Strong and Weak Convergence Theorems for Equilibrium Problems and Weak Relatively Uniformly Nonexpansive Multivalued Mappings in Banach Spaces

and Applied Analysis 3 Remark 1.6. The examples of weak relatively uniformly nonexpansive multivalued mapping can be found in Su 1 and Homaeipour and Razani 2 . Let E be a real Banach space, and let E∗ be the dual space of E. Let f be a bifunction from C × C to R. The equilibrium problem is to find x̂ ∈ C such that fx̂, y ≥ 0, ∀y ∈ C. 1.3 The set of solutions of 1.3 is denoted by EP f . Given a mapping T : C → E∗, let f x, y 〈Tx, y − x〉 for all x, y ∈ C. Then x̂ ∈ EP f if and only if 〈Tx̂, y − x̂〉 ≥ 0 for all y ∈ C, that is, x̂ is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of 1.3 . Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see 3–5 for details. In recent years, iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by many authors, see 2, 6–9 for details. In 2011, Homaeipour and Razani 2 introduced the concept of relatively nonexpansive multivalued mappings and proved some weak and strong convergence theorems to approximate a fixed point for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space E which improved and extended the corresponding results of Matsushita and Takahashi 10 . Very recently, Su 1 not only redefined relatively nonexpansive multivalued mappings, which was different from Homaeipour and Razani 2 ’s definition, but also introduced some interesting examples about the multivalued mappings. On the other hand, in 2009, Qin et al. 11 introduced an iterative algorithm for the equilibrium problem 1.3 and relatively nonexpansive mappings. Moreover, they proved a weak convergence theorem for finding a common element of the set of solutions to the equilibrium problem 1.3 and the common set of fixed points of two relatively nonexpansive mappings, which improved and extended the corresponding results of Takahashi and Zembayashi 12 . Motivated and inspired by the above facts, the purpose of this paper will introduce an iterative algorithm for the equilibrium problem 1.3 and two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, a weak convergence theorem will given for finding a common element of the set of solutions to the equilibrium problem 1.3 and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings in some Banach space. Our results improve and extend the corresponding results of Qin et al. 11 and Takahashi and Zembayashi 12 . 2. Preliminaries Let E be a real Banach space with norm ‖ · ‖, and let J be the normalized duality mapping from E into 2 ∗ given by Jx {x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖‖x∗‖, ‖x‖ ‖x∗‖}, 2.1 for all x ∈ E, where E∗ denotes the dual space of E and 〈·, ·〉 the generalized duality pairing between E and E∗. It is well known that if E∗ is uniformly convex, then J is uniformly continuous on bounded subsets of E. 4 Abstract and Applied Analysis As we all know 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. In this connection, Alber 13 introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. The generalized projection ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the Lyapunov 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 ) . 2.2 The existence and uniqueness of the operatorΠC follow from the properties of the Lyapunov functional φ x, y and strict monotonicity of the mapping J , see, for example, 13, 14 . In Hilbert spaces, ΠC PC. It is obvious from the definition of function φ that ∥y ∥ − ‖x‖2 ≤ φy, x ≤ ∥y∥ − ‖x‖2, ∀x, y ∈ E. 2.3 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. It is said to be uniformly convex if limn→∞‖xn − yn‖ 0 for any two sequences {xn} and {yn} in E such that ‖xn‖ ‖yn‖ 1 and limn→∞‖ xn yn /2‖ 1. Let U {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided by lim t→ 0 ∥x ty ∥ − ‖x‖ t , 2.4 which exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ E. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. We also need the following lemmas for the proof of our main results. Lemma 2.1 see 2 . Let E be a strictly convex and smooth Banach space, then φ x, y 0 if and only if x y. Lemma 2.2 see 2 . Let E be a uniformly convex and smooth Banach space and r > 0. Then, g ∥y − z∥ ≤ φy, z, 2.5 for all y, z ∈ Br 0 {x ∈ E : ‖x‖ ≤ r}, where g : 0,∞ → 0,∞ is a continuous, strictly increasing, and convex function with g 0 0. Lemma 2.3 see 11 . Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E. Then φ ( x,ΠCy ) φ ( ΠCy, y ) ≤ φx, y, ∀x ∈ C and y ∈ E. 2.6 Abstract and Applied Analysis 5 Lemma 2.4 see 11 . Let E be a uniformly convex Banach space and Br 0 be a closed ball of E. Then there exists a continuous strictly increasing convex function g : 0,∞ → 0,∞ with g 0 0 such that ∥ ∥λx μy γz ∥ ∥2 ≤ λ‖x‖ μ∥y∥2 γ‖z‖ − λμg∥x − y∥, 2.7and Applied Analysis 5 Lemma 2.4 see 11 . Let E be a uniformly convex Banach space and Br 0 be a closed ball of E. Then there exists a continuous strictly increasing convex function g : 0,∞ → 0,∞ with g 0 0 such that ∥ ∥λx μy γz ∥ ∥2 ≤ λ‖x‖ μ∥y∥2 γ‖z‖ − λμg∥x − y∥, 2.7 for all x, y, z ∈ Br 0 {x ∈ E : ‖x‖ ≤ r} and λ, μ, γ ∈ 0, 1 with λ μ γ 1. Lemma 2.5. Let E be a strictly convex and smooth Banach space, and let C be a closed convex subset of E. Suppose T : C → 2 is a weak relatively uniformly nonexpansive multivalued mapping. Then, F(T) is closed and convex. Proof. First, we show that F T is closed. Let {pn} be a sequence in F T such that pn → p as n → ∞. Since the multivalued operator T is uniformly weak relatively nonexpansive, one has φ ( pn, p̃ ) ≤ φpn, p ) , 2.8 for all p̃ ∈ Tp and for all n ∈ N. Therefore, φ ( p, p̃ ) lim n→∞ φ ( pn, p̃ ) ≤ lim n→∞ φ ( pn, p ) φ ( p, p ) . 2.9 Applying Lemma 2.1, one gets p p̃. Hence Tp {p}. Therefore, p ∈ F T . Next, we show that F T is convex. To this end, for arbitrary p1, p2 ∈ F T , t ∈ 0, 1 . Putting p tp1 1 − t p2, we prove that Tp {p}. Let q ∈ Tp, we have φ ( p, q ) ∥p ∥2 − 2p, Jq ∥q∥2 ∥p ∥2 − 2tp1 1 − t p2, Jq 〉 ∥q ∥2 ∥p ∥2 − 2tp1, Jq 〉 − 2 1 − t p2, Jq 〉 ∥q ∥2 ∥p ∥2 tφ ( p1, q ) 1 − t φp2, q ) − t∥p1 ∥2 − 1 − t ∥p2 ∥2 ≤ ∥p∥2 tφp1, p ) 1 − t φp2, p ) − t∥p1 ∥2 − 1 − t ∥p2 ∥2 ∥p ∥2 − 2tp1 1 − t p2, Jp 〉 ∥p ∥∥2 ∥p ∥2 − 2p, Jp ∥p∥2 0. 2.10 Using Lemma 2.1 again, we also obtain p q. Hence, T p {p}, that is, p ∈ F T . Therefore, F T is convex. For solving the equilibrium problem for a bifunction f : C×C → R, let us assume that f satisfies the following conditions: A1 f x, x 0 for all x ∈ C; A2 f is monotone, that is, f x, y f y, x ≤ 0 for all x, y ∈ C; 6 Abstract and Applied Analysis A3 for each x, y, z ∈ C, lim t↓0 f ( tz 1 − t x, y ≤ fx, y; 2.11 A4 for each x ∈ C, y → f x, y is convex and lower semicontinuous. Lemma 2.6 see 12 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r > 0 and x ∈ E. Then, there exists z ∈ C such that f ( z, y ) 1 r 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C. 2.12 Lemma 2.7 see 12 . Let C be a closed subset of a strictly convex, uniformly smooth, and reflexive Banach space E, and let f be a bifunction from C × C to R satisfying A1 – A4 . For all r > 0 and x ∈ E, define a mapping Tr : E → C as follows: Trx { z ∈ C : fz, y 1 r 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C } , 2.13 for all x ∈ E. Then, the following hold: 1 Tr is single-valued; 2 Tr is firmly nonexpansive mapping, that is, for all x, y ∈ E, 〈 Trx − Try, JTrx − JTry 〉 ≤ Trx − Try, Jx − Jy 〉 ; 2.14 3 F Tr EP f 4 EP f is closed and convex. Lemma 2.8 see 12 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to R satisfying A1 – A4 , and let r > 0 and x ∈ E and q ∈ F Tr , φ ( q, Trx ) φ Trx, x ≤ φ ( q, x ) . 2.15 Lemma 2.9 see 12 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let x ∈ E, and let z ∈ C. Then z ΠCx ⇐⇒ 〈 y − z, Jx − Jz ≤ 0, ∀y ∈ C. 2.16 3. Main Results In this section, we prove a weak convergence theorem for finding a common element of the set of solutions for an equilibrium problem and the set of fixed points of two weak relatively Abstract and Applied Analysis 7 uniformly nonexpansive multivalued mappings in a Banach space. Before proving the result, we need the following theorem. Theorem 3.1. Let C be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C × C to R satisfying A1 – A4 and let T, S : C → C be two weak relatively uniformly nonexpansive multivalued mappings such that F F T ∩ F S ∩ EP f / ∅. Let {xn} be a sequence generated by the following manner: xn ∈ C such that f ( xn, y ) 1 rn 〈 y − xn, Jxn − Jun 〉 ≥ 0, ∀y ∈ C, un 1 J−1 ( αnJxn βnJyn γnJzn ) , 3.1and Applied Analysis 7 uniformly nonexpansive multivalued mappings in a Banach space. Before proving the result, we need the following theorem. Theorem 3.1. Let C be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C × C to R satisfying A1 – A4 and let T, S : C → C be two weak relatively uniformly nonexpansive multivalued mappings such that F F T ∩ F S ∩ EP f / ∅. Let {xn} be a sequence generated by the following manner: xn ∈ C such that f ( xn, y ) 1 rn 〈 y − xn, Jxn − Jun 〉 ≥ 0, ∀y ∈ C, un 1 J−1 ( αnJxn βnJyn γnJzn ) , 3.1 where yn ∈ Txn, zn ∈ Sxn, and J are the duality mapping on E. Assume that {αn}, {βn}, and {γn} are three sequences in 0, 1 satisfying the following conditions: a αn βn γn 1; b lim infn→∞αnβn > 0, lim infn→∞αnγn > 0; c {rn} ⊂ a,∞ for some a > 0. Then {ΠFxn} converges strongly to z ∈ F, whereΠF is the generalized projection of E onto F. Proof. Let p ∈ F. Putting xn Trnun for all n ∈ N, it is well known that Trn is relatively nonexpansive, one has φ ( p, xn 1 ) φ ( p, Trnun 1 ) ≤ φp, un 1 ) φ ( p, J−1 ( αnJxn βnJyn γnJzn )) ∥p ∥2 − 2αn 〈 p, Jxn 〉 − 2βn 〈 p, Jyn 〉 − 2γn 〈 p, Jzn 〉 ∥αnJxn βnJyn γnJzn ∥2 ≤ ∥p∥2 − 2αn 〈 p, Jxn 〉 − 2βn 〈 p, Jyn 〉 − 2γn 〈 p, Jzn 〉 αn‖Jxn‖ βn ∥Jyn ∥2 γn‖Jzn‖ φ ( p, xn ) βnφ ( p, yn ) γnφ ( p, zn ) ≤ φp, xn ) . 3.2 Therefore, limn→∞φ p, xn exists. Since φ p, xn is bounded, {xn}, {yn}, and {zn} are bounded. Define vn ΠFxn for all n ∈ N. Then, from vn ∈ F and 3.2 , one gets φ vn, xn 1 ≤ φ vn, xn . 3.3 8 Abstract and Applied Analysis Since ΠF is the generalized projection, from Lemma 2.3, one sees φ vn 1, xn 1 φ ΠFxn 1, xn 1 ≤ φ vn, xn 1 − φ vn,ΠFxn 1 φ vn, xn 1 − φ vn, vn 1 ≤ φ vn, xn 1 . 3.4 Hence, from 3.3 , one has φ vn 1, xn 1 ≤ φ vn, xn . 3.5 Therefore, {φ vn, xn } is a convergent sequence. Applying 3.3 again, one also obtains that, for all m ∈ N, φ vn, xn m ≤ φ vn, xn . 3.6 From vn m ΠFxn m and Lemma 2.3, one has φ vn, vn m φ vn m, xn m ≤ φ vn, xn m ≤ φ vn, xn , 3.7