Volume 2008, Article ID 751383,7pages doi:10.1155/2008/751383
Research Article
Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces
Jing Zhao,1 Songnian He,1and Yongfu Su2
1College of Science, Civil Aviation University of China, Tianjin 300300, China
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Jing Zhao,[email protected] Received 25 August 2007; Accepted 16 December 2007
Recommended by Tomonari Suzuki
The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mappingT and a finite family of nonexpansive mappings{Ti}Ni1, respec- tively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.
Copyrightq2008 Jing Zhao 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
LetEbe a real Banach space, K a nonempty closed convex subset ofE, andT : K → Ka mapping. We useFTto denote the set of fixed points ofT, that is,FT {x∈K:Txx}.
Tis called nonexpansive ifTx−Ty ≤ x−yfor allx, y∈K. In this paper,and→denote weak and strong convergence, respectively. coAdenotes the closed convex hull ofA, where Ais a subset ofE.
In 2001, Xu and Ori1introduced the following implicit iteration scheme for common fixed points of a finite family of nonexpansive mappings{Ti}Ni1in Hilbert spaces:
xnαnxn−1 1−αn
Tnxn, n≥1, 1.1
whereTnTnmodN, and they proved weak convergence theorem.
In this paper, we introduce a new implicit iteration scheme:
xnαnxn−1βnTxn−1γnTxn, n≥1, 1.2
for fixed points of nonexpansive mappingT in Banach space and also prove weak and strong convergence theorems. Moreover, we introduce an implicit iteration scheme:
xnαnxn−1βnTnxn−1γnTnxn, n≥1, 1.3 where Tn TnmodN, for common fixed points of a finite family of nonexpansive mappings {Ti}Ni1in Banach spaces and also prove weak and strong convergence theorems.
Observe that if K is a nonempty closed convex subset of a real Banach spaceE and T : K → Kis a nonexpansive mapping, then for everyu ∈ K,α, β, γ ∈ 0,1, and positive integern, the operatorSSα,β,γ,n:K→Kdefined by
SxαuβTuγTx 1.4
satisfies
Sx−SyγTx−γTy ≤γx−y 1.5
for allx, y ∈ K. Thus, ifγ < 1 then Sis a contractive mapping. ThenShas a unique fixed pointx∗ ∈ K. This implies that, ifγn < 1, the implicit iteration scheme1.2and1.3can be employed for the approximation of fixed points of nonexpansive mapping and common fixed points of a finite family of nonexpansive mappings, respectively.
Now, we give some definitions and lemmas for our main results.
A Banach spaceEis said to satisfy Opial’s condition if, for any{xn} ⊂Ewithxn x∈E, the following inequality holds:
lim sup
n→∞
xn−x<lim sup
n→∞
xn−y, ∀y∈E, x /y. 1.6 LetDbe a closed subset of a real Banach spaceEand letT:D→Dbe a mapping.
Tis said to be demiclosed at zero ifTx00 whenever{xn} ⊂D,xn x0andTxn→0.
T is said to be semicompact if, for any bounded sequence{xn} ⊂ Dwith limn→∞xn− Txn0, there exists a subsequence{xnk} ⊂ {xn}such that{xnk}converges strongly tox∗∈D.
Lemma 1.1see2,3. LetEbe a uniformly convex Banach space, letKbe a nonempty closed convex subset ofE, and letT :K→Kbe a nonexpansive mapping. ThenI−Tis demiclosed at zero.
Lemma 1.2see 4. LetEbe a uniformly convex Banach space and leta, bbe two constants with 0 < a < b <1. Suppose that{tn} ⊂ a, bis a real sequence and{xn},{yn}are two sequences inE.
Then the conditions
n→∞limtnxn 1−tnynd, lim sup
n→∞
xn≤d, lim sup
n→∞
yn≤d 1.7
imply that limn→∞xn−yn0, whered≥0 is a constant.
2. Main results
Theorem 2.1. LetEbe a real uniformly convex Banach space which satisfies Opial’s condition, letKbe a nonempty closed convex subset ofE, and letT :K→Kbe a nonexpansive mapping with nonempty fixed points setF. Let{αn},{βn},{γn}be three real sequences in0,1satisfyingαnβnγn1 and 0< a≤γn≤b <1, wherea, bare some constants. Then implicit iteration process{xn}defined by1.2 converges weakly to a fixed point ofT.
Proof. Firstly, the condition ofTheorem 2.1impliesγn<1, so that1.2can be employed for the approximation of fixed point of nonexpansive mapping.
For any givenp∈F, we have
xn−pαnxn−1βnTxn−1γnTxn−p αn
xn−1−p βn
Txn−1−p γn
Txn−p
≤αnxn−1−pβnTxn−1−pγnTxn−p
≤
αnβnxn−1−pγnxn−p
2.1
which leads to
1−γnxn−p≤
αnβnxn−1−p
1−γnxn−1−p. 2.2 It follows from the conditionγn≤b <1 that
xn−p≤xn−1−p. 2.3 Thus limn→∞xn−pexists, and so let
n→∞limxn−pd. 2.4
Hence{xn}is a bounded sequence. Moreover, co{xn}is a bounded closed convex subset of K. We have
n→∞limxn−p lim
n→∞αn
xn−1−p βn
Txn−1−p γn
Txn−p lim
n→∞
1−γn
αn
1−γn
xn−1−p βn
1−γn
Txn−1−p γn
Txn−ptd, lim sup
n→∞
Txn−p≤lim sup
n→∞
xn−pd.
2.5 Again, it follows from the conditionαnβnγn1 that
lim sup
n→∞
αn
1−γn
xn−1−p βn
1−γn
Txn−1−p
≤lim sup
n→∞
αn
1−γn
xn−1−p βn
1−γn
Txn−1−p
≤lim sup
n→∞
αnβn
1−γn
xn−1−p d.
2.6
ByLemma 1.2, the condition 0< a≤γn≤b <1, and2.5–2.6, we get
n→∞lim αn
1−γn
xn−1−p βn
1−γn
Txn−1−p
−
Txn−p0. 2.7
This means that
n→∞lim αn
1−γnxn−1 βn
1−γnTxn−1−Txn
lim
n→∞
1 1−γn
αnxn−1βnTxn−1− 1−γn
Txn0.
2.8 Since 0< a≤γn≤b <1, we have 1/1−a≤1/1−γn≤1/1−b. Hence,
n→∞limαnxn−1βnTxn−1− 1−γn
Txn0. 2.9 Because
n→∞limαnxn−1βnTxn−1− 1−γn
Txn lim
n→∞xn−γnTxn− 1−γn
Txn lim
n→∞xn−Txn, 2.10
by2.9, we get
n→∞limxn−Txn0. 2.11
SinceEis uniformly convex, every bounded closed convex subset ofEis weakly com- pact, so that there exists a subsequence{xnk}of sequence{xn} ⊆co{xn}such thatxnk q∈ K. Therefore, it follows from2.11that
k→∞limTxnk−xnk0. 2.12
ByLemma 1.1, we know thatI−Tis demiclosed at zero; it is esay to see thatq∈F.
Now, we show thatxn q. In fact, this is not true; then there must exist a subsequence {xni} ⊂ {xn}such thatxni q1∈K,q1/q. Then, by the same method given above, we can also prove thatq1∈F.
Because, for anyp∈F, the limit limn→∞xn−pexists. Then we can let
n→∞limxn−qd1, lim
n→∞xn−q1d2. 2.13
SinceEsatisfies Opial’s condition, we have d1lim sup
k→∞
xnk−q<lim sup
k→∞
xnk−q1d2, d2lim sup
i→∞
xni−q1<lim sup
i→∞
xni−qd1. 2.14 This is a contradiction and henceq q1. This implies that{xn}converges weakly to a fixed pointqofT. This completes the proof.
From the proof ofTheorem 2.1, we give the following strong convergence theorem.
Theorem 2.2. Let Ebe a real uniformly convex Banach space, let K be a nonempty closed convex subset ofE, letT :K →Kbe a nonexpansive mapping with nonempty fixed points setF, and letT be semicompact. Let{αn},{βn},{γn}be three real sequences in0,1satisfyingαnβnγn1 and 0< a≤γn≤b <1, wherea, bare some constants. Then implicit iteration process{xn}defined by1.2 converges strongly to a fixed point ofT.
Proof. From the proof ofTheorem 2.1, we know that there exists subsequence {xnk} ⊂ {xn} such thatxnk q ∈Kand satisfies2.11. By the semicompactness ofT, there exists a subse- quence of{xnk}we still denote it by{xnk}such that limn→∞xnk−q 0. Because the limit limn→∞xn−qexists, thus we get limn→∞xn−q0. This completes the proof.
Next, we study weak and strong convergence theorems for common fixed points of a finite family of nonexpansive mappings{Ti}Ni1in Banach spaces.
Theorem 2.3. LetEbe a real uniformly convex Banach space which satisfies Opial’s condition, letK be a nonempty closed convex subset ofE, and let{Ti}Ni1 :K→KbeNnonexpansive mappings with nonempty common fixed points setF. Let{αn},{βn},{γn}be three real sequences in0,1satisfying αnβnγn 1, 0< a≤γn ≤b <1, andαn−βn > c >0, wherea,b,care some constants. Then implicit iteration process{xn}defined by1.3converges weakly to a common fixed point of{Ti}Ni1. Proof. Substituing Ti 1 ≤ i ≤ N toT in the proof of Theorem 2.1, we know that for all i 1≤i≤N,
n→∞limxn−Tnxn0. 2.15
Now we show that, for anyl1,2, . . . , N,
n→∞limxn−Tlxn0. 2.16
In fact,
xn−xn−1βnTnxn−1γnTnxn− βnγn
xn−1 βnTnxn−1−βnxnγnTnxn−γnxn
βnγn
xn−xn−1
≤βnTnxn−1−xnγnTnxn−xn
βnγnxn−xn−1
≤βnTnxn−1−TnxnβnTnxn−xnγnTnxn−xn
βnγnxn−xn−1
≤
βnγnTnxn−xn
2βnγnxn−xn−1
βnγnTnxn−xn
βn1−αnxn−xn−1.
2.17 By removing the second term on the right of the above inequality to the left, we get
αn−βnxn−xn−1≤
βnγnTnxn−xn. 2.18 It follows from the conditionαn−βn> c >0 and2.15that
n→∞limxn−xn−10. 2.19
So, for anyi1,2, . . . , N,
n→∞limxn−xni0. 2.20
Since, for anyi1,2,3, . . . , N,
xn−Tnixn≤xn−xnixni−TnixniTnixni−Tnixn
≤2xn−xnixni−Tnixni,
2.21
it follows from2.15and2.20that
n→∞limTnixn−xn0, i1,2,3, . . . , N. 2.22
BecauseTnTnmodN, it is easy to see, for anyl1,2,3, . . . , N, that
n→∞limTlxn−xn0. 2.23
SinceEis uniformly convex, so there exists a subsequence {xnk}of bounded sequence{xn} such thatxnk q∈K. Therefore, it follows from2.23that
k→∞limTlxnk−xnk0, ∀l1,2,3, . . . , N. 2.24 ByLemma 1.1, we know thatI−Tlis demiclosed, it is easy to see thatq∈FTl, so thatq∈F
Nl1FTl. BecauseEsatisfies Opial’s condition, we can prove that{xn}converges weakly to a common fixed pointqof{Tl}Nl1by the same method given in the proof ofTheorem 2.1.
Remark 2.4. IfN 1, implicit iteration scheme1.3becomes1.2, so fromTheorem 2.1, we know that assumptionαn−βn> c >0 inTheorem 2.3can be removed.
Theorem 2.5. LetEbe a real uniformly convex Banach space, letKbe a nonempty closed convex subset ofE, let{Ti}Ni1 : K → KbeNnonexpansive mappings with nonempty common fixed points setF, and there exists anl ∈ {1,2, . . . , N}such thatTl is semicompact. Let{αn},{βn},{γn}be three real sequences in0,1satisfyingαnβnγn 1, 0< a ≤γn ≤b < 1, andαn−βn > c > 0, wherea, b,care some constants. Then implicit iteration process{xn}defined by1.3converges strongly to a common fixed point of{Ti}Ni1.
Proof. From the proof ofTheorem 2.3, we know that there exists subsequence{xnk} ⊂ {xn}such that{xnk}converges weakly to someq ∈Kand satisfies2.23. By the semicompactness ofTl, there exists a subsequence of{xnk}we still denote it by{xnk}such that limn→∞xnk−q0.
Because the limit limn→∞xn−qexists, thus we get limn→∞xn−q 0. This completes the proof.
Acknowledgment
This research is supported by Tianjin Natural Science Foundation in China Grant no.
06YFJMJC12500.
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