Volume 2008, Article ID 548627,10pages doi:10.1155/2008/548627
Research Article
A New One-Step Iterative Process for Common
Fixed Points in Banach Spaces
Mujahid Abbas,1Safeer Hussain Khan,2 and Jong Kyu Kim3
1Mathematics Department, Lahore University of Management Sciences, Lahore 54792, Pakistan 2Department of Mathematics and Physics, Qatar University, P.O. Box 2713, Doha, Qatar
3Department of Mathematics Education, Kyungnam University, Masan, Kyungnam 631-701, South Korea
Correspondence should be addressed to Jong Kyu Kim,[email protected]
Received 26 September 2008; Accepted 22 October 2008
Recommended by Ram U. Verma
We introduce a new one-step iterative process and use it to approximate the common fixed points of two asymptotically nonexpansive mappings through some weak and strong convergence theorems. Our process is computationally simpler than the processes currently being used in literature for the purpose.
Copyrightq2008 Mujahid Abbas 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
Throughout this paper,Ndenotes the set of positive integers. LetEbe a real Banach space,Ca nonempty convex subset ofE.A mappingT :C → Cis called asymptotically nonexpansive if there is a sequence{kn} ⊂1,∞such that
Tnx−Tny ≤knx−y ∀x, y∈C, ∀n∈N, 1.1
where ∞k1kn−1<∞. A pointx∈Cis a fixed point ofT, provided thatTxx.
To approximate the common fixed points of two mappings, the following Ishikawa-type two-step iterative process is widely usedsee, e.g.,1–9, and references cited therein:
x1x∈C,
xn1 1−anxnanSnyn,
yn 1−bnxnbnTnxn, n∈N,
1.2
In this paper, we introduce a new one-step iterative process to compute the common fixed points of two asymptotically nonexpansive mappings. Let S, T : C → C be two asymptotically nonexpansive mappings. Then, our process reads as follows:
x1x∈C,
xn1anSnxn 1−anTnxn, n∈N,
1.3
where{an}is a sequence in0,1.
This process is computationally simpler than 1.2 to approximate common fixed points of two mappings. It is worth noting that our process is of independent interest. Neither
1.2implies1.3nor conversely. However, both1.2and1.3reduce to Mann-type iterative process whenT I, that is, the identity mapping is as follows:
x1x∈C,
xn1anSnxn 1−anxn, n∈N.
1.4
Remark 1.1. The question may arise that one needs two different sequences {sn} and {tn}
for the mappings S and T used in1.3, but it is readily answered when one takeskn
sup{sn, tn}. Henceforth, we will take only one sequence{kn}which works equally good for
both mappingsSandT.
Let us recall the following definitions.
A Banach spaceEis said to satisfy Opial’s condition11, if for any sequence{xn}in
E,xn ximplies that
lim sup
n→ ∞ xn−x<lim supn→ ∞ xn−y ∀y∈Ewithy /x. 1.5
Examples of Banach spaces satisfying this condition are Hilbert spaces and all spaces
lp 1< p <∞. On the other hand,Lp0,2π with 1< p /2fails to satisfy Opial’s condition.
A mappingT :C → Eis called demiclosed with respect toy∈E if for each sequence
{xn}inCand eachx∈E, xn xandTxn → yimply thatx∈CandTxy.
A Banach spaceEis said to satisfy the Kadec Klee property if for every sequence{xn} inEconverging weakly toxtogether withxnconverging strongly tox, {xn}converges strongly to x. Uniformly convex Banach spaces, Banach spaces of finite dimension, and reflexive locally uniform convex Banach spaces are some of the examples which satisfy the Kadec Klee property.
Next, we state the following useful lemmas.
Lemma 1.2see12. Let{δn}, {βn},and{γn}be three sequences of nonnegative numbers such thatβn≥1and
δn1≤βnδnγn ∀n∈N. 1.6
Lemma 1.3 see13. Suppose thatE is a uniformly convex Banach space and0 < p ≤ tn ≤
q <1for all positive integersn.Also, suppose that{xn}and{yn}are two sequences of Esuch that
lim supn→ ∞xn ≤ r, lim supn→ ∞yn ≤ r, andlimn→ ∞tnxn 1−tnyn r hold for some
r≥0.Then,limn→ ∞xn−yn0.
Lemma 1.4see14,15. LetEbe a uniformly convex Banach space and letCbe a nonempty closed convex subset ofE. LetTbe an asymptotically nonexpansive mapping ofCinto itself. Then,I−Tis demiclosed with respect to zero.
Lemma 1.5see16. LetCbe a convex subset of a uniformly convex Banach spaceE. Then, there is a strictly increasing and continuous convex functiong:0,∞ → 0,∞withg0 0such that for every Lipschitzian mapU:C → Cwith Lipschitz constantL≥1,the following inequality holds:
Utx 1−ty−tUx 1−tUy
≤Lg−1x−y −L−1Ux−Uy ∀x, y∈C, t∈0,1. 1.7
Letωw{xn}denote the set of all weak subsequential limits of a bounded sequence {xn}inE. Then, the following is actually Lemma 3.2 of Falset et al.16.
Lemma 1.6. LetEbe a uniformly convex Banach space with its dualE∗ satisfying the Kadec Klee property. Assume that{xn}is a bounded sequence such thatlimn→ ∞txn 1−tp1−p2exists for allt∈0,1and for allp1, p2∈ωw{xn}.Then,ωw{xn}is a singleton.
2. Some preparatory lemmas
In this section, we will prove the following important lemmas. In the sequel, we will write
FFS∩FTfor the set of all common fixed points of the mappingsSandT.
Lemma 2.1. LetCbe a nonempty closed convex subset of a normed spaceE. LetS, T :C → Cbe asymptotically nonexpansive mappings. Let{xn}be the process as defined in1.3, where{an}is a sequence inδ,1−δfor someδ∈0,1. IfF /φ,thenlimn→ ∞xn−x∗exists for allx∗∈F.
Proof. Letx∗∈F,then
xn1−x∗anSnxn 1−anTnxn−x∗
anSnxn−x∗ 1−anTnxn−x∗ ≤anSnx
n−x∗ 1−anTnxn−x∗ ≤anknxn−x∗ 1−anknxn−x∗ knxn−x∗.
2.1
Thus, byLemma 1.2, limn→ ∞xn−x∗exists for eachx∗∈F.
Lemma 2.2. LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceE. Let
1.3satisfying
xn−Snxn ≤ Snxn−Tnxn, n∈N. 2.2
IfF /φ, thenlimn→ ∞Sxn−xn0limn→ ∞Txn−xn.
Proof. ByLemma 2.1, limn→ ∞xn−x∗exists. Suppose that
lim
n→ ∞xn−x
∗c 2.3
for somec≥0.Then,Snxn−x∗ ≤knxn−x∗implies that
lim sup
n→ ∞ S
nx
n−x∗ ≤c. 2.4
Similarly, we have
lim sup
n→ ∞ T
nx
n−x∗ ≤c. 2.5
Further, limn→ ∞xn1−x∗cgives that
lim
n→ ∞anS
nx
n−x∗ 1−anTnxn−x∗c. 2.6
ApplyingLemma 1.3, we obtain that
lim
n→ ∞S
nx
n−Tnxn0. 2.7
But then by the conditionxn−Snxn ≤ Snxn−Tnxn,
lim sup
n→ ∞ xn−S
nxn ≤0. 2.8
That is,
lim
n→ ∞xn−S
nxn0. 2.9
Also, thenxn−Tnxn ≤ xn−SnxnSnxn−Tnxnimplies that
lim
n→ ∞xn−T
nxn0. 2.10
Now, by definition of{xn}, xn1−Tnxn ≤anSnxn−Tnxnso that
lim
n→ ∞xn1−T
Then,xn1−Snxn ≤ xn1−TnxnSnxn−Tnxnimplies
lim
n→ ∞xn1−S
nxn0. 2.12
Similarly, byxn1−xn ≤ xn1−Tnxnxn−Tnxn,we have
lim
n→ ∞xn1−xn0. 2.13
Next,
xn1−Sxn1 ≤ xn1−Sn1xn1Sn1xn1−Sn1xnSn1xn−Sxn1
≤ xn1−Sn1xn1kn1xn1−xnk1Snxn−xn1
2.14
yields
lim
n→ ∞xn−Sxn0. 2.15
Moreover,
Sxn1−Txn1 ≤ Sxn1−Sn1xn1Sn1xn1−Tn1xn1
Tn1x
n1−Tn1xnTn1xn−Txn1
≤k1xn1−Snxn1Sn1xn1−Tn1xn1
kn1xn1−xnk1Tnxn−xn1
≤k1xn1−SnxnSnxn−Snxn1
Sn1x
n1−Tn1xn1kn1xn1−xn
k1Tnxn−xn1
≤k1xn1−Snxnknxn−xn1
Sn1x
n1−Tn1xn1kn1xn1−xn
k1Tnxn−xn1
2.16
gives by2.7,2.11,2.12, and2.13that
lim
n→ ∞Sxn−Txn0. 2.17
In turn, by2.15and2.17, we get
lim
n→ ∞xn−Txn0. 2.18
Lemma 2.3. LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceE. Let
S, T :C → Cbe asymptotically nonexpansive mappings and{xn}as defined in1.3. Then, for any
p1, p2∈F, limn→ ∞txn 1−tp1−p2exists for allt∈0,1.
Proof. ByLemma 2.1, limn→ ∞xn−pexists for allp∈Fand so{xn}is bounded. Thus, there
exists a real number r > 0 such that{xn} ⊆ D ≡ Br0∩C,so that D is a closed convex
bounded nonempty subset of C.Put
unt txn 1−tp1−p2. 2.19
Notice that limn→ ∞un0 p1−p2and limn→ ∞un1 xn−p2exist as in the proof of Lemma 2.1.
DefineWn:D → Dby
WnxanSnx 1−anTnx. 2.20
It is easy to verify thatWnxnxn1, Wnppfor allp∈Fand
Wnx−Wny ≤knx−y ∀x, y∈C, n∈N. 2.21
Set
Rn,mWnm−1Wnm−2· · ·Wn, m∈N,
vn,mRn,mtxn 1−tp1−tRn,mxn 1−tp1.
2.22
Then,Rn,mx−Rn,my ≤nmjn−1kjx−y, Rn,mxnxnm,andRn,mppfor allp∈F.
ApplyingLemma 1.5withxxn, yp1, URn,m, and using the facts that
∞
k1kn− 1<∞and limn→ ∞xn−pexist for allp∈F,we obtainvn,m → 0 asn → ∞and for allm≥1.
Finally, from the inequality,
unmt txnm 1−tp1−p2
tRn,mxn 1−tp1−p2
≤vn,mRn,mtxn 1−tp1−p2
≤vn,m nm−1
jn
kjtxn 1−tp1−p2
vn,m nm−1
jn
kjunt,
it follows that
lim sup
n→ ∞ unt≤lim infn→ ∞ unt. 2.24
Hence, limn→ ∞txn 1−tp1−p2exists for allt∈0,1.
3. Common fixed point approximations by weak convergence
Here, we will approximate common fixed points of the mappingsSandTthrough the weak convergence of the process{xn}defined in 1.3. Our first result in this direction uses the Opial’s condition and the second one the Kadec Klee property.
Theorem 3.1. Let E be a uniformly convex Banach space satisfying the Opial’s condition and
C, S, T, and let{xn}be as inLemma 2.2. IfF /φ, then{xn}converges weakly to a common fixed point ofSandT.
Proof. Letx∗∈F,then as proved inLemma 2.1, limn→ ∞xn−x∗exists. Now, we prove that
{xn}has a unique weak subsequential limit inF.To prove this, letz1andz2be weak limits of the subsequences{xni}and{xnj}of{xn},respectively. ByLemma 2.2, limn→ ∞xn−Sxn0
andI−Sare demiclosed with respect to zero fromLemma 1.4. Therefore, we obtainSz1
z1.Similarly,Tz1z1.Again, in the same way, we can prove thatz2 ∈F.Next, we prove the uniqueness. For this, suppose thatz1/z2,then by the Opial’s condition
lim
n→ ∞xn−z1nlimi→ ∞xni−z1<nlimi→ ∞xni −z2nlim→ ∞xn−z2 lim
nj→ ∞xnj−z2<nlimj→ ∞xnj−z1nlim→ ∞xn−z1.
3.1
This is a contradiction. Hence,{xn}converges weakly to a point inF.
Theorem 3.2. LetEbe a uniformly convex Banach space with its dualE∗satisfying the Kadec Klee property. LetC, S, T,and{xn}be as inLemma 2.2. IfF /φ, then{xn}converges weakly to a common fixed point ofSandT.
Proof. By the boundedness of{xn}and reflexivity ofE,we have a subsequence{xni}of{xn}
that converges weakly to somep in C. ByLemma 2.2, we have limi→ ∞xni −Sxni 0
limi→ ∞xni−Txni. This givesp∈F.To prove that{xn}converges weakly top,suppose that {xnk}is another subsequence of{xn}that converges weakly to someqinC.Then, by Lemmas
2.2and1.4,p, q ∈ W∩F,whereW ωw{xn}.Since limn→ ∞txn 1−tp−qexists for
allt∈0,1byLemma 2.3, therefore,p qfromLemma 1.6. Consequently,{xn}converges weakly top∈Fand this completes the proof.
By puttingTI,the identity mapping, in Theorems3.1and3.2, we have the following corollaries. Note that the conditionxn−Snxn ≤ Snxn−Tnxn, n ∈ N,becomes trivially
true in this case.
Corollary 3.3. Let Ebe a uniformly convex Banach space satisfying the Opial’s condition and let
Corollary 3.4. Let Ebe a uniformly convex Banach space with dual E∗ satisfying the Kadec Klee property. LetC, Sbe as inLemma 2.1and{xn}as in1.4. IfFS/φ, then{xn}converges weakly to a fixed point ofS.
4. Common fixed point approximations by strong convergence
We first prove a strong convergence theorem in general real Banach spaces as follows.
Theorem 4.1. LetEbe a real Banach space andC, {xn},and letS, T be as inLemma 2.1. IfF /φ, then{xn}converges strongly to a common fixed point ofSandTif and only if
lim inf
n→ ∞Dxn, F 0, 4.1
whereDx, F inf{x−p:p∈F}.
Proof. Necessity is obvious. Conversely, suppose that
lim inf
n→ ∞Dxn, F 0. 4.2
As in the proof ofLemma 2.1, we have
xn1−p ≤knxn−p. 4.3
This gives
Dxn1, F≤knDxn, F, 4.4
so that limn→ ∞Dxn, Fexists; but by hypothesis
lim inf
n→ ∞Dxn, F 0, 4.5
we have limn→ ∞Dxn, F 0.
Next, we show that {xn} is a Cauchy sequence in C. Let > 0 be given. Since limn→ ∞Dxn, F 0, there exists a constantn0such that for alln≥n0, we have
Dxn, F<
4. 4.6
In particular, inf{xn0−p:p∈F}< /4.Hence, there existsp∗∈Fsuch that
xn0−p∗<
2. 4.7
Now, form, n≥n0, we have
xnm−xn ≤ xnm−p∗x
n−p∗ ≤2xn0−p∗<2
2
Hence{xn}is a Cauchy sequence in a closed subsetCof a Banach spaceE,therefore, it must converge inC. Let limn→ ∞xn q.Now, limn→ ∞Dxn, F 0 gives thatDq, F 0; but as
being well known,Fis closed, therefore,q∈F.
Fukhar-ud-din and Khan gave the following so-called conditionAin17.
Two mappingsS, T : C → C, whereCis a subset ofE,are said to satisfy condition
Aif there exists a nondecreasing functionf :0,∞ → 0,∞withf0 0, fr>0 for
allr ∈0,∞such that eitherx−Tx ≥fDx, Forx−Sx ≥ fDx, Ffor allx ∈C whereDx, F inf{x−x∗:x∗∈F}.
Our next theorem is an application ofTheorem 4.1and makes use of conditionA.
Theorem 4.2. LetEbe a uniformly convex Banach space, and letC,{xn}be as in Lemma 2.2. Let
S, T :C → Cbe two asymptotically nonexpansive mappings satisfying conditionA.IfF /φ,then
{xn}converges strongly to a common fixed point ofSandT.
Proof. By Lemma 2.1, limn→ ∞xn −x∗exists for allx∗ ∈ F. Let it bec for some c ≥ 0.If
c 0,there is nothing to prove. Supposec > 0.Now,xn1−x∗ ≤ knxn−x∗gives that
Dxn1, F ≤ knDxn, F and so limn→ ∞Dxn, F exists byLemma 1.2. By using condition A,either
lim
n→ ∞fDxn, F≤nlim→ ∞xn−Txn0 4.9
or
lim
n→ ∞fDxn, F≤nlim→ ∞xn−Sxn0. 4.10
In both the cases,
lim
n→ ∞fDxn, F 0. 4.11
Since f is a nondecreasing function and f0 0, limn→ ∞Dxn, F 0. Now, applying
Theorem 4.2, we get the result.
Remark 4.3. When T I, both of the above theorems remain valid for the Mann iterative
process1.4.
Remark 4.4. Above theorems can also be proved using our process with error terms:
x1x∈C,
xn1anSnxnbnTnxncnun, n∈N,
4.12
whereanbncn1, ∞n1cn<∞and{un}is a bounded sequence inC.
Remark 4.5. Non-self-asymptotically nonexpansive mappings case can also be dealt with
Acknowledgment
This work was supported by Kyungnam University Research Fund, 2008.
References
1 S. S. Chang, J. K. Kim, and S. M. Kang, “Approximating fixed points of asymptotically quasi-nonexpansive type mappings by the Ishikawa iterative sequences with mixed errors,” Dynamic Systems and Applications, vol. 13, no. 2, pp. 179–186, 2004.
2 S. H. Khan and W. Takahashi, “Approximating common fixed points of two asymptotically nonexpansive mappings,”Scientiae Mathematicae Japonicae, vol. 53, no. 1, pp. 143–148, 2001.
3 J. K. Kim, “Convergence of Ishikawa iterative sequences for accretive Lipschitzian mappings in Banach spaces,”Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 553–561, 2006.
4 J. K. Kim, K. H. Kim, and K. S. Kim, “Convergence theorems of modified three-step iterative sequences with mixed errors for asymptotically quasi-nonexpansive mappings in Banach spaces,”
Panamerican Mathematical Journal, vol. 14, no. 1, pp. 45–54, 2004.
5 J. K. Kim, K. S. Kim, and Y. M. Nam, “Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces,”Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 159–172, 2007.
6 J. K. Kim, Z. Liu, and S. M. Kang, “Almost stability of Ishikawa iterative schemes with errors forφ -strongly quasi-accretive andφ-hemicontractive operators,”Communications of the Korean Mathematical Society, vol. 19, no. 2, pp. 267–281, 2004.
7 J. Li, J. K. Kim, and N. J. Huang, “Iteration scheme for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in Banach spaces,”Taiwanese Journal of Mathematics, vol. 10, no. 6, pp. 1419–1429, 2006.
8 S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2306–2315, 2007.
9 B. Xu and M. A. Noor, “Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 444–453, 2002.
10 W. Takahashi, “Iterative methods for approximation of fixed points and their applications,”Journal of the Operations Research Society of Japan, vol. 43, no. 1, pp. 87–108, 2000.
11 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
12 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,”Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181–1191, 2000.
13 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
14 Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,”Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 707–717, 2004.
15 G. Li and J. K. Kim, “Demiclosedness principle and asymptotic behavior for nonexpansive mappings in metric spaces,”Applied Mathematics Letters, vol. 14, no. 5, pp. 645–649, 2001.
16 J. G. Falset, W. Kaczor, T. Kuczumow, and S. Reich, “Weak convergence theorems for asymptotically nonexpansive mappings and semigroups,”Nonlinear Analysis: Theory, Methods & Applications, vol. 43, no. 3, pp. 377–401, 2001.
