RESEARC H Open Access Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces Uthai Kamraksa 1 and Rabian Wangkeeree 1,2* * Correspondence: uthaikam@hotmail.com 1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Full list of author information is available at the end of the article Abstract We first prove the existence of a solution of the generalized equilibrium problem (GEP) using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings. AMS Subject Classification: 47H09, 47H10 Keywords: Banach space, Fixed point, Metric projection, Generalized equilibrium pro- blem, Nonexpansive mapping 1. Introduction Let E be a real Banach space with the dual E*andC beanonemptyclosedconvex subset of E.Wedenoteby N and R the sets of positive integers and re al numbers, respectively. Also, we denote by J the normalized duality mapping from E to 2 E* defined by Jx = { x ∗ ∈ E ∗ :  x, x ∗  = || x || 2 = || x ∗ || 2 } , ∀x ∈ E , where 〈·,·〉 denotes the generalized duality pairing. We know that if E is smooth, then J is single-valued and if E is uniformly smooth, then J is uniformly norm-to-norm con- tinuous on bounded subsets of E. We s hall still denote by J the single-valued duality mapping. Let f : C × C → R be a bifunction and A : C ® E* be a nonlinear mapping. We consider the following generalized equilibrium problem (GEP): Find u ∈ C such that f ( u, y ) + Au, y − u≥0, ∀y ∈ C . (1:1) The set of such u Î C is denoted by GEP (f), i.e., GEP ( f ) = {u ∈ C : f ( u, y ) + Au, y − u≥0, ∀y ∈ C} . Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 © 2011 Kamraksa and Wangkeeree; lice nsee Sp ringer. This is an Open Access article distr ibu ted under the terms of the Creati ve Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribu tion, and reproduction in any medium, provided the original work is properly cited. Whenever E = H aHilbertspace,theproblem(1.1)wasintroducedandstudiedby Takahashi and Takahashi [1]. Similar problems have been studied extensively recently. In the case of A ≡ 0, GEP (f) is denoted by EP (f). In the case of f ≡ 0, EP is also denoted by VI(C, A). Problem (1.1) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash eq uilibrium problem in noncooperative games, and others; see, e.g., [2,3]. A mapping T : C ® E is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y Î C.DenotebyF (T )thesetoffixedpointsofT ,thatis,F (T )={x Î C : Tx = x}. A mapping A : C ® E* is called a-inverse-strongly monotone, if there exists an a > 0 such that Ax − A y , x − y ≥α||Ax − A y || 2 , ∀x, y ∈ C . It is easy to see that if A : C ® E*isana-inverse-strongly monotone mapping, then it is 1/a- Lipschitzian. In 1953, Mann [4] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: x n+1 = α n x n + ( 1 − α n ) Tx n , ∀n ∈ N , (1:2) where the initial point x 0 is taken in C arbitrarily and {a n } is a sequence in [0, 1]. However, we note that Manns iteration process (1.2) has only weak convergence, in general; for instance, see [5-7]. Let C be a nonempty, closed, and convex subset o f a Banach space E and {T n }be sequence of mappings of C into itself such that  ∞ n =1 F(T n ) = ∅ .Then,{T n }issaidto satisfy the NST-condition if for each bounded sequence {z n } ⊂ C, lim n → ∞ ||z n − T n z n || = 0 implies ω w (z n ) ⊂  ∞ n =1 F(T n ) ,whereω w (z n ) is the set of all weak cluster points of {z n }; see [8-10]. In 2008, Takahash i et al. [11] ha s adapted Nakajo and Takahashi’s [12] idea to mod- ify the process (1.2) so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space: x 0 Î H, C 1 = C , u 1 = P C 1 x 0 and ⎧ ⎨ ⎩ y n = α n u n +(1− α n )T n u n , C n+1 = {z ∈ C n : ||y n − z|| ≤ ||u n − z||} , u n+1 = P C n+1 x 0 , n ∈ N , (1:3) where 0 ≤ a n ≤ a <1forall n ∈ N .Theyprovedthatif{T n } satisfies the NST- condition , then {u n } generated by (1.3) converges strongly to a common fixed point of T n . Recently, motivated by Nakajo and Takahashi [12] and Xu [13], Matsushita and Takahashi [14] introduced the iterative algorithm for finding fixed points of nonexpan- sive mappings in a uniformly convex and smooth Banach space: x 0 = x Î C and ⎧ ⎨ ⎩ C n = co{z ∈ C : ||z − Tz|| ≤ t n ||x n − Tx n ||} , D n = {z ∈ C : x n − z, J(x − x n )≥0}, x n+1 = P C n ∩D n x, n ≥ 0, (1:4) Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 2 of 12 where co D denotes the convex closure of the set D,{t n } is a sequence in (0,1) with t n ® 0, and P C n ∩D n is the metric projection from E ont o C n ∩ D n . They proved that {x n } generated by (1.4) converges strongly to a fixed point of T . Very recently, Kimura and Nakajo [15] investigated iterative schemes for finding com- mon fixed points of a family of nonexpansive mappings and proved strong convergence theorems by using the Mosco convergence technique in a uniformly convex and smooth Banach space. In particular, they proposed the following algorithm: x 1 = x Î C and ⎧ ⎨ ⎩ C n = co{z ∈ C : ||z − T n z|| ≤ t n ||x n − T n x n ||} , D n = {z ∈ C : x n − z, J(x − x n )≥0}, x n+1 = P C n ∩D n x, n ≥ 0, (1:5) where {t n } is a sequ ence in (0,1) with t n ® 0asn ® ∞. They proved that if {T n } satis- fies the NST-condition, then {x n } converges strongly to a common fixed point of T n . Motivated and inspired by N akajo and Takahashi [12], Takahash i et al. [11], Xu [13], Masushita and Takahashi [14], and Kimura and Nakajo [15], we introduce a hybrid projection algorithm for finding a common element in t he solution set of a GEP and the common fixed point set of a family of nonexpansive mappings in a B anach space setting. 2. Preliminaries Let E be a real Banach space and let U ={x Î E :||x|| = 1} be the unit sphere of E. A Banach space E is said to be strictly convex if for any x, y Î U, x = y implies —|x + y || < 2 . It is also said to be uniformly convex if for each ε Î (0, 2], there exists δ >0such that for any x, y Î U, ||x − y|| ≥ ε implies —|x + y|| < 2 ( 1 − δ ). It is known that a unifo rmly convex Banach space is reflexive and strictly convex. Define a function δ: [0, 2] ® [0, 1] called the modulus of convexity of E as follows: δ(ε)=inf  1 −|| x + y 2 || : x, y ∈ E, ||x|| = ||y|| =1, ||x − y|| ≥ ε  . Then, E is uniformly convex if and only if δ(ε) > 0 for all ε Î (0, 2]. A Banach space E is said to be smooth if the limit lim t→0 || x + ty || − || x || t (2:1) exists for all x, y Î U. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex and smooth Banach space E. Then, for any x Î E, there exists a unique point x 0 Î C such that ||x 0 − x|| ≤ min y ∈C ||y − x|| . The mapping P C : E ® C defined by P C × = x 0 is called t he metric projection from E onto C. Let x Î E and u Î C. Then, it is known that u = P C × if and only if u − y, J ( x − u ) ≥ 0 (2:2) Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 3 of 12 for all y Î C; see [16] for more details. It is well known that if P C is a metric projection from a real Hilbert space H onto a nonempty, closed, and convex subset C,thenP C is nonexpansive . However, in a genera l Banach space, this fact is not true. In the sequel, we will need the following lemmas. Lemma 2.1. [17]Let E be a uniformly convex Banach space,{a n } be a sequence of real numbers such that 0<b ≤ a n ≤ c <1for all n ≥ 1, and {x n } and {y n } be sequences in E such that lim sup n®∞ ||x n || ≤ d, lim sup n®∞ ||y n || ≤ dandlim n®∞ ||a n x n +(1-a n ) y n || = d. Then, lim n®∞ ||x n - y n || = 0. Lemma 2.2. [18]Let C be a bounded, closed, and convex subset of a uniformly convex Banach space E. Then, there exists a strictly increasing, convex, and continuous function g : [0, ∞) ® [0, ∞) such thatg (0) = 0 and γ       T  n  i=1 λ i x i  − n  i=1 λ i Tx i       ≤ max 1≤j≤k≤n (||x j − x k || − ||Tx j − Tx k || ) for all n ∈ N ,{x 1 , x 2 , , x n } ⊂ C,{l 1 , l 2 , , l n } ⊂ [0, 1] with  n i =1 λ i = 1 and nonexpan- sive mapping T of C into E. Following Bruck’s [19] idea, we know the following result for a convex combination of nonexpansive mappings which is considered by Aoyama et al. [20] and Kimura and Nakajo [15]. Lemma 2.3.[15]Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E and {S n } be a family of nonexpansive mappings of C into itself such that F =  ∞ n =1 F(S n ) = ∅ . Let {β k n } be a family of nonnegative numbers wi th indices n, k ∈ N with k ≤ n such that (i)  n k =1 β k n = 1 for every n ∈ N ; (ii) lim n→∞ β k n > 0 for every k ∈ N and let T n = α n I +(1− α n )  n k =1 β k n S k for all n ∈ N , where {a n } ⊂ [a, b] for some a, b Î (0, 1) with a ≤ b. Then,{T n } is a family of nonexpansive mappings of C into itself with  ∞ n =1 F(T n )= F and satisfies the NST-condition. Now, let us turn to following well-known concept and result. Definition 2.4. Let B be a subset of topological vector space X. A mapping G : B ® 2 X is called a KKM mapping if co{x 1 , x 2 , , x m }⊂  m i =1 G(x i ) for x i Î B and i = 1, 2, , m, where coA denotes the convex hull of the set A. Lemma 2.5. [21]Let B be a nonempty subset of a Hausdorff topological vector space × and let G : B ® 2 X be a KK M mapping. If G(x) is closed for all × Î B and is compact for at least one x Î B, then ⋂ xÎB G(x) ≠ ∅. 3. Existence results of gep Motivated by Takahashi and Zembayashi [22], and Ceng and Yao [23], we next prove the following crucial lemma concerning the GEP in a strictly convex, reflexive, and smooth Banach space. Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 4 of 12 Theorem 3.1. Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × Cto R satisfying (A1)-(A4), where (A1) f(x, x)=0for all x Î C; (A2) f is monotone, i.e. f(x, y)+f(y, x) ≤ 0 for all x, y Î C; (A3) for all y Î C, f(., y) is weakly upper semicontinuous; (A4) for all x Î C, f(x,.) is convex. Let A be a-inverse strongly monotone of C into E*. For all r >0and × Î E, define the mapping S r : E ® 2 C as follows: S r (x)={z ∈ C : f (z, y)+Az, y − z + 1 r y − z, J(z − x)≥0, ∀y ∈ C} . (3:1) Then, the following statements hold: (1) for each x Î E, S r (x) ≠ ∅; (2) S r is single-valued; (3) 〈S r (x)-S r (y), J(S r x - x)〉 ≤ 〈S r (x)-S r (y), J(S r y - y)〉 for all x, y Î E; (4) F (S r )=GEP (f); (5) GEP(f) is nonempty, closed, and convex. Proof.(1)Letx 0 be any gi ven point in E. For each y Î C, we define the mapping G : C ® 2 E by G(y)={z ∈ C : f (z, y)+Az, y − z + 1 r y − z, J(z − x 0 )≥0} for all y ∈ C . It is easily seen that y Î G(y), and hence G(y). ≠ ∅ (a) First, we will show that G is a KKM mapping. Suppose that there exists a finite subset {y 1 , y 2 , , y m }ofC and a i > 0 with  m i =1 α i = 1 such that ˆ x =  m i =1 α i y i ∈ G(y i ) for all i = 1, 2, , m. It follows that f ( ˆ x, y i )+A ˆ x, y i − ˆ x + 1 r y i − ˆ x, J( ˆ x − x 0 ) < 0, for all i =1,2, , m . By (A1) and (A4), we have 0=f ( ˆ x, ˆ x)+A ˆ x, ˆ x − ˆ x + 1 r  ˆ x − ˆ x, J( ˆ x − x 0 ) ≤ m  i =1  f ( ˆ x, y i )+A ˆ x, y i − ˆ x + 1 r y i − ˆ x, J( ˆ x − x 0 )  < 0 , which is a contradiction. Thus, G is a KKM mapping on C. (b) Next, we show that G(y)isclosedforally Î C.Let{z n }beasequenceinG(y) such that z n ® z as n ® ∞. It then follows from z n Î G(y) that, f (z n , y)+Az n , y − z n  + 1 r y − z n , J(z n − x)≥0 . (3:2) Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 5 of 12 By (A3), the continuity of J, and the lower semicontinuity of || · || 2 , we obtain from (3.2) that 0 ≤ lim inf n→∞ [f (z n , y)+Az n , y − z n  + 1 r y − z n , J(z n − x 0 )] ≤ lim sup n→∞ [f (z n , y)+Az n , y − z n  + 1 r y − x 0 , J(z n − x 0 ) + 1 r x 0 − z n , J(z n − x 0 )] = lim sup n→∞ [f (z n , y)+Az n , y − z n  + 1 r y − x 0 , J(z n − x 0 )− 1 r ||z n − x 0 || 2 ] ≤ lim sup n→∞ f (z n , y) + lim sup n→∞ Az n , y − z n  + 1 r lim sup n→∞ y − x 0 , J(z n − x 0 )− 1 r lim inf n→∞ ||z n − x 0 || 2 ≤ f(z, y)+Az, y − z + 1 r y − x 0 , J(z − x 0 )− 1 r ||z − x 0 || 2 = f(z, y)+Az, y − z + 1 r y − x 0 , J(z − x 0 )− 1 r z − x 0 , J(z − x 0 ) = f(z, y)+Az, y − z + 1 r y − z, J(z − x 0 ). This shows that z Î G(y), and hence G(y) is closed for all y Î C . (c) We prove that G(y) is weakly compact. We now equip E with the weak topology. Then, C, as closed, bounded convex subset in a reflexive space, is weakly compact. Hence, G(y) is also weakly compact. Using (a), (b), and (c) and Lemma 2.5, we have ⋂ xÎC G(y) ≠ ∅. It is easily seen that S r (x 0 )=  y ∈C G(y ) Hence, s r (x 0 ) ≠ ∅. Since x 0 is arbitrary, we can conclude that s r (x) ≠ ∅ for all x Î E. (2) We prove that S r is single-valued. In fact, for x Î C and r >0,letz 1 , z 2 Î S r (x). Then, f (z 1 , z 2 )+Az 1 , z 2 − z 1  + 1 r z 2 − z 1 , J(z 1 − x)≥0 . and f (z 2 , z 1 )+Az 2 , z 1 − z 2  + 1 r z 1 − z 2 , J(z 2 − x)≥0 . Adding the two inequalities and from the co ndition (A2) and monotonicity of A,we have 0 ≤ f(z 1 , z 2 )+f (z 2 , z 1 )+Az 1 , z 2 − z 1  + Az 2 , z 1 − z 2  + 1 r z 2 − z 1 , J(z 1 − x) − J(z 2 − x)  ≤Az 1 − Az 2 , z 2 − z 1  + 1 r z 2 − z 1 , J(z 1 − x) − J(z 2 − x) ≤−α||Az 1 − Az 2 || 2 + 1 r z 2 − z 1 , J(z 1 − x) − J(z 2 − x) ≤ 1 r z 2 − z 1 , J(z 1 − x) − J(z 2 − x), (3:3) and hence, z 2 − z 1 , J ( z 1 − x ) − J ( z 2 − x ) ≥0 . Hence, 0 ≤z 2 − z 1 , J ( z 1 − x ) − J ( z 2 − x )  =  ( z 2 − x ) − ( z 1 − x ) , J ( z 1 − x ) − J ( z 2 − x )  . Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 6 of 12 Since J is monotone and E is strictly convex, we obtain that z 1 - x = z 2 - x and hence z 1 = z 2 . Therefore S r is single-valued. (3) For x, y Î C, we have f (S r x, S r y)+AS r x, S r y − S r x + 1 r S r y − S r x, J(S r x − x)≥ 0 and f (S r y, S r x)+AS r y, S r x − S r y + 1 r S r x − S r y, J(S r y − y)≥0 . Again, adding the two inequalities, we also have AS r x − AS r y, S r y − S r x + S r y − S r x, J ( S r x − x ) − J ( S r y − y ) ≥0 . It follows from monotonicity of A that S r y − S r x, J ( S r x − x ) ≤S r y − S r x, J ( S r y − y )  . (4) It is easy to see that z ∈ F(S r ) ⇔ z = S r z ⇔ f (z, y)+Az, y − z + 1 r y − z, J(z − z)≥0, ∀y ∈ C ⇔ f (z, y)+Az, y − z≥0, ∀y ∈ C ⇔ z ∈ GEP ( f ) . Hence, F (S r )=GEP (f). (5) Finally, we claim that GEP (f) is nonempty, closed, and convex. For each y Î C, we define the mapping Θ : C ® 2 E by  ( y ) = {x ∈ C : f ( x, y ) + Ax, y − x≥0} . Since y Î Θ (y), we have Θ(y ) ≠ ∅ We prove that Θ is a KKM mapping on C. Supposethatthereexistsafinitesubset{z 1 , z 2 , , z m }ofC and a i >0with  m i =1 α i = 1 such that ˆ z =  m i =1 α i z i ∈ (z i ) for all i = 1, 2, , m. Then, f ( ˆz, z i ) + Aˆz, z i −ˆz < 0, i =1,2, , m . From (A1) and (A4), we have 0=f (ˆz, ˆz)+Aˆz, ˆz −ˆz≤ m  i =1 α i  f (ˆz, z i )+Aˆz, z i −ˆz  < 0 , which is a contradiction. Thus, Θ is a KKM mapping on C. Next, we prove that Θ (y)isclosedforeachy Î C. For any y Î C,let{x n }beany sequence in Θ (y)suchthatx n ® x 0 . We claim that x 0 Î Θ (y). Then, for each y Î C, we have f ( x n , y ) + Ax n , y − x n ≥0 . By (A3), we see that f (x 0 , y)+Ax 0 , y − x 0 ≥lim sup n →∞ f (x n , y) + lim n→∞ Ax n , y − x n ≥0 . Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 7 of 12 This shows that x 0 Î Θ (y)andΘ(y)isclosedforeachy Î C. Thus,  y ∈C (y)=GEP(f ) is also closed. We observe that Θ (y) is weakly compact. In fact, since C is bounded, closed, and conv ex, we also have Θ(y) is weakly compact in the weak topology. By Lemma 2.5, we can conclude that  y ∈C (y)=GEP(f ) = ∅ . Finally, we prove that GEP (f) is convex. In fact, let u, v Î F (S r ) and z t = tu+(1 - t)v for t Î (0, 1). From (3), we know that S r u − S r z t , J ( S r z t − z t ) − J ( S r u − u ) ≥0 . This yields that u − S r z t , J ( S r z t − z t ) ≥0 . (3:4) Similarly, we also have v − S r z t , J ( S r z t − z t ) ≥ 0 . (3:5) It follows from (3.4) and (3.5) that ||z t − S r z t || 2 = z t − S r z t , J(z t − S r z t ) = tu − S r z t , J(z t − S r z t ) +(1− t)v − S r z t , J(z t − S r z t )  ≤ 0. Hence, z t Î F (S r )=GEP (f) and hence GEP (f) is convex. This completes the proof. 4. Strong convergence theorem In this section, we prove a strong convergence theorem using a hybrid projection algo- rithm in a uniformly convex and smooth Banach space. Theorem 4.1. Let E be a uniformly convex and smooth Banach space and C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × Cto R satisfying (A1)-(A4), A an a-inverse strongly monotone mappin g of C into E*and {T n } ∞ n = 0 a sequence of nonexpansive mappings of C into itself such that  :=  ∞ n = 0 F(T n ) ∩ GEP(f ) = ∅ and suppose that {T n } ∞ n = 0 satisfies the NST-condition. Let {x n } be the sequence in C generated by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 0 ∈ C, D 0 = C, C n = co{z ∈ C : ||z − T n z|| ≤ t n ||x n − T n x n ||}, n ≥ 1, u n ∈ C such that f(u n , y)+Au n , y − u n  + 1 r n y − u n , J(u n − x n )≥0, ∀y ∈ C, n ≥ 0 , D n = {z ∈ D n−1 : u n − z, J(x n − u n )≥0}, n ≥ 1, x n+1 = P C n ∩D n x 0 , n ≥ 0, (4:1) where {t n } and {r n } are sequences which satisfy the following conditions: (C1) {t n } ⊂ (0, 1) and lim n®∞ t n =0; (C2) {r n } ⊂ (0, 1) and lim inf n®∞ r n >0. Then, the sequence {x n } converges strongly to P F x 0 . Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 8 of 12 Proof. First, we rewrite the algorithm (4.1) as the following: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x 0 ∈ C, D 0 = C, C n = co{z ∈ C : ||z − T n z|| ≤ t n ||x n − T n x n ||}, n ≥ 1 , D n = {z ∈ D n−1 : S r n x n − z, J(x n − S r n x n )≥0}, n ≥ 1 , x n+1 = P C n ∩D n x 0 , n ≥ 0, (4:2) where S r is the mapping defined by (3.1) for all r > 0. We first show that th e sequence {x n } is well defined. It is easy to verify that C n ∩ D n is closed and convex and Ω ⊂ C n for all n ≥ 0. Next, we prove that Ω ⊂ C n ∩ D n .SinceD 0 = C,wealso have Ω ⊂ C 0 ∩ D 0 .SupposethatΩ ⊂ C k -1 ∩ D k -1 for k ≥ 2. It follows from Lemma (3) that S r k x k − S r k u, J(S r k u − u) − J(S r k x k − x k )≥0 , for all u Î Ω. This implies that S r k x k − u, J(x k − S r k x k )≥0 , for all u Î Ω. Hence, Ω ⊂ D k . By the mathematical induction, we get that Ω ⊂ C n ∩ D n for each n ≥ 0 and hence {x n } is well defined. Let w = P F x 0 .SinceΩ ⊂ C n ∩ D n and x n+1 = P C n ∩D n x 0 , we have || x n+1 − x 0 || ≤ || w − x 0 || , n ≥ 0 . (4:3) Since {x n } is bounded, there exists a subsequenc e {x n i } of {x n } such that x n i  v ∈ C . Since x n+2 Î D n+1 ⊂ D n and x n+1 = P C n ∩D n x 0 , we have || x n+1 − x 0 || ≤ || x n+2 − x 0 ||. Since {x n - x 0 } is bounded, we have lim n®∞ ||x n - x 0 || = d for some a constant d. Moreover, by the convexity of D n , we also have 1 2 (x n+1 + x n+2 ) ∈ D n and hence | |x 0 − x n+1 || ≤    x 0 − x n+1 + x n+2 2    ≤ 1 2 ( ||x 0 − x n+1 || + ||x 0 − x n+2 || ) . This implies that lim n→∞     1 2 (x 0 − x n+1 )+ 1 2 (x 0 − x n+2 )     = lim n→∞    x 0 − x n+1 + x n+2 2    = d . By Lemma 2.1, we have lim n ®∞ ||x n - x n+1 || = 0. Next, we show that v ∈  ∞ n = 0 F(T n ) .Sincex n+1 Î C n and t n >0,thereexists m ∈ N , {l 0 , l 1 , , l m } ⊂ [0, 1] and {y 0 , y 1 , , y m } ⊂ C such that m  i=1 λ i =1,      x n+1 − m  i=0 λ i y i      < t n , and —|y i − T n y i || ≤ t n ||x n − T n x n | | Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 9 of 12 for each i = 0, 1, , m. Since C is bounded, by Lemma 2.2, we have ||x n − T n x n || ≤ ||x n − x n+1 || +      x n+1 − m  i=0 λ i y i      +      m  i=0 λ i y i − m  i=0 λ i T n y i      +      m  i=0 λ i T n y i − T n  m  i=0 λ i y i       +      T n  m  i=0 λ i y i  − T n x n      ≤ 2||x n − x n+1 || +(2+2M)t n + γ −1  max 0≤i≤j≤m (||y i − y j || − ||T n y i − T n y j ||)  ≤ 2||x n − x n+1 || +(2+2M)t n + γ −1  max 0≤i≤j≤m (||y i − T n y i || − ||y j − T n y j ||)  ≤ 2||x n − x n+1 || + ( 2+2M ) t n + γ −1 ( 4Mt n ) , where M =sup n≥ 0 ||x n - w||. It follows from (C1) that lim n ®∞ ||x n - T n x n || = 0. Since { T n } satisfies the NST-condition, we have v ∈  ∞ n = 0 F(T n ) . Next, we show that v Î GEP (f). By the construction of D n ,weseefrom(2.2)that S r n x n = P D n x n . Since x n+1 Î D n , we obtain | |x n − S r n x n || ≤ ||x n − x n+1 || → 0 , as n ® ∞. From (C2), we also have 1 r n   J(x n − S r n x n )   = 1 r n ||x n − S r n x n || → 0 , (4:4) as n ® ∞. Since {x n } is bounded, it has a subsequence {x n i } which weakly converges to some v Î E. By (4.4), we also have S r n i  v . By the definition of S r n j , for each y Î C, we obtain f (S r n i x n i , y)+AS r n i x n i , y − S r n i x n i  + 1 r n i y − S r n i x n i , J(S r n i x n i − x n i )≥0 . By (A3) and (4.4), we have f ( v, y ) + Av, y − v≥0, ∀y ∈ C . This shows that v Î GEP (f) and hence v ∈  :=  ∞ n = 0 F(T n ) ∩ GEP(f ) . Note that w = P Ω x 0 . Finally, we show that x n ® w as n ® ∞. By the weakly lower semicontinuity of the norm, it follows from (4.3) that | |x 0 − w|| ≤ ||x 0 − v|| ≤ lim inf i→∞ ||x 0 − x n i || ≤ lim sup i →∞ ||x 0 − x n i || ≤ ||x 0 − w|| . This shows that lim i →∞ ||x 0 − x n i || = ||x 0 − w|| = ||x 0 − v| | and v = w. Since E is uniformly convex, we obtain that x 0 − x n i → x 0 − w . It follows that x n i → w . Hence, we have x n ® w as n ® w. This completes the proof. 5. Deduced theorems If we take f ≡ 0 and A ≡ 0 in Theorem 4.1, then we obtain the following result. Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 10 of 12 [...]... Properties of fixed-point sets of nonexpansive mappings in Banach spaces Trans Am Math Soc 179, 251–262 (1973) 20 Aoyama, K, Kimura, Y, Takahashi, W, Toyoda, M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space Nonlinear Anal 67, 2350–2360 (2007) doi:10.1016/j.na.2006.08.032 21 Fan, K: A generalization of Tychonoffs fixed point theorem Math Ann 142,... Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mappings in a Hilbert space J Nonlinear Anal 69, 1025–1033 (2008) doi:10.1016/j.na.2008.02.042 2 Kohsaka, F, Takahashi, W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space Abstr Appl Anal 3, 239–249 (2004) 3 Martinet, B: Regularisation d’inequations... nonexpansive mappings Bull Aust Math Soc 74, 143–151 (2006) doi:10.1017/S0004972700047535 14 Matsushita, S, Takahashi, W: Approximating fixed points of nonexpansive mappings in a Banach space by metric projections Appl Math Comput 196, 422–425 (2008) doi:10.1016/j.amc.2007.06.006 15 Kimura, Y, Nakajo, K: Some characterizations for a family of nonexpansive mappings and convergence of a generated sequence to their... doi:10.1287/moor.26.2.248.10558 9 Nakajo, K, Shimoji, K, Takahashi, W: Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces J Nonlinear Convex Anal 8, 11–34 (2007) 10 Nakajo, K, Shimoji, K, Takahashi, W: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces Taiwan J Math 10, 339–360 (2006) 11 Takahashi, W, Takeuchi, Y, Kubota, R: Strong... methods for families of nonexpansive mappings in Hilbert spaces J Math Anal Appl 341, 276–286 (2008) doi:10.1016/j.jmaa.2007.09.062 12 Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups J Math Anal Appl 279, 372–379 (2003) doi:10.1016/S0022-247X(02)00458-4 13 Xu, H-K: Strong convergence of approximating fixed point sequences for nonexpansive mappings. ..Kamraksa and Wangkeeree Fixed Point Theory and Applications 2011, 2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11 Page 11 of 12 Theorem 5.1 Let E be a uniformly convex and smooth Banach space, C a nonempty, bounded, closed, and convex subset of E and {Tn }∞ a sequence of nonexpansive mapn=0 pings of C into itself such that := ∞ n=0 F(Tn ) = 0and suppose that {Tn }∞ satisfies... 186–201 (2008) doi:10.1016/j.cam.2007.02.022 doi:10.1186/1687-1812-2011-11 Cite this article as: Kamraksa and Wangkeeree: Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces Fixed Point Theory and Applications 2011 2011:11 Page 12 of 12 ... doi:10.1007/BF01353421 22 Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces J Nonlinear Anal 70, 45–57 (2009) doi:10.1016/j.na.2007.11.031 23 Ceng, L-C, Yao, J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems J Comput Appl Math 214, 186–201 (2008) doi:10.1016/j.cam.2007.02.022 doi:10.1186/1687-1812-2011-11... their common fixed point Fixed Point Theory Appl (2010) 16 Takahashi, W: Nonlinear Functional Analysis Yokohama Publishers (2000) 17 Zeidler, E: Nonlinear Functional Analysis and Its Applications, vol I: Fixed-Point Theorems Springer, New York (1986) 18 Bruck, RE: On the convex approximation property and the asymptotic behaviour of nonlinear contractions in Banach spaces Isr J Math 38, 304–314 (1981)... 1) and lim infn®∞ rn > 0, then {xn} converges strongly to PGEP (f)x0 If we take A ≡ 0 in Theorem 4.1, then we obtain the following result concerning an equilibrium problem in a Banach space setting Theorem 5.4 Let E be a uniformly convex and smooth Banach space and C be a nonempty, bounded, closed, and convex subset of E Let f be a bifunction from C × C to Rsatisfying (A1 )- (A4 ) and let {Tn }∞ be a sequence . RESEARC H Open Access Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces Uthai Kamraksa 1 and Rabian Wangkeeree 1,2* *. and Wangkeeree: Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces. Fixed Point Theory and Applications 2011. 251–262 (1973) 20. Aoyama, K, Kimura, Y, Takahashi, W, Toyoda, M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360