Oscillations of Volterra integral equations with delay (Q1319199)

Consider the Volterra integral equation with delays \(x(t)=f(t)-\int_ 0^ t K(t,s,x_ s)ds\), \(t\geq 0\), where \(f\in C(\mathbb{R}^ +,\mathbb{R})\), \(K(t,s,\varphi)\) is continuous in \(t\) and \(\varphi\), and maps bounded sets into bounded sets. For each bounded subset \(B\subseteq C\) the function \(s\to\sup_{\varphi\in B} K(t,s,\varphi)\) is measurable in \(s\) for fixed \(t\). Also \(K(t,s,\varphi)\geq 0\) if \(\varphi\geq 0\) and \(K(t,s,\varphi)\leq 0\) if \(\varphi\leq 0\). Then we investigate how the oscillation behavior of \(f\) is inherited by all solutions of the equation.