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- 1. Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2010, Article ID 349158, 14 pages doi:10.1155/2010/349158 Research Article A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems Watcharaporn Cholamjiak1, 2 and Suthep Suantai1, 2 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 2 PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Correspondence should be addressed to Suthep Suantai, scmti005@chiangmai.ac.th Received 26 January 2010; Accepted 21 April 2010 Academic Editor: Binggen Zhang Copyright q 2010 W. Cholamjiak and S. Suantai. 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 a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established. 1. Introduction Let D be a nonempty convex subset of a Banach spaces E. Let F be a bifunction from D × D to R, where R is the set of all real numbers. The equilibrium problem for F is to find x ∈ D such that F x, y ≥ 0 for all y ∈ D. The set of such solutions is denoted by EP F . The set D is called proximal if for each x ∈ E, there exists an element y ∈ D such that x − y d x, D , where d x, D inf{ x − z : z ∈ D}. Let CB D , K D , and P D denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of D, respectively. The Hausdorff metric on CB D is defined by H A, B max sup d x, B , sup d y, A 1.1 x∈A y∈B for A, B ∈ CB D . A single-valued map T : D → D is called nonexpansive if T x−T y ≤ x−y for all x, y ∈ D. A multivalued map T : D → CB D is said to be nonexpansive if H T x, T y ≤
- 2. 2 Discrete Dynamics in Nature and Society x − y for all x, y ∈ D. An element p ∈ D is called a fixed point of T : D → D resp., T : D → CB D if p T p resp., p ∈ T p . The set of fixed points of T is denoted by F T . The mapping T : D → CB D is called quasi-nonexpansive 1 if F T / ∅ and H T x, T p ≤ x − p for all x ∈ D and all p ∈ F T . It is clear that every nonexpansive multivalued map T with F T / ∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive; see 2 . The mapping T : D → CB D is called hemicompact if, for any sequence {xn } in D such that d xn , T xn → 0 as n → ∞, there exists a subsequence {xnk } of {xn } such that xnk → p ∈ D. We note that if D is compact, then every multivalued mapping T : D → CB D is hemicompact. A mapping T : D → CB D is said to satisfy Condition (I) if there is a nondecreasing function f : 0, ∞ → 0, ∞ with f 0 0, f r > 0 for r ∈ 0, ∞ such that d x, T x ≥ f d x, F T 1.2 for all x ∈ D. In 1953, Mann 3 introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: xn 1 αn xn 1 − αn T xn , ∀n ∈ N, 1.3 where the initial point x0 is taken in C arbitrarily and {αn } is a sequence in 0, 1 . However, we note that Mann’s iteration process 1.3 has only weak convergence, in general; for instance, see 4–6 . In 2003, Nakajo and Takahashi 7 introduced the method which is the so-called CQ method to modify the process 1.3 so that strong convergence is guaranteed. They also proved a strong convergence theorem for a nonexpansive mapping in a Hilbert space. Recently, Tada and Takahashi 8 proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H. In 2005, Sastry and Babu 9 proved that the Mann and Ishikawa iteration schemes for multivalued map T with a fixed point p converge to a fixed point q of T under certain conditions. They also claimed that the fixed point q may be different from p. More precisely, they proved the following result for nonexpansive multivalued map with compact domain. In 2007, Panyanak 10 extended the above result of Sastry and Babu 9 to uniformly convex Banach spaces but the domain of T remains compact. Later, Song and Wang 11 noted that there was a gap in the proofs of Theorem 3.1 10 and Theorem 5 9 . They further solved/revised the gap and also gave the affirmative answer to Panyanak 10 question using the following Ishikawa iteration scheme. In the main results, domain of T is still compact, which is a strong condition see 11, Theorem 1 and T satisfies condition I see 11, Theorem 1 . In 2009, Shahzad and Zegeye 2 extended and improved the results of Panyanak 10 , Sastry and Babu 9 , and Song and Wang 11 to quasi-nonexpansive multivalued maps. They also relaxed compactness of the domain of T and constructed an iteration scheme which removes the restriction of T , namely, T p {p} for any p ∈ F T . The results provided an affirmative answer to Panyanak 10 question in a more general setting. In the main results,
- 3. Discrete Dynamics in Nature and Society 3 T satisfies Condition I see 2, Theorem 2.3 and T is hemicompact and continuous see 2, Theorem 2.5 . A mapping A : D → H is called α-inverse-strongly monotone 12 if there exists a positive real number α such that Ax − Ay, x − y ≥ α Ax − Ay 2 , ∀x, y ∈ D. 1.4 Remark 1.1. It is easy to see that if A : D → H is α-inverse-strongly monotone, then it is a 1/α -Lipschitzian mapping. Let A : D → H be a mapping. The classical variational inequality problem is to find a u ∈ D such that Au, v − u ≥ 0, ∀v ∈ D. 1.5 The set of solutions of variational inequality 3.9 is denoted by V I D, A . Question. How can we construct an iteration process for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inequality problem, and the set of common fixed points of nonexpansive multivalued maps ? In the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of single-valued nonexpansive mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; for instance, see 8, 13–20 and the references cited theorems. In this paper, we introduce a monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space. 2. Preliminaries The following lemmas give some characterizations and a useful property of the metric projection PD in a Hilbert space. Let H be a real Hilbert space with inner product ·, · and norm · . Let D be a closed and convex subset of H. For every point x ∈ H, there exists a unique nearest point in D, denoted by PD x, such that x − PD x ≤ x − y , ∀y ∈ D. 2.1 PD is called the metric projection of H onto D. We know that PD is a nonexpansive mapping of H onto D.
- 4. 4 Discrete Dynamics in Nature and Society Lemma 2.1 see 21 . Let D be a closed and convex subset of a real Hilbert space H and let PD be the metric projection from H onto D. Given x ∈ H and z ∈ D, then z PD x if and only if the following holds: x − z, y − z ≤ 0, ∀y ∈ D. 2.2 Lemma 2.2 see 7 . Let D be a nonempty, closed and convex subset of a real Hilbert space H and PD : H → D the metric projection from H onto D. Then the following inequality holds: y − PD x 2 x − PD x 2 ≤ x − y 2, ∀x ∈ H, ∀y ∈ D. 2.3 Lemma 2.3 see 21 . Let H be a real Hilbert space. Then the following equations hold: i x−y 2 x 2 − y 2 − 2 x − y, y , for all x, y ∈ H; ii tx 1−t y 2 t x 2 1−t y 2 − t 1 − t x − y 2 , for all t ∈ 0, 1 and x, y ∈ H. Lemma 2.4 see 22 . Let D be a nonempty, closed and convex subset of a real Hilbert space H. Given x, y, z ∈ H and also given a ∈ R, the set v ∈D : y−v 2 ≤ x−v 2 z, v a 2.4 is convex and closed. For solving the equilibrium problem, we assume that the bifunction F : D × D → R satisfies the following conditions: A1 F x, x 0 for all x ∈ D; A2 F is monotone, that is, F x, y F y, x ≤ 0 for all x, y ∈ D; A3 for each x, y, z ∈ D, lim supt↓0 F tz 1 − t x, y ≤ F x, y ; A4 F x, · is convex and lower semicontinuous for each x ∈ D. Lemma 2.5 see 13 . Let D be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bifunction from D × D to R satisfying (A1)–(A4) and let r > 0 and x ∈ H. Then, there exists z ∈ D such that 1 F z, y y − z, z − x ≥ 0, ∀y ∈ D. 2.5 r Lemma 2.6 see 18 . For r > 0, x ∈ H, defined a mapping Tr : H → D as follows: 1 Tr x z ∈ D : F z, y y − z, z − x ≥ 0, ∀y ∈ D . 2.6 r Then the following holds: 1 Tr is a single value;
- 5. Discrete Dynamics in Nature and Society 5 2 Tr is firmly nonexpansive, that is, for any x, y ∈ H, Tr x − Tr y 2 ≤ Tr x − Tr y, x − y ; 2.7 3 F Tr EP F ; 4 EP F is closed and convex. In the context of the variational inequality problem, u ∈ V I D, A ⇐⇒ u PD u − λAu , ∀λ > 0. 2.8 A set-valued mapping T : H → 2H is said to be monotone if for all x, y ∈ H, f ∈ T x, and g ∈ T y imply that f − g, x − y ≥ 0. A monotone mapping T : H → H is said to be maximal 23 if the graph G T of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for x, f ∈ H × H, f − g, x − y ≥ 0, ∀ y, g ∈ G T imply that f ∈ T x. Let A : D → H be an inverse strongly monotone mapping and let ND v be the normal cone to D at v ∈ D, that is, ND v {w ∈ H : v − u, w ≥ 0, ∀u ∈ D}, 2.9 and define ⎧ ⎨Av ND v, v ∈ D, Tv 2.10 ⎩∅, ∈ v / D. Then T is maximal monotone and 0 ∈ T v if and only if v ∈ V I D, A see, e.g., 24 . In general, the fixed point set of a nonexpansive multivalued map T is not necessary to be closed and convex see 25, Example 3.2 . In the next Lemma, we show that F T is closed and convex under the assumption that T p {p} for all p ∈ F T . Lemma 2.7. Let D be a closed and convex subset of a real Hilbert space H. Let T : D → CB D be a nonexpansive multivalued map with F T / ∅ and T p {p} for each p ∈ F T . Then F T is a closed and convex subset of D. Proof. First, we will show that F T is closed. Let {xn } be a sequence in F T such that xn → x as n → ∞. We have d x, T x ≤ d x, xn d xn , T x ≤ d x, xn H T xn , T x 2.11 ≤ 2d x, xn .
- 6. 6 Discrete Dynamics in Nature and Society It follows that d x, T x 0, so x ∈ F T . Next, we show that F T is convex. Let p tp1 1− t p2 where p1 , p2 ∈ F T and t ∈ 0, 1 . Let z ∈ T p; by Lemma 2.3, we have p−z 2 t z − p1 1 − t z − p2 2 t z − p1 2 1 − t z − p2 2 − t 1 − t p 1 − p2 2 2 2 td z, T p1 1 − t d z, T p2 − t 1 − t p1 − p2 2 2 2 ≤ tH T p, T p1 1 − t H T p, T p2 − t 1 − t p 1 − p2 2 2.12 ≤ t p − p1 2 1 − t p − p2 2 − t 1 − t p 1 − p2 2 2 t 1−t p1 − p2 2 1 − t t2 p1 − p2 2 − t 1 − t p 1 − p2 2 0. Hence p z. Therefore, p ∈ F T . 3. Main Results In the following theorem, we introduce a new monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space, and we prove strong convergence theorem without the condition I . Theorem 3.1. Let D be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bifunction from D × D to R satisfying (A1)–(A4), let A : D → H be an α-inverse strongly monotone mapping, and let Ti : D → CB D be nonexpansive multivalued maps for all i ∈ N with ∞ Ω: i 1 F Ti ∩ EP F ∩ V I D, A / ∅ and Ti p {p}, ∀p ∈ ∞1 F Ti . Assume that αi,n ∈ 0, 1 i with lim supn → ∞ αi,n < 1 for all i ∈ N, {rn } ⊂ b, ∞ for some b ∈ 0, ∞ , and {λn } ⊂ c, d for some c, d ∈ 0, 2α . For an initial point x0 ∈ H with C1 D and x1 PC1 x0 , let {xn }, {yn }, {si,n }, and {un } be sequences generated by 1 F un , y y − un , un − xn ≥ 0, ∀y ∈ D, rn yn PD un − λn Aun , si,n αi,n yn 1 − αi,n zi,n , 3.1 Ci,n 1 z ∈ Ci,n : si,n − z ≤ yn − z ≤ xn − z , ∞ Cn 1 Ci,n 1 , i 1 xn 1 PCn 1 x0 , ∀n ∈ N, where zi,n ∈ Ti yn . Then, {xn }, {yn }, and {un } converge strongly to z0 PΩ x0 .
- 7. Discrete Dynamics in Nature and Society 7 Proof. We split the proof into six steps. Step 1. Show that PCn 1 x0 is well defined for every x0 ∈ H. Since 0 < c ≤ λn ≤ d < 2α for all n ∈ N, we get that PC I − λn A is nonexpansive for all n ∈ N. Hence, ∞ 1 F PC I − λn A n V I D, A is closed and convex. By Lemma 2.6 4 , we know that EP F is closed and convex. By Lemma 2.7, we also know that ∞1 F Ti is closed i ∞ and convex. Hence, Ω : i 1 F Ti ∩ EP F ∩ V I D, A is a nonempty, closed and convex set. By Lemma 2.4, we see that Ci,n 1 is closed and convex for all i, n ∈ N. This implies that Cn 1 is also closed and convex. Therefore, PCn 1 x0 is well defined. Let p ∈ Ω and i ∈ N. From un Trn xn , we have un − p Trn xn − Trn p ≤ xn − p 3.2 for every n ≥ 0. From this, we have si,n − p αi,n yn 1 − αi,n zi,n − p ≤ αi,n yn − p 1 − αi,n zi,n − p ≤ αi,n yn − p 1 − αi,n d zi,n , Ti p ≤ αi,n yn − p 1 − αi,n H Ti yn , Ti p 3.3 ≤ yn − p PD un − λn Aun − PD p − λn Ap ≤ un − p ≤ xn − p . So, we have p ∈ Ci,n 1 , hence Ω ⊂ Ci,n 1 , ∀i ∈ N. This shows that Ω ⊂ Cn 1 ⊂ Cn . Step 2. Show that limn → ∞ xn − x0 exists. Since Ω is a nonempty closed convex subset of H, there exists a unique v ∈ Ω such that z0 PΩ x0 . 3.4 From xn PCn x0 , Cn 1 ⊂ Cn and xn 1 ∈ Cn , ∀n ≥ 0, we get xn − x0 ≤ xn 1 − x0 , ∀n ≥ 0. 3.5 On the other hand, as Ω ⊂ Cn , we obtain xn − x0 ≤ z0 − x0 , ∀n ≥ 0. 3.6
- 8. 8 Discrete Dynamics in Nature and Society It follows that the sequence {xn } is bounded and nondecreasing. Therefore, limn → ∞ xn − x0 exists. Step 3. Show that xn → q ∈ D as n → ∞. For m > n, by the definition of Cn , we see that xm PCm x0 ∈ Cm ⊂ Cn . By Lemma 2.2, we get xm − xn 2 ≤ xm − x0 2 − xn − x0 2 . 3.7 From Step 2, we obtain that {xn } is Cauchy. Hence, there exists q ∈ D such that xn → q as n → ∞. Step 4. Show that q ∈ F. From Step 3, we get xn 1 − xn −→ 0 3.8 as n → ∞. Since xn 1 ∈ Cn 1 ⊂ Cn , we have si,n − xn ≤ si,n − xn 1 xn 1 − xn ≤ 2 xn 1 − xn −→ 0 3.9 as n → ∞ for all i ∈ N, yn − xn ≤ yn − xn 1 xn 1 − xn ≤ 2 xn 1 − xn −→ 0 3.10 as n → ∞. Hence, yn → q as n → ∞. It follows from 3.9 and 3.10 that 1 zi,n − yn si,n − yn −→ 0 3.11 1 − αi,n as n → ∞ for all i ∈ N. For each i ∈ N, we have d q, Ti q ≤ q − yn yn − zi,n d zi,n , Ti q ≤ q − yn yn − zi,n H Ti yn , Ti q 3.12 ≤ q − yn yn − zi,n yn − q . From 3.11 , we obtain d q, Ti q 0. Hence q ∈ F.
- 9. Discrete Dynamics in Nature and Society 9 Step 5. Show that q ∈ EP F . By the nonexpansiveness of PD and the inverse strongly monotonicity of A, we obtain yn − p 2 ≤ un − λn Aun − p − λn Ap 2 ≤ un − p 2 λn λn − 2α Aun − Ap 2 3.13 Trn xn − Trn p 2 λn λn − 2α Aun − Ap 2 ≤ xn − p 2 c d − 2α Aun − Ap 2 . This implies c 2α − d Aun − Ap 2 ≤ xn − p 2 − yn − p 2 3.14 ≤ xn − yn xn − p yn − p . It follows from 3.10 that lim Aun − Ap 0. 3.15 n→∞ Since PD is firmly nonexpansive, we have 2 2 yn − p PD un − λn Aun − PD p − λn Ap ≤ un − λn Aun − p − λn Ap , yn − p 1 2 un − λn Aun − p − λn Ap 2 2 2 yn − p − un − λn Aun − p − λn Ap − yn − p 3.16 1 2 2 2 ≤ un − p yn − p − un − yn − λn Aun − Ap 2 1 2 2 2 ≤ xn − p yn − p − un − yn 2λn un − yn , Aun − Ap 2 1 2 2 2 ≤ xn − p yn − p − un − yn 2λn un − yn Aun − Ap . 2 This implies that yn − p 2 ≤ xn − p 2 − un − yn 2 2λn un − yn Aun − Ap . 3.17
- 10. 10 Discrete Dynamics in Nature and Society It follows that un − yn 2 ≤ xn − yn xn − p yn − p 3.18 2d un − yn Aun − Ap . From 3.10 and 3.15 , we get lim un − yn 0. 3.19 n→∞ It follows from 3.10 and 3.19 that lim un − xn 0. 3.20 n→∞ Since un Trn xn , we have 1 F un , y y − un , un − xn ≥ 0, ∀y ∈ D. 3.21 rn From the monotonicity of F, we have 1 y − un , un − xn ≥ F y, un , ∀y ∈ D, 3.22 rn hence un − xn y − un , ≥ F y, un , ∀y ∈ D. 3.23 rn From 3.20 and condition A4 , we have 0 ≥ F y, q , ∀y ∈ D. 3.24 For t with 0 < t ≤ 1 and y ∈ D, let yt ty 1 − t q. Since y, q ∈ D and D is convex, then yt ∈ D and hence F yt , q ≤ 0. So, we have 0 F yt , yt ≤ tF yt , y 1 − t F yt , q ≤ tF yt , y . 3.25 Dividing by t, we obtain F yt , y ≥ 0, ∀y ∈ D. 3.26
- 11. Discrete Dynamics in Nature and Society 11 Letting t ↓ 0 and from A3 , we get F q, y ≥ 0, ∀y ∈ D. 3.27 Therefore, we obtain q ∈ EP F . Step 6. Show that q ∈ V I D, A . Since T is the maximal monotone mapping defined by 2.10 , ⎧ ⎨Ax ND x, x ∈ D, Tx 3.28 ⎩∅, ∈ x / D. For any given x, u ∈ G T , hence u − Ax ∈ ND x. It follows that x − yn , u − Ax ≥ 0. 3.29 On the other hand, since yn PD un − λn Aun , we have x − yn , yn − un − λn Aun ≥ 0, 3.30 and so yn − un x − yn , Aun ≥ 0. 3.31 λn From 3.29 , 3.31 , and the α-inverse monotonicity of A, we have x − yn , u ≥ x − yn , Ax yn − un ≥ x − yn , Ax − x − yn , Aun λn yn − un 3.32 x − yn , Ax − Ayn x − yn , Ayn − Aun − x − yn , λn yn − un ≥ x − yn , Ayn − Aun − x − yn , . λn It follows that lim x − yn , u x − q, u ≥ 0. 3.33 n→∞ Again since T is maximal monotone, hence 0 ∈ T q. This shows that q ∈ V I D, A .
- 12. 12 Discrete Dynamics in Nature and Society Step 7. Show that q z0 PΩ x0 . Since xn PCn x0 and Ω ⊂ Cn , we obtain x0 − xn , xn − p ≥ 0 ∀p ∈ Ω. 3.34 By taking the limit in 3.34 , we obtain x0 − q, q − p ≥ 0 ∀p ∈ Ω. 3.35 This shows that q PΩ x0 z0 . From Steps 3 to 5, we obtain that {xn }, {yn }, and {un } converge strongly to z0 PΩ x0 . This completes the proof. Theorem 3.2. Let D be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti : ∞ D → CB D be nonexpansive multivalued maps for all i ∈ N with Ω : i 1 F Ti ∩ V I D, A / ∅ ∞ and Ti p {p}, for all p ∈ i 1 F Ti . Assume that αi,n ∈ 0, 1 with lim supn → ∞ αi,n < 1 and {λn } ⊂ c, d for some c, d ∈ 0, 2α . For an initial point x0 ∈ H with C1 D and x1 PC1 x0 , let {xn }, {yn }, and {si,n } be sequences generated by yn PD xn − λn Axn , si,n αi,n yn 1 − αi,n zi,n , Ci,n 1 z ∈ Ci,n : si,n − z ≤ yn − z ≤ xn − z , ∞ 3.36 Cn 1 Ci,n 1 , i 1 xn 1 PCn 1 x0 , ∀n ∈ N, where zi,n ∈ Ti yn . Then, {xn } and {yn } converge strongly to z0 PΩ x0 . Proof. Putting F x, y 0 for all x, y ∈ D in Theorem 3.1, we obtain the desired result directly from Theorem 3.1. Theorem 3.3. Let D be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti : D → ∞ CB D be nonexpansive multivalued maps for all i ∈ N with Ω : i 1 F Ti / ∅ and Ti p {p}, for ∞ all p ∈ i 1 F Ti . Assume that αi,n ∈ 0, 1 with lim supn → ∞ αi,n < 1. For an initial point x0 ∈ H with C1 D and x1 PC1 x0 , let {xn } and {si,n } be sequences generated by si,n αi,n xn 1 − αi,n zi,n , Ci,n 1 {z ∈ Ci,n : sn − z ≤ xn − z }, ∞ 3.37 Cn 1 Ci,n 1 , i 1 xn 1 PCn 1 x0 , ∀n ∈ N, where zi,n ∈ Ti yn . Then, {xn } converge strongly to z0 PΩ x0 .
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