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≤kx − y, ∀x, y ∈ H, Fx− Fy,x − y≥ηx − 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 +2y,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. Yao, “Stability of iterative procedures with errors for approximating com- mon fixed points of a couple of q-contractive-like mappings in Banach spaces,” Journal of Math- ematical Analysis and Applications, vol. 321, no. 2, pp. 661–674, 2006. [2] Y C. Lin, N C. Wong, and J C. Yao, “Strong convergence theorems of Ishikawa iteration pro- cess with errors for fixed points of Lipschitz continuous mappings in Banach spaces,” Tai wa ne se Journal of Mathematics, vol. 10, no. 2, pp. 543–552, 2006. [3] L C. Zeng, N C. Wong, and J C. Yao, “Strong convergence theorems for strictly pseudocon- tractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 837–849, 2006. [4] Y C. Lin, “Three-step iterative convergence theorems with er rors in Banach spaces,” Ta iwa ne se Journal of Mathematics, vol. 10, no. 1, pp. 75–86, 2006. [5] L C. Zeng, G. M. Lee, and N C. Wong, “Ishikawa iteration with errors for approximating fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 1, pp. 87–99, 2006. [6] S. Schaible, J C. Yao, and L C. Zeng, “A proximal method for pseudomonotone type variational-like inequalities,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 497–513, 2006. [7] L C. Zeng, L. J. Lin, and J C. Yao, “Auxiliary problem method for mixed variational-like in- equalities,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 515–529, 2006. [8] L C. Zeng, N C. Wong, and J C. Yao, “On the convergence analysis of modified hybrid steepest-descent methods with variable par ameters for variational inequalities,” to appear in Journal of Optimization Theory and Applications. [9] L. C. Ceng, P. 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 convergence theorems for finitely many nonex- pansive mappings and applications,” to appear in Nonlinear Analysis. [11] L C. Zeng, S. Y. Wu, and J C. Yao, “Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems,” Taiwanese Journal of Mathematics, vol. 10, no. 6, pp. 1497–1514, 2006. [12] L. C. Ceng, C. Lee, and J C. Yao, “Strong weak convergence theorems of implicit hybrid steepest-descent methods for var iational inequalities,” to appear in Taiwanese Journal of Mathe- matics. [13] L C. Zeng and J C. Yao, “Implicit iteration scheme with 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 in Hilbert spaces,” Fixed Point The- ory and Applications, vol. 2007, Article ID 28619, 8 pages, 2007. [15] 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. [16] H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational in- equalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. 10 Fixed Point Theory and Applications [17] 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. [18] T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in gen- eral Banach spaces,” Fixed Point Theory and Applications, vol. 2005, no. 1, pp. 103–123, 2005. [19] H K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathe- matical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. [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