Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. (English) Zbl 1270.90100

The author introduces a new iteration method and proves strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of a monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on the parameters, the author shows that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.


90C48 Programming in abstract spaces
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[1] Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972) · Zbl 0302.49019
[2] Anh, P.N.: An LQP regularization method for equilibrium problems on polyhedral. Vietnam J. Math. 36, 209–228 (2008) · Zbl 1197.65075
[3] Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994) · Zbl 0888.49007
[4] Bre’zis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital., VI, 129–132 (1972) · Zbl 0264.49013
[5] Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Springer, Berlin (1995) · Zbl 0834.00044
[6] Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2004) · Zbl 0979.00025
[7] Korpelevich, G.M.: Extragradient method for finding saddle points and other problems. Èkon. Mat. Metody 12, 747–756 (1976) · Zbl 0342.90044
[8] Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2003) · Zbl 1069.49009
[9] Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms to equilibrium problems. J. Glob. Optim. 52, 139–159 (2012) · Zbl 1258.90088
[10] Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér. 4, 154–159 (1970)
[11] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053
[12] Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000) · Zbl 0986.49004
[13] Anh, P.N.: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math. Vietnam. 34, 183–200 (2009) · Zbl 1200.65044
[14] Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Appl. Math. 61, 2588–2595 (2011) · Zbl 1221.90083
[15] Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim., 46, 635–646 (2010) · Zbl 1198.47081
[16] Chen, J., Zhang, L.J., Fan, T.G.: Viscosity approximation methods for nonexpansive mappings and monotone mappings. J. Math. Anal. Appl. 334, 1450–1461 (2007) · Zbl 1137.47307
[17] Anh, P.N., Son, D.X.: A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions. J. Appl. Math. Inform. 29, 1179–1191 (2011) · Zbl 1232.65079
[18] Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953) · Zbl 0050.11603
[19] Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060
[20] 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) · Zbl 1122.47056
[21] Ceng, L.C., Schaible, S., Yao, J.C.: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J. Optim. Theory Appl. 139, 403–418 (2008) · Zbl 1163.47051
[22] Kim, J.K., Anh, P.N., Nam, J.M.: Strong convergence of an extragradient method for equilibrium problems and fixed point problems. J. Korean Math. Soc. 49, 187–200 (2012) · Zbl 1317.65146
[23] Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007) · Zbl 1147.47052
[24] Yao, Y., Liou, Y.C., Wu, Y.J.: An extragradient method for mixed equilibrium problems and fixed point problems. Fixed Point Theory Appl. (2009). doi: 10.1155/2009/632819 . Article ID 632819, 15 pages · Zbl 1186.47080
[25] Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 1–13 (2011)
[26] Suzuki, T.: Strong convergence of Krasnoselskii and Mann type sequences for one-parameter nonexpansive semi-groups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005) · Zbl 1068.47085
[27] Goebel, K., Kirk, W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) · Zbl 0708.47031
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