Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and K a nonempty, closed, and convex subset of E. Let T : K K be an asymptotically -strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I -T) is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann's algorithm for T under suitable control conditions. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the

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... ified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm. 2000 AMS Subject Classification: 47H09; 47H10.