Xu’s strong convergence theorem of iterative algorithms shows that for every nonexpansive mapping T defined on a nonempty closed bounded convex set C of a uniformly smooth Banach space X, the iteration sequence \((x_n)\) strongly converges to a fixed point of T, where \(x_{n+1} = \alpha _{n}u + (1-\alpha _n)Tx_n ~(n\ge 0)\) , where \(u,x_0\in C\) are arbitrary and \((\alpha _n)\subset [0,1]\) . In this paper, by weakening the uniform smoothness assumption of the entire space, we give this theorem and Xu’s another strong convergence theorem of iterative algorithm localized settings: the theorems still hold on a closed bounded convex set \(C\subset X\) whenever the norm of X is C-uniformly Gâteaux smooth. In particular, there is an equivalent norm \(\Vert |\cdot \Vert |\) on \(L_1[0,1]\) so that theorems mentioned above hold on every weakly compact convex set of \((L_1[0,1], \Vert |\cdot \Vert |)\) .
