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A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings. (English) Zbl 1207.47071

Let H be a Hilbert space and let E be a nonempty closed convex subset of H. Consider the following:

An infinite family {T n } n=1 of nonexpansive mappings T n :EE,n=1,2,, and denote by Fix (T n ) the set of all fixed points of T n , i.e.,

Fix(T n )=xE:T n (x)=x·

An equilibrium function F:E×E, which defines the following equilibrium problem: find xE such that F(x,y)0,yE. Denote by EP(F) the set of equilibrium points of this problem.

A nonlinear mapping B:EH which defines the following variational inequality: find xE such that Bx,y-x0,yE, and denote by VI(E,B) the set of all solutions of this variational problem.

In order to approximate an element

z n=1 Fix(T n )EP(F)VI(E,B),

a priori supposed to exist, the authors introduce a (very complicated) hybrid algorithm for which they state and prove a strong convergence theorem (Theorem 3.1).

Alas, no examples are given to illustrate that the hypotheses in Theorem 3.1 are feasible.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
References:
[1]Aslam Noor, M., Ottli, W.: On general nonlinear complementarity problems and quasi equilibria. Mathematics (Catania). 49, 313–331 (1994)
[2]Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
[3]Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[4]Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997) · doi:10.1016/S0025-5610(96)00071-8
[5]Ceng, L.C., Yao, J.C.: Iterative algorithms for generalized set-valued strong nonlinear mixed variational-like inequalities. J. Optim. Theory Appl. 124, 725–738 (2005) · Zbl 1067.49007 · doi:10.1007/s10957-004-1182-z
[6]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) · Zbl 1198.47082 · doi:10.1016/j.na.2008.04.035
[7]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) · Zbl 1141.47040 · doi:10.1016/j.jmaa.2008.02.041
[8]Deutsch, F., Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998) · Zbl 0913.47048 · doi:10.1080/01630569808816813
[9]Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005) · Zbl 1093.47058 · doi:10.1016/j.na.2003.07.023
[10]Jaiboon, C., Kumam, P.: A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Fixed Point Theory Appl. 374815, 32 (2009)
[11]Liu, F., Nashed, M.Z.: Regularization of nonlinear Ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998) · Zbl 0924.49009 · doi:10.1023/A:1008643727926
[12]Marino, G., Xu, H.K.: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[13]Plubtieng, S., Punpaenga, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336(1), 455–469 (2007) · Zbl 1127.47053 · doi:10.1016/j.jmaa.2007.02.044
[14]Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[15]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) · Zbl 1170.47044 · doi:10.1016/j.na.2007.10.025
[16]Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970) · doi:10.1090/S0002-9947-1970-0282272-5
[17]Su, Y., Shang, M., Qin, X.: A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings. J. Appl. Math. Comput. 28, 283–294 (2008) · Zbl 1170.47046 · doi:10.1007/s12190-008-0103-y
[18]Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[19]Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 5, 387–404 (2001)
[20]Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
[21]Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[22]Wangkeeree, R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fixed Point Theory Appl. 134148, 17 (2008)
[23]Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[24]Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[25]Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[26]Yamada, I.: The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithm for Feasibility and Optimization, pp. 473–504. Elsevier, Amsterdam (2001)
[27]Yao, J.C., Chadli, O.: Pseudomonotone complementarity problems and variational in-equalities. In: Crouzeix, J.P., Haddjissas, N., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501–558. Kluwer Academic, Dordrecht (2005)
[28]Yao, Y., Liou, Y.C., Yao, J.C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 64363, 12 (2007)