### Modified inertial subgradient extragradient method for a variational inequality problem in Hilbert space

#### Abstract

In this paper, we introduce and study strong convergence of modified inertial-type subgradient extragradient method for finding solution of variational inequality problem and common fixed point problem of an infinite family of demimetric mappings in a Hilbert space. Our results substantially improve and generalize some well-known results in the literature.

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S. Akashi, W. Takahashi, Weak convergence theorem for an infinite family of demimetric mappings in a Hilbert space, J. Nonlinear Convex Anal., 10(1) (2016), 2159 - 2169.

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm

for maximal monotone operators in Hilbert space, SIAM J. Optim. 14 (2004), 773 - 782.

F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization

of a nonlinear oscillator with damping, Set-Valued Anal. 9 (2001), 3 - 11.

J.-P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley, New York (1984)

C. Baiocchi, A. Capelo, Variational and Quasivariational Inequalities; Applications to Free Boundary

Problems, Wiley, New York (1984)

H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces,

CMS Books in Mathematics. Springer, New York (2011).

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

L. Q. Dong, Y. Y. Lu, J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality, Optimization 65 (2016), 2217 - 2226.

Z. Chbani, H. Riachi, Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequaliies, Optim. Lett., 7 (2013), 185-206, doi: 10.1007/211590-011-0407-y.

C. E. Chidume, S. Maruster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math. 234 (2010), 861 - 882.

C.E. Chidume, S.I. Ikechukwu and A. Adamu, Inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps,Fixed Point Theory and Applications (2018) 2018:9

Y. Censor, A. Gibali, S. Reich, A von Neumann alternating method for finding common solutions

to variational inequalities, Nonlinear Anal. 75 (2012), 4596 - 4603

Y. Censor, A. Gibali, S. Reich, S. Sabach, Common solutions to variational inequalities, Set-Valued Var. Anal. 20 (2012), 229 - 247.

Y. Censor, Gibali, S Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148 (2011), 318 - 335.

Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the

variational inequality problem in Hilbert space, Optim. Methods Softw. 26 (2011), 827 - 845.

Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich’s extragradient method for the variational

inequality problem in Euclidean space, Optimization 61 (2012), 1119 - 1132.

Q.L. Dong, K.R. Kazmi, R. Ali, X.H. Li, Inertial Krasnosel’skii–Mann type hybrid algorithms

for solving hierarchical fixed point problems, J. Fixed Point Theory Appl. 21, (2019), 57.

K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings.

Marcel Dekker, New York (1984)

A.R. Khan, G.C. Ugwunnadi, Z. G. Makukula and M. Abbas, Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space, Carpathian J. Math., 35 (2019), 327 - 338.

G. Kassay, S. Reich, S. Sabach, Iterative methods for solving systems of variational inequalities in

reflexive Banach spaces, SIAM J. Optim. 21 (2011), 1319 - 1344.

F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mapping in Banach space, SIAM. J. Optim. 19 (2008), 824 - 835.

F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), 166 - 177.

H. Komiya and W. Takahashi, Strong convergence theorem for an infinite family of demimetric mappings in a Hilbert space, J. Convex Analysis, 24 (2017), 1357 - 1373.

G. M. Korpelevi$check{c}$, An extragradient method for finding saddle points and for other problems. $grave{E}$kon. Mat. Metody 12 (1976), 747 - 756. (In Russian)

R. Kraikaew and S. Saejung, Strong Convergence of the Halpern Subgradient Extragradient Method for sovlving Variational Inequalities in Hilbert Spaces, J. Optim. Theory Appl. 163 (2014), 399 - 412.

P. E. Maing$acute{e}$, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899 - 912.

M.O. Osilike and D.I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl., 40 (2000), 559 - 567.

M. O. Osilike and F. O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Anal., 74 (2011), 1814 - 1822.

B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4 (5) (1964), 1 - 17.

Y. Song, Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces,J. Nonlinear Sci. Appl., 11 (2018), 198 - 217

G. Stampacchi; Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sciences, Paris, 258 (1964), 4413 - 4416.

W. Takahashi, Fixed point theorems for new nonlinear mapping in Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79 - 88.

W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama (2009).

W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Analysis, 24 (2017), 1015 - 1028.

W. Takahashi, C. F. Wen and J. C. Yao, The shrinking projection method for a finite family of demimetric mapping with variational inequality problems in a Hilbert space, Fixed Point Theory, 19 (2018), 407 - 420.

D. V. Thong and D. V. Hieu, Modified subgradient extragradient method for variational inequality problems, Numer. Algor. 79 (2018), 597 - 610.

D. V. Thong, N. T. Vinh, Y. J. Cho, A strong convergence theorem for Tseng's extragradient method for solving variational inequality problems,Optimization Letters, doi:10.1007/s11590-019-01391-3

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