## 3 Discussion and Applications  
Depending on the new criteria for the nonexistence of Kneser solutions, we introduced new criteria for oscillation of (1). When checking the behavior of positive solutions of DDE (1), we have three Cases \((1)\)-\((3)\). In order to illustrate the importance of the results obtained for Case \((3)\), we recall an existing criterion for a particular case of (1) with \(\alpha=\beta\):  
**Theorem 3** (Theorem 2.1 with \(n=4\), [25]). : _Assume that \(\alpha=\beta\),_  
\[\liminf_{l\to\infty}\int_{\tau(l)}^{l}q(\varrho)\frac{\tau^{3\alpha}(\varrho)} {a(\tau(\varrho))}\mathrm{d}\varrho>\frac{6^{\alpha}}{\mathrm{e}} \tag{28}\]  
_and there exists \(a\in C^{1}(l_{0},\mathbb{R}^{+})\) such that_  
\[\limsup_{l\to\infty}\int_{l_{0}}^{l}\Biggl{(}\delta_{0}^{\kappa}(\varrho)q( \varrho)\biggl{(}\frac{\lambda}{2!}\tau^{2}(\varrho)\biggr{)}^{\alpha}-\frac{ \alpha^{\alpha+1}}{(\alpha+1)^{(\alpha+1)}}\frac{1}{\delta(\varrho)a^{1/ \alpha}(\varrho)}\Biggr{)}\mathrm{d}\varrho=\infty, \tag{29}\]  
_for some \(\lambda\in(0,1)\). Then every solution of (1) is oscillatory or tends to zero._  
From the previous Theorems, we conclude under the assumptions of the Theorem that every positive solution \(x\) of (1) tends to zero, and hence \(x\) satisfies Case \((3)\). Therefore, conditions (28) and (29) ensure (3) without verifying the extra condition (2). In view of Theorems 1 and 3, we obtain the following:  
**Corollary 2**. : _Assume that (28) and (29) hold for some \(\lambda\in(0,1)\). If \((\mathbf{C}_{1})\), \((\mathbf{C}_{2})\) or \((\mathbf{C}_{3})\) holds, then (1) is oscillatory._  
Proof. : Suppose that \(x\) is a nonoscillatory solution of (1). Thus, we have three cases. From Theorem 3, we find (28) and (29) contradicts Case (1) and Case (2) respectively. For Case (3), using Theorem 1, if one of the conditions \((\mathbf{C}_{1})\)-\((\mathbf{C}_{3})\) holds, then we obtain a contradiction. The proof is complete. **Corollary 3**. : _Assume that (28) and (29) hold for some \(\lambda\in(0,1)\). If (25), (26) or (27) holds, then (1) is oscillatory._  
Proof. : Suppose the \(x\) is a nonoscillatory solution of (1). Thus, we have three cases. From Theorem 3, we find (28) and (29) contradict Case (1) and Case (2) respectively. For Case (3), using Corollary 1, if one of the conditions (25)-(27) holds, then we obtain a contradiction. The proof is complete. We state now an Example:  
**Example 1**. : _We have the fourth-order DDE_  
\[\left(\mathrm{e}^{al}x^{\prime\prime\prime}(l)\right)^{\prime}+q_{0}\mathrm{e}^{al }x^{\kappa}(\tau_{0}l)=0, \tag{30}\]  
_where \(l\geq 1\), \(\tau_{0}\in(0,1-1/\mathrm{e})\) and \(q_{0}>0\). Note that \(a(l)=\mathrm{e}^{al}\), \(q(l)=q_{0}\mathrm{e}^{al}\), \(\tau(l)=\tau_{0}l\). It is easy to conclude that \(\delta_{m}(l)=\mathrm{e}^{-l}\) for \(m=0,1,2\). Then, we see that (28) and (29) are satisfied for all \(q_{0}>0\)._  
_For condition \((\mathbf{C}_{1})\), we have_  
\[\eta=\left(\frac{q_{0}}{\alpha}\right)^{1/\alpha}>1. \tag{31}\]  
_By using the fact that \(\mathrm{e}^{y}>\mathrm{e}\) for \(y>0\), we get_  
\[\frac{\delta_{2}(\tau(l))}{\delta_{2}(l)}=\mathrm{e}^{(1-\tau_{0})l}>\mathrm{ e}(1-\tau_{0})l\geq\mathrm{e}(1-\tau_{0}):=h>1,\]  
_for \(l\geq 1\). Hence, Conditions \((\mathbf{C}_{2})\) or \((\mathbf{C}_{3})\) reduce to_  
\[\left(\frac{q_{0}}{\alpha}\right)^{1/\alpha}(\mathrm{e}(1-\tau_{0}))^{(q_{0}/ \alpha)^{1/\alpha}}>1 \tag{32}\]  
_and_  
\[q_{0}(\mathrm{e}(1-\tau_{0}))^{(q_{0}/\alpha)^{1/\alpha}}>\left(\frac{\alpha} {\alpha+1}\right)^{\alpha+1}, \tag{33}\]  
_respectively._  
_Thus, by Corollary, if (31), (32) or (33) holds, then (30) is oscillatory._  
**Remark 1**. : _To the best of our knowledge, the known related sharp criterion for (30) based on (Theorem 2.1, [38]) gives_  
\[q_{0}>\left(\frac{\alpha}{\alpha+1}\right)^{\alpha+1}. \tag{34}\]  
_Note firstly that our criteria (32) and (33) essentially take into account the influence of delay argument \(\tau(l)\), which has been neglected in all previous results of fourth-order equations.