The paper by Zdzisław Opiol, communicated by F. Browder, focuses on the weak convergence of the sequence of successive approximations for nonexpansive mappings in Hilbert and Banach spaces. The main result strengthens a previous finding by F. E. Browder and W. V. Petryshyn, showing that under certain conditions, the sequence $\{T^n x\}$ is not only weakly convergent but also converges weakly to a fixed point of the mapping $T$. The author provides a detailed proof of this result, using basic definitions and lemmas, and discusses the extension of these findings to Banach spaces with weakly continuous duality mappings. The paper also explores the application of these results to modified sequences of successive approximations and examines the limits of validity for key lemmas. Additionally, it addresses the conditions under which the weak limit characterization holds in general Banach spaces, highlighting that it does not extend to all uniformly convex Banach spaces.The paper by Zdzisław Opiol, communicated by F. Browder, focuses on the weak convergence of the sequence of successive approximations for nonexpansive mappings in Hilbert and Banach spaces. The main result strengthens a previous finding by F. E. Browder and W. V. Petryshyn, showing that under certain conditions, the sequence $\{T^n x\}$ is not only weakly convergent but also converges weakly to a fixed point of the mapping $T$. The author provides a detailed proof of this result, using basic definitions and lemmas, and discusses the extension of these findings to Banach spaces with weakly continuous duality mappings. The paper also explores the application of these results to modified sequences of successive approximations and examines the limits of validity for key lemmas. Additionally, it addresses the conditions under which the weak limit characterization holds in general Banach spaces, highlighting that it does not extend to all uniformly convex Banach spaces.