An explicit iteration for asymptotic centers of orbits of nonexpansive mappings in Banach spaces (Q1122732)

Let T be a nonexpansive mapping on a Banach space E and let \(x\in E\). The authors give an explicit iteration scheme for the calculation of the asymptotic center z of the orbit \(\{T^ ix\}^{\infty}_{i=0}\). If the space E is uniformly convex, uniformly smooth and admits a weakly continuous duality mapping then the presented algorithm converges weakly to z. As a particular case, it is obtained an iteration scheme which is calculated from a finite number of iterates \(\{T^ ix,0\leq i\leq n\}\) at each step. The obtained results partially extend those obtained by \textit{H. Brézis} and \textit{F. E. Browder} [Adv. Math. 25, 165-177 (1977; Zbl 0399.47058)].