New Mixed Equilibrium Problems and Iterative Algorithms for Fixed Point Problems in Banach Spaces

We first introduce a new mixed equilibrium problem with a relaxed monotone mapping in Banach spaces and prove the existence of solutions of the equilibrium problem. Then we introduce a new iterative algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a quasi- -nonexpansive mapping and prove some strong convergence theorems of the iteration. Our results extend and improve the corresponding ones given by Wang et al., Takahashi and Zembayashi, and some others. 1. Introduction Let be a Banach space with the dual space and let be a nonempty closed convex subset of . Let be a bifunction from to , where denotes the set of numbers. The equilibrium problem is to find such that The set of solutions of the above equilibrium problem is denoted by . In order to solve the equilibrium problem, the bifunction is usually to be assumed that following conditions are satisfied:(A1) for all ;(A2) is monotone; that is, for all ;(A3)for all , (A4)for all , is convex and lower semicontinuous. Recently, Takahashi and Zembayashi [1] extended the equilibrium problems and fixed point problems from Hilbert spaces to Banach spaces. More precisely, they gave the following iterative scheme: where is a relatively nonexpansive mapping from into itself such that , is the duality mapping on , and satisfies and for some . They proved that the sequence generated by (3) converges strongly to , where is the generalized projection of onto . Very recently, Qin et al. [2] introduced the following hybrid algorithm to solve the equilibrium problems and fixed point problems for quasi- -nonexpansive mappings in a uniformly convex and uniformly smooth Banach space with a nonempty closed convex subset of : where and are two quasi- -nonexpansive mappings from into itself such that , , , and are three sequences in satisfying some certain conditions. They proved that the sequence generated by (4) converges strongly to , where is the generalized projection of onto . In [3], Fang and Huang introduced a concept called a relaxed - -monotone mapping. A mapping is said to be relaxed？？ - -monotone？？if there exist a mapping and a function with for all and , where is a constant, such that Especially, if for all and , where and are two constants, then is said to be -monotone (see, e.g., [4–6]). They proved that, under some suitable assumptions, the following variational inequality is solvable: find such that where is a function from to . They also proved that the variational inequality (6) is equivalent to the following: find such that In this