In this chapter first we shall follow Meehan and O’Regan [213,215] and present results which guarantee the existence of nonnegative solutions of the Volterra integral equation $$y\left( t \right) = h\left( t \right) - \int_0^t {k\left( {t,s} \right)} g\left( {s,y\left( s \right)} \right)ds,t \in {\text{ }}\left[ 0 \right. \leqslant T \leqslant \infty $$ (22.1) .