Approximating fixed points of non-self asymptotically nonexpansive mappings in Banach spaces

Yongfu Su and Xiaolong Qin

International Journal of Stochastic Analysis, 2006, vol. 2006, 1-13

Abstract:

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be an asymptotically nonexpansive mapping with { k n } ⊂ [ 1 , ∞ ) such that ∑ n = 1 ∞ ( k n − 1 ) < ∞ and F ( T ) is nonempty, where F ( T ) denotes the fixed points set of T . Let { α n } , { α n ' } , and { α n '' } be real sequences in (0,1) and ε ≤ α n , α n ' , α n ' ' ≤ 1 − ε for all n ∈ ℕ and some ε > 0 . Starting from arbitrary x 1 ∈ K , define the sequence { x n } by x 1 ∈ K , z n = P ( α n ' ' T ( P T ) n − 1 x n + ( 1 − α n ' ' ) x n ) , y n = P ( α n ' T ( P T ) n − 1 z n + ( 1 − α n ' ) x n ) , x n + 1 = P ( α n T ( P T ) n − 1 y n + ( 1 − α n ) x n ) . (i) If the dual E * of E has the Kadec-Klee property, then { x n } converges weakly to a fixed point p ∈ F ( T ) ; (ii) if T satisfies condition (A), then { x n } converges strongly to a fixed point p ∈ F ( T ) .

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:021961

DOI: 10.1155/JAMSA/2006/21961

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