Abstract
Abstract In this paper, let E be a reflexive and strictly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We consider the hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in E. We prove the strong convergence of this method to a common fixed point of the infinite family of nonexpansive mappings, which solves a variational inequality on their common fixed point set. We also give a weak convergence theorem for the hybrid viscosity approximation method involving an infinite family of nonexpansive mappings in a Hilbert space. MSC:47H17, 47H09, 47H10, 47H05.
Highlights
Let C be a nonempty closed convex subset of a Banach space E, and let T : C → C be a nonlinear mapping
A self-mapping f : C → C is said to be a contraction on C if there exists a constant α in (, ) such that f (x) – f (y) ≤ α x – y, ∀x, y ∈ C
Let T : C → C be a nonexpansive mapping with F(T) = ∅, and let f ∈ C with a contractive constant α in (, )
Summary
Let C be a nonempty closed convex subset of a (real) Banach space E, and let T : C → C be a nonlinear mapping. Reich [ ] extends Browder’s result and proves that if E is a uniformly smooth Banach space, xt converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction u → Q(u) from C onto F(T). Kim and Xu [ ] proposed the following simpler modification of the original Mann’s iterative process: Let C be a nonempty closed convex subset of a Banach space E and T : C → C a nonexpansive mapping such that F(T) = ∅.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
