Zdzisław Opiál proved that for a nonexpansive mapping T on a Hilbert space, if T is asymptotically regular and has at least one fixed point, then the sequence of successive approximations {T^n x} is weakly convergent. This result was strengthened by showing that under the same assumptions, the sequence {T^n x} is necessarily weakly convergent. The paper also discusses the extension of this result to Banach spaces with weakly continuous duality mappings. It is shown that Lemma 1, which characterizes the weak limit of a weakly convergent sequence in Hilbert spaces, does not hold for all uniformly convex Banach spaces, but it does hold for a large class of uniformly convex Banach spaces with weakly continuous duality mappings. The paper also discusses the application of the result to modified sequences of successive approximations and the limits of validity of the first key lemma. The main theorem, Theorem 2, states that in a uniformly convex Banach space with a weakly continuous duality mapping, the sequence of successive approximations {T^n x} is weakly convergent to a fixed point of T. The paper also discusses the extension of the result to a modified sequence of successive approximations and the implications for Schaefer's conjecture. Finally, it is shown that Lemma 1 cannot be extended to all uniformly convex Banach spaces, and that certain spaces like L^p[0, 2π] do not have weakly continuous duality mappings.Zdzisław Opiál proved that for a nonexpansive mapping T on a Hilbert space, if T is asymptotically regular and has at least one fixed point, then the sequence of successive approximations {T^n x} is weakly convergent. This result was strengthened by showing that under the same assumptions, the sequence {T^n x} is necessarily weakly convergent. The paper also discusses the extension of this result to Banach spaces with weakly continuous duality mappings. It is shown that Lemma 1, which characterizes the weak limit of a weakly convergent sequence in Hilbert spaces, does not hold for all uniformly convex Banach spaces, but it does hold for a large class of uniformly convex Banach spaces with weakly continuous duality mappings. The paper also discusses the application of the result to modified sequences of successive approximations and the limits of validity of the first key lemma. The main theorem, Theorem 2, states that in a uniformly convex Banach space with a weakly continuous duality mapping, the sequence of successive approximations {T^n x} is weakly convergent to a fixed point of T. The paper also discusses the extension of the result to a modified sequence of successive approximations and the implications for Schaefer's conjecture. Finally, it is shown that Lemma 1 cannot be extended to all uniformly convex Banach spaces, and that certain spaces like L^p[0, 2π] do not have weakly continuous duality mappings.