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1. 1. Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2010, Article ID 349158, 14 pagesdoi:10.1155/2010/349158Research ArticleA Hybrid Method for a Countable Family ofMultivalued Maps, Equilibrium Problems, andVariational 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. IntroductionLet D be a nonempty convex subset of a Banach spaces E. Let F be a bifunction from D × Dto R, where R is the set of all real numbers. The equilibrium problem for F is to ﬁnd x ∈ Dsuch that F x, y ≥ 0 for all y ∈ D. The set of such solutions is denoted by EP F . The set Dis 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 familiesof nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximalbounded 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∈Bfor A, B ∈ CB D . A single-valued map T : D → D is called nonexpansive if T x−T y ≤ x−yfor all x, y ∈ D. A multivalued map T : D → CB D is said to be nonexpansive if H T x, T y ≤
2. 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 . Themapping T : D → CB D is called quasi-nonexpansive 1 if F T / ∅ and H T x, T p ≤ x − pfor all x ∈ D and all p ∈ F T . It is clear that every nonexpansive multivalued map T withF T / ∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are notnonexpansive; see 2 . The mapping T : D → CB D is called hemicompact if, for any sequence {xn } in Dsuch that d xn , T xn → 0 as n → ∞, there exists a subsequence {xnk } of {xn } such thatxnk → p ∈ D. We note that if D is compact, then every multivalued mapping T : D → CB Dis hemicompact. A mapping T : D → CB D is said to satisfy Condition (I) if there is a nondecreasingfunction f : 0, ∞ → 0, ∞ with f 0 0, f r > 0 for r ∈ 0, ∞ such that d x, T x ≥ f d x, F T 1.2for all x ∈ D. In 1953, Mann 3 introduced the following iterative procedure to approximate a ﬁxedpoint of a nonexpansive mapping T in a Hilbert space H: xn 1 αn xn 1 − αn T xn , ∀n ∈ N, 1.3where 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, ingeneral; for instance, see 4–6 . In 2003, Nakajo and Takahashi 7 introduced the method which is the so-called CQmethod to modify the process 1.3 so that strong convergence is guaranteed. They alsoproved a strong convergence theorem for a nonexpansive mapping in a Hilbert space. Recently, Tada and Takahashi 8 proposed a new iteration for ﬁnding a commonelement of the set of solutions of an equilibrium problem and the set of ﬁxed points of anonexpansive mapping T in a Hilbert space H. In 2005, Sastry and Babu 9 proved that the Mann and Ishikawa iteration schemesfor multivalued map T with a ﬁxed point p converge to a ﬁxed point q of T under certainconditions. 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 uniformlyconvex 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 aﬃrmativeanswer to Panyanak 10 question using the following Ishikawa iteration scheme. In the mainresults, domain of T is still compact, which is a strong condition see 11, Theorem 1 and Tsatisﬁes 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 whichremoves the restriction of T , namely, T p {p} for any p ∈ F T . The results provided anaﬃrmative answer to Panyanak 10 question in a more general setting. In the main results,
3. 3. Discrete Dynamics in Nature and Society 3T 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 apositive real number α such that Ax − Ay, x − y ≥ α Ax − Ay 2 , ∀x, y ∈ D. 1.4Remark 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 au ∈ D such that Au, v − u ≥ 0, ∀v ∈ D. 1.5The 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 theset of solutions of an equilibrium problem, the set of solutions of a variational inequalityproblem, 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 solutionsof equilibrium problems and the set of ﬁxed points of single-valued nonexpansive mappingsin the framework of Hilbert spaces and Banach spaces has been intensively studied by manyauthors; for instance, see 8, 13–20 and the references cited theorems. In this paper, we introduce a monotone hybrid iterative scheme for ﬁnding a commonelement of the set of a common ﬁxed points of a countable family of nonexpansivemultivalued maps, the set of variational inequality, and the set of solutions of an equilibriumproblem in a Hilbert space.2. PreliminariesThe following lemmas give some characterizations and a useful property of the metricprojection PD in a Hilbert space. Let H be a real Hilbert space with inner product ·, · and norm · . Let D be a closedand 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. 2.1PD is called the metric projection of H onto D. We know that PD is a nonexpansive mapping ofH onto D.
4. 4. 4 Discrete Dynamics in Nature and SocietyLemma 2.1 see 21 . Let D be a closed and convex subset of a real Hilbert space H and let PDbe the metric projection from H onto D. Given x ∈ H and z ∈ D, then z PD x if and only if thefollowing holds: x − z, y − z ≤ 0, ∀y ∈ D. 2.2Lemma 2.2 see 7 . Let D be a nonempty, closed and convex subset of a real Hilbert space H andPD : 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. 2.3Lemma 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 2.4is convex and closed. For solving the equilibrium problem, we assume that the bifunction F : D × D → Rsatisﬁes 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. LetF be a bifunction from D × D to R satisfying (A1)–(A4) and let r > 0 and x ∈ H. Then, there existsz ∈ D such that 1 F z, y y − z, z − x ≥ 0, ∀y ∈ D. 2.5 rLemma 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 rThen the following holds: 1 Tr is a single value;
5. 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. 2.8A set-valued mapping T : H → 2H is said to be monotone if for all x, y ∈ H, f ∈ T x, andg ∈ 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 monotonemapping. 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 stronglymonotone 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}, 2.9and 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 necessaryto be closed and convex see 25, Example 3.2 . In the next Lemma, we show that F T isclosed 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 anonexpansive multivalued map with F T / ∅ and T p {p} for each p ∈ F T . Then F T is a closedand 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 → xas 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. 6 Discrete Dynamics in Nature and SocietyIt 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 ResultsIn the following theorem, we introduce a new monotone hybrid iterative scheme for ﬁndinga common element of the set of a common ﬁxed points of a countable family of nonexpansivemultivalued maps, the set of variational inequality, and the set of solutions of an equilibriumproblem 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 bea bifunction from D × D to R satisfying (A1)–(A4), let A : D → H be an α-inverse stronglymonotone 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 iwith lim supn → ∞ αi,n < 1 for all i ∈ N, {rn } ⊂ b, ∞ for some b ∈ 0, ∞ , and {λn } ⊂ c, d forsome 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. 7. Discrete Dynamics in Nature and Society 7Proof. 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 forall n ∈ N. Hence, ∞ 1 F PC I − λn A n V I D, A is closed and convex. By Lemma 2.6 4 , weknow 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 convexset. By Lemma 2.4, we see that Ci,n 1 is closed and convex for all i, n ∈ N. This implies thatCn 1 is also closed and convex. Therefore, PCn 1 x0 is well deﬁned. Let p ∈ Ω and i ∈ N. Fromun Trn xn , we have un − p Trn xn − Trn p ≤ xn − p 3.2for 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 suchthat z0 PΩ x0 . 3.4From xn PCn x0 , Cn 1 ⊂ Cn and xn 1 ∈ Cn , ∀n ≥ 0, we get xn − x0 ≤ xn 1 − x0 , ∀n ≥ 0. 3.5On the other hand, as Ω ⊂ Cn , we obtain xn − x0 ≤ z0 − x0 , ∀n ≥ 0. 3.6
8. 8. 8 Discrete Dynamics in Nature and SocietyIt follows that the sequence {xn } is bounded and nondecreasing. Therefore, limn → ∞ xn − x0exists.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 . 3.7From Step 2, we obtain that {xn } is Cauchy. Hence, there exists q ∈ D such that xn → q asn → ∞.Step 4. Show that q ∈ F. From Step 3, we get xn 1 − xn −→ 0 3.8as n → ∞. Since xn 1 ∈ Cn 1 ⊂ Cn , we have si,n − xn ≤ si,n − xn 1 xn 1 − xn ≤ 2 xn 1 − xn −→ 0 3.9as n → ∞ for all i ∈ N, yn − xn ≤ yn − xn 1 xn 1 − xn ≤ 2 xn 1 − xn −→ 0 3.10as 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,nas 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. 9. Discrete Dynamics in Nature and Society 9Step 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 . 2This implies that yn − p 2 ≤ xn − p 2 − un − yn 2 2λn un − yn Aun − Ap . 3.17
10. 10. 10 Discrete Dynamics in Nature and SocietyIt 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 rnFrom the monotonicity of F, we have 1 y − un , un − xn ≥ F y, un , ∀y ∈ D, 3.22 rnhence un − xn y − un , ≥ F y, un , ∀y ∈ D. 3.23 rnFrom 3.20 and condition A4 , we have 0 ≥ F y, q , ∀y ∈ D. 3.24For t with 0 < t ≤ 1 and y ∈ D, let yt ty 1 − t q. Since y, q ∈ D and D is convex, thenyt ∈ 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 . 3.25Dividing by t, we obtain F yt , y ≥ 0, ∀y ∈ D. 3.26
11. 11. Discrete Dynamics in Nature and Society 11Letting t ↓ 0 and from A3 , we get F q, y ≥ 0, ∀y ∈ D. 3.27Therefore, 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. 3.29On the other hand, since yn PD un − λn Aun , we have x − yn , yn − un − λn Aun ≥ 0, 3.30and so yn − un x − yn , Aun ≥ 0. 3.31 λnFrom 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 , . λnIt 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. 12 Discrete Dynamics in Nature and SocietyStep 7. Show that q z0 PΩ x0 . Since xn PCn x0 and Ω ⊂ Cn , we obtain x0 − xn , xn − p ≥ 0 ∀p ∈ Ω. 3.34By taking the limit in 3.34 , we obtain x0 − q, q − p ≥ 0 ∀p ∈ Ω. 3.35This 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 directlyfrom 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 ∈ Hwith 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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