Fundamental in the numerical solution of any integral equation is the discretization process. Annunziato and Messina describe a heterogeneous process of discretization for the space and temporal variable. Using a contraction theorem invariant to a map, the existence of the solution through their discretization (in space) is proven. Moreover, the uniqueness of the solution is shown, although the last proof is completed taking into account the discretization in time. Quadratic convergence is achieved in numerical experiments.

The proofs are a very important part of this paper, since Annunziato and Messina give explicit hints on how they adapted some of the elements of similar proofs for existence, uniqueness, and convergence in their own case. This should encourage many researchers to review both the mentioned sources for the proofs and this work itself, in order to see other possible adaptations to Volterra equations, where the invariance to certain maps is fulfilled. The study of monotonicity and conservativity (in the sense of being time-flow invertible) is still pending, as the authors mention.

I would strongly recommend this paper for researchers of inverse problems that can be cast into Volterra equations, applied mathematicians, pure mathematicians interested in generalizing the exhibited proofs, and those studying proofs for numerical analysis.