Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Abstract

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type (r(t)(z'(t))^\gamma)' +\sum_{i=1}^m q_i(t)x^{\alpha_i}(\sigma_i(t))=0, \quad t\geq t_0, where $z(t)=x(t)+p(t)x(\tau(t))$. Under the assumption $\int^{\infty}(r(\eta))^{-1/\gamma} {\rm d}\eta=\infty$, we consider two cases when $\gamma>\alpha_i$ and $\gamma<\alpha_i$. Our main tool is Lebesgue's dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

Keywords

• oscillation
• non-oscillation
• neutral
• delay
• lebesgue's dominated convergence theorem