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Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. (English) Zbl 1205.65185
Summary: We introduce an iterative method for finding a common element of the set of solutions of the generalized equilibrium problems, the set of solutions for the systems of nonlinear variational inequalities problems and the set of fixed points of nonexpansive mappings in Hilbert spaces. Furthermore, we apply our main result to the set of fixed points of an infinite family of strict pseudo-contraction mappings. The results obtained in this paper are viewed as a refinement and improvement of the previously known results.
MSC:
65J15Equations with nonlinear operators (numerical methods)
47J15Abstract bifurcation theory
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
65K15Numerical methods for variational inequalities and related problems
References:
[1]Flores-Bazan, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case, SIAM J. Optim. 11, 675-690 (2000) · Zbl 1002.49013 · doi:10.1137/S1052623499364134
[2]Dien, N. H.: Some remarks on variational-like and quasivariational-like inequalities, Bull. aust. Math. soc. 46, 335-342 (1992) · Zbl 0773.90071 · doi:10.1017/S0004972700011941
[3]Noor, M. A.: Variational-like inequalities, Optim. 30, 323-330 (1994) · Zbl 0816.49005 · doi:10.1080/02331939408843995
[4]Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. student 63, 123-145 (1994) · Zbl 0888.49007
[5]Noor, M. A.; Oettli, W.: On general nonlinear complementarity problems and quasi equilibria, Matematiche (Catania) 49, 313-331 (1994) · Zbl 0839.90124
[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]Ceng, L. C.; Ansari, Q. H.; Yao, J. C.: Viscosity approximation methods for generalized equilibrium problems and fixed point problems, J. global optim. 43, 487-502 (2009) · Zbl 1172.47045 · doi:10.1007/s10898-008-9342-6
[8]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, No. 1, 20-30 (2009) · Zbl 1165.65027 · doi:10.1016/j.cam.2008.06.011
[9]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) · Zbl 1187.65058 · doi:10.1016/j.mcm.2007.12.008
[10]Tada, A.; Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, J. optim. Theory appl. 133, 359-370 (2007)
[11]Takahashi, S.; Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear anal. 69, 1025-1033 (2008) · Zbl 1142.47350 · doi:10.1016/j.na.2008.02.042
[12]Yao, Y.; Noor, M. A.; Zainab, S.; Liou, Y. C.: Mixed equilibrium problems and optimization problems, J. math. Anal. appl. 354, 319-329 (2009) · Zbl 1160.49013 · doi:10.1016/j.jmaa.2008.12.055
[13]Verma, R. U.: Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. optim. Theory appl. 121, 203-210 (2004) · Zbl 1056.49017 · doi:10.1023/B:JOTA.0000026271.19947.05
[14]Verma, R. U.: Generalized class of partial relaxed monotonicity and its connections, Adv. nonlinear var. Inequal. 7, 155-164 (2004) · Zbl 1079.49011
[15]Verma, R. U.: General convergence analysis for two-step projection methods and applications to variational problems, Appl. math. Lett. 18, 1286-1292 (2005) · Zbl 1099.47054 · doi:10.1016/j.aml.2005.02.026
[16]Verma, R. U.: On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. sci. Res. hot-line 3, 65-68 (1999) · Zbl 0970.49011
[17]Ceng, L. C.; Wang, C. Y.; Yao, J. C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. methods oper. Res. 67, 375-390 (2008) · Zbl 1147.49007 · doi:10.1007/s00186-007-0207-4
[18]Bruck, R. E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. amer. Math. soc. 179, 251-262 (1973) · Zbl 0265.47043 · doi:10.2307/1996502
[19]Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601 · doi:10.1007/BF00251595
[20]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) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[21]Xu, H. K.: Iterative algorithms for nonlinear operators, J. lond. Math. soc. 66, 240-256 (2002)
[22]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 · doi:10.1016/j.jmaa.2006.08.036
[23]Zhou, H. Y.: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear anal. 69, 456-462 (2008) · Zbl 1220.47139 · doi:10.1016/j.na.2007.05.032