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Puy chosuantai2

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Puy chosuantai2

  1. 1. Weak and strong convergence to common fixed points of a countable family of multivalued mappings in Banach spaces ∗ W. Cholamjiak1,2 , P. Cholamjiak3 , Y. J. Cho4 , S. Suantai1,2 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand 3 Department of Mathematics, Faculty of Science, University of Phayao, Phayao 56000, Thailand 4 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea Abstract In this paper, we introduce a modified Mann iteration for a countable family of multivalued mappings. We use the best approximation operator to obtain weak and strong convergence theorems in a Banach space. We apply the main results to the problem of finding a common fixed point of a countable family of nonexpansive multivalued mappings.Keywords: Nonexpansive multivalued mapping; fixed point; weak convergence; strong con-vergence.AMS Subject Classification: 47H10, 47H09. ∗ Corresponding author. Email addresses: c-wchp007@hotmail.com (W. Cholamjiak), prasitch2008@yahoo.com(P.Cholamjiak), yjcho@gnu.ac.kr (Y. J. Cho), scmti005@chiangmai.ac.th (S. Suantai) 1
  2. 2. 2 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. Suantai1 Introduction Let D be a nonempty convex subset of a Banach spaces E. The set D is calledproximinal 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), CCB(D), K(D) andP (D) denote the families of nonempty closed bounded subsets, nonempty closedconvex bounded subsets, nonempty compact subsets, and nonempty proximinalbounded subsets of D, respectively. The Hausdorff metric on CB(D) is definedby H(A, B) = max sup d(x, B), sup d(y, A) x∈A y∈Bfor 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 mapping T : D → CB(D)is said to be nonexpansive if H(T x, T y) ≤ x − y for all x, y ∈ D. An elementp ∈ D is called a fixed point of T : D → D (respectively, T : D → CB(D)) ifp = T p (respectively, 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[23] if F (T ) = ∅ andH(T x, T p) ≤ x − p for all x ∈ D and all p ∈ F (T ). It is clear that everynonexpansive multivalued mapping T with F (T ) = ∅ is quasi-nonexpansive. Butthere exist quasi-nonexpansive mappings that are not nonexpansive, see [22]. It isknown that if T is a quasi-nonexpansive multivalued mapping, then F (T ) is closed. Throughout this paper, we denote the weak convergence and strong convergenceare and →, respectively. The mapping T : D → CB(D) is called hemicompactif, for any sequence {xn } in D such that d(xn , T xn ) → 0 as n → ∞, there exists asubsequence {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 Banach space E is said to satisfy Opial’s property; see [16] if for each x ∈ Eand each sequence {xn } weakly convergent to x, the following condition holds forall x = y : lim inf xn − x < lim inf xn − y . n→∞ n→∞The mapping T : D → CB(D) is said to be dimi-closed if for every sequence{xn } ⊂ D and any yn ∈ T xn such that xn x and yn → y, we have x ∈ D andy ∈ T x.Remark 1.1. [11] If the space satisfies Opial’s property, then I − T is demi-closed at0, where T : D → K(D) is a nonexpansive multivalued mapping. For single valued nonexpansive mappings, in 1953, Mann [14] introduced the fol-lowing iterative procedure to approximate a fixed point of a nonexpansive mapping
  3. 3. Weak and strong convergence to common fixed points... 3T in a Hilbert space H: xn+1 = αn xn + (1 − αn )T xn , ∀n ∈ N, (1.1)where the initial point x1 is taken in D arbitrarily and {αn } is a sequence in [0,1]. However, we note that Mann’s iteration process (1.1) has only weak convergence,in general; for instance, see [2, 7, 18]. Subsequently, Mann iteration has extensively been studied by many authors, formore details, see in [3, 4, 6, 9, 10, 12, 24, 25, 27, 30] and many other results whichcannot be mentioned here. However, the studying of multivalued nonexpansivemappings is harder than that of single valued nonexpansive mappings in both Hilbertspaces and Banach spaces. The result of fixed points for multivalued contractionsand nonexpansive mappings by using the Hausdorff metric was initiated by Markin[13]. Later, different iterative processes have been used to approximate fixed pointsof multivalued nonexpansive mappings (see [1, 17, 19, 20, 29, 23, 26, 28]). In 2009, Song and Wang [28] proved strong and weak convergence for Manniteration of a multivalued nonexpansive mapping T in a Banach space. They gavea strong convergence of the modified Mann iteration which is independent of theimplicit anchor-like continuous path zt ∈ tu + (1 − t)T zt . Let D be a nonempty closed subset of a Banach space E, βn ∈ [0, 1], αn ∈ [0, 1]and γn ∈ (0, +∞) such that limn→∞ γn = 0.(A) Choose x0 ∈ D, xn+1 = (1 − αn )xn + αn yn , ∀n ≥ 0,where yn ∈ T xn such that yn+1 − yn ≤ H(T xn+1 , T xn ) + γn .(B) For fixed u ∈ D, the sequence of modified Mann iteration is defined by x0 ∈ D, xn+1 = βn u + αn xn + (1 − αn − βn )yn , ∀n ≥ 0,where yn ∈ T xn such that yn+1 − yn ≤ H(T xn+1 , T xn ) + γn . Very recently, Shahzad and Zegeye [22] obtained the strong convergence theo-rems for a quasi-nonexpansive multivalued mapping. They relaxed compactness ofthe domain of T and constructed an iteration scheme which removes the restrictionof T namely T p = {p} for any p ∈ F (T ). The results provided an affirmative answerto some questions in [17]. They introduced new iterations as follows: Let D be a nonempty convex subset of a Banach space E and T : D → CB(D)and let αn , αn ∈ [0, 1],
  4. 4. 4 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. Suantai(C) The sequence of Ishikawa iterates is defined by x0 ∈ D, yn = αn zn + (1 − αn )xn , xn+1 = αn zn + (1 − αn )xn , ∀n ≥ 0,where zn ∈ T xn and zn ∈ T yn .(D) Let T : D → P (D) and PT x = {y ∈ T x : x − y = d(x, T x)}, we denote PTis the best approximation operator. The sequence of Ishikawa iterates is defined byx0 ∈ D, yn = αn zn + (1 − αn )xn , xn+1 = αn zn + (1 − αn )xn , ∀n ≥ 0,where zn ∈ PT xn and zn ∈ PT yn . It is remarked that Hussain and Khan [8], in 2003, employed the best approx-imation operator PT to study fixed points of *-nonexpansive multivalued mappingT and strong convergence of its iterates to a fixed point of T defined on a closedand convex subset of a Hilbert space. Let D be a nonempty closed convex subset of a Banach space E. Let {Tn }∞ n=1be a family of multivalued mappings from D into 2D and PTn x = {yn ∈ Tn x : x − yn = d(x, Tn x)}, n ≥ 1. Let {αn } be a sequence in (0, 1).(E) The sequence of the modified Mann iteration is defined by x1 ∈ D and xn+1 ∈ αn xn + (1 − αn )PTn xn , ∀n ≥ 1. (1.2) In this paper, we modify Mann iteration by using the best approximation oper-ator PTn , n ≥ 1 to find a common fixed point of a countable family of nonexpansivemultivalued mappings {Tn }∞ , n ≥ 1. Then we obtain weak and strong conver- n=1gence theorems for a countable family of multivalued mappings in Banach spaces.Moreover, we apply our results to the problem of finding a common fixed point of afamily of nonexpansive multivalued mappings.2 Preliminaries In this section, we give some characterizations and properties of the metricprojection in a Hilbert space.
  5. 5. Weak and strong convergence to common fixed points... 5 Let H be a real Hilbert space with inner product ·, · and norm · . Let Dbe a closed and convex subset of H. If, for any point x ∈ H, there exists a uniquenearest point in D, denoted by PD x, such that x − PD x ≤ x − y , ∀y ∈ D,then PD is called the metric projection of H onto D. We know that PD is a nonex-pansive mapping of H onto D.Lemma 2.1. [31] Let D be a closed and convex subset of a real Hilbert space Hand PD be the metric projection from H onto D. Then, for any x ∈ H and z ∈ D,z = PD x if and only if the following holds: x − z, y − z ≤ 0, ∀y ∈ D. Using the proof line in Lemma 3.1.3 of [31], we obtain the following result.Proposition 2.2. Let D be a closed and convex subset of a real Hilbert space H.Let T : D → CCB(D) be a multivalued mapping and PT the best approximationoperator. Then, for any x ∈ D, z ∈ PT x if and only if the following holds: x − z, y − z ≤ 0, ∀y ∈ T x.Lemma 2.3. [31] Let H be a real Hilbert space. Then the following equations hold: (1) x − y 2 = x 2 − y 2 − 2 x − y, y for all x, y ∈ H; (2) 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. Next, we show that PT is nonexpansive under some suitable conditions imposedon T .Remark 2.4. Let D be a closed and convex subset of a real Hilbert space H. LetT : D → CCB(D) be a multivalued mapping. If T x = T y, ∀x, y ∈ D, then PT is anonexpansive multivalued mapping.Proof. Let x, y ∈ D. For each a ∈ PT x, we have d(a, PT y) ≤ a − b , ∀b ∈ PT y. (2.1)By Proposition 2.2, we have x − y − (a − b), a − b = x − a, a − b + y − b, b − a ≥ 0.
  6. 6. 6 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. SuantaiIt follows that 2 a−b = x − y, a − b + a − b − (x − y), a − b ≤ x − y, a − b ≤ x−y a−b . (2.2)This implies that a−b ≤ x−y . (2.3)From (2.1) and (2.3), we obtain d(a, PT y) ≤ x − yfor every a ∈ PT x. Hence supa∈PT x d(a, PT y) ≤ x − y . Similarly, we can showthat supb∈PT y d(PT x, b) ≤ x − y . This shows that H(PT x, PT y) ≤ x − y . It is clear that if a nonexpansive multivalued mapping T satisfies the conditionthat T x = T y, ∀x, y ∈ D, then PT is nonexpansive. The following example showsthat if T is a nonexpansive multivalued mapping satisfying the property that T x =T y, ∀x, y ∈ D, then T x is not a singleton for all x ∈ D. Example 1. Consider D = [0, 1] with the usual norm. Define T : D → K(D)by T x = [0, a], a ∈ (0, 1].For x, y ∈ D, we have H(T x, T y) = 0 ≤ x − y . Hence T is nonexpansive andF (T ) = [0, a]. Next, we show that there exists a nonexpansive multivalued mapping T whichPT is nonexpansive but T x = T y, ∀x, y ∈ D, x = y. Example 2. Consider D = [0, 1] with the usual norm. Define T : D → K(D)by T x = [0, x].For x, y ∈ D, x = y, we have PT x = {x} and PT y = {y}. Then H(T x, T y) = x − y = H(PT x, PT y). Hence T and PT are nonexpansive. Open Question: Can we remove the assumption that PT is nonexpansive ? Now, we give an example which T is not nonexpansive, but PT is nonexpansive.Example 3. Consider D = [0, 1] with the usual norm. Define T : D → K(D) by [0, x] , x ∈ [0, 1 ], 2 Tx = { 1 } , x ∈ ( 1 , 1]. 2 2
  7. 7. Weak and strong convergence to common fixed points... 7Since H(T ( 1 ), T ( 3 )) = H([0, 5 ], { 2 }) = 1 > 2 = 1 − 3 , T is not nonexpansive. 5 5 1 1 2 5 5 5However, PT is nonexpansive. In fact,Case 1, if x, y ∈ [0, 1 ] then H(PT x, PT y) = x − y . 2Case 2, if x ∈ [0, 2 ] and y ∈ ( 1 , 1] then H(PT x, PT y) = x − 1 ≤ x − y . 1 2 2 1Case 3, if x, y ∈ ( 2 , 1] then H(PT x, PT y) = 0. It would be interesting to study the convergence of a multivalued mapping T byusing the best approximation operator PT . In order to deal with a family of nonlinear mappings, we consider the followingconditions. Let E be a Banach space and D be a subset of E. (1) Let {Tn } and τ be two families of nonlinear mappings of D into itself with∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ ) n=1is the set of all common fixed points of τ . The family {Tn } is said to satisfy theNST-condition [15] with respect to τ if, for each bounded sequence {zn } in C, lim zn − Tn zn = 0 =⇒ lim zn − T zn = 0, ∀T ∈ τ. n→∞ n→∞ (2) Let {Tn } and τ be two families of multivalued mappings of D into 2D with∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ ) is n=1the set of all common fixed points of τ . The family {Tn } is said to satisfy the SC-condition with respect to τ if, for each bounded sequence {zn } in D and sn ∈ Tn zn , lim zn − sn = 0 =⇒ lim zn − cn = 0, ∃cn ∈ T zn , ∀T ∈ τ. n→∞ n→∞It is easy to see that, if the family {Tn } of nonexpansive mappings satisfies theN ST −condition, then {Tn } satisfies the SC −condition for single valued mappings. (3) Let T be a multivalued mapping from D into 2D with F (T ) = ∅. Themapping T 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 ))) for all x ∈ D. The following result can be found in [21].Lemma 2.5. [21] Let D be a bounded closed subset of a Banach space E. Supposethat a nonexpansive multivalued mapping T : D → P (D) has a nonempty fixedpoint set. If I − T is closed, then T satisfies Condition I on D. (4) Let {Tn } and τ be two families of multivalued mappings of D into 2D with∅ = F (τ ) = ∞ F (Tn ), where F (Tn ) is the set of all fixed points of Tn and F (τ ) is n=1the set of all common fixed points of τ . The family {Tn } is said to satisfy Condition
  8. 8. 8 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. Suantai(A) if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0for r ∈ (0, ∞) such that there exists T ∈ τ , d(x, T x) ≥ f (d(x, F (τ ))) for all x ∈ D. We will give the example of {Tn } which satisfies the SC-condition and Condition(A) in the last section. Now, we need the following lemmas to prove our main results.Lemma 2.6. [32] Let X be uniformly convex Banach space and Br (0) be a closedball of X. Then there exists a continuous strictly increasing convex function g :[0, ∞) → [0, ∞) with g(0) = 0 such that 2 2 2 λx + (1 − λ)y ≤λ x + (1 − λ) y − λ(1 − λ)g( x − y )for all x, y ∈ Br (0) and λ ∈ [0, 1].Lemma 2.7. [5] Let E be a uniformly convex Banach space and Br (0) = {x ∈ E : x ≤ r} be a closed ball of E. Then there exists a continuous strictly increasing con-vex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that, for any j ∈ {1, 2, · · · , m}, m m m 2 2 αj αi xi ≤ αi xi − αi g( xj − xi ) m−1 i=1 i=1 i=1 mfor all m ∈ N, xi ∈ Br (0) and αi ∈ [0, 1] for all i = 1, 2, · · · , m with i=1 αi = 1.3 Strong and weak convergence of the modified Mann iteration in Banach spaces In this section, we prove strong convergence theorems of the modified Manniteration for a countable family of multivalued mappings under the SC-conditionand Condition (A) and obtain weak convergence theorem under the SC-conditionin Banach spaces. Moreover, we apply the main result to obtain weak and strongconvergence theorems for a countable family of nonexpansive multivalued mappings.Theorem 3.1. Let D be a closed and convex subset of a uniformly convex Banachspace E which satisfies Opial’s property. Let {Tn } and τ be two families of multi-valued mappings from D into P (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a n=1sequence in (0, 1) such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } begenerated by (1.2). Assume that(A1) for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ );(A2) I − T is demi-closed at 0 for all T ∈ τ .If {Tn } satisfies the SC-condition, then the sequence {xn } converges weakly to anelement in F (τ ).
  9. 9. Weak and strong convergence to common fixed points... 9Proof. Since xn+1 ∈ αn xn + (1 − αn )PTn xn , there exists zn ∈ PTn xn such thatxn+1 = αn xn + (1 − αn )zn . We note that PTn p = {p} for all p ∈ F (τ ) and n ∈ N. Itfollows from (A1) that xn+1 − p ≤ αn xn − p + (1 − αn ) zn − p = αn xn − p + (1 − αn )d(zn , PTn p) ≤ αn xn − p + (1 − αn )H(PTn xn , PTn p) ≤ xn − p (3.1)for every p ∈ F (τ ). Then { xn −p } is a decreasing sequence and hence limn→∞ xn −p exists for every p ∈ F (τ ). Next, for p ∈ F (τ ), since {xn } and {zn } arebounded, by Lemma 2.6, there exists a continuous strictly increasing convex functiong : [0, ∞) → [0, ∞) with g(0) = 0 such that 2 2 xn+1 − p = αn (xn − p) + (1 − αn )(zn − p) 2 2 ≤ αn xn − p + (1 − αn ) zn − p − αn (1 − αn )g( xn − zn ) 2 = αn xn − p + (1 − αn )d(zn , PTn p)2 − αn (1 − αn )g( xn − zn ) 2 ≤ αn xn − p + (1 − αn )H(PTn xn , PTn p)2 − αn (1 − αn )g( xn − zn ) 2 ≤ xn − p − αn (1 − αn )g( xn − zn ).It follows that 2 αn (1 − αn )g( xn − zn ) ≤ xn − p − xn+1 − p 2 .Since limn→∞ xn − p exists and 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, lim g( xn − zn ) = 0. n→∞By the properties of g, we can conclude that lim xn − zn = 0. n→∞Since {Tn } satisfies the SC-condition, there exists cn ∈ T xn such that lim xn − cn = 0 (3.2) n→∞for every T ∈ τ . Since {xn } is bounded, there exists a subsequence {xnk } of {xn }converges weakly to some q1 ∈ D. It follows from (A2) and (3.2) that q1 ∈ T q1for every T ∈ τ . Next, we show that {xn } converges weakly to q1 , take anothersubsequence {xmk } of {xn } converging weakly to some q2 ∈ D. Again, as above, we
  10. 10. 10 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. Suantaican conclude that q2 ∈ T q2 for every T ∈ τ . Finally, we show that q1 = q2 . Assumeq1 = q2 . Then by the Opial’s property of E, we have lim xn − q1 = lim xnk − q1 n→∞ k→∞ < lim xnk − q2 k→∞ = lim xn − q2 n→∞ = lim xmk − q2 k→∞ < lim xmk − q1 k→∞ = lim xn − q1 , n→∞which is a contradiction. Therefore q1 = q2 . This shows that {xn } converges weaklyto a fixed point of T for every T ∈ τ . This completes the proof. Using the above results and Remark 1.1, we obtain the following:Corollary 3.2. Let D be a closed and convex subset of a uniformly convex Banachspace E which satisfies Opial’s property. Let {Tn } and τ be two families of nonex-pansive multivalued mappings from D into K(D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let n=1{αn } be a sequence in (0, 1) such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let{xn } be generated by (1.2). Assume that for each n ∈ N, H(PTn x, PTn p) ≤ x − p ,∀x ∈ D, p ∈ F (τ ). If {Tn } satisfies the SC-condition, then the sequence {xn }converges weakly to an element in F (τ ).Theorem 3.3. Let D be a closed and convex subset of a uniformly convex Banachspace E. Let {Tn } and τ be two families of multivalued mappings from D intoP (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a sequence in (0, 1) such that n=10 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } be generated by (1.2). Assumethat(B1) for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ );(B2) the best approximation operator PT is nonexpansive for every T ∈ τ ;(B3) F (τ ) is closed.If {Tn } satisfies the SC-condition and Condition (A), then the sequence {xn } con-verges strongly to an element in F (τ ).Proof. It follows from the proof of Theorem 3.1 that limn→∞ xn − p exists forevery p ∈ F (τ ) and limn→∞ xn − zn = 0 where zn ∈ PTn xn . Since {Tn } satisfiesthe SC-condition, there exists cn ∈ T xn such that lim xn − cn = 0 n→∞
  11. 11. Weak and strong convergence to common fixed points... 11for every T ∈ τ . This implies that lim d(xn , T xn ) ≤ lim d(xn , PT xn ) ≤ lim xn − cn = 0 n→∞ n→∞ n→∞for every T ∈ τ . Since that {Tn } satisfies Condition (A), we have limn→∞ d(xn , F (τ )) =0. It follows from (B3), there is subsequence {xnk } of {xn } and a sequence {pk } ⊂F (τ ) such that 1 xnk − pk < k (3.3) 2for all k. From 3.1, we obtain xnk+1 − p ≤ xnk+1 −1 − p ≤ xnk+1 −2 − p . . . ≤ xnk − pfor all p ∈ F (τ ). This implies that 1 xnk+1 − pk ≤ xnk − pk < . (3.4) 2kNext, we show that {pk } is a Cauchy sequence in D. From (3.3) and (3.4), we have pk+1 − pk ≤ pk+1 − xnk+1 + xnk+1 − pk 1 < k−1 . (3.5) 2This implies that {pk } is a Cauchy sequence in D and thus converges to q ∈ D.Since PT is nonexpansive for every T ∈ τ , d(pk , T q) ≤ d(pk , PT q) ≤ H(PT pk , PT q) ≤ pk − q (3.6)for every T ∈ τ . It follows that d(q, T q) = 0 for every T ∈ τ and thus q ∈ F (τ ). Itimplies by (3.3) that {xnk } converges strongly to q. Since limn→∞ xn − q exists,it follows that {xn } converges strongly to q. This completes the proof. We known that if T is a quasi-nonexpansive multivalued mapping, then F (T ) isclosed. So we have the following result.Corollary 3.4. Let D be a closed and convex subset of a uniformly convex Banachspace E. Let {Tn } and τ be two families of nonexpansive multivalued mappingsfrom D into P (D) with F (τ ) = ∩∞ F (Tn ) = ∅. Let {αn } be a sequence in (0, 1) n=1such that 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1. Let {xn } be generated by (1.2).
  12. 12. 12 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. SuantaiAssume that for each n ∈ N, H(PTn x, PTn p) ≤ x − p , ∀x ∈ D, p ∈ F (τ ) and thebest approximation operator PT is nonexpansive for every T ∈ τ . If {Tn } satisfiesthe SC-condition and Condition (A), then the sequence {xn } converges strongly toan element in F (τ ).4 Application Let E be a real Banach space and D be a nonempty closed convex subset ofE. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D into i=0CB(D) and {βi,n } ⊂ [0, 1] be such that N βi,n = 1 for all n ∈ N. We define a i=0mapping Sn : D → 2 D as follows: N Sn = βi,n PTi , (4.1) i=0where T0 = I, I denotes the identity mapping. We also show that the mapping Sndefined by (4.1) satisfies the condition imposed on our main theorem.Lemma 4.1. Let D be a closed and convex subset of a uniformly convex Banachspace E. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D i=1into P (D) and, let {βi,n }N be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤ i=0lim supn→∞ βi,n < 1 for all i ∈ {0, 1, · · · , N } and N βi,n = 1 for all n ∈ N. For i=0all n ∈ N, let Sn be the mapping defined by (4.1). Assume that the best approximationoperator PTi is nonexpansive for all i ∈ {1, 2, ..., N }. Then the followings hold: (1) ∩∞ F (Sn ) = ∩N F (Ti ); n=1 i=0 (2) {Sn } satisfies the SC-condition; (3) for each n ∈ N, H(PSn x, PSn p) ≤ x − p for all x ∈ D and p ∈ ∩N F (Ti ). i=0Proof. (1) It is easy to see that ∩N F (Ti ) ⊂ ∩∞ F (Sn ). Next, we show that i=0 n=1∩∞ F (Sn ) ⊂ ∩N F (Ti ). Let p ∈ ∩∞ F (Sn ) and x∗ ∈ ∩N F (Ti ). Then there n=1 i=1 n=1 i=0exists zi ∈ PTi p such that p = β0,n p + N βi,n zi for all n ∈ N. By Lemma 2.7, i=1there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with
  13. 13. Weak and strong convergence to common fixed points... 13g(0) = 0 such that N ∗ 2 ∗ p−x = β0,n (p − x ) + βi,n (zi − x∗ ) 2 i=1 N N β0,n ≤ β0,n p − x∗ 2 + βi,n zi − x∗ 2 − βi,n g( zi − p ) N i=1 i=1 N N β0,n = β0,n p − x∗ 2 + βi,n d(zi , PTi x∗ )2 − βi,n g( zi − p ) N i=1 i=1 N N ∗ 2 β0,n ∗ 2 ≤ β0,n p − x + βi,n H(PTi p, PTi x ) − βi,n g( zi − p ) N i=1 i=1 N ∗ 2 β0,n ≤ p−x − βi,n g( zi − p ) . N i=1By the properties of g, we can conclude that zi = p, ∀i = 1, 2, · · · , N.Hence p ∈ ∩N F (Ti ). This completes the proof. i=1 (2) Let {vn } ⊂ D be a bounded sequence and an ∈ Sn vn be such that limn→∞ an −vn = 0. Then there exists zi,n ∈ PTi vn , i = 1, 2, · · · , N such that an = β0,n vn + N i=1 βi,n zi,n . Since {vn } and {zi,n } are bounded, by Lemma 2.7, there exists acontinuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0such that N 2 2 an − p = β0,n (vn − p) + βi,n (zi,n − p) i=1 N N 2 2 β0,n ≤ β0,n vn − p + βi,n zi,n − p − βi,n g( zi,n − vn ) N i=1 i=1 N N 2 β0,n = β0,n vn − p + βi,n d(zi,n , PTi p)2 − βi,n g( zi,n − vn ) N i=1 i=1 N N 2 β0,n ≤ β0,n vn − p + βi,n H(PTi vn , PTi p)2 − βi,n g( zi,n − vn ) N i=1 i=1 N 2 β0,n ≤ vn − p − βi,n g( zi,n − vn ) , ∀p ∈ ∩N F (Ti ). i=1 N i=1This implies that Nβ0,n βi,n g( zi,n − vn ) ≤ vn − an vn − p + an − p , ∀p ∈ ∩N F (Ti ). i=1 N i=1
  14. 14. 14 W. Cholamjiak, P. Cholamjiak, Y. J. Cho, S. SuantaiBy our assumptions, we get limn→∞ g( zi,n − vn ) = 0 for all i = 1, 2, · · · , N . Bythe properties of g, we can conclude that lim zi,n − vn = 0, ∀i = 1, 2, · · · , N. n→∞Hence {Sn } satisfies the SC-condition. (3) For p ∈ ∩N F (Ti ), we know that PTi p = {p} for every i ∈ {1, 2, ..., N }. Let i=1n ∈ N. Then for each x ∈ D and an ∈ PSn x, there exists zi,n ∈ PTi x such thatan = β0,n x + N βi,n zi,n . It follows that i=1 N an − p ≤ β0,n x − p + βi,n zi,n − p i=1 N = β0,n x − p + βi,n d(zi,n , PTi p) i=1 N ≤ β0,n x − p + βi,n H(PTi x, PTi p) i=1 ≤ x−p . (4.2)This implies that d(PSn x, p) ≤ an − p ≤ x − p . It follows from 4.2 thatsupan ∈PSn x d(an , PSn p) ≤ x−p . Since PSn p = {p}, supp∈PSn p d(PSn x, p) ≤ x−p .This shows that H(PSn x, PSn p) ≤ x − p for all x ∈ D and p ∈ ∩N F (Ti ). This i=1completes the proof.Lemma 4.2. Let D be a closed and convex subset of a Banach space E. Let {Ti }N i=1be a finite family of nonexpansive multivalued mappings of D into P (D) and, let{βi,n }N be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1 i=0for all i ∈ {0, 1, · · · , N } and N βi,n = 1 for all n ∈ N. For each n ∈ N, let Sn be i=0the mapping defined by (4.1). Assume that there exists i0 ∈ {1, 2, ..., N } such thatF (Ti0 ) = ∩N F (Ti ) = ∅ and I − Ti0 is closed. Then {Sn } satisfies Condition (A). i=1Proof. Since I − Ti0 is closed, it follows by Lemma 2.5 that Ti0 satisfies Condition I.Then there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0for r ∈ (0, ∞) such that d(x, Ti0 x) ≥ f (d(x, F (Ti0 ))) = f (d(x, ∩N F (Ti )) i=1for all x ∈ D. This completes the proof. Using Lemma 4.1, we have the following:
  15. 15. Weak and strong convergence to common fixed points... 15Corollary 4.3. Let D be a closed and convex subset of a uniformly convex Banachspace E which satisfies Opial’s property. Let {Ti }N be a finite family of nonexpan- i=1sive multivalued mappings of D into P (D) with ∩N F (Ti ) = ∅ and, let {βi,n }N i=1 i=0be sequences in (0, 1) such that 0 < lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1 for all Ni ∈ {0, 1, · · · , N } and i=0 βi,n = 1 for all n ∈ N. For each n ∈ N, let Sn bethe mapping defined by (4.1). Assume that the best approximation operator PTi isnonexpansive for all i ∈ {1, 2, ..., N }. Let {xn } be generated by xn+1 ∈ αn xn + (1 − αn )PSn xn , n ≥ 1. (4.3)If 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, then the sequence {xn } convergesweakly to an element in ∩N F (Ti ). i=1Proof. Putting Tn = Sn for all n ≥ 1 in the Theorem 3.1, we obtain the desiredresult. Now, using Lemma 4.1 and Lemma 4.2, we have the following:Corollary 4.4. Let D be a closed and convex subset of a uniformly convex Banachspace E. Let {Ti }N be a finite family of nonexpansive multivalued mappings of D i=1 N F (T ) = ∅ and, let {β }N be sequences in (0, 1) such that 0 <into P (D) with ∩i=1 i i,n i=0lim inf n→∞ βi,n ≤ lim supn→∞ βi,n < 1 for all i ∈ {0, 1, · · · , N } and N βi,n = 1 i=0for all n ∈ N. For each n ∈ N, let Sn be the mapping defined by (4.1). Assumethat the best approximation operator PTi is nonexpansive for all i ∈ {1, 2, ..., N } andthere exists i0 ∈ {1, 2, ..., N } such that F (Ti0 ) = ∩N F (Ti ) with I − Ti0 is closed. i=1Let {xn } be generated by xn+1 ∈ αn xn + (1 − αn )PSn xn , n ≥ 1. (4.4)If 0 < lim inf n→∞ αn ≤ lim supn→∞ αn < 1, then the sequence {xn } convergesstrongly to an element in ∩N F (Ti ). i=1Proof. Putting Tn = Sn for all n ≥ 1 in the Theorem 3.3, we obtain the desiredresult.Acknowledgement. The authors would like to thank the referees for valuable sug-gestion on the manuscript. The first and fourth authors supported by the Center ofExcellence in Mathematics, the Commission on Higher Education and the GraduateSchool of Chiang Mai University for the financial support. The second author was
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