A Volterra integral equation of the form \(u(t)=\epsilon \int^{t}_{0}[\pi (t-s)]^{-1/2}F(u(s),s)ds\) is considered where \(F(0,t)\sim c_ 0t^{-r_ 0}\) and \(F_ u(0,t)\sim -c_ 1t^{-r_ 1}\) as \(t\to +\infty\) and \(r_ 1<1/2\). Writing \(u(t)\sim \epsilon u_ 0(t)+\Delta (\epsilon)v_ 0(\tau)+...,\) where \(\tau =\epsilon^{\alpha}t,\quad \alpha =2/(1-2r_ 1)\) is the large-time variable a method is suggested to obtain \(v_ 0\).