×

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.

MSC:

90C48 Programming in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.