Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007, Article ID 76971, 9 pages doi:10.1155/2007/76971 Research Article Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings Meijuan Shang, Yongfu Su, and Xiaolong Qin Received 23 May 2007; Accepted 2 August 2007 Recommended by J. R. L. Webb We modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend the results announced by many others. Copyright © 2007 Meijuan Shang et al. This is an open access article dist ributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries Let H be a real Hilbert space, C a nonempty closed convex subset of H,andT : C → C a mapping. Recall that T is nonexpansive if Tx − Ty≤x − y for all x, y ∈ C.A point x ∈ C is called a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T, that is, F(T) ={x ∈ C : Tx = x}. 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 for all x, y ∈ C.WeuseΠ C to denote the collection of all contract ions on C, that is, Π C = { f | f : C → C a contraction}.AnoperatorA is strongly positive if there exists a constant γ>0 with the property Ax, x≥γx 2 ∀x ∈ H. (1.1) Iterative methods for nonexpansive mappings have recently been applied to solve con- vex minimization problems (see, e.g., [1, 2] and the references therein). A typical prob- lem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H: min x∈C 1 2 Ax, x−x,b, (1.2) 2 Fixed Point Theory and Applications where C is the fixed point set of a nonexpansive mapping S,andb is a given point in H. In [2], it is proved that the s equence {x n } defined by the iterative method below, with the initial guess x 0 ∈ H chosen arbitrarily, x n+1 =  I − α n A  Sx n + α n b, n ≥ 0, (1.3) converges strongly to the unique solution of the minimization problem (1.2)provided the sequence {α n } satisfies certain conditions. Recently, Marino and Xu [1]introduceda new iterative scheme by the viscosity approximation method x n+1 =  I − α n A  Sx n + α n γf  x n  , n ≥ 0. (1.4) They proved that the sequence {x n } generated by the above iterative scheme converges strongly to the unique solution of the variational inequality (A − γf)x ∗ ,x − x ∗ ≥0, x ∈ C, which is the optimality condition for the minimization problem min x∈C 1 2 Ax, x−h(x), (1.5) where C is the fixed point set of a nonexpansive mapping S,andh is a potential function for γf (i.e., h  (x) = γf(x)forx ∈ H.) Mann’s iteration process [3] is often used to approximate a fixed point of a nonexpan- sive mapping, which is defined as x n+1 = α n x n +  1 − α n  Tx n , n ≥ 0, (1.6) where the initial guess x 0 is taken in C arbitrarily and the sequence {α n } ∞ n=0 is in the interval [0,1]. But Mann’s iteration process has only weak convergence, in general. For example, Reich [4] shows that if E is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence {α n } is such that  α n (1 − α n ) =∞, then the sequence {x n } generated by process (1.6) converges weakly to a point in F(T). Therefore, many authors try to modify Mann’s iteration process to have strong convergence. Kim and Xu [5] introduced the following iteration process: x 0 = x ∈ C arbitrarily chosen, y n = β n x n +  1 − β n  Tx n , x n+1 = α n u +  1 − α n  y n . (1.7) They proved that the sequence {x n } defined by (1.7) converges strongly to a fixed point of T provided the control sequences {α n } and {β n } satisfy a ppropriate conditions. Recently, Yao et al. [6] also modified Mann’s iterative scheme (1.7) and got a strong convergence theorem. They improved the results of Kim and Xu [5] to some extent. Meijuan Shang et al. 3 In this paper, we study the mapping W n defined by U n0 = I, U n1 = γ n1 T 1 U n0 +  1 − γ n1  I, U n2 = γ n2 T 2 U n1 +  1 − γ n2  I, . . . U n,N−1 = γ n,N−1 T N−1 U n,N−2 +  1 − γ n,N−1  I, W n = U nN = γ nN T N U n,N−1 +  1 − γ nN  I, (1.8) where {γ n1 },{γ n2 }, ,{γ nN }∈(0,1]. Such a mapping W n is called the W n -mapping gen- erated by T 1 ,T 2 , ,T N and {γ n1 },{γ n2 }, ,{γ nN }. Nonexpansivity of each T i ensures the nonexpansivity of W n .Itfollowsfrom[7, Lemma 3.1] that F(W n ) =∩ N i =1 F(T i ). Very recently, Xu [2] studied the following iterative scheme: x n+1 = α n u +  I − α n A  T n+1 x n , n ≥ 0, (1.9) where A is a linear bounded operator, T n = T n mod N and the mod function takes values in {1,2, ,N}. He proved that the sequence {x n } generated by the above iterative scheme converges strongly to the unique solution of the minimization problem (1.2)providedT n satisfy F  T N ···T 2 T 1  = F  T 1 T N ···T 3 T 2  =··· = F  T N−1 ···T 1 T n  , (1.10) and {α n }∈(0,1) satisfying the following control conditions: (C1) lim n→∞ α n = 0; (C2)  ∞ n=1 α n =∞; (C3)  ∞ n=1 |α n − α n+N | < ∞ or lim n→∞ α n /α n+N = 0. Remark 1.1. There are many nonexpansive mappings, which do not satisfy (1.10). For example, if X = [0,1] and T 1 , T 2 are defined by T 1 x = x/2+1/2andT 2 x = x/4, then F(T 1 T 2 ) ={4/7},whereasF(T 2 T 1 ) ={1/7}. In this paper, motivated by Kim and Xu [5], Marino and Xu [1], Xu [2], and Yao et al. [6], we introduce a composite iteration scheme as follows: x 0 = x ∈ C arbitrarily chosen, y n = β n x n +  1 − β n  W n x n , x n+1 = α n γf  x n  +  I − α n A  y n , (1.11) where f ∈ Π C is a contraction, and A is a linear bounded operator. We prove, under cer- tain appropriate assumptions on the sequences {α n } and {β n },that{x n } defined by (1.11) converges to a common fixed point of the finite family of nonexpansive mappings, which solves some variation inequality and is also the optimality condition for the minimization problem (1.5). 4 Fixed Point Theory and Applications Now, we consider some special cases of iterative scheme (1.11). When A = I and γ = 1 in (1.11), we have that (1.11) collapses to x 0 = x ∈ C arbitrarily chosen, y n = β n x n +  1 − β n  W n x n , x n+1 = α n f  x n  +  1 − α n  y n . (1.12) When A = I and γ = 1in(1.11), N = 1and{γ n1 }=1in(1.8), we have that (1.11)col- lapses to the iterative scheme which was considered by Yao et al. [6]. When A = I and γ = 1in(1.11), N = 1and{γ n1 }=1in(1.8), and f (y) = u ∈ C for all y ∈ C in (1.11), we have that (1.11)reducesto(1.7), which was considered by Kim and Xu [5]. In order to prove our main results, we need the following lemmas. Lemma 1.2. In a Hilber t space H, there holds the inequality x + y 2 ≤x 2 +2  y,(x + y)  , x, y ∈ H. (1.13) Lemma 1.3 (Suzuki [8]). Let {x n } and {y n } be bounded sequences in a Banach space X and let β n beasequencein[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. (1.14) Then lim n→∞ y n − x n =0. Lemma 1.4 (Xu [2]). Assume that {α n } is a sequence of nonnegative real number s such that α n+1 ≤  1 − γ n  α n + δ n , (1.15) where γ n isasequencein(0,1) and {δ n } isasequencesuchthat (i)  ∞ n=1 γ n =∞; (ii) limsup n→∞ δ n /γ n ≤ 0 or  ∞ n=1 |δ n | < ∞. Then lim n→∞ α n = 0. Lemma 1.5 (Marino and Xu [1]). Assume that A is a strongly positive linear bounded oper- ator on a Hilbert space H with coefficient γ>0 and 0 <ρ≤A −1 , then I − ρA≤1 − ργ. Lemma 1.6 (Marino and Xu [1]). Let H be a Hilbert space. Let A be a strongly positive linear bounded selfadjoint operator with coefficient γ>0. Assume that 0 <γ<γ/α.LetT : C → C be a nonexpansive mapping with a fixed point x t ∈ C of the contraction C  x → tγ f (x)+ (1 − tA)Tx. Then {x t } converges strongly as t → 0 to a fixed point x of T, which solves the variational inequality  (A − γf)x,z − x  ≤ 0, z ∈ F(T). (1.16) 2. Main results Theorem 2.1. Let C be a closed convex subset of a Hilbert space H.LetA be a strongly pos- itive linear bounded operator with coefficient γ>0 and W n is defined by (1.8). Assume that Meijuan Shang et al. 5 0 <γ< γ/α and F =∩ N i =1 F(T i ) =∅. Given a map f ∈ Π C , the initial guess x 0 ∈ C is chosen arbitrarily a nd given sequences {α n } ∞ n=0 and {β n } ∞ n=0 in (0,1), the following conditions are satisfied: (C1)  ∞ n=0 α n =∞; (C2) lim n→∞ α n = 0; (C3) 0 < liminf n→∞ β n ≤ limsup n→∞ β n < 1; (C4) lim n→∞ |γ n,i − γ n−1,i |=0, for all i = 1,2, ,N. Let {x n } ∞ n=1 be the c omposite process defined by (1.11). Then {x n } ∞ n=1 converges strongly to q ∈ F, which also solves the following variational inequality:  γf(q) − Aq, p − q  ≤ 0, p ∈ F. (2.1) Proof. First, we observe that {x n } ∞ n=0 is bounded. Indeed, take a point p ∈ F and notice that   y n − p   ≤ β n   x n − p   +  1 − β n    W n x n − p   ≤   x n − p   . (2.2) It follows that   x n+1 − p   =   α n  γf  x n  − Ap  +  I − α n A  y n − p    ≤  1 − α n (γ − γα)    x n − p   + α n   γf(p) − Ap   . (2.3) By simple inductions, we have x n − p≤max{x 0 − p,Ap− γf(p)/(γ − γα)}, which gives t hat the sequence {x n } is bounded, so are {y n } and {z n }. Next, we claim that   x n+1 − x n   −→ 0. (2.4) Put l n = (x n+1 − β n x n )/(1 − β n ). Now, we compute l n+1 − l n , that is, x n+1 =  1 − β n  l n + β n x n , n ≥ 0. (2.5) Observing that l n+1 − l n = α n+1 γf  x n+1  +  I − α n+1 A  y n+1 − β n+1 x n+1 1 − β n+1 − α n γf  x n  +  I − α n A  y n − β n x n 1 − β n = α n+1  γf  x n+1  − Ay n+1  1 − β n+1 − α n  γf  x n  − Ay n  1 − β n + W n+1 x n+1 − W n x n , (2.6) 6 Fixed Point Theory and Applications we have   l n+1 − l n   ≤ α n+1 1 − β n+1   γf  x n+1  − Ay n+1   + α n 1 − β n   Ay n − γf  x n    +   x n+1 − x n   +   W n+1 x n − W n x n   . (2.7) Next, we will use M to denote the possible different constants appearing in the fol l ow ing reasoning. It follows from the definition of W n that   W n+1 x n − W n x n   =   γ n+1,N T N U n+1,N−1 x n +  1 − γ n+1,N  x n − γ n,N T N U n,N−1 x n −  1 − γ n,N  x n   ≤   γ n+1,N − γ n,N     x n   +   γ n+1,N T N U n+1,N−1 x n − γ n,N T N U n,N−1 x n   ≤   γ n+1,N − γ n,N     x n   +   γ n+1,N  T N U n+1,N−1 x n − T N U n,N−1 x n    +   γ n+1,N − γ n,N     T N U n,N−1 x n   ≤ 2M   γ n+1,N − γ n,N   + γ n+1,N   U n+1,N−1 x n − U n,N−1 x n   . (2.8) Next, we consider   U n+1,N−1 x n − U n,N−1 x n   =   γ n+1,N−1 T N−1 U n+1,N−2 x n +  1 − γ n+1,N−1  x n − γ n,N−1 T N−1 U n,N−2 x n −  1 − γ n,N−1  x n   ≤   γ n+1,N−1 − γ n,N−1     x n   + γ n+1,N−1   T N−1 U n+1,N−2 y n − T N−1 U n,N−2 x n   +   γ n+1,N−1 − γ n,N−1     T N−1 U n,N−2 x n   ≤ 2M   γ n+1,N−1 − γ n,N−1   +   U n+1,N−2 x n − U n,N−2 x n   . (2.9) It follows that   U n+1,N−1 x n − U n,N−1 x n   ≤ 2M   γ n+1,N−1 − γ n,N−1   +2M   γ n+1,N−2 − γ n,N−2   +   U n+1,N−3 x n − U n,N−3 x n   ≤ 2M N−1  i=2   γ n+1,i − γ n,i   +   U n+1,1 x n − U n,1 x n   ≤ 2M N−1  i=1   γ n+1,i − γ n,i   . (2.10) Meijuan Shang et al. 7 Substituting (2.10)into(2.8) yields that   W n+1 x n − W n x n   ≤ 2M   γ n+1,N − γ n,N   +2γ n+1,N M N−1  i=1   γ n+1,i − γ n,i   ≤ 2M N  i=1   γ n+1,i − γ n,i   . (2.11) It follows that   l n+1 − l n   −   x n − x n−1   ≤ α n+1 1 − β n+1   γf  x n+1  − Ay n+1   + α n 1 − β n   Ay n − γf  x n    +2M N  i=1   γ n+1,i − γ n,i   . (2.12) Observing conditions (C1), (C4) and takeing the limits as n →∞,weget limsup n→∞    l n+1 − l n   −   x n+1 − x n    ≤ 0. (2.13) We can o btain lim n→∞ l n − x n =0 easily by Lemma 1.3.Sincex n+1 − x n = (1 − β n )(l n − x n ), we have that (2.4) holds. Obser ving that x n+1 − y n = α n (γf(x n ) − Ay n ), we can easily get lim n→∞ y n − x n+1 =0, which implies that   y n − x n   ≤   x n − x n+1   +   x n+1 − y n   , (2.14) that is, lim n→∞   y n − x n   = 0. (2.15) On the other hand, we have   W n x n − x n   ≤   x n − y n   +   y n − W n x n   ≤   x n − y n   + β n   x n − W n x n   , (2.16) which implies (1 − β n )W n x n − x n ≤x n − y n . From condition (C3) and (2.15), we obtain   W n x n − x n   −→ 0. (2.17) Next, we claim that limsup n→∞  γf(q) − Aq,x n − q  ≤ 0, (2.18) where q = lim t→0 x t with x t being the fixed point of the contraction x → tγ f (x)+(I − tA)W n x.Then,x t solves the fixed point equation x t = tγ f (x t )+(I − tA)W n x t .Thus,we 8 Fixed Point Theory and Applications have x t − x n =(I − tA)(W n x t − x n )+t(γf(x t ) − Ax n ).ItfollowsfromLemma 1.2 that   x t − x n   2 =   (I − tA)  W n x t − x n  + t  γf  x t  − Ax n    2 ≤ (1 − γt) 2   W n x t − x n   2 +2t  γf  x t  − Ax n ,x t − x n  ≤  1 − 2γt +(γt) 2    x t − x n   2 + f n (t) +2t  γf  x t  − Ax t ,x t − x n  +2t  Ax t − Ax n ,x t − x n  , (2.19) where f n (t) =  2   x t − x n   +   x n − W n x n      x n − W n x n   −→ 0, as n −→ 0. (2.20) It follows that  Ax t − γf  x t  ,x t − x n  ≤ γt 2  Ax t − Ax n ,x t − x n  + 1 2t f n (t). (2.21) Let n →∞in (2.21) and note that (2.20)yields limsup n→∞  Ax t − γf  x t  ,x t − x n  ≤ t 2 M, (2.22) where M>0 is a constant such that M ≥ γAx t − Ax n ,x t − x n  for all t ∈ (0,1) and n ≥ 1. Tak ing t → 0from(2.22), we have limsup t→0 limsup n→∞ Ax t − γf(x t ),x t − x n ≤0. Since H is a Hilbert space, the order of limsup t→0 and limsup n→∞ is exchangeable, and hence (2.18)holds.NowfromLemma 1.2,wehave   x n+1 − q   2 =    I − α n A  y n − q  + α n  γf  x n  − Aq    2 ≤    I − α n A  y n − q    2 +2α n  γf  x n  − Aq,x n+1 − q  ≤  1 − α n γ  2   x n − q   2 + α n γα    x n − q   2 +   x n+1 − q   2  +2α n  γf(q) − Aq,x n+1 − q  , (2.23) which implies that   x n+1 − q   2 ≤  1 − α n γ  2 + α n γα 1 − α n γα   x n − q   2 + 2α n 1 − α n γα  γf(q) − Aq,x n+1 − q  ≤  1 − 2α n (γ − αγ) 1 − α n γα    x n − q   2 + 2α n (γ − αγ) 1 − α n γα  1 γ − αγ  γf(q) − Aq,x n+1 − q  + α n γ 2 2(γ − αγ) M  . (2.24) Meijuan Shang et al. 9 Put l n =2α n (γ − α n γ)/(1 − α n αγ)andt n = 1/(γ − αγ) γf(q) − Aq,x n+1 − q + α n γ 2 /(2(γ − αγ))M, that is,   x n+1 − q   2 ≤  1 − l n    x n − q   + l n t n . (2.25) It follows from conditions (C1), (C2), and (2.22) that lim n→∞ l n = 0,  ∞ n=1 l n =∞,and limsup n→∞ t n ≤ 0. Apply Lemma 1.4 to (2.25)toconcludethatx n → q. This completes the proof.  Remark 2.2. Our results relax the condition of Kim and Xu [1] imposed on control se- quences and also improve the results of Yao et al. [6] from one single mapping to a finite family of nonexpansive mappings, respectively. References [1] G. Marino and H K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. [2] H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. [3] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical So- ciety, vol. 4, no. 3, pp. 506–510, 1953. [4] S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. [5] T H. Kim and H K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005. [6] Y. Yao, R. Chen, and J C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” to appear in Nonlinear Analysis: Theory, Methods & Applications . [7] W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibil- ity problems,” Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1463–1471, 2000. [8] T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Ap- plications, vol. 305, no. 1, pp. 227–239, 2005. Meijuan Shang: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China; Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China Email address: meijuanshang@yahoo.com.cn Yongfu Su: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: suyongfu@tjpu.edu.cn Xiaolong Qin: Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea Email address: qxlxajh@163.com . “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. [5] T H. Kim and H K. Xu, Strong convergence. conditions for modified Mann iteration,” to appear in Nonlinear Analysis: Theory, Methods & Applications . [7] W. Takahashi and K. Shimoji, Convergence theorems for nonexpansive mappings and feasibil- ity. method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. [2] H. K. Xu, “An iterative approach to quadratic optimization,”