Oscillatory and nonoscillatory delay equations with piecewise constant argument (Q1583943)

The authors consider the linear delay differential equation with piecewise constant deviating argument of the form \[ y'(t)+ a(t)y(t)+b(t)y([t-1])=0,\quad t\geq 0, \tag{1} \] where \(a(t),\;b(t)\) are continuous functions on \([-1,\infty),\;b(t)\geq 0(\not\equiv 0)\) for \(t\geq 0\), and \([\cdot ]\) denotes the greatest integer function. A new technique to analyze certain difference equations to obtain some new oscillation and nonoscillation criteria for equation (1) is introduced. Moreover, the authors prove that equation (1) can be classified as oscillatory or nonoscillatory, i.e., either all solutions are oscillatory or all solutions are nonoscillatory. The authors emphasize that their method yields ``best possible'' oscillation and nonoscillation criteria.