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Projection methods for solving split equilibrium problems

  • * Corresponding author: dangvanhieu@tdtu.edu.vn

    * Corresponding author: dangvanhieu@tdtu.edu.vn
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  • The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.

    Mathematics Subject Classification: Primary: 65K10, 65K15; Secondary: 90C33.


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  • Figure 1.  Algorithm 1 for $ (m, k) = (30, 20) $ and different sequences of $ \beta_n $. The number of iterations is 360,353,339,360,355,376, respectively

    Figure 2.  Algorithm 1 for (m; k) = (60; 40) and different sequences of βn. The number of iterations is 258,333,336,326,291,293, respectively

    Figure 3.  Algorithm 1 for (m; k) = (100; 50) and different sequences of βn. The number of iterations is 215,236,283,280,321,290, respectively

    Figure 4.  Algorithm 1 for $ (m, k) = (150,100) $ and different sequences of $ \beta_n $. The number of iterations is 161,188,219,209,245,264, respectively

    Figure 5.  Experiment for the algorithms with $ (m, k) = (30, 20) $. The number of iterations is 334,240,379,168,130, respectively

    Figure 6.  Experiment for the algorithms with (m; k) = (60; 40). The number of iterations is 326,221,292,129,108, respectively

    Figure 7.  Experiment for the algorithms with (m; k) = (100; 50). The number of iterations is 308,250,356,114, 89, respectively

    Figure 8.  Experiment for the algorithms with $ (m, k) = (150,100) $. The number of iterations is 254,192,271, 87, 69, respectively

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