Strong convergence of modified Halpern's iterations for a k-strictly pseudocontractive mapping

Abstract

In this paper, we discuss three modified Halpern iterations as follows: xn+1 = αnu + (1 - αn)((1 - δ)xn + δTxn), (I) xn+1 = αn((1 - δ)u + δxn) + (1 - αn)Txn, (II) xn+1 = αnu + βnxn + γnTxn, n ≥ 0, (III) and obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where {αn} satisfies the conditions: (C1) limn→∞ αn = 0 and (C2) ∑ ∞ n=1 αn = +∞, respectively. The results presented in this work improve the corresponding ones announced by many other authors. © 2013 Li et al.; licensee Springer.