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doi: 10.3934/jimo.2022075
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Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications

1. 

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

2. 

Department of Mathematics University of Calabar, Calabar Nigeria

3. 

Department of Mathematics University of Eswatini, Private Bag 4 Kwaluseni Eswatini

4. 

Department of Mathematics and Applied Mathematics Sefako Makgatho Health Sciences University, Pretoria South Africa

*Corresponding author: Jacob Ashiwere Abuchu

Received  January 2022 Revised  March 2022 Early access May 2022

In this paper, we study a modified relaxed inertial Mann-type iterative algorithm for solving split monotone variational inclusion problem without prior knowledge of the bounded linear operator norm in real Hilbert spaces. Under some appropriate assumptions on the parameters, we established a strong convergence result of the proposed algorithm. As applications, some special cases of the general problem are given. Finally, we also give some numerical illustrations of the proposed method in comparison with other methods in the literature to show the applicability and advantage of our results.

Citation: Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Ojen Kumar Narain. Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022075
References:
[1]

R. AhmadQ. H. Ansari and S. S. Irfan, Generalized variational inclusions and generalized resolvent equations in Banach spaces, Comput. Math. Appl., 49 (2005), 1825-1835.  doi: 10.1016/j.camwa.2004.10.044.

[2]

M. Alansari1, M. Farid and R. Ali, An iterative scheme for split monotone variational inclusion, variational inequality and fixed point problems, Adv. Difference Equ., (2020), Paper No. 485, 21 pp. doi: 10.1186/s13662-020-02942-0.

[3]

Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Austr. Math. Soc., 59 (1999), 433-442.  doi: 10.1017/S0004972700033116.

[4]

Q. H. Ansari and J. C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optimiz. Theory Appl., 108 (2001), 527-541.  doi: 10.1023/A:1017531323904.

[5]

K. AoyamaY. KimuraW. Takahashi and M. Toyoda, Approximation of common fixed point of a countable family of nonexpansive mapping in a Banach space, Nonlinear Anal., 67 (2007), 2350-2360.  doi: 10.1016/j.na.2006.08.032.

[6]

H. AttouchA. Cabot and Z. Chbani, Inertial forward-Backward algorithms with perturbations: Application to Tikhonov regularization, J. Optim. Theory Appl., 179 (2018), 1-36.  doi: 10.1007/s10957-018-1369-3.

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Noordhoff, 1976.

[8]

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. doi: 10.1007/978-1-4419-9467-7.

[9]

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.

[10]

A. Cegielski, Landweber-type operator and its properties. A panorama of mathematics: Pure and applied, Contemp. Math, 658 (2016), 139-148. 

[11]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unifed 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. 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.

[14]

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

[15]

Y. CensorA. GibaliS. Reich and S. Sabach, Common solutions to variational inequalities, Set-Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x.

[16]

C. S. Chuang, Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem, Optimization, 65 (2016), 859-876.  doi: 10.1080/02331934.2015.1072715.

[17]

P. L. Combettes, The convex feasibility problem in image recovery, Adv Imaging Electron Phys., 95 (1996), 155-270.  doi: 10.1016/S1076-5670(08)70157-5.

[18]

G. Crombez, A geometrical look at iterative methods for operators with fixed points, Numer. Funct. Anal. Optim., 26 (2005), 157-175.  doi: 10.1081/NFA-200063882.

[19]

T. H. CuongJ. C. Yao and N. D. Yen, Qualitative properties of the minimum sum-of-squares clustering problem, Optimization, 69 (2020), 2131-2154.  doi: 10.1080/02331934.2020.1778685.

[20]

Y. DangM. Ang and J. Sun, An inertial triple-projection algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., (2022), 120-132. 

[21]

Y. DangJ. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383-1394.  doi: 10.3934/jimo.2016078.

[22]

J. DeephoP. ThounthongP. Kumam and S. Phiangsungnoen, A new general iterative scheme for split variational inclusion and fixed point problems of k-strict pseudo-contraction mappings with convergence analysis, J. Comput. Appl. Math., 318 (2017), 293-306.  doi: 10.1016/j.cam.2016.09.009.

[23]

Q. L. DongY. J. Cho and T. M. Rassias, The projection and contraction methods for finding common solutions to variational inequality problems, Optim. Lett., 12 (2018), 1871-1896.  doi: 10.1007/s11590-017-1210-1.

[24]

Q. L. 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.

[25]

J. N. Ezeora and C. Izuchukwu, Iterative approximation of solution of split variational inclusion problems, Filomat., 32 (2018), 2921-2932.  doi: 10.2298/FIL1808921E.

[26]

J. L. Guan, L. C. Ceng and B. Hu, Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems, J. Inequal. Appl., 2018 (2018), Paper No. 311, 29 pp.. doi: 10.1186/s13660-018-1905-6.

[27]

D. V. Hieu, P. K. Quy and H. N. Duong, Strong convergence of double-projection method for variational inequality problems, Comput. Appl. Math., 40 (2021), Paper No. 73, 19 pp.. doi: 10.1007/s40314-021-01441-6.

[28]

C. Izuchukwu, J. N. Ezeora and J. Martinez-Moreno, A modified contraction method for solving certain class of split monotone variational inclusion problems with application, Comput. Appl. Math., 39 (2020), Paper No. 188, 20 pp. doi: 10.1007/s40314-020-01221-8.

[29]

C. Izuchukwu, F. O. Isiogugu and C. C. Okeke, A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces, J. Inequal. Appl., (2019), Paper No. 253, 33 pp. doi: 10.1186/s13660-019-2201-9.

[30]

C. IzuchukwuG. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 71 (2022), 583-611.  doi: 10.1080/02331934.2020.1808648.

[31]

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.

[32]

K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett., 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2.

[33]

S. A. Khan, S. Suantai and W. Cholamjiak, Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Math., 113 (2019), 645–656. doi: 10.1007/s13398-018-0504-1.

[34]

P. E. Maing$\acute{e}$, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[35]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[36]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete $CAT(0)$ spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.

[37]

P. Pawicha, K. Poom, J. C. Yeol, J. Wachirapong and S. Kanokwan, The modified inertial iterative algorithm for solving split variational inclusion problem for multi-valued quasi nonexpansive mappings with some applications, http:\www.mdpi.com/journal/mathematics, 7 (2019).

[38]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791-803. 

[39]

R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.  doi: 10.1090/S0002-9947-1970-0282272-5.

[40]

V. V. Semenov, Inertial hybrid splitting methods for operator inclusion problems, Cybernet. Systems Anal., 54 (2018), 936-943. 

[41]

Y. Shehu and D. F. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comp. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.

[42]

Y. Shehu and J. C. Yao, Rate of convergence for inertial iterative method for countable family of certain quasi-nonexpansive mappings, J. Nonlinear Convex Anal., 21 (2020), 533-541. 

[43]

P. Sunthrayuth, N. Pholasa and P. Cholamjiak, Mann-type algorithms for solving the monotone inclusion problem and the fixed point problem in reflexive Banach spaces, Ric. Mat., (2021). doi: 10.1007/s11587-021-00596-y.

[44]

W. Takahashi, Iterative method for the split common fixed points problem in Hilbert spaces, Pure Appl. Funct. Anal., 3 (2018), 349-369. 

[45]

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

[46]

G. C. UgwunnadiA. R. Khan and M. Abbas, Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space, CARPATHIAN J. Math., 35 (2019), 327-338.  doi: 10.37193/CJM.2019.03.07.

[47]

U. WitthayaratY. J. Cho and P. Cholamjiak, On solving split feasibility problems and applications, Ann. Funct. Anal., 9 (2018), 111-122.  doi: 10.1215/20088752-2017-0028.

[48]

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

[49]

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.  doi: 10.1016/j.jmaa.2004.04.059.

[50]

Y. YaoY. C. Liou and R. Chen, Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces, Math. Nachr., 282 (2009), 1827-1835.  doi: 10.1002/mana.200610817.

[51]

Y. Yao and S. Maruster, Strong convergence of an iterative algorithm for variational inequalities in Banach spaces, Math. Comput. Model., 54 (2011), 325-329.  doi: 10.1016/j.mcm.2011.02.016.

[52]

Y. YaoY. ShehuX. H. Li and Q. L. Dong, A method with inertial extrapolation step for split monotone inclusion problems, Optimization, 70 (2021), 741-761.  doi: 10.1080/02331934.2020.1857754.

[53]

T. ZhaoD. WangL. CengL. HeC. Wang and H. Fan, Quasi-inertial Tseng's extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 42 (2020), 69-90.  doi: 10.1080/01630563.2020.1867866.

show all references

References:
[1]

R. AhmadQ. H. Ansari and S. S. Irfan, Generalized variational inclusions and generalized resolvent equations in Banach spaces, Comput. Math. Appl., 49 (2005), 1825-1835.  doi: 10.1016/j.camwa.2004.10.044.

[2]

M. Alansari1, M. Farid and R. Ali, An iterative scheme for split monotone variational inclusion, variational inequality and fixed point problems, Adv. Difference Equ., (2020), Paper No. 485, 21 pp. doi: 10.1186/s13662-020-02942-0.

[3]

Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Austr. Math. Soc., 59 (1999), 433-442.  doi: 10.1017/S0004972700033116.

[4]

Q. H. Ansari and J. C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optimiz. Theory Appl., 108 (2001), 527-541.  doi: 10.1023/A:1017531323904.

[5]

K. AoyamaY. KimuraW. Takahashi and M. Toyoda, Approximation of common fixed point of a countable family of nonexpansive mapping in a Banach space, Nonlinear Anal., 67 (2007), 2350-2360.  doi: 10.1016/j.na.2006.08.032.

[6]

H. AttouchA. Cabot and Z. Chbani, Inertial forward-Backward algorithms with perturbations: Application to Tikhonov regularization, J. Optim. Theory Appl., 179 (2018), 1-36.  doi: 10.1007/s10957-018-1369-3.

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Noordhoff, 1976.

[8]

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. doi: 10.1007/978-1-4419-9467-7.

[9]

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.

[10]

A. Cegielski, Landweber-type operator and its properties. A panorama of mathematics: Pure and applied, Contemp. Math, 658 (2016), 139-148. 

[11]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unifed 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. 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.

[14]

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

[15]

Y. CensorA. GibaliS. Reich and S. Sabach, Common solutions to variational inequalities, Set-Valued Var. Anal., 20 (2012), 229-247.  doi: 10.1007/s11228-011-0192-x.

[16]

C. S. Chuang, Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem, Optimization, 65 (2016), 859-876.  doi: 10.1080/02331934.2015.1072715.

[17]

P. L. Combettes, The convex feasibility problem in image recovery, Adv Imaging Electron Phys., 95 (1996), 155-270.  doi: 10.1016/S1076-5670(08)70157-5.

[18]

G. Crombez, A geometrical look at iterative methods for operators with fixed points, Numer. Funct. Anal. Optim., 26 (2005), 157-175.  doi: 10.1081/NFA-200063882.

[19]

T. H. CuongJ. C. Yao and N. D. Yen, Qualitative properties of the minimum sum-of-squares clustering problem, Optimization, 69 (2020), 2131-2154.  doi: 10.1080/02331934.2020.1778685.

[20]

Y. DangM. Ang and J. Sun, An inertial triple-projection algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., (2022), 120-132. 

[21]

Y. DangJ. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383-1394.  doi: 10.3934/jimo.2016078.

[22]

J. DeephoP. ThounthongP. Kumam and S. Phiangsungnoen, A new general iterative scheme for split variational inclusion and fixed point problems of k-strict pseudo-contraction mappings with convergence analysis, J. Comput. Appl. Math., 318 (2017), 293-306.  doi: 10.1016/j.cam.2016.09.009.

[23]

Q. L. DongY. J. Cho and T. M. Rassias, The projection and contraction methods for finding common solutions to variational inequality problems, Optim. Lett., 12 (2018), 1871-1896.  doi: 10.1007/s11590-017-1210-1.

[24]

Q. L. 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.

[25]

J. N. Ezeora and C. Izuchukwu, Iterative approximation of solution of split variational inclusion problems, Filomat., 32 (2018), 2921-2932.  doi: 10.2298/FIL1808921E.

[26]

J. L. Guan, L. C. Ceng and B. Hu, Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems, J. Inequal. Appl., 2018 (2018), Paper No. 311, 29 pp.. doi: 10.1186/s13660-018-1905-6.

[27]

D. V. Hieu, P. K. Quy and H. N. Duong, Strong convergence of double-projection method for variational inequality problems, Comput. Appl. Math., 40 (2021), Paper No. 73, 19 pp.. doi: 10.1007/s40314-021-01441-6.

[28]

C. Izuchukwu, J. N. Ezeora and J. Martinez-Moreno, A modified contraction method for solving certain class of split monotone variational inclusion problems with application, Comput. Appl. Math., 39 (2020), Paper No. 188, 20 pp. doi: 10.1007/s40314-020-01221-8.

[29]

C. Izuchukwu, F. O. Isiogugu and C. C. Okeke, A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces, J. Inequal. Appl., (2019), Paper No. 253, 33 pp. doi: 10.1186/s13660-019-2201-9.

[30]

C. IzuchukwuG. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 71 (2022), 583-611.  doi: 10.1080/02331934.2020.1808648.

[31]

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.

[32]

K. R. Kazmi and S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett., 8 (2014), 1113-1124.  doi: 10.1007/s11590-013-0629-2.

[33]

S. A. Khan, S. Suantai and W. Cholamjiak, Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Math., 113 (2019), 645–656. doi: 10.1007/s13398-018-0504-1.

[34]

P. E. Maing$\acute{e}$, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[35]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[36]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete $CAT(0)$ spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.

[37]

P. Pawicha, K. Poom, J. C. Yeol, J. Wachirapong and S. Kanokwan, The modified inertial iterative algorithm for solving split variational inclusion problem for multi-valued quasi nonexpansive mappings with some applications, http:\www.mdpi.com/journal/mathematics, 7 (2019).

[38]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Mat. Fiz., 4 (1964), 791-803. 

[39]

R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75-88.  doi: 10.1090/S0002-9947-1970-0282272-5.

[40]

V. V. Semenov, Inertial hybrid splitting methods for operator inclusion problems, Cybernet. Systems Anal., 54 (2018), 936-943. 

[41]

Y. Shehu and D. F. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comp. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.

[42]

Y. Shehu and J. C. Yao, Rate of convergence for inertial iterative method for countable family of certain quasi-nonexpansive mappings, J. Nonlinear Convex Anal., 21 (2020), 533-541. 

[43]

P. Sunthrayuth, N. Pholasa and P. Cholamjiak, Mann-type algorithms for solving the monotone inclusion problem and the fixed point problem in reflexive Banach spaces, Ric. Mat., (2021). doi: 10.1007/s11587-021-00596-y.

[44]

W. Takahashi, Iterative method for the split common fixed points problem in Hilbert spaces, Pure Appl. Funct. Anal., 3 (2018), 349-369. 

[45]

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

[46]

G. C. UgwunnadiA. R. Khan and M. Abbas, Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space, CARPATHIAN J. Math., 35 (2019), 327-338.  doi: 10.37193/CJM.2019.03.07.

[47]

U. WitthayaratY. J. Cho and P. Cholamjiak, On solving split feasibility problems and applications, Ann. Funct. Anal., 9 (2018), 111-122.  doi: 10.1215/20088752-2017-0028.

[48]

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

[49]

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.  doi: 10.1016/j.jmaa.2004.04.059.

[50]

Y. YaoY. C. Liou and R. Chen, Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces, Math. Nachr., 282 (2009), 1827-1835.  doi: 10.1002/mana.200610817.

[51]

Y. Yao and S. Maruster, Strong convergence of an iterative algorithm for variational inequalities in Banach spaces, Math. Comput. Model., 54 (2011), 325-329.  doi: 10.1016/j.mcm.2011.02.016.

[52]

Y. YaoY. ShehuX. H. Li and Q. L. Dong, A method with inertial extrapolation step for split monotone inclusion problems, Optimization, 70 (2021), 741-761.  doi: 10.1080/02331934.2020.1857754.

[53]

T. ZhaoD. WangL. CengL. HeC. Wang and H. Fan, Quasi-inertial Tseng's extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators, Numer. Funct. Anal. Optim., 42 (2020), 69-90.  doi: 10.1080/01630563.2020.1867866.

Figure 1.  Plot of Errors against Iterations number (n): Case Ⅰ
Figure 2.  Plot of Errors against Iterations number (n): Case Ⅱ
Figure 3.  Plot of Errors against Iterations number (n): Case Ⅲ
Figure 4.  Plot of Errors against Iterations number (n): Case Ⅳ
Table 1.  Numerical Result for Example 5.1
Cases Algorithm (1.10) Algorithm 3.2
Case Ⅰ Sec. 32.2782 21.8243
No of Iter. 24 9
Case Ⅱ Sec. 35.5980 11.1867
No of Iter. 24 9
Case Ⅲ Sec. 67.1801 36.5033
No of Iter. 27 10
Case Ⅳ Sec. 58.9075 27.9240
No of Iter. 30 10
Cases Algorithm (1.10) Algorithm 3.2
Case Ⅰ Sec. 32.2782 21.8243
No of Iter. 24 9
Case Ⅱ Sec. 35.5980 11.1867
No of Iter. 24 9
Case Ⅲ Sec. 67.1801 36.5033
No of Iter. 27 10
Case Ⅳ Sec. 58.9075 27.9240
No of Iter. 30 10
[1]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[2]

Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022060

[3]

Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021160

[4]

Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187

[5]

Preeyanuch Chuasuk, Ferdinard Ogbuisi, Yekini Shehu, Prasit Cholamjiak. New inertial method for generalized split variational inclusion problems. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3357-3371. doi: 10.3934/jimo.2020123

[6]

Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021185

[7]

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

[8]

Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2873-2902. doi: 10.3934/jimo.2021095

[9]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383

[10]

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, 2021  doi: 10.3934/naco.2021037

[11]

Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial and Management Optimization, 2022, 18 (1) : 239-265. doi: 10.3934/jimo.2020152

[12]

Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial and Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749

[13]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046

[14]

Yixuan Yang, Yuchao Tang, Meng Wen, Tieyong Zeng. Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Problems and Imaging, 2021, 15 (4) : 787-825. doi: 10.3934/ipi.2021014

[15]

Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021210

[16]

Jiawei Dou, Lan-sun Chen, Kaitai Li. A monotone-iterative method for finding periodic solutions of an impulsive competition system on tumor-normal cell interaction. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 555-562. doi: 10.3934/dcdsb.2004.4.555

[17]

Ken-Ichi Nakamura, Toshiko Ogiwara. Periodically growing solutions in a class of strongly monotone semiflows. Networks and Heterogeneous Media, 2012, 7 (4) : 881-891. doi: 10.3934/nhm.2012.7.881

[18]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial and Management Optimization, 2022, 18 (2) : 773-794. doi: 10.3934/jimo.2020178

[19]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems and Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004

[20]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

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