Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2011, Article ID 976505, 23 pages doi:10.1155/2011/976505 Research Article Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings Wiyada Kumam,1, 2, 3 Prapairat Junlouchai,1 and Poom Kumam2, 3 1 Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand 2 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand 3 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 20 February 2011; Accepted 3 May 2011 Academic Editor: Jianshe Yu Copyright q 2011 Wiyada Kumam 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 introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inversestrongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., 2008 and many others. 1. Introduction Variational inequalities are known to play a crucial role in mathematics as a unified framework for studying a large variety of problems arising, for instance, in structural analysis, engineering sciences and others.Roughly speaking, they can be recast as fixed-point problems, and most of the numerical methods related to this topic are based on projection methods. Let H be a real Hilbert space with inner product ·, · and · , and let E be a nonempty, closed, convex subset of H. A mapping A : E → H is called α-inverse-strongly monotone if there exists a positive real number α > 0 such that 2 Ax − Ay, x − y ≥ αAx − Ay , ∀x, y ∈ E 1.1 2 Discrete Dynamics in Nature and Society see 1, 2. It is obvious that every α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S : E → E is called nonexpansive if Sx − Sy ≤ x − y, ∀x, y ∈ E. 1.2 We denote by FS the set of fixed points of S and by PE the metric projection of H onto E. Recall that the classical variational inequality, denoted by V IA, E, is to find an x∗ ∈ E such that Ax∗ , x − x∗ ≥ 0, ∀x ∈ E. 1.3 The set of solutions of V IA, E is denoted by Γ. The variational inequality has been widely studied in the literature; see, for example, 3–6 and the references therein. For finding an element of FS ∩ Γ, Takahashi and Toyoda 7 introduced the following iterative scheme: xn1 αn xn 1 − αn SPE xn − λn Axn , 1.4 for every n 0, 1, 2, . . ., where x0 x ∈ E, {αn } is a sequence in 0, 1, and {λn } is a sequence in 0, 2α. On the other hand, for solving the variational inequality problem in the finitedimensional Euclidean space Rn , Korpelevich 1976 8 introduced the following so-called extragradient method: x0 x ∈ E, yn PE xn − λn Axn , xn1 PE xn − λn Ayn , 1.5 for every n 0, 1, 2, . . ., where λn ∈ 0, 1/k. Many authors using extragradient method for approximating common fixed points and variational inequality problems see also 9, 10. Recently, Nadezhkina and Takahashi 11 and Zeng et al. 12 proposed some iterative schemes for finding elements in FS ∩ Γ by combining 1.4 and 1.5. Further, these iterative schemes are extended in Y. Yao and J. C. Yao 13 to develop a new iterative scheme for finding elements in FS ∩ Γ. Consider the following problem of finding x∗ , y∗ ∈ E × E such that see cf. Ceng et al. 14: λAy∗ x∗ − y∗ , x − x∗ ≥ 0, μBx∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ E, ∀x ∈ E, 1.6 Discrete Dynamics in Nature and Society 3 which is called general system of variational inequalities GSVI, where λ > 0 and μ > 0 are two constants. In particular, if A B, then problem 1.6 reduces to finding x∗ , y∗ ∈ E × E such that λAy∗ x∗ − y∗ , x − x∗ ≥ 0, μAx∗ y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ E, ∀x ∈ E, 1.7 which is defined by 15, 16, and is called the new system of variational inequalities. Further, if x∗ y∗ , then problem 1.7 reduces to the classical variational inequality VIA, E, that is, find x∗ ∈ E such that Ax∗ , x − x∗ ≥ 0, for all x ∈ E. We can characteristic problem, if x∗ ∈ FS ∩ V IA, E, then it follows that x∗ Sx∗ ∗ PE x − ρAx∗ , where ρ > 0 is a constant. In 2008, Ceng et al. 14 introduced a relaxed extragradient method for finding solutions of problem 1.6. Let the mappings A, B : E → H be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let S : E → E be a nonexpansive mapping. Suppose x1 u ∈ E and {xn } is generated by xn1 yn PE xn − μBxn , αn u βn xn γn SPE yn − λn Ayn , 1.8 where λ ∈ 0, 2α, μ ∈ 0, 2β, and {αn }, {βn }, {γn } are three sequences in 0, 1 such that αn βn γn 1, for all n ≥ 1. First, problem 1.6 is proven to be equivalent to a fixed point problem of a nonexpansive mapping. In this paper, motivated by what is mentioned above, we consider generalized system of variational inequalities as follows. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let A, B, C : E → H be three mappings. We consider the following problem of finding x∗ , y∗ , z∗ ∈ E × E × E such that λAy∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ E, μBz∗ y∗ − z∗ , x − y∗ ≥ 0, ∀x ∈ E, τCx∗ z∗ − x∗ , x − z∗ ≥ 0, ∀x ∈ E, 1.9 which is called a general system of variational inequalities where λ > 0, μ > 0 and τ > 0 are three constants. In particular, if A B C, then problem 1.9 reduces to finding x∗ , y∗ , z∗ ∈ E × E × E such that λAy∗ x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ E, μAz∗ y∗ − z∗ , x − y∗ ≥ 0, ∀x ∈ E, τAx∗ z∗ − x∗ , x − z∗ ≥ 0, ∀x ∈ E. 1.10 4 Discrete Dynamics in Nature and Society Next, we consider some special classes of the GSVI problem 1.9 reduce to the following GSVI. i If τ 0, then the GSVI problems 1.9 reduce to GSVI problem 1.6. ii If τ μ 0, then the GSVI problems 1.9 reduce to classical variational inequality VIA,E problem. The above system enters a class of more general problems which originated mainly from the Nash equilibrium points and was treated from a theoretical viewpoint in 17, 18. Observe at the same time that, to construct a mathematical model which is as close as possible to a real complex problem, we often have to use constraints which can be expressed as one several subproblems of a general problem. These constrains can be given, for instance, by variational inequalities, by fixed point problems, or by problems of different types. This paper deals with a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under classical conditions, we prove a strong convergence theorem for this method. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems. 2. Preliminaries Let E be a nonempty, closed, convex subset of a real Hilbert space H. For every point x ∈ H, there exists a unique nearest point in E, denoted by PE x, such that x − PE x ≤ x − y, ∀y ∈ E. 2.1 PE is call the metric projection of H onto E. Recall that, PE x is characterized by following properties: PE x ∈ E and x − PE x, y − PE x ≤ 0, x − y2 ≥ x − PE x2 PE x − y2 , 2.2 for all x ∈ H and y ∈ E. Lemma 2.1 see cf. Zhang et al. 19. The metric projection PE has the following properties: i PE : H → E is nonexpansive; ii PE : H → E is firmly nonexpansive, that is, PE x − PE y2 ≤ PE x − PE y, x − y , ∀x, y ∈ H; 2.3 ∀y ∈ E. 2.4 iii for each x ∈ H, z PE x ⇐⇒ x − z, z − y ≥ 0, Discrete Dynamics in Nature and Society 5 Lemma 2.2 see Osilike and Igbokwe 20. Let E, ·, · be an inner product space. Then for all x, y, z ∈ E and α, β, γ ∈ 0, 1 with α β γ 1, one has αx βy γz2 αx2 βy2 γz2 − αβx − y2 − αγx − z2 − βγ y − z2 . 2.5 Lemma 2.3 see Suzuki 21. Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn yn βn xn for all integers n ≥ 0 and lim supn → ∞ yn1 − yn − xn1 − xn ≤ 0. Then, limn → ∞ yn − xn 0. Lemma 2.4 see Xu 22. Assume {an } is a sequence of nonnegative real numbers such that an1 ≤ 1 − αn an δn , n ≥ 0, 2.6 where {αn } is a sequence in 0, 1 and {δn } is a sequence in R such that i ∞ n1 αn ∞, ii lim supn → ∞ δn /αn ≤ 0 or ∞ n1 |δn | < ∞. Then, limn → ∞ an 0. Lemma 2.5 Goebel and Kirk 23. Demiclosedness Principle. Assume that T is a nonexpansive self-mapping of a nonempty, closed, convex subset E of a real Hilbert space H. If T has a fixed point, then I − T is demiclosed; that is, whenever {xn } is a sequence in E converging weakly to some x ∈ E (for short, xn x ∈ E), and the sequence {I − T xn } converges strongly to some y (for short, I − T xn → y), it follows that I − T x y. Here, I is the identity operator of H. The following lemma is an immediate consequence of an inner product. Lemma 2.6. In a real Hilbert space H, there holds the inequality x y2 ≤ x2 2 y, x y , ∀x, y ∈ H. 2.7 Remark 2.7. We also have that, for all u, v ∈ E and λ > 0, I − λAu − I − λAv2 u − v − λAu − Av2 u − v2 − 2λu − v, Au − Av λ2 Au − Av2 ≤ u − v2 λλ − 2αAu − Av2 . So, if λ ≤ 2α, then I − λA is a nonexpansive mapping from E to H. 2.8 6 Discrete Dynamics in Nature and Society 3. Main Results In this section, we introduce an iterative precess by the relaxed extragradient approximation method for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of the variational inequality problem for three inverse-strongly monotone mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to a common element of the above two sets. In order to prove our main result, the following lemmas are needed. Lemma 3.1. For given x∗ , y∗ , z∗ ∈ E × E × E, x∗ , y∗ , z∗ is a solution of problem 1.9 if and only if x∗ is a fixed point of the mapping G : E → E defined by Gx PE PE PE x − τCx − μBPE x − τCx −λAPE PE x − τCx − μBPE x − τCx , ∀x ∈ E, 3.1 where y∗ PE z∗ − μBz∗ and z∗ PE x∗ − τCx∗ . Proof. λAy∗ x∗ − y∗ , x − x∗ ≥ 0, μBz∗ y∗ − z∗ , x − y∗ ≥ 0, τCx∗ z∗ − x∗ , x − z∗ ≥ 0, ∀x ∈ E, ∀x ∈ E, 3.2 ∀x ∈ E, ⇔ −y∗ λAy∗ x∗ , x − x∗ ≥ 0, ∗ −z μBz∗ y∗ , x − y∗ ≥ 0, −x∗ τCx∗ z∗ , x − z∗ ≥ 0, ∀x ∈ E, ∀x ∈ E, 3.3 ∀x ∈ E, ⇔ y∗ − λAy∗ − x∗ , x∗ − x ≥ 0, ∗ z − μBz∗ − y∗ , y∗ − x ≥ 0, x∗ − τCx∗ − z∗ , z∗ − x ≥ 0, ∀x ∈ E, ∀x ∈ E, 3.4 ∀x ∈ E, ⇔ x∗ PE y∗ − λAy∗ , y∗ PE z∗ − μBz∗ , z∗ PE x∗ − τCx∗ , ⇔ x∗ PE PE z∗ − μBz∗ − λAPE z∗ − μBz∗ . 3.5 Discrete Dynamics in Nature and Society 7 Thus, x∗ PE PE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ −λAPE PE x∗ −τCx∗ − μBPE x∗ − τCx∗ . 3.6 Lemma 3.2. The mapping G defined by Lemma 3.1 is nonexpansive mappings. Proof. For all x, y ∈ E, Gx − G y PE PE PE x − τCx − μBPE x − τCx −λAPE PE x − τCx − μBPE x − τCx − PE PE PE y − τCy − μBPE y − τCy −λAPE PE y − τCy − μBPE y − τCy ≤ PE x − τCx − μBPE x − τCx − λAPE PE x − τCx − μBPE x − τCx − PE y − τCy − μBPE y − τCy −λAPE PE y − τCy − μBPE y − τCy I − λA PE x − τCx − μBPE x − τCx −I − λA PE y − τCy − μBPE y − τCy ≤ PE x − τCx − μBPE x − τCx − PE y − τCy − μBPE y − τCy I − μB PE x − τCx − I − μB PE y − τCy ≤ PE x − τCx − PE y − τCy ≤ x − τCx − y − τCy I − τCx − I − τC y ≤ x − y. 3.7 This shows that G : E → E is a nonexpansive mapping. Throughout this paper, the set of fixed points of the mapping G is denoted by Υ. Now, we are ready to proof our main results in this paper. 8 Discrete Dynamics in Nature and Society Theorem 3.3. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A, B, C : E → H be α-inverse-strongly monotone, β-inverse-strongly monotone, and γ-inversestrongly monotone, respectively. Let S be a nonexpansive mapping of E into itself such that FS ∩ Υ / ∅. Let f be a contraction of H into itself and given x1 ∈ H arbitrarily and {xn } is generated by zn PE xn − τCxn , xn1 yn PE zn − μBzn , αn fxn βn xn γn SPE yn − λAyn , 3.8 n ≥ 0, where λ ∈ 0, 2α, μ ∈ 0, 2β, τ ∈ 0, 2γ, and {αn }, {βn }, {γn } are three sequences in 0, 1 such that i αn βn γn 1, ii limn → ∞ αn 0 and ∞ n1 αn ∞, iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Then, {xn } converges strongly to x ∈ FS ∩ Γ, where x PFS∩Γ fx and x, y, z is a solution of problem 1.9, where y PE z − μBz , z PE x − τCx. 3.9 Proof. Let x∗ ∈ FS ∩ Γ. Then, x∗ Sx∗ and x∗ Gx∗ , that is, x∗ PE PE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ − λAPE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ . 3.10 Put x∗ PE y∗ −λAy∗ and tn PE yn −λAyn . Then, x∗ PE PE z∗ −μBz∗ −λAPE z∗ −μBz∗ implies that y∗ PE z∗ − μBz∗ , where z∗ PE x∗ − τCx∗ . Since I − λA, I − μB and I − τC are nonexpansive mappings. We obtain that tn − x∗ PE yn − λAyn − x∗ PE yn − λAyn − PE y∗ − λAy∗ ≤ yn − λAyn − y∗ − λAy∗ I − λAyn − I − λAy∗ ≤ yn − y∗ yn − PE z∗ − μBz∗ PE zn − μBzn − PE z∗ − μBz∗ 3.11 Discrete Dynamics in Nature and Society 9 ≤ I − μB zn − I − μB z∗ ≤ zn − z∗ , 3.12 zn − z∗ PE xn − τCxn − PE x∗ − τCx∗ ≤ xn − τCxn − x∗ − τCx∗ I − τCxn − I − τCx∗ 3.13 ≤ xn − x∗ . Substituting 3.13 into 3.12, we have tn − x∗ ≤ xn − x∗ , 3.14 yn − y∗ ≤ xn − x∗ . 3.15 and by 3.11 we also have Since xn1 αn fxn βn xn γn Stn and by Lemma 2.2, we compute xn1 − x∗ αn fxn βn xn γn Stn − x∗ αn fxn − x∗ βn xn − x∗ γn Stn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn Stn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn tn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn xn − x∗ αn fxn − x∗ 1 − αn xn − x∗ αn fxn − fx∗ fx∗ − x∗ 1 − αn xn − x∗ ≤ αn fxn − fx∗ αn fx∗ − x∗ 1 − αn xn − x∗ ≤ αn kxn − x∗ αn fx∗ − x∗ 1 − αn xn − x∗ 10 Discrete Dynamics in Nature and Society αn k 1 − αn xn − x∗ αn fx∗ − x∗ 1 − αn 1 − kxn − x∗ αn fx∗ − x∗ ∗ 1 − αn 1 − k xn − x αn 1 − k fx∗ − x∗ 1 − k . 3.16 By induction, we get xn1 − x∗ ≤ M, 3.17 where M max{x0 − x∗ 1/1 − kfx∗ − x∗ }, n ≥ 0. Therefore, {xn } is bounded. Consequently, by 3.11, 3.12 and 3.13, the sequences {tn }, {Stn }, {yn }, {Ayn }, {zn }, {Bzn }, {Cxn }, and {fxn } are also bounded. Also, we observe that zn1 − zn PE xn1 − τCxn1 − PE xn − τCxn ≤ I − τCxn1 − I − τCxn 3.18 ≤ xn1 − xn , tn1 − tn PE yn1 − λAyn1 − PE yn − λAyn ≤ yn1 − λAyn1 − yn − λAyn I − λAyn1 − I − λAyn ≤ yn1 − yn 3.19 PE zn1 − μBzn1 − PE zn − μBzn ≤ zn1 − zn ≤ xn1 − xn . Let xn1 1 − βn wn βn xn . Thus, we get xn1 − βn xn αn fxn γn SPC yn − λn Ayn αn u γn Stn wn 1 − βn 1 − βn 1 − βn 3.20 Discrete Dynamics in Nature and Society 11 It follows that wn1 − wn αn1 fxn1 γn1 Stn1 αn fxn γn Stn − 1 − βn1 1 − βn αn1 fxn1 γn1 Stn1 αn1 fxn αn1 fxn αn fxn γn Stn − − − 1 − βn1 1 − βn1 1 − βn1 1 − βn1 1 − βn 1 − βn γn1 Stn1 γn Stn αn1 αn αn1 fxn − − fxn1 − fxn 1 − βn1 1 − βn1 1 − βn 1 − βn1 1 − βn αn1 αn αn1 − fxn fxn1 − fxn 1 − βn1 1 − βn1 1 − βn γn1 Stn1 γn1 Stn γn1 Stn γn Stn − − 1 − βn1 1 − βn1 1 − βn1 1 − βn αn1 αn αn1 − fxn fxn1 − fxn 1 − βn1 1 − βn1 1 − βn γn γn1 γn1 − Stn Stn1 − Stn 1 − βn1 1 − βn1 1 − βn αn1 αn αn1 − fxn fxn1 − fxn 1 − βn1 1 − βn1 1 − βn γn1 αn αn1 − Stn Stn1 − Stn 1 − βn1 1 − βn 1 − βn1 αn αn1 αn1 − fxn Stn fxn1 − fxn 1 − βn1 1 − βn1 1 − βn γn1 Stn1 − Stn . 1 − βn1 3.21 Combining 3.19 and 3.21, we obtain αn1 fxn1 − fxn αn1 − αn fxn Stn wn1 − wn − xn1 − xn ≤ 1 − βn1 1 − βn1 1 − βn γn1 Stn1 − Stn − xn1 − xn 1 − βn1 αn1 kxn1 − xn αn1 − αn fxn Stn ≤ 1 − βn1 1 − βn1 1 − βn γn1 tn1 − tn − xn1 − xn 1 − βn1 αn1 αn1 αn fxn Stn kxn1 − xn ≤ − 1 − βn1 1 − βn1 1 − βn 12 γn1 1−β αn1 1−β n1 n1 Discrete Dynamics in Nature and Society xn1 − xn − xn1 − xn kxn1 − xn αn1 1 − β n1 − αn fxn Stn 1 − βn γn1 − 1 βn1 xn1 − xn 1 − βn1 αn1 kxn1 − xn αn1 − αn fxn Stn 1 − βn1 1 − βn1 1 − βn αn1 xn1 − xn . 1 − βn1 3.22 This together with i, ii, and iii implies that lim supwn1 − wn − xn1 − xn ≤ 0. n→∞ 3.23 Hence, by Lemma 2.3, we have lim wn − xn 0. 3.24 lim xn1 − xn lim 1 − βn wn − xn 0. 3.25 n→∞ Consequently, n→∞ n→∞ From 3.18 and 3.19, we also have zn1 − zn → 0tn1 − tn → 0 and yn1 − yn → 0 as n → ∞. Since xn1 − xn αn fxn βn xn γn Stn − xn αn fxn − xn γn Stn − xn , 3.26 it follows by ii and 3.25 that lim xn − Stn 0. n→∞ 3.27 Discrete Dynamics in Nature and Society 13 Since x∗ ∈ FS ∩ Γ, from 3.15 and Lemma 2.2, we get 2 xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn Stn − x∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2 2 2 αn fxn − x∗ βn xn − x∗ 2 γn PE yn − λAyn − PE y∗ − λAy∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn yn − λAyn − y∗ − λAy∗ 2 2 αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ − λ Ayn − Ay∗ 2 αn fxn − x∗ βn xn − x∗ 2 2 γn yn − y∗ 2 − 2λ yn − y∗ , Ayn − Ay∗ λ2 Ayn − Ay∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 2 2 2 γn yn − y∗ − 2λαAyn − Ay∗ λ2 Ayn − Ay∗ 2 2 2 αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ λλ − 2αAyn − Ay∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 λλ − 2αAyn − Ay∗ 2 2 αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2 2 αn fxn − x∗ βn γn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2 2 αn fxn − x∗ 1 − αn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2 2 ≤ αn fxn − x∗ xn − x∗ 2 γn λλ − 2αAyn − Ay∗ . 3.28 Therefore, we have 2 2 −γn λλ − 2αAyn − Ay∗ ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2 2 αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2 ≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 . From ii, iii, and xn1 − xn → 0, as n → ∞, we get Ayn − Ay∗ → 0 as n → ∞. 3.29 14 Discrete Dynamics in Nature and Society Since x∗ ∈ FS ∩ Υ, from 3.11 and Lemma 2.2, we get 2 xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ 2 2 αn fxn − x∗ βn xn − x∗ 2 γn PE zn − μBzn − PE z∗ − μBz∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − μBzn − z∗ − μBz∗ 2 2 αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ − μBzn − μBz∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ 2 μ μ − 2β Bzn − Bz∗ 2 2 ≤ αn fxn − x∗ xn − x∗ 2 γn μ μ − 2β Bzn − Bz∗ 2 . 3.30 Thus, we also have 2 −γn μ μ − 2β Bzn − Bz∗ 2 ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2 2 αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2 ≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 . 3.31 By again ii, iii, and 3.25, we also get Bzn − Bz∗ → 0 as n → ∞. Let x∗ ∈ FS ∩ Υ; again from 3.12, 3.13 and Lemma 2.2, we get 2 xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − τCxn − x∗ − τCx∗ 2 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 τ τ − 2γ Cxn − Cx∗ 2 2 ≤ αn fxn − x∗ xn − x∗ 2 γn τ τ − 2γ Cxn − Cx∗ 2 . 3.32 Discrete Dynamics in Nature and Society 15 Again, we have 2 −γn τ τ − 2γ Cxn − Cx∗ 2 ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2 2 αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2 ≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 . 3.33 Similarly again by ii, iii, and xn − xn1 → 0 as n → ∞, and from 3.33, we also that Cxn − Cx∗ → 0. On the other hand, we compute that zn − z∗ 2 PE xn − τCxn − PE x∗ − τCx∗ 2 ≤ xn − τCxn − x∗ − τCx∗ , PE xn − τCxn − PE x∗ − τCx∗ xn − τCxn − x∗ − τCx∗ , zn − z∗ 1 xn − τCxn − x∗ − τCx∗ 2 zn − z∗ 2 2 −xn − τCxn − x∗ − τCx∗ − zn − z∗ 2 1 I − τCxn − I − τCx∗ 2 zn − z∗ 2 2 −xn − τCxn − x∗ − τCx∗ − zn − z∗ 2 3.34 1 xn − x∗ 2 zn − z∗ 2 − xn − zn − τCxn − Cx∗ − x∗ − z∗ 2 2 1 xn − x∗ 2 zn − z∗ 2 − xn − zn − x∗ − z∗ − τCxn − Cx∗ 2 2 1 xn − x∗ 2 zn − z∗ 2 − xn − zn − x∗ − z∗ 2 2 2τxn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 Cxn − Cx∗ 2 . ≤ So, we obtain zn − z∗ 2 ≤ xn − x∗ 2 − xn − zn − x∗ − z∗ 2 2τxn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 Cxn − Cx∗ 2 . Hence, by 3.12, it follows that 2 xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn Stn − x∗ 2 3.35 16 Discrete Dynamics in Nature and Society ≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn tn − x∗ 2 ≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn zn − z∗ 2 ≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2 2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 γn Cxn − Cx∗ 2 αn kxn − x∗ 2 1 − αn xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2 2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 γn Cxn − Cx∗ 2 ≤ αn kxn − x∗ 2 xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2 2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ ≤ αn kxn − x∗ 2 xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2 2τγn xn − zn − x∗ − z∗ Cxn − Cx∗ , 3.36 which implies that γn xn − zn − x∗ − z∗ 2 ≤ αn kxn − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2 2τγn xn − zn − x∗ − z∗ Cxn − Cx∗ ∗ 2 ∗ ∗ ∗ 3.37 ≤ αn kxn1 − x 2γn τxn − zn − x − z Cxn − Cx xn − xn1 xn − x∗ xn1 − x∗ . By ii, iii, xn − xn1 → 0, and Cxn − Cx∗ → 0 as n → ∞, from 3.37 and we get xn − zn − x∗ − z∗ → 0 as n → ∞. Now, observe that 2 zn − tn x∗ − z∗ 2 zn − PE yn − λAyn PE y∗ − λAy∗ − z∗ zn − PE yn − λAyn PE y∗ − λAy∗ 2 −z∗ μBzn − μBzn μBz∗ − μBz∗ zn − μBzn − z∗ − μBz∗ 2 − PE yn − λAyn − PE y∗ − λAy∗ μBzn − Bz∗ Discrete Dynamics in Nature and Society 17 2 ≤ zn − μBzn − z∗ − μBz∗ − PE yn − λAyn − PE y∗ − λAy∗ 2μ Bzn − Bz∗ , zn − μBzn − z∗ − μBz∗ − PE yn − λAyn − PE y∗ − λAy∗ μBzn − Bz∗ 2 zn − μBzn − z∗ − μBz∗ − PE yn − λAyn − PE y∗ − λAy∗ 2μBzn − Bz∗ , zn − tn x∗ − z∗ 2 ≤ zn − μBzn − z∗ − μBz∗ 2 − PE yn − λn Ayn − PE y∗ − λn Ay∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ 2 ≤ zn − μBzn − z∗ − μBz∗ 2 − SPE yn − λn Ayn − SPE y∗ − λn Ay∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ 2 zn − μBzn − z∗ − μBz∗ − Stn − Sx∗ 2 2μBzn − Bz∗ zn − tn x∗ − z∗ ≤ zn − μBzn − z∗ − μBz∗ − Stn − x∗ × zn − μBzn − z∗ − μBz∗ Stn − x∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ . 3.38 Since Stn − xn → 0, xn − zn − x∗ − z∗ → 0, and Bzn − Bz∗ → 0, as n → ∞, it follows that zn − tn x∗ − z∗ → 0, as n → ∞. 3.39 Since Stn − tn ≤ Stn − xn xn − zn − x∗ − z∗ zn − tn x∗ − z∗ , 3.40 we obtain lim Stn − tn 0. n→∞ 3.41 18 Discrete Dynamics in Nature and Society Next, we show that lim sup fx − x, xn − x ≤ 0, n→∞ 3.42 where x PFS∩Υ fx. Indeed, since {tn } and {Stn } are two bounded sequence in E, we can choose a subsequence {tni } of {tn } such that tni of tn such that tni z ∈ E and lim sup fx − x, Stn − x lim fx − x, Stni − x . i→∞ n→∞ 3.43 Since limn → ∞ Stn − tn 0, we obtain Stni z as i → ∞. Now, we claim that z ∈ FS ∩ Υ. First by Lemma 2.5, it is easy to see that z ∈ FS. Since Stn − tn → 0, Stn − xn → 0, and tn − xn tn − Stn Stn − xn ≤ tn − Stn Stn − xn 3.44 Stn − tn Stn − xn , we conclude that tn − xn → 0 as n → ∞. Furthermore, by Lemma 3.2 that G is nonexpansive, then tn − Gtn Gxn − Gtn ≤ xn − tn . 3.45 Thus limn → ∞ tn − Gtn 0. According to Lemma 2.5, we obtain z ∈ Υ. Therefore, there holds z ∈ FS ∩ Υ. On the other hand, it follows from 2.2 that lim sup fx − x, xn − x lim sup fx − x, Stn − x n→∞ n→∞ lim fx − x, Stni − x n→∞ fx − x, z − x ≤ 0. 3.46 Discrete Dynamics in Nature and Society 19 Finally, we show that xn → x, and by 3.14 that 2 xn1 − x2 αn fxn βn xn γn Stn − x 2 ≤ βn xn − x γn Stn − x 2αn fxn − x, xn1 − x 2 ≤ βn xn − x γn Stn − x 2αn fxn − fx, xn1 − x 2αn fx − x, xn1 − x ≤ βn xn − x2 γn Stn − x2 2αn fxn − fxxn1 − x 2αn fx − x, xn1 − x ≤ βn xn − x2 γn tn − x2 2αn αxn − xxn1 − x 3.47 2αn fx − x, xn1 − x ≤ 1 − αn 2 xn − x2 αn α xn − x2 xn1 − x2 2αn fx − x, xn1 − x which implies that xn1 − x2 ≤ α2n 21 − ααn 1− xn − x2 xn − x2 1 − ααn 1 − ααn 2αn fx − x, xn1 − x 1 − ααn : 1 − σn xn − x2 δn , 3.48 n ≥ 0, where σn 21−ααn /1−ααn and δn α2n /1−ααn xn −x2 2αn /1−ααn fx−x, xn1 −x. Therefore, by 3.46 and Lemma 2.4, we get that {xn } converges to x, where x PFS∩Υ fx. This completes the proof. Setting A B C, we obtain the following corollary. Corollary 3.4. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A : E → H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such that FS ∩ Υ / ∅. Let f be a contraction of H into itself and given x0 ∈ H arbitrarily and {xn } is generated by xn1 zn PE xn − τAxn , yn PE zn − μAzn , αn fxn βn xn γn SPE yn − λAyn , 3.49 n ≥ 1, where λ, μ, τ ∈ 0, 2α and {αn }, {βn }, {γn } are three sequences in 0, 1 such that 20 Discrete Dynamics in Nature and Society i αn βn γn 1, ii limn → ∞ αn 0 and ∞ n1 αn ∞, iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Then, {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ fx and x, y, z is a solution of problem 1.10, where y PE z − μAz , 3.50 z PE x − τAx. Setting A ≡ B ≡ 0 the zero operators, we obtain the following corollary for solving the foxed points problem and the classical variational inequality problems. Corollary 3.5. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A : E → H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such that FS ∩ V IA, E / ∅. Let f be a contraction of H into itself and given x0 ∈ H arbitrarily and {xn } is generated by xn1 αn fxn βn xn γn SPE xn − λAxn , n ≥ 1, 3.51 where λ ∈ 0, 2α and {αn }, {βn }, {γn } are three sequences in 0, 1 such that i αn βn γn 1, ii limn → ∞ αn 0 and ∞ n1 αn ∞, iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Then, {xn } converges strongly to x ∈ FS ∩ V IA, E, where x PFS∩V IA,E fx. 4. Some Applications We recall that a mapping T : E → E is called strictly pseudocontractive if there exists some k with 0 ≤ k < 1 such that T x − T y2 ≤ x − y2 kI − T x − I − T y2 , ∀x, y ∈ E. 4.1 For recent convergence result for strictly pseudocontractive mappings, put A I − T . Then, we have I − Ax − I − Ay2 ≤ x − y2 kAx − Ay2 . 4.2 I − Ax − I − Ay2 ≤ x − y2 Ax − Ay2 − 2 x − y, Ax − Ay . 4.3 On the other hand, Discrete Dynamics in Nature and Society 21 Hence, we have 1 − k Ax − Ay2 . x − y, Ax − Ay ≥ 2 4.4 Consequently, if T : E → E is a strictly pseudocontractive mapping with constant k, then the mapping A I − T is 1 − k/2-inverse-strongly monotone. Setting A I − T , B I − V , and C I − W, we obtain the following corollary. Theorem 4.1. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let T, V, W be strictly pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive mapping of E into itself such that FS ∩ Υ / ∅. Let f be a contraction of H into itself and given x0 ∈ H arbitrarily and {xn } is generated by zn I − τxn τWxn , yn I − μ zn μV zn , xn1 αn fxn βn xn γn S 1 − λyn λT yn , 4.5 n ≥ 1, where λ ∈ 0, 2α, μ ∈ 0, 2β, and τ ∈ 0, 2γ and {αn }, {βn }, {γn } are three sequences in 0, 1 such that i αn βn γn 1, ii limn → ∞ αn 0 and ∞ n1 αn ∞, iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Then {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ fx and x, y, z is a solution of problem 1.9, where y PE z − μBz , z PE x − τCx. 4.6 Proof. Since A I − T , B I − V , and C I − W, we have PE xn − τCxn I − τxn τWxn , PE yn − λAyn I − λyn λT yn , 4.7 PE zn − μBzn I − μ zn μV zn . Thus, the conclusion follows immediately from Theorem 3.3. If fx x0 , for all x ∈ E, and T V W in Theorem 4.1, we obtain the following corollary. 22 Discrete Dynamics in Nature and Society Corollary 4.2. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let T be strictly pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive mapping of E into itself such that FS ∩ Υ / ∅. Given x0 ∈ H arbitrarily and {xn } is generated by xn1 zn I − τxn τT xn , yn I − μ zn μT zn , αn x0 βn xn γn S 1 − λyn λT yn , 4.8 n ≥ 1, where λ ∈ 0, 2α, μ ∈ 0, 2β, and τ ∈ 0, 2γ and {αn }, {βn }, {γn } are three sequences in 0, 1 such that αn βn γn 1, lim αn 0 and n→∞ ∞ αn ∞, n1 4.9 0 < lim inf βn ≤ lim sup βn < 1. n→∞ n→∞ Then, {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ x and x, y, z is a solution of problem 1.10, where y PE z − μAz z PE x − τAx. 4.10 Acknowledgments W. Kumam would like to give thanks to the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, for their financial support for Ph.D. program at KMUTT. 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