Let \(E\) be a real Banach space, \(I=[t_ 0,t_ 0+a]\) \((a>0)\), \(f\in C(I\times E\times E\times E;E)\), and \(u_ 0\in E\). The authors discuss the existence of minimal and maximal solutions to the initial value problem \(u'=f(t,u,Tu,Su)\), \(u(t_ 0)=u_ 0\), where \(Tu(t)=\int^ t_{t_ 0}k(t,s)u(s)ds\), \(Su(t)=\int^{t_ 0+a}_{t_ 0}h(t,s)u(s)ds\). The kernels \(k:\{(t,s):t_ 0\leq s\leq t\leq t_ 0+a\}\to\mathbb{R}\), and \(h:I\times I\to\mathbb{R}\) are assumed to be continuous and nonnegative.