Volterra integral equations are intrinsically related to the study of initial value problems of ordinary differential equations, and they appear in very different models of applications of mathematics. In this article we present some existence theorems for solutions of nonlinear Volterra and Fredholm-Hammerstein integral equations in the context of \(\mathcal {L}^2\) -spaces. In both cases, we propose an average-based iterative algorithm to approximate the solutions of these equations, proving its convergence when the integrand satisfies sufficient conditions. The second part of the paper describes the solution of linear and nonlinear systems of equations by means of the same algorithm. The presented iterative procedure generalizes the fixed point method for the approximation of solutions of systems of equations. The complexity, number of iterations to reach a given precision and stability are analyzed in some specific cases. An empirical procedure to find a solution close to a given point in the case of multiple solutions is also proposed. In the last part of the article, the proved results are applied to the numerical solution of nonlinear integral equations by means of quadratures, involving the solution of a system of nonlinear equations.
