Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak${}^{\star }$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\mathrm{\Phi }}$ with gauge $\varphi$. Let $f$ be an {\em $\alpha$-contraction} and $\{T_n\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes x_{n+1} = \alpha_n f(x_n) + (1-\alpha_n) T_n x_n with a general theorem and then recover and improve some specific cases studied in the literature