Abstract
In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized equilibrium problem and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced and finally we apply our results to solving optimization problems and obtaining zeroes of maximal monotone operators and co-coercive mappings.
Similar content being viewed by others
References
Ali, B.: Fixed points approximation and solutions of some equilibrium and variational inequality problems. Int. J. Math. Math. Sci. 2009, p. 13, Article ID 656534 (2009)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Browder F.E.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965)
Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)
Bruck R.E.: On weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)
Bruck, R.E.: Asymptotic behaviour of nonexpansive mappings. In: Sine, R.C. (ed) Contemporary Mathematics, 18, Fixed Points and Nonexpansive Mappings, AMS, Providence (1980)
Byrne C.: A unified treatment of some iterative algorithms in signal processing and image construction. Inverse Probl. 20, 103–120 (2004)
Chang, S.S., Cho, Y.J., Kim, J.K.: Approximation methods of solutions for equilibrium problem in Hilbert spaces. Dynam. Systems Appl. (in print)
Cho Y.J., Qin X., Kang J.I.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009)
Combettes P.L., Hirstoaga S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Flam S.D., Antipin A.S.: Equilibrium programming using proximal-like algorithms. Math. Program 78, 29–41 (1997)
Iiduka H., Takahashi W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005)
Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, Dublin (1980)
Liu, Y.: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. doi:10.1016/j.na.2009.03.060
Lim T.C., Xu H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. TMA 2, 1345–1355 (1994)
Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008)
Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer (1999)
Nadezhkina N., Takahashi W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)
Noor M.A.: General variational inequalities and nonexpansive mappings. J. Math. Anal. Appl. 331, 810–822 (2007)
Peng J.W., Yao J.C.: Strong convergence theorems of an iterative scheme based on extragradient method for mixed equilibrium problems and fixed points problems. Math. Com. Model. 49, 1816–1828 (2009)
Plubtieng S., Punpaeng R.: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 197, 548–558 (2008)
Plubtieng S., Kumam P.: Weak convergence theorems for monotone mappings and countable family of nonexpansive mappings. Com. Appl. Math. 224, 614–621 (2009)
Podilchuk C.I., Mammone R.J.: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. A 7, 517–521 (1990)
Qin, X., Shang, M., Su, Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal (2007). doi:10.1016/j.na.2007.10.025
Qin X., Shang M., Su Y.: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Com. Model. 48, 1033–1046 (2008)
Rockafellar R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Shioji S., Takahashi W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. proc. Am. Math. Soc. 125, 3641–3645 (1997)
Su Y., Shang M., Qin X.: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69, 2709–2719 (2008)
Suzuki T.: Strong convergence of Krasnoselkii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)
Takahashi S., Takahashi W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)
Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–518 (2007)
Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
Wangkeeree, R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. J. Fixed Point Theory Appl. 2008, p. 17, Article ID 134148 (2008)
Xu H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66(2), 1–17 (2002)
Yao Y., Yao J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186(2), 1551–1558 (2007)
Youla D.: On deterministic convergence of iterations of related projection operators. J. Vis. Commun. Image Represent 1, 12–20 (1990)
Yao, Y., Liou, Y., Marino, G.: Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2009, p. 7, Article ID 279058 (2009)
Zhao, J., He, S.: A new iterative method for equilibrium problems and fixed points problems for infinite nonexpansive mappings and monotone mappings. Appl. Math. Com. doi:10.1016/j.amc.2009.05.041
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shehu, Y. Fixed point solutions of variational inequality and generalized equilibrium problems with applications. Ann. Univ. Ferrara 56, 345–368 (2010). https://doi.org/10.1007/s11565-010-0102-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-010-0102-4