Journal of the
Korean Mathematical Society JKMS

We consider forced second order differential equation with $p$-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of \begin{equation*} \left( p(t)\phi _{\gamma }\left( x^{\prime }(t)\right) \right) ^{\prime }+q_{0}\left( t\right) \phi _{\gamma }\left( x(t)\right) +\int_{0}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left( x(t)\right) d\zeta \left( s\right) =e(t), \end{equation*} where $\phi _{\alpha }\left( u\right) :=\left\vert u\right\vert ^{\alpha } \mbox{${\rm sgn}\,$}u$, $\gamma ,\ b\in \left( 0,\infty \right) ,$ $\alpha \in C\left[ 0,b\right) $ is strictly increasing such that $0\leq \alpha \left( 0\right) <\gamma <\alpha \left( b-\right) $, $p,\ q_{0},\ e\in C\left( [t_{0},\infty ),\mathbb{R}\right) $ with $p\left( t\right) >0$ on $ [t_{0},\infty )$, $q\in C\left( \left[ 0,\infty \right) \times \left[ 0,b\right) \right) $, and $\zeta :\left[ 0,b\right) \rightarrow \mathbb{R}$ is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature.

Keywords: interval criteria, forced oscillation, $p$-Laplacian, nonlinear differential equations

MSC numbers: 34C10, 34C15