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New inertial method for generalized split variational inclusion problems

  • * Corresponding author: F. U. Ogbuisi

    * Corresponding author: F. U. Ogbuisi 
The work of the second author is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant Numbers: 111992). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF. This work was supported by Thailand Science Research and Innovation grant no. IRN62W0007 and Thailand Research Fund and University of Phayao grant no. RSA6180084
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  • The purpose of this paper is to introduce a new inertial iterative method for solving split variational inclusion problems in real Hilbert spaces. We prove that the generated sequence converges weakly to the solution of the considered problem under some mild conditions. The major contributions of our results are: (ⅰ) to increase the rate of convergence of the method for solving split variational inclusion problem through the inertial extrapolation step, (ⅱ) to relax the choice of the inertial factor and show the inertial factor can be chosen greater than 1/3 unlike what is previously known before for inertial proximal point method in the literature (ⅲ) to show the numerical efficiency and superiority of our proposed method through some test example.

    Mathematics Subject Classification: Primary: 49J53, 65K10; Secondary: 49M37, 90C25.

    Citation:

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  • Table 1.  The results of Algorithm 3.1 with the difference of $\gamma_{n}$

    NN.P. $\gamma_{n}$ Average iteration Average times
    10 10 $\frac{n}{2(n+1)}$ 1385 36.2813
    10 10 1 734 18.9375
    10 10 2 387 9.5625
    10 10 3 265 6.6094
    10 10 $\frac{3(n+1)}{(n+2)}$ 202 5.0625
     | Show Table
    DownLoad: CSV

    Table 2.  The results computed on Algorithm 3.1 and the method in [47]

    N N.P. Average iteration Average times
    Algorithm 3.1 Method in [47] Algorithm 3.1 Method in [47]
    10 10 202 921 5.0625 30.4219
    20 10 114 14319 3.0781 403.1563
    30 10 72 23114 2.6250 969.2813
    50 10 43 26894 1.5625 1000.0190
    100 10 17 45443 0.4844 3000.4453
     | Show Table
    DownLoad: CSV

    Table 3.  The results of Algorithm 3.1 with the difference of $\theta_{n}$

    $\theta_{n}$ Average iteration Average times
    0 431 4.9200
    0.1 236 2.4900
    0.25 223 2.300
    0.5 80 0.6600
    0.75 71 0.5538
    0.9 54 0.4600
    1 41 0.3700
     | Show Table
    DownLoad: CSV

    Table 4.  The results computed on Algorithm 3.1 and the method in [43]

    N Average iteration Average times
    Algorithm 3.1 Method in [43] Algorithm 3.1 Method in [43]
    10 96 206 0.7400 1.9600
    20 77 150 0.6100 1.3300
     | Show Table
    DownLoad: CSV

    Table 5.  The results computed in Algorithm 3.1 and the method in [44]

    N Average iteration Average times
    Algorithm 3.1 Method in [44] Algorithm 3.1 Method in [44]
    10 38 150 0.7600 4.1100
    20 10 129 0.2200 2.3600
     | Show Table
    DownLoad: CSV
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