Strong convergence and control condition of modified Halpern iterations in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T ∈ ΓC and f ∈ΠC. Assume that {xt} converges strongly to a fixed point z of T as t→ 0, where xt is the unique element of C which satisfies xt = t f (xt) + (1− t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1) limn→∞αn = 0; (C2) ∑∞ n=0αn =∞; (C6) 0 < liminfn→∞βn ≤ limsupn→∞βn < 1. For arbitrary x0 ∈ C, let the sequence {xn} be defined iteratively by yn = αn f (xn) + (1−αn)Txn, n≥ 0, xn+1 = βnxn + (1− βn)yn, n≥ 0. Then {xn} converges strongly to a fixed point of T .