Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

and Applied Analysis 3 Let E be a real Banach space and let C be a nonempty closed and convex subset of E, and A : C → E∗ be a mapping. The classical variational inequality problem for a mapping A is to find x∗ ∈ C such that 〈 Ax∗, y − x∗〉 ≥ 0, ∀y ∈ C. 1.7 The set of solutions of 1.4 is denoted by VI A,C . Recall that A is called i monotone if 〈 Ax −Ay, x − y〉 ≥ 0, ∀x, y ∈ C, 1.8 ii an α-inverse-strongly monotone if there exists a constant α > 0 such that 〈 Ax −Ay, x − y〉 ≥ α∥∥x − y∥∥2, ∀x, y ∈ C. 1.9 Such a problem is connected with the convex minimization problem, the complementary problem, and the problem of finding a point x∗ ∈ E satisfying Ax∗ 0. Let f be a bifunction from C × C to R, where R denotes the set of real numbers. The equilibrium problem for short, EP is to find x∗ ∈ C such that f ( x∗, y ) ≥ 0, ∀y ∈ C. 1.10 The set of solutions of 1.10 is denoted by EP f . Given a mapping T : C → E∗, let f x, y 〈Tx, y − x〉 for all x, y ∈ C. Then x∗ ∈ EP f if and only if 〈Tx∗, y − x∗〉 ≥ 0 for all y ∈ C; that is, x∗ is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of 1.10 . Some methods have been proposed to solve the equilibrium problem; see, for instance, 9–11 . Let C be a closed convex subset of E; a mapping T : C → C is said to be nonexpansive if ∥ ∥Tx − Ty∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C. 1.11 A point x ∈ C is a fixed point of T provided that 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 12 ifC 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 F̂ T . A mapping T from C into itself is said to be relatively nonexpansive 13–15 if F̂ 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 16–18 . 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 x ∈ C and p ∈ F T . A mapping T in a Banach space E is closed if xn → x and Txn → y, then Tx y. 4 Abstract and Applied Analysis Remark 1.3. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings 16–19 which requires the strong restriction F T F̂ T . In Hilbert spaces H, Iiduka et al. 20 proved that the sequence {xn} defined by: x1 x ∈ C and xn 1 PC xn − λnAxn , 1.12 where PC is themetric projection ofH ontoC, and {λn} is a sequence of positive real numbers, and converges weakly to some element of VI A,C . It is well know that if C is a nonempty closed and 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. In this connection, Alber 4 recently introduced a generalized projection mapping ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. In 2008, Iiduka and Takahashi 21 introduced the following iterative scheme for finding a solution of the variational inequality problem for inverse-strongly monotoneA in a 2-uniformly convex and uniformly smooth Banach space E: x1 x ∈ C and xn 1 ΠCJ−1 Jxn − λnAxn 1.13 for every n 1, 2, 3, . . . ,whereΠC is the generalized metric projection from E onto C, J is the dualitymapping from E into E∗, and {λn} is a sequence of positive real numbers. They proved that the sequence {xn} generated by 1.13 converges weakly to some element of VI A,C . Matsushita and Takahashi 22 introduced the following iteration: a sequence {xn} defined by xn 1 ΠCJ−1 αnJxn 1 − αn JTxn , 1.14 where the initial guess element x0 ∈ C is arbitrary, {αn} is a real sequence in 0, 1 , T is a relatively nonexpansive mapping, and ΠC denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {xn} converges weakly to a fixed point of T . Abstract and Applied Analysis 5 In 2005, Matsushita and Takahashi 19 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:and Applied Analysis 5 In 2005, Matsushita and Takahashi 19 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 : φ(z, yn ) ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ∏