Oscillatory and asymptotic behavior for third order differential equations with piecewise constant argument (Q2858059)

The paper deals with the oscillation and asymptotic properties of a third order nonlinear differential equation with piecewise constant argument of the form NEWLINE\[NEWLINE(r(t)x''(t))'+f(t,x([t]))=0,\eqno{(1)}NEWLINE\]NEWLINE where \(r\) is a positive and continuous function on \([0,\infty)\) and \(\int_a^{\infty}r^{-1}(t)dt=\infty\) for every \(a>0\). In addition, \(xf(t,x)>0\) for \(x\neq 0\) and \(t\geq 0\). The authors derive several conditions which ensure that every solution of the equation (1) is oscillatory or tends to zero as \(t\to\infty\).