The main result in this note slightly generalizes Theorem 3.4 in [S¸ t. Maruster and I. A. Rus, ˘ Kannan contractions and strongly demicontractive mappings, Creat. Math. Inform., 24 (2015), No. 3, 171–180]. We also significantly improve its proof. Both results in the paper are based on the concepts of demicontractivity and quasi-expansivity and involve relatively weak conditions that guarantee the convergence of Krasnoselskij method (the same conditions are not sufficient for the convergence of Picard iteration). Thus, our note is a satisfactory answer to the following question: if for a given mapping some specific conditions for the convergence of Picard iteration are not satisfied (and presumptively the Picard iteration fails to converge), what are the conditions which still ensure the convergence of the Mann iteration?