Theorem 1 can be applied to study oscillatory properties of differential equations. We are going to outline one such application based on comparison principle.

#### Corollary 1

*Noncanonical differential equation* (*E*) *can be written in canonical form as*

$$\begin{array}{}{\displaystyle \frac{1}{{\pi}_{321}(t)}{\left(\frac{{r}_{3}{\pi}_{321}^{2}}{\mathit{\Omega}}{\left(\frac{{r}_{2}{\mathit{\Omega}}^{2}}{{\pi}_{321}{\pi}_{123}}{\left(\frac{{r}_{1}{\pi}_{123}^{2}}{\mathit{\Omega}}{\left(\frac{y}{{\pi}_{123}}\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)+p(t)y(\tau (t))=0.}\end{array}$$

Setting *z*(*t*) = *y*(*t*)/*π*_{123}(*t*) we get the following comparison result that reduces oscillation of strongly noncanonical equation to that of canonical equation.

#### Corollary 2

*Noncanonical equation* (*E*) *is oscillatory if and only if the canonical equation*

$$\begin{array}{}{\displaystyle {\left(\frac{{r}_{3}{\pi}_{321}^{2}}{\mathit{\Omega}}{\left(\frac{{r}_{2}{\mathit{\Omega}}^{2}}{{\pi}_{321}{\pi}_{123}}{\left(\frac{{r}_{1}{\pi}_{123}^{2}}{\mathit{\Omega}}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)+{\pi}_{321}(t){\pi}_{123}(\tau (t))p(t)z(\tau (t))=0.}\end{array}$$(Ec)

*is oscillatory*.

#### Example 3

*Let us consider the fourth order differential equation*

$$\begin{array}{}{\displaystyle {\left({t}^{\gamma}{\left({t}^{\beta}{\left({t}^{\alpha}{y}^{\prime}(t)\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}+p(t)y(\tau (t))=0}\end{array}$$(3.1)

*with* *α* > 1, *β* > 1, *γ* > 1. *By Corollary 2*, *this equation is oscillatory if and only if the canonical equation*

$$\begin{array}{}{\displaystyle {\left({t}^{2-\alpha}{\left({t}^{2-\beta}{\left({t}^{2-\gamma}{z}^{\prime}(t)\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}+(t\tau (t){)}^{3-\alpha -\beta -\gamma}p(t)z(\tau (t))=0}\end{array}$$(3.2)

*is oscillatory. It is more convenient to study oscillation of* (3.12) *instead of* (3.1).

Now, we are ready to study the properties of (*E*) with the help of (*E*_{c}). Without loss of generality, we can consider only with the positive solutions of (*E*_{c}). The following result is a modification of Kiguradze’s lemma [3].

Let us denote

$$\begin{array}{}{\displaystyle {q}_{1}(t)=\frac{{r}_{1}(t){\pi}_{123}^{2}(t)}{\mathit{\Omega}(t)},\phantom{\rule{1em}{0ex}}{q}_{2}(t)=\frac{{r}_{2}(t){\mathit{\Omega}}^{2}(t)}{{\pi}_{321}(t){\pi}_{123}(t)},\phantom{\rule{1em}{0ex}}{q}_{3}(t)=\frac{{r}_{3}(t){\pi}_{321}^{2}(t)}{\mathit{\Omega}(t)}}\end{array}$$

and

$$\begin{array}{}{\displaystyle P(t)={\pi}_{321}(t){\pi}_{123}(\tau (t))p(t).}\end{array}$$

Then (*E*_{c}) can be rewritten as

$$\begin{array}{}{\displaystyle {\left({q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)+P(t)z(\tau (t))=0.}\end{array}$$

#### Lemma 3

*Assume that z*(*t*) *is an eventually positive solution of* (*E*_{c}), *then either*

$$\begin{array}{}{\displaystyle z(t)\in {\mathcal{N}}_{1}\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}{z}^{\prime}(t)>0,\phantom{\rule{1em}{0ex}}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}(t)<0,\phantom{\rule{1em}{0ex}}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)>0}\end{array}$$

*or*

$$\begin{array}{}{\displaystyle z(t)\in {\mathcal{N}}_{3}\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}{z}^{\prime}(t)>0,\phantom{\rule{1em}{0ex}}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}(t)>0,\phantom{\rule{1em}{0ex}}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)>0}\end{array}$$

Consequently, the set 𝒩 of all positive solutions of (*E*) has the decomposition

$$\begin{array}{}{\displaystyle \mathcal{N}={\mathcal{N}}_{1}\cup {\mathcal{N}}_{3}.}\end{array}$$

To obtain oscillation of studied equation (*E*), we need to eliminate both cases of possible non-oscillatory solutions.

Let us denote

$$\begin{array}{}{\displaystyle {Q}_{1}(t)=\left(\frac{1}{{q}_{2}(t)}\underset{t}{\overset{\mathrm{\infty}}{\int}}\frac{1}{{q}_{3}(u)}\underset{u}{\overset{\mathrm{\infty}}{\int}}P(s)\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u\right)\underset{{t}_{1}}{\overset{\tau (t)}{\int}}\frac{1}{{q}_{1}(u)}\mathrm{d}u}\end{array}$$

and

$$\begin{array}{}{\displaystyle {Q}_{2}(t)=P(t)\underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{1}({s}_{1})}\underset{{t}_{1}}{\overset{{s}_{1}}{\int}}\frac{1}{{q}_{2}(u)}\underset{{t}_{1}}{\overset{s}{\int}}\frac{1}{{q}_{3}(s)}\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}{s}_{1}.}\end{array}$$

#### Theorem 2

*Let* (1.2) *hold. Assume that both first*-*order delay differential equations*

$$\begin{array}{}{\displaystyle {x}^{\prime}(t)+{Q}_{1}(t)x(\tau (t))=0}\end{array}$$(3.3)

*and*

$$\begin{array}{}{\displaystyle {x}^{\prime}(t)+{Q}_{2}(t)x(\tau (t))=0.}\end{array}$$(3.4)

*are oscillatory. Then* (*E*) *is oscillatory*.

#### Proof

Assume that *y*(*t*) is an eventually positive solution of (*E*), say for *t* ≥ *t*_{1}. Then by Corollary 2, *z*(*t*) = *y*(*t*)/*π*_{123}(*t*) is a solution of (*E*_{c}).

It follows from Lemma 3 that either *z*(*t*) ∈ 𝒩_{1} or *z*(*t*) ∈ 𝒩_{3}. At first, we admit that *z*(*t*) ∈ 𝒩_{1}. Noting that *q*_{1}(*t*)*z*′(*t*) is decreasing, we see that

$$\begin{array}{}{\displaystyle z(t)\ge \underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{1}(u)}{q}_{1}(u){z}^{\prime}(u)\phantom{\rule{thinmathspace}{0ex}}du\ge {q}_{1}(t){z}^{\prime}(t)\underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{1}(u)}\phantom{\rule{thinmathspace}{0ex}}du.}\end{array}$$(3.5)

Integrating (*E*_{c}) from *t* to ∞, we have

$$\begin{array}{}{\displaystyle {q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)\ge \underset{t}{\overset{\mathrm{\infty}}{\int}}P(s)z(\tau (s))\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s.}\end{array}$$(3.6)

Taking into account that *z*(*τ*(*t*)) is increasing, the last inequality yields

$$\begin{array}{}{\displaystyle {\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)\ge z(\tau (t))\phantom{\rule{thinmathspace}{0ex}}\frac{1}{{q}_{3}(t)}\phantom{\rule{thinmathspace}{0ex}}\underset{t}{\overset{\mathrm{\infty}}{\int}}P(s)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s.}\end{array}$$(3.7)

Integrating once more, we are led to

$$\begin{array}{}{\displaystyle -{\left({q}_{1}{z}^{\prime}\right)}^{\prime}(t)\ge z(\tau (t))\frac{1}{{q}_{2}(t)}\underset{t}{\overset{\mathrm{\infty}}{\int}}\frac{1}{{q}_{3}(u)}\underset{u}{\overset{\mathrm{\infty}}{\int}}P(s)\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u.}\end{array}$$(3.8)

Combining the last inequality with (3.5), one gets

$$\begin{array}{}{\displaystyle -{\left({q}_{1}{z}^{\prime}\right)}^{\prime}(t)\ge \left({z}^{\prime}(\tau (t))\frac{{q}_{1}(\tau (t))}{{q}_{2}(t)}\underset{t}{\overset{\mathrm{\infty}}{\int}}\frac{1}{{q}_{3}(u)}\underset{u}{\overset{\mathrm{\infty}}{\int}}P(s)\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u\right)\underset{{t}_{1}}{\overset{\tau (t)}{\int}}\frac{1}{{q}_{1}(u)}\mathrm{d}u}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}={q}_{1}(\tau (t)){z}^{\prime}(\tau (t){Q}_{1}(t).}\end{array}$$(3.9)

Thus, the function *x*(*t*) = *q*_{1}(*t*)*z*′(*t*) is a positive solution of the differential inequality

$$\begin{array}{}{\displaystyle {x}^{\prime}(t)+{Q}_{1}(t)x(\tau (t))\le 0.}\end{array}$$

Hence, by Philos theorem [4], we conclude that the corresponding differential equation (3.3) also has a positive solution, which contradicts the assumptions of the theorem.

Now, we shall assume that *z*(*t*) ∈ 𝒩_{3}. Since $\begin{array}{}{\displaystyle {q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}}\end{array}$ is decreasing, we are led to

$$\begin{array}{}{\displaystyle {q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}(t)\ge \underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{3}(s)}{q}_{3}(s){\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(s)\mathrm{d}s}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\ge {q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)\underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{3}(s)}\mathrm{d}s.}\end{array}$$

Integrating the above inequality, one can verify that

$$\begin{array}{}{\displaystyle {z}^{\prime}(t)\ge {q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}(t)\frac{1}{{q}_{1}(t)}\underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{2}(u)}\underset{{t}_{1}}{\overset{s}{\int}}\frac{1}{{q}_{3}(s)}\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u}\end{array}$$

Integrating once more, we see that *x*(*t*) = $\begin{array}{}{\displaystyle {q}_{3}{\left({q}_{2}{\left({q}_{1}{z}^{\prime}\right)}^{\prime}\right)}^{\prime}}\end{array}$(*t*) satisfies

$$\begin{array}{}{\displaystyle z(t)\ge x(t)\underset{{t}_{1}}{\overset{t}{\int}}\frac{1}{{q}_{1}({s}_{1})}\underset{{t}_{1}}{\overset{{s}_{1}}{\int}}\frac{1}{{q}_{2}(u)}\underset{{t}_{1}}{\overset{s}{\int}}\frac{1}{{q}_{3}(s)}\mathrm{d}s\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}u\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}{s}_{1}.}\end{array}$$

Setting the last estimate into (*E*_{c}), we see that *x*(*t*) is a positive solution of the differential inequality

$$\begin{array}{}{\displaystyle {x}^{\prime}(t)+{Q}_{2}(t)x(\tau (t))\le 0,}\end{array}$$

which in view of Philos theorem in [4] guarantees that the corresponding differential equation (3.4) has also a positive solution. This is a contradiction and the proof is complete now.□

Applying suitable criteria for oscillation of (3.3) and (3.4), we immediately obtain the criteria for oscillation of (E). We use the one which is due to Ladde et al. [5].

#### Corollary 3

*Let* (1.2) *hold. Assume that for* *i* = 1, 2

$$\begin{array}{}{\displaystyle \underset{t\to \mathrm{\infty}}{lim\u2006inf}\underset{\tau (t)}{\overset{t}{\int}}{Q}_{i}(s)\mathrm{d}s>\frac{1}{\mathrm{e}}}\end{array}$$(3.10)

*hold. Then* (*E*) *is oscillatory*.

We support our results by another example.

#### Example 4

*Let us consider the general Euler delay differential equation*

$$\begin{array}{}{\displaystyle {\left({t}^{\gamma}{\left({t}^{\beta}{\left({t}^{\alpha}{y}^{\prime}(t)\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}+a\phantom{\rule{thinmathspace}{0ex}}{t}^{\alpha +\beta +\gamma -4}y(\lambda t)=0}\end{array}$$(3.11)

*with* *α* > 1, *β* > 1, *γ* > 1, *a* > 0, *and* 0 < *λ* < 1. *By Corollary 2 and Example 3*, *this equation is oscillatory if and only if the canonical equation*

$$\begin{array}{}{\displaystyle {\left({t}^{2-\alpha}{\left({t}^{2-\beta}{\left({t}^{2-\gamma}{z}^{\prime}(t)\right)}^{\prime}\right)}^{\prime}\right)}^{\prime}+a{\lambda}^{\alpha +\beta +\gamma -3}{t}^{\alpha +\beta +\gamma -4}z(\lambda t)=0}\end{array}$$(3.12)

*is oscillatory. The straightforward computation yields that*

$$\begin{array}{}{\displaystyle {Q}_{1}(t)\sim \frac{a{\lambda}^{2-\alpha -\beta}}{(\alpha +\beta +\gamma -3)(\beta +\gamma -2)(\gamma -1)}\phantom{\rule{thinmathspace}{0ex}}{t}^{-1}}\end{array}$$

*and*

$$\begin{array}{}{\displaystyle {Q}_{2}(t)\sim \frac{a{\lambda}^{3-\alpha -\beta -\gamma}}{(\alpha +\beta +\gamma -3)(\alpha +\beta -2)(\alpha -1)}\phantom{\rule{thinmathspace}{0ex}}{t}^{-1}.}\end{array}$$

*By Corollary 4 considered equation is oscillatory*, *provided that both conditions*

$$\begin{array}{}{\displaystyle \frac{a{\lambda}^{2-\alpha -\beta}}{(\alpha +\beta +\gamma -3)(\beta +\gamma -2)(\gamma -1)}\phantom{\rule{thinmathspace}{0ex}}\mathrm{ln}\left(\frac{1}{\lambda}\right)>\frac{1}{\mathrm{e}}}\end{array}$$

*and*

$$\begin{array}{}{\displaystyle \frac{a{\lambda}^{3-\alpha -\beta -\gamma}}{(\alpha +\beta +\gamma -3)(\alpha +\beta -2)(\alpha -1)}\phantom{\rule{thinmathspace}{0ex}}\mathrm{ln}\left(\frac{1}{\lambda}\right)>\frac{1}{\mathrm{e}}}\end{array}$$

*are satisfied*.

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