## 5 Oscillation Criteria  
Based on the fact that there are only two cases for the corresponding function \(z\), we can use the results in the previous two sections to infer new criteria for oscillation of all solutions of Equation (1). Any of the criteria (23), (24) and (26) ensures that \(\widetilde{\aleph}=\emptyset\), whereas one of the criteria (27), (31) and (34) ensures that \(\aleph=\emptyset\). This guarantees that \(\aleph=\emptyset=\emptyset\), and we can ensure that there are no non-oscillatory solutions. Hence through these results we proved the following oscillation Theorem:  
**Theorem 6**.: _Assume the non-linear NDDE of third order (1). Then:__._  
1. _If_ \(r_{i}^{\prime}\left(t\right)>0\) _for_ \(i=1,2\)_._ * _If there exists a function_ \(\psi\in C\left(\left[t_{0},\infty\right),\left(0,\infty\right)\right)\) _satisfying_ \(g\left(t\right)<\psi\left(t\right)\) _and_ \(\tau^{-1}\left(\psi\left(t\right)\right)<t\)_, such that the first-order delay differential Equations (_24_) and (_27_) are oscillatory, then (_1_) is oscillatory;_ * _If (_23_) and (_31_) hold, then (_1_) is oscillatory;_ * _If (_23_) and (_34_) hold, then (_1_) is oscillatory._
2. _If there exists a positive function_ \(\rho(t)\)_such that (_26_) and_ * _(_31_) hold, then (_1_) is oscillatory;_ * _(_34_) hold, then (_1_) is oscillatory._  
_Example_  
Consider the third-order NDDE  
\[\left(\left(\left(x\left(t\right)+p_{0}x\left(\delta t\right)\right)^{\prime \prime}\right)^{a_{2}}\right)^{\prime}+\frac{q_{0}}{t^{2a_{2}+1}}x^{a_{2}} \left(\lambda t\right)=0,\ \ t\geq 1, \tag{35}\]  
where \(p_{0}\) and \(q_{0}\) are positive constants and \(\delta,\lambda\in\left(0,1\right)\). Please note that \(r_{1}\left(t\right)=r_{2}\left(t\right)=1\), \(a_{1}=1\), \(\beta=a_{2}\) and \(f\left(x\right)=x^{a_{2}}\). It is easy to verify that \(R_{i}\left(t,u\right)=t-u\) for \(i=1,2\),  
\[\widetilde{R}\left(v,u\right) = \int_{u}^{v}\left(v-s\right)\mathrm{d}s=\frac{1}{2}\left(v-u \right)^{2},\] \[\theta\left(t\right) = q_{0}\left(1-p_{0}\right)^{a_{2}}\frac{1}{t^{2a_{2}+1}},\]  
and \(Q\left(t\right)=q_{0}/t^{2a_{2}+1}\). Then, we have  
\[\liminf_{t\to\infty}\left(\widetilde{R}\left(g\left(t\right),t_{0} \right)\right)^{\beta}\int_{t}^{\infty}\theta\left(s\right)ds = \frac{q_{0}\left(1-p_{0}\right)^{a_{2}}}{2^{a_{2}+1}}\liminf_{t \to\infty}\frac{\left(\lambda t-t_{0}\right)^{2a_{2}}}{t^{2a_{2}}}\] \[= \frac{q_{0}\left(1-p_{0}\right)^{a_{2}}}{2^{a_{2}+1}}\lambda^{2a _{2}}.\]  
Thus, the condition (23) becomes  
\[q_{0}>\frac{2^{a_{2}+1}a_{2}^{a_{2}+1}}{\lambda^{2a_{2}}\left(1-p_{0}\right)^{ a_{2}}\left(a_{2}+1\right)^{a_{2}+1}}. \tag{36}\]  
Next, by choosing \(\psi\left(t\right)=\left(\lambda+\delta\right)t/2\), condition (31) reduces to  
\[\liminf_{t\to\infty}\int_{\tau^{-1}\left(\psi\left(t\right)\right) }^{t}Q\left(s\right)\left(\widetilde{R}\left(\psi\left(s\right),g\left(s \right)\right)\right)^{\beta}\mathrm{d}s = \liminf_{t\to\infty}\int_{\left(\delta+\lambda/2\right)t}^{t} \frac{kq_{0}}{2^{a_{2}}}\left(\frac{\delta-\lambda}{2}\right)^{2a_{2}}\frac{1 }{s}\mathrm{d}s\] \[= \frac{kq_{0}}{2^{a_{2}}}\left(\frac{\delta-\lambda}{2}\right)^{2a _{2}}\ln\frac{2\delta}{\delta+\lambda}\]  
or equivalently,  
\[\frac{kq_{0}}{2^{a_{2}}}\left(\frac{\delta-\lambda}{2}\right)^{2a_{2}}\ln\frac{ 2\delta}{\delta+\lambda}>\frac{\tau_{0}+p_{0}^{a_{2}}}{\mathrm{e}\mu\tau_{0}}. \tag{37}\]  
Hence, by Theorem 6, every solution of Equation (35) is oscillatory if (36) and (37) hold.  
**Remark 2**.: _By using our results, we obtain sufficient conditions to ensure that all solutions of (1) are oscillatory. Whereas, the related results [37; 38; 40; 41; 43] created conditions that ensure that solutions are either oscillatory or tend to zero. So, our new criteria improve and complement a number of existing results._