Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces (Q884245)

Let \(C\) be a bounded closed convex subset of a uniformly smooth Banach space. Let \(\{ T_n: n \in \mathbb{N} \}\) be an infinite family of commuting nonexpansive mappings on \(C\). Let \(\{ \alpha_n \}\) and \(\{ t_n \}\) be sequences in \((0, 1/2)\), \(\lim_{n} t_n=\lim_{n} \alpha_n/ t_n^l=0\) for \(l \in {\mathbb{N}}\). Fix \(u \in C\) and define a sequence \(\{ u_n \}\) in \(C\) by \(u_n=(1-\alpha_n) ((1-\sum_{k=1}^n t_n^k)T_1 u_n+ \sum_{k=1}^n T_{k+1} u_n)+\alpha_n u\) for \(n \in {\mathbb{N}}\). Then \(\{ u_n \}\) converges strongly to \(Pu\), where \(P\) is the unique sunny nonexpansive retraction from \(C\) onto \(\cap_{n=1}^{\infty}F(T_n)\). Here, \(F(T)\) is the set of fixed points of \(T\).