Viscosity approximation methods for nonexpansive mapping in Banach spaces

Let C be a closed convex subset of a uniformly smooth Banach space E and let T : C → C be a nonexpansive mapping such that F(T) = ∅. The initial guess x0 ∈ C is chosen arbitrarily and given sequences {αn}∞n=0 in (0,1) and {βn}∞n=0 and {γn}∞n=0 in [0,1], the following conditions are satisfied: (i) ∞ n=0 αn = ∞, αn → 0; (ii) (1 + βn)γn − 2βn > a, for some a ∈ [0, 1); (iii) ∞ n=0 |αn+1 − αn| < ∞, ∞ n=0 |βn+1 − βn| < ∞ and ∞ n=0 |γn+1 − γn| < ∞. Let {xn}∞n=1 be the composite process defined by ⎧ ⎪⎨ ⎪⎩ zn = γnxn + (1 − γn)T xn yn = βnxn + (1 − βn)T zn xn+1 = αnf(xn) + (1 − αn)yn. Then {xn}∞n=1 converges strongly to a fixed point of T which solves some variational inequality. 1.