# Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Open Mathematics (2007)

• Volume: 5, Issue: 2, page 345-357
• ISSN: 2391-5455

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## Abstract

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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 Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = x ∈ K: T 1x = T 2x = T 3 x = xdenotes 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 $\left\{\begin{array}{c}{z}_{n}=P\left({\alpha }_{n}^{\text{'}\text{'}}{T}_{3}{\left(P{T}_{3}\right)}^{n-1}{x}_{n}+\left(1-{\alpha }_{n}^{\text{'}\text{'}}\right){x}_{n}\right),\\ {y}_{n}=P\left({\alpha }_{n}^{\text{'}}{T}_{2}{\left(P{T}_{2}\right)}^{n-1}{z}_{n}+\left(1-{\alpha }_{n}^{\text{'}}\right){x}_{n}\right),\\ {x}_{n+1}=P\left({\alpha }_{n}{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{y}_{n}+\left(1-{\alpha }_{n}\right){x}_{n}\right).\end{array}\right\$ (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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