Iterative solutions via some variants of extragradient approximants in Hilbert spaces

Yasir Arfat; Muhammad Aqeel Ahmad Khan; Poom Kumam; Wiyada Kumam; Kanokwan Sitthithakerngkiet; Yasir Arfat; Muhammad Aqeel Ahmad Khan; Poom Kumam; Wiyada Kumam; Kanokwan Sitthithakerngkiet

doi:10.3934/math.2022768

AIMS Mathematics

2022, Volume 7, Issue 8: 13910-13926. doi: 10.3934/math.2022768

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Iterative solutions via some variants of extragradient approximants in Hilbert spaces

1.
KMUTT Fixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
3.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
4.
Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan
5.
Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand
6.
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok(KMUTNB), 1518, Wongsawang, Bangsue, Bangkok 10800, Thailand

Received: 25 February 2022 Revised: 18 April 2022 Accepted: 25 April 2022 Published: 25 May 2022
MSC : 47H05, 47H10, 47J25, 49M30, 54H25

This paper provides iterative solutions, via some variants of the extragradient approximants, associated with the pseudomonotone equilibrium problem (EP) and the fixed point problem (FPP) for a finite family of $ \eta $-demimetric operators in Hilbert spaces. The classical extragradient algorithm is embedded with the inertial extrapolation technique, the parallel hybrid projection technique and the Halpern iterative methods for the variants. The analysis of the approximants is performed under suitable set of constraints and supported with an appropriate numerical experiment for the viability of the approximants.
- inertial extrapolation technique,
- parallel hybrid projection technique equilibrium problem,
- fixed point problem,
- strong convergence
Citation: Yasir Arfat, Muhammad Aqeel Ahmad Khan, Poom Kumam, Wiyada Kumam, Kanokwan Sitthithakerngkiet. Iterative solutions via some variants of extragradient approximants in Hilbert spaces[J]. AIMS Mathematics, 2022, 7(8): 13910-13926. doi: 10.3934/math.2022768

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Abstract

This paper provides iterative solutions, via some variants of the extragradient approximants, associated with the pseudomonotone equilibrium problem (EP) and the fixed point problem (FPP) for a finite family of $ \eta $-demimetric operators in Hilbert spaces. The classical extragradient algorithm is embedded with the inertial extrapolation technique, the parallel hybrid projection technique and the Halpern iterative methods for the variants. The analysis of the approximants is performed under suitable set of constraints and supported with an appropriate numerical experiment for the viability of the approximants.

References

[1]	P. N. Anh, A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems, B. Malays. Math. Sci. Soc., 36 (2013), 107–116.
[2]	Y. Arfat, P. Kumam, P. S. Ngiamsunthorn, M. A. A. Khan, An inertial based forward-backward algorithm for monotone inclusion problems and split mixed equilibrium problems in Hilbert spaces, Adv. Differ. Equ., 2020 (2020), 453. https://doi.org/10.1186/s13662-020-02915-3 doi: 10.1186/s13662-020-02915-3
[3]	Y. Arfat, P. Kumam, P. S. Ngiamsunthorn, M. A. A. Khan, H. Sarwar, H. F. Din, Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces, Adv. Differ. Equ., 2020 (2020), 512. https://doi.org/10.1186/s13662-020-02956-8 doi: 10.1186/s13662-020-02956-8
[4]	Y. Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn, A. Kaewkhao, An inertially constructed forward-backward splitting algorithm in Hilbert spaces, Adv. Differ. Equ., 2021 (2021), 124. https://doi.org/10.1186/s13662-021-03277-0 doi: 10.1186/s13662-021-03277-0
[5]	Y. Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn, An accelerated projection based parallel hybrid algorithm for fixed point and split null point problems in Hilbert spaces, Math. Meth. Appl. Sci., 2021, 1–19. https://doi.org/10.1002/mma.7405 doi: 10.1002/mma.7405
[6]	Y. Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn, Parallel shrinking inertial extragradient approximants for pseudomonotone equilibrium, fixed point and generalized split null point problem, Ric. Mat., 2021. https://doi.org/10.1007/s11587-021-00647-4 doi: 10.1007/s11587-021-00647-4
[7]	Y. Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn, Shrinking approximants for fixed point problem and generalized split null point problem in Hilbert spaces, Optim. Lett., 2021. https://doi.org/10.1007/s11590-021-01810-4 doi: 10.1007/s11590-021-01810-4
[8]	Y. Arfat, , P. Kumam, M. A. A. Khan, O. S. Iyiola, Multi-inertial parallel hybrid projection algorithm for generalized split null point problems, J. Appl. Math. Comput., 2021. https://doi.org/10.1007/s12190-021-01660-4 doi: 10.1007/s12190-021-01660-4
[9]	Y. Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn, An accelerated Visco-Cesaro means Tseng type splitting method for fixed point and monotone inclusion problems, Carpathian J. Math., 38 (2022), 281–297. https://doi.org/10.37193/CJM.2022.02.02 doi: 10.37193/CJM.2022.02.02
[10]	J. P. Aubin. Optima and equilibria: An introduction to nonlinear analysis, Springer, New York, NY, 1998.
[11]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2 Eds., Springer, Berlin, 2017.
[12]	M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31–43. https://doi.org/10.1007/BF02192244 doi: 10.1007/BF02192244
[13]	E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123–145.
[14]	L. C. Ceng, A. Petrusel, X. Qin, 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. https://doi.org/10.24193/fpt-ro.2020.1.07 doi: 10.24193/fpt-ro.2020.1.07
[15]	L. C. Ceng, A. Petrusel, X. Qin, J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, 70 (2021), 1337–1358. https://doi.org/10.1080/02331934.2020.1858832 doi: 10.1080/02331934.2020.1858832
[16]	L. C. Ceng, Q. Yuan, Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems, Optimization, 70 (2021), 1337–1358. https://doi.org/10.1186/s13660-019-2229-x doi: 10.1186/s13660-019-2229-x
[17]	P. Cholamjiak, Y. Shehu, Inertial forward-backward splitting method in Banach spaces with application to compressed sensing, Appl. Math., 64 (2019), 409–435. https://doi.org/10.21136/AM.2019.0323-18 doi: 10.21136/AM.2019.0323-18
[18]	P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex A., 6 (2005), 117–136.
[19]	P. Daniele, F. Giannessi, A. Maugeri, Equilibrium problems and variational models, Kluwer Academic Publisher, 2003.
[20]	B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957–961. https://doi.org/10.1090/S0002-9904-1967-11864-0 doi: 10.1090/S0002-9904-1967-11864-0
[21]	D. V. Hieu, L. D. Muu, P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, 73 (2016), 197–217. https://doi.org/10.1007/s11075-015-0092-5 doi: 10.1007/s11075-015-0092-5
[22]	D. V. Hieu, H. N. Duong, B. H. Thai, Convergence of relaxed inertial methods for equilibrium problems, J. Appl. Numer. Optim., 3 (2021), 215–229. https://doi.org/10.23952/jano.3.2021.1.13 doi: 10.23952/jano.3.2021.1.13
[23]	M. A. A. Khan, Y. Arfat, A. R. Butt, A shrinking projection approach to solve split equilibrium problems and fixed point problems in Hilbert spaces, Sci. Bull. Ser. A, 80 (2018), 33–46.
[24]	G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekon. Mat. Meto., 12 (1976), 747–756.
[25]	L. Liu, S. Y. Cho, J. C. Yao, Convergence analysis of an inertial Tseng's extragradient algorithm for solving pseudomonotone variational inequalities and applications, J. Nonlinear Var. Anal., 5 (2021), 627–644.
[26]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.1090/S0002-9939-1953-0054846-3 doi: 10.1090/S0002-9939-1953-0054846-3
[27]	A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2003), 447–454. https://doi.org/10.1016/S0377-0427(02)00906-8 doi: 10.1016/S0377-0427(02)00906-8
[28]	F. U. Ogbuisi, O. S. Iyiola, J. M. T. Ngnotchouye, T. M. M. Shumba, On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 2021 (2021). https://doi.org/10.23952/jnfa.2021.4 doi: 10.23952/jnfa.2021.4
[29]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Comput. Math. Math. Phys., 4 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
[30]	T. D. Quoc, L. D. Muu, N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749–776. https://doi.org/10.1007/s10898-011-9693-2 doi: 10.1007/s10898-011-9693-2
[31]	Y. Shehu, P. Cholamjiak, Another look at the split common fixed point problem for demicontractive operators, RACSAM, 110 (2016), 201–218. https://doi.org/10.1007/s13398-015-0231-9 doi: 10.1007/s13398-015-0231-9
[32]	Y. Shehu, P. T. Vuong, P. Cholamjiak, A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems, J. Fixed Point Theory A., 21 (2019), 1–24. https://doi.org/10.1007/s11784-019-0684-0 doi: 10.1007/s11784-019-0684-0
[33]	A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, J. Nonlinear Convex A., 2006,609–617. https://doi.org/10.1007/s10957-007-9187-z doi: 10.1007/s10957-007-9187-z
[34]	W. Takahashi, H. K. Xu, J. C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set Valued Var. Anal., 23 (2015), 205–221. https://doi.org/10.1007/s11228-014-0285-4 doi: 10.1007/s11228-014-0285-4
[35]	W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Conv. Anal., 24 (2017), 1015–1028.
[36]	W. Takahashi, C. F. Wen, J. C. Yao, The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Fixed Point Theory, 19 (2018), 407–419. https://doi.org/10.24193/fpt-ro.2018.1.32 doi: 10.24193/fpt-ro.2018.1.32
[37]	J. Tiel, Convex analysis: An introductory text, Wiley, Chichester, 1984.

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