In a recent paper Rhoades [6] has shown, for a selfmap $T$ of a Banach space satisfying the contractive definitions of Ćirić [1] or of Pal and Maiti [5], that if the sequence of Mann iterates converges then it converges to a fixed point of $T$. In this note we propose to draw the same conclusion in some of these cases even for subsequential limit points, i.e., every subsequential limit point of the sequence of Mann iterates will be a fixed point of $T$. Further we shall derive the conclusions of Rhoades in the case of mappings satisfying even weaker conditions. Our final result will be concerned with the extension of a result of Maiti and Babu [4] to mappings satisfying conditions similar to those in Rhoades [6, Theorem 3]. This is closed in spirit to the main result of Diaz and Metcalf [2].