Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 37513, 10 pages doi:10.1155/2007/37513 Research Article Strong Convergence to Common Fixed Points of a Finite Family of Nonexpansive Mappings Yeong-Cheng Liou, Yonghong Yao, and Kenji Kimura Received 24 December 2006; Accepted 2 May 2007 Recommended by Yeol Je Cho We suggest and analyze an iterative algorithm for a finite family of nonexpansive map- pings T 1 ,T 2 , ,T r . Further, we prove that the proposed iterative algorithm converges strongly to a common fixed point of T 1 ,T 2 , ,T r . Copyright © 2007 Yeong-Cheng Liou et al. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let C be a closed convex subset of a Banach space E.AmappingT of C into itself is called nonexpansive if Tx− Ty≤x − y for all x, y ∈ C. We denote by F(T) the set of fixed points of T.LetT 1 ,T 2 , ,T r be a finite family of nonexpansive mappings satisfying that the set F =  r i =1 F(T i ) of common fixed points of T 1 ,T 2 , ,T r is nonempty. The problem of finding a common fixed point has been investigated by many researchers; see, for ex- ample, Atsushiba and Takahashi [1], Bauschke [2], Lions [3], Shimizu and Takahashi [4], Takahashi et al. [5], Zeng et al. [6]. To solve this problem, the iterative scheme x 1 ∈ C and x n+1 = α n x 1 +  1 −α n  T n x n , n ∈ N, (1.1) where T n+r = T n and 0 <α n , is used. Wittmann [7] dealt with the iterative scheme for the case r = 1; see originally Halpern [8]. Bauschke [2] dealt with the iterative scheme for a finite family of nonexpansive mappings under the restriction that F = F  T r T r−1 ···T 1  = F  T 1 T r ···T 2  =···= F  T r−1 ···T 1 T r  . (1.2) 2 Journal of Inequalities and Applications Recently, Kimura et al. [9] dealt with an iteration scheme which is more general than that of Wittmann’s result. They proved the following theorems. Theorem 1.1 (see [9, Theorem 4]). Let E be a uniformly convex Banach space whose norm is uniformly G ˆ ateaux differentiable and let C be a closed convex subset of E.LetT 1 ,T 2 , ,T r be nonexpansive mappings of C into itself such that the set F =  r i =1 F(T i ) of common fixed points of T 1 ,T 2 , ,T r is nonempty. Let {α n } and {β n } be t wo sequences in [0,1] which satisfy the following control conditions: (C1) lim n→∞ α n = 0; (C2)  ∞ n=1 α n =∞; (C3)  ∞ n=1 |α n+1 − α n | < ∞; (C4) lim n→∞ β i n = β i and  r i =1 β i n = 1, n ∈ N for some β i ∈ (0,1); (C5)  ∞ n=1  r i =1 |β i n+1 − β i n | < ∞. Let x ∈ C and define a sequence {x n } by x 1 ∈ C and x n+1 = α n x +  1 −α n  r  i=1 β i n T i x n , n ∈ N. (1.3) Then {x n } converges strongly to the point Px,whereP is a sunny nonexpansive retraction of C onto F. Theorem 1.2 (see [9, Theorem 5]). Let E be a uniformly convex Banach space whose norm is uniformly G ˆ ateaux differentiable and let C beaclosedconvexsubsetofE.LetS, T be nonexpansive mappings of C into itself such that the set F(S) ∩ F(T) of common fixed points of S and T is nonempty. Let x ∈ C and let {x n } beasequencegeneratedby x n+1 = α n x +  1 −α n  β n Sx n +  1 −β n  Tx n  , n ∈ N. (1.4) Assume (C1) and ( C2) hold and the following conditions are satisfied: (C3  )lim n→∞ (α n /α n+1 ) = 1; (C4  )lim n→∞ β n = β ∈ (0,1); (C5  )  ∞ n=1 |β n+1 − β n | < ∞. Then {x n } converges st rongly to the point Px,whereP is a sunny nonexpansive retraction of C onto F(S) ∩ F(T). We remark that the control conditions (C3) and (C3  ) were introduced initially by Wittmann [7]andXu[10], respectively. On the other hand, we have to remark that con- ditions (C1) and (C2) are necessary for the strong convergence of algorithms (1.3)and (1.4) for nonexpansive mappings. It is unclear if they are sufficient. The objective of this paper is to show another generalization of Mann and Halpern iterative algorithm to a setting of a finite family of nonexpansive mappings. We deal with the iterative scheme x 0 ∈ C and x n+1 = α n f  x n  + β n x n + γ n r  i=1 τ i n T i x n , n ∈ N. (1.5) Yeo ng -C he ng L i o u e t al . 3 Using this iterative scheme, we can find a common fixed point of a finite family of non- expansive mappings u nder some type of control conditions. 2. Preliminaries Let E be a Banach space with norm ·and let E ∗ be the dual of E. Denote by ·,· the duality product. The normalized duality mapping J from E to E ∗ is defined by J(x) =  x ∗ ∈ E ∗ :  x, x ∗  = x 2 =   x ∗   2  (2.1) for x ∈ E. ABanachspaceE is said to be strictly convex if (x + y)/2 < 1forallx, y ∈ E with x=y=1andx = y.Itisalsosaidtobeuniformlyconvexiflim n→∞ x n − y n =0for any two sequences {x n }, {y n } in E such that x n =y n =1andlim n→∞ (x n + y n )/2= 1. Let U ={x ∈ E : x=1} betheunitsphereofE. Then the Banach space E is said to be smooth provided that lim t→0 x + ty−x t (2.2) exists for each x, y ∈ U. In this case, the norm of E is said to be G ˆ ateaux differentiable. It is said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U.Thenormof E is said to be uniformly G ˆ ateaux differentiable if for any y ∈ U the limit exists uniformly for all x ∈ U. It is known that if the norm of E is uniformly G ˆ ateaux differentiable, then the normalized duality mapping J is norm to weak star uniformly continuous on any bounded subsets of E. Let C be a closed convex subset of a Banach space E and let D be a subset of C.Recall that a self-mapping f : C → C is a contraction on C if there exists a constant α ∈ (0,1) such that  f (x) − f (y)≤αx − y, x, y ∈ C.AmappingP : C → D is said to be sunny if P(Px + t(x − Px)) = Px whenever Px + t(x − Px) ∈ C for x ∈ C and t ≥ 0. If P 2 = P, then P is called a retraction. We know that a retraction P of C onto D is sunny and nonexpansive if and only if x − Px,J(y − Px)≤0forally ∈ D. From this inequality, it is easy to show that there exists at most one sunny nonexpansive retraction of C onto D. If there is a sunny nonexpansive retraction of C onto D,thenD is said to be a sunny nonexpansive retraction of C. Now, we introduce several lemmas for our main results in this paper. Lemma 2.1 (see [11]). Let C be a nonempty closed convex subset of a str ictly convex Banach space. For each r ∈ N,letT r be a nonexpansive mapping of C into E.Let{τ r } be a sequence of positive real numbers s uch that  ∞ r=1 τ r = 1.If  ∞ r=1 F(T r ) is nonempty, then the mapping T =  ∞ r=1 τ r T r is well-defined and F(T) =  ∞ r=1 F(T r ). Lemma 2.2 (see [12]). Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0,1] with 0 < liminf n→∞ β n ≤ limsup n→∞ β n < 1.Supposex n+1 = (1 − β n )y n + β n x n for all integers n ≥ 0 and limsup n→∞ (y n+1 − y n −x n+1 − x n ) ≤ 0. Then, lim n→∞ y n − x n =0. 4 Journal of Inequalities and Applications Lemma 2.3 (see [10]). Assume {a n } is a sequence of nonnegative real numbers such that a n+1 ≤ (1 − γ n )a n + δ n ,where{γ n } isasequencein(0,1) and {δ n } is a sequence such that (1)  ∞ n=1 γ n =∞; (2) limsup n→∞ δ n /γ n ≤ 0 or  ∞ n=1 |δ n | < ∞. Then lim n→∞ a n = 0. 3. Main results First, we consider the following iterative scheme: x n+1 = α n f  x n  + β n x n + γ n  τ n Sx n +  1 − τ n  Tx n  , n ≥ 0, (3.1) where {α n }, {β n }, {γ n },and{τ n } are sequences in [0,1]. Theorem 3.1. Let E be a strictly c onvex Banach space whos e norm is uniformly G ˆ ateaux differentiable and let C be a closed convex subset of E.LetS and T be nonexpansive mappings of C into itself such that F(S) ∩ F(T) =∅.Let f : C → C be a fixed contractive mapping. Assume that {z t } convergesstronglytoafixedpointz of U as t → 0,wherez t is the unique element of C which satisfies z t = tf(z t )+(1− t)Uz t , U = τS+(1− τ)T, 0 <τ<1.Let{α n }, {β n }, {γ n },and{τ n } be four real sequences in [0,1] such that α n + β n + γ n = 1. Assume {α n } satisfies conditions (C1) and (C2) and assume the following control conditions hold: (D3) 0 < liminf n→∞ β n ≤ limsup n→∞ β n < 1; (D4) lim n→∞ τ n = τ. For arbitr ary x 0 ∈ C, then the sequence {x n } defined by (3.1) converges strongly to a common fixed point of S and T. Proof. We show first that {x n } is bounded. To end this, by taking a fixed element p ∈ F(S) ∩ F(T) and using (3.1), we have   x n+1 − p   ≤ α n   f  x n  − p   + β n   x n − p   + γ n  τ n   Sx n − p   +  1 − τ n    Tx n − p    ≤ α n α   x n − p   + α n   f (p) − p   +  β n + γ n    x n − p   =  1 − α n + αα n    x n − p   + α n   f (p) − p   ≤ max    x n − p   , 1 1 − α   f (p) − p    . (3.2) By induction, we get   x n − p   ≤ max    x 0 − p   , 1 1 − α   f (p) − p    (3.3) for all n ≥ 0. This shows that {x n } is bounded, so are {Tx n }, {Sx n },and{ f (x n )}. We show then that x n+1 − x n →0(n →∞). Define a sequence {y n } which satisfies x n+1 =  1 − β n  y n + β n x n . (3.4) Yeo ng -C he ng L i o u e t al . 5 Observe that y n+1 − y n = α n+1 1 − β n+1  f  x n+1  − f  x n  +  α n+1 1 − β n+1 − α n 1 − β n  f  x n  + γ n+1 τ n+1 1 − β n+1  Sx n+1 − Sx n  +  γ n+1 τ n+1 1 − β n+1 − γ n τ n 1 − β n  Sx n + γ n+1  1 − τ n+1 ) 1 − β n+1  Tx n+1 − Tx n  +  γ n+1  1 − τ n+1  1 − β n+1 − γ n  1 − τ n  1 − β n  Tx n = α n+1 1 − β n+1  f  x n+1  − f  x n  +  α n+1 1 − β n+1 − α n 1 − β n  f  x n  + γ n+1 τ n+1 1 − β n+1  Sx n+1 − Sx n  + γ n+1  1 − τ n+1  1 − β n+1  Tx n+1 − Tx n  + γ n+1 1 − β n+1  τ n+1 − τ n  Sx n +  γ n+1 1 − β n+1 − γ n 1 − β n  τ n Sx n + γ n+1 1 − β n+1  τ n − τ n+1  Tx n +  γ n+1 1 − β n+1 − γ n 1 − β n   1 − τ n  Tx n . (3.5) It follows that   y n+1 − y n   −   x n+1 − x n   ≤ αα n+1 1 − β n+1   x n+1 − x n   +     α n+1 1 − β n+1 − α n 1 − β n       f  x n    +     γ n+1 1 − β n+1 − 1       x n+1 − x n   + τ n     γ n+1 1 − β n+1 − γ n 1 − β n       Sx n   +  1 − τ n      γ n+1 1 − β n+1 − γ n 1 − β n       Tx n   + γ n+1 1 − β n+1   τ n − τ n+1      Sx n   +   Tx n    ≤ (1 + α)α n+1 1 − β n+1   x n+1 − x n   + γ n+1 1 − β n+1   τ n − τ n+1      Sx n   +   Tx n    +     α n+1 1 − β n+1 − α n 1 − β n        f  x n    + τ n   Sx n   +  1 − τ n    Tx n    . (3.6) Since {x n }, {Tx n }, {Sx n },and{ f (x n )} are bounded, we obtain limsup n→∞    y n+1 − y n   −   x n+1 − x n    ≤ 0. (3.7) 6 Journal of Inequalities and Applications Hence, by Lemma 2.2 we know that y n − x n →0asn →∞. Consequently, lim n→∞ x n+1 − x n =lim n→∞ (1 − β n )y n − x n =0. Define U = τS+(1− τ)T.Then,byLemma 2.1, F(U) = F(S) ∩ F(T). Observing that   x n − Ux n   ≤   x n+1 − x n   +   x n+1 − Ux n   ≤   x n+1 − x n   + α n   f  x n  − Ux n   + β n   x n − Ux n   + γ n   τ − τ n      Sx n   +   Tx n    (3.8) and using control conditions (C1), (D3), and (D4) on {α n }, {β n },and{τ n },weconclude that lim n→∞ Ux n − x n =0. We next show that limsup n→∞  z − f (z), j  z − x n  ≤ 0. (3.9) Let x t be the unique fixed point of the contraction mapping U t given by U t x = tf(x)+(1− t)Ux. (3.10) Then x t − x n = t  f  x t  − x n  +(1− t)  Ux t − x n  . (3.11) We compute as follows:   x t − x n   2 ≤ (1 − t) 2   Ux t − x n   2 +2t  f  x t  − x n , j  x t − x n  ≤ (1 − t) 2    Ux t − Ux n   +   Ux n − x n    2 +2t  f  x t  − x t , j  x t − x n  +2t   x t − x n   2 ≤ (1 − t) 2   x t − x n   2 + a n (t)+2t   x t − x n   2 +2t  f  x t  − x t , j  x t − x n  , (3.12) where a n (t) =Ux n − x n (2x t − x n  + Ux n − x n ) → 0asn →∞. The last inequality implies  x t − f  x t  , j  x t − x n  ≤ t 2   x t − x n   2 + 1 2t a n (t). (3.13) It follows that limsup n→∞  x t − f  x t  , j  x t − x n  ≤ t 2 M 2 . (3.14) Yeo ng -C he ng L i o u e t al . 7 Letting t → 0, we obtain limsup t→0 limsup n→∞  x t − f  x t  , j  x t − x n  ≤ 0. (3.15) Moreover , we have  z − f (z), j  z − x n  =  z − f (z), j  z − x n  −  z − f (z), j  x t − x n  +  z − f (z), j  x t − x n  −  x t − f (z), j  x t − x n  +  x t − f (z), j  x t − x n  − x t − f  x t  , j  x t − x n  +  x t − f  x t  , j  x t − x n  =  z − f (z), j  z − x n  − j  x t − x n  +  z − x t , j  x t − x n  +  f  x t  − f (z), j  x t − x n  +  x t − f  x t  , j  x t − x n  . (3.16) Then, we obtain limsup n→∞  z − f (z), j  z − x n  ≤ sup n∈N  z − f (z), j  z − x n  − j  x t − x n  +   z − x t   limsup n→∞   x t − x n   +   f  x t  − f (z)   limsup n→∞   x t − x n   +limsup n→∞  x t − f  x t  , j  x t − x n  ≤ sup n∈N  z − f (z), j  z − x n  − j  x t − x n  +(1+α)   z − x t   limsup n→∞   x t − x n   +limsup n→∞  x t − f  x t  , j  x t − x n  . (3.17) By hypothesis x t → z ∈F(S)∩F(T)ast→ 0andj is norm-to-weak ∗ uniformly continuous on bounded subset of E,weobtain lim t→0 sup n∈N  z − f (z), j  z − x n  − j  x t − x n  = 0. (3.18) Therefore, we have limsup n→∞  z − f (z), j  z − x n  = limsup t→0 limsup n→∞  z − f (z), j  z − x n  ≤ limsup t→0 limsup n→∞  x t − f  x t  , j  x t − x n  ≤ 0. (3.19) 8 Journal of Inequalities and Applications Finally, we have   x n+1 − z   2 ≤   β n  x n − z  + γ n  τ n Sx n +  1 − τ n  Tx n − z    2 +2α n  f  x n  − z, j  x n+1 − z  ≤ β 2 n   x n − z   2 + γ 2 n   τ n  Sx n − z  +  1 − τ n  Tx n − z    2 +2β n γ n   x n − z     τ n  Sx n − z  +  1 − τ n  Tx n − z    +2α n  f  x n  − f (z), j  x n+1 − z  +2α n  f (z) − z, j  x n+1 − z  ≤ β 2 n   x n − z   2 + γ 2 n   x n − z   2 +2β n γ n   x n − z     x n − z   +2α n  f  x n  − f (z), j  x n+1 − z  +2α n  f (z) − z, j  x n+1 − z  ≤  1 − α n  2   x n − z   2 + αα n    x n − z   2 +   x n+1 − z   2  +2α n  f (z) − z, j  x n+1 − z  . (3.20) It follows that   x n+1 − z   2 ≤ 1 − (2 − α)α n 1 − αα n   x n − z   2 + 2α n 1 − αα n  f (z) − z, j  x n+1 − z  + α 2 n 1 − αα n   x n − z   2 ≤  1 − 2(1 − α)α n    x n − z   2 +2(1− α)α n  1 (1− α)  1− αα n    f (z) − z, j  x n+1 − z  + α n 2   x n − z   2  . (3.21) Noting that  ∞ n=0 [2(1 − α)α n ] =∞and limsup n→∞  1 (1 − α)  1 − αα n    f (z) − z, j  x n+1 − z  + α n 2   x n − z   2  ≤ 0. (3.22) Apply Lemma 2.3 to (3.21)toconcludethatx n → z as n →∞. This completes the proof.  Remark 3.2. We note that every uniformly smooth Banach space has a uniformly G ˆ ateaux differentiable norm. By Xu [13, Theorem 4.1], we know that {z t } converges strongly to a fixed point of U as t → 0, where z t is the unique element of C which satisfies z t = tf(z t )+ (1 − t)Uz t . Corollar y 3.3. Let E be a strictly convex and uniformly smooth Banach space whose norm is uniformly G ˆ ateaux differentiable and let C be a closed convex subset of E.LetS and T be nonexpansive mappings of C into itself such that F(S) ∩ F(T) =∅.Let f : C → C be a fixed contractive mapping. Let {α n }, {β n }, {γ n },and{τ n } be four real sequences in [0,1] such that α n + β n + γ n = 1. Assume the control conditions (C1), (C2), (D3), and (D4) are sat isfied. For arbitrary x 0 ∈ C, then the sequence {x n } defined by (3.1) converges strongly to a common fixed point of S and T. We can obtain the following results from Takahashi and Ueda [14]whichisrelatedto the existence of sunny nonexpansive retr a ctions. Yeo ng -C he ng L i o u e t al . 9 Corollar y 3.4. Let E be a uniformly convex Banach space whose norm is uniformly G ˆ ateaux differentiable and let C be a closed convex subset of E.LetS and T be nonexpansive mappings of C into itself such that F(S) ∩ F(T) =∅.Letu ∈ C beagivenpoint.Let{α n }, {β n }, {γ n },and{τ n } be four real sequences in [0,1] such that α n + β n + γ n = 1. Assume the control conditions (C1), (C2), (D3), and (D4) are satisfied. For arbitrary x 0 ∈ C,letthe sequence {x n } be defined by x n+1 = α n u + β n x n + γ n  τ n Sx n +  1 − τ n  Tx n  , n ≥ 0. (3.23) Then {x n } converges strongly to the point Pu,whereP is a sunny nonex pansive retraction of C onto F(S) ∩ F(T). We can also obtain the following theorems for a finite family of nonexpansive map- pings. The proof is similar to that of Theorem 3.1, the details of the proof, therefore, are omitted. Theorem 3.5. Let E be a strictly c onvex Banach space whos e norm is uniformly G ˆ ateaux differentiable and let C be a closed convex subset of E.LetT 1 ,T 2 , ,T r beafinitefamily of nonexpansive mappings of C into itself such that the set F =  r i =1 F(T i ) of common fixed points of T 1 ,T 2 , ,T r is nonempty. Let f : C → C be a fixed contractive mapping. Assume that {z t } convergesstronglytoafixedpointz of U as t → 0,wherez t is the unique element of C which sat isfies z t = tf(z t )+(1− t)Uz t , U =  r i =1 τ i T i , 0 <τ i < 1,and  r i =1 τ i n = 1.Let {α n }, {β n }, {γ n },and{τ i n } be real sequences in [0,1] such that α n + β n + γ n = 1. Assume the control conditions (C1), (C2), and (D3) hold. Assume {τ i n } satisfies the condition (D4  ): lim n→∞ τ i n = τ i , i = 1,2, ,r, r  i=1 τ i n = 1. (3.24) For arbitr ary x 0 ∈ C,letthesequence{x n } be defined by x n+1 = α n f  x n  + β n x n + γ n r  i=1 τ i n T i x n , n ≥ 0. (3.25) Then {x n } converges strongly to a common fixed point of T 1 ,T 2 , ,T r . Theorem 3.6. Let E be a strictly convex and uniformly smooth Banach space and let C be a closed convex subset of E.LetT 1 ,T 2 , ,T r be a finite family of nonexpansive mappings of C into itself such that the set F =  r i =1 F(T i ) of common fixed p oints of T 1 ,T 2 , ,T r is nonempty. Let f : C → C be a fixed contractive mapping. Let {α n }, {β n }, {γ n },and{τ i n } be real sequences in [0,1] such that α n + β n + γ n = 1. Assume the control conditions (C1), (C2), (D3), and (D4  ) are satisfied. For arbitrary x 0 ∈ C, then the sequence {x n } defined by (3.25) converges strong ly to a common fixed point of T 1 ,T 2 , ,T r . Acknowledgment The research was partially supposed by Grant NSC 95-2622-E-230-005CC3. 10 Journal of Inequalities and Applications References [1] S. Atsushiba and W. 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Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan Email addresses: simplex liou@hotmail.com; ycliou@csu.edu.tw Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: yuyanrong@tjpu.edu.cn Kenji Kimura: Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan Email address: kimura@math.nsysu.edu.tw . mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997. [5] W. Takahashi, T. Tamura, and M. Toyoda, “Approximation of common fixed points of a family of. points of families of nonexpansive mappings,” to appear in Taiwanese Journal of Mathematics. [7] R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathe- matik, vol objective of this paper is to show another generalization of Mann and Halpern iterative algorithm to a setting of a finite family of nonexpansive mappings. We deal with the iterative scheme x 0 ∈ C and x n+1 =