<p>This paper provides new insights to the search and approximation of fixed points of non-contractive operators. It analyzes the role of the concepts of demiclosedness and demicompactness in the existence of critical points, giving sufficient conditions for their existence. Both notions are related to the construction of approximation sequences of the fixed points, that provide iterative algorithms for the resolution of equations of all kinds. In this paper, two different algorithms are proposed, and their convergence is studied in the framework of normed and quasi-normed spaces. The sequences are composed of weighted averages of the variable and its image by the operator. They aim to be an alternative to Picard’s method, when this procedure does not converge. These algorithms are applied to the numerical solution of a Fredholm integral equation of second kind, that appears in a great number of problems of physics, engineering and applied mathematics. The methods are checked in a particular case of Fredholm integral equation with exact solution, in order to compute the errors committed by the different approximations, and illustrate the convergence of the iterations to the real solution.</p>