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

Author

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  • S. Takahashi

    (Yokohama Publishers)

  • W. Takahashi

    (Tokyo Institute of Technology
    National Sun Yat-sen University)

  • M. Toyoda

    (Tamagawa University)

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.

Suggested Citation

  • S. Takahashi & W. Takahashi & M. Toyoda, 2010. "Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 27-41, October.
    • Handle: RePEc:spr:joptap:v:147:y:2010:i:1:d:10.1007_s10957-010-9713-2
    DOI: 10.1007/s10957-010-9713-2
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    References listed on IDEAS

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    1. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
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    Cited by:

    1. Kim, Jong Kyu & Tuyen, Truong Minh, 2016. "Approximation common zero of two accretive operators in banach spaces," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 265-281.
    2. Prasit Cholamjiak & Suparat Kesornprom & Nattawut Pholasa, 2019. "Weak and Strong Convergence Theorems for the Inclusion Problem and the Fixed-Point Problem of Nonexpansive Mappings," Mathematics, MDPI, vol. 7(2), pages 1-19, February.
    3. W. Takahashi, 2013. "Strong Convergence Theorems for Maximal and Inverse-Strongly Monotone Mappings in Hilbert Spaces and Applications," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 781-802, June.
    4. Mohammad Akram & Mohammad Dilshad & Aysha Khan & Sumit Chandok & Izhar Ahmad, 2023. "Convergence Analysis for Generalized Yosida Inclusion Problem with Applications," Mathematics, MDPI, vol. 11(6), pages 1-19, March.
    5. Wataru Takahashi, 2020. "A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space," Mathematics, MDPI, vol. 8(3), pages 1-15, March.
    6. Mohammad Akram & Mohammad Dilshad & Arvind Kumar Rajpoot & Feeroz Babu & Rais Ahmad & Jen-Chih Yao, 2022. "Modified Iterative Schemes for a Fixed Point Problem and a Split Variational Inclusion Problem," Mathematics, MDPI, vol. 10(12), pages 1-17, June.
    7. Yonghong Yao & Yeong-Cheng Liou & Ngai-Ching Wong, 2013. "Superimposed optimization methods for the mixed equilibrium problem and variational inclusion," Journal of Global Optimization, Springer, vol. 57(3), pages 935-950, November.
    8. Marwan A. Kutbi & Abdul Latif & Xiaolong Qin, 2019. "Convergence of Two Splitting Projection Algorithms in Hilbert Spaces," Mathematics, MDPI, vol. 7(10), pages 1-13, October.
    9. Adamu, A. & Kitkuan, D. & Padcharoen, A. & Chidume, C.E. & Kumam, P., 2022. "Inertial viscosity-type iterative method for solving inclusion problems with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 445-459.

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