Báo cáo hóa học: " An extragradient-like approximation method for variational inequalities and fixed point problems" ppt

RESEA R C H Open Access An extragradient-like approximation method for variational inequalities and fixed point problems Lu-Chuan Ceng 1,2 , Qamrul Hasan Ansari 3 , Ngai-Ching Wong 4* and Jen-Chih Yao 4,5 * Correspondence: wong@math. nsysu.edu.tw 4 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan Full list of author information is available at the end of the article Abstract The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solu tions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings. 1991 MSC: 47H09; 47J20. Keywords: extragradient-like approximation method, modified Mann iteration pro- cess, variational inequality, asymptotically strict pseudocontractive mapping in the intermediate sense, fixed point, monotone mapping, strong convergence, demiclo- sedness principle 1. Introduction Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and || · ||, respectively, and let C be a nonempty closed convex subset of H. Corresponding to an operator A : C ® H and set C, the variational inequality problem VIP(A, C)is defined as follows: Find ¯ x ∈ C such that A ¯ x, y − ¯ x≥0, ∀ y ∈ C . (1:1) The set of solutions of VIP(A, C) is denoted by Ω.ItiswellknownthatifA is a strongly monotone and Lipschitz-co ntinuous mapping on C, then the VIP(A, C) has a unique solution. Not only the existence and uniqueness of a solution are important topics in the st udy of the VIP(A, C) but al so how to compute a solution of the VIP(A, C) is important. For applications and further details on VIP(A, C), we re fer to [1-4] and the references therein. The set of fixed points of a mapping S is denoted by Fix(S), that is, Fix(S)={x Î H : Sx = x}. For finding an element of F(S) ∩ Ω under the assumpti on that a set C ⊂ H is none- mpty, closed and convex, a mapping S : C ® C is nonexpansive and a mapping A : C Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 © 2011 Ceng et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creat ivecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ® H is b-inverse-strongly monotone, Takahashi and Toyoda [5] proposed an iterative scheme and proved that the sequence generated by the proposed scheme converges weakly to a point z Î F(S) ∩ Ω if F(S) ∩ Ω ≠ ∅. Recently, motivated by the idea of Korpelevich’s extragradient method [6], Nadezh- kina and Takahashi [7] introduced an iterative scheme, called extragradient method, for finding an element of F(S) ∩ Ω and established the weak convergence result. Very recently, inspired by t he work in [7], Zeng and Yao [8] introduced an iterative scheme for finding an element of F(S) ∩ Ω and obtained the weak convergence r esult. The viscosity approximation method for finding a fixed point of a given nonexpansive map- ping was proposed by Moudafi [9]. He proved the strong convergence of the sequence generat ed by the proposed method to a unique solution of some variational inequality. Xu [10] extended the results of [9] to the more general version. Later on, Ceng and Yao [11] also introduced an extragradient-like approximation method, which is b ased on the above extragradient method and viscosity approximation method, and proved the strong convergence result under certain conditions. An iterative method for the approximation of f ixed points of asymptotically nonex- pansive mappings was developed by Schu [12]. Iterative methods for the approximation of fixed points of asymptotically nonexpansive mappings have been further studied in [13,14] and the references therein. The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [15]. The iterative methods for t he approximation of fixed points of such types of non-Lipschitzian mappings have been further studied in [16-18]. On the other hand, Kim and Xu [19] introduced the concept of asymptotically -strict pseudocontractive m appings in a Hilbert space and studied t he weak and strong convergence theorems fo r this class of mapping s. Sahu et al. [20] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian. They proposed modi- fied Mann iteration p rocess and proved its weak convergence for an asymptotically -strict pseudocontractive mapping in the intermediate sense. Very recently, Ceng et al. [21] established the strong convergence of viscosity approximation method for a modified Mann iteration process for asymptotically strict pseudocontractive mappings in intermediate sense and then proved the strong conver- gence of general CQ algorithm for asymptotically strict pseudocontractive mappings in intermediate sense. They extended the concept of asymptotically strict pseudocontrac- tive mappings in intermediate sense to Banach space setting, called nearly asymptoti- cally -strict pseudocontractive mapping in intermediate sense. They also established the weak convergence theorems for a fixed point of a nearly asymptotically  -strict pseudocontractive mapping in intermediate sense which is not necessarily Lipschitzian. In this paper, we propose and study an extragradient-like iterative algorithm that is based on the extragradient-like approximation method in [11] and the modified Mann iteration process in [20]. We apply the extragradient-like iterative algorithm to design- ing an iterative scheme for finding a common fixed point of two nonlinear mappings. Here, we remind the reader of the following facts: (i) the modified Mann iteration pro- cess in [[ 20], Theorem 3.4] is extended to develop the extragradient-li ke iterative algo- rithm for finding an element of F(S) ∩ Ω; (ii) the extragradient-like iterative algorithm Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 2 of 18 is very different from the extragradient-like iterative scheme in [11] since the class of mappings S in our scheme is more general than the class of nonexpansive mappings. 2. Preliminaries Throughout the paper, unless otherwise specified, we assume that H is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and || · ||, respectively, and C isanonemptyclosedconvexsubsetofH. The set of fixed points of a mapping S is denoted by Fix(S ), that is, Fix(S)={x Î H : Sx = x}. We write x n ⇀ x to indicate that the sequence {x n } converges weakly to x. The sequence {x n } converges strongly to x is denoted by x n ® x. Recall that a mapping S : C ® C is said to be L-Lipschitzian if there exists a constant L ≥ 0 such that ||Sx - Sy|| ≤ L||x-y||, ∀x, y Î C. In particul ar, if L Î [0, 1), then S is called a contraction on C;ifL = 1, then S is called a nonexpansive mapping on C. The mapping S : C ® C is called pseudocontractive if | |Sx − Sy|| 2 ≤||x − y|| 2 + || ( I −S ) x − ( I − S ) y|| 2 , ∀x, y ∈ C . A mapping A : C ® H is called (i) monotone if Ax − A y , x − y ≥0, ∀x, y ∈ C ; (ii) b-inverse-strongly monotone [22,23] if there exists a positive constant b such that Ax − A y , x − y ≥β||Ax −A y || 2 , ∀x, y ∈ C . It is obvious that if A is b-inverse-strongly monotone, then A is monotone and Lipschitz continuous. It is easy to see that if a mapping S : C ® C is nonexpa nsive, then the mapping A = I-Sis 1/2-inverse-strongly monotone; moreover, F(S)=Ω (see, e.g., [5]). At the same time, if a mapping S : C ® C is pseudocontractive and L-Lipschitz continuous, then the mapping A =(I - S) is monotone and L + 1-Lipschitz continuous; moreover, F(S)= Ω (see, e.g., [[24], proof of Theorem 4.5]). Definition 2.1.LetC be a nonempty subset of a normed space X.AmappingS : C ® C is said to be (a) asymptotically nonexpansive [25] if there exists a sequence {k n } of positive num- bers such that lim n®∞ K n = 1 and | |S n x − S n y || ≤ k n ||x − y ||, ∀n ≥ 1, ∀x, y ∈ C ; (b) asymptot ically nonexpansive in the intermediate sense [15] provided S is uni- formly continuous and lim sup n→∞ sup x, y ∈C (||S n x − S n y|| − ||x −y||) ≤ 0 ; (c) uniformly Lipschitzian if there exists a constant L > 0 such that | |S n x − S n y || ≤ L||x − y ||, ∀n ≥ 1, ∀x, y ∈ C . Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 3 of 18 It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goe- bel and Kirk [25] as below: Theorem 2.1. [[25], Theor em 1] If C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping S : C ® C has a fixed point in C. Definition 2.2.[19]AmappingS : C ® C is said to be an asymptotically -strict pseudocontractive mapping with sequence {g n } if there exist a constant  Î [0, 1) and a sequence {g n } in [0, ∞) with lim n®∞ g n = 0 such that ||S n x − S n y|| 2 ≤ ( 1+γ n ) ||x − y|| 2 + κ||x −S n x − ( y − S n y ) || 2 , ∀n ≥ 1, ∀x, y ∈ C . (2:1) It is important to note that every asymptotically -strict pseudocontractive mapping with sequence {g n }isauniformlyL-Lipschitzian mapping with L =sup  κ+ √ 1+(1−κ)γ n 1+κ : n ≥ 1  . Definition 2.3.[20]AmappingS : C ® C is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence {g n } if there exist a constant  Î [0, 1) and a sequence {g n } in [0, ∞) with lim n®∞ g n = 0 such that lim sup n→∞ sup x, y ∈C (||S n x − S n y|| 2 − (1 + γ n )||x − y|| 2 − κ||x −S n x − (y −S n y)|| 2 ) ≤ 0 . (2:2) Put c n := max  0, sup x,y∈C (||S n x − S n y|| 2 − (1 + γ n )||x − y|| 2 − κ||x − S n x − (y − S n y)|| 2 )  . Then, c n ≥ 0(∀n ≥ 1), c n ® 0(n ® ∞) and (2.2) reduces to the relation ||S n x −S n y|| 2 ≤ ( 1+γ n ) ||x −y|| 2 + κ||x −S n x − ( y −S n y ) || 2 + c n , ∀n ≥ 1, ∀x, y ∈ C . (2:3) Whenever c n =0foralln ≥ 1in(2.3),thenS is an asymptotically -strict pseudo- contractive mapping with sequence {g n }. For every point x Î H, there exists a unique nearest point in C, denoted by P C x, such that | |x −P C x|| ≤ ||x − y ||, ∀ y ∈ C. P C is called the metric projection of H onto C. Recall that the inequality holds x − P C x, P C x − y ≥0, ∀x ∈ H, y ∈ C . (2:4) Moreover, it is equivalent to ||P C x − P C y || 2 ≤P C x − P C y , x − y , ∀x, y ∈ H ; it is also equivalent to ||x − y || 2 ≥||x − P C x|| 2 + || y − P C x|| 2 , ∀x ∈ H, y ∈ C . (2:5) Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 4 of 18 It is easy to see th at P C is a nonexpansive mapping from H onto C; see, e.g., [26] for further detail. Lemma 2.1. Let A : C ® H be a monotone mapping. Then, u ∈  ⇔ u = P C ( u − λAu ) , ∀λ>0 . Lemma 2.2. Let H be a real Hilbert space. Then, the following hold: | |x − y || 2 = ||x|| 2 −|| y || 2 − 2x − y , y , ∀x, y ∈ H . Lemma 2.3. [[20], Lemma 2.6] Let S : C ® Cbeanasymptotically-strict pseudo- contractive mapping in the intermediate sense with sequence {g n }. Then, | |S n x − S n y|| ≤ 1 1 − κ  κ||x −y|| +  (1+(1− κ)γ n )||x − y|| 2 +(1− κ)c n  for all x, y Î C and n ≥ 1. Lemma 2.4. [[20], Lemma 2.7] Let S : C ® C be a uniformly c ontinuous asymptoti- cally -strict pseudocontractive mapping in the intermediate sense with sequence {g n }. Let {x n } be a sequence in C such that ||x n - x n+1 || ® 0 and ||x n -S n x n || ® 0 as n ® ∞. Then,||x n -Sx n || ® 0 as n ® ∞. Proposition 2.1 (Demiclosedness Principle). [[20], Proposit ion 3.1] Let S : C ® Cbe a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence {g n }.Then,I-Sisdemiclosedatzerointhesensethatif{ x n } is a sequence in C such that x n ⇀ x Î Candlim sup m® ∞ lim sup n® ∞ ||x n S m x n || = 0, then (I-S)x =0. Proposition 2.2. [[20], Proposition 3.2] Let S : C ® C be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence {g n } such that F (S) ≠ ∅. Then, F(S) is closed and convex. Remark 2.1. Pr opositions 2.1 and 2.2 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence {g n }. Moreover, Proposition 2.1 extends the demiclosedness principles studied for certain classes of nonlinear mappings in [19,27-29]. Lemma 2.5. [30]Let (X, 〈·,·〉) be an inner product space. Then, for all x, y, z Î Xand all a, b, g Î [0, 1] with a + b + g =1, we have ||αx + β y + γ z|| 2 = α||x|| 2 + β|| y || 2 + γ ||z|| 2 − αβ||x − y || 2 −αγ ||x −z|| 2 −βγ|| y −z|| 2 . Lemma 2.6. [[31], Lemma 2.5] Let {s n } be a sequence of nonnegative real numbers satisfying s n+1 ≤ ( 1 −¯α n ) s n + ¯α n ¯ β n + ¯γ n , ∀n ≥ 1 , where { ¯α n } , { ¯ β n } , and {¯ γ n } satisfy the conditions: (i) {¯α n }⊂[0, 1],  ∞ n =1 ¯α n = ∞ , or equivalently,  ∞ n =1 (1 −¯α n )= 0 ; (ii) lim sup n → ∞ ¯ β n ≤ 0 ; (iii) ¯γ n ≥ 0(n ≥ 1),  ∞ n =1 ¯γ n < ∞ . Then, lim n®∞ s n =0. Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 5 of 18 Lemma 2.7 . [32]Let {x n } and {z n } be bounded sequences in a Banach space X and let {ϱ n } be a sequence in [0, 1] with 0 < lim inf n®∞ ϱ n ≤ li m sup n®∞ ϱ n ≤ 1. Suppose that x n+1 = ϱ n x n +(1-ϱ n )z n for all integers n ≥ 1 and lim sup n®∞ (||z n+1 - z n || - | |x n+1 - x n ||) ≤ 0. Then, lim n®∞ ||z n -x n || = 0. The following lemma can be easily proved, and therefore, we omit the proof. Lemma 2.8. In a real Hilbert space H, there holds the inequality | |x + y || 2 ≤||x|| 2 +2 y , x + y , ∀x, y ∈ H . A set-valued mapping T : H ® 2 H is called monotone if for all x , y Î H, f Î Tx and g Î Ty imply 〈x-y, f-g〉 ≥ 0. A monotone mapping T : H ® 2 H is maximal if its graph G(T) is not pr operl y contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal i f and only if for (x, f) Î H × H, 〈x- y, f-g〉 ≥ 0forall(y, g) Î G(T) implies f Î Tx.LetA : C ® H be a monotone, L- Lipschitz continuous mapping and let N C v be the normal cone to C at v Î C, i.e., N C v ={w Î H : 〈v-u, w〉 ≥ 0, ∀u Î C}. Define Tv =  Av + N C v if v ∈ C , ∅ if v ∈ C . It is known that in this case T is maximal monotone, and 0 Î Tv if and only if v Î Ω; see [33]. 3. Extragradient-like approximation method and strong convergence results Let A : C ® H be a monotone and L-Lipschitz continuous mapping, f : C ® C be a contraction with contractive c onstant a Î (0, 1) and S : C ® C be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence {g n }. In this pap er, we introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method in [11] and the modified Mann iteration pro- cess in [20]: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x 1 = x ∈ C chosen arbitrary, y n =(1− μ n )x n + μ n P C (x n − λ n Ax n ), t n = P C (x n − λ n Ay n ), x n+1 =(1− α n − β n − ν n )x n + α n f (y n )+β n t n + ν n S n t n , ∀n ≥ 1 , (3:1) where {l n } is a sequence in (0, 1) with  ∞ n =1 λ n < ∞ , and {a n }, {b n }, {μ n } and {ν n }are sequences in [0, 1] satisfying the following conditions: (A1) a n + b n + ν n ≤ 1 for all n ≥ 1; (A2) lim n®∞ a n =0,  ∞ n =1 α n = ∞ ; (A3)  < lim inf n®∞ b n ≤ lim sup n®∞ b n <1; (A4)  ∞ n =1 ν n = ∞ . The following re sult shows the strong convergence of the sequences {x n }, {y n }gener- ated by the scheme (3.1) to the same point q = P F(S)∩ Ω f (q) if and only if {Ax n }is bounded, ||(I-S n )x n || ® 0 and lim inf n®∞ 〈Ax n , y-x n 〉 ≥ 0 for all y Î C. Theorem 3.1. Let A : C ® H be a monotone and L-Lipschitz continuou s mapping, f : C ® C be a contraction with contractive constant a Î (0, 1) and S : C ® Cbeauni- formly continuo us asymptotically -strict pseudocontractive mapping in the Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 6 of 18 intermediate sense with sequence {g n } such that F(S) ∩ Ω ≠ ∅ and  ∞ n =1 γ n < ∞ . Let {x n }, {y n } be th e sequences generated by (3.1),where{l n } is a sequence in (0, 1) with  ∞ n =1 λ n < ∞ ,and{a n }, {b n }, {μ n } and{y n } are sequences in [0, 1] satisfying the condi- tions (A1)-(A4). Then, the sequences {x n }, {y n } converge strongly to the same point q = P F(S)∩Ω f (q) if and only if { Ax n } is bounded,||(I-S n )x n || ® 0 and lim inf n®∞ 〈Ax n , y- x n 〉 ≥ 0 for all y Î C. Proof. “Necessity” . Suppose that the sequences {x n }, {y n } converge strongly to the same point q = P F(S)∩Ω f (q). Then from the L-Lipschitz continuity of A , it f ollows that {Ax n } is bounded, and for each y Î C: |Ax n , y − x n −Aq, y − q| ≤|Ax n , y − x n −Ax n , y − q| + |Ax n , y − q−Aq, y − q | = |Ax n , q − x n | + |Ax n − Aq, y − q| ≤||Ax n ||||q − x n || + ||Ax n − Aq||||y −q|| ≤||Ax n |||| q − x n || + L||x n − q |||| y − q || → 0, which implies that lim n → ∞ Ax n , y − x n  = Aq, y − q≥0, ∀y ∈ C due to q Î Ω. Furthermore, utilizing Lemma 2.3, we have | |S n x n − q|| ≤ 1 1 − κ  κ||x n − q|| +  (1+(1−κ)γ n )||x n − q|| 2 +(1− κ)c n  → 0 due to x n ® q, g n ® 0 and c n ® 0. Consequently, we conclude that for each y Î C ||S n x n − x n || ≤ ||S n x n − q || + ||x n − q || → 0 . That is, ||(I-S n )x n || ® 0. “Sufficiency”. Suppose that { Ax n } is bounded, ||(I-S n )x n || ® 0andliminf n®∞ 〈Ax n , y-x n 〉 ≥ 0 for all y Î C. Note that lim inf n®∞ b n >. Hence, we may assume, without loss of generality, that b n > for all n ≥ 1. Next, we divide the proof of the sufficiency into several steps. STEP 1. We claim that {x n } is bounded. Indeed, put t n = P C (x n - l n Ay n ) for all n ≥ 1. Let x* Î F(S) ∩ Ω.Then,x* = P C ( x* - l n Ax*). Putting x = x n - l n Ay n and y = x* in (2.5), we obtain | |t n − x ∗ || 2 ≤||x n − λ n Ay n − x ∗ || 2 −||x n − λ n Ay n − t n || 2 = ||x n − x ∗ || 2 − 2λ n Ay n , x n − x ∗  + λ 2 n ||Ay n || 2 −||x n − t n || 2 +2λ n Ay n , x n − t n −λ 2 n ||Ay n || 2 = ||x n − x ∗ || 2 +2λ n Ay n , x ∗ − t n −||x n − t n || 2 = ||x n − x ∗ || 2 −||x n − t n || 2 − 2λ n Ay n − Ax ∗ , y n − x ∗  − 2λ n Ax ∗ , y n − x ∗  +2λ n A y n , y n − t n . (3:2) Since A is monotone and x* is a solution of VIP(A, C), we have A y n − Ax ∗ , y n − x ∗ ≥0andAx ∗ , y n − x ∗ ≥0 . Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 7 of 18 It follows from (3.2) that ||t n − x ∗ || 2 ≤||x n − x ∗ || 2 −||x n − t n || 2 +2λ n Ay n , y n − t n  = ||x n − x ∗ || 2 −||(x n − y n )+(y n − t n )|| 2 +2λ n Ay n , y n − t n  = ||x n − x ∗ || 2 −||x n − y n || 2 − 2x n − y n , y n − t n −||y n − t n || 2 +2λ n Ay n , y n − t n  = ||x n − x ∗ || 2 −||x n − y n || 2 −|| y n − t n || 2 +2x n − λ n A y n − y n , t n − y n  . (3:3) Note that x n Î C for all n ≥ 1 and that y n =(1-μ n )x n + μ n P C (x n - l n Ax n ). Hence, we have 2x n − λ n A y n − y n , t n − y n  ≤ 2||x n − λ n Ay n − y n ||||t n − y n || ≤ ||x n − λ n Ay n − y n || 2 + ||t n − y n || 2 = ||x n − y n || 2 − 2λ n Ay n , x n − y n  + λ 2 n ||Ay n || 2 + ||t n − y n || 2 = ||x n − y n || 2 + ||t n − y n || 2 +2λ n μ n Ay n , P C (x n − λ n Ax n ) − P C x n  + λ 2 n ||Ay n || 2 ≤||x n − y n || 2 + ||t n − y n || 2 +2λ n μ n ||Ay n ||||P C (x n − λ n Ax n ) − P C x n || + λ 2 n ||Ay n || 2 ≤||x n − y n || 2 + ||t n − y n || 2 +2λ 2 n μ n ||Ay n ||||Ax n || + λ 2 n ||Ay n || 2 . (3:4) Since {Ax n } is bounded and A is L-Lipschitz continuous, we have ||Ay n − Ax n || ≤ L||y n − x n || = Lμ n ||P C ( x n − λ n Ax n ) − P C x n || ≤ L||Ax n || , and h ence ||Ay n || ≤ (1+ L)||Ax n ||, which implies that {Ay n } is bounded. Hence, we may as sume that there exists a constant M ≥ sup{||Ax n || + ||Ay n || + ||Ax*||: n ≥ 1}. Then, it follows from (3.4) that 2x n − λ n Ay n − y n , t n − y n ≤||x n − y n || 2 + ||t n − y n || 2 + λ 2 n (||Ax n || + ||Ay n ||) 2 ≤||x n − y n || 2 + ||t n − y n || 2 + λ 2 n M 2 . This together with (3.3) implies that ||t n − x ∗ || 2 ≤||x n − x ∗ || 2 −||x n − y n || 2 −||y n − t n || 2 +2x n − λ n Ay n − y n , t n − y n  ≤||x n − x ∗ || 2 −||x n − y n || 2 −||y n − t n || 2 + ||x n − y n || 2 + ||t n − y n || 2 + λ 2 n M 2 = ||x n − x ∗ || 2 + λ 2 n M 2 . (3:5) Observe that | |f (y n ) − x ∗ || 2 ≤ (||f (y n ) − f(x ∗ )|| + ||f (x ∗ ) − x ∗ ||) 2 ≤ (α||y n − x ∗ || + ||f(x ∗ ) − x ∗ ||) 2 =  α||y n − x ∗ || +(1− α) ||f (x ∗ ) − x ∗ || 1 − α  2 ≤ α||y n − x ∗ || 2 + ||f (x ∗ ) − x ∗ || 2 1 − α = α||(1 − μ n )(x n − x ∗ )+μ n (P C (x n − λ n Ax n ) − P C (x ∗ − λ n Ax ∗ )|| 2 + ||f (x ∗ ) − x ∗ || 2 1 − α ≤ α[(1 − μ n )||x n − x ∗ || 2 + μ n ||P C (x n − λ n Ax n ) − P C (x ∗ − λ n Ax ∗ )|| 2 ]+ ||f (x ∗ ) − x ∗ || 2 1 − α ≤ α[(1 − μ n )||x n − x ∗ || 2 + μ n ||(x n − x ∗ ) − λ n (Ax n − Ax ∗ )|| 2 ]+ ||f (x ∗ ) − x ∗ || 2 1 − α = α[(1 − μ n )||x n − x ∗ || 2 + μ n (||x n − x ∗ || 2 − 2λ n x n − x ∗ , Ax n − Ax ∗  +λ 2 n ||Ax n − Ax ∗ || 2 ]+ ||f (x ∗ ) − x ∗ || 2 1 − α ≤ α[(1 − μ n )||x n − x ∗ || 2 + μ n (||x n − x ∗ || 2 + λ 2 n ||Ax n − Ax ∗ || 2 ]+ ||f (x ∗ ) − x ∗ || 2 1 − α ≤ α||x n − x ∗ || 2 + λ 2 n M 2 + ||f (x ∗ ) − x ∗ || 2 1 − α . (3:6) Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 8 of 18 Putting τ n = a n + b n + ν n and utilizing Lemma 2.5, we obtain from (3.5) and (3.6) ||x n+1 − x ∗ || 2 = ||(1 − α n − β n − ν n )(x n − x ∗ )+α n (f (y n ) −x ∗ )+β n (t n − x ∗ )+ν n (S n t n − x ∗ )|| 2 ≤ (1 − τ n )||x n − x ∗ || 2 + τ n || α n τ n (f (y n ) −x ∗ )+ β n τ n (t n − x ∗ )+ ν n τ n (S n t n − x ∗ )|| 2 ≤ (1 − τ n )||x n − x ∗ || 2 + τ n  α n τ n ||f (y n ) −x ∗ || 2 + β n τ n ||t n − x ∗ || 2 + ν n τ n ||S n t n − x ∗ || 2 − β n ν n τ 2 n ||t n − S n t n || 2  =(1−τ n )||x n − x ∗ || 2 + α n ||f (y n ) −x ∗ || 2 + β n ||t n − x ∗ || 2 + ν n ||S n t n − x ∗ || 2 − β n ν n τ n ||t n − S n t n || 2 ≤ (1 − τ n )||x n − x ∗ || 2 + α n ||f (y n ) −x ∗ || 2 + β n ||t n − x ∗ || 2 +ν n [(1 + γ n )||t n − x ∗ || 2 + κ||t n − S n t n || 2 + c n ] − β n ν n τ n ||t n − S n t n || 2 =(1−τ n )||x n − x ∗ || 2 + α n ||f (y n ) −x ∗ || 2 +(β n + ν n + ν n γ n )||t n − x ∗ || 2 +ν n (κ − β n τ n )||t n − S n t n || 2 + ν n c n ≤ (1 − τ n )||x n − x ∗ || 2 + α n ||f (y n ) −x ∗ || 2 +(β n + ν n + γ n )||t n − x ∗ || 2 + ν n c n ≤ (1 − τ n )||x n − x ∗ || 2 + α n  α||x n − x ∗ || 2 + λ 2 n M 2 + ||f (x ∗ ) −x ∗ || 2 1 −α  +(β n + ν n + γ n )(||x n − x ∗ || 2 + λ 2 n M 2 )+ν n c n =(1−(1 − α)α n + γ n )||x n − x ∗ || 2 +(α n + β n + ν n + γ n )λ 2 n M 2 +(1 −α)α n ||f (x ∗ ) −x ∗ || 2 (1 −α) 2 + ν n c n ≤ (1 − (1 − α)α n + γ n )max  ||x n − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 −α) 2  +(1+γ n )λ 2 n M 2 +(1 −α)α n max  ||x n − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2  + ν n c n ≤ (1 + γ n )max  ||x n − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 ( 1 −α ) 2  +2M 2 λ 2 n + ν n c n . (3:7) Now, let us show that for all n ≥ 1 ||x n+1 −x ∗ || 2 ≤ ⎛ ⎝ n  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2   . (3:8) As a matter of fact, whenever n = 1, from (3.7), we have ||x 2 − x ∗ || 2 ≤ (1 + γ 1 )max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 −α) 2  +2M 2 λ 2 1 + ν 1 c 1 ≤ (1 + γ 1 )  max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 −α) 2  +2M 2 λ 2 1 + ν 1 c 1  = ⎛ ⎝ 1  j=1 (1 + γ j ) ⎞ ⎠  1  i=1  2M 2 λ 2 i + ν i c i  +max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 −α) 2   . Assume that (3.8) holds for some n ≥ 1. Consider the case of n + 1. From (3.7), we obtain Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 9 of 18 | |x n+2 − x ∗ || 2 ≤ (1 + γ n+1 )max  ||x n+1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2  +2M 2 λ 2 n+1 + ν n+1 c n+1 ≤ (1 + γ n+1 )  max  ||x n+1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2  +2M 2 λ 2 n+1 + ν n+1 c n+1  ≤ (1 + γ n+1 ) ⎛ ⎝ max ⎧ ⎨ ⎩ ⎛ ⎝ n  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2   ||f (x ∗ ) −x ∗ || 2 (1 −α) 2  +2M 2 λ 2 n+1 + ν n+1 c n+1  ≤ (1 + γ n+1 ) ⎛ ⎝ ⎛ ⎝ n  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2   +2M 2 λ 2 n+1 + ν n+1 c n+1  = ⎛ ⎝ n+1  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2   +(1 + γ n+1 )(2M 2 λ 2 n+1 + ν n+1 c n+1 ) ≤ ⎛ ⎝ n+1  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 −α) 2   + ⎛ ⎝ n+1  j=1 (1 + γ j ) ⎞ ⎠  2M 2 λ 2 n+1 + ν n+1 c n+1  = ⎛ ⎝ n+1  j=1 (1 + γ j ) ⎞ ⎠  n+1  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) −x ∗ || 2 (1 −α) 2   . This shows that (3.8) holds for the case of n +1.Byinduction,weknowthat(3.8) holds f or all n ≥ 1. Sinc e  ∞ n =1 γ n < ∞ ,  ∞ n =1 λ 2 n < ∞ and  ∞ n =1 ν n c n < ∞ ,from(3.8) we deduce that for all n ≥ 1 ||x n+1 − x ∗ || 2 ≤ ⎛ ⎝ n  j=1 (1 + γ j ) ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 − α) 2   ≤ exp ⎛ ⎝ n  j=1 γ j ⎞ ⎠  n  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 − α) 2   ≤ exp ⎛ ⎝ ∞  j=1 γ j ⎞ ⎠  ∞  i=1 (2M 2 λ 2 i + ν i c i )+max  ||x 1 − x ∗ || 2 , ||f (x ∗ ) − x ∗ || 2 (1 − α) 2   . This implies that {x n } is bounded. STEP 2. We claim that lim n®∞ ||x n+1 - x n || = 0. Indeed, observe that ||t n+1 − t n || = ||P C (x n+1 − λ n+1 Ay n+1 ) −P C (x n − λ n Ay n )| | ≤||(x n+1 − λ n+1 Ay n+1 ) −(x n − λ n Ay n )|| ≤||x n+1 − x n || + λ n+1 ||Ay n+1 || + λ n ||Ay n || ≤||x n+1 − x n || + ( λ n + λ n+1 ) M (3:9) Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 10 of 18 [...]... extragradient method for finding saddle points and other problems Matecon 12, 747–756 (1976) Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings J Optim Theory Appl 128, 191–201 (2006) doi:10.1007/s10957-005-7564-z Zeng, LC, Yao, JC: Strong convergence theorem by an extragradient method for fixed point problems and variational. .. References 1 Baiocchi, C, Capelo, A: Variational and Quasivariational Inequalities, Application to Free Boundary Problems Wiley, New York (1984) 2 Kinderlehrer, D, Stampacchia, G: An Introduction to Variational Inequalities and their Applications Academic Press, New York (1980) Ceng et al Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22... λn < ∞, and {an} , {bn}, {μn} and {νn} are four sequences in [0, 1] satisfying n=1 the conditions (A1)-(A4) Then, the sequences {xn}, {yn} converge strongly to the same Page 15 of 18 Ceng et al Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 B point q = PA−1 0∩B−1 0 f (q)if and only if {Ax n } is bounded, ||(I − (Jr )n )xn || → 0and lim... inequality problems Taiwanese J Math 10(5), 1293–1303 (2006) Moudafi, A: Viscosity approximation methods for fixed- points problems J Math Anal Appl 241, 46–55 (2000) doi:10.1006/jmaa.1999.6615 Xu, HK: Viscosity approximation methods for nonexpansive mappings J Math Anal Appl 298, 279–291 (2004) doi:10.1016/j.jmaa.2004.04.059 Ceng, LC, Yao, JC: An extragradient-like approximation method for variational inequality... mapping, and f : C ® C be a contraction with contractive constant a Î (0, 1) Let Ω ≠ ∅ Let {xn}, {yn} Ceng et al Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 be the sequences generated by (3.21), where {l n } is a sequence in (0, 1) with ∞ n=1 λn < ∞, and {an} , {bn} and {μn} are three sequences in [0, 1] satisfying the conditions: (B1) an. .. λn < ∞, and {an} , {bn} and {μn} are three n=1 sequences in [0, 1] satisfying the conditions (B1)-(B3) Then, the sequences {xn}, {yn} converge strongly to the same point q = PA−1 0 f (q)if and only if {Axn} is bounded and lim infn®∞ 〈Axn, y - xn〉 ≥ 0 for all y Î C Proof In Theorem 3.1, put C = H, νn = 0 (∀n ≥ 1) and S = I the identity mapping of H Then, we know that  = 0, gn = 0 and cn = 0 for all... 1) with ∞ λn < ∞, n=1 and {an} , {bn}, {μn} and {νn} are four sequences in [0, 1] satisfying the conditions (A1)(A4) Then, the sequences {x n }, {y n } converge strongly to the same point q = PF(S)∩A−1 0 f (q)if and only if {Ax n } is bounded, ||(I - S n )x n || ® 0 and lim inf n®∞ 〈Axn, y - xn〉 ≥ 0 for all y Î H Let B : H ® 2H be a maximal monotone mapping Then, for any x Î H and r > 0, B B consider... variational inequality problems and fixed point problems Appl Math Comput 190, 205–215 (2007) doi:10.1016/j.amc.2007.01.021 Schu, J: Iterative construction of fixed points of asymptotically nonexpansive mapping J Math Anal Appl 159, 407–413 (1991) Ceng, LC, Xu, HK, Yao, JC: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces Nonlinear Anal 69(4), 1402–1412 (2008)... nonlinear ill-posed variational inequalities and convergence rates Set-Valued Anal 6, 313–344 (1998) doi:10.1023/A:1008643727926 Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings SIAM J Optim 16, 1230–1241 (2006) doi:10.1137/050624315 Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings... convergence for fixed points for mappings of asymptotically nonexpansive type Nonlinear Anal 16, 1139–1146 (1991) doi:10.1016/0362-546X(91)90201-B Osilike, MO, Igbokwe, DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations Comput Math Appl 40, 559–567 (2000) doi:10.1016/S0898-1221(00)00179-6 Xu, HK: Iterative algorithms for nonlinear . variational inequalities and fixed point problems. Fixed Point Theory and Applications 2011 2011:22. Ceng et al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page. C H Open Access An extragradient-like approximation method for variational inequalities and fixed point problems Lu-Chuan Ceng 1,2 , Qamrul Hasan Ansari 3 , Ngai-Ching Wong 4* and Jen-Chih Yao 4,5 *. al. Fixed Point Theory and Applications 2011, 2011:22 http://www.fixedpointtheoryandapplications.com/content/2011/1/22 Page 17 of 18 3. Konnov, IV: Combined Relaxation Methods for Variational Inequalities,