Oscillations and asymptotic stability of solutions of second order NEPCA (Q1355069)

The author considers \[ {d^2\over dt^2} \Biggl(x(t)+ px\Biggl(t+{1\over 2}\Biggr)\Biggr)= qx\Biggl(\Biggr[t+{1\over 2}\Biggr]\Biggr). \] It is proved that (i) the zero solution is asymptotically stable if and only if \((p,q)= (0,8)\); (ii) if \(p=1\) and \(q<0\) then every solution oscillates; (iii) if \(p=0\), \(q<0\) or \(q\geq 8\) then every solution oscillates; (iii) if \(p\neq 0\) and \(p\neq 1\) then every solution oscillates provided that either \[ p\in(-\infty,-2]\cup \Biggl(0,{1\over 4}\Biggr),\quad {16p^2+8\over 1-4p}<q \] or \(p\in(-2,0)\) and \(q>8\).