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 {y_t} ⊂ K satisfying y_t = (1 - t) f (y_t) + tT (y_t). Furthermore, 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, 2, ..., 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, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family T_i, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.