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Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces

Journal of Optimization Theory and Applications volume 147, pages 27–41 (2010)Cite this article

Abstract

Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an α-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F (T)∩(A+B)−10, where F (T) is the set of fixed points of T and (A+B)−10 is the set of zero points of A+B. Then, we prove a strong convergence theorem, which is different from the results of Halpern’s type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.

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Authors and Affiliations

  1. Yokohama Publishers, 101, 6-27, Satsukigaoka, Aoba-ku, Yokohama, 227-0053, Japan

    S. Takahashi

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    W. Takahashi

  3. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    W. Takahashi

  4. Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida-Shi, Tokyo, 194-8610, Japan

    M. Toyoda

Authors
  1. S. Takahashi
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  2. W. Takahashi
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  3. M. Toyoda
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Correspondence to W. Takahashi.

Additional information

Communicated by J.-C. Yao.

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Takahashi, S., Takahashi, W. & Toyoda, M. Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces. J Optim Theory Appl 147, 27–41 (2010). https://doi.org/10.1007/s10957-010-9713-2

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  • DOI: https://doi.org/10.1007/s10957-010-9713-2

Keywords

  • Nonexpansive mapping
  • Maximal monotone operator
  • Inverse strongly-monotone mapping
  • Fixed point
  • Iteration procedure
  • Equilibrium problem