Singularly perturbed Volterra integral equations with weakly singular kernels (Q1608163)

The author considers the following Volterra integral equation \[ \varepsilon u(t)=f(t)+\frac{1}{\Gamma(\beta)}\int\limits_0^t \frac{k(t,s)}{(t-s)^{1-\beta}} ds, 0\leq t\leq T, \] where \(0<\varepsilon \ll 1\) and \(0<\beta< 1\) with continuous known function \(f(t)\). It is supposed that \(k(t,s)\) is also continuous and \(k(t,t)=-1\). The principal term \(U_0 (t, \varepsilon)\) of the asymptotic solution such that \(|u(t, \varepsilon)-U_0 (t, \varepsilon)|=O(\varepsilon)\) as \(\varepsilon \to 0\) is found.