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Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. (English) Zbl 1194.47089

Summary: Let K be a nonempty closed and convex subset of a real Banach space E. Let TKE be a nonexpansive weakly inward mapping with nonempty fixed point set Fix (T) and fKK be a contraction. Then, for t(0,1), there exists a sequence {y t }K satisfying

y t =(1-t)f(y t )+tT(y t )·

If E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {y t } 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 {T i } i=1 r is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {T i } i=1 r and to a solution of a certain variational inequality is constructed. In the above setting, the family {T i } i=1 r is not required to satisfy the condition

i=1 r Fix (T i )= Fix (T r T r-1 T 1 )= Fix (T 1 T r T 2 )== Fix (T r-1 T r-2 T 1 T r )·

47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties