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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 167535, 14 pages doi:10.1155/2008/167535 Research Article Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces Jong Soo Jung Department of Mathematics, Dong-A University, Busan 604-714, South Korea Correspondence should be addressed to Jong Soo Jung, jungjs@mail.donga.ac.kr Received 13 January 2008; Revised 5 April 2008; Accepted 3 May 2008 Recommended by Mohammed Khamsi Let E be a reflexive Banach space with a uniformly G ˆ ateaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f : C → C a contractive mapping or a weakly contractive mapping,andT : C → C nonexpansive mapping with the fixed point set FT /  ∅.Let{x n } be generated by a new composite iterative scheme: y n  λ n fx n 1−λ n Tx n , x n1 1−β n y n β n Ty n , n ≥ 0.Itisprovedthat{x n } converges strongly to a point in FT, which is a solution of certain variational inequality provided that the sequence {λ n }⊂0, 1 satisfies lim n→∞ λ n  0and  ∞ n1 λ n  ∞, {β n }⊂0,a for some 0 <a<1 and the sequence {x n } is asymptotically regular. Copyright q 2008 Jong Soo Jung. 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 Let E be a real Banach space and let C be a nonempty closed convex subset of E.Recallthata mapping f : C → C is a contraction on C if there exists a constant k ∈ 0, 1 such that fx − fy≤kx − y,x,y∈ C. We use Σ C to denote the collection of mappings f verifying the above inequality. That is, Σ C  {f : C → C | f is a contraction with constant k}. Note that each f ∈ Σ C has a unique fixed point in C. Now let T : C → C be a nonexpansive mapping recall that a mapping T : C → C is nonexpansive if Tx − Ty≤x − y,x,y∈ C and FT denote the set of fixed points of T;that is, FT{x ∈ C : x  Tx}. We consider the iterative scheme: for T nonexpansive mapping, f ∈ Σ C and λ n ∈ 0, 1, x n1  λ n f  x n    1 − λ n  Tx n ,n≥ 0. 1.1 As a special case of 1.1, the following iterative scheme: z n1  λ n u   1 − λ n  Tz n ,n≥ 0, 1.2 2 Fixed Point Theory and Applications where u, z 0 ∈ C are arbitrary but fixed, has been investigated by many authors; see, for example, Cho et al. 1, Halpern 2,Lions3,Reich4, 5, Shioji and Takahashi 6, Wittmann 7,andXu8. The authors above showed that the sequence {z n } generated by 1.2 converges strongly to a point in the fixed point set FT under appropriate conditions on {λ n } in either Hilbert spaces or certain Banach spaces. Recently, many authors also considered the iterative scheme 1.2 for finite or countable families of nonexpansive mappings {T i } i∈{1,2, ,r or∞} ; see, for instance, 9–14. The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in a Hilbert space was proposed by Moudafi 15see 16 for finding hierarchically a fixed point. In 2004, Xu 17 extended Theorem 2.2 of Moudafi 15 for the iterative scheme 1.1 to a Banach space setting using the following conditions on {λ n }: H1 lim n→∞ λ n  0;  ∞ n0 λ n  ∞ or, equivalently,  ∞ n0 1 − λ n 0; H2  ∞ n0 |λ n1 − λ n | < ∞ or lim n→∞ λ n /λ n1 1. We also refer to 18–23 for the iterative scheme 1.1 for finite of countable families of nonexpansive mappings {T i } i∈{1,2, ,r or ∞} . For the iterative scheme 1.1 with generalized contractive mappings instead of contractions, see 22, 24.Wecanreferto25 for the general iteration method for finding a zero of accretive operator. Recently, Kim and Xu 26 provided a simpler modification of Mann iterative scheme 1.3 in a uniformly smooth Banach space as follows: x 0  x ∈ C, y n  β n x n   1 − β n  Tx n , x n1  α n u   1 − α n  y n , 1.3 where u ∈ C is an arbitrary but fixed element, and {α n } and {β n } are two sequences in 0, 1. They proved that {x n } generated by 1.3 converges to a fixed point of T under the control conditions: i lim n→∞ α n  0, lim n→∞ β n  0; ii  ∞ n0 α n  ∞, or equivalently,  ∞ n0 1 − α n 0,  ∞ n0 β n  ∞; iii  ∞ n0 |α n1 − α n | < ∞,  ∞ n0 |β n1 − β n | < ∞. In this paper, motivated by the above-mentioned results, as the viscosity approximation method, we consider a new composite iterative scheme for nonexpansive mapping T: x 0  x ∈ C, y n  λ n f  x n    1 − λ n  Tx n , x n1   1 − β n  y n  β n Ty n , IS where {β n }, {λ n }⊂0, 1. First, we prove the strong convergence of the sequence {x n } generated by IS under the suitable conditions on the control parameters {β n } and {λ n } and the asymptotic regularity on {x n } in reflexive Banach space with a uniformly G ˆ ateaux differentiable norm together with the assumption that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Moreover, we show that the strong Jong Soo Jung 3 limit is a solution of certain variational inequality. Next, we study the viscosity approximation with the weakly contractive mapping to a fixed point of nonexpansive mapping in the same Banach space. The main results improve and complement the corresponding results of 1– 8, 15, 17. In particular, if β n  0, for all n ≥ 0, then IS reduces to 1.1. We point out that the iterative scheme IS is a new one for finding a fixed point of T. 2. Preliminaries and lemmas Let E be a real Banach space with norm · and let E ∗ be its dual. The value of f ∈ E ∗ at x ∈ E will be denoted by x, f. When {x n } is a sequence in E,thenx n → x resp., x n x will denote strong resp., weak convergence of the sequence {x n } to x. The normalized duality mapping J from E into the family of nonempty by Hahn-Banach theorem weak-star compact subsets of its dual E ∗ is defined by Jx  f ∈ E ∗ : x, f  x 2  f 2  2.1 for each x ∈ E 27. The norm of E is said to be G ˆ ateaux differentiable and E is said to be smooth if lim t→0 x  ty−x t 2.2 exists for each x, y in its unit sphere U  {x ∈ E : x  1}. The norm is said to be uniformly G ˆ ateaux differentiable if for y ∈ U, the limit is attained uniformly for x ∈ U. The space E is said to have a uniformly Fr ´ echet differentiable norm and E is said to be uniformly smooth if the limit in 2.2 is attained uniformly for x, y ∈ U × U.ItisknownthatE is smooth if and only if each duality mapping J is single-valued. It is also well known that if E has a uniformly G ˆ ateaux differentiable norm, J is uniformly norm to weak continuous on each bounded subset of E 27. Let C be a nonempty closed convex subset of E. C is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset D of C has a fixed point in D. Let D be a subset of C. Then, a mapping Q : C → D is said to be a retraction from C onto D if Qx  x for all x ∈ D. A retraction Q : C → D is said to be sunny if QQx  tx − Qx  Qx for all x ∈ C and t ≥ 0withQxtx−Qx ∈ C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. In a smooth Banach space E,itiswellknown28, page 48 that Q is a sunny nonexpansive retraction from C onto D if and only if the following condition h olds:  x − Qx, Jz − Qx  ≤ 0,x∈ C, z ∈ D. 2.3 We need the following lemmas for the proof of our main results. Lemma 2.1 was also given by Jung and Morales 29 and Lemma 2.2 is essentially Lemma 2 of Liu 30also see 8. Lemma 2.1. Let X be a real Banach space and let J be the duality mapping. Then, for any given x,y ∈ X, one has x  y 2 ≤x 2  2  y, jx  y  2.4 for all jx  y ∈ Jx  y. 4 Fixed Point Theory and Applications Lemma 2.2. Let {s n } be a sequence of nonnegative real numbers satisfying s n1 ≤ 1 − α n s n  α n γ n  δ n ,n≥ 0, 2.5 where {α n }, {γ n }, and {δ n } satisfy the following conditions: i {α n }⊂0, 1 and  ∞ n0 α n  ∞ or, equivalently,  ∞ n0 1 − α n 0, ii lim sup n→∞ γ n ≤ 0 or  ∞ n1 α n γ n < ∞, iii δ n ≥ 0 n ≥ 0,  ∞ n0 δ n < ∞. Then, lim n→∞ s n  0. Recall that a mapping A : C → C is said to be weakly contractive if Ax − Ay≤x − y−ψ  x − y  , ∀x, y ∈ C, 2.6 where ψ : 0, ∞ → 0, ∞ is a continuous and strictly increasing function such that ψ is positive on 0, ∞ and ψ00. As a special case, if ψt1 − kt for t ∈ 0, ∞,where k ∈ 0, 1, then the weakly contractive mapping A is a contraction with constant k. Rhoades 31  obtained the following result for weakly contractive mapping. Lemma 2.3 see 31,Theorem2. Let X, d be a complete metric space, and A a weakly contractive mapping on X.Then,A has a unique fixed point p in X. Moreover, for x ∈ X, {A n x} converges strongly to p. The following lemma was given in 32, 33. Lemma 2.4. Let {s n } and {γ n } be two sequences of nonnegative real numbers and {λ n } a sequence of positive numbers satisfying the conditions i  ∞ n0 λ n  ∞ or, equivalently,  ∞ n0 1 − λ n 0, ii lim n→∞ γ n /λ n 0. Let the recursive inequality s n1 ≤ s n − λ n ψ  s n   γ n ,n 0, 1, 2, , 2.7 be given where ψt is a continuous and strict increasing function on 0, ∞ with ψ00.Then, lim n→∞ s n  0. Finally, the sequence {x n } in E is said to be asymptotically regular if lim n→∞ x n1 − x n   0. 2.8 3. Main results First, using the asymptotic regularity, we study a strong convergence theorem for a composite iterative scheme for the nonexpansive mapping with the contractive mapping. Jong Soo Jung 5 For T : C → C nonexpansive and so for any t ∈ 0, 1 and f ∈ Σ C , tf 1 − tT : C → C defines a strict contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point x f t satisfying x f t  tf  x f t  1 − tTx f t . R For simplicity, we will write x t for x f t provided no confusion occurs. In 2006, the following result was given by Jung 18see also Xu 17 for the result in uniformly smooth Banach spaces. Theorem J see Jung 18. Let E be a reflexive Banach space with a uniformly G ˆ ateaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E and T nonexpansive mapping from C into itself with FT /  ∅.Then,{x t } defined by R converges strongly to a point in FT.If one defines Q : Σ C → FT by Qf : lim t→0  x t ,f∈ Σ C , 3.1 then Qf solves a variational inequality  I − f  Qf  ,J  Qf − p  ≤ 0,f∈ Σ C ,p∈ FT. 3.2 Remark 3.1. In Theorem J, if fxu ∈ C is a constant, then 3.2 becomes  Qu − u, J  Qu − p  ≤ 0,u∈ C, p ∈ FT. 3.3 Hence by 2.3, Q reduces to the sunny nonexpansive retraction from C to FT. Namely, FT is a sunny nonexpansive retraction of C. Using Theorem J and the asymptotic regularity on the sequence {x n },wehavethe following result. Theorem 3.2. Let E be a reflexive Banach space with a uniformly G ˆ ateaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E and T nonexpansive mappings from C into itself with FT /  ∅.Let{β n } and {λ n } be sequences in 0, 1 which satisfies the conditions: B1 β n ∈ 0,a for some 0 <a<1 for all n ≥ 0, C1 lim n→∞ λ n  0;  ∞ n0 λ n  ∞. Let f ∈ Σ C and x 0 ∈ C be chosen arbitrarily. Let {x n } be the sequence generated by x 0  x ∈ C, y n  λ n f  x n    1 − λ n  Tx n , x n1   1 − β n  y n  β n Ty n ,n≥ 0. IS If {x n } is asymptotically regular, then {x n } converges strongly to Qf ∈ FT,whereQf is the unique solution of the variational inequality  I − f  Qf  ,J  Qf − p  ≤ 0,f∈ Σ C ,p∈ FT. 3.4 6 Fixed Point Theory and Applications Proof. We notice that by Theorem J, there exists a solution Qf of a variational inequality  I − f  Qf  ,J  Qf − p  ≤ 0,f∈ Σ C ,p∈ FT. 3.5 Namely, Qflim t→0  x t ,wherex t is defined by R. We will show that x n → Qf. We proceed with the following steps. Step 1. We show that x n −z≤max{x 0 −z, 1/1−kfz−z} for all n ≥ 0andallz ∈ FT and so {x n }, {y n }, {fx n }, {Tx n }, and {Ty n } are bounded. Indeed, let z ∈ FT. Then, we have   y n − z      λ n  f  x n  − z    1 − λ n  Tx n − z    ≤ λ n   f  x n  − z     1 − λ n    x n − z   ≤ λ n    f  x n  − fz      fz − z      1 − λ n    x n − z   ≤ λ n k   x n − z    λ n   fz − z     1 − λ n    x n − z     1 −  1 − kλ n    x n − z    λ n   fz − z   ≤ max    x n − z   , 1 1 − k   fz − z    ,   x n1 − z       1 − β n  y n − z   β n  Ty n − z    ≤  1 − β n    y n − z    β n   y n − z      y n − z   ≤ max    x n − z   , 1 1 − k   fz − z    . 3.6 Using an induction, we obtain   x n − z   ≤ max    x 0 − z   , 1 1 − k   fz − z    3.7 for all n ≥ 0. Hence, {x n } is bounded, and so are {y n }, {Tx n }, {Ty n }, and {fx n }. Moreover, it follows from condition C1 that   y n − Tx n    λ n   f  x n  − Tx n   −→ 0 as n −→ ∞ . 3.8 Step 2. We show that lim n →∞ x n1 − y n   0 and lim n→∞ x n − y n   0. Indeed, by the condition B1   x n1 − y n    β n   Ty n − y n   ≤ β n    Ty n − Tx n      Tx n − y n    ≤ a    y n − x n      Tx n − y n    ≤ a    y n − x n1      x n1 − x n      Tx n − y n    3.9 which implies that   x n1 − y n   ≤ a 1 − a    x n1 − x n      Tx n − y n    . 3.10 Jong Soo Jung 7 So, by asymptotic regularity of {x n } and 3.8,wehavex n1 − y n →0, and also   x n − y n   ≤   x n − x n1      x n1 − y n   −→ 0 as n −→ ∞ . 3.11 Step 3. We show that lim n→∞ y n − Ty n   0. By 3.8 and Step 2,wehave   y n − Ty n   ≤   y n − Tx n      Tx n − Ty n   ≤   y n − Tx n      x n − y n   −→ 0. 3.12 Step 4. We show that lim sup n→∞ Qf − fQf,JQf − y n ≤0. To prove this, let a subsequence {y n j } of {y n } be such that lim sup n→∞  Qf − f  Qf  ,J  Qf − y n   lim j→∞  Qf − f  Qf  ,J  Qf − y n j  3.13 and y n j pfor some p ∈ E. From Step 3, it follows that lim j→∞ y n j − Ty n j   0. Now let Qflim t→0  x t ,wherex t  tfx t 1 − tTx t . Then, we can write x t − y n j  t  f  x t  − y n j  1 − t  Tx t − y n j  . 3.14 Putting a j t1 − t 2   Ty n j − y n j    2   x t − y n j      Ty n j − y n j    −→ 0 j −→ ∞ 3.15 by Step 3 and using Lemma 2.1,weobtain x t − y n j  2 ≤ 1 − t 2 Tx t − y n j  2  2t  f  x t  − y n j ,J  x t − y n j  ≤ 1 − t 2  Tx t − Ty n j   Ty n j − y n j   2  2t  f  x t  − x t ,J  x t − y n j   2tx t − y n j  2 ≤ 1 − t 2 x t − y n j  2  a j t2t  f  x t  − x t ,J  x t − y n j   2tx t − y n j  2 . 3.16 The last inequality implies  x t − f  x t  ,J  x t − y n j  ≤ t 2 x t − y n j  2  1 2t a j t. 3.17 It follows that lim sup j→∞  x t − f  x t  ,J  x t − y n j  ≤ t 2 M, 3.18 where M>0 is a constant such that M ≥x t − y n  2 for all n ≥ 0andt ∈ 0, 1.Takingthe lim sup as t → 0in3.18 and noticing the fact that the two limits are interchangeable due to the fact that J is uniformly continuous on bounded subsets of E from the strong topology of E to the weak ∗ topology of E ∗ ,wehave lim sup j→∞  Qf − f  Qf  ,J  Qf − y n j  ≤ 0. 3.19 8 Fixed Point Theory and Applications Indeed, letting t → 0, from 3.18 we have lim sup t→0 lim sup j→∞  x t − f  x t  ,J  x t − y n j  ≤ 0. 3.20 So, for any ε>0, there exists a positive n umber δ 1 such that for any t ∈ 0,δ 1 , lim sup j→∞  x t − f  x t  ,J  x t − y n j  ≤ ε 2 . 3.21 Moreover, since x t → Qf as t → 0, the set {x t − y n j } is bounded and the duality mapping J is norm-to-weak ∗ uniformly continuous on bounded subset of E, there exists δ 2 > 0 such that, for any t ∈ 0,δ 2 ,    Qf − f  Qf  ,J  Qf − y n j  −  x t − f  x t  ,J  x t − y n j        Qf − f  Qf  ,J  Qf − y n j  − J  x t − y n j    Qf − f  Qf  −  x t − f  x t  ,J  x t − y n j    ≤    Qf − f  Qf  ,J  x t − y n j  − J  Qf − y n j       Qf − f  Qf  −  x t − f  x t      x t − y n j   < ε 2 . 3.22 Choose δ  min{δ 1 ,δ 2 }, we have for all t ∈ 0,δ and j ∈ N,  Qf − f  Qf  ,J  Qf − y n j  <  x t − f  x t  ,J  x t − y n j   ε 2 , 3.23 which implies that lim sup j→∞  Qf − f  Qf  ,J  Qf − y n j  ≤ lim sup j→∞  x t − f  x t  ,J  x t − y n j   ε 2 . 3.24 Since lim sup j→∞ x t − fx t ,Jx t − y n j ≤ε/2, we have lim sup j→∞  Qf − f  Qf  ,J  Qf − y n j  ≤ ε. 3.25 Since ε is arbitrary, we obtain that lim sup j→∞  Qf − f  Qf  ,J  Qf − y n j  ≤ 0. 3.26 Step 5. We show that lim n→∞ x n − Qf  0. By using IS,wehave   x n1 − Qf   ≤   y n − Qf      λ n  f  x n  − Qf    1 − λ n  Tx n − Qf    . 3.27 Jong Soo Jung 9 Applying Lemma 2.1,weobtain   x n1 − Qf   2 ≤   y n − Qf   2 ≤  1 − λ n  2   Tx n − Qf   2  2λ n  f  x n  − Qf,J  y n − Qf  ≤  1 − λ n  2 x n − Qf 2  2λ n  f  x n  − f  Qf  ,J  y n − Qf   2λ n  f  Qf  − Qf,J  y n − Qf  ≤  1 − λ n  2   x n − Qf   2  2kλ n x n − Qf     y n − Qf    2λ n  f  Qf  − Qf,J  y n − Qf  ≤  1 − λ n  2   x n − Qf   2  2kλ n   x n − Qf   2  2λ n  f  Qf  − Qf,J  y n − Qf  . 3.28 It then follows that   x n1 − Qf   2 ≤  1 − 21 − kλ n  λ 2 n    x n − Qf   2  2λ n  Qf − f  Qf  ,J  Qf − y n  ≤  1 − 2 − kλ n    x n − Qf   2  λ 2 n M 2  2λ n  Qf − f  Qf  ,J  Qf − y n  , 3.29 where M  sup n≥0 x n − Qf.Put α n  21 − kλ n , γ n  λ n 21 − k M 2  1 1 − k  Qf − f  Qf  ,J  Qf − y n  . 3.30 From the condition C1 and Step 4, it follows that α n → 0,  ∞ n0 α n  ∞, and lim sup n→∞ γ n ≤ 0. Since 3.29 reduces to x n1 − Qf 2 ≤ 1 − α n x n − Qf 2  α n γ n , 3.31 from Lemma 2.2 with δ n  0, we conclude that lim n→∞ x n − Qf  0. This completes the proof. Corollary 3.3. Let E be a uniformly smooth Banach space. Let C, T, f, {β n }, {λ n },f,x 0 ,and{x n } be thesameasinTheorem 3.2. Then, the conclusion of Theorem 3.2 still holds. Proof. Since E is a uniformly smooth Banach space, E is reflexive and the norm is uniformly G ˆ ateaux differentiable norm and its every nonempty weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Thus, the conclusion of Corollary 3.3 follows from Theorem 3.2 immediately. 10 Fixed Point Theory and Applications Remark 3.4. 1 If {β n } and {λ n } in Theorem 3.2 satisfy the conditions B2  ∞ n0 |β n1 − β n | < ∞, C1 lim n→∞ λ n  0,  ∞ n0 λ n  ∞, C2  ∞ n0 |λ n1 − λ n | < ∞,or C3 lim n→∞ λ n /λ n1 1, or C4 |λ n1 − λ n |≤◦λ n1 σ n ,  ∞ n0 σ n < ∞ the perturbed control condition, then the sequence {x n } generated by IS is asymptotically regular. Now, we only give the proof in case when {β n } and {λ n } satisfy the conditions B2, C1,andC4.Indeed,fromIS, we have for every n ≥ 1, y n  λ n f  x n    1 − λ n  Tx n , y n−1  λ n−1 f  x n−1    1 − λ n−1  Tx n−1 , 3.32 and so, for every n ≥ 1, we have   y n − y n−1       1 − λ n  Tx n − Tx n−1   λ n  f  x n  − f  x n−1    λ n − λ n−1  f  x n−1  − Tx n−1    ≤  1 − λ n    x n − x n−1    L   λ n − λ n−1    kλ n   x n − x n−1     1 − 1 − kλ n    x n − x n−1    L   λ n − λ n−1   , 3.33 where L  sup{fx n  − Tx n  : n ≥ 0}. On the other hand, by IS,wealsohaveforeveryn ≥ 1, x n1   1 − β n  y n  β n Ty n , x n   1 − β n−1  y n−1  β n−1 Ty n−1 . 3.34 Simple calculations show that x n1 − x n   1 − β n  y n − y n−1   β n  Ty n − Ty n−1    β n − β n−1  Ty n−1 − y n−1  , 3.35 then it follows that   x n1 − x n   ≤  1 − β n    y n − y n−1    β n   y n − y n−1      β n − β n−1     Ty n−1 − y n−1   . 3.36 Substituting 3.33 into 3.36 and using the condition C4,wederive   x n1 − x n   ≤  1 − 1 − kλ n    x n − x n−1    L   λ n − λ n−1    M   β n − β n−1   ≤  1 − 1 − kλ n    x n − x n−1    L  ◦  λ n   σ n−1   M   β n − β n−1   , 3.37 where M  sup{Ty n − y n  : n ≥ 0}.Bytakings n1  x n1 − x n , α n 1 − kλ n , α n γ n  L◦ λ n , and δ n  Lσ n−1  M|β n − β n−1 |,wehave s n1 ≤  1 − α n  s n  α n γ n  δ n . 3.38 [...]... ¸ nonexpansive mappings with generalized contraction mappings, ” Nonlinear Analysis: Theory, Methods & Applications In press 23 W Takahashi, “Voscosity approximation methods for countable families of nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications In press 24 N C Wong, D R Sahu, and J.-C Yao, “Solving variational inequalities involving nonexpansive type mappings, ”... methods for a family of finite nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol 64, no 11, pp 2536–2552, 2006 19 J S Jung, Convergence theorems of iterative algorithms for a family of finite nonexpansive mappings, ” Taiwanese Journal of Mathematics, vol 11, no 3, pp 883–902, 2007 20 J S Jung, Iterative approximation to common fixed points of a sequence of nonexpansive. .. “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol 75, no 1, pp 287–292, 1980 5 S Reich, “Approximating fixed points of nonexpansive mappings, ” Panamerican Mathematical Journal, vol 4, no 2, pp 23–28, 1994 6 N Shioji and W Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach. .. fixed points of families of nonexpansive mappings, ” Taiwanese Journal of Mathematics, vol 12, no 2, 2008 12 J S Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol 302, no 2, pp 509–520, 2005 13 J S Jung, Y J Cho, and R P Agarwal, Iterative schemes with some control conditions for a family of finite nonexpansive. .. nonexpansive mappings in Banach spaces,” Journal of Computational and Applied Mathematics In press 21 P.-E Maing´ , “Approximation methods for common fixed points of nonexpansive mappings in e Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 325, no 1, pp 469–479, 2007 14 Fixed Point Theory and Applications 22 A Petrusela and J.-C Yao, “Viscosity approximation to common fixed points of... compact convex subset of E has the fixed point property for nonexpansive mappings Let C be a nonempty closed convex subset of E and T nonexpansive mappings from C into itself with F T / ∅ Let {βn } and {λn } be sequences in 0, 1 which satisfy the conditions (B1), (B2), (C1), and (C4) (or the conditions (B1), (B2), (C1), and (C2), or the conditions (B1), (B2), (C1), and (C3)) Let A : C → C be a weakly contractive... of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol 202, no 1, pp 150–159, 1996 10 L C Ceng, P Cubiotti, and J.-C Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 5, pp 1464–1473, 2007 11 L C Ceng, P Cubiotti, and J.-C Yao, “Approximation of common... supported by research funds from Dong-A University References 1 Y J Cho, S M Kang, and H Zhou, “Some control conditions on iterative methods,” Communications on Applied Nonlinear Analysis, vol 12, no 2, pp 27–34, 2005 2 B Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol 73, no 6, pp 957–961, 1967 3 P.-L Lions, “Approximation de points fixes de contractions,”... family of finite nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol 2005, no 2, pp 125–135, 2005 14 J S Jung and T H Kim, Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces,” Bulletin of the Korean Mathematical Society, vol 34, no 1, pp 93–102, 1997 15 A Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal... Even βn 0 in IS , Corollary 3.5 generalizes the corresponding results by Halpern 2 , Lions 3 , Reich 4, 5 , Shioji and Takahashi 6 , Wittmann 7 , and Xu 8 to the viscosity methods along with the perturb control condition C4 Next, we consider the viscosity approximation method with the weakly contractive mapping for the nonexpansive mapping Theorem 3.7 Let E be a reflexive Banach space with a uniformly . 2.8 3. Main results First, using the asymptotic regularity, we study a strong convergence theorem for a composite iterative scheme for the nonexpansive mapping with the contractive mapping. Jong Soo. C. Wong, D. R. Sahu, and J C. Yao, “Solving variational inequalities involving nonexpansive type mappings, ” Nonlinear Analysis: Theory, Methods & Applications. In press. 25 P E. Maing ´ e,. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 167535, 14 pages doi:10.1155/2008/167535 Research Article Convergence on Composite Iterative Schemes