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June  2022, 12(2): 255-278. doi: 10.3934/naco.2021004

A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings

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

* Corresponding author: Oluwatosin Temitope Mewomo

Received  May 2020 Revised  January 2021 Published  June 2022 Early access  February 2021

Fund Project: The third author is supported by International Mathematical Union (IMU) Breakout Graduate Fellowship and the fourth author is supported by National Research Foundation (NRF), South Africa, grant 119903

In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm.

Citation: Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004
References:
[1]

H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 6, 20 pp. doi: 10.1287/moor.20.2.449.

[2]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020152.

[3]

T. O. Alakoya, L. 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, 2020. doi: 10.1080/02331934.2020.1723586.

[4]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math., 53 (2020), 208-224.  doi: 10.1515/dema-2020-0013.

[5]

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.

[6]

K. O. AremuH. A. AbassC. Izuchukwu and O. T. Mewomo, A viscosity-type algorithm for an infinitely countable family of $(f, g)$-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Analysis, 40 (2020), 19-37.  doi: 10.1515/anly-2018-0078.

[7]

K. O. Aremu, C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, Multi-step Iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020063.

[8]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejer-monotone method in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.

[9]

R. I. Bot and E. R. Csetnek, A hybrid proximal extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36 (2015), 951-963.  doi: 10.1080/01630563.2015.1042113.

[10]

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.

[11]

L. Q. DongY. Y. Lu and J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality, Optimization, 65 (2016), 2217-2226.  doi: 10.1080/02331934.2016.1239266.

[12]

G. Fichera, Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ, Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34 (1963), 138-142. 

[13]

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. 175. doi: 10.1007/s00025-020-01306-0.

[14]

K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, New York, 1984.

[15]

D. HieuK. P. Anh and L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932.  doi: 10.1007/s10589-019-00093-x.

[16]

D. V. Hieu, Y. J. Cho and Y-B. Xiao, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 2020. doi: 10.1007/s10013-020-00447-7.

[17]

D. V. HieuP. K. Anh and L. D. Muu, An explicit extragradient algorithm for solving variational inequalities, J. Optim. Theory Appl., 158 (2020), 476-503.  doi: 10.1007/s10957-020-01661-6.

[18]

D. V. Hieu, P. K. Anh and L. D. Muu, Strong convergence of subgradient extragradient method with regularization for solving variational inequalities, Optim. Eng., 2020. doi: 10.1007/s11081-020-09540-9.

[19]

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.

[20]

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.

[21]

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 mappings of the demicontractive type, Ukra?n. Mat. Zh., 71 (2019), 1480-1501. 

[22]

C. IzuchukwuG. C. Ugwunnadi and O. T. Mewomo, 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.

[23]

L. O. JolaosoT. O. AlakoyaA. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo Ⅱ, 69 (2020), 711-735.  doi: 10.1007/s12215-019-00431-2.

[24]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 2020. doi: 10.1080/02331934.2020.1716752.

[25]

L. O. Jolaoso, K. O. Oyewole, K. O. Aremu and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math., 2020. doi: 10.1080/00207160.2020.1856823.

[26]

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. doi: 10.1007/s40314-019-1014-2.

[27]

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.  doi: 10.1007/s10957-020-01672-3.

[28]

G. KassayS. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.  doi: 10.1137/110820002.

[29]

S. H. Khan, T. O. Alakoya and O. T. Mewomo, Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach spaces, Math. Comput. Appl., 25 (2020), Art. 54. doi: 10.3390/mca25030054.

[30]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika I Matematicheskie Metody, 12 (1976), 747-756. 

[31]

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.

[32]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[33]

P. E. Maingé and M. L. Gbinddass, 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]

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

[35]

G. Marino and H. K. Xu, Weak and strong convergence theorems for pseudo-contraction in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346.  doi: 10.1016/j.jmaa.2006.06.055.

[36]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241.  doi: 10.1137/050624315.

[37]

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.

[38]

C. C. OkekeC. Izuchukwu and O. T. Mewomo, Strong convergence results for convex minimization and monotone variational inclusion problems in Hilbert space, Rend. Circ. Mat. Palermo (2), 69 (2020), 675-693.  doi: 10.1007/s12215-019-00427-y.

[39]

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

[40]

O. K. Oyewole and O. T. Mewomo, Subgradient extragradient algorithm for solving split equilibrium and fixed point problems in reflexive Banach spaces, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 39, 19 pp. doi: 10.1186/s13660-020-2295-0.

[41]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4 (1964), 1-17. 

[42]

Y. Song and Y. J. Cho, Some note on ishikawa iteration for multivalued mappings, Bull. Korean Math. Soc., 48 (2011), 575-584.  doi: 10.4134/BKMS.2011.48.3.575.

[43]

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

[44]

A. Taiwo, T. 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, 2020. doi: 10.1007/s11075-020-00937-2.

[45]

A. Taiwo, L.O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), Art. 77. doi: 10.1007/s40314-019-0841-5.

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A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert Spaces, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020092.

[47]

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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W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mapping and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560.

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D. V. ThongN. T. Vinh and Y. J. Cho, Accelerated subgradient extragradient methods for variational inequality problems, J. Sci. Comput., 80 (2019), 1438-1462.  doi: 10.1007/s10915-019-00984-5.

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D. V. Thong and D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80-98.  doi: 10.1016/j.cam.2018.03.019.

[53]

D. V. Thong and D. V. Hieu, Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algorithms, 82 (2019), 761-789.  doi: 10.1007/s11075-018-0626-8.

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D. V. Thong, Viscosity approximation method for solving fixed point problems and split common fixed point problems, J. Fixed Point Theory Appl., 16 (2017), 1481-1499.  doi: 10.1007/s11784-016-0323-y.

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M. Tian and B. N. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Ineq. Appl., 2017 (2017), 1-17.  doi: 10.1186/s13660-017-1397-9.

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

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L. C. Zeng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan J. Math., 10 (2006), 1293-1303.  doi: 10.11650/twjm/1500557303.

show all references

References:
[1]

H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 6, 20 pp. doi: 10.1287/moor.20.2.449.

[2]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020152.

[3]

T. O. Alakoya, L. 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, 2020. doi: 10.1080/02331934.2020.1723586.

[4]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math., 53 (2020), 208-224.  doi: 10.1515/dema-2020-0013.

[5]

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.

[6]

K. O. AremuH. A. AbassC. Izuchukwu and O. T. Mewomo, A viscosity-type algorithm for an infinitely countable family of $(f, g)$-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Analysis, 40 (2020), 19-37.  doi: 10.1515/anly-2018-0078.

[7]

K. O. Aremu, C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, Multi-step Iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020063.

[8]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejer-monotone method in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.

[9]

R. I. Bot and E. R. Csetnek, A hybrid proximal extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36 (2015), 951-963.  doi: 10.1080/01630563.2015.1042113.

[10]

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.

[11]

L. Q. DongY. Y. Lu and J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality, Optimization, 65 (2016), 2217-2226.  doi: 10.1080/02331934.2016.1239266.

[12]

G. Fichera, Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ, Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34 (1963), 138-142. 

[13]

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. 175. doi: 10.1007/s00025-020-01306-0.

[14]

K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, New York, 1984.

[15]

D. HieuK. P. Anh and L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932.  doi: 10.1007/s10589-019-00093-x.

[16]

D. V. Hieu, Y. J. Cho and Y-B. Xiao, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 2020. doi: 10.1007/s10013-020-00447-7.

[17]

D. V. HieuP. K. Anh and L. D. Muu, An explicit extragradient algorithm for solving variational inequalities, J. Optim. Theory Appl., 158 (2020), 476-503.  doi: 10.1007/s10957-020-01661-6.

[18]

D. V. Hieu, P. K. Anh and L. D. Muu, Strong convergence of subgradient extragradient method with regularization for solving variational inequalities, Optim. Eng., 2020. doi: 10.1007/s11081-020-09540-9.

[19]

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.

[20]

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.

[21]

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 mappings of the demicontractive type, Ukra?n. Mat. Zh., 71 (2019), 1480-1501. 

[22]

C. IzuchukwuG. C. Ugwunnadi and O. T. Mewomo, 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.

[23]

L. O. JolaosoT. O. AlakoyaA. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo Ⅱ, 69 (2020), 711-735.  doi: 10.1007/s12215-019-00431-2.

[24]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 2020. doi: 10.1080/02331934.2020.1716752.

[25]

L. O. Jolaoso, K. O. Oyewole, K. O. Aremu and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math., 2020. doi: 10.1080/00207160.2020.1856823.

[26]

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. doi: 10.1007/s40314-019-1014-2.

[27]

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.  doi: 10.1007/s10957-020-01672-3.

[28]

G. KassayS. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.  doi: 10.1137/110820002.

[29]

S. H. Khan, T. O. Alakoya and O. T. Mewomo, Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach spaces, Math. Comput. Appl., 25 (2020), Art. 54. doi: 10.3390/mca25030054.

[30]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika I Matematicheskie Metody, 12 (1976), 747-756. 

[31]

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.

[32]

P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.

[33]

P. E. Maingé and M. L. Gbinddass, 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]

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

[35]

G. Marino and H. K. Xu, Weak and strong convergence theorems for pseudo-contraction in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346.  doi: 10.1016/j.jmaa.2006.06.055.

[36]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16 (2006), 1230-1241.  doi: 10.1137/050624315.

[37]

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.

[38]

C. C. OkekeC. Izuchukwu and O. T. Mewomo, Strong convergence results for convex minimization and monotone variational inclusion problems in Hilbert space, Rend. Circ. Mat. Palermo (2), 69 (2020), 675-693.  doi: 10.1007/s12215-019-00427-y.

[39]

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

[40]

O. K. Oyewole and O. T. Mewomo, Subgradient extragradient algorithm for solving split equilibrium and fixed point problems in reflexive Banach spaces, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 39, 19 pp. doi: 10.1186/s13660-020-2295-0.

[41]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4 (1964), 1-17. 

[42]

Y. Song and Y. J. Cho, Some note on ishikawa iteration for multivalued mappings, Bull. Korean Math. Soc., 48 (2011), 575-584.  doi: 10.4134/BKMS.2011.48.3.575.

[43]

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Figure 1.  From Top to Bottom: Case 1 - Case 4
Table 1.  Numerical results
Alg. 3.1 Non-inertial
Case 1 CPU time (sec) 1.9051 2.1221
No of Iter. 4 6
Case 2 CPU time (sec) 1.8332 1.9000
No. of Iter. 4 6
Case 3 CPU time (sec) 1.9193 2.1708
No of Iter. 4 6
Case 4 CPU time (sec) 16.5580 24.9989
No of Iter. 4 5
Alg. 3.1 Non-inertial
Case 1 CPU time (sec) 1.9051 2.1221
No of Iter. 4 6
Case 2 CPU time (sec) 1.8332 1.9000
No. of Iter. 4 6
Case 3 CPU time (sec) 1.9193 2.1708
No of Iter. 4 6
Case 4 CPU time (sec) 16.5580 24.9989
No of Iter. 4 5
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