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Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

The author is supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2017.305

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  • In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.

    Mathematics Subject Classification: Primary: 46N10, 47H09; Secondary: 47J25, 47N10.

    Citation:

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  • Table 1.  Computational results by the first method with (28)-(30)

    $k$ $x_1^{k+1}$ $ x_2^{k+1}$ $x_3^{k+1}$ $ x_4^{k+1}$ $ x_5^{k+1}$
    10 0.0478319229 0.0597899037 0.0717478844 0.0837058651 0.1610527635
    20 0.0065913104 0.0082391379 0.0098869655 0.0115347931 0.0455817835
    30 0.0011746538 0.0014683173 0.0017619807 0.0020556442 0.0239721493
    40 0.0002336513 0.0002920642 0.0003504770 0.0004088898 0.0167508304
    50 0.0000494154 0.0000617693 0.0000741233 0.0000864770 0.0131756678
     | Show Table
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    Table 2.  Computational results by the second method with (30)-(31)

    $k$ $x_1^{k+1}$ $ x_2^{k+1}$ $x_3^{k+1}$ $ x_4^{k+1}$ $ x_5^{k+1}$
    10 0.0162388209 0.0127985261 0.0153582313 0.0179179365 0.0874825391
    20 0.0003020202 0.0003775252 0.0004530303 0.0005285353 0.0325388347
    30 0.0000115214 0.0000144018 0.0000172821 0.0000201625 0.0215356201
    40 0.0000004906 0.0000006132 0.0000007354 0.0000008585 0.0162614503
    50 0.0000000222 0.0000000277 0.0000000333 0.0000000389 0.0130719537
     | Show Table
    DownLoad: CSV
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