Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

✍ Scribed by Habtu Zegeye; Naseer Shahzad

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 218 KB
• Volume: 68
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.01.027

✦ Synopsis

Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping J ϕ for some gauge ϕ. Let T i : K → K , i = 1, 2, . . . , be a family of quasi-nonexpansive mappings with F := ∩ i≥1 F(T i ) = ∅ which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x 0 ∈ K , let {x n } be generated by the algorithm

where f : K → K is a contraction mapping and {α n } ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {x n } satisfies condition (A). Then it is proved that {x n } converges strongly to a common fixed point x = Q f ( x) of a family T i , i = 1, 2, . . . . Moreover, x is the unique solution in F to a certain variational inequality.