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 $J_{\Phi}$ with gauge $\phi$. Let $f$ be an {\em $\alpha$-contraction} and $\{T_n\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes \begin{equation} x_{n+1} = \alpha_n f(x_n) + (1-\alpha_n) T_n x_n \end{equation} with a general theorem and then recover and improve some specific cases studied in the literatur