Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces

• Published in: Journal of inequalities and applications Vol. 2011; no. 1; pp. 1 - 10
• Main Authors: Naraghirad, Eskandar, Wong, Ngai-Ching, Yao, Jen-Chih
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 23.11.2011; Springer Nature B.V; BioMed Central Ltd
• Subjects: Analysis; Applications of Mathematics; Approximation; Asymptotic properties; Convergence; Hilbert space; Inequalities; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Theorems; monotone mapping; hybrid extragradient-like approximation method; variational inequality; asymptotically nonexpansive mapping in the intermediate sense; fixed point; strong convergence
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/1029-242X-2011-119

Let C be a nonempty closed convex subset of a real Hilbert space H . Let S : C → C be an asymptotically nonexpansive map in the intermediate sense with the fixed point set F ( S ). Let A : C → H be a Lipschitz continuous map, and VI ( C, A ) be the set of solutions u ∈ C of the variational inequality ⟨ A u , v - u ⟩ ≥ 0 , ∀ v ∈ C . The purpose of this study is to introduce a hybrid extragradient-like approximation method for finding a common element in F ( S ) and VI ( C, A ). We establish some strong convergence theorems for sequences produced by our iterative method. AMS subject classifications : 49J25; 47H05; 47H09.