The authors extend the formal methodology for the asymptotic analysis of singularly perturbed Volterra integral equations developed by themselves [ibid. 47, 1-14 (1987; Zbl 0616.45009)] to several problems of the form \[ \epsilon (a(\epsilon)u'(t)+b(\epsilon)u(t))=\int^{t}_{0}k(t,s;\epsilon)f[u(s),s ;\epsilon]\quad ds+f(t;\epsilon),\quad t\geq 0. \] The singularity is due to the first \(\epsilon\) in the equation. For several of the examples considered an ansatz of the form \(u(t)=[y_ 0(t)+z_ 0(\tau)]+\epsilon^{m_ 1}[y_ 1(t)+z_ 1(\tau)]+...,\tau =t/\epsilon^{\gamma}\), is appropriate, but they also study some examples where the correct ansatz is \(u(t)=y_ 0(t)z_ 0(\tau)+\epsilon^{m_ 1}y_ 1(t)z_ 1(\tau)+...,\tau =t/\epsilon^{\gamma}\).