Abstract :
Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly
Gâteaux differentiable norm and T :K → K be a nonexpansive mapping with F(T ) := {x ∈ K:
T x = x} = ∅. For a fixed δ ∈ (0, 1), define S :K →K by Sx := (1 − δ)x + δT x, ∀x ∈ K. Assume
that {zt } converges strongly to a fixed point z of T as t →0, where zt is the unique element of K
which satisfies zt = tu+(1−t)T zt for arbitrary u ∈ K. Let {αn} be a real sequence in (0, 1) which
satisfies the following conditions: C1: limαn = 0; C2: αn =∞. For arbitrary x0 ∈ K, let the
sequence {xn} be defined iteratively by
xn+1 = αnu+(1−αn)Sxn.
Then, {xn} converges strongly to a fixed point of T .
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