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An approximation method for continuous pseudocontractive mappings

Abstract

Let be a closed convex subset of a real Banach space, is continuous pseudocontractive mapping, and is a fixed-Lipschitzian strongly pseudocontractive mapping. For any, let be the unique fixed point of. We prove that if has a fixed point and has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then converges to a fixed point of as approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).

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References

  1. Chang S-S, Cho YJ, Zhou H: Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science, New York; 2002:xiv+459.

    MATH  Google Scholar 

  2. Gao J: Modulus of convexity in Banach spaces. Applied Mathematics Letters 2003,16(3):273–278. 10.1016/S0893-9659(03)80043-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Martin RH Jr.: Differential equations on closed subsets of a Banach space. Transactions of the American Mathematical Society 1973, 179: 399–414.

    Article  MathSciNet  MATH  Google Scholar 

  4. Megginson RE: An Introduction to Banach Space Theory, Graduate Texts in Mathematics. Volume 183. Springer, New York; 1998:xx+596.

    Book  MATH  Google Scholar 

  5. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000:iv+276.

    MATH  Google Scholar 

  6. Takahashi W, Ueda Y: On Reich's strong convergence theorems for resolvents of accretive operators. Journal of Mathematical Analysis and Applications 1984,104(2):546–553. 10.1016/0022-247X(84)90019-2

    Article  MathSciNet  MATH  Google Scholar 

  7. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Yisheng Song.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Song, Y., Chen, R. An approximation method for continuous pseudocontractive mappings. J Inequal Appl 2006, 28950 (2006). https://doi.org/10.1155/JIA/2006/28950

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  • DOI: https://doi.org/10.1155/JIA/2006/28950

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