On a new approach of enriched operators

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.