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A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems. (English) Zbl 1250.47078
Summary: We introduce an iterative process for finding an element in the common fixed point set of a finite family of closed relatively quasi-nonexpansive mappings, common solutions of a finite family of equilibrium problems and common solutions of a finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
References:
[1]Kinderlehrer, D.; Stampaccia, G.: An iteration to variational inequalities and their applications, (1990)
[2]Lions, J. L.; Stampacchia, G.: Variational inequalities, Comm. pure appl. Math. 20, 493-517 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[3]Iiduka, H.; Takahashi, W.: Strong convergence studied by a hybrid type method for monotone operators in a Banach space, Nonlinear anal. 68, 3679-3688 (2008) · Zbl 1220.47095 · doi:10.1016/j.na.2007.04.010
[4]Kamimura, S.; Takahashi, W.: Strong convergence of proximal-type algorithm in a Banach space, SIAM J. Optim. 13, 938-945 (2002) · Zbl 1101.90083 · doi:10.1137/S105262340139611X
[5]Zegeye, H.; Ofoedu, E. U.; Shahzad, N.: Convergence theorems for equilibrium problem, variotional inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. math. Comput. 216, 3439-3449 (2010) · Zbl 1198.65100 · doi:10.1016/j.amc.2010.02.054
[6]Zegeye, H.; Shahzad, N.: Strong convergence for monotone mappings and relatively weak nonexpansive mappings, Nonlinear anal. 70, 2707-2716 (2009) · Zbl 1223.47108 · doi:10.1016/j.na.2008.03.058
[7]Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems, (1990)
[8]Takahashi, W.: Nonlinear functional analysis, (1988)
[9]Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications, Leture notes in pure and appl. Math. 178, 15-50 (1996) · Zbl 0883.47083
[10]Reich, S.: A weak convergence theorem for the alternating method with Bergman distance, Leture notes in pure and appl. Math. 178, 313-318 (1996) · Zbl 0943.47040
[11]Qin, X.; Cho, Y. J.; Kang, S. M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spces, J. comput. Appl. math. (2008)
[12]Su, Y.; Wang, D.; Shang, M.: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed point theory appl. (2008) · Zbl 1203.47078 · doi:10.1155/2008/284613
[13]Butanriu, D.; Reich, S.; Zaslavski, A. J.: Asymtotic behavior of relatively nonexpansive operators in Banach spaces, J. appl. Anal. 7, 151-174 (2001) · Zbl 1010.47032 · doi:10.1515/JAA.2001.151
[14]Kohasaka, F.; Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in Banach spaces, Abstr. appl. Anal. 3, 239-249 (2004) · Zbl 1064.47068 · doi:10.1155/S1085337504309036
[15]Mann, M. R.: Mean value methods in iteration, Proc. amer. Math. soc. 4, 503-510 (1953) · Zbl 0050.11603 · doi:10.2307/2032162
[16]Reich, S.: Weak convergence theorems for nonexpansive mappings, J. math. Anal. appl. 67, 274-276 (1979) · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6
[17]Bauschke, H. H.; Matouskova, E.; Reich, S.: Projection and proximal point methods: convergence results and counterexamples, Nonlinear anal. 56, 715-738 (2004) · Zbl 1059.47060 · doi:10.1016/j.na.2003.10.010
[18]Genel, A.; Lindenstress, J.: An example concerning fixed points, Israel J. Math. 22, 81-86 (1975) · Zbl 0314.47031 · doi:10.1007/BF02757276
[19]Bauschke, H. H.; Combettes, P. L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. oper. Res. 26, 248-264 (2001) · Zbl 1082.65058 · doi:10.1287/moor.26.2.248.10558
[20]Nakajo, K.; Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semi-groups, J. math. Anal. appl. 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[21]Matsushita, S. Y.; Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. Theory 134, 257-266 (2005) · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007
[22]Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. stud. 63, 123-145 (1994) · Zbl 0888.49007
[23]Moudafi, A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. nonlinear convex anal. 9, 37-43 (2008) · Zbl 1167.47049
[24]Kumam, P.: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive, Nonlinear anal. Hybrid syst. (2008)
[25]Tada, A.; Takahashi, W.: Weak and strong convergence theorems for nonexpansive mappings and equilbrium problems, J. optim. Theory appl. 133, 359-370 (2007) · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z
[26]Takahashi, W.; Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear anal. 70, 45-57 (2009) · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031
[27]Nadezhkina, N.; Takahashi, W.: Strong convergence theorems by a hybrid method for nonexpansive mappings and mappings, SIAM J. Optim. 16, 1230-1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315