Oscillatory behaviour of a first-order neutral differential equation in relation to an old open problem (Q779981)

Summary: In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation \((y(t) - \sum_{j=1}^k p_j (t) y (r_j (t)))' +q(t) G(y(g(t))) -u(t) H(y(h(t))) =f(t)\), where \(p_j\) and \(r_j\) for each \(j\) and \(q\), \(u\), \(G\), \(H\), \(g\), \(h\), and \(f\) are all continuous functions and \(q\geq 0\), \(u\geq 0\), \(h(t)<t\), \(g(t)<t\), and \(r_j(t)<t\) for each \(j\). Further, each \(r_j(t)\), \(g(t)\), and \(h(t)\longrightarrow \infty\) as \(t\longrightarrow \infty\). This paper improves and generalizes some known results.