Let E be a real Banach space. Let T : E longrightarrow E be a map with F(T) : = { x $in$ E : Tx = x} $ eq$ 0 and satisfying the accretive-type condition $lambda$mid$x-Tx$mid$^2$</TEX>, for all $xinE,;x^*inf(T);and;lambda >0$. We prove some necessary and sufficient conditions for the convergence of the Mann iterative sequence to a fixed point of T.