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

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