An investigation is made of the asymptotic behavior of the solutionu(t;e) to the Volterra integral equation $$\varepsilon u(t;\varepsilon ) = \pi ^{ - \tfrac{1}{2}} \int\limits_0^t {(t - s)^{ - \tfrac{1}{2}} [f(s) - u^n (s;\varepsilon )]} ds, t \geqslant 0, n \geqslant 1$$ , in the limit as e→0. This investigation involves a singular perturbation analysis. For the linear problem (n=1) an infinite, uniformly valid asymptotic expansion ofu(t;e) is obtained. For the nonlinear problem (n≥2), the leading two terms of a uniformly valid expansion are found