Iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces Original Research Article

Let EE be a uniformly convex real Banach space with a uniformly Gâteaux differentiable norm. Let KK be a closed, convex and nonempty subset of EE. Let View the MathML source{Ti}i=1∞ be a family of nonexpansive self-mappings of KK. For arbitrary fixed δ∈(0,1)δ∈(0,1), define a family of nonexpansive maps View the MathML source{Si}n=1∞ by Si≔(1−δ)I+δTiSi≔(1−δ)I+δTi where II is the identity map of KK. Let View the MathML sourceF≔∩i=1∞F(Ti)≠0̸. It is proved that an iterative sequence {xn}{xn} defined by x0∈K,xn+1=αnu+∑i≥1σi,tnSixn,n≥0x0∈K,xn+1=αnu+∑i≥1σi,tnSixn,n≥0, converges strongly to a common fixed point of the family View the MathML source{Ti}i=1∞, where {αn}{αn} and {σi,tn}{σi,tn} are sequences in (0,1)(0,1) satisfying appropriate conditions, in each of the following cases: (a) E=lp,1