## 2 Nonexistence of Increasing Positive Solutions  
In this section, we obtain nonexistence criteria for increasing positive solutions of odd-order delay differential Equation (1).  
**Lemma 3**.: _Assume that \(x\in X_{I}^{+}\). Then,_  
\[x\left(\psi\left(\xi,s\right)\right)\geq\eta_{k}\left(\psi\left(\xi,s\right) \right)x^{\left(n-1\right)}\left(\psi\left(\xi,s\right)\right), \tag{4}\]  
_where_  
\[\eta_{0}\left(\xi\right):=\frac{\delta_{0}}{\left(n-1\right)!}\xi^{n-1}\]  
_and_  
\[\eta_{k+1}\left(\xi\right):=\left(\frac{\delta_{k}r^{1/\alpha}\left(\xi \right)}{\left(n-2\right)!}\right)\int_{\xi_{1}}^{\xi}v^{n-2}\left(\frac{1}{r \left(v\right)}\exp\left(\int_{v}^{\xi}\frac{\rho_{k}\left(u\right)}{r\left(u \right)}\mathrm{d}u\right)\right)^{1/\alpha}\mathrm{d}v,\]  
_for all \(\delta_{k}\in\left(0,1\right)\) and \(k=0,1,...\)._  
Proof.: Assume that \(x\in X_{I}^{+}\). Then, there exists a \(\xi_{1}\geq\xi_{0}\) such that \(x\left(\xi\right)>0\) and \(x\left(\psi\left(\xi,s\right)\right)>0\) for all \(\xi\geq\xi_{1}\). Now, we will use the induction to prove Equation (4). For \(k=0\), using Lemma 1, we obtain  
\[x\left(\psi\left(\xi,s\right)\right)\geq\frac{\delta_{0}}{\left(n-1\right)!} \psi^{n-1}\left(\xi,s\right)x^{\left(n-1\right)}\left(\psi\left(\xi,s\right) \right)\geq\eta_{0}\left(\psi\left(\xi,s\right)\right)x^{\left(n-1\right)} \left(\psi\left(\xi,s\right)\right).\]  
Next, we assume that \(x\left(\psi\left(\xi,s\right)\right)\geq\eta_{k}\left(\psi\left(\xi,s\right) \right)x^{\left(n-1\right)}\left(\psi\left(\xi,s\right)\right)\), for \(k>1\). Then  
\[x^{\alpha}\left(\psi\left(\xi,s\right)\right)\geq\eta_{k}^{\alpha}\left(\psi \left(\xi,s\right)\right)\left(x^{\left(n-1\right)}\left(\psi\left(\xi,s \right)\right)\right)^{\alpha}\]  
or equivalently,  
\[f\left(x\left(\psi\left(\xi,s\right)\right)\right)\geq\varrho\eta_{k}^{\alpha} \left(\psi\left(\xi,s\right)\right)\left(x^{\left(n-1\right)}\left(\psi\left( \xi,s\right)\right)\right)^{\alpha}. \tag{5}\]  
Combining Equations (1) and (5), we get  
\[\left(r\left(\xi\right)\left(x^{\left(n-1\right)}\left(\xi\right)\right)^{ \alpha}\right)^{\prime}\leq-\varrho\int_{a}^{b}\rho\left(\xi,s\right)\eta_{k}^ {\alpha}\left(\psi\left(\xi,s\right)\right)\left(x^{\left(n-1\right)}\left( \psi\left(\xi,s\right)\right)\right)^{\alpha}\mathrm{d}s.\]  
Since \(x^{\left(n\right)}<0\) and \(\psi\left(\xi,a\right)<\xi\), we have  
\[\left(r\left(\xi\right)\left(x^{\left(n-1\right)}\left(\xi\right)\right)^{ \alpha}\right)^{\prime}\leq-\varrho\left(x^{\left(n-1\right)}\left(\xi\right) \right)^{\alpha}\int_{a}^{b}\rho\left(\xi,s\right)\eta_{k}^{\alpha}\left(\psi \left(\xi,s\right)\right)\mathrm{d}s. \tag{6}\]  
**Corollary 1**.: _Assume that \(\eta_{k}\) is defined as in Lemma 3. If, for some \(\delta_{k}\in(0,1)\) and some \(k\in\mathbb{N}\),_  
\[\liminf_{\xi\rightarrow\infty}\int_{\psi(\xi,a)}^{\xi}\frac{\rho_{k}\left(u \right)}{r\left(\psi\left(u,a\right)\right)}\mathrm{d}u>\frac{1}{\mathrm{e}}, \tag{11}\]  
_then \(X_{I}^{+}\) is empty._  
Proof.: Applying a well-known criterion Theorem 2 in [13] for first-order delay differential Equation (9) to be oscillatory, we obtain immediately the criterion (11).  
**Example 1**.: _Consider the third-order differential equation_  
\[x^{\prime\prime\prime}+\int_{\epsilon}^{1}\frac{\rho_{0}}{\xi^{3}}x\left( \epsilon\xi\right)\mathrm{d}\xi=0, \tag{12}\]  
_where \(\xi\geq 1\), \(\alpha=1\), \(n=3\), \(r=1\), \(\rho\left(\xi,s\right)=\rho_{0}/\xi^{3}\), \(\rho_{0}>0\), \(a=e\), \(b=1\), \(\psi\left(\xi,s\right)=e\xi\) and \(e\in(0,2/3)\). Then, we get that_  
\[\eta_{0}\left(\xi\right):=\frac{\delta_{0}}{2}\epsilon^{2}\xi^{2}.\]  
_It is easy to verify that condition (11) with \(k=0\) reduces to_  
\[\varrho\rho_{0}\delta_{0}\epsilon^{2}\left(1-\epsilon\right)\ln\frac{1}{ \epsilon}>\frac{2}{\mathrm{e}}. \tag{13}\]  
_From Corollary 1, we see that \(X_{I}^{+}\) is empty if (13) satisfies._