Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces |
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Authors: | Xiaolong Qin Yongfu Su Meijuan Shang |
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Affiliation: | (1) Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, PR China;(2) Department of Mathematics, Shijiazhuang University, Shijiazhuang, 050035, PR China |
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Abstract: | Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with {k n }, {l n } and {j n }. [1, ∞) such that Σ n=1 ∞ (k n − 1) < ∞, Σ n=1 ∞ (l n − 1) < ∞ and Σ n=1 ∞ (j n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T 1x = T 2x = T 3 x} = x} denotes the common fixed points set of T 1, T 2 and T 3. Let {α n }, {α′ n } and {α″ n } be real sequences in (0, 1) and ∈ ≤ {α n }, {α′ n }, {α″ n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence {x n } by (i) If the dual E* of E has the Kadec-Klee property then {x n } converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x n } converges strongly to a common fixed point p ∈ F. |
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Keywords: | Asymptotically nonexpansive non-self map Kadec-Klee property Uniformly convex |
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