Volume 2010, Article ID 385986,11pages doi:10.1155/2010/385986
Research Article
Approximating Fixed Points of Some Maps in
Uniformly Convex Metric Spaces
Abdul Rahim Khan,
1Hafiz Fukhar-ud-din,
2and Abdul Aziz Domlo
31Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dahran 31261, Saudi Arabia
2Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan 3Department of Mathematics, Taibah University, P.O. Box 30002, Madinah Munawarah, Saudi Arabia
Correspondence should be addressed to Abdul Rahim Khan,[email protected]
Received 1 October 2009; Accepted 19 January 2010
Academic Editor: Mohamed A. Khamsi
Copyrightq2010 Abdul Rahim Khan 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.
We study strong convergence of the Ishikawa iterates of qasi-nonexpansive generalized nonexpansive maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.
1. Introduction and Preliminaries
LetCbe a nonempty subset of a metric spaceX, dand letT : C → Cbe a map. Denote the set of fixed points of T,{x ∈ C : Tx x} by F. The map T is said to bei quasi-nonexpansive ifF /φanddTx, p ≤ dx, pfor allx ∈Candp ∈ F,iik-Lipschitz if for somek >0,we havedTx, Ty≤kdx, yfor allx, y∈C; fork1,it becomes nonexpansive, andiiigeneralized nonexpansivecf.1and the references thereinif
dTx, Ty≤adx, ybdx, Tx dy, Tycdx, Tydy, Tx ∗
for allx, y∈Cwherea, b, c≥0 witha2b2c≤1.
Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive maps have been extensively studied in uniformly convex Banach spaces 1, 3–6. Senter and Dotson7established convergence of Mann type iterates of quais-nonexpansive maps under a condition in uniformly convex Banach spaces. In 1973, Goebel et al. 8 proved that generalized nonexpansive self maps have fixed points in uniformly convex Banach spaces. Based on their work, Bose and Mukerjee1proved theorems for the convergence of Mann type iterates of generalized nonexpansive maps and obtained a result of Kannan9under relaxed conditions. Maiti and Ghosh6generalized the results of Bose and Mukerjee1for Ishikawa iterates by using modified conditions of Senter and Dotson7 see, also10. For the sake of completeness, we state the result of Kannan9and its generalization by Bose and Mukerjee1.
Theorem 1.1 see9. LetCbe a nonempty, bounded, closed, and convex subset of a uniformly
convex Banach space. LetTbe a map ofCinto itself such that
iTx−Ty ≤1/2x−Tx 1/2y−Tyfor allx, y∈C,
iisupz∈Kz−Tz ≤δK/2,whereKis any nonempty convex subset ofCwhich is mapped
into itself byTandδKis the diameter ofK.
Then the sequence{xn}defined byxn1 1/2xn1/2Txnconverges to the fixed point ofT,where
x1is any arbitrary point ofC.
Theorem 1.2 see1. LetCbe a nonempty, bounded, closed, and convex subset of a uniformly
convex Banach space. LetTbe a map ofCinto itself such that
Tx−Ty≤ax−yb
x−Txy−Tycx−Tyy−Tx 1.1
for allx, y∈Cwherea, b, c≥0and3a2b4c≤1.Define a sequence{xn}inCforx1∈C, xn1
1−αnxnαnTxn, for alln≥1, where0< β≤αn≤γ <1.Then{xn}converges to a fixed point of
T.
InTheorem 1.2, taking a c 0, b 1/2, andαn 1/2 for all n ≥ 1,it becomes
Theorem 1.1without requiring conditionii.
In 1970, Takahashi11introduced a notion of convexity in a metric spaceX, das follows: a mapW:X×X×I → Xis a convex structure inXif
du, Wx, y, λ≤λdu, x 1−λdu, y 1.2
In the sequel, we assume thatCis a nonempty convex subset of a convex metric space
XandTis a selfmap onC. For an initial valuex1∈C,we define the Ishikawa iteration scheme
inCas follows:
x1∈C,
xn1WTyn, xn, αn
,
ynW
Txn, xn, βn
∀n≥1,
1.3
where{αn}and{βn}are control sequences in0,1.
If we chooseβn0,then1.3reduces to the following Mann iteration scheme:
x1∈C, xn1WTxn, xn, αn, ∀n≥1, 1.4
where{αn}is a control sequence in0,1.
IfXis a normed space withCas its convex subset, thenWx, y, λ λx 1−λyis a convex structure inX; consequently1.3and1.4, respectively, become
x1∈C, xn1 1−αnxnαnTyn,
yn
1−βn
xnβnTxn ∀n≥1.
x1∈C, xn1 1−αnxnαnTxn, ∀n≥1,
1.5
where{αn}and{βn}are control sequences in0,1.
A convex metric spaceX is said to be uniformly convex11if for arbitrary positive numbersandr, there existsα>0 such that
d
z, W
x, y,1
2
≤r1−α 1.6
wheneverx, y, z∈X, dz, x≤r, dz, y≤randdx, y≥r.
In 1989, Maiti and Ghosh6generalized the two conditions due to Senter and Dotson
7. We state all these conditions in convex metric spaces:
LetT be a map with nonempty fixed point setFanddx, F infp∈Fdx, p.ThenT is said
to satisfy the following Condotions.
Condition 1. If there is a nondecreasing function f : 0,∞ → 0,∞with f0 0 and
fr>0 for allr >0 such thatdx, Tx≥fdx, Fforx∈C.
Condition 2. If there exists a real numberk >0 such thatdx, Tx≥kdx, Fforx∈C.
Condition 3. If there is a nondecreasing functionf:0,∞ → 0,∞withf0 0 andfr>
0 for allr >0 such thatdx, Ty≥fdx, Fforx∈Cand all correspondingyWTx, x, t
where 0≤t≤β <1.
Condition 4. If there exists a real numberk >0 such thatdx, Ty≥kdx, Fforx∈Cand all
Note that ifT satisfies Condition 1resp., 3, then it satisfies Condition2 resp., 4. We also note that Conditions1 and 2 become Conditions A and B, respectively, of Senter and Dotson7while Conditions3and4become Conditions I and II, respectively, of Maiti and Ghosh6in a normed space. Further, Conditions3and4reduce to Conditions1and2, respectively, whent0.
In this note, we present results under relaxed control conditions which generalize the corresponding results of Kannan9, Bose and Mukerjee1, and Maiti and Ghosh6from uniformly convex Banach spaces to uniformly convex metric spaces. We present sufficient conditions for the convergence of Ishikawa iterates ofk−Lipschitz maps to their fixed points
in convex metric spaces and improve3, Lemma 2. A necessary and sufficient condition is obtained for the convergence of a sequence to fixed point of a generalized nonexpansive map in metric spaces.
We need the following fundamental result for the developmant of our results.
Theorem 1.3 see 20. Let X be a uniformly convex metric space with a continuous convex
structureW :X×X×0,1 → X.Then for arbitrary positive numbersandr, there existsα>0
such that
dz, Wx, y, λ≤r1−2 min{λ,1−λ}α 1.7
for allx, y, z∈X, dz, x≤r, dz, y≤r, dx, y≥randλ∈0,1.
2. Convergence Analysis
We prove a lemma which plays key role to establish strong convergence of the iterative schemes1.3and1.4.
Lemma 2.1. LetXbe a uniformly convex metric space. LetCbe a nonempty closed convex subset
ofX, T : C → Ca quasi-nonexpansive map and{xn}as in 1.3. If ∞n0αn1−αn ∞and
0≤βn≤β <1,thenlim infn→ ∞dxn, Tyn 0.
Proof. Forp∈F, we consider
dxn1, pdp, WTyn, xn, αn
≤αnd
p, Tyn
1−αnd
p, xn
≤αnd
p, yn
1−αnd
p, xn
αnd
p, WTxn, xn, βn
1−αnd
p, xn
≤αnβnd
p, Txn
αn
1−βn
dp, xn
1−αnd
p, xn
≤αnβnd
p, xn
αn
1−βn
dp, xn
1−αnd
p, xn
dxn, p
.
2.1
This implies that the sequence {dxn, p} is nonincreasing and bounded below. Thus
For anyp∈F, we have that
dxn, Tyn
≤dxn, p
dTyn, p
≤dxn, p
dyn, p
dxn, p
dp, WTxn, xn, βn
≤dxn, p
βnd
Txn, p
1−βn
dxn, p
≤dxn, p
βnd
xn, p
1−βn
dxn, p
2dxn, p
.
2.2
Since limn→ ∞dxn, p exists, so dxn, Tyn is bounded and hence infn≥1dxn, Tyn
exists. We show that infn≥1dxn, Tyn 0. Assume that infn≥1dxn, Tyn σ >0.
Then
dxn, Tyn
≥dxn, p
· σ
dxn, p
≥dxn, p
· σ
dx1, p
.
2.3
Hence byTheorem 1.3, there existsασ/dx1, p>0 such that
dxn−1,p
dWTyn, xn, αn
, p
≤dxn, p
1−2 min{αn,1−αn}α
≤dxn, p
1−2αn1−αnα.
2.4
That is,
2cαn1−αnα≤d
xn, p
−dxn1, p
. 2.5
Takingm1 and summing up them1terms on the both sides in the above inequality, we have
2cα
m
n1
αn1−αn≤d
p, x1
−dp, xm
∀m≥1. 2.6
Letm → ∞. Then, we have
∞ ≤dp, x1
<∞. 2.7
In the light of above result, we can construct subsequences{xni}and{yni}of{xn}and {yn}, respectively, such that limi→ ∞dxni, Tyni 0 and hence lim infn→ ∞dxn, Tyn 0.
Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.
Theorem 2.2. LetXbe a uniformly convex complete metric space with continuous convex structure
and let Cbe its nonempty closed convex subset. LetT be a continuous quasi-nonexpansive map ofC
into itself satisfying Condition3. If{xn}is as in1.3, where ∞n0αn1−αn ∞and0≤βn≤β <
1, then{xn}converges to a fixed point ofT.
Proof. In Lemma 2.1, we have shown that dxn1,p ≤ dxn, p. Therefore dxn1, F ≤
dxn, F. This implies that the sequence{dxn, F}is nonincreasing and bounded below. Thus
limn→ ∞dxn, Fexists. Now by Condition3, we have
lim inf
n→ ∞ fdxn, F≤lim infn→ ∞ d
Tyn, xn
0. 2.8
Using the properties off,we have lim infn→ ∞dxn, F 0. As limn→ ∞dxn, Fexists,
therefore limn→ ∞dxn, F 0.
Now, we show that{xn}is a Cauchy sequence. For >0,there exists a constantn0such
that for alln≥n0,we havedxn, F< /4.In particular,dxn0, F< /4.That is, inf{dxn0, p:
p∈F}< /4.There must existp∗∈Fsuch thatdxn0, p∗< /2.Now, form, n≥n0, we have
that
dxnm, xn≤d
xnm, p∗
dxn, p∗
≤2dxn0, p ∗
< .
2.9
This proves that {xn} is a Cauchy sequence in C. Since C is a closed subset of a
complete metric spaceX,therefore it must converge to a pointqinC. Finally, we prove thatqis a fixed point ofT.
Since
dq, F≤dq, xn
dxn, F, 2.10
thereforedq, F 0.AsFis closed, soq∈F.
Chooseβn 0 for alln ≥ 1,in the above theorem; it reduces to the following Mann
type convergence result.
Theorem 2.3. LetXbe a uniformly convex complete metric space with continuous convex structure
and let Cbe its nonempty closed convex subset. LetT be a continuous quasi-nonexpansive map of
Cinto itself satisfying Condition1. If{xn}is as in1.4, where n∞0αn1−αn ∞, then{xn}
converges to a fixed point ofT.
Theorem 2.4. LetXandCbe as inTheorem 2.3. LetTbe a continuous generalied nonexpansive map
ofCinto itself with at least one fixed point. If{xn}is as in1.3, where ∞n0αn1−αn ∞and
0≤βn≤β <1,then{xn}converges to a fixed point ofT.
Proof. Letpbe any fixed point ofT.Then settingypin∗, we have
dTx, p≤acdx, pbdx, Tx cdTx, p
≤abcdx, p bcdTx, p, 2.11
which implies
dTx, p≤ abc
1−b−cd
x, p≤dx, p. 2.12
ThusTis quasi-nonexpansive.
For anyy∈C,we also observe that
dTy, p≤acdy, pbdy, TycdTy, p. 2.13
IfyWTx, x, t,where 0≤t≤β <1,then
dy, pdWTx, x, t, p
≤tdTx, p 1−tdx, p
≤dx, p,
2.14
dy, xdWTx, x, t, x
≤tdx, Tx 1−tdx, x
tdx, Tx
≤tdx, pdTx, p
≤2tdx, p.
2.15
Using2.14in2.13, we have
dTy, p≤acdy, pbdy, TycdTy, p
≤acdy, pcdx, pdx, Tybdx, ydx, Ty
≤a2cdx, pbdx, y bcdx, Ty.
2.16
Also it is obvious that
Combining2.16and2.17, we get that
bdx, y 1bcdx, Ty≥1−a−2cdx, p
≥2bdx, p. 2.18
Now inserting2.15in2.18, we derive
1bcdx, Ty≥2bdx, p−bdx, y
≥2b1−tdx, p. 2.19
That is,
dx, Ty≥ 2b1−t
1bcd
x, p≥ 2b
1−β
1bc d
x, p, 2.20
where 2b1−t/1bc > 0.ThusT satisfies Condition4 and hence Condition3. The result now follows fromTheorem 2.2.
Remark 2.5. In the above theorem, we have assumed that the generalied nonexpansive map
T has a fixed point. It remains an open questions: what conditions ona, b, andcin∗are sufficient to guarantee the existence of a fixed point ofT even in the setting of a metric space.
Chooseβn0 for alln≥1 inTheorem 2.4to get the following Mann type convergence
result.
Theorem 2.6. LetX, C,andT be as inTheorem 2.4. If{xn}is as in1.4, where ∞n0αn1−αn
∞,then{xn}converges to a fixed point ofT.
Proof. For βn 0 for all n ≥ 1, y WTx, x,0 x,the inequality 2.20in the proof of
Theorem 2.4becomes
dx, Tx≥ 2b
1bcd
x, p. 2.21
Thus T satisfies Condition 2 and hence Condition 1 and so the result follows from Theorem 2.3.
The analogue of Kannan result in uniformly convex metric space can be deduced from Theorem 2.6by takingac0, b1/2, andαn1/2 for alln≥1as follows.
Theorem 2.7. LetXbe a uniformly convex complete metric space with continuous convex structure
and let Cbe its nonempty closed convex subset. LetT be a continuous map ofCinto itself with at
least one fixed point such thatdTx, Ty≤ 1/2dx, Tx 1/2dy, Tyfor allx, y ∈C. Then
the sequence{xn}wherex1∈Candxn1 WTxn, xn,1/2converges to a fixed point ofT.
Theorem 2.8. Let X, dbe a convex metric space and let C be its nonempty convex subset. Let
T be ak-Lipschitz selfmap of C.Let{xn}be the sequence as in 1.3, where{αn}and {βn}satisfy
i0≤αn, βn ≤1for alln≥1iilim infn→ ∞αn >0andiiilim infn→ ∞βn< k−1.Ifdxn1, xn
αndxn, Tynandxn → p,thenpis a fixed point ofT.
Proof. Letp∈C.Then
dp, Tp≤dxn, p
dxn, Tyn
dTyn, Tp
dxn, p
1
andxn1, xn d
Tyn, Tp
≤dxn, p
1
andxn1, xn kd
yn, p
dxn, p
1
andxn1, xn kd
WTxn, xn, βn
, p
≤dxn, p
1
andxn1, xn k
βnd
Txn, p
1−βn
dxn, p
≤dxn, p
1
andxn1, xn kβn
dTxn, Tp
dp, Tp
k1−βn
dxn, p
.
2.22
That is,
1−kβn
dp, Tp≤1k2βnk
1−βn
dxn, p
1
andxn1, xn,
2.23
Since lim infan>0,therefore there existsa >0 such thatan> afor alln≥1.This implies that
1−kβn
dp, Tp≤1k2β
nk
1−βn
dxn, p
1
adxn1, xn, 2.24
Taking lim sup on both the sides in the above inequality and using the condition lim infn→ ∞βn< k−1, we havedp, Tp 0.
Finally, using a generalized nonexpansive mapT on a metric spaceX, we provide a necessary and sufficient condition for the convergence of an arbitrary sequence{xn}inX to
a fixed point ofTin terms of the approximating sequence{dxn, Txn}.
Theorem 2.9. Suppose thatCis a closed subset of a complete metric spaceX, dandT :C → Cis
a continuous map such that for somea, b, c≥0, a2c <1, the following inequality holds:
dTx, Ty≤adx, ybdx, Tx dy, Tycdx, Tydy, Tx 2.25
for all x, y ∈ C. Then a sequence {xn} in C converges to a fixed point of T if and only if
Proof. Suppose that limn→ ∞dxn, Txn 0.First we show that{xn}is a Cauchy sequence in
C.To acheive this goal, consider:
dTxn, Txm≤adxn, xm b{dxn, Txn dxm, Txm}
c{dxn, Txm dxm, Txn}
≤adxn, xm b{dxn, Txn dxm, Txm}
c{dxn, xm dxm, Txm dxm, xn dxn, Txn}
a2cdxn, xm
bc{dxn, Txn dxm, Txm}
≤ab3c{dxn, Txn dxm, Txm}
a2cdTxn, Txm.
2.26
That is,
1−a−2cdTxn, Txm≤ab3c{dxn, Txn dxm, Txm}. 2.27
Since limn→ ∞dxn, Txn 0 and a2c < 1,therefore from the above inequality, it
follows that {Txn} is a Cauchy sequence in C. In view of closedness of C, this sequence
converges to an elementpofC.Also limn→ ∞dxn, Txn 0 gives that limn→ ∞xn p.Now
using the continuity ofT,we haveTp Tlimn→ ∞xn limn→ ∞Txn p.Hencep is a
fixed point ofT.
Conversely, suppose that{xn}converges to a fixed pointpofT.Using the continuity
ofT,we have that limn→ ∞Txnp.Thus limn→ ∞dxn, Txn 0.
Remark 2.10. Theorem 2.8improves Lemma 2 in3from real line to convex metric space
setting.Theorem 2.9is an extension of Theorem 4 in21to metric spaces. If we choosec0 inTheorem 2.9, it is still an improvement of21, Theorem 4.
Remark 2.11. We have proved our results2.1–2.8in convex metric space setting. All these
results, in particular, hold in Banach spaces if we setWx, y, λ λx 1−λy.
Acknowledgment
The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals for support during this research.
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