If there exist positive mutually conjugate numbers* *l*, {l}^{\ast}, *and positive functions* \rho (t), \phi (t) *such that*

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}\rho (s)Q(s)-\frac{1}{{(\alpha +1)}^{\alpha +1}}\frac{\rho (s)r(\tau (s))}{{({\tau}^{\prime}(s))}^{\alpha}}\hfill \\ \phantom{\rule{1em}{0ex}}\times [{l}^{\alpha -1}{\left(\frac{{\rho}_{+}^{\prime}(s)}{\rho (s)}\right)}^{\alpha +1}+{\left({l}^{\ast}\right)}^{\alpha -1}\frac{{p}_{0}^{\alpha}\phi (s)}{{\tau}_{0}}{(\frac{{\rho}^{\prime}(s)}{\rho (s)}+{\left(\frac{{p}_{0}^{\alpha}\phi (s)}{{\tau}_{0}}\right)}^{\prime}\frac{{\tau}_{0}}{{p}_{0}^{\alpha}\phi (s)})}_{+}^{\alpha +1}]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty},\hfill \end{array}

(26)

*then* (1) *is oscillatory*.

*Proof* The proof is he same as the proof of Corollary 1. We just use Theorem 2 instead of Theorem 1. □

**Example 4** Consider (2) with {\lambda}_{2}\ge {\lambda}_{1}. We choose the functions *ρ* and *φ* as in Example 1 and find that (1) is oscillatory if

\beta >{\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{{(1+{p}_{0}{\lambda}_{1})}^{\alpha}}{{\lambda}_{1}^{\alpha}}.

(27)

Let us compare this result with (4). The inequalities \alpha \ge 1, {p}_{0}\ge 0, and {\lambda}_{1}<1 imply

{2}^{\alpha -1}+{2}^{\alpha -1}\frac{{p}_{0}^{\alpha}}{{\lambda}_{1}}\ge {2}^{\alpha -1}+{2}^{\alpha -1}{p}_{0}^{\alpha}{\lambda}_{1}^{\alpha}=f(2)\ge {(1+{p}_{0}{\lambda}_{1})}^{\alpha},

where *f* is defined by (22) and {(1+{p}_{0}{\lambda}_{1})}^{\alpha} is a global minimum of *f* on (1,\mathrm{\infty}). Hence

\frac{{2}^{\alpha -1}{\alpha}^{\alpha +1}}{{(\alpha +1)}^{\alpha +1}{\lambda}_{1}^{\alpha}}(1+\frac{{p}_{0}}{{\lambda}_{1}})\ge {\left(\frac{\alpha}{\alpha +1}\right)}^{\alpha +1}\frac{{(1+{p}_{0}{\lambda}_{1})}^{\alpha}}{{\lambda}_{1}^{\alpha}}

and (27) is sharper than (4).

The following corollary is a variant of Corollary 2 for \sigma (t)\ge \tau (t).

**Corollary 4** *Suppose that* p(t)\equiv {p}_{0}, (8), (14), \tau (t)\le t *and* \sigma (t)\ge \tau (t) *hold*. *If* (26) *holds for some mutually conjugate numbers* *l*, {l}^{\ast} *and positive functions* \rho (t), \phi (t), *then every solution of* (1) *is either oscillatory*, *or the first derivative of this solution is oscillatory*.

*Proof* The proof is the same as the proof of Corollary 2; we only replace Theorem 1 by Theorem 2 and Corollary 1 by Corollary 3. □

**Remark 3** There are two main approaches how to handle Riccati type transformation in the oscillation theory of neutral differential equations. The first applies if 0\le p(t)<1 and the shift in the differential term is handled by utilizing the estimate z(t)(1-p(t))\le x(t); see *e.g.* [7–9]. Thus the results of this type depend on term (1-p(\sigma (t))). Another frequent approach which has been used in [3, 10] and also in this paper is summing up the equation at *t* and \tau (t) and working with the resulting sum. Since it is necessary to take out common factor, the oscillation criteria usually contain term min\{q(t),q(\tau (t))\}. Since both q(t) and q(\tau (t)) may differ significantly, we developed in this paper a method which replaces this term with the term min\{q(t),\phi (t)q(\tau (t))\}, where the function \phi (t) is in some sense arbitrary and may have influence on the final oscillation criterion. We also showed on examples in previous section that this idea produces nonempty extension of known results. We conjecture that a similar idea can be used to obtain new results also in the case of a series of papers by Baculíková and Džurina [5, 11, 12], where a sum of two equations (in the original variable and in the shifted variable) is used to derive a certain first-order delay differential equation and the oscillation criteria are formulated in terms of this first-order equation. However, this idea exceeds the scope of this paper and will be examined in other research.