Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces

• Main Authors: A. Kaewkhao, K. Sokhuma
• Format: Journal
• Published: 2018
• Subjects: Mathematics
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=79251579274&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/50976
• Institution: Chiang Mai University

Let E be a nonempty compact convex subset of a uniformly convex Banach space X, and let t: E → E and T: E → K C (E) be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that Fix (t) ∩ Fix (T) ≠ θ and Tw = {w} for all w ε Fix (t) ∩ Fix (T). We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary x0 ε by yn = (1 -βn) xn + βn zn, xn + 1 = (1 - n) xn + an ty n, where zn Txn and {an}, {βn} are sequences of positive numbers satisfying 0 < a ≤ an, βn b < 1, converges strongly to a common fixed point of t and T; that is, there exists x ε E such that x = tx ε Tx. Copyright © 2010 K. Sokhuma and A. Kaewkhao.