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doi: 10.3934/naco.2021037
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Inertial method for split null point problems with pseudomonotone variational inequality problems

1. 

School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

2. 

Mathematics Institute African University of Science and Technology, Abuja Nigeria

3. 

Department of Mathematics, Federal University Dutsinma, Kastina State Nigeria

4. 

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94 Medunsa 0204, Pretoria, South Africa

*Corresponding author: Christian Chibueze Okeke

Received  April 2021 Revised  September 2021 Early access September 2021

This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.

Citation: Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021037
References:
[1]

H. AttouchJ. Peypouquet and P. Redont, A dynamical approach to an inertial forward backward algorithm for convex minimization, SIAM J. Optim., 24 (2014), 232-256.  doi: 10.1137/130910294.

[2]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[3]

R. I. Bot E. R. Csetnek, An inertial alternating direction method of multipliers, Minimax Theory Appl., 1, (2016) 29-49.

[4]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. doi: 10.1088/0266-5611/18/2/310.

[5]

C. ByrneY. CensorA. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775. 

[6]

L.C. CengA. PetruselX. Qin and J. C. Yao, A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.

[7]

L. C. CengA. PetruselJ. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39.

[8]

L. C. CengA. PetruselX. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832.

[9]

L. C. Ceng, and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., 33 (2020), 19 pp. doi: 10.1186/s13660-020-2306-1.

[10]

L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203.

[11]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[12]

R. W. Cottle and J. C. Yao, Pseudomonotone of complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[13]

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

[14]

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

[15]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.

[16]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[17]

S. V. DensisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.

[18]

M. Eslamian and J. Vahidi, Split common fixed point problem of nonexpansive semigroup, Mediterr J. Math., 13 (2016), 1177-1195.  doi: 10.1007/s00009-015-0541-3.

[19]

A. Gibali, Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Var. Anal., 2 (2018), 49-61. 

[20]

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

[21]

B. S. HeZ. H. Yang and X. M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374.  doi: 10.1016/j.jmaa.2004.04.068.

[22]

A. N. Iusem and B. F. Savaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321.  doi: 10.1080/02331939708844365.

[23]

P. Jailoka and S. Suantai, Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim. Methods Softw., 34 (2019), 248-263.  doi: 10.1080/10556788.2017.1359265.

[24]

S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.

[25]

E. V. Khoroshilova, Extragradient-type method for optimal control problem with linear constraints and convex objective function, Optim. Lett., 7 (2013), 1193-1214.  doi: 10.1007/s11590-012-0496-2.

[26]

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

[27]

P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.

[28]

P. E. Mainge and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168.  doi: 10.1007/s10957-016-0972-4.

[29]

Y. V. Malitsky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 148 (2014), 318-355.  doi: 10.1007/s10559-014-9614-8.

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an Extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[32]

M. S. Nikol'skii, Convergence of the gradient projection method in optimal control problems, Comp. Math. Model., 18 (2007), 148-156.  doi: 10.1007/s10598-007-0015-y.

[33]

M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.

[34]

C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 223-248. 

[35]

C. C. Okeke and O. T. Mewomo, Regularized gradient-projection algorithm for solving one-parameter nonexpansive semigroup, constrained convex minimization and generalized equilibrium problems, Buletinul Academiei Ştiiţe Republicii Moldova Matematica, 88 (2018), 32-56.

[36]

S. Reich, D. V. Thong, Q. Li Dong, X-H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algo., 87 (2021), 52-549. doi: 10.1007/s11075-020-00977-8.

[37]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.

[38]

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

[39]

S. SuantaiN. Pholasa and P. Cholamijiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.

[40]

S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), 281-287.  doi: 10.1080/02331934.2015.1020943.

[41]

W. Takahashi, The split common null point problem in Banach spaces, Arch Math., 104 (2015), 357-365.  doi: 10.1007/s00013-015-0738-5.

[42]

D. V. ThongY. Shehu and O. S. Iyiola, A new iterative methods for solving pseudomonotone variational inequality with non-Lipschitz operators, Comp. Appl. Math., 39 (2020), 108.  doi: 10.1007/s40314-020-1136-6.

[43]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[44]

P. T. Vuong, On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities, J. Optim. Theory Appl., 176 (2018), 399-409.  doi: 10.1007/s10957-017-1214-0.

[45]

P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms, 81 (2019), 26-291.  doi: 10.1007/s11075-018-0547-6.

[46]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.

[47]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019). doi: 10.1007/s10092-018-0300-5.

[48]

Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algo., 76 (2017), 259-282.  doi: 10.1007/s11075-016-0253-1.

[49]

S. SuantaiY. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problems in Banach spaces, Optim. Meth. Softw., 34 (2019), 853-874.  doi: 10.1080/10556788.2018.1472257.

[50]

K. SitthithakerngkietJ. DeephoJ. Martínez-Moreno and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems, Numer. Algorithms, 79 (2018), 801-824.  doi: 10.1007/s11075-017-0462-2.

[51]

Y. YaoG. Marino and L. Muglia, A modified Korpelevich's method convergent to minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569.  doi: 10.1080/02331934.2012.674947.

[52]

Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[53]

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

[54]

H. K. Xu, A variable Krasonosel'skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006) 2021-2034. doi: 10.1088/0266-5611/22/6/007.

[55]

H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 105018, 17 p. doi: 10.1088/0266-5611/26/10/105018.

[56]

T. Y. Zhao, D. Q. Wang, L. C. Ceng et al., Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90. doi: 10.1007/s11075-018-0583-2.

show all references

References:
[1]

H. AttouchJ. Peypouquet and P. Redont, A dynamical approach to an inertial forward backward algorithm for convex minimization, SIAM J. Optim., 24 (2014), 232-256.  doi: 10.1137/130910294.

[2]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.

[3]

R. I. Bot E. R. Csetnek, An inertial alternating direction method of multipliers, Minimax Theory Appl., 1, (2016) 29-49.

[4]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. doi: 10.1088/0266-5611/18/2/310.

[5]

C. ByrneY. CensorA. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775. 

[6]

L.C. CengA. PetruselX. Qin and J. C. Yao, A Modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21 (2020), 93-108.  doi: 10.24193/fpt-ro.

[7]

L. C. CengA. PetruselJ. C. Yao and Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501.  doi: 10.24193/fpt-ro.2018.2.39.

[8]

L. C. CengA. PetruselX. Qin and J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337-1358.  doi: 10.1080/02331934.2020.1858832.

[9]

L. C. Ceng, and M. Shang, Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems, J. Inequal. Appl., 33 (2020), 19 pp. doi: 10.1186/s13660-020-2306-1.

[10]

L. C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70 (2021), 715-740.  doi: 10.1080/02331934.2019.1647203.

[11]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[12]

R. W. Cottle and J. C. Yao, Pseudomonotone of complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295.  doi: 10.1007/BF00941468.

[13]

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

[14]

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

[15]

Y. CensorA. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132.  doi: 10.1080/02331934.2010.539689.

[16]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[17]

S. V. DensisovV. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765.  doi: 10.1007/s10559-015-9768-z.

[18]

M. Eslamian and J. Vahidi, Split common fixed point problem of nonexpansive semigroup, Mediterr J. Math., 13 (2016), 1177-1195.  doi: 10.1007/s00009-015-0541-3.

[19]

A. Gibali, Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Var. Anal., 2 (2018), 49-61. 

[20]

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

[21]

B. S. HeZ. H. Yang and X. M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374.  doi: 10.1016/j.jmaa.2004.04.068.

[22]

A. N. Iusem and B. F. Savaiter, A variant of Korpelevich's method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321.  doi: 10.1080/02331939708844365.

[23]

P. Jailoka and S. Suantai, Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim. Methods Softw., 34 (2019), 248-263.  doi: 10.1080/10556788.2017.1359265.

[24]

S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization, 68 (2019), 2365-2391.  doi: 10.1080/02331934.2019.1638389.

[25]

E. V. Khoroshilova, Extragradient-type method for optimal control problem with linear constraints and convex objective function, Optim. Lett., 7 (2013), 1193-1214.  doi: 10.1007/s11590-012-0496-2.

[26]

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

[27]

P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.

[28]

P. E. Mainge and M. L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168.  doi: 10.1007/s10957-016-0972-4.

[29]

Y. V. Malitsky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 148 (2014), 318-355.  doi: 10.1007/s10559-014-9614-8.

[30]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.

[31]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an Extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.

[32]

M. S. Nikol'skii, Convergence of the gradient projection method in optimal control problems, Comp. Math. Model., 18 (2007), 148-156.  doi: 10.1007/s10598-007-0015-y.

[33]

M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.

[34]

C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 223-248. 

[35]

C. C. Okeke and O. T. Mewomo, Regularized gradient-projection algorithm for solving one-parameter nonexpansive semigroup, constrained convex minimization and generalized equilibrium problems, Buletinul Academiei Ştiiţe Republicii Moldova Matematica, 88 (2018), 32-56.

[36]

S. Reich, D. V. Thong, Q. Li Dong, X-H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numer. Algo., 87 (2021), 52-549. doi: 10.1007/s11075-020-00977-8.

[37]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.

[38]

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

[39]

S. SuantaiN. Pholasa and P. Cholamijiak, The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.

[40]

S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), 281-287.  doi: 10.1080/02331934.2015.1020943.

[41]

W. Takahashi, The split common null point problem in Banach spaces, Arch Math., 104 (2015), 357-365.  doi: 10.1007/s00013-015-0738-5.

[42]

D. V. ThongY. Shehu and O. S. Iyiola, A new iterative methods for solving pseudomonotone variational inequality with non-Lipschitz operators, Comp. Appl. Math., 39 (2020), 108.  doi: 10.1007/s40314-020-1136-6.

[43]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.

[44]

P. T. Vuong, On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities, J. Optim. Theory Appl., 176 (2018), 399-409.  doi: 10.1007/s10957-017-1214-0.

[45]

P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms, 81 (2019), 26-291.  doi: 10.1007/s11075-018-0547-6.

[46]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.

[47]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019). doi: 10.1007/s10092-018-0300-5.

[48]

Y. Shehu and O. S. Iyiola, Strong convergence result for monotone variational inequalities, Numer. Algo., 76 (2017), 259-282.  doi: 10.1007/s11075-016-0253-1.

[49]

S. SuantaiY. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problems in Banach spaces, Optim. Meth. Softw., 34 (2019), 853-874.  doi: 10.1080/10556788.2018.1472257.

[50]

K. SitthithakerngkietJ. DeephoJ. Martínez-Moreno and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems, Numer. Algorithms, 79 (2018), 801-824.  doi: 10.1007/s11075-017-0462-2.

[51]

Y. YaoG. Marino and L. Muglia, A modified Korpelevich's method convergent to minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559-569.  doi: 10.1080/02331934.2012.674947.

[52]

Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[53]

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

[54]

H. K. Xu, A variable Krasonosel'skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl., 22 (2006) 2021-2034. doi: 10.1088/0266-5611/22/6/007.

[55]

H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 105018, 17 p. doi: 10.1088/0266-5611/26/10/105018.

[56]

T. Y. Zhao, D. Q. Wang, L. C. Ceng et al., Inertial method for split null point problems with pseudomonotone variational inequality problems, Numer. Funct. Anal. Appl., 42 (2020), 69-90. doi: 10.1007/s11075-018-0583-2.

Figure 1.  Example 5.1, Top Left: Case I; Top Right: Case II; Bottom: Case III
Figure 2.  Example 5.2, Top Left: Case I; Top Right: Case II; Bottom: Case III
Table 1.  Computational result for Example 5.1
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 7 8 8
CPU time (sec) 0.034 0.0050 0.0034
Algo. 3.3 ($ \alpha =3 $) No of Iter. 13 14 14
CPU time (sec) 0.0109 0.0160 0.0076
NUMA No of Iter. 51 54 58
CPU time (sec) 0.0249 0.0296 0.0288
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 7 8 8
CPU time (sec) 0.034 0.0050 0.0034
Algo. 3.3 ($ \alpha =3 $) No of Iter. 13 14 14
CPU time (sec) 0.0109 0.0160 0.0076
NUMA No of Iter. 51 54 58
CPU time (sec) 0.0249 0.0296 0.0288
Table 2.  Computational result for Example 5.2
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 8 16 90
CPU time (sec) 0.0096 0.0223 0.0425
Algo. 3.3 ($ \alpha =3 $) No of Iter. 8 19 323
CPU time (sec) 0.0035 0.00089 0.4394
NUMA No of Iter. 8 17 69
CPU time (sec) 0.0051 0.0084 0.0257
Case I Case II Case III
Algo. 3.3 ($ \alpha =5 $) No of Iter. 8 16 90
CPU time (sec) 0.0096 0.0223 0.0425
Algo. 3.3 ($ \alpha =3 $) No of Iter. 8 19 323
CPU time (sec) 0.0035 0.00089 0.4394
NUMA No of Iter. 8 17 69
CPU time (sec) 0.0051 0.0084 0.0257
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