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doi: 10.3934/jimo.2021133
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Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

1. 

Institute of Theoretical and Applied Research, Hanoi, 100000

2. 

Faculty of Information Technology, Duy Tan University, Da Nang, 550000, Vietnam

3. 

Vietnam Academy of Science and Technology, Institute of Information Technology, 18, Hoang Quoc Viet, Hanoi, Vietnam

Received  December 2020 Revised  April 2021 Early access September 2021

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

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.

Citation: Nguyen Buong. Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021133
References:
[1]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[2]

T. O. AlakoyaA. TaiwoO. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type, Ann. Univ. Ferrara Sez. Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2.  Google Scholar

[3]

A. Aleyner and S. Reich, Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces, J. Math. Anal. Appl., 343 (2008), 427-435.  doi: 10.1016/j.jmaa.2008.01.087.  Google Scholar

[4]

S. Antman, The influence of elasticity in analysis: Modern developments, Bull. Amer. Math. Soc., 9 (1983), 267-291.  doi: 10.1090/S0273-0979-1983-15185-6.  Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011.  Google Scholar

[6]

M. Brook, Y. Censor and A. Gibali, Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning, Intern. Trans. Op. Res., (2020), 1–25. Google Scholar

[7]

N. Buong and L. T. T. Duong, An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 151 (2011), 513-524.  doi: 10.1007/s10957-011-9890-7.  Google Scholar

[8]

N. Buong and N. T. Q. Anh, An implicit iteration method for variational inequqlities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Appl., 2011 (2011), 276859.  doi: 10.1155/2011/276859.  Google Scholar

[9]

N. Buong and N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399-411.  doi: 10.1007/s10957-013-0350-4.  Google Scholar

[10]

N. BuongN. T. H. Phuong and N. T. T. Thuy, Explicit iteration methods for a class of variational inequalities in Banach spaces, Russian Math. (Iz. VUZ), 59 (2015), 16-22.  doi: 10.3103/S1066369X15100023.  Google Scholar

[11]

N. BuongN. S. Ha and N. T. T. Thuy, A new explicit iteration method for a class of variational inequalities, Numer. Algorithms, 72 (2016), 467-481.  doi: 10.1007/s11075-015-0056-9.  Google Scholar

[12]

N. BuongV. X. Quynh and N. T. T. Thuy, A steepest-descent Krasnosel'skii - Mann algorithm for a class of variational inequalities in Banach spaces, J. Fixed Point Theory Appl., 18 (2016), 519-532.  doi: 10.1007/s11784-016-0290-3.  Google Scholar

[13]

N. Buong, Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces, Numer. Algorithms, 76 (2017), 783-798.  doi: 10.1007/s11075-017-0282-4.  Google Scholar

[14]

N. Buong, Steepest - descent proximal point algorithms for a class of variational inequalities in Banach spaces, Math. Nachr., 291 (2018), 1191-1207.  doi: 10.1002/mana.201600240.  Google Scholar

[15]

N. Buong and P. T. T. Hoai, Iterative methods for zeros of a monotone variational inclusion in Hilbert spaces, Calcolo, 55 (2018), 18 pp. doi: 10.1007/s10092-018-0250-y.  Google Scholar

[16]

N. BuongT. V. Dinh and T. T. Huong, New algorithms for a class of accretive variational inequalities in Banach spaces, Acta Math. Vietnamica, 45 (2020), 767-781.  doi: 10.1007/s40306-019-00355-0.  Google Scholar

[17]

N. BuongN.Q. Anh and K. T. Binh, Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces, Filomat, 34 (2020), 1557-1569.  doi: 10.2298/FIL2005557B.  Google Scholar

[18]

A. CegielskiS. Reich and R. Zalas, Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators, Optimization, 69 (2020), 605-636.  doi: 10.1080/02331934.2019.1598407.  Google Scholar

[19]

A. CegielskiA. GibaliS. Reich and R. Zalas, Outer approximation methods for solving variational inequalities over the solution set of a split convex feasibility problem, Numer. Funct. Anal. Optim., 41 (2020), 1089-1108.  doi: 10.1080/01630563.2020.1737938.  Google Scholar

[20]

A. CegielskiS. Reich and R. Zalas, Regular sequences of quasi-nonexpansive operators and their applications, SIAM J. Optim., 28 (2018), 1508-1532.  doi: 10.1137/17M1134986.  Google Scholar

[21]

A. Cegielski and F. Al-Musallam, Strong convergence of a hybrid steepest descent method for the split common fixed point problem, Optimization, 65 (2016), 1463-1476.  doi: 10.1080/02331934.2016.1147038.  Google Scholar

[22]

A. Cegielski, General method for solving the split common fixed point problem, J. Optim. Theory Appl., 165 (2015), 385-304.  doi: 10.1007/s10957-014-0662-z.  Google Scholar

[23]

A. Cegielski, Landweber-type operator and its properties, A Panorama of Mathematics: Pure and Applied, Contemp. Math., Amer. Math. Soc., Providence, RI, 658 (2016), 139-148.  doi: 10.1090/conm/658/13139.  Google Scholar

[24]

A. Cegielski and R. Zalas, Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 34 (2013), 255-283. doi: 10.1080/01630563.2012.716807.  Google Scholar

[25]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, Vol. 2057, Springer, Heidelberg, 2012.  Google Scholar

[26]

Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert spaces, Optim. Methods Solftw., 26 (2011), 827-845.  doi: 10.1080/10556788.2010.551536.  Google Scholar

[27]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

[28]

G. Fichera, La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent'anni, Accademia Nazionale dei Lincei, 114 (1995), 47-53.   Google Scholar

[29]

A. Gibali, A new non-Lipschitzian projection method for solving variational inequalities in Euclidian spaces, J. Nonlinear Anal. Optim. Theory Appl., 6 (2015), 41-51.   Google Scholar

[30]

A. GibaliS. Reich and R. Zalas, Outer approximation methods for solving variational inequalities in Hilbert spaces, Optimization, 66 (2017), 417-437.  doi: 10.1080/02331934.2016.1271800.  Google Scholar

[31]

A. A. Goldstein, Convex programming in Hilbert space, Bull. Am. Math. Soc., 70 (1964), 709-710.  doi: 10.1090/S0002-9904-1964-11178-2.  Google Scholar

[32]

B. Halpern, Fixed points of nonexpanding maps, Bull. Am. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0.  Google Scholar

[33]

S. He and H. K. Xu, Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators, Fixed Point Theory, 10 (2009), 245-258.   Google Scholar

[34]

S. He, L. Liu and A. Gibali, Self-adaptive iterative method for solving boundedly Lipschitzian and strongly monotone variational inequalities, J. Ineq. Appl., 2018 (2018), 12pp. doi: 10.1186/s13660-018-1941-2.  Google Scholar

[35]

C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 23pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

[36]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

[37]

G. M. Korpelevich, Extragradient method for finding saddle points and for other problems, Èkonom. i Mat. Metody, 12 (1976), 747-756.   Google Scholar

[38]

J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[39]

P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Var. Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar

[40]

P. E. Maingé, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59 (2010), 74-79.  doi: 10.1016/j.camwa.2009.09.003.  Google Scholar

[41]

G. J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J., 29 (1962), 314-346.   Google Scholar

[42]

S. Reich and R. Zalas, A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert spaces, Numer. Algorithms, 72 (2016), 297-323.  doi: 10.1007/s11075-015-0045-z.  Google Scholar

[43]

Y. ShehuX. H. Li and Q. L. Dong, An efficient projection-type method for monotone variational inequalities in Hilbert spaces, Numer. Algorithms, 84 (2020), 365-388.  doi: 10.1007/s11075-019-00758-y.  Google Scholar

[44]

G. Stampacchia, Formes bilineares coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[45]

H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Func. Anal. Optim., 22 (2001), 767-773.  doi: 10.1081/NFA-100105317.  Google Scholar

[46]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

[47]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Stud. Comput. Math., Vol. 8, North-Holland, Amsterdam, 2001,473–504. doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar

[48]

J. Yang and H. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert spaces, Numer. Algorithms, 80 (2019), 741-752.  doi: 10.1007/s11075-018-0504-4.  Google Scholar

[49]

J. YangH. Liu and G. Li, Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities, Numer. Algorithms, 84 (2020), 389-405.  doi: 10.1007/s11075-019-00759-x.  Google Scholar

[50]

L. C. Zeng and J. C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonl. Anal., 64 (2006), 2507-2515.  doi: 10.1016/j.na.2005.08.028.  Google Scholar

[51]

H. Zhou and P. Wang, A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 716-727.  doi: 10.1007/s10957-013-0470-x.  Google Scholar

show all references

References:
[1]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574.  doi: 10.1080/02331934.2020.1723586.  Google Scholar

[2]

T. O. AlakoyaA. TaiwoO. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type, Ann. Univ. Ferrara Sez. Sci. Mat., 67 (2021), 1-31.  doi: 10.1007/s11565-020-00354-2.  Google Scholar

[3]

A. Aleyner and S. Reich, Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces, J. Math. Anal. Appl., 343 (2008), 427-435.  doi: 10.1016/j.jmaa.2008.01.087.  Google Scholar

[4]

S. Antman, The influence of elasticity in analysis: Modern developments, Bull. Amer. Math. Soc., 9 (1983), 267-291.  doi: 10.1090/S0273-0979-1983-15185-6.  Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011.  Google Scholar

[6]

M. Brook, Y. Censor and A. Gibali, Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning, Intern. Trans. Op. Res., (2020), 1–25. Google Scholar

[7]

N. Buong and L. T. T. Duong, An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 151 (2011), 513-524.  doi: 10.1007/s10957-011-9890-7.  Google Scholar

[8]

N. Buong and N. T. Q. Anh, An implicit iteration method for variational inequqlities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Appl., 2011 (2011), 276859.  doi: 10.1155/2011/276859.  Google Scholar

[9]

N. Buong and N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399-411.  doi: 10.1007/s10957-013-0350-4.  Google Scholar

[10]

N. BuongN. T. H. Phuong and N. T. T. Thuy, Explicit iteration methods for a class of variational inequalities in Banach spaces, Russian Math. (Iz. VUZ), 59 (2015), 16-22.  doi: 10.3103/S1066369X15100023.  Google Scholar

[11]

N. BuongN. S. Ha and N. T. T. Thuy, A new explicit iteration method for a class of variational inequalities, Numer. Algorithms, 72 (2016), 467-481.  doi: 10.1007/s11075-015-0056-9.  Google Scholar

[12]

N. BuongV. X. Quynh and N. T. T. Thuy, A steepest-descent Krasnosel'skii - Mann algorithm for a class of variational inequalities in Banach spaces, J. Fixed Point Theory Appl., 18 (2016), 519-532.  doi: 10.1007/s11784-016-0290-3.  Google Scholar

[13]

N. Buong, Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces, Numer. Algorithms, 76 (2017), 783-798.  doi: 10.1007/s11075-017-0282-4.  Google Scholar

[14]

N. Buong, Steepest - descent proximal point algorithms for a class of variational inequalities in Banach spaces, Math. Nachr., 291 (2018), 1191-1207.  doi: 10.1002/mana.201600240.  Google Scholar

[15]

N. Buong and P. T. T. Hoai, Iterative methods for zeros of a monotone variational inclusion in Hilbert spaces, Calcolo, 55 (2018), 18 pp. doi: 10.1007/s10092-018-0250-y.  Google Scholar

[16]

N. BuongT. V. Dinh and T. T. Huong, New algorithms for a class of accretive variational inequalities in Banach spaces, Acta Math. Vietnamica, 45 (2020), 767-781.  doi: 10.1007/s40306-019-00355-0.  Google Scholar

[17]

N. BuongN.Q. Anh and K. T. Binh, Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces, Filomat, 34 (2020), 1557-1569.  doi: 10.2298/FIL2005557B.  Google Scholar

[18]

A. CegielskiS. Reich and R. Zalas, Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators, Optimization, 69 (2020), 605-636.  doi: 10.1080/02331934.2019.1598407.  Google Scholar

[19]

A. CegielskiA. GibaliS. Reich and R. Zalas, Outer approximation methods for solving variational inequalities over the solution set of a split convex feasibility problem, Numer. Funct. Anal. Optim., 41 (2020), 1089-1108.  doi: 10.1080/01630563.2020.1737938.  Google Scholar

[20]

A. CegielskiS. Reich and R. Zalas, Regular sequences of quasi-nonexpansive operators and their applications, SIAM J. Optim., 28 (2018), 1508-1532.  doi: 10.1137/17M1134986.  Google Scholar

[21]

A. Cegielski and F. Al-Musallam, Strong convergence of a hybrid steepest descent method for the split common fixed point problem, Optimization, 65 (2016), 1463-1476.  doi: 10.1080/02331934.2016.1147038.  Google Scholar

[22]

A. Cegielski, General method for solving the split common fixed point problem, J. Optim. Theory Appl., 165 (2015), 385-304.  doi: 10.1007/s10957-014-0662-z.  Google Scholar

[23]

A. Cegielski, Landweber-type operator and its properties, A Panorama of Mathematics: Pure and Applied, Contemp. Math., Amer. Math. Soc., Providence, RI, 658 (2016), 139-148.  doi: 10.1090/conm/658/13139.  Google Scholar

[24]

A. Cegielski and R. Zalas, Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 34 (2013), 255-283. doi: 10.1080/01630563.2012.716807.  Google Scholar

[25]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, Vol. 2057, Springer, Heidelberg, 2012.  Google Scholar

[26]

Y. CensorA. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert spaces, Optim. Methods Solftw., 26 (2011), 827-845.  doi: 10.1080/10556788.2010.551536.  Google Scholar

[27]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

[28]

G. Fichera, La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent'anni, Accademia Nazionale dei Lincei, 114 (1995), 47-53.   Google Scholar

[29]

A. Gibali, A new non-Lipschitzian projection method for solving variational inequalities in Euclidian spaces, J. Nonlinear Anal. Optim. Theory Appl., 6 (2015), 41-51.   Google Scholar

[30]

A. GibaliS. Reich and R. Zalas, Outer approximation methods for solving variational inequalities in Hilbert spaces, Optimization, 66 (2017), 417-437.  doi: 10.1080/02331934.2016.1271800.  Google Scholar

[31]

A. A. Goldstein, Convex programming in Hilbert space, Bull. Am. Math. Soc., 70 (1964), 709-710.  doi: 10.1090/S0002-9904-1964-11178-2.  Google Scholar

[32]

B. Halpern, Fixed points of nonexpanding maps, Bull. Am. Math. Soc., 73 (1967), 957-961.  doi: 10.1090/S0002-9904-1967-11864-0.  Google Scholar

[33]

S. He and H. K. Xu, Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators, Fixed Point Theory, 10 (2009), 245-258.   Google Scholar

[34]

S. He, L. Liu and A. Gibali, Self-adaptive iterative method for solving boundedly Lipschitzian and strongly monotone variational inequalities, J. Ineq. Appl., 2018 (2018), 12pp. doi: 10.1186/s13660-018-1941-2.  Google Scholar

[35]

C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 23pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

[36]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

[37]

G. M. Korpelevich, Extragradient method for finding saddle points and for other problems, Èkonom. i Mat. Metody, 12 (1976), 747-756.   Google Scholar

[38]

J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[39]

P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Var. Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar

[40]

P. E. Maingé, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59 (2010), 74-79.  doi: 10.1016/j.camwa.2009.09.003.  Google Scholar

[41]

G. J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J., 29 (1962), 314-346.   Google Scholar

[42]

S. Reich and R. Zalas, A modular string averaging procedure for solving the common fixed point problem for quasi-nonexpansive mappings in Hilbert spaces, Numer. Algorithms, 72 (2016), 297-323.  doi: 10.1007/s11075-015-0045-z.  Google Scholar

[43]

Y. ShehuX. H. Li and Q. L. Dong, An efficient projection-type method for monotone variational inequalities in Hilbert spaces, Numer. Algorithms, 84 (2020), 365-388.  doi: 10.1007/s11075-019-00758-y.  Google Scholar

[44]

G. Stampacchia, Formes bilineares coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[45]

H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Func. Anal. Optim., 22 (2001), 767-773.  doi: 10.1081/NFA-100105317.  Google Scholar

[46]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.  Google Scholar

[47]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Stud. Comput. Math., Vol. 8, North-Holland, Amsterdam, 2001,473–504. doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar

[48]

J. Yang and H. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert spaces, Numer. Algorithms, 80 (2019), 741-752.  doi: 10.1007/s11075-018-0504-4.  Google Scholar

[49]

J. YangH. Liu and G. Li, Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities, Numer. Algorithms, 84 (2020), 389-405.  doi: 10.1007/s11075-019-00759-x.  Google Scholar

[50]

L. C. Zeng and J. C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonl. Anal., 64 (2006), 2507-2515.  doi: 10.1016/j.na.2005.08.028.  Google Scholar

[51]

H. Zhou and P. Wang, A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 716-727.  doi: 10.1007/s10957-013-0470-x.  Google Scholar

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
$k$ $x_1^{k+1}$ $ 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
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
$k$ $x_1^{k+1}$ $ 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
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