Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi--Asymptotically Nonexpansive Mappings

and Applied Analysis 3 where J is the duality mapping from E into E∗. It is well known that if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function φ that ‖x‖ − ∥y∥2 ≤ φx, y ≤ ‖x‖ ∥y∥2, ∀x, y ∈ E. 1.6 If E is a Hilbert space, then φ x, y ‖x − y‖, for all x, y ∈ E. On the other hand, the generalized projection Alber 6 ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φ x, y , that is, ΠCx x, where x is the solution to the minimization problem φ x, x inf y∈C φ ( y, x ) , 1.7 and existence and uniqueness of the operatorΠC follow from the properties of the functional φ x, y and strict monotonicity of the mapping J see, for example, 6, 7, 30–32 . Remark 1.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0 then x y. From 1.5 , we have ‖x‖ ‖y‖. This implies that 〈x, Jy〉 ‖x‖2 ‖Jy‖2. From the definition of J, one has Jx Jy. Therefore, we have x y; see 31, 32 for more details. Let C be a closed convex subset of E; a mapping T : C → C is said to be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. A point x ∈ C is a fixed point of T provided Tx x. Denote by F T the set of fixed points of T ; that is, F T {x ∈ C : Tx x}. Recall that a point p in C is said to be an asymptotic fixed point of T 33 if C contains a sequence {xn} which converges weakly to p such that limn→∞‖xn − Txn‖ 0. The set of asymptotic fixed points of T will be denoted by F̃ T . A mapping T from C into itself is said to be relatively nonexpansive 34–36 if F̃ T F T and φ p, Tx ≤ φ p, x for all x ∈ C and p ∈ F T . The asymptotic behavior of a relatively nonexpansive mapping was studied in 37–39 . T is said to be φ-nonexpansive, if φ Tx, Ty ≤ φ x, y for x, y ∈ C. T is said to be relatively quasi-nonexpansive if F T / ∅ and φ p, Tx ≤ φ p, x for all x ∈ C and p ∈ F T . T is said to be quasi-φ-asymptotically nonexpansive if F T / ∅ and there exists a real sequence {kn} ⊂ 1,∞ with kn → 1 such that φ p, Tx ≤ knφ p, x for all n ≥ 1x ∈ C and p ∈ F T . A mapping T is said to be closed if for any sequence {xn} ⊂ C with xn → x and Txn → y, Tx y. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasi-φ-nonexpansive mappings as a subclass and the class of quasi-φ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true see more details 37–41 . A Banach space E is said to be strictly convex if ‖ x y /2‖ < 1 for all x, y ∈ E with ‖x‖ ‖y‖ 1 and x / y. Let U {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit limt→ 0 ‖x ty‖ − ‖x‖ /t exists for each 4 Abstract and Applied Analysis x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U. Let E be a Banach space. The modulus of convexity of E is the function δ : 0, 2 → 0, 1 defined by δ ε inf{1 − ‖ x y /2‖ : x, y ∈ E, ‖x‖ ‖y‖ 1, ‖x − y‖ ≥ ε}. A Banach space E is uniformly convex if and only if δ ε > 0 for all ε ∈ 0, 2 . Let p be a fixed real number with p ≥ 2. A Banach space E is said to be p-uniformly convex if there exists a constant c > 0 such that δ ε ≥ cε for all ε ∈ 0, 2 ; see 42 for more details. Observe that every p-uniform convex is uniformly convex. One should note that no Banach space is puniform convex for 1 < p < 2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each p > 1, the generalized duality mapping Jp : E → 2E is defined by Jp x {x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖p, ‖x∗‖ ‖x‖p−1} for all x ∈ E. In particular, J J2 is called the normalized duality mapping. If E is a Hilbert space, then J I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. The following basic properties can be found in Cioranescu 31 . i If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E. ii If E is a reflexive and strictly convex Banach space, then J−1 is norm-weak∗continuous. iii If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : E → 2E is single-valued, one-to-one, and onto. iv A Banach space E is uniformly smooth if and only if E∗ is uniformly convex. v Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {xn} ⊂ E, if xn ⇀ x ∈ E and ‖xn‖ → ‖x‖, xn → x. In 2005, Matsushita and Takahashi 40 proposed the following hybrid iteration method it is also called the CQ method with generalized projection for relatively nonexpansive mapping T in a Banach space E: x0 ∈ C chosen arbitrarily, yn J−1 αnJxn 1 − αn JTxn , Cn { z ∈ C : φz, yn ) ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0},