Common Fixed Points for Countable Families of Nonexpansive Mappings

Various authors have studied iterative schemes similar to that of Bauschke (Theorem BSK, Chapter 15) in more general Banach spaces on one hand and using various conditions on the sequence {λ_n} on the other hand (see, for example, Colao et al. [192], Yao [532], Takahashi and Takahashi [482], Plubtieng and Punpaeng [385], Ceng et al. [193], Chidume and Ali [125], Jung [264], Jung et al. [265], O'Hara et al. [361], Zhou et al. [559]). Most of the results in these references are proved for finite families of nonexpansive mappings defined in Hilbert spaces.

Convergence theorems have also been proved for common fixed points of countable infinite families of nonexpansive mappings. Before we proceed, we first state the following important theorem.