Abstract
In this paper, we introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, of the set of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in the framework of real Hilbert spaces. Strong convergence of the proposed iterative algorithm is obtained. As an application, we utilize the main results which improve the corresponding results announced in Chang et al. (Nonlinear Anal, 70:3307–3319, 2009), Colao et al. (J Math Anal Appl, 344:340–352, 2008), Plubtieng and Punpaeng (Appl Math Comput, 197:548–558, 2008) to study the optimization problem.
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Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Ceng L.C., Yao J.C.: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput. 198, 729–741 (2008)
Ceng L.C., Yao J.C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Meth. Oper. Res. 67, 375–390 (2008)
Chang S.S., Lee H.W.J., Chan C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009)
Colao V., Marino G., Xu H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)
Combettes P.L., Hirstoaga S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Deutsch F., Yamada I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)
Iiduka H., Takahashi W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005)
Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008)
Marino G., Xu H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)
Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 595–597 (1967)
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)
Qin X., Shang M., Zhou H.: Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces. Appl. Math. Comput. 200, 242–253 (2008)
Qin X., Shang M., Su Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69, 3897–3909 (2008)
Qin X., Shang M., Su Y.: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Modelling 48, 1033–1046 (2008)
Qin X., Cho Y.J., Kang S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)
Rockafellar R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)
Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J. Math. Anal. Appl. 305, 227–239 (2005)
Su Y., Shang M., Qin X.: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69, 2709–2719 (2008)
Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)
Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)
Takahashi W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
Takahashi S., Takahahsi W.: Strong convergence theorem of a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)
Verma R.U.: Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv. Nonlinear Var. Inequal. 4, 117–124 (2001)
Xu H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)
Xu H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)
Yao Y., Noor M.A., Liou Y.C.: On iterative methods for equilibrium problems. Nonlinear Anal. 70, 497–509 (2009)
Yao Y., Yao J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186, 1551–1558 (2007)
Yamada I., Ogura N., Yamashita Y., Sakaniwa K.: Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optim. 19, 165–190 (1998)
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Qin, X., Cho, S.Y. & Kang, S.M. Iterative algorithms for variational inequality and equilibrium problems with applications. J Glob Optim 48, 423–445 (2010). https://doi.org/10.1007/s10898-009-9498-8
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DOI: https://doi.org/10.1007/s10898-009-9498-8