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Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces. (English) Zbl 1258.90083
Summary: We introduce two iterative schemes (one implicit and one explicit one) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of S. Li, L. Li and Y. Su [Nonlinear Anal., Theory Methods Appl. 70, No. 9, A, 3065–3071 (2009; Zbl 1177.47075)] and F. Cianciaruso, G. Marino and L. Muglia [J. Optim. Theory Appl. 146, No. 2, 491–509 (2010; Zbl 1210.47080)] and many others.
MSC:
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
References:
[1]Baillon J.B.: On theoreme de type ergodique pour les contractions onlineaires dans un espace de Hilbert. C. R. Acad. Sci. Paris 280, 1511–1514 (1975)
[2]Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
[3]Browder F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banch spaces. Arch. Ration. Mech. Anal. 24, 82–89 (1967) · Zbl 0148.13601 · doi:10.1007/BF00251595
[4]Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[5]Chadli O., Schaible S., Yao J.C.: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 121, 571–596 (2004) · Zbl 1107.91067 · doi:10.1023/B:JOTA.0000037604.96151.26
[6]Cianciaruso F., Marino G., Muglia L.: Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. J. Optim. Theory Appl. 146, 491–509 (2010) · Zbl 1210.47080 · doi:10.1007/s10957-009-9628-y
[7]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) · Zbl 0913.47048 · doi:10.1080/01630569808816813
[8]Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2001)
[9]Kaczor W., Kuczumow T., Reich S.: A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense. J. Math. Anal. Appl. 246, 1–27 (2000) · Zbl 0981.47037 · doi:10.1006/jmaa.2000.6733
[10]Konnov I.V., Schaible S., Yao J.C.: Combined relaxation method for mixed e quilibrium problems. J. Optim. Theory Appl. 126, 309–322 (2005) · Zbl 1110.49028 · doi:10.1007/s10957-005-4716-0
[11]Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008)
[12]Li S., Li L., Su Y.: General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space. J. Nonlinear Anal. 70, 3065–3071 (2009) · Zbl 1177.47075 · doi:10.1016/j.na.2008.04.007
[13]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
[14]Marino G., Xu H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[15]Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000) · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615
[16]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
[17]Plubtieng S., Punpaeng R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007) · Zbl 1127.47053 · doi:10.1016/j.jmaa.2007.02.044
[18]Reich S.: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 24, 269–272 (1978) · Zbl 0404.47032 · doi:10.1016/0021-9045(78)90012-6
[19]Reich S.: A note on the mean ergodic theorem for nonlinear semigroups. J. Math. Anal. Appl. 91, 547–551 (1983) · Zbl 0521.47034 · doi:10.1016/0022-247X(83)90168-3
[20]Shimizu T., Takahashi W.: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 211, 71–83 (1997) · Zbl 0883.47075 · doi:10.1006/jmaa.1997.5398
[21]Suzuki T.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 135, 99–106 (2007) · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8
[22]Wangkeeree, R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. In: Fixed Point Theory and Applications, Article ID 134148, 17 pp (2008). doi: 10.1155/2008/134148
[23]Wangkeeree R., Kamraksa U.: An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems. Nonlinear Anal. Hybrid Syst. 3, 615–630 (2009) · Zbl 1219.49009 · doi:10.1016/j.nahs.2009.05.005
[24]Wangkeeree, R., Kamraksa, U.: A general iterative method for solving the variational inequality problem and fixed point Problem of an infinite family of nonexpansive mappings in Hilbert spaces. In: Fixed Point Theory and Applications, vol. 2009, Article ID 369215, 23 pp. doi: 10.1155/2009/369215
[25]Wangkeeree, R., Petrot, N., Wangkeeree, R.: The general iterative methods for nonexpansive mappings in Banach spaces. J. Glob. Optim. doi: 10.1007/s10898-010-9617-6
[26]Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[27]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
[28]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
[29]Zeng L.C., Wu S.Y., Yao J.C.: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwan. J. Math. 10(6), 1497–1514 (2006)