Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings.

Let K be a nonempty closed convex subset of a real Banach space E, let S : K → K be nonexpansive, and let T : K → K be Lipschitz strongly pseudocontractive mappings such that p ∈ F(S) ∩ F(T) = {x ∈ K : Sx = Tx = x} and ∥x - Sy∥ ≤ ∥Sx - Sy∥ and ∥x - Ty∥ ≤ ∥Tx - Ty∥ for all x, y, ∈ K. Let {βn} be a sequence in [0, 1] satisfying (i) ∑n = 1∞ βn = ∞; (ii) limn → ∞βn = 0. For arbitrary x0 ∈ K, let {xn} be a sequence iteratively defined by xn = syn, yn = (1 - βn)xn-1 + βnTxn, n ≥ 1. Then the sequence {xn} converges strongly to a common fixed point p of S and T. [ABSTRACT FROM AUTHOR]