This study focuses on iteration methods based on the Banach fixed point theorem and a posteriori error estimates of Ostrowski. Their application for systems of linear simultaneous equations, bounded linear operators, as well as integral and differential equations is considered. The study presents a new version of the Picard–Lindelöf method for ordinary differential equations (ODEs) supplied with guaranteed and explicitly computable upper bounds of the approximation error. The estimates derived in the thesis take into account interpolation and integration errors and, therefore, provide objective information on the accuracy of computed approximations.