Oscillations of first order neutral delay differential equations
Consider the neutral delay differential equation \[ (*)\quad (d/dt)[y(t)+py(t-\tau)]+qy(t-\sigma)=0,\quad t\geq t_ 0 \] where \(\tau\), q and \(\sigma\) are positive constants, while \(p\in (-\infty,-1)\cup (0,+\infty)\). Theorem 1. Assume \(p<-1\). Then every nonoscillatory solution y(t) of (*) tends to \(\pm \infty\) as \(t\to \infty\). Theorem 2. Assume \(p<-1\) and \(q/(1+p)(\sigma -\tau)>1/e\). Then every solution of (*) oscillates. Theorem 3. Assume \(p>0\). Then every nonoscillatory solution y(t) of (*) tends to zero as \(t\to \infty\). Theorem 4. Assume \(p>0\). Then a necessary condition for all solutions of (*) to oscillate is that \(\sigma >\tau\). Theorem 5. Assume \(p>0\) and \(q/(1+p)(\sigma -\tau)>1/e\). Then every solution of (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.
first order neutral delay differential equations
scientific article; zbMATH DE number 3903107
scientific article