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CLC number: O177.91

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Received: 2007-03-13

Revision Accepted: 2007-04-29

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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.10 P.1691~1694

http://doi.org/10.1631/jzus.2007.A1691


Viscosity approximation methods with weakly contractive mappings for nonexpansive mappings


Author(s):  WANG Ya-qin

Affiliation(s):  Mathematics and Sciences College, Shanghai Normal University, Shanghai 200234, China

Corresponding email(s):   wangyaqin0579@sohu.com

Key Words:  Viscosity approximation methods, Weakly contractive mapping, Fixed point, Weakly sequentially continuous duality mapping


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WANG Ya-qin. Viscosity approximation methods with weakly contractive mappings for nonexpansive mappings[J]. Journal of Zhejiang University Science A, 2007, 8(10): 1691~1694.

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%DOI 10.1631/jzus.2007.A1691

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T1 - Viscosity approximation methods with weakly contractive mappings for nonexpansive mappings
A1 - WANG Ya-qin
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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2007.A1691


Abstract: 
Let K be a closed convex subset of a real reflexive Banach space E, T:KK be a nonexpansive mapping, and f:KK be a fixed weakly contractive (may not be contractive) mapping. Then for any t∈(0, 1), let xt(K be the unique fixed point of the weak contraction xtf(x)+(1−t)Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E*, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in (Xu, 2004).

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Reference

[1] Alber, Y.I., Guerre-Delabriere, S., 1997. Principle of weakly contractive maps in Hilbert spaces. Operator Theory, Advances and Applications, 98:7-22.

[2] Chen, R.D., Song, Y.S., Zhou, H.Y., 2006. Viscosity approximation methods for continuous pseudocontractive mappings. Acta Math. Sinica (Chinese Series), 49(6):1275-1278.

[3] Gossez, J.P., Lami Dozo, E., 1972. Some geometric properties related to the fixed point theory for nonexpansive mapping. Pacific J. Math., 40:565-573.

[4] Jung, J.S., 2005. Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl., 302:509-520.

[5] Rhoades, B.E., 2001. Some theorems on weakly contractive maps. Nonl. Anal., 47:2683-2693.

[6] Song, Y.S., Chen, R.D., 2007. Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. Nonl. Anal.: Theory, Methods & Appl., 67(2):486-497.

[7] Xu, H.K., 2004. Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl., 298:279-291.

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