RESEARC H Open Access Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems 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 infinite family of nonexpansive mapping and infinite real numbers. By means of the new mapping, we prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of nonexpansive mappings and the set of a finite family of variational inclusion problems in Hilbert space. In the last section, we apply our main result to prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of strictly pseudo- contractive mappings and the set of finite family of variational inclusion problems. Keywords: nonexpansive mapping, strict pseudo contraction, strongly positive operator, variational inclusion problem, fixed point 1 Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let A : C ® H be a nonlinear mapping and let F : C × C ® ℝ be a bifunction. A mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y Î H. We denote by F (T) the set of fixed points of T (i.e. F(T)={x Î H : Tx = x}). Goebel and Kirk [1] showed that F(T) is always closed convex and also nonempty provided T has a bounded trajectory. The problem for finding a common fixed point of a family of nonexpansive map- pings has been studied by many authors. The well-known convex f easibility problem reduces to finding a point in the intersection of the fixed point sets of a family of non- expansive mappings (see, e.g., [2,3]). A bounded linear operator A on H is called strongly positive with coefficient ¯ γ if there exists a constant ¯ γ > 0 with the property Ax, x≥ ¯ γ  x 2 . A mapping A of C into H is called inverse-strongly monotone, 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. The variational inequality problem is to find a point u Î C such that  v − u, Au  ≥ 0forallv ∈ C . (1:1) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 © 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. The set of solutions of (1.1) is denoted by VI( C, A). Many authors have studied methods for finding solution of variational inequality problems (see, e.g., [5-8]). In 2008, Qin et al. [9] introduced the following iterative scheme:  y n = P C (I − s n A)x n x n+1 = α n γ f (W n x n )+(I − α n B)W n P C (I − r n A)y n , ∀n ∈ N , (1:2) where W n is the W-mapping generated by a finite family o f nonexpansive mappings and real numbers, A : C ® H is relaxed (u,v) cocoercive and μ-Lipschitz continuous, and P C isametricprojectionH onto C. Under suitable conditions of {s n }, {r n }{a n }, g,they proved that {x n } converges strongly to an element of the set of variational inequality pro- blem and the set of a common fixed point of a finite family of nonexpansive mappings. In 2006, Marino and Xu [10] introduced the iterative scheme as follows: x 0 ∈ H, x n+1 = ( I − α n A ) Sx n + α n γ f ( x n ) , ∀n ≥ 0 , (1:3) where S is a nonexp ansive mapping, f is a contraction with the coefficient a Î (0, 1), A is a stron gly positive bounded linear self-adjoint operator with the coefficient ¯ γ , and g is a constant such that 0 <γ < γ a .Theyprovedthat{x n } generated by the above iterative scheme converges strongly to the unique solution of the variational inequality:  ( A − γ f ) x ∗ , x − x ∗ ≥0, x ∈ F ( S ) . We know that a mapping B : H ® H is said to be monotone, if for each x, y Î H,wehave  Bx − B y , x − y ≥ 0 . A set-valued mapping M : H ® 2 H is called monotone if for all x, y Î H, f Î Mx and g Î My imply 〈x - y, f - g〉 ≥ 0. A monotone mapping M : H ® 2 H is maximal if the graph of Graph(M) of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x, f) Î H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) Î Graph(M) implies f Î Mx. Next, we consider the following so-called variational inclusion problem: Find a u Î H such that θ ∈ Bu + M u (1:4) where B : H ® H, M : H ® 2 H are two nonlinear mappings, and θ is zero vector in H (see, for instance, [11-16]). The set of the solution of (1.4) is denoted by VI(H, B, M). Let C be a nonempty closed convex subset of Banach space X. Let {T n } ∞ n = 1 be an infi- nite family of nonexpansive mappings of C into itself, and let l 1 , l 2 , , be real numbers in [0, 1]; then we define the mapping K n : C ® C as follows: U n,0 = I U n,1 = λ 1 T 1 U n,0 +(1− λ 1 )U n,0 , U n,2 = λ 2 T 2 U n,1 +(1− λ 2 )U n,1 , U n,3 = λ 3 T 3 U n,2 +(1− λ 3 )U n,2 , . . . U n,k = λ k T k U n,k−1 +(1− λ k )U n,k−1 U n,k+1 = λ k+1 T k+1 U n,k +(1− λ k+1 )U n,k . . . U n,n−1 = λ n−1 T n−1 U n,n−2 +(1− λ n−1 )U n,n−2 K n = U n,n = λ n T n U n,n−1 + ( 1 − λ n ) U n,n−1 . Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 2 of 16 Such a mapping K n is called the K-mapping generated by T 1 , T 2 , , T n and l 1 , l 2 , , l n . Let x 1 Î H and {x n } be the sequence generated by x n+1 = α n γ f ( x n ) + β n x n + (( 1 − β n ) I − α n A )( γ n K n x n + ( 1 − γ n ) Sx n ), (1:5) where A is a strongly positive linear-bounded self-adjoint operator with the coeffi- cient 0 < γ < 1 , S : C ® C is the S - mapping generated by G 1 , G 2 , , G N and ν 1 , ν 2 , , ν N ,whereG i : H ® H is a mapping defined by J M i, η (I − ηB i )x = G i x for every x Î H, and h Î (0, 2δ i ) for every i = 1, 2, , N, f : H ® H is contractive mapping with coeffi- cient θ Î (0, 1) and 0 <γ < γ θ ,{a n }, { b n }, {g n } are sequences in [0, 1]. In this article, by motivation of (1.3), we prove a s trong convergence theorem of the proposed algorithm scheme (1.5) to an element z ∈  ∞ i =1 F( T i )  N i =1 V(H, B i , M i ) , under suitable conditions of {a n }, {b n }, { g n }. 2 Preliminaries In this section, we provide some useful lemmas that will be used for our 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., 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. (see [17]) Given x Î HandyÎ C . Then P C x = yifandonlyifthere holds the inequality x − y , y − z≥0 ∀z ∈ C . Lemma 2.2. (see [18]) 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 ≤ 0or ∞  n =1 |α n β n | < ∞ . Then lim n®∞ s n =0. Lemma 2.3.(see[19])LetCbeaclosedconvexsubsetofastrictlyconvexBanach 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. (see [20]) 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 - Sis demi-closed at zero. Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 3 of 16 Lemma 2.5. (see [21]) 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. Suppose x n+1 = b n x n +(1 - b 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. In 2009, Kangtunykarn and Suantai [5] introdu ced the S-mappin g generated by a finite family of nonexpansive mappings and real numbers as follows: Definition 2.1. 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. For each j = 1, 2, , N, let α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I where I Î [0, 1] and α j 1 + α j 2 + α j 3 = 1 , define the mapping S : C ® C as follows: U 0 = I U 1 = α 1 1 T 1 U 0 + α 1 2 U 0 + α 1 3 I U 2 = α 2 1 T 2 U 1 + α 2 2 U 1 + α 2 3 I U 3 = α 3 1 T 3 U 2 + α 3 2 U 2 + α 3 3 I . . . U N−1 = α N−1 1 T N−1 U N−2 + α N−1 2 U N−2 + α N−1 3 I S = U N = α N 1 T N U N−1 + α N 2 U N−1 + α N 3 I. (2:1) This mapping is called the S-mapping generated by T 1 , , T N and a 1 , a 2 , , a N . Lemma 2.6. (see [5]) Let C be a nonempty closed convex subset of strictly convex. Let {T i } N i = 1 be a finite family of nonexpanxive mappings of C into itself with  N i =1 F( T i ) = ∅ and let α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I , j = 1,2,3, , N, where I = [0, 1], α j 1 ∈ (0, 1 ) , α j 1 ∈ (0, 1 ) for all j = 1,2, , N-1, α N 1 ∈ (0, 1]α j 2 , α j 3 ∈ [0, 1 ) for all j = 1,2, , N. Let S be the mapping generated by T 1 , , T N and a 1 , a 2 , , a N . Then F( S)=  N i =1 F( T i ) . Lemma 2.7. (see [5]) Let C be a nonempty closed convex subset of Banach space. Let {T i } N i = 1 be a finite family of nonexpansive mappings of C into itself and α j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I , 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 and α j 1 + α j 2 + α j 3 = 1 such that α n,j i → α j i ∈ [0, 1 ] as n ® ∞ for i = 1,3 and j = 1,2,3, , N. Moreover, for every n Î N, let S and S n be the S-mappings generated by T 1 , T 2 , , T N and a 1 , a 2 , , a N and T 1 , T 2 , , T N and α ( n ) 1 , α ( n ) 2 , , α ( n ) N , respectively. Then lim n®∞ ||S n x - Sx || = 0 for every x Î C. Definition 2.2.(see[11])Let M : H ® 2 H be a multi-valued maximal monotone mapping, then the single-valued mapping J M,l : H ® H defined by J M,λ ( u ) = ( I + λM ) −1 ( u ) , ∀u ∈ H , is called the resolvent operator associated with M, where l is any positive number and I is identity mapping. Lemma 2.8. (see [11]) u Î H is a solution of variational inclusion (1.4) if and only if u = J M, l (u - lBu), ∀l >0, i.e., VI ( H, B, M ) = F ( J M,λ ( I − λB )) , ∀λ>0 . Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 4 of 16 Further, if l Î (0, 2a], then V I(H, B, M) is closed convex subset in H. Lemma 2.9. (see [22]) The resolvent operator J M,l associated with M is single-valued, nonexpansive for all l >0 and 1-inverse-strongly monotone. Lemma 2.10. In a strictly convex Banach space E, if | |x|| = ||y|| = ||λx + ( 1 − λ ) y| | for all x, y Î E and l Î (0, 1), then x = y. Lemma 2.11. LetCbeanonemptyclosedconvexsubsetofastrictlyconvexBanach space. Let {T i } ∞ i= 1 be an infinite family of nonexpanxive mappings of C into itself with  ∞ i =1 F( T i ) = ∅ and let l 1 , l 2 , , be real numbers such that 0 < l i <1 for every i = 1, 2, , and  ∞ i =1 λ i < ∞ . For every n Î N, let K n be the K-mapping generated by T 1 , T 2 , , T n and l 1 , l 2 , , l n . Then for every x Î C and k Î N, lim n®∞ K n x exits. Proof. Let x Î C. Then for k, n Î N, we have  U n+1,k x − U n,k x  = λ k T k U n+1,k−1 x +(1− λ k )U n+1,k−1 x − λ k T k U n,k−1 x − (1 − λ k )U n,k−1 x  = λ k (T k U n+1,k−1 x − T k U n,k−1 x)+(1− λ k )(U n+1,k−1 x − U n,k−1 x)  ≤ λ k  T k U n+1,k−1 x − T k U n,k−1 x  +(1 − λ k )  U n+1,k−1 x − U n,k−1 x  ≤ λ k  U n+1,k−1 x − U n,k−1 x  +(1 − λ k )  U n+1,k−1 x − U n,k−1 x  = U n+1,k−1 x − U n,k−1 x  = λ k−1 T k−1 U n+1,k−2 x +(1− λ k−1 )U n+1,k−2 x − λ k−1 T k−1 U n,k−2 x − (1 − λ k−1 )U n,k−2 x  ≤ λ k−1  T k−1 U n+1,k−2 x − T k−1 U n,k−2 x  +(1 − λ k−1 )  U n+1,k−2 x − U n,k−2 x  ≤ U n+1,k−2 x − U n,k−2 x  . . . ≤ U n+1,1 x − U n,1 x  = λ 1 T 1 U n+1,0 x +(1− λ 1 )U n+1,0 x − λ 1 T 1 U n,0 x − (1 − λ 1 )U n,0 x  = λ 1 T 1 x +(1− λ 1 )x − λ 1 T 1 x − (1 − λ 1 )x  =0 , (2:2) which implies that U n+1,k = U n, k for every k, n Î N.Hence,K n = U n, n = U n+1,n . Since K n+1 x = U n+1,n+1 x = l n+1 T n+1 K n x +(1-l n+1 )K n x, we have K n+1 x − K n x = λ n+1 ( T n+1 K n x − K n x ). (2:3) Let x ∗ ∈  ∞ i =1 F( T i ) and x Î C. For each n Î N, we have  K n x − x ∗  = λ n T n U n,n−1 x +(1− λ n )U n,n−1 x − x ∗  ≤ λ n  T n U n,n−1 x − x ∗  +(1 − λ n )  U n,n−1 x − x ∗  ≤ U n,n−1 x − x ∗  = λ n−1 T n−1 U n,n−2 x +(1− λ n−1 )U n,n−2 x − x ∗  ≤ λ n−1  T n−1 U n,n−2 x − x ∗  +(1 − λ n−1 )  U n,n−2 x − x ∗  ≤ U n,n−2 x − x ∗  . . . ≤U n,1 x − x ∗  =  λ 1 T 1 U n,0 x +(1− λ 1 )U n,0 x − x ∗  ≤ λ 1  T 1 x − x ∗  +(1 − λ 1 )  x − x ∗  =  x − x ∗  , (2:4) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 5 of 16 which implies that {K n x} is bounded, and so is {T n K n x}. For m ≥ n, by (2.3) we have  K m x − K n x  = K m x − K m−1 x + K m−1 x − K m−2 x + K m−2 x − + ··· − K n+1 x + K n+1 x − K n x  ≤ K m x − K m−1 x  +  K m−1 x − K m−2 x  +  K m−2 x − K m−3 x  + ·· · +  K n+2 − K n+1 x  +  K n+1 x − K n x  = λ m  T m K m−1 x − K m−1 x  +λ m−1  T m−1 K m−2 x − K m−2 x  + ··· + λ n+1  T n+1 K n x − K n x  ≤ M m  k = n +1 λ k , (2:5) where M =sup nÎN {||T n+1 K n x - K n x||}. This implies that {K n x} is Cauchy sequence. Hence lim n®∞ K n x exists. From Lemma 2.11, we can define a mapping K : C ® C as follows: Kx = lim n → ∞ K n x, x ∈ C . Such a mapping K is called the K-mapping generated by T 1 , T 2 , , and l 1 , l 2 , Remark 2.12. It is easy to see that for each n Î N, K n is nonexpansive mappings. Let x, y Î C, then we have  Kx − Ky = lim n ⇒ ∞  K n x − K n y ≤  x − y  . (2:6) By (2.6), we have K : C ® C is nonexpansive mapping. Next, we will show that lim n®∞ sup xÎD ||K n x - Kx|| = 0 for every bounded subset D of C. To show this, let x, y Î C and D be a bounded subset of C. By (2.5), for m ≥ n, we have  K m x − K n x ≤ M m  k = n +1 λ k . By letting m ® ∞, for any x Î D, we have  Kx − K n x ≤ M ∞  k=n+1 λ k . Since  ∞ n =1 λ n < ∞ , we have lim n→∞ sup x ∈ D  Kx − K n x =0 . By the next lemma, we will show that F( K)=  ∞ i =1 F( T i ) Lemma 2.13. LetCbeanonemptyclosedconvexsubsetofastrictlyconvexBanach space. Let {T i } ∞ i = 1 be an infinite family of nonexpansive mappings of C into itself with  ∞ i =1 F( T i ) = ∅ , and let l 1 , l 2 , , be real numbers such that 0<l i <1for every i = 1, 2, with  ∞ i =1 λ i < ∞ . Let K n and K be the K-mapping generated by T 1 , T 2 , T n and l 1 , l 2 , l n and T 1 , T 2 , and l 1 , l 2 , , respectively. Then F( K)=  ∞ i =1 F( T i ) . Proof. It is easy to see that  ∞ i =1 F( T i ) ⊆ F(K ) . Next, we show that F( K) ⊆  ∞ i =1 F( T i ) . Let x 0 Î F (K) and x ∗ ∈  ∞ i =1 F( T i ) . Let k Î N be fixed. Since Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 6 of 16  K n x 0 − x ∗  = λ n T n U n,n−1 x 0 +(1− λ n )U n,n−1 x 0 − x ∗  = λ n (T n U n,n−1 x 0 − x ∗ )+(1− λ n )(U n,n−1 x 0 − x ∗ )  ≤ λ n  T n U n,n−1 x 0 − x ∗  +(1 − λ n )  U n,n−1 x 0 − x ∗  ≤ U n,n−1 x 0 − x ∗  = λ n−1 (T n−1 U n,n−2 x 0 − x ∗ )+(1− λ n−1 )U n,n−2 (x 0 − x ∗ )  ≤ λ n−1  T n−1 U n,n−2 x 0 − x ∗  +(1 − λ n−1 )  U n,n−2 x 0 − x ∗  ≤ U n,n−2 x 0 − x ∗  . . . . . ≤ U n,k x 0 − x ∗  = λ k (T k U n,k−1 x 0 − x ∗ )+(1− λ k )(U n,k−1 x 0 − x ∗ )  ≤ λ k  T k U n,k−1 x 0 − x ∗  +(1 − λ k )  U n,k−1 x 0 − x ∗  ≤ U n,k−1 x 0 − x ∗  . . . ≤ U n,1 x 0 − x ∗  = λ 1 (T 1 x 0 − x ∗ )+(1− λ 1 )(x 0 − x ∗ )  ≤ λ 1  T 1 x 0 − x ∗  +(1 − λ 1 )  x 0 − x ∗  ≤  x 0 − x ∗  , (2:7) we have  x 0 − x ∗  = lim n→∞  K n x 0 − x ∗ ≤λ 1 (T 1 x 0 − x ∗ )+(1− λ 1 )(x 0 − x ∗ )  ≤ λ 1  T 1 x 0 − x ∗  +(1 − λ 1 )  x 0 − x ∗  ≤  x 0 − x ∗  , this implies that  x 0 − x ∗  =  T 1 x 0 − x ∗  =  λ 1 ( T 1 x 0 − x ∗ ) + ( 1 − λ 1 )( x 0 − x ∗ )  . By Lemma 2.10, we have T 1 x 0 = x 0 , that is x 0 Î F (T 1 ). It follows that U n,1 x 0 = x 0 .By (2.7), we have  K n x 0 − x ∗ ≤U n,2 x 0 − x ∗ =  λ 2 (T 2 U n,1 x 0 − x ∗ )+(1− λ 2 )(U n,1 x 0 − x ∗ )  = λ 2 (T 2 x 0 − x ∗ )+(1− λ 2 )(x 0 − x ∗ )  ≤ λ 2  T 2 x 0 − x ∗  +(1 − λ 2 )  x 0 − x ∗  ≤  x 0 − x ∗  . It follows that  x 0 − x ∗  = lim n→∞  K n x 0 − x ∗  ≤ λ 2 (T 2 x 0 − x ∗ )+(1− λ 2 )(x 0 − x ∗ )  ≤ λ 2  T 2 x 0 − x ∗  +(1 − λ 2 )  x 0 − x ∗  ≤  x 0 − x ∗  , Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 7 of 16 which implies  x 0 − x ∗ =  T 2 x 0 − x ∗ =  λ 2 ( T 2 x 0 − x ∗ ) + ( 1 − λ 2 )( x 0 − x ∗ )  . By Lemma 2.10, we obtain that T 2 x 0 = x 0 , that is x 0 Î F (T 2 ). It follows that U n,2 x 0 = x 0 . By using the same argument, we can conclude that T i x 0 = x 0 and U i x 0 = x 0 for i = 1, 2, , k - 1. By (2.7), we have  K n x 0 − x ∗ ≤U n,k x 0 − x ∗  = λ k (T k U n,k−1 x 0 − x ∗ )+(1− λ k )(U n,k−1 x 0 − x ∗ )  = λ k (T k x 0 − x ∗ )+(1− λ k )(x 0 − x ∗ )  ≤ λ k  T k x 0 − x ∗  +(1 − λ k )  x 0 − x ∗  ≤  x 0 − x ∗  . It follows that  x 0 − x ∗  = lim n→∞  K n x 0 − x ∗  = λ k (T k x 0 − x ∗ )+(1− λ k )(x 0 − x ∗ )  ≤ λ k  T k x 0 − x ∗  +(1 − λ k )  x 0 − x ∗  ≤  x 0 − x ∗  , (2:8) which implies  x 0 − x ∗ =  T k x 0 − x ∗ =  λ k ( T k x 0 − x ∗ ) + ( 1 − λ k )( x 0 − x ∗ )  . (2:9) By Lemma 2.10, we have T k x 0 = x 0 ,thatisx 0 Î F (T k ). This implies that x 0 ∈  ∞ i =1 F( T i ) . 3 Main result Theorem 3.1. LetHbearealHilbertspace,andletM i : H ® 2 H be maximal mono- tone mappings for every i = 1, 2, , N. Let B i : H ® Hbeaδ i - inverse strongly mono- tone mapping for every i = 1, 2, , Nand {T i } ∞ i= 1 an infinite family of nonexpansive mappings from H into itself. Let A be a strongly positive linear-bounded self-adjoint operator with the coefficient 0 < γ < 1 . Let G i : H ® Hbedefinedby J M i, η (I − ηB i )x = G i x for every x Î Handh Î (0, 2δ i ) for every i = 1, 2, , Nandlet ν j =(α j 1 , α j 2 , α j 3 ) ∈ I × I × I , j = 1, 2, 3, , N, where I =[0,1], α j 1 + α j 2 + α j 3 = 1 , α j 1 ∈ (0, 1 ) for all j = 1,2, , N-1, α N 1 ∈ (0, 1]α j 2 , α j 3 ∈ [0, 1 ) for all j =1,2, ,N Let S : C ® C be the S-mapping generated by G 1 , G 2 , , G N and ν 1 , ν 2 , , ν N . Let l 1 , l 2 , , be real numbers such that 0 < l i <1 for every i = 1, 2, , with  ∞ i =1 λ i < ∞ , and let K n be the K-mapping generated by T 1 , T 2 , , T n and l 1 , l 2 , , l n , and let K be the K-mapping generated by T 1 , T 2 , , and l 1 , l 2 , , i.e., Kx = lim n → ∞ K n x for every x Î C. Assume that F =  ∞ i =1 F( T i )  N i =1 V(H, B i , M i ) = ∅ . For every n Î N, i = 1, 2, , N, let x 1 Î H and {x n } be the sequence generated by x n+1 = α n γ f ( x n ) + β n x n + (( 1 − β n ) I − α n A )( γ n K n x n + ( 1 − γ n ) Sx n ), (3:1) where f : H ® H is contractive mapping with coefficient θ Î (0, 1) and 0 <γ < γ θ . Let {a n }, {b n }, {g n } be sequences in [0, 1], satisfying the following conditions: Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 8 of 16 (i) lim n →∞ α n = 0 and  ∞ n = 0 α n = ∞ (ii) 0 < lim inf n→∞ β n ≤ lim sup n → ∞ β n < 1, (iii) lim n →∞ γ n = c ∈ (0, 1 ) Then {x n } converges strongly to z ∈ F , which solves uniquely the following variational inequality:  ( A − γ f ) z, z − x ∗ ≤0, ∀x ∗ ∈ F . (3:2) Equivalently, we have P F (I − A + γ f)z = z . Proof. Let z be the unique solution of (3.2). First, we will show that the mapping G i is a nonexpansive mapping for every i = 1, 2, , N.Letx, y Î H,sinceB i is δ i -inverse strongly monotone mapping and 0 < h <2δ i , for every i = 1, 2, , N, we have  (I − ηB i )x − (I − ηB i )y 2 = x − y − η(B i x − B i y) 2 = x − y 2 − 2ηx − y, B i x − B i y + η 2  B i x − B i y 2 ≤ x − y 2 − 2δ i η  B i x − B i y 2 + η 2  B i x − B i y 2 = x − y 2 + η(η − 2δ i )  B i x − B i y 2 ≤ x − y  2 . (3:3) Thus, (I - hB i ) is a nonexpansive mapping for every i = 1, 2, , N . By Lemma 2.9, we have G i = J M i, η (I − ηB i ) is a nonexpansive mappings for every i = 1, 2, , N.Let x ∗ ∈ F ; by Lemma 2.8, we have x ∗ = G i x ∗ = J M i ,η (I − ηB i )x ∗ , ∀i =1,2, N . (3:4) Let e n = g n K n x n +(1-g n )Sx n .SinceG i is a nonexpansive mapping for ev ery i =1, 2, , N, we have that S is a nonexpansive mapping. By nonexpansiveness of K n we have  e n − x ∗  = γ n (K n x n − x ∗ )+(1− γ n )(Sx n − x ∗ )  ≤ γ n  K n x n − x ∗  +(1 − γ n )  Sx n − x ∗  ≤ γ n  x n − x ∗  +(1 − γ n )  x n − x ∗  ≤  x n − x ∗  . (3:5) Without loss of generality, by conditions (i)and(ii), we have a n ≤ (1 - b n )||A|| -1 . Since A is a strongly positive linear-bounded self-adjoint operator, we have  A  =su p {|Ax, x| : x ∈ H,  x  =1} . (3:6) For each x Î C with ||x|| = 1, we have  (( 1 − β n ) I − α n A ) x, x =1− β n − α n Ax, x≥1 − β n − α n  A ≥0 , (3:7) then (1 - b n )I -a n A is positive. By (3.6) and (3.7), we have  (1 − β n )I − α n A  =sup{((1 − β n )I − α n A)x, x : x ∈ C,  x  =1 } =sup{1 − β n − α n Ax, x : x ∈ C,  x  =1} ≤ 1 − β n − α n Ax, x ≤ 1 − β n − α n γ . (3:8) We shall divide our proof into six steps. Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 9 of 16 Step 1. We will show that the sequence {x n } is bounded. Let x ∗ ∈ F ,by(3.5)and (3.8), we have  x n+1 − x ∗  = α n γ f (x n )+β n x n + ((1 − β n )I − α n A)e n − x ∗  = α n γ f (x n ) − α n Ax ∗ + α n Ax ∗ − β n x ∗ + β n x ∗ + β n x n + ((1 − β n )I − α n A)e n − x ∗  ≤ α n  γ f (x n ) − Ax ∗  +β n  x n − x ∗  +  ((1 − β n )I − α n A)(e n − x ∗ )  ≤ α n  γ f (x n ) − Ax ∗  +β n  x n − x ∗  + ((1 − β n )I − α n γ )  x n − x ∗  ≤ α n ( γ f (x n ) − γ f (x ∗ )  +  γ f (x ∗ ) − Ax ∗ )+(1− α n γ )  x n − x ∗  ≤ α n γθ  x n − x ∗  + α n  γ f(x ∗ ) − Ax ∗  +(1− α n γ )  x n − x ∗  = α n  γ f (x ∗ ) − Ax ∗  +(1− α n (γ − γθ))  x n − x ∗  ≤ max{ x n − x ∗ ,  γ f (x ∗ ) − Ax ∗  γ − γ θ }. By induction, we can prove that {x n } is bounded, and so are {e n }, {K n x n }, {Sx n } and {G i (x n )} for every i = 1, 2, , N. Without loss of generality, we can assume that there exists a bounded set D ⊂ H such that e n , x n , Sx n , K n x n , G i x n ∈ D, ∀n ∈ N andi =1,2, , N . (3:9) Step 2. We will show that lim n→∞  x n+1 − x n  = 0 . Define sequence {z n }by z n = 1 1 − β n (x n+1 − β n x n ) . Then x n+1 = b n x n +(1- b n )z n . Since {x n } is bounded, we have | |z n+1 − z n || = || x n+2 − β n+1 x n+1 1 − β n+1 −  x n+1 − β n x n 1 − β n  || = || α n+1 γ f (x n+1 )+  (1 − β n+1 )I − α n+1 A  e n+1 1 − β n+1 −  α n γ f (x n )+  (1 − β n )I − α n A  e n 1 − β n  || ≤ α n+1 || γ f (x n+1 ) − Ae n+1 1 − β n+1 || + ||e n+1 − e n || + α n || γ f (x n ) − Ae n 1 − β n ||. (3:10) By definition of e n and nonexpansiveness of S, we have  e n+1 − e n  =   γ n+1 K n+1 x n+1 +(1− γ n+1 )Sx n+1 − γ n K n x n − (1 − γ n )Sx n   =   γ n+1 K n+1 x n+1 +(1− γ n+1 )Sx n+1 − γ n+1 K n x n + γ n+1 K n x n −(1 − γ n+1 )Sx n +(1− γ n+1 )Sx n − γ n K n x n − (1 − γ n )Sx n   =   γ n+1 (K n+1 x n+1 − K n x n )+(1− γ n+1 )(Sx n+1 − Sx n ) +(γ n+1 − γ n )K n x n +(γ n − γ n+1 )Sx n   ≤ γ n+1  K n+1 x n+1 − K n x n  +(1− γ n+1 )  x n+1 − x n  + | γ n+1 − γ n | K n x n  + | γ n − γ n+1 | Sx n  ≤ γ n+1  K n+1 x n+1 − K n x n  +(1− γ n+1 )  x n+1 − x n  +2| γ n+1 − γ n |M, Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 10 of 16 [...]... Convergence theorems of fixed points for k-strict pseudocontractions in Hilbert spaces Nonlinear Anal 69, 456–462 (2008) doi:10.1016/j.na.2007.05.032 doi:10.1186/1687-1812-2011-38 Cite this article as: Kangtunyakarn: Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems Fixed Point Theory and Applications 2011 2011:38... doi:10.1007/s10483-008-0502-y 12 Chang, SS: Set- valued variational inclusions in Banach spaces J Math Anal Appl 248, 438–454 (2000) doi:10.1006/ jmaa.2000.6919 13 Noor, MA, Noor, KI: Sensitivity analysis for quasi -variational inclusions J Math Anal Appl 236, 290–299 (1999) doi:10.1006/jmaa.1999.6424 14 Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings J Optim Theor Appl 118,... doi:10.1023 /A: 1025407607560 15 Li, Y, Wu, C: On the Convergence for an iterative method for quasivariational inclusions Fixed Point Theory Appl 2010, 11 (2010) Article ID 278973 16 Hao, Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications Appl Math Comput 217(7), 3000–3010 (2010) doi:10.1016/j.amc.2010.08.033 17 Takahashi, W: Nonlinear Functional Analysis Yokohama Publishers,... loss of generality, we may assume that {xn j } converses weakly to some q Î H By nonexpansiveness of S and K, (3.20) and Lemma 2.3, we have that Q is nonexpansive mapping and F(Q) = F(K) F(S) (3:25) Kangtunyakarn Fixed Point Theory and Applications 2011, 2011:38 http://www.fixedpointtheoryandapplications.com/content/2011/1/38 Page 13 of 16 Since JMi,η (I − ηBi )x = Gi x for every x Î H and i = 1, 2,... 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Publishers, Yokohama (2000) 18 Xu, HK: Iterative algorithms for nonlinear opearators J Lond Math Soc 66, 240–256 (2002) doi:10.1112/ S0024610702003332 19 Bruck, RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces Trans Am Math Soc 179, 251–262 (1973) 20 Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces Proc Sympos Pure Math 18, 78–81 (1976) . this article as: Kangtunyakarn: Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems. Fixed Point Theory and. RESEARC H Open Access Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems Atid Kangtunyakarn Correspondence: beawrock@hotmail.com Department. {r n } {a n }, g,they proved that {x n } converges strongly to an element of the set of variational inequality pro- blem and the set of a common fixed point of a finite family of nonexpansive mappings. In