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

Abstract EN

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∈FS∩FT=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-βnxn-1+βnTxn, n≥1.

Then the sequence xn converges strongly to a common fixed point p of S and T.