Necessary and sufficient conditions for oscillation of a mixed neutral equation (Q1322316)

The author considers the following scalar neutral differential equation with mixed arguments: \[ (d/dt) \bigl[y(t) - py(t-\tau) \bigr] + q_ 1y(t-\sigma_ 1) + q_ 2y (t+\sigma_ 2) = 0, \tag{1} \] where \(p, q_ 1, q_ 2, \tau, \sigma_ 1, \sigma_ 2\) are positive real numbers. The obtained result is that every solution of (1) is oscillatory iff the characteristic equation \(\lambda + \lambda pe^{-\lambda \tau} + q_ 1e^{\lambda \sigma_ 1} + q_ 2 e^{\lambda \sigma_ 2} = 0\) has no real root.