RESEARC H Open Access Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions Atid Kangtunyakarn Correspondence: beawrock@hotmail.com Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Abstract In this article, we introduce a new mapping generated by an infinite family of  i - strict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo- contractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces. Keywords: nonexpansive mappings, strongly positive operator, generalized equili- brium problem, strict pseudo-contraction, fixed point 1 Introduction Let C be a closed convex subset of a real Hilbert space H, and let G : C × C ® ℝ be a bifunction. We know that the equilibrium problem for a bifunction G is to find x Î C such that G ( x, y ) ≥ 0 ∀y ∈ C . (1:1) The set of solutions of (1.1) is denoted by EP(G). Given a mapping T : C ® H,letG (x, y)=〈Tx, y - x〉 fo r all x, y Î. Then, z Î EP(G) if and only if 〈Tz, y - z〉 ≥ 0 for all y Î C,i.e.,z is a solution of the variational inequality. Let A : C ® H beanonlinear mapping. The variational inequality problem is to find a u Î C such that  v − u, A u  ≥ 0 (1:2) for all v Î C. The set of solutions of the variational inequality is denoted by VI(C, A). Now, we consider the following generalized equilibrium problem: Find z ∈ C such that G ( z, y ) + Az, y − z≥0, ∀y ∈ C . (1:3) The set of such z Î C is denoted by EP(G, A), i.e., EP ( G, A ) = {z ∈ C : G ( z, y ) + Az, y − z≥0, ∀y ∈ C . Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 © 2011 Kangtunyakarn; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use , distribution, and reproduction in any medium, provided the original work is properly cited. In the case of A ≡ 0, EP(G, A) is denoted by EP(G). In th e case of G ≡ 0, EP(G, A)is also denoted by VI(C, A). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics reduce to find a solution of (1.3) (see, for instance, [1]-[3]). A mapping A of C into H is called inverse-strongly monot one (see [4]), if there exists a positive real number a such that x − y , Ax − A y ≥α||Ax − A y || 2 for all x, y Î C. A mapping T with domain D(T) and range R(T) is called nonexpansive if || Tx − T y|| ≤ || x − y|| (1:4) for all x, y Î D(T) and T is said to be - strict pseudo-contration if there exist  Î [0, 1) such that ||Tx − Ty|| 2 ≤||x − y|| 2 + κ|| ( I − T ) x − ( I − T ) y|| 2 , ∀x, y ∈ D ( T ). (1:5) We know that the class of -strict pseudo-contractions includes class of nonexpan- sive mappings. If  = 1, then T is said to be pseudo-contractive. T is strong pseudo-con- traction if there exists a positive constant l Î (0, 1) such that T + lI is pseudo- contractive. In a real Hilbert space H (1.5) is equivalent to Tx − Ty, x − y≤ ||x − y|| 2 − 1 − κ 2 ||(I − T)x − (I − T)y|| 2 ∀x, y ∈ D(T) . (1:6) T is pseudo-contractive if and only if Tx − Ty , x − y≤ ||x − y|| 2 ∀x, y ∈ D ( T ). Then, T is strongly pseudo-contractive, if there exists a positive constant l Î (0, 1) such that Tx − Ty , x − y≤ ( 1 − λ ) x − y 2 , ∀x, y ∈ D ( T ). The class of -strict pseudo-contractions fall into the one between classes of nonex- pansive mappings and pseudo-contractions, and the class of strong pseudo-contrac- tions is independent of the class of -strict pseudo-contractions. We denote by F(T) the set of fixed points of T.IfC ⊂ H is bounded, closed and con- vex and T is a nonexpansive mapping of C into itself, then F(T) is nonem pty; for instance, see [5]. Recently, Tada and Takahashi [6] and Takahashi and Takahashi [7] considered iterative methods for finding an element of EP(G) ∩ F(T). Browder and Pet- ryshyn [8] showed that if a -strict pseudo-contraction T has a fixed point in C, then starting with an initial x 0 Î C, the sequence {x n } generated by the recursive formula: x n+1 = αx n + ( 1 − α ) Tx n , (1:7) where a isaconstantsuchthat0< a <1, converges weakly to a fixed point of T. Marino and Xu [9] extended Browder and Petryshyn’s above mentioned result by prov- ing that the sequence {x n } generated by the following Manns algorithm [10]: x n+1 = α n x n + ( 1 − α n ) Tx n (1:8) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 2 of 16 converges weakly to a fixed point of T provided the control sequence {α n } ∞ n = 0 satisfies the conditions that  < a n <1 for all n and  ∞ n = 0 (α n − κ)(1 − α n )= ∞ . Recently, in 2009, Qin et al. [11] introd uced a general iterative method for finding a common element of EP(F, T), F(S), and F(D). They defined {x n } as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 , u ∈ C, Fu n , y + Tx n , y − u n  + 1 r y − u n , u n − x n , ∀y ∈ C, y n = P C (x n − ηBx n ), v n = P C (y n − λAy n ), x n+1 = α n u + β n x n + γ n (μ 1 S k x n + μ 2 u n + μ 3 v n ), ∀n ∈ N , (1:9) where the mapping D : C ® C is defined by D(x)=P C (P C (x - hBx)-lAP C (x - hBx)), S k is the mapping defined by S k x = kx +(1-k)Sx, ∀x Î C, S : C ® C is a -strict pseudo-contraction, and A, B : C Î H are a-inverse-strongly monotone mapping and b-inverse-strongly monotone mappings, respectively. Under suitable conditions, they proved strong convergence of {x n } defined by (1.9) to z = P EP(F, T)∩F(S) ∩F(D) u. Let C be a nonempty convex subset of a real Hilbe rt sp ace. Let T i , i = 1, 2, be map- pings of C into itself. For each j = 1, 2, , let α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I where I = [0, 1] and α j 1 + α j 2 + α j 3 = 1 .Foreveryn Î N, we define the mapping S n : C ® C as follows: U n,n+1 = I U n,n = α n 1 T n U n,n+1 + α n 2 U n,n+1 + α n 3 I U n,n−1 = α n−1 1 T n−1 U n,n + α n−1 2 U n,n + α n−1 3 I . . . U n,k+1 = α k+1 1 T k+1 U n,k+2 + α k+1 2 U n,k+2 + α k+1 3 I U n,k = α k 1 T k U n,k+1 + α k 2 U n,k+1 + α k 3 I . . . U n,2 = α 2 1 T 2 U n,1 + α 2 2 U n,1 + α 2 3 I S n = U n,1 = α 1 1 T 1 U n,2 + α 1 2 U n,2 + α 1 3 I. This mapping is called S-mapping generated by T n , , T 1 and a n , a n-1 , , a 1 . Question. H ow can we define an iterative method for finding an element in F =  ∞ i =1 F( T i )  N i =1 EF(F i , A i )  N i =1 F( G i ) ? In this article, motivated by Qin et al. [11], by using S-mapping, we int roduce a new iteration method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Our iteration scheme is define as follows. For u, x 1 Î C, let {x n } be generated by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ F i v i n , v + A i x n , v − v i n  + 1 r i v − v i n , v i n − x n , ∀v ∈ C, i =1,2, , N . y n =  N i=1 δ i v i n x n+1 = α n u + β n x n + γ n (a n S n x n + b n Bx n + c n y n ), ∀n ∈ N. For i = 1, 2, , N,letF i : C × C ® ℝ be bifunction, A i : C ® H be a i -inverse strongly monotone and let G i : C ® C be defined by G i (y)=P C (I - l i A i )y, ∀y Î C with Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 3 of 16 (0, 1] ⊂ (0, 2 a i ) such that F =  ∞ i =1 F( T i )  N i =1 EF(F i , A i )  N i =1 F( G i ) = ∅ , where B is the K-mapping generated by G 1 , G 2 , , G N and b 1 , b 2 , , b N . We prove a strong convergence theorem of purposed iterative sequence {x n }toa point z ∈ F and z is a solution of (1.10) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x − z, A 1 z≥0 x − z, A 2 z≥0 . . . x − z, A N z≥0, ∀x ∈ C and λ i ∈ ( 0, 1] i =1,2, , N . (1:10) 2 Preliminaries In this section, we coll ect and provide someusefullemmasthatwillbeusedforour main result in the next section. Let C be a closed convex subset of a real Hilbert space H,andletP C be the metric projection of H onto C i.e., so that for x Î H, P C x satisfies the property: | x − P C x|| =m i n y ∈C ||x − y|| . The following characterizes the projection P C . Lemma 2.1 [5]. Given x Î HandyÎ C. Then, P C x = y if and only if there holds the inequality x − y , y − z≥0 ∀z ∈ C . Lemma 2.2 [12]. Let {s n } be a sequence of nonnegative real number satisfying s n+1 = ( 1 − α n ) s n + α n β n , ∀n ≥ 0 where {a n }, {b n } satisfy the conditions (1) {α n }⊂[0, 1], ∞  n=1 α n = ∞; (2) lim sup n→∞ β n ≤ 0 or ∞  n =1 |α n β n | < ∞ . Then lim n®∞ s n =0. Lemma 2.3 [13]. Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n Î N} be a sequence of nonexpansive mappings on C. Suppose  ∞ n =1 F( T n ) is nonempty. Let {l n } be a sequence of positive numbers with  ∞ n =1 λ n = 1 . Then, a mapping S on C defined by S(x)= ∞ n =1 λ n T n x n for x Î C is well defined, nonexpansive and F( S)=  ∞ n =1 F( T n ) hold. Lemma 2.4 [14] . Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : C ® C be a nonexpansive mapping. Then, I - Sisdemi- closed at zero. Lemma 2.5 [15]. Let {x n } and {z n } be bounded sequences in a Banach space X and let {b n } be a sequence in 0[1]with 0 <lim inf n®∞ b n ≤ lim sup n®∞ b n <1. Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 4 of 16 Suppose x n+1 = β n x n + ( 1 − β n ) z n for all integer n ≥ 0 and lim sup n →∞ (||z n+1 − z n || − ||x n+1 − x n ||) ≤ 0 . Then lim n®∞ ||x n - z n || = 0. 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∀x Î C; (A2) F is monotone, i.e. F(x, y)+F(y, x) ≤ 0, ∀x, y Î C; (A3) ∀x , y, z Î C, lim t → 0 + F( tz +(1− t)x, y) ≤ F(x, y) ; (A4) ∀x Î C, y ↦ F(x, y) is convex and lower semicontinuous. The following lemma appears implicitly in [1]. Lemma 2.6 [1]. LetCbeanonemptyclosedconvexsubsetofH,andletFbea bifunction of C × C into ℝ satisfying (A1) - (A4). Let r >0 and x Î H. Then, there exists z Î C such that F( z , y)+ 1 r y − z, z − x  (2:1) for all x Î C. Lemma 2.7 [16]. Assume that F : C × C ® ℝ satisfies (A1) - (A4). For r >0 and x Î H, define a mapping T r : H ® C as follows. T r (x)={z ∈ C : F(z, y)+ 1 r y − z, z − x≥0, ∀y ∈ C} . for all z Î H. Then, the following hold. (1) T r is single-valued, (2) T r is firmly nonexpansive i.e T r ( x ) − T r ( y )  2 ≤T r ( x ) − T r ( y ) , x − y∀x, y ∈ H ; (3) F(T r )=EP (F ); (4) EP (F) is closed and convex. Definition 2.1 [17]. Let C be a nonempty convex subset of real Banach space. Let {T i } N i = 1 be a finite family of nonexpanxive mappings of C into itself, and let l 1 , ,l N be real num- bers such that 0 ≤ l i ≤ 1 for every i = 1, , N . We define a mapping K : C ® Casfollows. U 1 = λ 1 T 1 +(1− λ 1 )I, U 2 = λ 2 T 2 U 1 +(1− λ 2 )U 1 , U 3 = λ 3 T 3 U 2 +(1− λ 3 )U 2 , . . . U N−1 = λ N−1 T N−1 U N−2 +(1− λ N−1 )U N−2 , K = U N = λ N T N U N−1 + ( 1 − λ N ) U N−1 . (2:3) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 5 of 16 Such a mapping K is called the K-mapping generated by T 1 , , T N and l 1 , , l N . Lemma 2.8 [17]. LetCbeanonemptyclosedconvexsubsetofastrictlyconvex Banach space. Let {T i } N i = 1 be a finite family of nonexpanxive mappings of C in to itself with  N i =1 F( T i ) = ∅ and let l 1 , ,l N be real numbers such that 0 < l i <1 for every i = 1, , N -1and 0 < l N ≤ 1. Let K be the K-mapping generated by T 1 , , T N and l 1 , , l N . Then F( K)=  N i =1 F( T i ) . Lemma 2.9 [9]. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH and S : C ® C be a self-mapping of C. If S is a -strict pseudo-contraction mapping, then S satisfies the Lipschitz condition. | |Sx − Sy|| ≤ 1+κ 1 − κ ||x − y||, ∀x, y ∈ C . Lemma 2.10. Let C be a nonempty closed convex subset of a real Hilbert space. Let {T i } N i = 1 be  i -strict pseudo-contr action mappings of C into itself with  ∞ i =1 F( T i ) = ∅ and  =sup i  i and let α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I , where I = [0, 1], α j 1 + α j 2 ≤ b < 1 , α j 1 + α j 2 ≤ b < 1 , and α j 1 , α j 2 , α j 3 ∈ (κ,1 ) for all j = 1, 2, For every n Î N, let S n be S-mapping generated by T n , ,T 1 and a n , a n-1 , , a 1 . Then, for every x Î C and k Î N, lim n®∞ U n , k x exists. Proof. Let x Î C and y ∈  ∞ i =1 F( T i ) . Fix k Î N, then for every n Î N with n ≥ k, we have U n+1,k x − U n,k x 2 = α k 1 T k U n+1,k+1 x + α k 2 U n+1,k+1 x + α k 3 x − α k 1 T k U n,k+1 x −α k 2 U n,k+1 x − α k 3 x 2 = α k 1 (T k U n+1,k+1 x − T k U n,k+1 x)+α k 2 (U n+1,k+1 x − U n,k+1 x) 2 ≤ α k 1 T k U n+1,k+1 x − T k U n,k+1 x 2 + α k 2 U n+1,k+1 x − U n,k+1 x 2 −α k 1 α k 2 T k U n+1,k+1 x − T k U n,k+1 x − U n+1,k+1 x + U n,k+1 x 2 ≤ α k 1 (U n+1,k+1 x − U n,k+1 x 2 + κ(I − T k )U n+1,k+1 x −(I − T k )U n,k+1 x 2 )+α k 2 U n+1,k+1 x − U n,k+1 x 2 −α k 1 α k 2 (I − T k )U n,k+1 x − (I − T k )U n+1,k+1 x 2 ≤ (1 − α k 3 )U n+1,k+1 x − U n,k+1 x 2 . . . ≤  n j=k (1 − α j 3 )U n+1,n+1 x − U n,n+1 x 2 =  n j=k (1 − α j 3 )α n+1 1 T n+1 U n+1,n+2 x + α n+1 2 U n+1,n+2 x + α n+1 3 x − x 2 =  n j=k (1 − α j 3 )α n+1 1 T n+1 x +(1− α n+1 1 )x − x 2 =  n j=k (1 − α j 3 )α n+1 1 (T n+1 x − x) 2 ≤  n j=k (1 − α j 3 )(T n+1 x − y + y − x) 2 ≤  n j=k (1 − α j 3 )  1+κ 1 − κ x − y + y − x  2 ≤  n j=k (1 − α j 3 )  2 1 − κ x − y  2 ≤ b n−(k−1)  2 1 − κ x − y   2 . It follows that ||U n+1,k x − U n,k x|| ≤ b n − (k − 1) 2  2 1 − κ ||x − y||  Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 6 of 16 = b n 2 b k−1 2  2 1 − κ ||x − y||  = a n a k−1 M, (2:4) where a = b 1 2 ∈ ( 0, 1 ) and M = 2 1 − κ ||x − y| | For any k, n, p Î N, p>0, n ≥ k, we have  U n+p,k x − U n,k x  ≤  U n+p,k x − U n+p−1,k x  +  U n+p−1,k x − U n+p−2,k x  + . +U n+1,k x − U n,k x =  n+p−1 j=n U j+1,k x − U j,k x ≤  n+p−1 j=n a j a k−1 M ≤ a n ( 1 − a ) a k−1 M. (2:5) Since a Î (0, 1), we have lim n®∞ a n =0.From(2.5),wehavethat{U n , k x}isaCau- chy sequence. Hence lim n®∞ U n,k x exists. □ For every k Î N and x Î C, we define mapping U ∞,k and S : C Î C as follows: lim n → ∞ U n,k x = U ∞,k x (2:6) and lim n → ∞ S n x = lim n → ∞ U n,1 x = S x (2:7) Such a mapping S is called S-mapping generated by T n , T n-1 , and a n , a n-1 , Remark 2.11. For each n Î N, S n is nonexpansive and lim n®∞ sup xÎD ||S n x - Sx|| = 0 for every bounded subset D of C. To show this, let x, y Î C and D be a bounded subset of C. Then, we have S n x − S n y 2 = α 1 1 (T 1 U n,2 x − T 1 U n,2 y)+α 1 2 (U n,2 x − U n,2 y)+α 1 3 (x − y) 2 ≤ α 1 1 T 1 U n,2 x − T 1 U n,2 y 2 + α 1 2 U n,2 x − U n,2 y 2 + α 1 3 x − y 2 −α 1 1 α 1 2 T 1 U n,2 x − T 1 U n,2 y − U n,2 x + U n,2 y 2 ≤ α 1 1 (U n,2 x − U n,2 y 2 + κ(I − T 1 )U n,2 x − (I − T 1 )U n,2 y 2 ) +α 1 2 U n,2 x − U n,2 y 2 + α 1 3 x − y 2 − α 1 1 α 1 2 (I − T 1 )U n,2 y − (I − T 1 )U n,2 x 2 ≤ (1 − α 1 3 )U n,2 x − U n,2 y 2 + α 1 3 x − y 2 ≤ (1 − α 1 3 )((1 − α 2 3 )U n,3 x − U n,3 y 2 + α 2 3 x − y 2 )+α 1 3 x − y) 2 =(1− α 1 3 )(1 − α 2 3 )U n,3 x − U n,3 y 2 + α 2 3 (1 − α 1 3 )x − y 2 + α 1 3 x − y) 2 =  2 j=1 (1 − α j 3 )||U n,3 x − U n,3 y|| 2 +(1−  2 j=1 (1 − α j 3 ))||x − y|| 2 . . . ≤  n j=1 (1 − α j 3 )||U n,n+1 x − U n,n+1 y|| 2 +(1−  n j=1 (1 − α j 3 ))||x − y|| 2 = ||x − y || 2 . Then, we have that S : C ® C is also nonexpansive indeed, observe that for each x, y Î C | Sx − Sy|| = lim n → ∞ ||S n x − S n y|| ≤ ||x − y|| . By (2.8), we have || S n+1 x − S n x|| = ||U n+1,1 x − U n,1 x| | ≤ a n M. Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 7 of 16 This implies that for m>nand x Î D, | |S m x − S n x|| ≤  m−1 j=n ||S j+1 x − S j x| | ≤  m−1 j=n a j M ≤ a n 1 − a M. By letting m ® ∞, for any x Î D, we have | |Sx − S n x|| ≤ a n 1 − a M . (2:8) It follows that lim n → ∞ sup x∈D ||S n x − Sx|| =0 . (2:9) Lemma 2.12. Let C be a nonempty closed convex subset of a real Hilbert space. Let {T i } ∞ i = 1 be  i -strict pseudo-contraction mappings of C into itself with  ∞ i =1 F( T i ) = ∅ and  =sup iÎ  i and let α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I , where I = [0, 1], α j 1 + α j 2 + α j 3 = 1 , α j 1 , α j 2 , α j 3 ∈ (κ,1 ) and α j 1 , α j 2 , α j 3 ∈ (κ,1 ) for all j = 1, For every n Î N, let S n and S be S-mappings generated by T n , , T 1 and a n , a n-1 , , a 1 and T n , T n-1 , , and a n , a n- 1 , , respectively. Then F( S)=  ∞ i =1 F( T i ) . Proof. It is evident that  ∞ i =1 F( T i ) ⊆ F(S ) . For every n, k Î N, with n ≥ k,letx 0 Î F (S) and x ∗ ∈  ∞ i =1 F( T i ) , we have S n x 0 − x ∗  2 = α 1 1 (T 1 U n,2 x 0 − x ∗ )+α 1 2 (U n,2 x 0 − x ∗ )+α 1 3 (x 0 − x ∗ ) 2 ≤ α 1 1 T 1 U n,2 x 0 − x ∗  2 + α 1 2 U n,2 x 0 − x ∗  2 + α 1 3 x 0 − x ∗  2 −α 1 1 α 1 2 T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 ≤ α 1 1 (U n,2 x 0 − x ∗  2 + κ(I − T 1 )U n,2 x 0  2 )+α 1 2 U n,2 x 0 − x ∗  2 +α 1 3 x 0 − x ∗  2 − α 1 1 α 1 2 T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 =(1− α 1 3 )U n,2 x 0 − x ∗  2 + α 1 3 x 0 − x ∗  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 ≤ (1 − α 1 3 )((1 − α 2 3 )U n,3 x 0 − x ∗  2 + α 2 3 x 0 − x ∗  2 −α 2 1 (α 2 2 − κ)T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 U n,3 x 0 − x 0  2 )+α 1 3 x 0 − x ∗  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 =(1− α 1 3 )(1 − α 2 3 )U n,3 x 0 − x ∗  2 + α 2 3 (1 − α 1 3 )x 0 − x ∗  2 + α 1 3 x 0 − x ∗  2 −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 =  2 j=1 (1 − α j 3 )U n,3 x 0 − x ∗  2 +(1−  2 j=1 (1 − α j 3 )x 0 − x ∗  2 −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 ≤  2 j=1 (1 − α j 3 )((1 − α 3 3 )U n,4 x 0 − x ∗  2 + α 3 3 x 0 − x ∗  2 −α 3 1 (α 3 2 − κ)T 3 U n,4 x 0 − U n,4 x 0  2 − α 3 2 α 3 3 U n,4 x 0 − x 0  2 ) +(1 −  2 j=1 (1 − α j 3 )x 0 − x ∗  2 − α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 −α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 − α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 −α 1 2 α 1 3 U n,2 x 0 − x 0  2 =  2 j=1 (1 − α j 3 )(1 − α 3 3 )U n,4 x 0 − x ∗  2 + α 3 3  2 j=1 (1 − α j 3 )x 0 − x ∗  2 −α 3 1 (α 3 2 − κ) 2 j=1 (1 − α j 3 )T 3 U n,4 x 0 − U n,4 x 0  2 −α 3 2 α 3 3  2 j=1 (1 − α j 3 )U n,4 x 0 − x 0  2 +(1−  2 j=1 (1 − α j 3 )x 0 − x ∗  2 −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 =  3 j=1 (1 − α j 3 )U n,4 x 0 − x ∗  2 +(1−  3 j=1 (1 − α j 3 )x 0 − x ∗  2 −α 3 1 (α 3 2 − κ) 2 j=1 (1 − α j 3 )T 3 U n,4 x 0 − U n,4 x 0  2 −α 3 2 α 3 3  2 j=1 (1 − α j 3 )U n,4 x 0 − x 0  2 − α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 −α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 − α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 −α 1 2 α 1 3 U n,2 x 0 − x 0  2 . . . (2:10) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 8 of 16 . . . ≤  k+1 j=1 (1 − α j 3 )||U n,k+2 x 0 − x ∗ || 2 +(1−  k+1 j=1 (1 − α j 3 )||x 0 − x ∗ || 2 − α k+1 1 (α k+1 2 − κ) k j=1 (1 − α j 3 )||T k+1 U n,k+2 x 0 − U n,k+2 x 0 || 2 − α k+1 2 α k+1 3  k j=1 (1 − α j 3 )||U n,k+2 x 0 − x 0 || 2 − α k 1 (α k 2 − κ) k−1 j=1 (1 − α j 3 )||T k U n,k+1 x 0 − U n,k+1 x 0 || 2 − α k 2 α k 3  k−1 j =1 (1 − α j 3 )||U n,k+1 x 0 − x 0 || 2 (2:11) . . . −α 3 1 (α 3 2 − κ) 2 j=1 (1 − α j 3 )T 3 U n,4 x 0 − U n,4 x 0  2 − α 3 2 α 3 3  2 j=1 (1 − α j 3 )U n,4 x 0 − x 0  2 −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 . . . ≤  n j=1 (1 − α j 3 )U n,n+1 x 0 − x ∗  2 +(1−  n j=1 (1 − α j 3 )x 0 − x ∗  2 −α n 1 (α n 2 − κ) n−1 j=1 (1 − α j 3 )T n U n,n+1 x 0 − U n,n+1 x 0  2 −α n 2 α n 3  n−1 j=1 (1 − α j 3 )U n,n+1 x 0 − x 0  2 . . . −α k+1 1 (α k+1 2 − κ) k j=1 (1 − α j 3 )T k+1 U n,k+2 x 0 − U n,k+2 x 0  2 −α k+1 2 α k+1 3  k j=1 (1 − α j 3 )U n,k+2 x 0 − x 0  2 −α k 1 (α k 2 − κ) k−1 j=1 (1 − α j 3 )T k U n,k+1 x 0 − U n,k+1 x 0  2 −α k 2 α k 3  k−1 j=1 (1 − α j 3 )U n,k+1 x 0 − x 0  2 −α k−1 1 (α k−1 2 − κ) k−2 j=1 (1 − α j 3 )T k−1 U n,k x 0 − U n,k x 0  2 −α k−1 2 α k−1 3  k−2 j=1 (1 − α j 3 )U n,k x 0 − x 0  2 . . . −α 3 1 (α 3 2 − κ) 2 j =1 (1 − α j 3 )T 3 U n,4 x 0 − U n,4 x 0  2 − α 3 2 α 3 3  2 j =1 (1 − α j 3 )U n,4 x 0 − x 0  2 (2:12) −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 = x 0 − x ∗  2 −α n 1 (α n 2 − κ) n−1 j=1 (1 − α j 3 )T n U n,n+1 x 0 − U n,n+1 x 0  2 . . . −α k+1 1 (α k+1 2 − κ) k j=1 (1 − α j 3 )T k+1 U n,k+2 x 0 − U n,k+2 x 0  2 −α k+1 2 α k+1 3  k j=1 (1 − α j 3 )U n,k+2 x 0 − x 0  2 −α k 1 (α k 2 − κ) k−1 j=1 (1 − α j 3 )T k U n,k+1 x 0 − U n,k+1 x 0  2 −α k 2 α k 3  k−1 j=1 (1 − α j 3 )U n,k+1 x 0 − x 0  2 −α k−1 1 (α k−1 2 − κ) k−2 j=1 (1 − α j 3 )T k−1 U n,k x 0 − U n,k x 0  2 −α k−1 2 α k−1 3  k−2 j=1 (1 − α j 3 )U n,k x 0 − x 0  2 . . . −α 3 1 (α 3 2 − κ) 2 j=1 (1 − α j 3 )T 3 U n,4 x 0 − U n,4 x 0  2 − α 3 2 α 3 3  2 j=1 (1 − α j 3 )U n,4 x 0 − x 0  2 −α 2 1 (α 2 2 − κ)(1 − α 1 3 )T 2 U n,3 x 0 − U n,3 x 0  2 − α 2 2 α 2 3 (1 − α 1 3 )U n,3 x 0 − x 0  2 −α 1 1 (α 1 2 − κ)T 1 U n,2 x 0 − U n,2 x 0  2 − α 1 2 α 1 3 U n,2 x 0 − x 0  2 . (2:13) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 9 of 16 For k Î N and (2.12), we have α k−1 2 α k−1 3  k−2 j=1 (1 − α j 3 )||U n,k x 0 − x 0 || 2 ≤||x 0 − x ∗ || 2 −||S n x 0 − x ∗ || 2 , (2:14) as n ® ∞. This implies that U ∞ , k x 0 = x 0 , ∀k Î N. Again by (2.12), we have α k 1 (α k 2 − κ) k−1 j =1 (1 − α j 3 )||T k U n,k+1 x 0 − U n,k+1 x 0 || 2 ≤||x 0 − x ∗ || 2 −||S n x 0 − x ∗ || 2 , (2:15) as n ® ∞. Hence α k 1 (α k 2 − κ) k−1 j =1 (1 − α j 3 )||T k U ∞,k+1 x 0 − U ∞,k+1 x 0 || 2 ≤ 0 . (2:16) From U ∞,k x 0 = x 0 , ∀k Î N, and (2.15), we obtain that T k x 0 = x 0 , ∀k Î N. This implies that x 0 ∈  ∞ i =1 F( T i ) . □ Lemma 2.13. Let C be a closed convex subs et of Hilbert space H. Let A i : C ® Hbe mappings and let G i : C ® CbedefinedbyG i (y)=P C (I - l i A i )ywithl i >0, ∀ i =1,2, N. Then x ∗ ∈  N i =1 VI(C, A i ) if and only if x ∗ ∈  N i =1 F( G i ) . Proof. For given x ∗ ∈  N i =1 VI(C, A i ) ,wehavex* Î VI(C, A i ), ∀ i =1,2, ,N.Since 〈A i x*, x - x*〉 ≥ 0, we have 〈l i A i x*, x - x*〉 ≥ 0, ∀l i >0, i = 1, 2, , N. It follows that x ∗ − ( I − λ i A i ) x ∗ , x − x ∗  = λ i A i x ∗ , x − x ∗ ≥0, ∀x ∈ C, i =1,2, , N . (2:17) Hence, x*=P C (I - l i A i )x*=G i (x*), ∀x Î C, i =1,2, ,N. Therefore, we have x ∗ ∈  N i =1 F( G i ) . For the converse, let x ∗ ∈  N i =1 F( G i ) ; then, we have for every i = 1, , N, x*=G i (x*) = P C (I - l i A i )x*, ∀l i >0, i = 1, 2, , N. It implies that x ∗ − ( I − λ i A i ) x ∗ , x − x ∗  = λ i A i x ∗ , x − x ∗ ≥0, ∀i =1,2, , N, ∀x ∈ C . (2:18) Hence, 〈A i x*, x-x*〉 ≥ 0, ∀x Î C,sox* Î VI (C, A i ), ∀i =1,2, ,N. Hence, x ∗ ∈  N i =1 VI(C, A i ) . □ 3 Main results Theorem 3.1. Let C be a close d convex subset of Hilbert space H. For every i = 1, 2, , N, let F i : C × C ® ℝ be a bifunction satisfying (A 1 )-(A 4 ), let A i : C ® Hbea i -inverse strongly monotone and let G i : C ® C be defined by G i (y)=P C (I - l i A i )y, ∀y Î C with l i Î (0, 1] ⊂ (0, 2a i ). Let B : C ® C be the K-mapping generated by G 1 , G 2 , , G N and b 1 , b 2 , , b N where b i Î (0, 1), ∀i = 1, 2, 3, , N-1, b N Î (0, 1] and let { T i } ∞ i = 1 be  i - strict pseudo-contraction mappings of C into itself with  = sup i  i and let ρ j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I ,whereI=[0,1], α j 1 + α j 2 + α j 3 = 1 , α j 1 + α j 2 ≤ b < 1 , and α j 1 , α j 2 , α j 3 ∈ (κ,1 ) for all j = 1, 2, . For every n Î N, let S n and S are S-mapping gener- ated by T n , ,T 1 and r n , r n-1 , , r 1 and T n , T n-1 , , and r n , r n-1 , ,respectively. Assume that F =  ∞ i =1 F( T i )  N i =1 EF(F i , A i )  N i =1 F( G i ) = ∅ . For every n Î N, i =1, 2, , N, let {x n } and {v i n } be generated by x 1 , u Î C and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ F i v i n , v + A i x n , v − v i n  + 1 r i v − v i n , v i n − x n ≥0, ∀v ∈ C , i =1,2, , N. y n =  N i=1 δ i v i n x n+1 = α n u + β n x n + γ n (a n S n x n + b n Bx n + c n y n ), ∀n ∈ N, (3:1) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 10 of 16 [...]... problems of finite family of nonexpansive mappings Nonlinear Anal 71, 4448–4460 (2009) doi:10.1016/j.na.2009.03.003 doi:10.1186/1687-1812-2011-23 Cite this article as: Kangtunyakarn: Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions Fixed Point Theory and Applications 2011 2011:23 Page 16 of 16... Ces’aro means for nonexpansive mappings and inverse-strongly monotone mappings J Nonlinear Convex Anal 7, 105–113 (2006) 5 Takahashi, W: Nonlinear Functional Analysis Yokohama Publishers, Yoko-hama (2000) 6 Tada, A, Takahashi, W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping J Optim Theory Appl 133, 359–370 (2007) doi:10.1007/s10957-007-9187-z 7 Takahashi, S, Takahashi,... Dynamical Approaches to Equilibrium Problems In Lecture Notes in Economics and Mathematical Systems, vol 477, pp 187–201 Springer (1999) 3 Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space Nonlinear Anal 69, 1025–1033 (2008) doi:10.1016/j.na.2008.02.042 4 Iiduka, H, Takahashi, W: Weak convergence theorem by Ces’aro... 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Iterative algorithms for nonlinear operators J Lond Math Soc 66, 240–256 (2002) doi:10.1112/ S0024610702003332 13 Bruck, RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces Trans Am Math Soc 179, 251–262 (1973) 14 Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces Proc Sympos Pure Math 18, 78–81 (1976) 15 Suzuki, T: Strong convergence of Krasnoselskii... converge strongly to z = P u Proof From Theorem 3.1, choose N = 1 and let A1 = A, l1 = l Then, we have B(y) = G1(y) = PC(I - lA)y, ∀y Î C Choose v1 = vn, a = an, b = bn, c = cn for all n Î N, and n 1 let T ≡ S1 : C ® C be S-mapping generated by T1 and r1 with T1 = T and α1 = κ, and then we obtain the desired result from Theorem 3.1 □ Acknowledgements The authors would like to thank Professor Dr Suthep Suantai... ) 2 Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:23 http://www.fixedpointtheoryandapplications.com/content/2011/1/23 Page 15 of 16 It follows that xn+1 − z 2 ≤ 2αn u − z, xn+1 − z + (1 − αn ) xn − z 2 (3:26) From Step 4, (3.26), and Lemma 2.2, we have limn®∞ xn = z, where z = P u The proof is complete □ 4 Applications From Theorem 3.1, we obtain the following strong convergence theorems . RESEARC H Open Access Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions Atid Kangtunyakarn Correspondence: beawrock@hotmail.com Department. Kangtunyakarn: Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions. Fixed Point Theory and Applications. element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo- contractions.