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

Abstract

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=tf(xt)+(1−t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitrary x0∈C, let the sequence {xn} be defined iteratively by yn=αnf(xn)+(1−αn)Txn, n≥0, xn+1=βnxn+(1−βn)yn, n≥0. Then {xn} converges strongly to a fixed point of T.