In this paper we investigate a class of Volterra integral equations for existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results. More precisely, we give a suitable integral representation of the solutions of the considered Volterra integral equation. Then, we construct two operators for which any fixed point of their sum is a solution of the considered Volterra integral equation.
