RESEA R C H Open Access Convergence and weaker control conditions for hybrid iterative algorithms Shuang Wang Correspondence: wangshuang19841119@163.com School of Mathematical Sciences, Yancheng Teachers University, Yancheng, 224051 Jiangsu, PR China, Abstract Very recently, Yao et al. (Appl. Math. Comput. 216, 822-829, 2010) have proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this article, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. It is worth pointing out that we use a new method to prove our results. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Furthermore, we also give a weak convergence theorem for their iterative algorithm involving an infinite family of nonexpansive mappings in a Hilbert space. MSC: 47H05, 47H09, 47H10 Keywords: Strong convergence, Variational inequality, Fixed points, k-Lipschitzian, η-strongly monotone, Hilbert space 1 Introduction Let H be a real Hilbert space and C be a nonempty, closed, convex subset of H,letF : H ® H be a nonlinear operator. The variational inequality problem is formulated as finding a point x* Î C such that  Fx ∗ , v − x ∗  ≥ 0, ∀v ∈ C . In 1964, Stampacchia [1] introduced and studied variational inequality initiall y. It is now well known that variational inequalities cover as diverse disciplines as partial dif- ferential equations, optimal control, op timization, mathematical programming, mechanics, and finance [1-5]. It is known that a mapping T : H ® H is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y||, ∀x, y Î H. We use F (T ) to denote the set of fixed points of T, that is F (T)={x Î H : Tx = x}. Yamada [2] introduced the fol lowing hybrid i terative method for solving the varia- tional inequality: x n+1 = Tx n − μλ n F ( Tx n ) , n ≥ 0 , (1:1) Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 © 2011 Wang; 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 p ermits unrestricte d use, distribution, and reproduction in any medium, provided the original work is properly cited. where F is a k-Lipschitzian and h-strongly monotone operator with k >0,h > 0 and 0<μ <2h /k 2 . Let a sequence {l n } of real numbers in (0,1) satisfy the conditions below: (C1) lim n®∞ l n =0, (C2)  ∞ n = 0 λ n = ∞ , (C3) lim n→∞ (λ n − λ n+1 )/λ 2 n +1 = 0 . He has proved that {x n } generated by (1.1) converges strongly to the unique solution of the variational inequality: F ˜ x, x − ˜ x≥0, ∀x ∈ F ( T ). An example of sequence {l n } which satisfies conditions (C1)-(C3) is given by l n =1/ n s , where 0 <s < 1. We note that the condition (C3) was first used by Lions [3]. It was observed that Lion’s conditions on the sequence {l n } exclud ed the canonical choice l n = 1/n. This was overcome in 2003 by Xu and Kim [4], if {l n } satisfies conditions (C1), (C2), and (C4) (C4) : lim n →∞ λ n /λ n+1 = 1, or equivalently, lim n →∞ (λ n − λ n+1 )/λ n+1 = 0 who proved th e strong convergenc e of {x n } to the unique solution u*ofthevaria- tional inequality 〈Fu*, v - u*〉 ≥ 0, ∀v Î C. It is e asy to see that the condition (C4) is strictly weaker than condition (C3), coupled with conditions (C1) an d (C2). Moreover , (C4) includes the important and natural choice {1/n} of {l n }. Very recently, motivated by Xu and Kim [4], Yao et al. [5] considered the following algorithms: for x 0 Î H arbitrarily,  y n = x n − λ n F(x n ), x n+1 =(1− α n )y n + α n W n y n , n ≥ 0 , (1:2) where F is a k-Lipschitzian and h-strongly monotone operator on H, and W n is a W- mapping defined by (2.3) cited later. Take k Î [1, ∞), h Î (0, 1), and {l n } satisfying the conditions (C1) and (C2). If a sequence {a n } satisfying (C5) (C5) : α n ∈  γ , 1 2  for some γ>0 , then they proved that the sequences {x n }and{y n } defined by (1.2) converge strongly to x ∗ ∈∩ ∞ n =1 F(T n ) , which solves the following variational inequality: Fx ∗ , x − x ∗ ≥0, ∀x ∈∩ ∞ n =1 F(T n ). We remind the reader of the fact that in order to guarantee the strong convergence of the iterative sequence {x n }, there is at least one pa rameter sequence converging to zero (i.e., l n ® 0) as a result of Yamada [2], Xu and Kim [4, Theorem 3.1, and Theo- rem3.2]andYaoetal.[5,Theorem3.2].Inaddition,h Î (0, 1) and (C5) are quite restrictive assumptions in Yao et al. [5]. In this article, under the convergence of no parameter sequences to zero and the weaker conditions on a n and h,weprovethatthesequence{y n } generated by the Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 2 of 14 iterative algorithm (1.2) converges to a common fixed point of an infinite family of nonexpansive mappings, which solves the variational inequality 〈Fx*, u - x*〉 ≥ 0, ∀u ∈∩ ∞ n =1 F(T n ) . In the meantime, we illustrate that this result is more general than Theorem 3.2 of Yao et a l. [5]. That is, we give an appropriate example such that all conditions of this result are satisfied and the conditions h Î (0, 1), (C1), and (C5) in Yao et al. [5, Theorem 3.2] are not satisfied. Furthermore, we also give a weak co nver- gence theo rem for hybrid iterative algorithm (1.2) involving an infinite family of non- expansive mappings in a Hilbert space H. It i s worth pointing out that we use a new method to prove our main results. The results presented in this article can be viewed as the improvement, supplement, and extension of the results obtained in [2-5]. 2 Preliminaries Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. For the sequence {x n }inH,wewritex n ⇀ x to indicate that the sequence {x n } converges weakly to x. x n ® x implies that {x n } converges strongly to x.Wedenotebyω w (x n ) the weak ω-limit set of {x n }, that is ω w (x n )={x ∈ H : x n i  x for some subsequence {x n i } of {x n }} . AmappingF : H ⇀ H is called k-Lipschitzian if there exists a positive constant k such that | |Fx − F y || ≤ k||x − y ||, ∀x, y ∈ H . (2:1) F is said to be h-strongly monotone if there exists a positive constant h such that Fx − F y , x − y ≥η||x, y || 2 , ∀x, y ∈ H . (2:2) It is known that X satisfies Opial’s property [6] provided, for each sequence {x n }inX, the condition x n ⇀ x implies lim sup n →∞ ||x n − x|| < lim sup n →∞ ||x n − y||, ∀y ∈ X, y = x . It is known that each l p (1 ≤ p < ∞) enjoys this property, while L p does not unless p =2. Finally, it is known that in a Hilbert space, there holds the following equality | |λx + ( 1 − λ ) y|| 2 = λ||x|| 2 + ( 1 − λ ) ||y|| 2 − λ ( 1 − λ ) ||x − y|| 2 for all x, y Î H and l Î [0,1] (see Takahashi [7]). In order to prove our main results, we need the following lemmas: Lemma 2.1.[8].Let H be a Hilbert space, C a closed, convex subset of H, a nd T : C ® C a nonexpansive mapping with F (T ) ≠ ∅; if {x n } is a sequence in C weakly converging to × and if {(I - T )x n } converges strongly to y, then (I - T )x = y. Lemma 2.2.[9].Let {x n } and {z n } be bounded sequences in Banach space E and { g n } be a sequence in [0,1] which satisfies the following condition: 0 < lim inf n→∞ γ n ≤ lim sup n → ∞ γ n < 1. Suppose that x n+1 = g n x n +(1-g n )z n , n ≥ 0, and lim sup n®∞ (||z n+1 - z n || - ||x n+1 - x n ||) ≤ 0. Then, lim n®∞ ||z n - x n || = 0. Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 3 of 14 Lemma 2.3. [10,11]. Let {s n } be a sequence of non-negative real numbers satisfying s n+1 ≤ ( 1 − λ n ) s n + λ n δ n + γ n , n ≥ 0 , where {l n } and {g n } satisfy the following conditions: (i) {l n } ⊂ [0,1] and  ∞ n = 0 λ n = ∞ , (ii) lim sup n® ∞ or  ∞ n = 0 λ n δ n < ∞ , (iii) g n ≥ 0(n ≥ 0),  ∞ n = 0 γ n < ∞ . Then lim n®∞ S n =0. Lemma 2.4. [12]. Let {a n } and {b n } be sequences of nonnegative real numbers such that  ∞ n = 0 b n < ∞ and a n+1 ≤ a n + b n for all n ≥ 0. Then lim n®∞ a n exists. Lemma 2.5.[13].Let F be a k-Lipschitzian and h -strong ly monotone operator on a Hilbert space H with 0<h ≤ k and 0<t <h/k 2 . Then S =(I - tF ):H ® H is a contrac- tion with contraction coefficient τ t =  1 − t(2η − tk 2 ) . Let {T i : H ® H}.beafamilyofinfinitelynonexpansivemappings,{ξ i }beareal sequence such that 0 <ξ i ≤ b <1,∀i ≥ 1. For any n ≥ 1, define a mapping W n : H ® H as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U n,n+1 = I, U n,n = ξ n T n U n,n+1 +(1− ξ n )I, U n,n−1 = ξ n−1 T n−1 U n,n +(1− ξ n−1 )I , ··· U n,k = ξ k T k U n,k+1 +(1− ξ k )I U n,k−1 = ξ k−1 T k−1 U n,k +(1− ξ k−1 )I ··· U n,2 = ξ 2 T 2 U n,3 +(1− ξ 2 )I, W n = U n,1 = ξ 1 T 1 U n,2 + ( 1 − ξ 1 ) I. (2:3) Such a mapping W n : H ® H is called a W-mapping generated by T n , T n-1 , , T 1 and ξ n , ξ n-1 , , ξ 1 . We have the following crucial conclusion concerning W n . We can find them in [14-17]. Now we only need the following similar version in Hilbert spaces: Lemma 2.6. Let H be a real Hilbert space,{T i : H ® H} be a family of infinitely non- expansive mappings with ∩ ∞ i =1 F(T i ) = ∅ ,{ξ i } be a real sequence such that 0<ξ i ≤ b <1, ∀i ≥ 1. Then, (1) W n is a nonexpansive and F(W n )=∩ n i =1 F(T i ) for each n ≥ 1; (2) For every x Î H and k Î N, the limit lim n®∞ U n, k x exists; (3) If we define a mapping W : H ® H as Wx =lim n®1 W n x, and W n x =lim n®∞ U n,1 x, for every Î H, then, F(W)=∩ ∞ i =1 F(T i ) ; (4) For any bounded sequence {x n } ⊂ H, we have lim n®∞ ||Wx n - W n x n || = 0. 3 Main results Let F be a k-Lipschitzian and h-strongly monotone operator on H with 0 <k, T : H ® H be a nonexpansive mapping. Let t Î (0,h/k 2 )and τ t =  1 − t(2η − tk 2 ) ,andcon- sider a mapping S t on H defined by S t x = T [ ( I − tF ) x], x ∈ H . Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 4 of 14 It is easy to see that S t is a contraction. Indeed, from Lemma 2.5, we have | |S t x − S t y|| ≤ ||T[(I − tF)x] − T[(I − tF)y]| | ≤||(I − tF)x − (I − tF)y|| ≤ τ t ||x − y ||, for all x, y Î H. Hence, it has a unique fixed point, denoted as x t , which uniquely solves the fixed point equation x t = T[ ( I − tF ) x t ], x t ∈ H . (3:1) Theorem 3.1. Let H be a real Hilbert space. Let T : H ® H be a nonexpansive map- ping such that F (T ) ≠ ∅,. Let F be a k-Lipschitzian and h-strongly monotone operator on H with 0<h ≤ k . For each t Î (0, h/k 2 ), let the net {x t } be generated by (3.1). Then, as t ® 0, the net {x t } converges strongly to a fixed point x*ofT which solves the varia- tional inequality: Fx ∗ , x ∗ − u≤0, ∀u ∈ F ( T ). (3:2) Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of F. Suppose x* Î F (T ) and ˜ x ∈ F ( T ) both are solutions to (3.2), then  Fx ∗ , x ∗ − ˜ x  ≤ 0 , (3:3) and  F ˜ x, ˜ x − x ∗  ≤ 0 . (3:4) Adding up (3.3) and (3.4) yields  Fx ∗ − F ˜ x, x ∗ − ˜ x  ≤ 0 . The strong monotonicity of F implies that x ∗ = ˜ x and the uniqueness is proved. Later, we use x* Î F (T ) to denote the unique solution of (3.2). Next, we prove that {x t } is bounded. Take u Î F (T ), from (3.1) and using Lemma 2.5, we have ||x t − u|| = ||T[(I − tF)x t ] − Tu|| ≤||(I − tF)x t − u|| ≤||(I − tF)x t − (I − tF)u − tFu|| ≤||(I − tF)x t − (I − tF)u|| + t||Fu| | ≤ τ t || x t − u || + t || Fu || , that is, ||x t − u|| ≤ t 1 − τ t ||Fu|| . (3:5) Observe that lim t→0 + t 1 − τ t = 1 η . Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 5 of 14 From t ® 0, we may assume, w ithout loss of generality, that t ≤ η k 2 − ε ,whereisa arbitrarily small positive number. Thus, we have t 1 − τ t is continuous, ∀ t ∈ [0, η k 2 − ε ] . Therefore, we obtain sup  t 1 − τ t : t ∈ (0, η k 2 − ε]  < +∞ . (3:6) From (3.5) and (3.6), we have that {x t } is bounded and so is {Fx t }. On the other hand, from (3.1), we obtain | |x t − Tx t || = ||T[ ( I − tF ) x t ] − Tx t || ≤ || ( I − tF ) x t − x t || = t||Fx t || → 0 ( t → 0 ). (3:7) To prove that x t ® x*. For a given u Î F (T ), using Lemma 2.5, we have ||x t − u|| 2 = ||T[(I − tF)x t ] − Tu|| 2 ≤||(I − tf )x t − (I − tF)u − tFu|| 2 ≤ τ t 2 ||x t − u|| 2 + t 2 ||Fu|| 2 +2t(I − tF)u − (I − tF)x t , Fu ≤ τ t ||x t − u|| 2 + t 2 ||Fu|| 2 +2tu − x t , Fu +2t 2 Fx t − Fu, Fu ≤ τ t || x t − u || 2 + t 2 || Fu || 2 +2t  u − x t , Fu  +2t 2 k || x t − u || || Fu ||. Therefore, | |x t − u|| 2 ≤ t 2 1 − τ t ||Fu 2 || + 2t 1 − τ t u − x t , Fu + 2t 2 k 1 − τ t ||x t − u|| ||Fu|| . (3:8) From τ t =  1 − t(2η − tk 2 ) ,wehave lim t→0 t 2 1 − τ t = 0 and lim t→0 2t 2 k 1 − τ t = 0 . Observe that, if x t ⇀ u, we have lim t→0 2t 1 − τ t u − x t , Fu = 0 . Since {x t } is bounded, we see that if {t n } is a sequence in (0, η k 2 − ε ] such that t n ® 0 and x t n  ˜ x , then by (3.8), w e see x t n → ˜ x . Moreover, by (3.7) and using Lemma 2.1, we have ˜ x ∈ F ( T ) .Wenextprovethat ˜ x solves the variational inequality (3.2). From (3.1) and u Î F (T ), we have ||x t − u|| 2 ≤||(I − tF)x t − u 2 = ||x t − u 2 || + t 2 ||Fx t || 2 − 2tFx t , x t − u , that is, Fx t , x t − u≤ t 2 ||Fx t || 2 . (3:9) Now replacing t in (3.9) with t n and letting n ® ∞ , we have  F ˜ x, ˜ x − u  ≤ 0 . That is ˜ x ∈ F ( T ) is a solution of (3.2), hence ˜ x = x ∗ by uniqueness. In a nutshell, we have shown that each cluster point of {x t }(att® 0) equals x*. Therefore, x t ® x*as t ® 0. Theorem 3.2. Let H be a real Hilbert space. Let F be a k-Lipschitzian and h-strongly monotone operator on H with 0<h ≤ k. Let {T n } ∞ n =1 : H → H be an infinite family of Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 6 of 14 nonexpansive mappings such that ∩ ∞ n=1 F(T n ) = ∅ and W n be a W-mapping defined by (2.3). Let {l n } be a sequence in [0, ∞) and {a n } be a sequence in [0,1], ε be a arbitrarily small positive number. Assume that the control conditions (C2),(C1)’,and(C5)’ hold for {l n } and {a n }, (C1)’: 0 <λ n ≤ η k 2 − ε , ∀n ≥ n 0 for some integer n 0 ≥ 0, and (C5) ‘:0<g ≤ lim inf n®∞ a n lim sup n®∞ a n <1for some gÎ(0, 1). For x 0 Î H arbitrarily, let the sequence {y n } be generated by (1.2). Then, y n → x ∗ ⇔ λ n F ( x n ) → 0 ( n →∞ ), where x ∗ ∈∩ ∞ n =1 F(T n ) solves the variational inequality Fx ∗ , x ∗ − u≤0, u ∈∩ ∞ n =1 F(T n ) . Proof. On the one hand, suppose that l n F(x n ) ® 0(n ® ∞). We proceed with the fol- lowing steps: Step 1. We claim that {x n } is bounded. In fact, let u ∈∩ ∞ n =1 F(T n ) , from (1.2), (C1)’ and using Lemma 2.5, we have | |y n − u|| = ||x n − λ n F(x n ) − u|| ≤||(I − λ n F)x n − (I − λ n F)u − λ n Fu| | ≤ τ λ n ||x n − u|| + λ n ||Fu||, (3:10) ∀n ≥ n 0 for some integer n 0 ≥ 0, where τ λ n =  1 − λ n (2η − λ n k 2 ) ∈ (0, 1 ) .Then, from (1.2) and (3.10), we obtain | | x n+1 − u|| = ||(1 − α n )(y n − u)+α n (W n y n − u)|| ≤||y n − u|| ≤ [1 − (1 − τ λ n )] ||x n − u || + λ n ||Fu|| ≤ max  ||x n − u|| , ||λ n Fu|| 1 − τ λ n  . By induction, we have || x n − u || ≤ max {|| x 0 − u || , M 1 || Fu ||} , ∀n ≥ n 0 for some integer n 0 ≥ 0, where M 1 =sup{ λ n 1 − τ λ n :0<λ n ≤ η k 2 − ε} < + ∞ . Therefore, {x n } is bounded. We also obtain that {y n }, {W n y n } and {Fx n } are bounded. Step 2. We claim that lim n®∞ ||x n+1 - x n || = 0. To this end, define x n+1 =(1-a n )x n + a n z n . We observe that ||z n+1 − z n || =     x n+2 − (1 − α n+1 )x n+1 α n+1 − x n+1 − (1 − α n )x n α n     ≤     (1 − α n+1 )y n+1 + α n+1 W n+1 y n+1 − (1 − α n+1 )x n+1 α n+1 − (1 − α n )y n + α n W n y n − (1 − α n )x n α n     ≤     α n+1 W n+1 y n+1 − (1 − α n+1 )λ n+1 F(x n+1 ) α n+1 − α n W n y n − (1 − α n )λ n F(x n ) α n     ≤ 1 − α n+1 α n+1 ||λ n+1 F(x n+1 )|| + 1 − α n α n ||λ n F(x n )|| + ||W n+1 y n+1 − W n y n || ≤ 1 − γ γ ||λ n+1 F(x n+1 )|| + 1 − γ γ ||λ n F(x n )|| + ||W n+1 y n+1 − W n y n ||. (3:11) Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 7 of 14 From (2.3), for u ∈∩ ∞ n=1 F(T n ) , we have | |W n+1 y n − W n y n || = ||ξ 1 T 1 U n+1,2 y n − ξ 1 T 1 U n,2 y n n|| ≤ ξ 1 ||U n+1,2 y n − U n,2 y n n|| = ξ 1 ||ξ 2 T 2 U n+1,3 y n − ξ 2 T 2 U n,3 y n || ≤ ξ 1 ξ 2 ||U n+1,3 y n − U n,3 y n || ≤··· ≤ ξ 1 ξ 2 ···ξ n ||U n+1,n+1 y n − U n,n+1 y n || = ξ 1 ξ 2 ···ξ n ||ξ n+1 T n+1 y n +(1− ξ n+1 )y n − y n || ≤  n+1  i=1 ξ i  (||T n+1 y n − u|| + ||u − y n ||) ≤  n+1  i=1 ξ i  (2||y n − u||) ≤ M 2 n+1  i =1 ξ i , (3:12) where M 2 = sup{2 ||y n - u||, n ≥ 0}. By (1.2) and (3.12), we have || W n+1 y n+1 − W n y n || ≤ || W n+1 y n+1 − W n+1 y n || + || W n+1 y n − W n y n || ≤||y n+1 − y n || + || W n+1 y n − W n y n || ≤||x n+1 − λ n+1 F(x n+1 ) − x n + λ n F(x n ) || + M 2 n+1  i=1 ξ i ≤||x n+1 − x n || + || λ n+1 F(x n+1 ) || + || λ n F(x n ) || + M 2 n+1  i =1 ξ i . (3:13) Substituting (3.13) into (3.11), we have | |z n+1 − z n || ≤ 1 − γ γ ||λ n+1 F(x n+1 )|| + 1 − γ γ ||λ n F(x n )|| + ||x n+1 − x n || + ||λ n+1 F(x n+1) )| | + ||λ n F(x n )|| + M 2 n+1  i=1 ξ i = 1 γ ||λ n+1 F(x n+1 )|| + 1 γ ||λ n F(x n )|| + ||x n+1 − x n || + M 2 n+1  i =1 ξ i , that is, | |z n+1 − z n || − || x n+1 − x n || ≤ 1 γ ||λ n+1 F(x n+1 ) || + 1 γ ||λ n F(x n ) || + M 2 n+ 1  i =1 ξ i . Observing l n F(x n ) ® 0(n ® ∞) and 0 <ξ i ≤ b < 1, it follows that lim sup n →∞ (||z n+1 − z n || − || x n+1 − x n ||) ≤ 0 . (3:14) By (C5)’ and using Lemma 2.2, we have lim n®∞ ||z n - x n || = 0. Therefore, lim n → ∞ || x n+1 − x n || = lim n → ∞ α n ||z n − x n || =0 . Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 8 of 14 Step 3. We claim that lim n®∞ ||x n - W n x n || = 0. Observe that || x n − W n x n || ≤ || x n − x n+1 || + || x n+1 − W n x n || ≤||x n − x n+1 || +(1− α n ) ||y n − W n x n || + α n || W n y n − W n x n || ≤||x n − x n+1 || +(1− α n ) ||y n − x n || +(1− α n ) ||x n − W n x n || + α n || y n − x n | | = || x n − x n+1 || + || y n − x n || + ( 1 − α n ) || x n − W n x n ||, that is, | |x n − W n x n || ≤ 1 α n (||x n+1 − x n || + || y n − x n ||) ≤ 1 γ (|| x n+1 − x n || + || λ n F(x n ) ||) → 0(n →∞) . (3:15) Step 4. We claim that lim n®∞ ||x n - Wx n || = 0. Indeed, we have || x n − Wx n || ≤ || x n − W n x n || + || W n x n − Wx n ||. (3:16) By (3.15), (3.16) and using Lemma 2.6, we obtain lim n → ∞ || x n − Wx n || =0 . Step 5. We claim that lim sup n®∞ 〈Fx*, x*-x n 〉 ≤ 0, where x*=lim n®∞ x t and x t defined by x t = W[(1 - tF)x t ]. Since x n is bounded, there exists a subsequence {x n k } of {x n } which converges wea kly to ω.FromStep4,weobtain Wx n k  ω . From Lemma 2.1, we have ω Î F (W). Hence, by Theorem 3.1, we have lim sup n →∞ Fx ∗ , x ∗ − x n  = lim k→∞ Fx ∗ , x ∗ − x n k  = Fx ∗ , x ∗ − ω≤0 . Step 6. We claim that {x n } converges strongly to x ∗ ∈∩ ∞ n =1 F(T n ) . From (1.2), we have || x n+1 − x ∗ || 2 ≤ (1 − α n )|| y n − x ∗ || 2 + α n || W n y n − x ∗ || 2 ≤||y n − x ∗ || 2 = ||x n − λ n F(x n ) − x ∗ || 2 ≤||(I − λ n F)x n − (I − λ n F)x ∗ − λ n Fx ∗ || 2 ≤ τ 2 λ n ||x n − x ∗ || 2 + λ 2 n || Fx ∗ || 2 +2λ n (I − λ n F)x ∗ − (I − λ n F)x n , Fx ∗  ≤ τ λ n || x n − x ∗ || 2 + λ 2 n || Fx ∗ || 2 +2λ n x ∗ − x n , Fx ∗  +2λ n λ n Fx n , Fx ∗ −2λ 2 n ||Fx ∗ || 2 ≤ [1 − (1 − τ λ n )]|| x n − x ∗ || 2 +2λ n x ∗ − x n , Fx ∗  +2λ n || λ n Fx n || ||Fx ∗ || − λ 2 n || Fx ∗ || 2 ≤ [1 − (1 − τ λ n )]|| x n − x ∗ || 2 +(1− τ λ n )  2λ n 1 − τ λ n x ∗ − x n , Fx ∗  + λ n M 3 1 − τ λ n ||λ n Fx n ||  ≤ [1 − (1 − τ λ n )]|| x n − x ∗ || 2 +(1− τ λ n )[2M 1 x ∗ − x n , Fx ∗  + M 1 M 3 || λ n Fx n ||], ∀n ≥ n 0 for some integer n 0 ≥ 0, where M 3 =2||Fx*||. For every n ≥ n 0 ,put μ n =1− τ λ n and δ n =2M 1 〈x*-x n , Fx*〉 +M 1 M 3 ||l n Fx n ||. It follows that | |x n+1 − x ∗ || 2 ≤ ( 1 − μ n ) ||x n − x ∗ || 2 + μ n δ n , ∀n ≥ n 0 . It is easy to see that  ∞ n =1 μ n = ∞ and lim sup n®∞ δ n ≤ 0. Hence, by Lemma 2.3, the sequence {x n } converges strongly to x ∗ ∈∩ ∞ n =1 F(T n ) . Observe that ||y n − x ∗ || ≤ ||y n − x n || + || x n − x ∗ || ≤ ||λ n F ( x n ) + || x n − x ∗ || → 0 ( n →∞ ). It follows t hat the sequence {y n }convergesstronglyto x ∗ ∈∩ ∞ n =1 F(T n ) .Fromx*= lim t®0 x t and Theorem 3.1, we have x* is the unique solution of the variational inequality: 〈Fx*, x*-u〉 ≤ 0, ∀ u ∈∩ ∞ n =1 F(T n ) . Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 9 of 14 On the other hand, suppose that y n → x ∗ ∈∩ ∞ n=1 F(T n ) as n ® ∞,where x ∗ ∈∩ ∞ n =1 F(T n ) solves the variational inequality: Fx ∗ , x ∗ − u≤0, u ∈∩ ∞ n =1 F(T n ) . From (1.2), we have | |x n+1 − x ∗ || = ||(1 − α n )(y n − x ∗ )+α n (W n y n − x ∗ ) | | ≤ (1 − α n )|| y n − x ∗ || + α n || y n − x ∗ || = ||y n − x ∗ → 0 ( n →∞ ) , (3:17) that is, x n → x ∗ ∈∩ ∞ n =1 F(T n ) . Again from (1.2), we obtain that ||λ n F ( x n ) || = ||y n − x n || ≤ ||y n − x ∗ || + || x n − x ∗ || . Since y n → x ∗ ∈∩ ∞ n =1 F(T n ) and x n → x ∗ ∈∩ ∞ n =1 F(T n ) ,wegetl n F(x n ) ® 0. This com- pletes the proof. Remark 3.3. It is clear that condition (C1)’ is strictly weaker than condition (C1). In the meantime, condition (C5)’ is also strictly weaker than condition (C5). Corollary 3.4. (Yao et al. [5, Theorem 3.2]). Let H be a real Hilbert space. Let F : H ® H be k-Lipschitzian and h-strongly monotone operator with k Î [1, ∞) and h Î (0, 1). Let {T n } ∞ n =1 : H → H be an infinite family of nonexpansive mappings such that ∩ ∞ n =1 F(T n ) = ∅ and {W n } be W-mapping defined by (2.3). Let {l n } be a sequenc e in [0, ∞) and {a n } be a sequence in [0,1]. Assume that (C1) lim n®1 l n =0; (C2)  ∞ n = 0 λ n = ∞ ; (C5) α n ∈  γ , 1 2  for some g >0. Then, the sequence {x n } and {y n } generated by (1.2) converge strongly to x ∗ ∈∩ ∞ n =1 F(T n ) , which solves the following va riational inequality 〈Fx*, x*-x〉 ≤ 0, x ∗ ∈∩ ∞ n =1 F(T n ) . Proof. Since lim n®∞ l n = 0, it i s easy to see that λ n ≤ η k 2 − ε , ∀n ≥ n 0 for some inte- ger n 0 ≥ 0. Without loss of generality, we assume that 0 <λ n ≤ η k 2 − ε , ∀n ≥ n 0 for some integer n 0 ≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we know that {x n } is bounded, and so are the sequence {y n }and{F( x n )}. Therefore, w e have l n F(x n ) ® 0. From α n ∈  γ , 1 2  for some g >0,wehave0<g ≤ lim inf n®∞ a n ≤ lim sup n®∞ a n < 1forsomeg Î (0, 1). Therefore, all conditions of Theorem 3.2 are satisfied. Henc e, using Theorem 3.2, we have that {y n } converges strongly to x ∗ ∈∩ ∞ n =1 F(T n ) which solves the following variational inequality 〈Fx*, x*-x〉 ≤ 0, x ∗ ∈∩ ∞ n =1 F(T n ) . It follows from (3.17) that {x n } also converges strongly to x ∗ ∈∩ ∞ n =1 F(T n ) .Thiscompletesthe proof. Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 10 of 14 [...]... Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings Fixed Point Theory Appl 2007, 12 (2007) (Article ID 64363) doi:10.1186/1687-1812-2011-3 Cite this article as: Wang: Convergence and weaker control conditions for hybrid iterative algorithms Fixed Point Theory and Applications 2011 2011:3 Submit your manuscript to a journal and benefit... 1357–1359 (1977) 4 Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities J Optim Theory Appl 119, 185–201 (2003) 5 Yao, YH, Noor, MN, Liou, YC: A new hybrid iterative algorithm for variational inequalities Appl Math Comput 216, 822–829 (2010) doi:10.1016/j.amc.2010.01.087 6 Opial, Z: Weak convergence of successive approximations for nonexpansive mappings Bull Am... 1 h = 1 and αn ∈ [γ , 2 ], the conditions that ln ® 0, αn ∈ γ , for some g > 0 and h Î 2 (0, 1) of Yao et al [5, Theorem 3.2] are not satisfied Next, we give a weak convergence theorem for hybrid iterative algorithm (1.2) involving an infinite family of nonexpansive mappings in a Hilbert space Theorem 3.7 Let H be a real Hilbert space Let F : H ® H be k-Lipschitzian and h-strongly monotone operator... Example 3.6 Let H = R the set of real numbers and Tn ≡ T Define a nonexpansive mapping T : H ® H and an operator F : H ® H as follows: Tx = 0 and F(x) = x, ∀x ∈ R It is easy to see that F(T) = {0}, ∩∞ F(Tn ) = {0 }and Wn x = (1 - ξ 1 )x, ∀x Î R Let n=1 2 1 1 1 ξ1 = , we have Wn x = x, ∀x Î R Given sequences {an} and {λn } : αn = , λn = for 3 2 2 2 all n ≥ 0 For an arbitrary x0 Î H, let {xn} defined as... ∀n ≥ n0 for some integer n0 ≥ 0; 2 ∞ ∞ 1 (B2) λn = = ∞; n=0 n=0 2 1 2 1 (B3) 0 < ≤ lim infn→∞ αn = = lim supn→∞ αn < 1for some constant γ = 2 3 2 (B1) 0 < λn = Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 12 of 14 1 Hence, there is no doubt that all conditions of Theorem 3.2 are satisfied Since λn = , 2 1 1 h = 1 and αn ∈... Wang, S, Hu, CS: Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces Fixed Point Theory Appl 2010, 12 (2010) (Article ID 852030) 14 Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems Math Comput Model 32, 1463–1471 (2000) doi:10.1016/S0895-7177(00)00218-1 15 Shimoji, K, Takahashi, W: Strong convergence to common fixed...Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 Page 11 of 14 Remark 3.5 Theorem 3.2 is more general than Theorem 3.2 of Yao et al [5] The following example shows that all conditions of Theorem 3.2 are satisfied However, the 1 conditions ln ® 0, h Î (0, 1) and αn ∈ γ , for some g > 0 in [5, Theorem 3.2] are 2 not... 258, 4413–4416 (1964) 2 Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings In: Butnariu D, Censor Y, Reich S (eds.) Stud Comput Math, vol 8, pp 473–504 Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), North-Holland, Amsterdam (2001) 3 Lions, PL: Approximation... Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications Taiwan J Math 5, 387–404 (2001) Page 13 of 14 Wang Fixed Point Theory and Applications 2011, 2011:3 http://www.fixedpointtheoryandapplications.com/content/2011/1/3 16 Chang, S-S: A new method for solving equilibrium problem and variational inequality problem with application to optimization Nonlinear Anal... Nonlinear and Convex Analysis Yokohama Publisher, Yokohama (2009) 8 Geobel, K, Kirk, WA: Topics in Metric Fixed point Theory In Cambridge Stud Adv Math, vol 28, pp 473–504.Cambridge University Press, Cambridge (1990) 9 Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral J Math Anal Appl 35, 227–239 (2005) 10 Liu, LS: Iterative . article as: Wang: Convergence and weaker control conditions for hybrid iterative algorithms. Fixed Point Theory and Applications 2011 2011:3. Submit your manuscript to a journal and benefi t from: 7. RESEA R C H Open Access Convergence and weaker control conditions for hybrid iterative algorithms Shuang Wang Correspondence: wangshuang19841119@163.com School. a sequence in [0, ∞) and {a n } be a sequence in [0,1], ε be a arbitrarily small positive number. Assume that the control conditions (C2),(C1)’ ,and( C5)’ hold for {l n } and {a n }, (C1)’: 0 <λ n ≤ η k 2 − ε ,