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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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. 

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[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.

[46]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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. 

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[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.

[46]

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

[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.

[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.

[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.

[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.

[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.

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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