Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T:K→E be a nonexpansive weakly inward mapping with F(T)≠∅ and f:K→K be a contraction. Then for t∈(0,1), there exists a sequence {yt}⊂K satisfying yt=(1-t)f(yt)+tT(yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F(T) to a certain variational inequality. Moreover, if {Ti,i=1,2,…,r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti,i=1,2,…,r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti,i=1,2,…,r need not satisfy the requirment that ⋂i=1rF(Ti)=F(TrTr-1,…,T1)=F(T1Tr,…,T2)=,⋯,=F(Tr-1Tr-2,…,T1Tr).