Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces

Let EE be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let KK be a nonempty closed convex subset of EE. Suppose that every nonempty closed convex bounded subset of KK has the fixed point property for nonexpansive mappings. Let T1,T2,…,TNT1,T2,…,TN be a family of nonexpansive self-mappings of KK, with F≔⋂i=1NFix(Ti)≠0̸,F=Fix(TNTN−1…T1)=Fix(T1TN…T2)=…=Fix(TN−1TN−2…T1TN). Let {λn}{λn} be a sequence in (0,1)(0,1) satisfying the following conditions: C1:limλn=0;C2:∑λn=∞C1:limλn=0;C2:∑λn=∞. For a fixed δ∈(0,1)δ∈(0,1), define Sn:K→KSn:K→K by Snx≔(1−δ)x+δTnx∀x∈KSnx≔(1−δ)x+δTnx∀x∈K where Tn=TnmodNTn=TnmodN. For an arbitrary fixed u,x0∈Ku,x0∈K, let B≔{x∈K:TNTN−1…T1x=γx+(1−γ)u, for some γ>1}B≔{x∈K:TNTN−1…T1x=γx+(1−γ)u, for some γ>1} be bounded, and let the sequence {xn}{xn} be defined iteratively by xn+1=λn+1u+(1−λn+1)Sn+1xn,for n≥0. Assume that limn→∞‖Tnxn−Tn+1xn‖=0limn→∞‖Tnxn−Tn+1xn‖=0. Then, {xn}{xn} converges strongly to a common fixed point of the family T1,T2,…,TNT1,T2,…,TN. This convergence theorem is also proved for non-self maps.