Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007, Article ID 29091, 10 pages doi:10.1155/2007/29091 Research Article Iteration Scheme with Perturbed Mapping for Common Fixed Points of a Finite Family of Nonexpansive Mappings Yeong-Cheng Liou, Yonghong Yao, and Rudong Chen Received 17 December 2006; Revised 6 February 2007; Accepted 6 February 2007 Recommended by H ´ el ` ene Frankowska We propose a n iteration scheme with perturbed mapping for a pproximation of common fixed points of a finite family of nonexpansive mappings {T i } N i =1 . We show that the pro- posed iteration scheme converges to the common fixed point x ∗ ∈ N i =1 Fix(T i ) which solves some variational inequality. 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 and preliminaries Let H be a real Hilbert space with inner product ·,· and norm ·, respectively. A mapping T with domain D(T) and range R(T)inH is called nonexpansive if Tx− Ty≤x − y, ∀x, y ∈ D(T). (1.1) Let {T i } N i =1 be a finite family of nonexpansive self-maps of H. Denote the common fi xed points set of {T i } N i =1 by N i =1 Fix(T i ). Let F : H → H be a mapping such that for some constants k,η>0, F is k-Lipschitzian and η-strongly monotone. Let {α n } ∞ n=1 ⊂ (0,1), {λ n } ∞ n=1 ⊂ [0,1) and take a fixed number μ ∈ (0,2η/k 2 ). The iterative schemes concern- ing nonlinear operators have been studied extensively by many authors, you may refer to [1–12]. Especially, in [13], Zeng and Yao introduced the following implicit iteration process with perturbed mapping F. For an arbitrary initial point x 0 ∈ H, the sequence {x n } ∞ n=1 is generated a s follows: x n = α n x n−1 + 1 − α n T n x n − λ n μF T n x n , n ≥ 1, (1.2) where T n := T nmodN . 2 Fixed Point Theory and Applications Using this iteration process, they proved the following weak and strong convergence theorems for nonexpansive mappings in Hilbert spaces. Theorem 1.1 (see [13]). Let H be a real Hilber t space and let F : H → H be a mapping such that for some constants k,η>0, F is k-Lipschitzain vcommentand η-strongly mono- tone. Let {T i } N i =1 be N nonexpansive self-mappings of H such that N i =1 Fix(T i ) =∅.Let μ ∈ (0,2η/k 2 ) and x 0 ∈ H.Let{λ n } ∞ n=1 ⊂ [0,1) and {α n } ∞ n=1 ⊂ (0,1) satisfying the condi- tions ∞ n=1 λ n < ∞ and α ≤ α n ≤ β, n ≥ 1,forsomeα,β ∈ (0,1). Then the sequence {x n } ∞ n=1 defined by (1.2) converges weakly to a common fixed point of the mappings {T i } N i =1 . Theorem 1.2 (see [13]). Let H be a real Hilbert space and let F : H → H be a mapping such that for some constants k,η>0, F is k-Lipschitzain and η-strongly monotone. Let {T i } N i =1 be N nonexpansive self-mappings of H such that N i =1 Fix(T i ) =∅.Letμ ∈ (0,2η/k 2 ) and x 0 ∈ H.Let{λ n } ∞ n=1 ⊂ [0,1) and {α n } ∞ n=1 ⊂ (0,1) satis fying the conditions ∞ n=1 λ n < ∞ and α ≤ α n ≤ β, n ≥ 1,forsomeα,β ∈ (0,1). Then the sequence {x n } ∞ n=1 defined by (1.2) converges strongly to a common fixed point of the mappings {T i } N i =1 if and only if liminf n→∞ d x n , N i=1 Fix T i = 0. (1.3) Very recently, Wang [14] considered an explicit iterative scheme with perturbed map- ping F and obtained the following result. Theorem 1.3. Let H be a Hilbert space, let T : H → H be a nonexpansive mapping with F(T) =∅, and let F : H → H be an η-strongly monotone and k-Lipschitzian mapping. For any given x 0 ∈ H, {x n } is defined by x n+1 = α n x n + 1 − α n T λ n+1 x n , n ≥ 0, (1.4) where T λ n+1 x n = Tx n − λ n+1 μF(Tx n ), {α n } and {λ n }⊂[0,1) satisfy the following condi- tions: (1) α ≤ α n ≤ β for some α,β ∈ (0,1); (2) ∞ n=1 λ n < ∞; (3) 0 <μ<2η/k 2 . Then (1) {x n } converges weakly to a fixed point of T, (2) {x n } convergesstronglytoafixedpointofT if and only if liminf n→∞ d x n ,F(T) = 0. (1.5) This natural ly brings us the following questions. Questions 1.4. Let T i : H → H (i = 1,2, ,N) be a finite family of nonexpansive mappings and F is k-Lipschitzain and η-strongly monotone. (i) Could we construct an explicit iterative algorithm to approximate the common fixed points of the mappings {T i } N i =1 ? (ii) Could we remove the assumption (2) imposed on the sequence {x n }? Yeon g-Che ng L io u et a l. 3 Motivated and inspired by the above research work of Zeng and Yao [13] and Wang [14], in this paper, we will propose a new explicit iteration scheme with perturbed map- ping for approximation of common fixed points of a finite family of nonexpansive self- mappings of H. We will establish strong convergence theorem for this explicit iteration scheme. To be more specific, let α n1 ,α n2 , , α nN ∈ (0,1], n ∈ N. Given the mappings T 1 ,T 2 , , T N , following [15], one can define, for each n,mappingsU n1 ,U n2 , , U nN by U n1 = α n1 T 1 + 1 − α n1 I, U n2 = α n2 T 2 U n1 + 1 − α n2 I, . . . U n,N−1 = α n,N−1 T N−1 U n,N−2 + 1 − α n,N−1 I, W n := U nN = α nN T N U n,N−1 + 1 − α nN I. (1.6) Such a mapping W n is called the W-mapping generated by T 1 , , T N and α n1 , , α nN . First we introduce the following explicit iteration scheme with perturbed mapping F. For an arbitrary initial point x 0 ∈ H, the sequence {x n } ∞ n=1 is generated iteratively by x n+1 = βx n +(1− β) W n x n − λ n μF W n x n , n ≥ 0, (1.7) where {λ n } is a sequence in (0,1), β is a constant in (0,1), F is k-Lipschitzian and η- strongly monotone, and W n is the W-mapping defined by (1.6). We have the following crucial conclusion concerning W n . Proposition 1.5 (see [15]). Let C be a nonempt y closed convex subset of a Banach space E.LetT 1 ,T 2 , , T N be nonexpansive mappings of C into itself such that N i =1 Fix(T i ) is nonempty, and let α n1 ,α n2 , , α nN be real numbers such that 0 <α ni ≤ b<1 for any i ∈ N. For any n ∈ N,letW n be the W-mapping of C into itself generated by T N ,T N−1 , , T 1 and α nN ,α n,N−1 , , α n1 . Then W n is nonexpansive. Further, if E is strictly convex, then Fix(W n ) = N i =1 Fix(T i ). Now we recall some basic notations. Let T : H → H be nonexpansive mapping and F : H → H be a mapping such that for some constants k,η>0, F is k-Lipschitzian and η-strongly monotone; that is, F satisfies the following conditions: Fx− Fy≤kx − y, ∀x, y ∈ H, Fx− Fy,x − y≥ηx − y 2 , ∀x, y ∈ H, (1.8) respectively. We may assume, without loss of generality, that η ∈ (0,1) and k ∈ [1,∞). Under these conditions, it is well know n that the variational inequality problem—find x ∗ ∈ N i =1 Fix(T i )suchthat VI F, N i=1 Fix T i : F x ∗ ,x − x ∗ ≥ 0, ∀x ∈ N i=1 Fix T i , (1.9) 4 Fixed Point Theory and Applications has a unique solution x ∗ ∈ N i =1 Fix(T i ). [Note: the unique existence of the solution x ∗ ∈ N i =1 Fix(T i ) is guaranteed automatically because F is k-Lipschitzian and η-strongly monotone over N i =1 Fix(T i ).] For any given numbers λ ∈ [0,1) and μ ∈ (0,2η/k 2 ), we define the mapping T λ : H → H by T λ x := Tx− λμF(Tx), ∀x ∈ H. (1.10) Concerning the corresponding result of T λ x, you can find it in [16]. Lemma 1.6 (see [16]). If 0 ≤ λ<1 and 0 <μ<2η/k 2 , then there holds for T λ : H → H, T λ x − T λ y ≤ (1 − λτ)x − y, ∀x, y ∈ H, (1.11) where τ = 1 − 1 − μ(2η − μk 2 ) ∈ (0,1). Next, let us state four preliminary results which will be needed in the sequel. Lemma 1.7 is very interesting and important, you may find it in [17], the original prove can be found in [18]. Lemmas 1.8 and 1.9 well-known demiclosedness principle and subdiffer- ential inequality, respectively. Lemma 1.10 is basic and important result, please consult it in [19]. Lemma 1.7 (see [17]). Le t {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. (1.12) Suppose x n+1 = 1 − β n y n + β n x n , (1.13) 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 1.8 (see [20]). Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H.IfT has a fixed point, the n I − T is demiclosed. That is, when- ever {x n } isasequenceinC weakly converging to some x ∈ C and the sequence {(I − T)x n } strongly converges to some y, it follows that (I − T)x = y.Here,I is the identity operator of H. Lemma 1.9 (see [21]). x + y 2 ≤x 2 +2y,x + y for all x, y ∈ H. Lemma 1.10 (see [19]). Assume that {a n } is a sequence of nonnegative real numbers such that a n+1 ≤ 1 − γ n a n + δ n , (1.15) Yeon g-Che ng L io u et a l. 5 where {γ n } isasequencein(0,1) and {δ n } isasequencesuchthat (1) ∞ n=1 γ n =∞, (2) limsup n→∞ δ n /γ n ≤ 0 or ∞ n=1 |δ n | < ∞. Then lim n→∞ a n = 0. 2. Main result Now we state and prove our main result. Theorem 2.1. Let H be a real Hilbert space and let F : H → H be a k-Lipschitzian and η-strongly monotone mapping. Let {T i } N i =1 be a finite family of nonexpansive self-mappings of H such that N i =1 Fix(T i ) =∅.Letμ ∈ (0,2η/k 2 ). Suppose the sequences {α n,i } N i =1 sat- isfy lim n→∞ (α n,i − α n−1,i ) = 0,foralli = 1,2, ,N.If{λ n } ∞ n=1 ⊂ [0,1) satisfy the following conditions: (i) lim n→∞ λ n = 0; (ii) ∞ n=0 λ n =∞, then the sequence {x n } ∞ n=1 defined by (1.7) converges strongly to a common fixed point x ∗ ∈ N i =1 Fix(T i ) which solves the variational inequality (1.9). Proof. Let x ∗ be an arbitrary element of N i =1 Fix(T i ). Observe that x n+1 − x ∗ = βx n +(1− β)W λ n n x n − x ∗ ≤ β x n − x ∗ +(1− β) W λ n n x n − x ∗ , (2.1) where W λ n n x := W n x − λ n μF(W n x). Note that W λ n n x ∗ = x ∗ − λ n μF x ∗ . (2.2) Utilizing Lemma 1.6,wehave W λ n n x n − x ∗ = W λ n n x n − W λ n n x ∗ + W λ n n x ∗ − x ∗ ≤ W λ n n x n − W λ n n x ∗ + W λ n n x ∗ − x ∗ ≤ 1 − λ n τ x n − x ∗ + λ n μ F x ∗ . (2.3) From (2.1)and(2.3), we have x n+1 − x ∗ ≤ β +(1− β) 1 − λ n τ x n − x ∗ +(1− β)λ n μ F x ∗ = 1 − (1 − β)λ n τ x n − x ∗ +(1− β)λ n μ F x ∗ ≤ max x 0 − x ∗ , μ τ F x ∗ . (2.4) Hence, {x n } is bounded. We also can obtain that {W n x n }, {T i U n, j x n }(i = 1, ,N; j = 1, , N), and {F(W n x n )} are all bounded. We w ill use M to denote the possible different constants appearing in the following reasoning. 6 Fixed Point Theory and Applications We note that W λ n+1 n+1 x n+1 − W λ n n x n = W n+1 x n+1 − W n x n − λ n+1 μF W n+1 x n+1 + λ n μF W n x n ≤ W n+1 x n+1 − W n x n + λ n+1 μ F W n+1 x n+1 + λ n μ F W n x n ≤ W n+1 x n+1 − W n+1 x n + W n+1 x n − W n x n + λ n+1 + λ n M ≤ x n+1 − x n + W n+1 x n − W n x n + λ n+1 + λ n M. (2.5) From (1.6), since T N and U n,N are nonexpansive, W n+1 x n − W n x n = α n+1,N T N U n+1,N−1 x n + 1 − α n+1,N x n − α n,N T N U n,N−1 x n − 1 − α n,N x n ≤ α n+1,N T N U n+1,N−1 x n − α n,N T N U n,N−1 x n + α n+1,N − α n,N x n ≤ α n+1,N T N U n+1,N−1 x n − T N U n,N−1 x n + α n+1,N − α n,N T N U n,N−1 x n + α n+1,N − α n,N x n ≤ α n+1,N U n+1,N−1 x n − U n,N−1 x n +2M α n+1,N − α n,N . (2.6) Again, from (1.6), we have U n+1,N−1 x n − U n,N−1 x n = α n+1,N−1 T N−1 U n+1,N−2 x n + 1 − α n+1,N−1 x n − α n,N−1 T N−1 U n,N−2 x n − 1 − α n,N−1 x n ≤ α n+1,N−1 T N−1 U n+1,N−2 x n − α n,N−1 T N−1 U n,N−2 x n + α n+1,N−1 − α n,N−1 x n ≤ α n+1,N−1 − α n,N−1 x n + α n+1,N−1 − α n,N−1 M + α n+1,N−1 T N−1 U n+1,N−2 x n − T N−1 U n,N−2 x n ≤ 2M α n+1,N−1 − α n,N−1 + α n+1,N−1 U n+1,N−2 x n − U n,N−2 x n ≤ 2M α n+1,N−1 − α n,N−1 + U n+1,N−2 x n − U n,N−2 x n . (2.7) Yeon g-Che ng L io u et a l. 7 Therefore, we have U n+1,N−1 x n − U n,N−1 x n ≤ 2M α n+1,N−1 − α n,N−1 +2M α n+1,N−2 − α n,N−2 + U n+1,N−3 x n − U n,N−3 x n ≤ 2M N−1 i=2 α n+1,i − α n,i + U n+1,1 x n − U n,1 x n = α n+1,1 T 1 x n + 1 − α n+1,1 x n − α n,1 T 1 x n − 1 − α n,1 x n +2M N−1 i=2 α n+1,i − α n,i , (2.8) then U n+1,N−1 x n − U n,N−1 x n ≤ α n+1,1 − α n,1 x n + α n+1,1 T 1 x n − α n,1 T 1 x n +2M N−1 i=2 α n+1,i − α n,i ≤ 2M N−1 i=1 α n+1,i − α n,i . (2.9) Substituting (2.9)into(2.6), we have W n+1 x n − W n x n ≤ 2M α n+1,N − α n,N +2α n+1,N M N−1 i=1 α n+1,i − α n,i ≤ 2M N i=1 α n+1,i − α n,i . (2.10) Substituting (2.10)into(2.5), we have W λ n+1 n+1 x n+1 − W λ n n x n ≤ x n+1 − x n +2M N i=1 α n+1,i − α n,i + λ n+1 + λ n M, (2.11) which implies that limsup n→∞ W λ n+1 n+1 x n+1 − W λ n n x n − x n+1 − x n ≤ 0. (2.12) We note that x n+1 = βx n +(1− β)W λ n n x n and 0 <β<1, then from Lemma 1.7 and (2.12), we hav e lim n→∞ W λ n n x n − x n =0. It follows that lim n→∞ x n+1 − x n = lim n→∞ (1 − β) W λ n n x n − x n = 0. (2.13) On the other hand, x n − W n x n ≤ x n+1 − x n + x n+1 − W n x n ≤ x n+1 − x n + β x n − W n x n +(1− β)λ n μ F W n x n , (2.14) 8 Fixed Point Theory and Applications that is, x n − W n x n ≤ 1 1 − β x n+1 − x n + λ n μ F W n x n , (2.15) this together with (i) and (2.13)imply lim n→∞ x n − W n x n = 0. (2.16) We next show that limsup n→∞ − F x ∗ ,x n − x ∗ ≤ 0. (2.17) To prove this, we pick a subsequence {x n i } of {x n } such that limsup n→∞ − F x ∗ ,x n − x ∗ = lim i→∞ − F x ∗ ,x n i − x ∗ . (2.18) Without loss of generality, we may further assume that x n i → z weakly for some z ∈ H. By Lemma 1.8 and (2.16), we have z ∈ Fix W n , (2.19) this together with Proposition 1.5 imply that z ∈ N i=1 Fix T i . (2.20) Since x ∗ solves the v ariational inequality (1.9), then we obtain limsup n→∞ − F x ∗ ,x n − x ∗ = − F x ∗ ,z − x ∗ ≤ 0. (2.21) Finally, we show that x n → x ∗ . Indeed, from Lemma 1.9,wehave x n+1 − x ∗ 2 = β x n − x ∗ +(1− β) W λ n n x n − W λ n n x ∗ +(1− β) W λ n n x ∗ − x ∗ 2 ≤ β x n − x ∗ +(1− β) W λ n n x n − W λ n n x ∗ 2 +2(1− β) W λ n n x ∗ − x ∗ ,x n+1 − x ∗ ≤ β x n − x ∗ +(1− β) W λ n n x n − W λ n n x ∗ 2 +2(1− β)λ n μ − F x ∗ ,x n+1 − x ∗ ≤ β x n − x ∗ +(1− β) 1 − λ n τ x n − x ∗ 2 +2(1− β)λ n μ − F x ∗ ,x n+1 − x ∗ ≤ 1 − (1 − β)τλ n x n − x ∗ 2 +(1− β)τλ n 2 μ τ − F x ∗ ,x n+1 − x ∗ . (2.22) Now applying Lemma 1.10 and (2.21)to(2.22) concludes that x n → x ∗ (n →∞). This completes the proof. Yeon g-Che ng L io u et a l. 9 Acknowledgments The authors thank the referees for their suggestions and comments which led to the present version. The research was partially supposed by Grant NSC 95-2221-E-230-017. References [1] L C. Zeng and J C. 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[20] K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. [21] G. Marino and H K. Xu, “Convergence of generalized proximal point algorithms,” Communi- cations on Pure and Applied Analysis, vol. 3, no. 4, pp. 791–808, 2004. Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan Email address: simplex liou@hotmail.com Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianji 300160, China Email address: yuyanrong@tjpu.edu.cn Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianji 300160, China Email address: chenrd@tjpu.edu.cn . H ´ el ` ene Frankowska We propose a n iteration scheme with perturbed mapping for a pproximation of common fixed points of a finite family of nonexpansive mappings {T i } N i =1 . We show that the pro- posed. perturbed mapping for common fixed points of a finite family of nonexpansive mappings,” Nonlinear Analysis, vol. 64, no. 11, pp. 2507–2515, 2006. [14] L. Wang, “An iteration method for nonexpansive mappings. Cubiotti, and J C. Yao, “Approximation of common fixed points of families of nonexpansive mappings,” to appear in Taiwanese Journal of Mathematics. [10] L. C. Ceng, P. Cubiotti, and J C. Yao, “Strong