Convergence theorems of common elements for mixed equilibrium problems, variational inequality problems and fixed point problems in Banach spaces
Kriengsak Wattanawitoon, Usa Hamphries, Poom Kumam · International journal of applied mathematics and computation · 2010
In this paper, a hybrid iterative scheme is introduced for the approximation method for finding a common element of the set of mixed equilibrium problems, variational inequality problems and fixed point problems of two quasi-$\phi$-nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space. Then, we obtain a strong convergence theorem for the sequence generated by those process in Banach spaces. Moreover, we obtain new result for finding a zero point of maximal monotone operators in a Banach space. Our results improve and extend the corresponding results announced by Takahashi and Zembayashi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45--57.] Qin, Cho and Kang [Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), 20--30.] Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal: Hybrid Systems. 3 (2009) 11--20.] Cholamjiak [A hybrid iterative scheme for equilibrium problems, variational inequality problems and fixed point problems in Banach spaces, Fixed Point Theory Appl., (2009), Article ID 719360, 18 pages] and many authors.