A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

Fei Ma; Jun Yang; Min Yin; Fei Ma; Jun Yang; Min Yin

doi:10.3934/math.2022279

AIMS Mathematics

2022, Volume 7, Issue 4: 5015-5028. doi: 10.3934/math.2022279

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A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

School of Mathematics and Statistics, Xianyang Normal University, Xianyang 712000, Shaanxi, China

Received: 18 October 2021 Revised: 06 December 2021 Accepted: 22 December 2021 Published: 29 December 2021
MSC : 65J15, 47C25, 90C33, 90C52

In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.
- variational inequalities,
- subgradient extragradient method,
- fixed point problem,
- Banach space,
- strong convergence
Citation: Fei Ma, Jun Yang, Min Yin. A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces[J]. AIMS Mathematics, 2022, 7(4): 5015-5028. doi: 10.3934/math.2022279

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Abstract

In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.

References

[1]	S. Antonio, Questioni di elasticita nonlinearizzata e semilinearizzata, Rend. Mat. Appl., 18 (1959), 95–139.
[2]	G. Stampacchia, Formes bilinaires coercitives sur les ensembles convexes, C. R. Acad. Sci., 258 (1964), 4413–4416.
[3]	P. Hartman, G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271–310. https://doi.org/10.1007/BF02392210 doi: 10.1007/BF02392210
[4]	J. P. Aubin, I. Ekeland, Applied nonlinear analysis, New York: Wiley, 1984.
[5]	C. Baiocchi, A. Capelo, Variational and quasivariational inequalities, applications to free boundary problems, New York: Wiley, 1984.
[6]	G. M. Korpelevich, The extragradient method for finding saddle points and other problem, Ekonomikai Matematicheskie Metody, 12 (1976), 747–756.
[7]	Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318–335. https://doi.org/10.1007/s10957-010-9757-3 doi: 10.1007/s10957-010-9757-3
[8]	J. Yang, H. W. Liu, Z. X. Liu, Modified subgradient extragradient algorithms for solving monotone variational inequalities, Optimization, 67 (2018), 2247–2258. https://doi.org/10.1080/02331934.2018.1523404 doi: 10.1080/02331934.2018.1523404
[9]	J. Yang, H. W. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer. Algorithms, 80 (2019), 741–752. https://doi.org/10.1007/s11075-018-0504-4 doi: 10.1007/s11075-018-0504-4
[10]	L. Liu, B. Tan, S. Y. Cho, On the resolution of variational inequality problems with a double-hierarchical structure, J. Nonlinear Convex Anal., 21 (2020), 377–386.
[11]	L. Liu, S. Y. Cho, J. C. Yao, Convergence analysis of an inertial Tsengs extragradient algorithm for solving pseudomonotone variational inequalities and applications, J. Nonlinear Var. Anal., 5 (2021), 627–644.
[12]	F. E. Browder, Nonlinear mappings of nonexpansive and accretive-type in Banach spaces, Bull. Amer. Math Soc., 73 (1967), 875–882.
[13]	V. T. Duong, V. H. Dang, Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems, Optimization, 67 (2018), 83–102. https://doi.org/10.1080/02331934.2017.1377199 doi: 10.1080/02331934.2017.1377199
[14]	G. Cai, A. Gibali, O. S. Iyiola, Y. Shehu, A new double-projection method for solving variational inequalities in Banach spaces, J. Optim. Theory Appl., 178 (2018), 219–239. https://doi.org/10.1007/s10957-018-1228-2 doi: 10.1007/s10957-018-1228-2
[15]	S. Chang, C. F. Wen, J. C. Yao, Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces, Optimization, 67 (2018), 1183–1196. https://doi.org/10.1080/02331934.2018.1470176 doi: 10.1080/02331934.2018.1470176
[16]	Y. Yao, M. Postolache, J. C. Yao, Iterative algorithms for pseudomonotone variational inequalities and fixed point problems of pseudocontractive operators, Mathematics, 7 (2019), 1189. https://doi.org/10.3390/math7121189 doi: 10.3390/math7121189
[17]	G. Cai, S. Yekini, O. S. Iyiola, Strong convergence theorems for fixed point problems for strict pseudo-contractions and variational inequalities for inverse-strongly accretive mappings in uniformly smooth Banach spaces, J. Fixed Point Theory Appl., 21 (2019), 41. https://doi.org/10.1007/s11784-019-0677-z doi: 10.1007/s11784-019-0677-z
[18]	L. C. Ceng, A. Petrusel, J. C. Yao, On Mann viscosity subgradient extragradient algorithms for fixed point problems of finitely many strict pseudocontractions and variational inequalities, Mathematics, 7 (2019), 925. https://doi.org/10.3390/math7100925 doi: 10.3390/math7100925
[19]	Y. Liu, H. Kong, Strong convergence theorems for relatively nonexpansive mappings and Lipschitz-continuous monotone mapping in Banach spaces, Indian J. Pure Appl. Math., 50 (2019), 1049–1065. https://doi.org/10.1007/s13226-019-0373-0 doi: 10.1007/s13226-019-0373-0
[20]	S. S. Chang, C. F. Wen, J. C. Yao, Zero point problem of accretive operators in Banach spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), 105–118. https://doi.org/10.1007/s40840-017-0470-3 doi: 10.1007/s40840-017-0470-3
[21]	H. W. Liu, J. Yang, Weak convergence of iterative methods for solving quasimonotone variational inequalities, Comput. Optim. Appl., 77 (2020), 491–508. https://doi.org/10.1007/s10589-020-00217-8 doi: 10.1007/s10589-020-00217-8
[22]	F. Ma, A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces, J. Inequal. Appl., 2020 (2020), 26. https://doi.org/10.1186/s13660-020-2295-0 doi: 10.1186/s13660-020-2295-0
[23]	J. Yang, H. W. Liu, The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space, Optim. Lett., 14 (2020), 1803–1816. https://doi.org/10.1007/s11590-019-01474-1 doi: 10.1007/s11590-019-01474-1
[24]	L. C. Ceng, A. Petrusel, X. Qin, J. C. Yao, Pseudomonotone variational inequalities and fixed point, Fixed Point Theory, 22 (2021), 543–558. https://doi.org/10.24193/fpt-ro.2021.2.36 doi: 10.24193/fpt-ro.2021.2.36
[25]	Y. I. Alber, Metric and generalized projection operator in Banach spaces: Properties and applications, In: Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 1996.
[26]	K. Aoyama, F. Kohsaka, Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings, Fixed Point Theory Appl., 2014 (2014), 95. https://doi.org/10.1186/1687-1812-2014-95 doi: 10.1186/1687-1812-2014-95
[27]	P. E. Mainge, The viscosity approximation process for quasi-nonexpansive mapping in Hilbert space, Comput. Math. Appl., 59 (2010), 74–79. https://doi.org/10.1016/j.camwa.2009.09.003 doi: 10.1016/j.camwa.2009.09.003
[28]	H. K. Xu, Iterative algorithm for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256. https://doi.org/10.1112/S0024610702003332 doi: 10.1112/S0024610702003332
[29]	N. Hadjisavvas, S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90 (1996), 95–111. https://doi.org/10.1007/BF02192248 doi: 10.1007/BF02192248
[30]	K. Rapeepan, S. Satit, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399–412. https://doi.org/10.1007/s10957-013-0494-2 doi: 10.1007/s10957-013-0494-2

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