# Variational inequalities and fixed point problems : a survey

## DOI:

https://doi.org/10.14419/ijamr.v3i3.2987## Published:

2014-08-12## Abstract

The variational inequality problem provides a broad unifying setting for the study of optimization, equilibrium and related problems and serves as a useful computational framework for the solution of a host of problems in very diverse applications. Variational inequalities have been a classical subject in mathematical physics, particularly in the calculus of variations associated with the minimization of infinite-dimensional functionals. This paper presents a survey of main results related to variational inequalities and fixed point problems defined on real Hilbert spaces and Banach spaces.

**Keywords**: Fixed Point Problem, Inverse-Strongly-Monotone Mappings, Monotone Mappings, Projection Mappings, Variational Inequality Problem.

## References

A. Bunyawat and S. Suantal, Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings, fixed point theory and applications 2011: 47 (2011). http://dx.doi.org/10.1186/1687-1812-2011-47.

A. Kangtunyakarn, A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces, Fixed Point Theory Appl. (2013).

A. Kangtunyakarn, Iterative scheme for a nonexpansive mapping, an

A. Kangtunyakarn, An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudocontractions and of the set of solutions for a modified system of variational inequalities, Fixed Point Theory Appl. (2013). http://dx.doi.org/10.1186/1687-1812-2013-143.

A. Moudafi, Viscosity approximation methods for fixed point problems, Journal of Mathematical Analysis and Applications 241(1) (2000) 46-55. http://dx.doi.org/10.1006/jmaa.1999.6615.

A. Nagurney, Network economics: a variational inequality approach, Kluwer Academic Publishers, Boston, 1993.

C. Baiocchi, A. Capelo, Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems. J. Wiley, New York, 1984.

Ceng et al., An extragradient-like approximation method for variational inequalities and fixed point problems, fixed point theory and applications, 2011: 22. http://dx.doi.org/10.1186/1687-1812-2011-22.

C. E. Chidume, N. Shahzad, H. Zegeye, Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense, Numer Funct Anal Optim. 25, (2004) 239-257. http://dx.doi.org/10.1081/NFA-120039611.

D. Kinderlehrer, G. Stampacchia, an Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.

D. R. Sahu, H. K. Xu, J. C.Yao, Asymptotically strict pseudocontractive mappings in the intermediate sense, Nonlinear Anal. 70, 3502-3511. http://dx.doi.org/10.1016/j.na.2008.07.007.

F. E. Browder and W. V. Petryshyn, Construction of Fixed Points of non-linear mapping in Hilbert Space, Journal of Mathematics Analysis and Applications, Vol.20, (1967) 197-228. http://dx.doi.org/10.1016/0022-247X (67)90085-6.

F. Liu, M. Z. Nashed, Regularization of Non-linear Ill- posed Variational inequalities and convergence rates,Set-valued Analysis 6 (1998) 313-344. http://dx.doi.org/10.1023/A:1008643727926.

G. E. Kim, T. H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comput Math Appl. 42 (2001) 1565-1570. http://dx.doi.org/10.1016/S0898-1221 (01)00262-0.

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon. 12 (1976) 747-756.

G. Stampaccha, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258(1964) 4413-4416.

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings, Nonlinear Analysis: Theory, Methods and Applications 61(3) (2005) 341-350.

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive nonself mappings and inverse strongly monotone mappings, J of Convex Analysis 11(1) (2004) 69-79.

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications 298(1), (2004) 279-291. http://dx.doi.org/10.1016/j.jmaa.2004.04.059.

H. Wang and Y. Song, An iteration scheme for nonexpansive mappings and variational inequalities, Bull. Korean Math. Soc. 48(5) (2011) 991-1002. http://dx.doi.org/10.4134/BKMS.2011.48.5.991.

H. Zhou, Convergence theorems of fixed points for k-strict pseudocontraction in Hilbert spaces, Nonlinear Anal. 69(2008) 456-462. http://dx.doi.org/10.1016/j.na.2007.05.032.

J. C. Jung, A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces, J of inequalities and applications (2010) 16 pages.

J. Chen, L. Zhang and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, Journal of Mathematical Analysis and Applications 334(2) (2007) 1450-1461. http://dx.doi.org/10.1016/j.jmaa.2006.12.088.

J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mapping, Journal of Mathematical Analysis and Applications 159 (1991) 407-413. http://dx.doi.org/10.1016/0022-247X (91)90245-U.

K. Aoyama, H. Iiduka and W. Takahashi, Strong Convergence of Halpern's sequence for accretive operators in a Banach space, Panamer. Math. J. 17(2007) 75-89.

K. Aoyama, H. Iiduka and W. Takahashi, Weak Convergence of an iterative sequence for accretive operators in a Banach spaces, Fixed Point Theory Appl. (2006) Art. ID 35390. http://dx.doi.org/10.1155/FPTA/2006/35390.

K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360. http://dx.doi.org/10.1016/j.na.2006.08.032.

K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc Am Math Soc 35(1) (1972) 171-174. http://dx.doi.org/10.1090/S0002-9939-1972-0298500-3.

L. C.Ceng, A. Petrusel, C. Lee, M. M. Wong, Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudocontractions, Taiwanese j of Math 13 (2009) 607-632.

L. C. Ceng, C. Y. Wang and J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper, Res. 67 (2008) 375-390. http://dx.doi.org/10.1007/s00186-007-0207-4.

L. C. Ceng, H. K. Xu, J. C. Yao, The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces, nonlinear anal., 69(4) (2008) 1402-1412.

L. C. Ceng and J. C. Yao, extragradient-like approximation method for variational inequalities and fixed point problems, Appl. Math. Comput. 190(2007) 205-215. http://dx.doi.org/10.1016/j.amc.2007.01.021.

L. C. Ceng, L. Abdul and Y. J. Chih, on solutions of a system of variational inequalities and fixed point problems in Banach spaces, Fixed Point Theory Appl. (2013).

L. C. Ceng, N. C. Wong, J. C. Yao, Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption, Comput Math Appl. 56 (2008) 2312-2322. http://dx.doi.org/10.1016/j.camwa.2008.05.002.

L. C. Ceng, Q. H. Ansari, J. C. Yao, Strong and weak convergence theorems for asymptotically strict pseudocontraction mappings in the intermediate sense, J Nonlinear Convex Anal. 11(2) (2008) 283-308.

L. C. Zeng, J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J Math, 10(5) (2006) 1293-1303.

L. C. Zeng and J. C. Yao, extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput, in press.

M. A. Noor, Y. Yao, Three step iterations for variational inequalities and nonexpansive mappings, Appl. Math. Comput. 190(2007) 1312-1321. http://dx.doi.org/10.1016/j.amc.2007.02.013.

M. A. Noor, Y. Yao, R. Chen, Y. C. Liou, An iterative method for fixed point problems and variational inequality problems, Mathematical communications 12(2007) 121-132.

M. Pappalardo, M. Passacantando, Stability for equilibrium problems from variational inequalities to dynamical systems, J.O.T.A. 113 (2002).

M. Shang, S. U. Yongfu, X. Qin, Three step iterations for nonexpansive mappings and inverse strongly monotone mappings, Jri Syst Sci and Complexity (2009) 333-344.

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone Mappings, Journal of Optimization Theory and Applications 128 (2006) 191-201. http://dx.doi.org/10.1007/s10957-005-7564-z.

N. Shahzad, A. Udomene, Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Analysis, 64(2006), 558-567. http://dx.doi.org/10.1016/j.na.2005.03.114.

P. Kumam and P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. 3(2009) 475-486.

P. T. Harker, J. S. Pang, Finite dimensional variational inequalities and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming 48 (1990) 161-220. http://dx.doi.org/10.1007/BF01582255.

R. Chugh, R. Rani, a new iterative scheme for nonexpansive and inverse strongly monotone mappings, International J. of Math. Sci. and Engg. Appls. 8(1) (January, 2014) 189-205.

R. E. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq Math, 65(1993) 169-179.

R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J Nonlinear Convex Anal. 8(1), (2007) 61-79.

R. T. Rockafellar, Monotone operators and proximal point algorithm, SIAM Journal on Control and Optimization 14 (1976) 877-898. http://dx.doi.org/10.1137/0314056.

R. T. Rockafellar, On the Maximality of Sums of nonlinear Monotone Operators," Transactions of the American Mathematical Society 149 (1970) 75-88. http://dx.doi.org/10.1090/S0002-9947-1970-0282272-5.

R. Wangkeeree, U. Kamraksa, A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces, Fixed point theory and applications (2009), Article ID 369215, 23 pages.

S.Matsushita and Kuroiwa, Approximation of fixed points of nonexpansive nonself mappings,Sci. Math. Jpn. 57(2003) 171-176.

S. Y. Cho, Approximation of solutions of a generalized variational inequality problem based on iterative methods, Commun Korean Math. Soc. 25 (2010) 207-214. http://dx.doi.org/10.4134/CKMS.2010.25.2.207.

T. H. Kim, H. K. Xu, Convergence of modified Mann's iteration method for asymptotically strict pseudocontractions, Nonlinear Anal. 68 (2008) 2828-2836. http://dx.doi.org/10.1016/j.na.2007.02.029.

W. Takahashi, Non-linear functional Analysis, Yokohama Publisher, Yokohama Japan, 2000.

W. Takahashi, T. Tamura, Convergence Theorems for a pair of nonexpansive Mapping, Journal of Convex Analysis 5 (1998) 45-56.

W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone Mappings, Journal of Optimization Theory and Applications 118 (2003) 417-428. http://dx.doi.org/10.1023/A:1025407607560.

X. Qin, S. M. Kang, S. U. Yongfu, M. Shang, Strong convergence of an iterative method for variational inequality problems and fixed point problems, Archivum Mathematicum(BRNO) Tomus 45 (2009) 147-158.

X. Qin, Y. J. Cho and S. M. Kang, Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications, J. Comput. Appl. Math. 233(2009) 231-240. http://dx.doi.org/10.1016/j.cam.2009.07.018.

Y. Chen, H. He and R. Chen, Fixed point solutions of variational inequalities for asymptotically pseudocontractive mappings in Banach spaces, Applied Mathematical Sciences 1 (2007) 2735-2745.

Y. C. Liou, Y. Yao, C. W. Tseng, H. T. Lin and P. X. Yang, Iterative algorithms approach to variational inequalities and fixed point problems, Abstract and applied analysis(2012) Article ID 949141, 15 pages.

Y. Qing, S. Y. Cho, X. Qin, Convergence of iterative sequences for strict pseudocontractions and inverse strongly monotone mappings, Advanced modeling and optimization 13, 2(2011).

Y. Yao and J. C. Yao On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186(2007) 1551-1558. http://dx.doi.org/10.1016/j.amc.2006.08.062.

Y. Yao, M. A. Noor, K. I. Noor and Y. C. Loiu, On an iterative algorithm for variational inequalities in Banach spaces, Mathematical communications, 16(2011) 95-104.

Y. Yao and M. A. Noor, on viscosity iterative methods for variational inequalities, J. Math. Anal. Appl. 325(2007) 776-787. http://dx.doi.org/10.1016/j.jmaa.2006.01.091.

Y. Yao, M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed point problems, J Optim Theory Appl (2012)155: 273-287. http://dx.doi.org/10.1007/s10957-012-0055-0.

Y. Yao, Y. C. Liou, J. C.Yao, An extragradient method for fixed point problems and variational inequality problems, J of inequalities and applications (2007) 12 pages.

Z. Opial,Weak convergence of the sequence of successive Approximations of Nonexpensive Mappings, Bulletin of the American Mathematical Society 73 (1967) 591-597. http://dx.doi.org/10.1090/S0002-9904-1967-11761-0.

## License

Authors who publish with this journal agree to the following terms:

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [1]Â Â Â Â Â Â Â Â Â Â Â **Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under aÂ Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [2]Â Â Â Â Â Â Â Â Â Â Â **Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [3]Â Â Â Â Â Â Â Â Â Â Â **Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (SeeÂ The Effect of Open Access).