A composite iterative algorithm for common fixed points for a finite family of nonexpansive mappings and applications

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk, andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0, p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

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Common Fixed Points Iteration Processes for a Finite Family of Asymptotically Nonexpansive Mappings

Georgian Mathematical Journal ◽

10.1515/gmj.2004.83 ◽

2004 ◽

Vol 11 (1) ◽

pp. 83-92

Author(s):

Jui-Chi Huang

Keyword(s):

Fixed Points ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Nonempty Closed Convex Subset ◽

Convex Banach Space ◽

Finite Family ◽

Uniformly Convex ◽

Asymptotically Nonexpansive ◽

Iteration Processes ◽

Suitable Sequence

Abstract Let 𝐸 be a uniformly convex Banach space which satisfies Opial's condition or its dual 𝐸* has the Kadec–Klee property, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇𝑗 : 𝐶 → 𝐶 an asymptotically nonexpansive mapping for each 𝑗 = 1, 2, . . . , 𝑟. Suppose {𝑥𝑛} is generated iteratively by where 𝑈𝑛(0) = 𝐼, 𝐼 is the identity map and {α 𝑛(𝑗)} is a suitable sequence in [0, 1]. If the set of common fixed points of is nonempty, then weak convergence of {𝑥𝑛} to some is obtained.

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Convergence to common fixed points of a finite family of nonexpansive mappings

Archiv der Mathematik ◽

10.1007/s00013-004-1175-z ◽

2005 ◽

Vol 84 (4) ◽

pp. 350-363 ◽

Cited By ~ 11

Author(s):

Yasunori Kimura ◽

Wataru Takahashi ◽

Masashi Toyoda

Keyword(s):

Fixed Points ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Finite Family

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An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

Fixed Point Theory and Applications ◽

10.1155/2011/276859 ◽

2011 ◽

Vol 2011 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Nguyen Buong ◽

Nguyen Thi Quynh Anh

Keyword(s):

Variational Inequalities ◽

Fixed Points ◽

Hilbert Spaces ◽

Iteration Method ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Finite Family ◽

Implicit Iteration

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An approximation method for common fixed points of a finite family of asymptotic pointwise nonexpansive mappings

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2012-108 ◽

2012 ◽

Vol 2012 (1) ◽

Author(s):

Bancha Nanjaras ◽

Bancha Panyanak

Keyword(s):

Fixed Points ◽

Approximation Method ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Finite Family

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Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

Abstract and Applied Analysis ◽

10.1155/2009/728510 ◽

2009 ◽

Vol 2009 ◽

pp. 1-14 ◽

Cited By ~ 3

Author(s):

S. Imnang ◽

S. Suantai

Keyword(s):

Fixed Points ◽

Banach Spaces ◽

Iterative Scheme ◽

Nonexpansive Mappings ◽

Common Fixed Points ◽

Iteration Scheme ◽

Finite Family ◽

Special Cases

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

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