On the oscillation of nonlinear delay differential equations and their applications

The oscillation of nonlinear differential equations is used in many applications of mathematical physics, biological and medical physics, engineering, aviation, complex networks, sociophysics and econophysics. The goal of this study is to create some new oscillation criteria for fourth-order differential equations with delay and advanced terms ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + ∑ j = 1 r β j ( x ) w k ( γ j ( x ) ) = 0 , {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+\mathop{\sum }\limits_{j=1}^{r}{\beta }_{j}(x){w}^{k}({\gamma }_{j}(x))=0, and ( a 1 ( x ) ( w ‴ ( x ) ) n ) ′ + a 2 ( x ) h ( w ‴ ( x ) ) + β ( x ) f ( w ( γ ( x ) ) ) = 0 . {({a}_{1}(x){({w}^{\prime\prime\prime }(x))}^{n})}^{^{\prime} }+{a}_{2}(x)h({w}^{\prime\prime\prime }(x))+\beta (x)f(w(\gamma (x)))=0. The method is based on the use of the comparison technique and Riccati method to obtain these criteria. These conditions complement and extend some of the results published on this topic. Two examples are provided to prove the efficiency of the main results.