**3. Oscillation Criteria**

In this section, we create a criterion that ensures that solutions to Equation (4) oscillate. The next theorem is a restatement of Theorem 2.1 in [28], when *α* = *β*.

**Theorem 1** ([28])**.** *Assume that*

$$\int\_{\mu\_0}^{\infty} \left( \int\_{u}^{\infty} (\mu - u)^{n-3} \left( \frac{1}{a(\mu)} \int\_{u\_1}^{\mu} q(v) \mathrm{d}v \right) \mathrm{d}v \right) \mathrm{d}u = \infty$$

*and*

$$\limsup\_{u \to \infty} \int\_{u\_0}^{u} \left( \mathcal{E}\_0(\upsilon) q(\upsilon) (1 - p(\underline{\chi}(\upsilon))) \left( \frac{\eta\_0 g^{\eta - 2}(\upsilon)}{(n - 2)!} \right) - \frac{1}{4a(\upsilon) \mathcal{E}\_0(\upsilon)} \right) d\upsilon = \infty,\tag{11}$$

*for some constant η*<sup>0</sup> ∈ (0, 1)*. If the DDE*

$$\frac{\mathbf{d}}{\mathbf{d}u}\boldsymbol{\Psi} + \boldsymbol{q} \cdot \frac{\eta\_1 (1 - [\boldsymbol{p} \circ \boldsymbol{g}]) \cdot \mathbf{g}^{n-1} (\boldsymbol{u})}{(n-1)! \, [\boldsymbol{a} \circ \boldsymbol{g}])} \cdot [\boldsymbol{\Psi} \circ \boldsymbol{g}] = 0 \tag{12}$$

*is oscillatory for some constant η*<sup>1</sup> ∈ (0, 1)*, then every solution of (3), with α* = *β*, *is either oscillatory or converges to zero as u* → ∞*.*

**Lemma 6.** *Assume that* lim*x*→<sup>0</sup> *<sup>x</sup> <sup>F</sup>*(*x*) = *K* < ∞*, and (8) holds. If*

$$\begin{split} \limsup\_{u \to \infty} \left[ \mathcal{E}\_{n-2}(\mathcal{g}(u)) \int\_{u\_0}^{\mathcal{g}(u)} \mathcal{Q}(v) \mathrm{d}v + \int\_{\mathcal{S}(u)}^u \mathcal{E}\_{n-2}(v) \mathcal{Q}(v) \mathrm{d}v \\ + F \Big( \mathcal{E}\_{n-2}^{-1}(\mathcal{g}(u)) \Big) \int\_u^{\infty} \mathcal{E}\_{n-2}(v) \mathcal{Q}(v) F(\mathcal{E}\_{n-2}(\mathcal{g}(v))) \mathrm{d}v \Big) > K, \tag{13} \end{split} \tag{13}$$

*then ω satisfies case* (D2) *in Lemma 1.*
