Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 350483, 10 pages doi:10.1155/2008/350483 Research Article An Implicit Iterative Scheme for an Infinite Countable Family of Asymptotically Nonexpansive Mappings in Banach Spaces Shenghua Wang, Lanxiang Yu, and Baohua Guo School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com Received 6 May 2008; Accepted 24 August 2008 Recommended by William Kirk Let K be a nonempty closed convex subset of a reflexive Banach space E with a weakly continuous dual mapping, and let {T i } ∞ i1 be an infinite countable family of asymptotically nonexpansive mappings with the sequence {k in } satisfying k in ≥ 1foreachi  1, 2, , n  1, 2, ,and lim n→∞ k in  1foreachi  1, 2, In this paper, we introduce a new implicit iterative scheme generated by {T i } ∞ i1 and prove that the scheme converges strongly to a common fixed point of {T i } ∞ i1 , which solves some certain variational inequality. Copyright q 2008 Shenghua Wang et al. This is an open access article distributed 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 E be a Banach space and let K be a nonempty closed convex subset of E.LetT : K→K be a mapping. Then T is called nonexpansive if Tx − Ty≤x − y 1.1 for all x, y ∈ K. T is called asymptotically nonexpansive if there exists a sequence {k n }⊂ 1, ∞ that converges to 1 as n→∞ such that   T n x − T n y   ≤ k n x − y 1.2 for all x, y ∈ K and all n ≥ 1. Obviously, a nonexpansive mapping is asymptotically nonexpansive. In 1, Goebel and Kirk originally introduced the concept of asymptotically nonexpansive mappings and proved that if E is a uniformly convex Banach space and K is a nonempty closed convex bounded subset of E, then every asymptotically nonexpansive 2 Fixed Point Theory and Applications self-mapping on K has a fixed point. After that, many authors began to study the convergence of the iterative scheme generated by asymptotically nonexpansive mappings 2–12. In 8, the authors introduced an iterative scheme generated by a finite family of asymptotically nonexpansive mappings: x n  α n x n−1   1 − α n  T l n 1 r n x n ,n≥ 1, 1.3 where {α n } is a sequence in 0, 1, {T i } N i1 : K→K are N asymptotically nonexpansive mappings, where K is a nonempty closed convex subset of a uniformly convex Banach space satisfying Opial’s condition 13, and where n  l n N  r n for some integers l n ≥ 0 and 1 ≤ r n ≤ N. Then the authors proved that if ∩ N i1 FT i  /  φ, then {x n } generated by 1.3 strongly converges to a common fixed point of {T i } N i1 . Let K be a nonempty closed convex subset of a uniformly convex Banach space E.Let S : K→K be a nonexpansive mapping and let T : K→K be an asymptotically nonexpansive mapping. In 10, the authors introduced the following modified Ishikawa iteration sequence with errors with respect to S and T: y n  a  n Sx n  b  n T n x n  c  n v n , x n1  a n Sx n  b n T n y n  c n u n , ∀n ≥ 1, 1.4 where {a  n }, {b  n }, {c  n } are three real numbers sequences in 0, 1 satisfying a  n  b  n  c  n  1, {a n }, {b n }, {c n } are also three real numbers sequences in 0, 1 satisfying a n  b n  c n  1, and {u n } and {v n } are given bounded sequences in K. Then the authors proved that the sequence {x n } generated by 1.4 strongly converges to a common fixed point of S and T if some certain conditions are satisfied. Let K be a nonempty closed convex subset of a Banach space E and let f : K→K be a contraction with efficient λ 0 <λ<1 such that   fx − fy   ≤ λx − y 1.5 for all x, y ∈ K. Shahzad and Udomene 9 studied the following implicit and explicit iterative schemes for an asymptotically nonexpansive mapping T with the sequence {k n } in a uniformly smooth Banach space: x n   1 − t n k n  f  x n   t n k n T n x n , x n1   1 − t n k n  f  x n   t n k n T n x n , 1.6 where {t n } is a sequence in 0, 1. They proved that the sequence {x n } converges strongly to the unique solution of some variational inequality if the sequence {t n } satisfies some certain conditions and the mapping T satisfies Tx n − x n →0asn→∞. Quite recently, Ceng et al. 12 introduced the following two implicit and explicit iterative schemes generated b y a finite family of asymptotically nonexpansive mappings Shenghua Wang et al. 3 {T i } N i1 with the same sequence {k n } in a reflexive Banach space with a weakly continuous duality map: x n   1 − 1 k n  x n  1 − t n k n f  x n   t n k n T n r n x n , x n1   1 − 1 k n  x n  1 − t n k n f  x n   t n k n T n r n x n , 1.7 where r n  n mod N and {t n } is a sequence in 0, 1. T hen they proved that if the control sequence {t n } satisfies some certain condition and T i x n −x n →0asn→∞ for each i  1, 2, ,N, then both schemes 1.7 strongly converge a common fixed point x ∗ of {T i } N i1 which solves the variational inequality  I − fx ∗ ,J  p − x ∗  ≥ 0,p∈ N  i1 F  T i  , 1.8 where FT i  denotes the set of fixed points of the mapping T i for each i  1, 2, ,N. Let E be a Banach space and let E ∗ be the dual space of E. Given a continuous strictly increasing function ϕ : R  →R  such that ϕ00 and lim t→∞ ϕt∞, we associate a possibly multivalued generalized duality map J ϕ : E→2 E ∗ , defined as J ϕ x  x ∗ ∈ E ∗ : x ∗ xxϕ  x  , x ∗   ϕ  x  1.9 for every x ∈ E. We call the function ϕ a gauge. If ϕtt for all t ≥ 0, then we call J ϕ a normalized duality mapping and write it as J. A Banach space E is said to have a weakly continuous generalized duality map if there exists a continuous strictly increasing function ϕ : R  →R  such that ϕ00, lim t→∞ ϕt∞, and J ϕ is single valued and sequentially continuous from E with the weak topology to E ∗ with the weak ∗ topology. For instance, every l p -space 1 <p<∞ has a weakly continuous generalized duality map for ϕtt p−1 . For each t ≥ 0, let Φt  t 0 ϕxdx. The following property may be seen in many literatures. Property 1.1. Let E be a real Banach space and let J ϕ be the duality map associated with the gauge ϕ. Then for all x, y ∈ E and jx  y ∈ J ϕ x  y one holds Φ  x  y  ≤ Φ  x    y, jx  y  . 1.10 One also holds x  y 2 ≤x 2  2  y, jx  y  1.11 for all x, y ∈ E and jx  y ∈ Jx  y. 4 Fixed Point Theory and Applications Lemma 1.2 see 14. Let E be a Banach space satisfying a weakly continuous duality map and let K be a nonempty closed convex subset of E. Let T : K→K be an asymptotically nonexpansive mapping with fixed point. Then I − T is demiclosed at zero. 2. Strong convergence results In this section, let E be a reflexive Banach space with a weakly continuous duality map J ϕ , where ϕ is a gauge and let K be a nonempty closed convex subset of E.Let{T i } ∞ i1 : K→K be an infinite countable family of asymptotically nonexpansive mappings such that   T n i x − T n i y   ≤ k in x − y 2.1 for all x, y ∈ K, where the sequence {k in }⊂1, ∞ and lim n→∞ k in  1 for each i  1, 2, For each n  1, 2, ,let b  n  sup{k in | i  1, 2, } and assume sup  b  n | n  1, 2,  < ∞, lim n→∞ b  n  b<∞. 2.2 Taking b n  max{b  n ,b} for each n  1, 2, ,obviously, we have lim n→∞ b n  b ≥ 1, b   sup  b n | n  1, 2,  < ∞. 2.3 Moreover, the following inequality   T n i x − T n i y   ≤ b n x − y 2.4 holds for all x, y ∈ K and each i  1, 2 Take an integer r>1 arbitrarily. For each n ≥ 1, define the mapping S ni : K→K by S ni  T n−1ri 2.5 for each i  1, 2, ,r,that is, S 11  T 1 , , S 1r  T r ,S 21  T r1 , ,S 2r  T 2r , 2.6 Shenghua Wang et al. 5 For each i  1, 2, ,r,let{α ni }⊂0, 1 be a sequence real numbers. For each n ≥ 1, define the mapping W n of K into itself by W n  U nr  α nr S n nr U nr−1   1 − α nr  I, 2.7 where U n1  α n1 S n n1   1 − α n1  I, U n2  α n2 S n n2 U n1   1 − α n2  I, . . . U nr−1  α nr−1 S n nr−1 U nr−2   1 − α nr−1  I. 2.8 We call W n a W-mapping generated by S n1 ,S n2 , ,S nr and α n1 ,α n2 , ,α nr. Let f : K→K be a λ-contraction with 0 <λ<1/b  r . Take a sequence of real numbers{t n }⊂0,b such that lim n→∞ t n  0,t n < b1 − b r n λ 1 − λb r n ,n≥ 1. 2.9 Note that since λ<1/b  r , one has 0 <b1 − b r n λ/1 − λb r n ≤ b. Therefore, the sequence {t n } can be taken easily to satisfy the condition 2.9, for example, t n 1/nb1− b r n λ/1−λb r n . Then, we introduce an implicit iterative scheme x n   1 − b b r1 n  x n  b − t n b r1 n f  W n x n   t n b r1 n W n x n ,n≥ 1. 2.10 By using the following lemmas, we will prove that the implicit scheme 2.10 is well defined. Lemma 2.1. Let {T i } ∞ i1 : K→K be an infinite countable family of asymptotically nonexpansive mappings with the sequences {k in } and let W n be a W-mapping generated by 2.7 for each n  1, 2, If ∩ ∞ i1 FT i  /  φ,then∩ ∞ i1 FT i  ⊂ FW n  for each n  1, 2, Proof. The conclusion is obtained directly from the definition of W n . Lemma 2.2. Let {T i } ∞ i1 : K→K with the sequences {k in } and let W n be the W-mapping generated by 2.7 for each n  1, 2, Then one holds   W n x − W n y   ≤ b r n x − y 2.11 for all n ≥ 1 and all x, y ∈ K. 6 Fixed Point Theory and Applications Proof. For any x, y ∈ K all n ≥ 1, we first see noting that b n ≥ 1   U n1 x − U n1 y       α n1 S n n1   1 − α n1  I  x −  α n1 S n n1   1 − α n1  I  y   ≤ α n1   S n n1 x − S n n1 y     1 − α n1  x − y  α n1   T n n−1r1 x − T n n−1r1 y     1 − α n1  x − y ≤ α n1 k n−1r1n x − y   1 − α n1  x − y ≤ α n1 b n x − y   1 − α n1  x − y ≤ α n1 b n x − y   1 − α n1  b n x − y  b n x − y,   U n2 x − U n2 y       α n2 S n n2 U n1   1 − α n2  I  x −  α n2 S n n2 U n1   1 − α n2  I  y   ≤ α n2   S n n2 U n1 x − S n n2 U n1 y     1 − α n2  x − y  α n2   T n n−1r2 U n1 x − T n n−1r2 U n1 y     1 − α n2  x − y ≤ α n2 k n−1r2n   U n1 x − U n1 y     1 − α n2  x − y ≤ α n2 b n   U n1 x − U n1 y     1 − α n2  x − y ≤ α n2 b 2 n x − y   1 − α n1  b 2 n x − y  b 2 n x − y. 2.12 Similarly, for each i  3, ,r − 1, we have U ni x − U ni y≤b i n x − y. 2.13 Hence,   W n x − W n y       α nr S n nr U nr−1   1 − α nr  I  x −  α nr S n nr U nr−1   1 − α nr  I  y   ≤ α nr   S n nr U nr−1 x − S n nr U nr−1 y     1 − α nr  x − y ≤ b r n x − y. 2.14 This completes the proof. Now we prove that the implicit scheme 2.10 is well defined. Since 0 <t n <b1 − b r n λ/1 − λb r n ,weobtain 0 < 1 − b b r1 n  b − t n b n λ  t n b n < 1. 2.15 Hence, the mapping x → Tx :  1 − b b r1 n  x  b − t n b r1 n f  W n x   t n b r1 n W n x 2.16 Shenghua Wang et al. 7 is a contraction on K. In fact, to see this, taking any x,y ∈ K,byLemma 2.2 we have Tx − Ty       1 − b b r1 n  x − y b − t n b r1 n  fW n x − f  W n y   t n b r1 n  W n x − W n y      ≤  1 − b b r1 n  x − y  b − t n λb r n b r1 n x − y  t n b r1 n b r n x − y   1 − b b r1 n  b − t n b n λ  t n b n  x − y ≤x − y, 2.17 which implies that the implicit scheme 2.10  is well defined. For the implicit scheme 2.10, we have strong convergence as follows. Theorem 2.3. Assume 2.9, FT∩ ∞ i1 FT i  /  φ and lim n→∞ x n −T i x n   0 for each i  1, 2, Then {x n } converges strongly to a common fixed point x ∈ FT,wherex solves the variational inequality  I − fx, Jp − x  ≥ 0,p∈ FT. 2.18 Proof. First, we prove that {x n } is bounded. By using Property 1.1, Lemmas 2.1, 2.2, for any z ∈ FT, we have noting 0 < 1 − b/b r1 n b − t n /b n λ  t n /b n < 1 x n − z 2       1 − b b r1 n   x n − z   b − t n b r1 n  f  W n x n  − fz   t n b r1 n  W n x n − z   b − t n b r1 n  fz − z      2 ≤      1 − b b r1 n   x n − z   b − t n b r1 n  f  W n x n  − fz   t n b r1 n  W n x n − z      2  2b − t n  b r1 n fz − z, jx n − z  ≤  1 − b b r1 n    x n − z    b − t n b r1 n   f  W n x n  − f  W n z     t n b r1 n   W n x n − W n z    2  2b − t n  b r1 n  fz − z, j  x n − z  ≤  1 − b b r1 n  b − t n λ b n  t n b n  2   x n − z   2  2b − t n  b r1 n  fz − z, j  x n − z  ≤  1 − b b r1 n  b − t n λ b n  t n b n    x n − z   2  2b − t n  b r1 n  fz − z, j  x n − z    1 − η n    x n − z   2  2b − t n  b r1 n  fz − z, j  x n − z  , 2.19 8 Fixed Point Theory and Applications where η n  b b r1 n − b − t n b n λ − t n b n > 0. 2.20 It follows from 2.19 that   x n − z   2 ≤ 2b − t n  η n b r1 n  fz − z, jx n − z  . 2.21 Since lim n→∞ b n  b, lim n→∞ t n  0, we have lim n→∞ b − t n η n b r1 n  1 1 − λb r . 2.22 Hence, {x n } is bounded. Now we prove that {x n } strongly converges to a common fixed point x ∈ FT.Tosee this, we assume that x is a weak limit point of {x n } and a subsequence {x n j } of {x n } converges weakly to x. Then by the assumption of the theorem and Lemma 1.2, we have x ∈ FT i  for every i  1, 2, In 2.21, replacing x n with x n j and z with x, respectively, and then taking the limit as j→∞, we obtain by the weak continuity of the duality map J lim j→∞ x n j − x  0. 2.23 Therefore, x n j →x. We further show that x solves the variational inequality  I − fx, Jp − x  ≥ 0,p∈ FT. 2.24 To see this result, taking any p ∈ FT, then by using Property 1.1, Lemmas 2.1 and 2.2 we compute Φ    x n − p    Φ       1 − b b r1 n   x n − p   b − t n b r1 n  x n − p   t n b r1 n  W n x n − p   b − t n b r1 n  f  W n x n  − x n       ≤ Φ       1 − t n b r1 n   x n − p   t n b r1 n  W n x n − p        b − t n b r1 n  f  W n x n  − x n ,J ϕ  x n − p  ≤  1 − t n b r1 n  t n  Φ    x n − p     b − t n b r1 n  fW n x n  − x n ,J ϕ  x n − p  , 2.25 Shenghua Wang et al. 9 which implies that  x n − f  W n x n  ,J ϕ  x n − p  ≤ b r1 n − 1t n b − t n Φ  x n − p  . 2.26 Now in 2.26, replacing x n with x n j and noting lim n→∞ b n  b and lim n→∞ t n  0, we obtain  x − fx,J ϕ x − p   lim j→∞  x n j − f  W n j x n j  ,J ϕ  x n j − p  ≤ lim sup j→∞ b r1 n j − 1t n j b − t n j Φ    x n j − p     0, 2.27 which implies that x is a solution to 2.24. Finally, we prove that the sequence {x n } strongly converges to x.Itsuffices t o prove that the variational inequality 2.24 can have only one solution. To see this, assuming that both u ∈ FT and v ∈ FT are solutions to 2.24, we have  I − fu, Ju − v  ≤ 0,  I − fv, Jv − u  ≤ 0. 2.28 Adding them yields  I − fu − I − fv, Ju − v  ≤ 0. 2.29 However, since f is a λ-contraction, we have that 1 − λ u − v 2 ≤  I − fu − I − fv, Ju − v  , 2.30 which implies that u  v. This completes the proof. Remark 2.4. In Theorem 2.3, the condition that lim n→∞ T i x n − x n   0 for each i  1, 2, is necessary see 9, 12. This theorem shows that if for each n  1, 2, , the supremum of the sequence {k in }, that is, sup{k in | i  1, 2, }, is finite and the limit of the sequence sup {k in | i  1, 2, } ∞ n1 exists, then by choosing the contraction constant λ and the control sequence {t n } we can obtain the common fixed point of {T i } ∞ i1 . Corollary 2.5. Let {T i } N i1 K→K be a finite family of asymptotically nonexpansive mappings with the sequences {k in } and let W n be a W-mapping generated by T 1 ,T 2 , ,T N and α n1 ,α n2 , ,α nN for each n  1, 2, Let the sequence {t n }⊂0, 1 and satisfy t n < 1−k N n λ/1−λk N n and t n →0,where k n  max{k 1n ,k 2n , ,k Nn } for e ach n  1, 2, Assume that k  sup{k n | n  1, 2, } < ∞.Let f be a contraction with λ0 <λ<1/k N . Consider the implicit iterative scheme x n   1 − 1 k N1 n  x n  1 − t n k N1 n f  W n x n   t n k N1 n W n x n . 2.31 10 Fixed Point Theory and Applications If {T i } N i1 satisfy the condition ∩ N i1 FT i  /  φ and T i x n − x n →0 as n→∞ for each i  1, 2, ,N, then {x n } converges strongly to a common fixed point x ∈∩ N i1 FT i ,wherex solves the variational inequality  I − fx, Jp − x  ≥ 0,p∈ N  i1 F  T i  . 2.32 Proof. In Theorem 2.3, take b n  k n ,b lim n→∞ k n  1,b   k, and r  N. 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Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. 14 T C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994. . for an In nite Countable Family of Asymptotically Nonexpansive Mappings in Banach Spaces Shenghua Wang, Lanxiang Yu, and Baohua Guo School of Mathematics and Physics, North China Electric Power. subset of a reflexive Banach space E with a weakly continuous dual mapping, and let {T i } ∞ i1 be an in nite countable family of asymptotically nonexpansive mappings with the sequence {k in }. 1.2 for all x, y ∈ K and all n ≥ 1. Obviously, a nonexpansive mapping is asymptotically nonexpansive. In 1, Goebel and Kirk originally introduced the concept of asymptotically nonexpansive mappings