Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 626159, 22 pages doi:10.1155/2011/626159 Research Article Finding Common Solutions of a Variational Inequality, a General System of Variational Inequalities, and a Fixed-Point Problem via a Hybrid Extragradient Method Lu-Chuan Ceng,1 Sy-Ming Guu,2 and Jen-Chih Yao3 Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China Department of Business Administration, College of Management, Yuan-Ze University, Taoyuan Hsien, Chung-Li City 330, Taiwan Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan Correspondence should be addressed to Sy-Ming Guu, iesmguu@saturn.yzu.edu.tw Received 25 September 2010; Accepted 20 December 2010 Academic Editor: Jong Kim Copyright q 2011 Lu-Chuan Ceng et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We propose a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of a general system of variational inequalities, and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space Our hybrid method is based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method By constrasting with other methods, our hybrid approach drops the requirement of boundedness for the domain in which various mappings are defined Furthermore, under mild conditions imposed on the parameters we show that our algorithm generates iterates which converge strongly to a common element of these three problems Introduction Let H be a real Hilbert space with inner product ·, · and norm · Let C be a nonempty closed convex subset of H and S : C → C be a self-mapping on C We denote by Fix S the set of fixed points of S and by PC the metric projection of H onto C Moreover, we also denote by R the set of all real numbers For a given nonlinear operator A : C → H, we consider the following variational inequality problem of finding x∗ ∈ C such that Ax∗ , x − x∗ ≥ 0, ∀x ∈ C 1.1 Fixed Point Theory and Applications The solution set of the variational inequality 1.1 is denoted by VI A, C Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems See, for example, 1–21 and the references therein For finding an element of Fix S ∩ VI A, C when C is closed and convex, S is nonexpansive and A is α-inverse strongly monotone, Takahashi and Toyoda 22 introduced the following Mann-type iterative algorithm: xn αn xn − αn SPC xn − λn Axn , ∀n ≥ 0, 1.2 where PC is the metric projection of H onto C, x0 x ∈ C, {αn } is a sequence in 0, , and {λn } is a sequence in 0, 2α They showed that, if Fix S ∩ VI A, C / ∅, then the sequence {xn } converges weakly to some z ∈ Fix S ∩ VI A, C Nadezhkina and Takahashi 23 and Zeng and Yao 24 proposed extragradient methods motivated by Korpeleviˇ 25 for finding c a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality problem Further, these iterative methods were extended in 26 to develop a new iterative method for finding elements in Fix S ∩ VI A, C Let B1 , B2 : C → H be two mappings Now we consider the following problem of finding x∗ , y∗ ∈ C × C such that μ B1 y ∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ C, μ2 B2 x ∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ C, 1.3 which is called a general system of variational inequalities where μ1 > and μ2 > are two constants The set of solutions of problem 1.3 is denoted by GSVI B1 , B2 , C In particular, if B1 B2 A, then problem 1.3 reduces to the problem of finding x∗ , y∗ ∈ C × C such that μ1 Ay∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ C, μ2 Ax∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ C, 1.4 which was defined by Verma 27 see also 28 and is called the new system of variational y∗ additionally, then problem 1.4 reduces to the classical inequalities Further, if x∗ variational inequality problem 1.1 Ceng et al 29 studied the problem 1.3 by transforming it into a fixed-point problem Precisely and for easy reference, we state their results in the following lemma and theorem Lemma CWY see 29 For given x, y ∈ C, x, y is a solution of problem 1.3 if and only if x is a fixed point of the mapping G : C → C defined by G x PC PC x − μ2 B2 x − μ1 B1 PC x − μ2 B2 x , ∀x ∈ C, 1.5 where y PC x − μ2 B2 x In particular, if the mapping Bi : C → H is μi -inverse strongly monotone for i 1, 2, then the mapping G is nonexpansive provided μi ∈ 0, 2μi for i 1, Fixed Point Theory and Applications Throughout this paper, the fixed-point set of the mapping G is denoted by Γ Utilizing Lemma CWY, they introduced and studied a relaxed extragradient method for solving problem 1.3 Theorem CWY see 29, Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let the mapping Bi : C → H be βi -inverse strongly monotone for i 1, Let S : C → C be a nonexpansive mapping with Fix S ∩ Γ / ∅ Suppose x1 u ∈ C and {xn } is generated by yn xn where μi ∈ 0, 2βi for i i αn βn βn xn γn SPC yn − μ1 B1 yn , 1.6 1, 2, and {αn }, {βn }, {γn } are three sequences in 0, such that γn ii limn → ∞ αn αn u PC xn − μ2 B2 xn , 1, for all n ≥ 1; 0, ∞ n αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < Then {xn } converges strongly to x y PC x − μ B x PFix S ∩ Γu and x, y is a solution of problem 1.3 , where It is clear that the above result unifies and extends some corresponding results in the literature Based on the relaxed extragradient method and viscosity approximation method, Yao et al 30 proposed and analyzed an iterative algorithm for finding a common element of the solution set of the general system 1.3 of variational inequalities and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space H Theorem YLK see 30, Theorem 3.2 Let C be a nonempty bounded closed convex subset of a real Hilbert space H Let the mapping Bi : C → H be μi -inverse strongly monotone for i 1, Let S : C → C be a k-strictly pseudocontractive mapping such that Fix S ∩ Γ / ∅ Let Q : C → C be a ρ-contraction with ρ ∈ 0, 1/2 For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn }, and {zn } be generated iteratively by zn αn Qxn yn xn where μi ∈ 0, 2βi for i i βn γn δn ii limn → ∞ αn PC xn − μ2 B2 xn , βn xn − αn PC zn − μ1 B1 zn , γn PC zn − μ1 B1 zn δn Syn , 1.7 ∀n ≥ 0, 1, and {αn }, {βn }, {γn }, {δn } are four sequences in 0, such that and γn and ∞ n δn k ≤ γn < − 2ρ δn for all n ≥ 0; αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < and lim infn → ∞ δn > 0; iv limn → ∞ γn / − βn − γn / − βn Then the sequence {xn } generated by 1.7 converges strongly to x PFix S ∩ Γ · Qx and x, y is a solution of the general system 1.3 of variational inequalities, where y PC x − μ2 B2 x Fixed Point Theory and Applications Motivated by the above work, in this paper, we introduce an iterative algorithm for finding a common element of the solution set of the variational inequality 1.1 , the solution set of the general system 1.3 and the fixed-point set of the strictly pseudocontractive mapping S : C → C via a hybrid extragradient method based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method, that is, zn yn αn Qxn PC xn − λn Axn , − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , xn βn xn γn yn δn Syn , 1.8 ∀n ≥ 0, where {λn } ⊂ 0, ∞ , {αn }, {βn }, {γn }, {δn } ⊂ 0, such that βn γn δn for all n ≥ Moreover, we prove that the studied iterative algorithm converges strongly to an element of Fix S ∩ Γ ∩ VI A, C under some mild conditions imposed on algorithm parameters Our method improves and extends Yao et al 30, Theorem 3.2 in the following aspects: i the problem of finding an element of Fix S ∩ Γ in 30, Theorem 3.2 is extended to the the problem of finding an element of Fix S ∩ Γ ∩ VI A, C ; ii the requirement of boundedness of C in 30, Theorem 3.2 is removed; iii the condition γn δn k ≤ γn < − 2ρ δn , for all n ≥ in 30, Theorem 3.2 is replaced by the one γn δn k ≤ γn , for all n ≥ 0; iv the argument of Step in the proof of 30, Theorem 3.2 is simplified under the lack of the condition γn < − 2ρ δn , for all n ≥ 0; v our iterative algorithm is similar to but different from the one of 30, Theorem 3.2 because the problem of finding an element of Fix S ∩ Γ ∩ VI A, C is more challenging than the problem of finding an element of Fix S ∩ Γ in 30, Theorem 3.2 Preliminaries In this section, we collect some notations and lemmas Let C be a nonempty closed convex subset of a real Hilbert space H A mapping A : C → H is called monotone if Ax − Ay, x − y ≥ 0, ∀x, y ∈ C 2.1 A mapping A : C → H is called Lipschitz continuous if there exists a real number L > such that Ax − Ay ≤ L x − y , ∀x, y ∈ C 2.2 Recall that a mapping A : C → H is called α-inverse strongly monotone if there exists a real number α > such that Ax − Ay, x − y ≥ α Ax − Ay , ∀x, y ∈ C 2.3 Fixed Point Theory and Applications It is clear that every inverse strongly monotone mapping is a monotone and Lipschitz continuous mapping Also, recall that a mapping S : C → C is said to be k-strictly pseudocontractive if there exists a constant ≤ k < such that Sx − Sy ≤ x−y k I−S x− I −S y , ∀x, y ∈ C 2.4 For such a case, we also say that S is a k-strict pseudo-contraction 31 It is clear that, in a real Hilbert space H, inequality 2.4 is equivalent to the following: Sx − Sy, x − y ≤ x − y − 1−k I −S x− I−S y , ∀x, y ∈ C 2.5 This immediately implies that if S is a k-strictly pseudocontractive mapping, then I − S is − k /2-inverse strongly monotone; see 32 for more details We use Fix S to denote the set of fixed points of S It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings which are mappings S : C → C such that Sx − Sy ≤ x − y , for all x, y ∈ C A mapping Q : C → C is called a contraction if there exists a constant ρ ∈ 0, such that Qx − Qy ≤ ρ x − y for all x, y ∈ C For every point x ∈ H, there exists a unique nearest point in C, denoted by PC x such that x − PC x ≤ x − y , ∀y ∈ C 2.6 The mapping PC is called the metric projection of H onto C It is well known that PC is a nonexpansive mapping and satisfies x − y, PC x − PC y ≥ PC x − PC y , ∀x, y ∈ H 2.7 It is known that PC x is characterized by the following property: x − PC x, y − PC x ≤ 0, ∀x ∈ H, y ∈ C 2.8 In order to prove the main result in this paper, we will need the following lemmas in the sequel Lemma 2.1 see 33 Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be − βn yn βn xn a sequence in 0, with < lim infn → ∞ βn ≤ lim supn → ∞ βn < Suppose xn for all integers n ≥ and lim supn → ∞ yn − yn − xn − xn ≤ Then, limn → ∞ yn − xn Lemma 2.2 see 34, Proposition 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a self-mapping of C i If S is a k-strict pseudocontractive mapping, then S satisfies the Lipschitz condition Sx − Sy ≤ k x−y , 1−k ∀x, y ∈ C 2.9 Fixed Point Theory and Applications ii If S is a k-strict pseudocontractive mapping, then the mapping I − S is demiclosed at 0, that is, if {xn } is a sequence in C such that xn → x weakly and I − S xn → strongly, then I − S x iii If S is k-(quasi-)strict pseudo-contraction, then the fixed-point set Fix S of S is closed and convex so that the projection PFix S is well defined Lemma 2.3 see 9, Lemma 2.1 Let {sn } be a sequence of nonnegative real numbers satisfying the condition sn ≤ − αn sn αn βn , ∀n ≥ 0, 2.10 where {αn }, {βn } are sequences of real numbers such that ∞ n i {αn } ⊂ 0, and αn ∞ ∞, or equivalently, − αn : n n lim n→∞ − αk 0; 2.11 k ii lim supn → ∞ βn ≤ 0; or ii ∞ n αn βn is convergent Then, limn → ∞ sn Lemma 2.4 see 30 Let C be a nonempty closed convex subset of a real Hilbert space H Let S : C → C be a k-strictly pseudocontractive mapping Let γ and δ be two nonnegative real numbers Assume γ δ k ≤ γ Then γ x−y δ Sx − Sy ≤ γ δ x−y , ∀x, y ∈ C 2.12 The following lemma is an immediate consequence of an inner product Lemma 2.5 In a real Hilbert space H, there holds the inequality x y ≤ x 2 y, x y , ∀x, y ∈ H 2.13 Let A be a monotone mapping of C into H In the context of the variational inequality problem the characterization of projection 2.8 implies that u ∈ VI A, C ⇐⇒ u PC u − λAu , ∀λ > 2.14 It is also known that a set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, f ∈ T x and g ∈ T y imply that x − y, f − g ≥ A monotone set-valued mapping T : H → 2H is maximal if its graph Gph T is not properly contained in the graph of any other monotone set-valued mapping It is known that a monotone set-valued mapping T : H → 2H is maximal if and only if for x, f ∈ H × H, x − y, f − g ≥ for every y, g ∈ Gph T implies that f ∈ T x Let A be a Fixed Point Theory and Applications monotone and Lipschitz continuous mapping of C into H Let NC v be the normal cone to C at v ∈ C, that is, NC v {w ∈ H : v − u, w ≥ 0, ∀ ∈ C} 2.15 Define ⎧ ⎨Av Tv NC v if v ∈ C, ⎩∅ 2.16 if v ∈ C / It is known that in this case the mapping T is maximal monotone, and ∈ T v if and only if v ∈ VI A, C ; see [35] for more details Main Results The main idea for showing strong convergence of the sequence {xn } generated by 1.8 to an element of VI A, C is first to transform the variational inequality problem 1.1 into the zero point problem of a maximal monotone mapping T and then to derive the strong convergence of {xn } to a zero of T by using the technique in 10 We are now in a position to state and prove the main result in this paper Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be α-inverse strongly monotone and Bi : C → H be βi -inverse strongly monotone for i 1, Let S : C → C be a k-strictly pseudocontractive mapping such that Fix S ∩ Γ ∩ VI A, C / ∅ Let Q : C → C be a ρ-contraction with ρ ∈ 0, 1/2 For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn yn αn Qxn − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , xn where μi ∈ 0, 2βi for i i βn γn δn ii limn → ∞ αn PC xn − λn Axn , βn xn γn yn δn Syn , 3.1 ∀n ≥ 0, 1, 2, {λn } ⊂ 0, 2α and {αn }, {βn }, {γn }, {δn } ⊂ 0, such that and γn and ∞ n δn k ≤ γn for all n ≥ 0; αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < and lim infn → ∞ δn > 0; iv limn → ∞ γn / − βn − γn / − βn 0; v < lim infn → ∞ λn ≤ lim supn → ∞ λn < 2α and limn → ∞ |λn − λn | Then the sequence {xn } generated by 3.1 converges strongly to x PFix S is a solution of the general system 1.3 of variational inequalities, where y Proof We divide the proof into several steps Step {xn } is bounded ·Qx and x, y PC x − μ2 B2 x ∩ Γ ∩ VI A,C Fixed Point Theory and Applications Indeed, take x∗ ∈ Fix S ∩ Γ ∩ VI A, C arbitrarily Then Sx∗ x∗ , x∗ PC x∗ − λn Ax∗ and x∗ PC PC x ∗ − μ B2 x ∗ − μ B1 PC x ∗ − μ B2 x ∗ 3.2 Since A : C → H be α-inverse strongly monotone and < λn ≤ 2α, we have for all n ≥ 0, zn − x∗ PC xn − λn Axn − PC x∗ − λn Ax∗ ≤ xn − λn Axn − x∗ − λn Ax∗ xn − x∗ − λn Axn − Ax∗ ≤ xn − x∗ 2 2 3.3 − λn 2α − λn Axn − Ax∗ ≤ xn − x∗ For simplicity, we write y∗ PC x∗ − μ2 B2 x∗ and un PC zn − μ2 B2 zn for all n ≥ Since Bi : C → H be βi -inverse strongly monotone for i 1, and < μi < 2βi for i 1, 2, we know that for all n ≥ 0, − x∗ PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn −PC PC x∗ − μ2 B2 x∗ − μ1 B1 PC x∗ − μ2 B2 x∗ ≤ PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn − PC x ∗ − μ B x ∗ − μ B P C x ∗ − μ B x ∗ PC zn − μ2 B2 zn − PC x∗ − μ2 B2 x∗ −μ1 B1 PC zn − μ2 B2 zn − B1 PC x∗ − μ2 B2 x∗ ≤ PC zn − μ2 B2 zn − PC x∗ − μ2 B2 x∗ − μ1 2β1 − μ1 ≤ zn − x∗ − μ2 B2 zn − B2 x∗ 2 3.4 − μ1 2β1 − μ1 − μ1 2β1 − μ1 ≤ zn − x∗ − μ2 2β2 − μ2 ≤ xn − x∗ − λn 2α − λn Axn − Ax∗ ≤ xn − x∗ B1 PC zn − μ2 B2 zn − B1 PC x∗ − μ2 B2 x∗ zn − μ2 B2 zn − x∗ − μ2 B2 x∗ − μ2 2β2 − μ2 B2 zn − B2 x∗ B2 zn − B2 x∗ 2 B un − B y ∗ B un − B y ∗ − μ1 2β1 − μ1 B un − B y ∗ − μ1 2β1 − μ1 B un − B y ∗ 2 Fixed Point Theory and Applications Hence we get yn − x∗ αn Qxn − x∗ − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn ≤ αn Qxn − x∗ − αn ≤ αn ρ xn − x∗ Qx∗ − x∗ xn − x∗ − − ρ αn xn − x∗ , ≤ max PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn − x∗ − x∗ − αn xn − x∗ − ρ αn Qx∗ − x∗ 1−ρ Qx∗ − x∗ 1−ρ 3.5 Since γn δn k ≤ γn for all n ≥ 0, utilizing Lemma 2.4 we obtain from 3.5 δn Syn − x∗ γn yn − x∗ δn Syn − x∗ ≤ βn xn − x∗ γn yn − x∗ ≤ βn xn − x∗ xn − x∗ βn xn − x∗ γn ≤ βn xn − x∗ ≤ max γn xn − x∗ , δn yn − x∗ Qx∗ − x∗ xn − x∗ , 1−ρ δn max Qx∗ − x∗ 1−ρ 3.6 By induction, we obtain that for all n ≥ xn − x∗ ≤ max x0 − x∗ , Qx∗ − x∗ 1−ρ 3.7 Hence, {xn } is bounded Consequently, we deduce immediately that {zn }, {yn }, {Syn }, and {un } are bounded, where un PC zn − μ2 B2 zn for all n ≥ Now, put tn : PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , ∀n ≥ 3.8 Then it is easy to see that {tn } is bounded because PC , B1 , and B2 are Lipschitz continuous and {zn } is bounded Step limn → ∞ xn − xn 10 Fixed Point Theory and Applications Indeed, define xn wn γn yn γn yn 1 − yn 1 − βn xn − βn xn − δn Syn 1 − βn δn 1 − βn Since γn − βn wn for all n ≥ It follows that − βn xn − βn xn − wn βn xn − γn yn δn Syn − βn Syn δn − βn − Syn γn 1 − βn 1 γn − yn − βn 3.9 δn Syn − βn − δn k ≤ γn for all n ≥ 0, utilizing Lemma 2.4 we have γn yn − yn Next, we estimate yn zn − zn PC xn ≤ δn Syn − λn Axn − Syn ≤ γn δn yn − yn − yn Observe that 1 xn − λn Axn 1 − xn − λn Axn − Axn λn − λn xn − xn − λn Axn − Axn |λn ≤ xn tn − tn − xn |λn PC PC zn 1 PC zn 1 Axn − λn | Axn , − μ2 B2 zn − μ2 B2 zn − μ2 B2 zn −μ1 B1 PC zn ≤ PC zn 1 − μ1 B1 PC zn − μ1 B1 PC zn 1 − μ2 B2 zn − μ1 2β1 − μ1 ≤ PC zn ≤ 1 − μ2 B2 zn 1 − μ2 B2 zn zn − zn − μ2 B2 zn ≤ zn − zn ≤ zn − zn − B1 PC zn − μ2 B2 zn − μ2 B2 zn − zn − μ2 B2 zn − B2 zn − μ2 2β2 − μ2 B2 zn 2 2 − B1 PC zn − μ2 B2 zn − PC zn − μ2 B2 zn zn 1 − PC zn − μ2 B2 zn B1 PC zn − μ2 B2 zn − PC zn − μ2 B2 zn − μ2 B2 zn − μ2 B2 zn 3.11 − λn | Axn − PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn PC zn 3.10 − xn − λn Axn −PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn ≤ − PC xn − λn Axn xn ≤ − B2 zn 3.12 Fixed Point Theory and Applications 11 Combining 3.11 with 3.12 , we get tn PC PC zn − tn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn 1 −PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn ≤ xn − xn |λn 3.13 − λn | Axn This together with 3.13 implies that yn tn ≤ tn 1 − yn ≤ xn αn − tn Qxn − xn − tn Qxn αn 1 |λn − tn − αn Qxn − tn 1 − tn αn Qxn − tn − λn | Axn αn Qxn 1 3.14 − tn αn Qxn − tn Hence it follows from 3.9 , 3.10 , and 3.14 that wn − wn ≤ γn yn γn 1 − βn ≤ 1 − yn − γn − βn γn δn yn − βn yn 1 − xn γn 1 − βn |λn − Syn 1 − Syn δn 1 − βn yn γn 1 − βn − yn γn 1 − βn − yn ≤ xn δn − βn − γn − βn − λn | Axn γn − βn yn − − δn − βn γn − βn Syn Syn yn 3.15 Syn yn αn Syn Qxn − tn αn Qxn − tn Since {xn }, {yn }, and {tn } are bounded, it follows from conditions ii , iv , v that lim sup wn n→∞ − wn − xn ≤ lim sup |λn n→∞ 1 − xn − λn | Axn γn 1 − βn − γn − βn Hence by Lemma 2.1 we get limn → ∞ wn − xn lim xn n→∞ − xn αn Qxn yn − tn Syn αn Qxn − tn 3.16 0 Thus, lim − βn n→∞ wn − xn 3.17 12 Fixed Point Theory and Applications 0, limn → ∞ B1 un − B1 y∗ and limn → ∞ Axn − Ax∗ Step limn → ∞ B2 zn − B2 x∗ ∗ ∗ ∗ PC x − μ B2 x where y Indeed, utilizing Lemma 2.4 and the convexity of · , we get from 3.1 and 3.4 xn − x∗ βn xn − x∗ γn yn − x∗ ≤ βn xn − x∗ ≤ βn xn − x∗ δn Syn − x∗ γn yn − x∗ γn δn γn δn ≤ βn xn − x∗ γn δn αn Qxn − x∗ ≤ βn xn − x∗ αn Qxn − x∗ γn δn ≤ βn xn − x∗ αn Qxn − x∗ γn 0, δn Syn − x∗ δn xn − x∗ × yn − x∗ B un − B y ∗ αn Qxn − x∗ δn 2 × λn 2α − λn Axn − Ax∗ tn − x ∗ 2 − μ2 2β2 − μ2 B2 zn − B2 x∗ 2 − γn δn B2 zn − B2 x∗ −μ2 2β2 − μ2 − αn tn − x∗ − λn 2α − λn Axn − Ax∗ −μ1 2β1 − μ1 xn − x∗ γn − μ1 2β1 − μ1 B un − B y ∗ 3.18 Therefore, γn δn λn 2α − λn Axn − Ax∗ B un − B y ∗ μ1 2β1 − μ1 ≤ xn − x ≤ Since αn → 0, xn − xn 2α, we have lim Axn − Ax∗ n→∞ ∗ xn − x∗ ∗ − xn −x xn − x∗ μ2 2β2 − μ2 B2 zn − B2 x∗ 2 3.19 αn Qxn − x xn − xn ∗ αn Qxn − x∗ → 0, lim infn → ∞ γn δn > and < lim infn → ∞ λn ≤ lim supn → ∞ λn < lim B1 un − B1 y∗ 0, Step limn → ∞ Syn − yn n→∞ 0, lim B2 zn − B2 x∗ n→∞ 3.20 Fixed Point Theory and Applications 13 Indeed, noticing the firm nonexpansivity of PC we have zn − x∗ PC xn − λn Axn − PC x∗ − λn Ax∗ 2 xn − λn Axn − x∗ − λn Ax∗ , zn − x∗ ≤ xn − x∗ − λn Axn − Ax∗ zn − x∗ 2 − xn − x∗ − λn Axn − Ax∗ − zn − x∗ xn − x∗ zn − x∗ − ≤ xn − x∗ zn − x∗ − xn − zn xn − zn − λn Axn − Ax∗ 2λn xn − zn , Axn − Ax∗ − λ2 Axn − Ax∗ n ≤ xn − x∗ zn − x∗ − xn − zn 3.21 2 2λn xn − zn Axn − Ax∗ , that is, zn − x∗ ≤ xn − x∗ − xn − zn Axn − Ax∗ 2λn xn − zn 3.22 Similarly to the above argument, we obtain un − y∗ PC zn − μ2 B2 zn − PC x∗ − μ2 B2 x∗ 2 zn − μ2 B2 zn − x∗ − μ2 B2 x∗ , un − y∗ ≤ zn − x∗ − μ2 B2 zn − B2 x∗ − un − y ∗ 2 zn − x∗ − μ2 B2 zn − B2 x∗ − un − y∗ 2 zn − x∗ un − y ∗ − ≤ zn − x∗ un − y ∗ − zn − un − x∗ − y∗ 3.23 zn − un − μ2 B2 zn − B2 x∗ − x∗ − y∗ 2 2μ2 zn − un − x∗ − y∗ , B2 zn − B2 x∗ − μ2 B2 zn − B2 x∗ 2 , that is, un − y∗ ≤ zn − x∗ − zn − un − x∗ − y∗ 2μ2 zn − un − x∗ − y∗ B2 zn − B2 x∗ 3.24 14 Fixed Point Theory and Applications Substituting 3.22 in 3.24 , we have un − y ∗ ≤ xn − x∗ − xn − zn Axn − Ax∗ 2λn xn − zn 3.25 − zn − un − x∗ − y∗ 2μ2 zn − un − x∗ − y∗ B2 zn − B2 x∗ Further, similarly to the above argument, we derive tn − x∗ P C u n − μ B un − P C y ∗ − μ B y ∗ un − μ1 B1 un − y∗ − μ1 B1 y∗ , tn − x∗ ≤ u n − y ∗ − μ B un − B y ∗ − tn − x ∗ u n − y ∗ − μ B un − B y ∗ − t n − x ∗ un − y ∗ tn − x ∗ − ≤ un − y ∗ tn − x ∗ − un − tn 2μ1 un − tn 2 3.26 u n − t n − μ B un − B y ∗ x∗ − y ∗ x∗ − y ∗ 2 x ∗ − y ∗ , B un − B y ∗ − μ B un − B y ∗ , that is, tn − x∗ ≤ un − y ∗ − un − tn x∗ − y ∗ 2μ1 un − tn x∗ − y ∗ B un − B y ∗ 3.27 Substituting 3.25 in 3.27 , we have tn − x ∗ ≤ xn − x∗ − xn − zn 2λn xn − zn Axn − Ax∗ − zn − un − x∗ − y∗ 2μ2 zn − un − x∗ − y∗ x∗ − y ∗ 2μ1 un − tn − un − tn x∗ − y ∗ B2 zn − B2 x∗ B un − B y ∗ 3.28 Fixed Point Theory and Applications 15 Thus from 3.1 and 3.28 , it follows that xn − x∗ βn xn − x∗ γn yn − x∗ δn Syn − x∗ ≤ βn xn − x∗ γn βn xn − x∗ − βn ≤ βn xn − x∗ − βn αn Qxn − x∗ ≤ βn xn − x∗ αn Qxn − x∗ − βn ≤ βn xn − x∗ αn Qxn − x∗ − βn xn − x∗ × yn − x∗ δn yn − x∗ − xn − zn x∗ − y ∗ xn − x∗ 2μ1 un − tn − − βn 2 2 Axn − Ax∗ B2 zn − B2 x∗ x∗ − y ∗ B un − B y ∗ − βn 2μ2 zn − un − x∗ − y∗ x∗ − y ∗ xn − zn tn − x ∗ 2μ1 un − tn Axn − Ax∗ × 2λn xn − zn − αn tn − x∗ 2μ2 zn − un − x∗ − y∗ 2 αn Qxn − x∗ 2 2λn xn − zn − zn − un − x∗ − y∗ − un − tn B2 zn − B2 x∗ B un − B y ∗ zn − un − x∗ − y∗ un − tn x∗ − y ∗ , 3.29 which hence implies that − βn xn − zn ≤ xn − x∗ zn − un − x∗ − y∗ − xn × 2λn xn − zn xn − x∗ xn x∗ − y ∗ × 2λn xn − zn 2μ1 un − tn − x∗ un − tn αn Qxn − x∗ Axn − Ax∗ 2μ1 un − tn ≤ − x∗ 2 x∗ − y ∗ − βn 2μ2 zn − un − x∗ − y∗ B2 zn − B2 x∗ B un − B y ∗ xn − xn Axn − Ax∗ x∗ − y ∗ 3.30 αn Qxn − x∗ 2μ2 zn − un − x∗ − y∗ B un − B y ∗ − βn B2 zn − B2 x∗ 16 Fixed Point Theory and Applications Since lim supn → ∞ βn < 1, < λn ≤ 2α, αn → 0, Axn − Ax∗ → 0, B2 zn − B2 x∗ → 0, B1 un − B1 y∗ → and xn − xn → 0, it follows from the boundedness of {xn }, {zn }, {un }, and {tn } that lim xn − zn n→∞ 0, lim zn − un − x∗ − y∗ 0, n→∞ lim un − tn n→∞ x∗ − y ∗ 3.31 Consequently, it immediately follows that lim zn − tn lim xn − tn 0, n→∞ n→∞ 3.32 This together with yn − tn ≤ αn Qxn − tn → implies that lim xn − yn n→∞ 3.33 Since δn Syn − xn ≤ xn lim Syn − xn 0, − xn γn yn − xn , 3.34 it follows that n→∞ lim Syn − yn n→∞ 3.35 Step lim supn → ∞ Qx − x, xn − x ≤ 0, where x PFix S ∩ Γ ∩ VI A,C · Qx Indeed, since {xn } is bounded, there exists a subsequence {xni } of {xn } such that lim sup Qx − x, xn − x n→∞ lim Qx − x, xni − x 3.36 i→∞ Also, since H is reflexive and {yn } is bounded, without loss of generality we may assume that yni → p weakly for some p ∈ C First, it is clear from Lemma 2.2 that p ∈ Fix S Now let us show that p ∈ Γ We note that yn − G yn ≤ αn Qxn − G yn − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn αn Qxn − G yn − αn G zn − G yn ≤ αn Qxn − G yn − αn xn − yn − G yn −→ 3.37 Fixed Point Theory and Applications 17 According to Lemma 2.2 we obtain p ∈ Γ Further, let us show that p ∈ VI A, C As a matter of fact, since xn − zn → and xn − yn → 0, we deduce that xni → p weakly and zni → p weakly Let Tv ⎧ ⎨Av NC v if v ∈ C, ⎩∅ 3.38 if v ∈ C, / where NC v is the normal cone to C at v ∈ C In this case, the mapping T is maximal monotone, and ∈ T v if and only if v ∈ VI A, C ; see 10 for more details Let Gph T be the graph of T and let v, w ∈ Gph T Then, we have w ∈ T v Av NC v and hence w − Av ∈ NC v So, we have v − t, w − Av ≥ for all t ∈ C On the other hand, from zn PC xn − λn Axn and v ∈ C we have xn − λn Axn − zn , zn − v ≥ 3.39 and hence v − zn , zn − xn λn Axn ≥ 3.40 From v − t, w − Av ≥ for all t ∈ C and zni ∈ C, we have v − zni , w ≥ v − zni , Av ≥ v − zni , Av − v − zni , v − zni , Av − Azni zni − xni λni Axni v − zni , Azni − Axni − v − zni , ≥ v − zni , Azni − Axni − v − zni , zni − xni λni 3.41 zni − xni , λni Hence, we obtain v − p, w ≥ as i → ∞ Since T is maximal monotone, we have p ∈ T −1 and hence p ∈ VI A, C Therefore, p ∈ Fix S ∩ Γ ∩ VI A, C Hence it follows from 2.8 and 3.36 that lim sup Qx − x, xn − x n→∞ lim Qx − x, xni − x i→∞ Qx − x, p − x 3.42 ≤ Step limn → ∞ xn x Indeed, since G : C → C is nonexpansive, we have tn − x G zn − G x ≤ xn − x 3.43 18 Fixed Point Theory and Applications Note that Qxn − x, yn − x Qxn − x, xn − x Qxn − x, yn − xn Qxn − Qx, xn − x ≤ ρ xn − x Qx − x, xn − x Qx − x, xn − x Qxn − x, yn − xn Qxn − x yn − xn Utilizing Lemmas 2.4 and 2.5, we obtain from 3.4 and the convexity of · xn −x βn xn − x γn yn − x ≤ βn xn − x ≤ βn xn − x δn Syn − x 3.44 2 δn γn δn yn − x ≤ βn xn − x γn δn − αn tn − x 2αn Qxn − x, yn − x ≤ βn xn − x γn δn − αn xn − x 2αn Qxn − x, yn − x xn − x xn − x δn 2αn ρ xn − x ≤ − γn δn αn δn αn ≤ − − 2ρ γn γn δn δn Syn − x − γn γn γn γn yn − x γn γn δn 2αn Qxn − x, yn − x Qx − x, xn − x δn αn xn − x − 2ρ γn δn αn xn − x δn αn Note that lim infn → ∞ − 2ρ γn clear that lim sup n→∞ Qxn − x yn − xn 2 Qx − x, xn − x Qxn − x − 2ρ δn > It follows that Qx − x, xn − x yn − xn δn 2αn Qx − x, xn − x − − 2ρ γn Qxn − x 3.45 Qxn − x − 2ρ ∞ n yn − xn − 2ρ γn yn − xn ≤0 δn αn ∞ It is 3.46 because lim supn → ∞ Qx − x, xn − x ≤ and limn → ∞ xn − yn Therefore, all conditions of Lemma 2.3 are satisfied Consequently, we immediately deduce that xn → x This completes the proof Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be α-inverse strongly monotone and Bi : C → H be βi -inverse strongly monotone for i 1, Let Fixed Point Theory and Applications 19 S : C → C be a k-strictly pseudocontractive mapping such that Fix S ∩ Γ ∩ VI A, C / ∅ For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn }, and {zn } be generated iteratively by zn yn − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , αn u xn where μi ∈ 0, 2βi for i i βn γn δn ii limn → ∞ αn PC xn − λn Axn , βn xn γn yn δn Syn , 3.47 ∀n ≥ 0, 1, 2, {λn } ⊂ 0, 2α and {αn }, {βn }, {γn }, {δn } ⊂ 0, such that and γn ∞ n 0 and δn k ≤ γn for all n ≥ 0; αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < and lim infn → ∞ δn > 0; iv limn → ∞ γn / − βn − γn / − βn 0; v < lim infn → ∞ λn ≤ lim supn → ∞ λn < 2α and limn → ∞ |λn − λn | Then the sequence {xn } converges strongly to x PFix S ∩ Γ ∩ VI A,C · Qx and x, y is a solution of the general system 1.3 of variational inequalities, where y PC x − μ2 B2 x Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be α-inverse strongly monotone and Bi : C → H be βi -inverse strongly monotone for i 1, Let S : C → C be a nonexpansive mapping such that Fix S ∩ Γ ∩ VI A, C / ∅ Let Q : C → C be a ρ-contraction with ρ ∈ 0, 1/2 For given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by PC xn − λn Axn , zn yn αn Qxn − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , xn where μi ∈ 0, 2βi for i i βn γn δn ii limn → ∞ αn βn xn γn yn δn Syn , 3.48 ∀n ≥ 0, 1, 2, {λn } ⊂ 0, 2α and {αn }, {βn }, {γn }, {δn } ⊂ 0, such that for all n ≥ 0; and ∞ n αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < and lim infn → ∞ δn > 0; iv limn → ∞ γn / − βn − γn / − βn 0; v < lim infn → ∞ λn ≤ lim supn → ∞ λn < 2α and limn → ∞ |λn − λn | Then the sequence {xn } converges strongly to x PFix S ∩ Γ ∩ VI A,C · Qx and x, y is a solution of the general system 1.3 of variational inequalities, where y PC x − μ2 B2 x Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H be α-inverse strongly monotone and Bi : C → H be βi -inverse strongly monotone for i 1, Let 20 Fixed Point Theory and Applications S : C → C be a nonexpansive mapping such that Fix S ∩ Γ ∩ VI A, C / ∅ For fixed u ∈ C and given x0 ∈ C arbitrarily, let the sequences {xn }, {yn } and {zn } be generated iteratively by zn yn αn u − αn PC PC zn − μ2 B2 zn − μ1 B1 PC zn − μ2 B2 zn , xn where μi ∈ 0, 2βi for i i βn γn δn ii limn → ∞ αn PC xn − λn Axn , βn xn γn yn δn Syn , 3.49 ∀n ≥ 0, 1, 2, {λn } ⊂ 0, 2α and {αn }, {βn }, {γn }, {δn } ⊂ 0, such that for all n ≥ 0; and ∞ n αn ∞; iii < lim infn → ∞ βn ≤ lim supn → ∞ βn < and lim infn → ∞ δn > 0; iv limn → ∞ γn / − βn − γn / − βn 0; v < lim infn → ∞ λn ≤ lim supn → ∞ λn < 2α and limn → ∞ |λn − λn | Then the sequence {xn } converges strongly to x PFix S ∩ Γ ∩ VI A,C u and x, y is a solution of the general system 1.3 of variational inequalities, where y PC x − μ2 B2 x Acknowledgments This research was partially supported by the National Science Foundation of China 10771141 , Ph.D Program Foundation of Ministry of Education of China 20070270004 , Science and Technology Commission of Shanghai 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A, C Nadezhkina and Takahashi 23 and Zeng and Yao 24 proposed extragradient methods motivated by Korpeleviˇ 25 for finding c a common element of the fixed point set of a nonexpansive mapping and. .. Mathematical Sciences Research Hot-Line, vol 3, no 8, pp 65–68, 1999 28 R U Verma, “Iterative algorithms and a new system of nonlinear quasivariational inequalities,? ?? Advances in Nonlinear Variational. .. approximation method, Yao et al 30 proposed and analyzed an iterative algorithm for finding a common element of the solution set of the general system 1.3 of variational inequalities and the fixed-point