Hybrid method for the equilibrium problem and a family of generalized nonexpansive mappings in Banach spaces
Authors
Chakkrid Klineam
 Department of Mathematics, Faculty of Science, Naresuan University, , Thailand., Phitsanulok, 65000, Thailand.
 Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok, 65000, Thailand.
Prondanai Kaskasem
 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand.
Suthep Suantai
 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand.
Abstract
We introduce a hybrid method for finding a common element of the set of solutions of an equilibrium
problem defined on the dual space of a Banach space and the set of common fixed points of a family
of generalized nonexpansive mappings and prove strong convergence theorems by using the new hybrid
method. Using our main results, we obtain some new strong convergence theorems for finding a solution
of an equilibrium problem and a fixed point of a family of generalized nonexpansive mappings in a Banach
space.
Keywords
 Hybrid method
 generalized nonexpansive mapping
 NSTcondition
 equilibrium problem
 fixed point problem
 Banach space.
MSC
References

[1]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123145

[2]
P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117136

[3]
T. Ibaraki, W. Takahashi, A new projection and convergence theorems for the projections in Banach spaces, J. Approx. Theory, 149 (2007), 114

[4]
S. Kamimura, W. Takahashi, Strong convergence of a proximaltype algorithm in a Banach space, SIAM J. Optim., 13 (2002), 938945

[5]
F. Kohsaka, W. Takahashi, Generalized nonexpansive retractions and a proximaltype algorithm in Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 197209

[6]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506510

[7]
K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces, J. Nonlinear Convex Anal., 6 (2007), 1134

[8]
K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372379

[9]
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 247276

[10]
W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publ., Yokohama (2000)

[11]
W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276286

[12]
W. Takahashi, K. Zembayashi, A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space, Fixed point theory and its applications, Yokohama Publ., Yokohama, 197209 (2008),

[13]
C. Zălinescu, On uniformly convex functions, J. Math. Anal. Appl., 95 (1983), 344374