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June  2022, 12(2): 373-393. doi: 10.3934/naco.2021011

A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  November 2020 Revised  March 2021 Published  June 2022 Early access  March 2021

Fund Project: The first author acknowledge with thanks the bursary and financial support from African Institute for Mathematical Sciences (AIMS), South Africa. The second author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF CoE-MaSS) Doctoral Bursary. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903)

In this paper, we introduce and study a modified extragradient algorithm for approximating solutions of a certain class of split pseudo-monotone variational inequality problem in real Hilbert spaces. Using our proposed algorithm, we established a strong convergent result for approximating solutions of the aforementioned problem. Our strong convergent result is obtained without prior knowledge of the Lipschitz constant of the pseudo-monotone operator used in this paper, and with minimized number of projections per iteration compared to other results on split variational inequality problem in the literature. Furthermore, numerical examples are given to show the performance and advantage of our method as well as comparing it with related methods in the literature.

Citation: Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 373-393. doi: 10.3934/naco.2021011
References:
[1]

H. A. AbassC. IzuchukwuF. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136. 

[2]

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 (2020), 545-574.  doi: 10.1080/02331934.2020.1723586.

[3]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, A general iterative method for finding common fixed point of finite family of demicontractive mappings with accretive variational inequality problems in Banach spaces, Nonlinear Stud., 27 (2020), 1-24.  doi: 10.1007/s40314-019-1014-2.

[4]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat., 2021. doi: 10.1007/s13370-020-00869-z.

[5]

T. O. Alakoya, L.O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1895154.

[6]

T. O. Alakoya, A. Taiwo, O. 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 mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.

[7]

R. I. BotE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.

[8]

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.  doi: 10.1016/0022-247X(67)90085-6.

[9]

C. Byrne, A unified treatment for some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[10]

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.

[11]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. 

[12]

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

[13]

Y. CensorT. ElfvingN. Kopf and T. Bortfield, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[14]

L. C. CengN. Hadjisavvas and N-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646.  doi: 10.1007/s10898-009-9454-7.

[15]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[16]

C. E. Chidume and M. O. Nnakwe, Iterative algorithms for split variational inequalities and generalized split feasibility problems with applications, J. Nonlinear Var. Anal., 3 (2019), 127-140. 

[17]

H. DehghanC. IzuchukwuO. T. MewomoD. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., 43 (2020), 975-998.  doi: 10.2989/16073606.2019.1593255.

[18]

G. Fichéra, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 34 (1963), 138-142. 

[19]

A. Gibali, L. O. Jolaoso, O. T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math., 75 (2020), Art. No. 179, 36 pp. doi: 10.1007/s00025-020-01306-0.

[20]

E. C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 2020. doi: 10.1007/s40574-020-00.

[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]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.

[23]

C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.

[24]

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), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.

[25]

C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 2020. doi: 10.1080/02331934.2020.1808648.

[26]

C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed point problems for countable families of multivalued type-one demicontractive-type mappings, Ukran. Mat. Zh., 71 (2019), 1480-1501. 

[27]

C. IzuchukwuG.C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.

[28]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Art. No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.

[29]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766. 

[30]

P. D. Khanh and P. T. Vuong, Modified projection method for strongly pseudo-monotone variational inequalities, J. Global Optim., 58 (2014), 341-350. 

[31]

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

[32]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.

[33]

P. E. Maing$\acute{e}$ 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.

[34]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Anal., 72 (2010), 2086-2099.  doi: 10.1016/j.na.2009.10.009.

[35]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 127-152.  doi: 10.36045/bbms/1590199308.

[36]

G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 2021. doi: 10.1007/s11075-021-01081-1.

[37]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, On $\theta$-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math., 53 (2020), 95-111.  doi: 10.1515/dema-2020-0006.

[38]

A. O.-E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 2021. doi: 10.3934/naco.2021004.

[39]

O. K. Oyewole, H. A. Abass and O. T. Mewomo, Strong convergence algorithm for a fixed point constraint split null point problem, Rend. Circ. Mat. Palermo II, 2020. doi: 10.1007/s12215-020-00505-6.

[40]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.

[41]

Y. Song and X. Liu, Convergence comparison of several iteration algorithms for the common fixed point problems, Fixed Point Theory Appl., 2009 (2009), Art. ID 824374, 13 pp. doi: 10.1155/2009/824374.

[42]

G. Stampacchia, Variational inequalities, in Theory and Appli-cations of Monotone Operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy (Edizioni Odersi, Gubbio, Italy, (1968), 102–192.

[43]

A. Taiwo, T. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with applications, Asian-Eur. J. Math.. doi: 10.1142/S1793557121501370.

[44]

A. TaiwoT. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, 86 (2021), 1359-1389.  doi: 10.1007/s11075-020-00937-2.

[45]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Inertial-type algorithm for solving split common fixed-point problem in Banach spaces, J. Sci. Comput., 86 (2021), Art. No. 12. doi: 10.1007/s10915-020-01385-9.

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A. TaiwoL. O. JolaosoO. T. Mewomo and A. Gibali, On generalized mixed equilibrium problem with $\alpha$-$\beta$-$\mu$ bifunction and $\mu$-$\tau$ monotone mapping, J. Nonlinear Convex Anal., 21 (2020), 1381-1401. 

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A. Taiwo, A. O. E. Owolabi, L. O. Jolaoso, O. T. Mewomo and A. Gibali, A new approximation scheme for solving various split inverse problems, Afr. Mat., 2020. doi: 10.1007/s13370-020-00832-y.

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G. C. UgwunnadiC. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT(0) spaces, Afr. Mat., 30 (2019), 151-169.  doi: 10.1007/s13370-018-0633-x.

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show all references

References:
[1]

H. A. AbassC. IzuchukwuF. U. Ogbuisi and O. T. Mewomo, An iterative method for solution of finite families of split minimization problems and fixed point problems, Novi Sad J. Math., 49 (2019), 117-136. 

[2]

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 (2020), 545-574.  doi: 10.1080/02331934.2020.1723586.

[3]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, A general iterative method for finding common fixed point of finite family of demicontractive mappings with accretive variational inequality problems in Banach spaces, Nonlinear Stud., 27 (2020), 1-24.  doi: 10.1007/s40314-019-1014-2.

[4]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat., 2021. doi: 10.1007/s13370-020-00869-z.

[5]

T. O. Alakoya, L.O. Jolaoso, A. Taiwo and O. T. Mewomo, Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, 2021. doi: 10.1080/02331934.2021.1895154.

[6]

T. O. Alakoya, A. Taiwo, O. 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 mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.

[7]

R. I. BotE. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60.  doi: 10.1016/j.ejor.2020.04.035.

[8]

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.  doi: 10.1016/0022-247X(67)90085-6.

[9]

C. Byrne, A unified treatment for some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[10]

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.

[11]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. 

[12]

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

[13]

Y. CensorT. ElfvingN. Kopf and T. Bortfield, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[14]

L. C. CengN. Hadjisavvas and N-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646.  doi: 10.1007/s10898-009-9454-7.

[15]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[16]

C. E. Chidume and M. O. Nnakwe, Iterative algorithms for split variational inequalities and generalized split feasibility problems with applications, J. Nonlinear Var. Anal., 3 (2019), 127-140. 

[17]

H. DehghanC. IzuchukwuO. T. MewomoD. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., 43 (2020), 975-998.  doi: 10.2989/16073606.2019.1593255.

[18]

G. Fichéra, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 34 (1963), 138-142. 

[19]

A. Gibali, L. O. Jolaoso, O. T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math., 75 (2020), Art. No. 179, 36 pp. doi: 10.1007/s00025-020-01306-0.

[20]

E. C. Godwin, C. Izuchukwu and O.T. Mewomo, An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 2020. doi: 10.1007/s40574-020-00.

[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]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66 (2017), 75-96.  doi: 10.1007/s10589-016-9857-6.

[23]

C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.

[24]

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), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.

[25]

C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 2020. doi: 10.1080/02331934.2020.1808648.

[26]

C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed point problems for countable families of multivalued type-one demicontractive-type mappings, Ukran. Mat. Zh., 71 (2019), 1480-1501. 

[27]

C. IzuchukwuG.C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.

[28]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Art. No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.

[29]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766. 

[30]

P. D. Khanh and P. T. Vuong, Modified projection method for strongly pseudo-monotone variational inequalities, J. Global Optim., 58 (2014), 341-350. 

[31]

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

[32]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.

[33]

P. E. Maing$\acute{e}$ 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.

[34]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Anal., 72 (2010), 2086-2099.  doi: 10.1016/j.na.2009.10.009.

[35]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 127-152.  doi: 10.36045/bbms/1590199308.

[36]

G. N. Ogwo, C. Izuchukwu and O. T. Mewomo, Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, 2021. doi: 10.1007/s11075-021-01081-1.

[37]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, On $\theta$-generalized demimetric mappings and monotone operators in Hadamard spaces, Demonstr. Math., 53 (2020), 95-111.  doi: 10.1515/dema-2020-0006.

[38]

A. O.-E. Owolabi, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 2021. doi: 10.3934/naco.2021004.

[39]

O. K. Oyewole, H. A. Abass and O. T. Mewomo, Strong convergence algorithm for a fixed point constraint split null point problem, Rend. Circ. Mat. Palermo II, 2020. doi: 10.1007/s12215-020-00505-6.

[40]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.

[41]

Y. Song and X. Liu, Convergence comparison of several iteration algorithms for the common fixed point problems, Fixed Point Theory Appl., 2009 (2009), Art. ID 824374, 13 pp. doi: 10.1155/2009/824374.

[42]

G. Stampacchia, Variational inequalities, in Theory and Appli-cations of Monotone Operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy (Edizioni Odersi, Gubbio, Italy, (1968), 102–192.

[43]

A. Taiwo, T. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with applications, Asian-Eur. J. Math.. doi: 10.1142/S1793557121501370.

[44]

A. TaiwoT. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, 86 (2021), 1359-1389.  doi: 10.1007/s11075-020-00937-2.

[45]

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Figure 1.  The behavior of $ \mbox{TOL}_n $ with $ \varepsilon = 10^{-3} $ for Example 4.1: Top Left: Case 1; Top Right: Case 2; Bottom Left: Case 3; Bottom Right: Case 4
Figure 2.  The behavior of $ \mbox{TOL}_n $ with $ \varepsilon = 10^{-5} $ for Example 4.2: Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: Case Ⅲ; Bottom Right: Case Ⅳ
Table 1.  Numerical results for Example 4.1 with $ \varepsilon = 10^{-3} $
Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
1 CPU
Iter.
0.8353
10
1.3347
15
1.7686
20
6.0884
37
2 CPU
Iter.
0.8529
8
1.0366
13
1.1735
17
3.7446
30
3 CPU
Iter.
7.3932
11
10.0706
16
11.5588
20
117.5588
38
4 CPU
Iter.
9.1057
15
13.6644
19
14.2521
23
181.3317
45
Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
1 CPU
Iter.
0.8353
10
1.3347
15
1.7686
20
6.0884
37
2 CPU
Iter.
0.8529
8
1.0366
13
1.1735
17
3.7446
30
3 CPU
Iter.
7.3932
11
10.0706
16
11.5588
20
117.5588
38
4 CPU
Iter.
9.1057
15
13.6644
19
14.2521
23
181.3317
45
Table 2.  Numerical results for Example 4.2 with $\varepsilon = 10^{-5}$
Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
1 CPU
Iter.
0.1683
15
1.0294
23
1.0699
36
1.1343
71
2 CPU
Iter.
0.0124
12
0.1060
17
1.0093
34
1.0293
66
3 CPU
Iter.
0.0146
9
0.1125
10
1.0122
29
1.1248
55
4 CPU
Iter.
0.0124
14
1.0118
22
1.1103
35
1.2245
69
Cases Algorithm 3.1 Algorithm (41) Algorithm (11) Algorithm (42)
1 CPU
Iter.
0.1683
15
1.0294
23
1.0699
36
1.1343
71
2 CPU
Iter.
0.0124
12
0.1060
17
1.0093
34
1.0293
66
3 CPU
Iter.
0.0146
9
0.1125
10
1.0122
29
1.1248
55
4 CPU
Iter.
0.0124
14
1.0118
22
1.1103
35
1.2245
69
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