A New Iterative Scheme for Countable Families of Weak Relatively Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems

and Applied Analysis 3 R1 F T / ∅; R2 φ p, Tx ≤ φ p, x , for all x ∈ C, p ∈ F T ; R3 F T F̂ T . Definition 1.4. A point p ∈ C is said to be an strong asymptotic fixed point of T if C contains a sequence {xn}n 0 which converges strongly to p and limn→∞‖xn − Txn‖ 0. The set of strong asymptotic fixed points of T is denoted by F̃ T . We say that a mapping T is weak relatively nonexpansive see, e.g., 12, 13 if the following conditions are satisfied: R1 F T / ∅; R2 φ p, Tx ≤ φ p, x , for all x ∈ C, p ∈ F T ; R3 F T F̃ T . Definition 1.3 Definition 1.4 is a special form of Definition 1.1 Definition 1.2 as Tn ≡ T , for all n ≥ 0. Furthermore, Su et al. 5 gave an example which is a countable family of weak relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping T : C → C, we have F T ⊂ F̃ T ⊂ F̂ T . Therefore, if T is relatively nonexpansive mapping, then F T F̃ T F̂ T . Kang et al. 12 gave an example of a weak relatively nonexpansive mapping which is not relatively nonexpansive. Let F : C×C → be a bifunction,A : C → E∗ amapping and φ : C → a real-valued function. The generalizedmixed equilibrium problem is to find x ∈ C see, e.g., 14–19 such that F ( x, y ) 〈Ax, y − x〉 φ(y) − φ x ≥ 0, 1.5 for all y ∈ C. We will denote the solutions set of 1.5 by GMEP F, φ . Thus GMEP ( F,A, φ ) : { x∗ ∈ C : F(x∗, y) 〈Ax∗, y − x∗〉 φ(y) − φ x∗ ≥ 0, ∀y ∈ C}. 1.6 If φ 0 and A 0, then problem 1.5 reduces to an equilibrium problem studied by many authors see, e.g., 20–28 , which is to find x∗ ∈ C such that