In this book the solutions of a class of nonlinear Volterra integral equations and their discretized counterparts are considered. The equations have at least trivial solutions. We give a proof of existence and uniqueness of the positive solutions in an order interval of a cone in the appropriate Banach spaces. The concept of discrete approximations in Banach spaces has been used to prove the convergence of the solutions of the discrete equations to the solution of the continuous problem. In contrast to the usual approach we have used nonlinear restriction and prolongation operators. Numerical case studies confirm the theoretical results.