On the Solution of a Volterra Integral Equation with a Weakly Singular Kernel

The solution $x(t)$ of the Volterra integral equation of the second kind x(t) = f₁ (t) + √ t f₂ (t) + ∫ ₀ᵗ g(t,s,x(s))(t - s)^ - 1 / 2 ds is examined. It is shown that x(t) = u(t) + √ t v(t), where $u(t)$ and $v(t)$ are smooth under appropriate smoothness conditions on f₁ (t), f₂ (t) and $g(t,s,x)$ and satisfy a system of Volterra integral equations of the second kind.