In the first section of the paper, the author considers the nonlinear delay differential equation (1) \(y'(t)+ p(t) f(y(\sigma(t)))= q(t)\) under the following hypotheses: \(p(t)\in C([t_0, \infty); [0, \infty))\), \(q(t)\in C([t_0, \infty); \mathbb{R}^1)\); \(f(s)\in C(\mathbb{R}^1; \mathbb{R}^1)\), \(f(s)\geq 0\), for \(s\geq 0\), \(f(- s)= - f(s)\) for \(s> 0\), and \(f(s)\) is nondecreasing for \(s\geq 0\); \(\sigma(t)\in C([t_0, \infty); \mathbb{R}^1)\), \(\lim_{t\to \infty}\sigma(t)= \infty\), \(\sigma(t)\leq t\) for \(t\geq t_0\), and \(\sigma(s)\) is nondecreasing for \(s\geq 0\). Sufficient conditions are obtained under which all solutions of (1) are oscillatory.
In the second section the more general equation \[ y'(t)+ a(t) y(t)+ \sum^k_{i= 1} b_i(t) y(\rho_i(t))+ p(t) f(y(\sigma(t)))= q(t)\tag{2} \] is considered. Applying the results on equation (1), oscillation criteria for equation (2) are established. Examples are also given that illustrate the main results of the author.