Volterra integral equations: The singular case (Q1424056)

The authors are concerned with the investigation of singular Volterra integral equations of the form \[ y(t)= \int^t_0 k(t, s) f(s,y(s))\,ds,\quad t\in [0,T]. \] The singularity feature appears in the nonlinearity \(f(t,y)\), which may admit a nonregular behavior at \(y= 0\). By means of a truncation procedure, the singularity is eliminated, and then by a limiting process, the result is validated in the singular case. The special case investigated is \(f(t,y) =g(y) +h(y)\). Two special cases are incorporated, when \(g(y)= | y|^{-\alpha}\).