Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 327878, 9 pages.

Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Chang-He Xiang, Jiang-Hua Zhang, and Zhe Chen

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Abstract

Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T:CC is a Lipschitzian mapping, and x*C is a fixed point of T. For given x0C, suppose that the sequence {xn}C is the Mann iterative sequence defined by xn+1=(1-αn)xn+αnTxn,n0, where {αn} is a sequence in [0, 1], n=0αn2<, n=0αn=. We prove that the sequence {xn} strongly converges to x* if and only if there exists a strictly increasing function Φ:[0,)[0,) with Φ(0)=0 such that limsupninfj(xn-x*)J(xn-x*){Txn-x*,j(xn-x*)-xn-x*2+Φ(xn-x*)}0.

Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 327878, 9 pages.

Dates
First available in Project Euclid: 3 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357178244

Digital Object Identifier
doi:10.1155/2012/327878

Mathematical Reviews number (MathSciNet)
MR2979447

Zentralblatt MATH identifier
1325.47135

Citation

Xiang, Chang-He; Zhang, Jiang-Hua; Chen, Zhe. Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings. J. Appl. Math. 2012, Special Issue (2012), Article ID 327878, 9 pages. doi:10.1155/2012/327878. https://projecteuclid.org/euclid.jam/1357178244


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