# Approximation of Solutions of Monotone Variational Inequality Problems with Applications in Real Hilbert Spaces

## Keywords:

Fixed Point Problem, Hilbert Space, Monotone Mappings, Variational Inequality Problem, Approximation Method## Abstract

In this paper, variational inequality problem which hinges on operators of monotone type is studied, and an iterative algorithm which is a modification of extragradient method is proposed for approximation of solution (assuming existence) of the variational inequality problem. Weak and strong convergence theorems are obtained and as applications, the iterative scheme proposed is shown to also approximate fixed points of pseudocontractive mappings, zeros of monotone mappings and solutions of equilibrium problems. A numerical example is given to show the functionality of the scheme studied. The results obtained improve and unify the corresponding results of several authors.

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## References

\rm \bibitem{Bai} C. Baiocchi and A. Capello \emph{Variational and Quasivariational inequalities. Applications to free Boundary Problems}, Wiley, New York, 1984.

\rm \bibitem{Bau} H. H. Bauschke and P. L. Combettes, \emph{Convex Analysis and Monotone Operator Theory in Hilbert spaces},Springer, New York, 2011.

\bibitem{Blum} E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,

{\it The Mathematics Student} {\bf 63} Nos. 1-4 (1994), 123-145.

\bibitem{Censor} Y. Censor, A. Gibali, and S. Reich, \emph{The subgradient method for solving variational inequalities in Hilbert spaces}, J. Optim. Theory Appl., 148 (2011), pp. 318-335.

\bibitem{Comb} Combettes \rm \bibitem{Fac} F. Facchinei and J. S. Pang, \emph{Finite-Dimensional Variational inequalities and Complementarity Problems}, Springer-Verlag, New York, 2003.

\rm \bibitem{Iusem} A. N. Iusem and B. F. Svaiter, \emph{A variant of Korpelevich's method for variational inequalities with a new search strategy}, Optimization, 42 (1997), pp.309-321.

\rm \bibitem{Kho} E. N. Khobotov, \emph{Modification of the extra-gradient method for solving variational inequalities and certain optimization problems}, USSR Computational Mathematics and Mathematical Physics, Elsevier, 1987, 27, 120-127.

\rm \bibitem{Konnov} I. V. Konnov, \emph{Equilibrium models and variational inequalities}, Elsevier, 2007.

\rm \bibitem{Korp} G. Korpelevich, \emph{The extragradient method for finding saddle points and other problems}, Matecon, 1976, 12, 747-756.

\rm \bibitem{Mali} Y. V. Malitsky, and V. Semenov, \emph{An extragradient algorithm for monotone variational inequalities}, Cybernetics and Systems Analysis, Springer Science \& Business Media, 2014.

\rm \bibitem{Mal} Y. V. Malitsky, \emph{Projected reflected gradient methods for monotone variational inequalities}, Siam J. Optim. vol. 25, No. 1, pp. 502-520.

\rm \bibitem{Marcotte} P. Marcotte, \emph{Application of Khobotov?s algorithm to variational inequalities and network equilibrium problems}, INFOR: Information Systems and Operational Research, Taylor \& Francis, 1991, 29, 258-270.

\bibitem{EY} E. U. Ofoedu, A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory, {\it International Journal of Analysis} {\bf 2013}, Article ID 762831, 18 pages.

\bibitem{EYE} E. U. Ofoedu, Y. Shehu and J. N. Ezeora, Solution by iteration of nonlinear

variational inequalities involving finite family of asymptotically nonexpansive mappings and monotone

mappings, {\it PanAmer. Math. J.} {\bf 18} (2008) (4), 61-75.

\bibitem{Opial} Z. Opial, Weak Convergence of the sequence of successive approximation for nonexpansive mappings, \emph{Bulletin of American Mathematical Society} {\bf 73} (1967) (4), 591-597.

\rm \bibitem{Popov} Popov L. D. Popov, \emph{A modification of the Arrow-Hurwicz method for search of saddle points}, Mathematical notes of the Academy of Sciences of the USSR, Springer, 1980, 28, 845-848.

\rm \bibitem{Solodov} M. V. Solodov, and B. F. Svaiter, \emph{A new projection method for variational inequality problems}, SIAM Journal on Control and Optimization, SIAM, 1999, 37, 765-776.

\rm \bibitem{Sun} \rm \bibitem{Tseng} P. Tseng, \emph{On linear convergence of iterative methods for the variational inequality problem}, Journal of Computational and Applied Mathematics, Elsevier, 1995, 60, 237-252.

\rm \bibitem{Tseng1} P. Tseng, \emph{A modified forward-backward splitting method for maximal monotone mappings} SIAM Journal on Control and Optimization, SIAM, 2000, 38, 431-446.

\rm \bibitem{Zeidler} E. Zeidler, \emph{Nonlinear functional analysis and its applications: III: variational methods and optimization} Springer-Verlag, New York, 1985.

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*MathLAB Journal*,

*2*(1), 17-34. Retrieved from https://purkh.com/index.php/mathlab/article/view/253