Necessary and sufficient conditions for oscillation of second-order differential equation with several delays

Abstract

In this paper, necessary and sufficient conditions are establish of the solutions to second-order delay differential equations of the form \[\Big(r(t)\big(x'(t)\big)^\gamma\Big)' +\sum_{i=1}^m q_i(t)f_i\big(x(\sigma_i(t))\big)=0, \text{ for } t \geq t_0,\notag\]

We consider two cases when \(f_i(u)/u^\beta\) is non-increasing for \(\beta<\gamma\), and non-decreasing for \(\beta>\gamma\) where \(\beta\) and \(\gamma\) are the quotient of two positive odd integers. Our main tool is Lebesgue's Dominated Convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.