Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces
Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, {Tk}k=1∞:C→C an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂k=1∞Fix(Tk), and f:C→C a contraction. We introduce an explicit iterative algorithm xn+1=αnf(xn)+(1-αn)Lnxn, wher...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/787419 |
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| Summary: | Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, {Tk}k=1∞:C→C an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂k=1∞Fix(Tk), and f:C→C a contraction. We introduce an explicit iterative algorithm xn+1=αnf(xn)+(1-αn)Lnxn, where Ln=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,
and wk>0 with ∑k=1∞ωk=1. Under certain appropriate conditions on {αn}, we prove that {xn} converges strongly to a common fixed point x* of {Tk}k=1∞, which solves the following variational inequality: 〈x*-f(x*),J(x*-p)〉≤0, p∈⋂k=1∞Fix(Tk), where J is the (normalized) duality mapping of X. This algorithm is brief and needs less computational work, since it does not involve W-mapping. |
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| ISSN: | 1110-757X 1687-0042 |