Volume 2008, Article ID 562130,9pages doi:10.1155/2008/562130
Research Article
On Coincidence and Fixed-Point Theorems in Symmetric Spaces
Seong-Hoon Cho,1Gwang-Yeon Lee,1and Jong-Sook Bae2
1Department of Mathematics, Hanseo University, Chungnam 356-706, South Korea
2Department of Mathematics, Moyngji University, Youngin 449-800, South Korea
Correspondence should be addressed to Seong-Hoon Cho,[email protected] Received 28 August 2007; Revised 4 February 2008; Accepted 5 March 2008 Recommended by Lech Gorniewicz
We give an axiomC.Cin symmetric spaces and investigate the relationships betweenC.Cand axiomsW3,W4, andH.E. We give some results on coinsidence and fixed-point theorems in symmetric spaces, and also, we give some examples for the results of Imdad et al.2006.
Copyrightq2008 Seong-Hoon Cho 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
In 1, the author introduced the notion of compatible mappings in metric spaces and proved some fixed-point theorems. This concept of compatible mappings was frequently used to show the existence of common fixed points. However, the study of the existence of common fixed points for noncompatible mappings is, also, very interesting. In2, the author initially proved some common fixed-point theorems for noncompatible mappings.
In 3, the authors gave a notion E-A which generalizes the concept of noncompatible mappings in metric spaces, and they proved some common fixed-point theorems for noncompatible mappings under strict contractive conditions. In 4, the authors proved some common fixed-point theorems for strict contractive noncompatible mappings in metric spaces. Recently, in5the authors extended the results of3,4to symmetricsemimetric spaces under tight conditions. In 6, the author gave a common fixed-point theorem for noncompatible self-mappings in a symmetric spaces under a contractive condition of integral type.
In this paper, we give some common fixed-point theorems in symmetricsemimetric spaces and give counterexamples for the results of Imdad et al.5.
In order to obtain common fixed-point theorems in symmetric spaces, some axioms are needed. In5, the authors assumed axiomW3, and in6the author assumed axioms W3,W4, andH.E; seeSection 2for definitions.
We give another axiom for symmetric spaces and study their relationships inSection 2.
We give common fixed-point theorems of four mappings in symmetric spaces and give some examples which justifies the necessity of axioms inSection 3.
2. Axioms on symmetric spaces
A symmetric on a setXis a functiond:X×X→0,∞satisfying the following conditions:
idx, y 0,if and only ifxyforx, y∈X, iidx, y dy, x,for allx, y∈X.
Letdbe a symmetric on a setX. Forx∈Xand >0, letBx, {y∈X:dx, y< }.
A topologyτdonXdefined as follows:U∈τdif and only if for eachx∈U, there exists an >0 such thatBx, ⊂ U. A subsetSofXis a neighbourhood ofx ∈X if there exists U∈τdsuch thatx∈U⊂S. A symmetricdis a semimetric if for eachx∈Xand each >0, Bx, is a neighbourhood ofxin the topologyτd.
A symmetricresp., semimetricspaceX, dis a topological space whose topologyτd onXis induced by symmetricresp., semi-metricd.
The difference of a symmetric and a metric comes from the triangle inequality. Actually a symmetric space need not be Hausdorff. In order to obtain fixed-point theorems on a symmetric space, we need some additional axioms. The following axioms can be found in7.
W3for a sequence{xn}in X, x, y ∈ X, limn→∞dxn, x 0 and limn→∞dxn, y 0 implyxy.
W4for sequences{xn},{yn}inXandx∈X, limn→∞dxn, x 0 and limn→∞dyn, xn
0 imply limn→∞dyn, x 0.
Also the following axiom can be found in6.
H.E for sequences {xn},{yn} in X and x ∈ X, limn→∞dxn, x 0 and limn→∞dyn, x 0 imply limn→∞dxn, yn 0.
Now, we add a new axiom which is related to the continuity of the symmetricd.
C.C for sequences {xn} in X and x, y ∈ X, limn→∞dxn, x 0 implies limn→∞dxn, y dx, y.
Note that ifdis a metric, thenW3,W4,H.E, andC.Care automatically satisfied.
And ifτdis Hausdorff, thenW3is satisfied.
Proposition 2.1. For axioms in symmetric spaceX, d, one has 1(W4)⇒(W3),
2(C.C)⇒(W3).
Proof. Let{xn}be a sequence inXandx, y∈Xwith limn→∞dxn, x 0 and limn→∞dxn, y 0.
1By puttingyn yfor eachn∈N, we have limn→∞dxn, yn limn→∞dxn, y 0.
ByW4, we have 0limn→∞dyn, x dy, x.
2ByC.C, limn→∞dxn, x 0 impliesdx, y limn→∞dxn, y 0.
The following examples show that other relationships inProposition 2.1do not hold.
Example 2.2. W4H.EandW4C.Cand soW3H.EandW3C.Cby Proposition 2.11.
LetX 0,∞and let
dx, y
⎧⎪
⎨
⎪⎩
|x−y| x /0, y /0, 1
x x /0. 2.1
Then,X, dis a symmetric space which satisfiesW4but does not satisfyH.Efor xnn, ynn1. AlsoX, ddoes not satisfyC.C.
Example 2.3. H.EW3, and soH.EW4andH.EC.C.
LetX 0,1∪ {2}and let
dx, y
⎧⎨
⎩
|x−y| 0≤x≤1,0≤y≤1,
|x| 0< x≤1, y2 2.2
andd0,2 1.
Then, X, d is a symmetric space which satisfies H.E. Let xn 1/n. Then, limn→∞dxn,0 limn→∞dxn,2 0.Butd0,2/0 and hence the symmetric spaceX, d does not satisfyW3.
Example 2.4. C.CW4and soW3W4byProposition 2.12.
LetX{1/n:n1,2, . . .} ∪ {0}, and letd0,1/n 1/n nis odd,d0,1/n 1n is evenand
d 1
m,1 n
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩ 1
m− 1 n
mn is even, 1
m− 1 n
mn is odd and |m−n|1 , 1 mn is odd and|m−n|>2.
2.3
Then, the symmetric spaceX, dsatisfiesC.C but does not satisfyW4for xn 1/2n1andyn1/2n.
Example 2.5. C.CH.E.
LetX{1/n:n1,2, . . .} ∪ {0}, and let
d 1
m,1 n
⎧⎨
⎩ 1
m− 1 n
|m−n| ≥2 ,
1 |m−n|1 2.4
and d1/n,0 1/n. Then, X, d is a symmetric space which satisfies C.C. Let xn 1/n, yn 1/n1. Then, limn→∞dxn,0 limn→∞dyn,0 0.But limn→∞dxn, yn/0.
Hence, the symmetric spaceX, ddoes not satisfyH.E.
3. Common fixed points of four mappings
LetX, dbe a symmetricor semimetricspace and letf, gbe self-mappings ofX. Then, we say that the pairf, gsatisfies propertyE-A 3if there exists a sequence{xn}inX and a pointt∈Xsuch that limn→∞dfxn, t limn→∞dgxn, t 0.
A subsetSof a symmetric spaceX, dis said to bed-closed if for a sequence{xn} inS and a pointx ∈ X, limn→∞dxn, x 0 implies x ∈ S. For a symmetric spaceX, d, d-closedness impliesτd-closedness, and ifdis a semimetric, the converse is also true.
At first, we prove coincidence point theorems of four mappings satisfying the property E-Aunder some contractive conditions.
Theorem 3.1. Let X, d be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let A, B, S,andTbe self-mappings ofXsuch that
1AX⊂TXandBX⊂SX,
2the pairB, Tsatisfies property (E-A) (resp.,A, Ssatisfies property (E-A)), 3for anyx, y∈X,dAx, By≤mx, y, where
mx, y max{dSx, Ty,min{dAx, Sx, dBy, Ty},min{dAx, Ty, dBy, Sx}}, 3.1 4SXis ad-closed (τd-closed) subset ofX (resp.,TX is ad-closed(τd-closed) subset of
X).
Then, there existu, w∈Xsuch thatAuSuBwTw.
Proof. From 2, there exist a sequence {xn} in X, and a point t ∈ X such that limn→∞dTxn, t limn→∞dBxn, t 0.
From 1, there exists a sequence {yn} in X such that Bxn Syn and hence limn→∞dSyn, t 0. ByH.E, limn→∞dBxn, Txn limn→∞dSyn, Txn 0.
From4, there exists a pointu∈Xsuch thatSut.
From3, we have d Au, Bxn
≤max
d Su, Txn
,min
dAu, Su, d Bxn, Txn
,min
d Au, Txn
, d Bxn, Su . 3.2 By takingn→ ∞, we have limn→∞dAu, Bxn 0.ByW3, we getAuSu.
SinceAX⊂TX, there exists a pointw∈Xsuch thatAuTw.
We show thatTwBw.From3, we have dAu, Bw
≤max
dSu, Tw,min
dAu, Su, dBw, Tw ,min
dAu, Tw, dBw, Su max
dTw, Tw,min
dAu, Au, dBw, Tw ,min
dAu, Au, dBw, Su 0.
3.3 Hence,AuBwand henceAuSuBwTw.
For the existence of a common fixed point of four self-mappings of a symmetric space, we need an additional condition, so-called weak compatibility.
Recall that for self-mappings f and g of a set, the pair f, g is said to be weakly compatible8iffgx gfx, wheneverfx gx. Obviously, iff and g are commuting, the pairf, gis weakly compatible.
Theorem 3.2. Let X, d be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let A, B, S,andTbe self-mappings ofXsuch that
1AX⊂TXandBX⊂SX,
2the pairB, Tsatisfies property (E-A)(resp.,A, Ssatisfies property (E-A)), 3the pairsA, SandB, Tare weakly compatible,
4for anyx, y∈Xx /y, dAx, By< mx, y,
5SXis ad-closed(τd-closed) subset ofX(resp.,TXis ad-closed (τd-closed) subset of X).
Then,A, B, S,andThave a unique common fixed point inX.
Proof. FromTheorem 3.1, there exist u, w ∈ X such thatAu Su Tw Bw. From 3, ASuSAu,AAuASuSAuSSuandBTwTBwTTwBBw.
IfAu /w, then from4we have dAu, AAu
dAAu, Bw
<max
dSAu, Tw,min
dAAu, SAu, dBw, Tw ,min
dAAu, Tw, dBw, SAu max
dAAu, Au,0, dAAu, Au dAAu, Au
3.4 which is a contradiction.
Similarly, ifu /Bw, we have a contradiction. Thus,AuwSuTwBwu,and wis a common fixed point ofA, B, S,andT.
For the uniqueness, letzbe another common fixed point of A, B, S,andT. Ifw /z, then from4we get
dz, w dAz, Bw
<max
dSz, Tw,min
dAz, Sz, dBw, Tw ,min
dAz, Tw, dBw, Sz max
dz, w,min
dz, z, dw, w ,min
dz, w, dw, z dz, w
3.5
which is a contradiction. Hence,wz.
Remark 3.3. In the case ofABg andS T f inTheorem 3.1resp.,Theorem 3.2, we can show thatfandghave a coincidence pointresp.,fandghave a unique common fixed pointwithout making the assumptiongX⊂fX.
Recently, R. P. Pant and V. Pant4obtained the existence of a common fixed point of the pair off, gin a metric spaceX, dsatisfying the condition
P.Pfor anyx, y∈X, dgx, gy<max
dfx, fy,k 2
dfx, gx dfy, gy ,1
2
dfy, gx dfx, gy , 3.6 where 1≤k <2.
Also in5, the authors tried to extend the result of 4to symmetric spaces which satisfy axiomW3.
Now, we will extend R. P. Pant and V. Pant’s result to symmetric spaces which satisfy additional conditionsH.EandC.C.
Theorem 3.4. Let X, d be a symmetric(semimetric) space that satisfies (H.E) and (C.C) and let A, B, S,andTbe self-mappings ofXsuch that
1AX⊂TXandBX⊂SX,
2the pairB, Tsatisfies property (E-A) (resp.,A, Ssatisfies property (E-A)),
3for any x, y ∈ X,dAx, By ≤ m1x, y, where m1x, y max{dSx, Ty,k/2 {dAx, Sx dBy, Ty},k/2{dAx, Ty dBy, Sx}}, 0< k <2,
4SXis ad-closed (τd-closed) subset ofX(resp.,TXis ad-closed (τd-closed) subset of X).
Then, there existu, w∈Xsuch thatAuSuBwTw.
Proof. As in the proof ofTheorem 3.1, there exist sequences{xn},{yn}inXand a pointt∈X such that limn→∞dTxn, t limn→∞dBxn, t0 andBxnSyn. Hence, limn→∞dSyn, t0.
From4, there exists a pointu∈Xsuch thatSut.
We showAuSu.From3,we have d Au, Bxn
≤max
d Su, Txn ,k
2
dAu, Su d Bxn, Txn ,k
2
d Au, Txn
d Bxn, Su 3.7 In the above inequality, we taken→ ∞, byC.CandH.E, we have
dAu, Su≤max
0,k
2dAu, Su,k 2
dAu, Su
k
2dAu, Su.
3.8
Since 0< k/2<1, we getdAu, Su 0 and henceAuSu.
SinceAX⊂TX, there exists a pointw∈Xsuch thatAuTw.
We show thatTwBw.From3,we have dTw, Bw dAu, Bw
≤max
dSu, Tw,k 2
dAu, Su dBw, Tw ,k
2
dAu, Tw dBw, Su
max
dTw, Tw,k 2
dAu, Au dBw, Tw ,k
2
dAu, Au dBw, Su
max k
2dBw, Tw,k
2dBw, Su
k
2dBw, Tw.
3.9 Since 0 < k/2 < 1, we getdTw, Bw 0 and henceTw Bw. Therefore, we have AuSuBwTw.
Theorem 3.5. X, dbe a symmetric(semimetric) space that satisfies (H.E) and (C.C) and letA, B, S, andT be self-mappings ofXsuch that
1AX⊂TXandBX⊂SX,
2the pairB, Tsatisfies property (E-A) (resp.,A, Ssatisfies property (E-A)), 3the pairsA, SandB, Tare weakly compatible,
4for any x, y ∈ Xx /y, dAx, By < m2x, y,where m2x, y max{dSx, Ty, k/2{dAx, Sx dBy, Ty},1/2{dAx, Ty dBy, Sx}}, 0< k <2.
5SX is a d-closed (τd-closed) subset ofX (resp., TX is a d-closed(τd-closed) subset ofX).
ThenA, B, S,andThave a unique common fixed point inX.
Proof. FromTheorem 3.4, there exist pointsu, w ∈Xsuch thatAuSuTw Bw,AAu ASuSAuSSu,andBTwTBwTTwBBw.
We show thatAuw.IfAu /w, then from4we have dAu, AAu
dAAu, Bw
<max
dSAu, Tw,k 2
dAAu, SAu dBw, Tw ,1
2
dAAu, Tw dBw, SAu max
dAAu, Au,0, dAAu, Au
d AAu, Au .
3.10 which is a contradiction.
Similarly, ifu /Bw, we have a contradiction. ThusAuwSuTwBwu.
For the uniqueness, letw be another common fixed point ofA, B, S,andT. Ifw /z, then from4we get
dz, w dAz, Bw
<max
dSz, Tw,k 2
dAz, Sz dBw, Tw ,1
2
dAz, Tw dBw, Sz
max
dz, w,k 2
dz, z dw, w ,1
2
dz, w dw, z max
dz, w,0, dw, z
dz, w.
3.11 which is a contradiction. Hencewz.
Example 3.6. LetX 0,1and dx, y x−y2. Define self-mappingsA, B, S,andT by AxBx 1/2xandSxTxxfor allx∈X. Then, we have the following:
0 X, dis a symmetric space satisfying the propertiesH.EandC.C, 1AX⊂TXandBX⊂SX,
2the pairB, Tsatisfies propertyE-Afor the sequencexn1/n, n1,2,3, . . . , 3the pairsA, SandB, Tare weakly compatible,
4for anyx, y∈Xx /y,dAx, By< dSx, Ty≤mix, y, i1,2, 5SXis ad-closedτd-closedsubset ofX,
6A0B0S0T00.
Remark 3.7. In the case ofABg andS T f inTheorem 3.4resp.,Theorem 3.5, we can show thatfandghave a coincidence pointresp.,fandghave a unique common fixed pointwithout the condition1, that is,gX⊂fX.
The following example shows that the axiomsH.EandC.Ccannot be dropped in Theorem 3.4.
Example 3.8. LetX, dbe the symmetric space as inExample 2.2. Then, the symmetricddoes not satisfy bothH.EandC.C.
LetST fandABgbe self-mappings ofXdefined as follows:
fxxx≥0, gx
⎧⎪
⎪⎨
⎪⎪
⎩ 1
3x x >0, 1
3 x0.
3.12
Then, the condition3 resp.,4ofTheorem 3.4resp.,Theorem 3.5is satisfied fork1.
To show this, letn1x, y max{dfx, fy,1/2{dfx, gx dfy, gy},1/2{dfy, gx dfx, gy}}. We consider two cases.
Case 1. x0, y >0, n1x, y max
d0, y,1 2
d
0,1
3
d
y,1 3y
,1
2
d
0,1 3y
d
y,1
3
max 1
y,1 2
3 2
3y
,1 2
3 y
y−1 3
≥ 1 2
32
3y
y 3 3
2 > 1
3|y−1|d 1
3,1 3y
dgx, gy.
3.13
Case 2. x >0, y >0x /y,
n1x, y≥dfx, fy |x−y|> 1
3|x−y|dgx, gy. 3.14 Thus, the condition 3 resp.,4 of Theorem 3.4 resp.,Theorem 3.5 is satisfied.
Note thatfXis ad-closedτd-closedsubset ofX. Also, the pairf, gsatisfies property E-Aforxn n, but the pairf, ghas no coincidence points, and also the pairf, ghas no common fixed points.
Remark 3.9. Example 3.6satisfies all conditions of5, Theorems 2.1 and 2.2and satisfies also all conditions of5, Theorem 2.3.
Letφ:R→Rbe a function such that φ1φis nondecreasing onR, φ20< φt< tfor allt∈0,∞.
Note that fromφ1andφ2, we haveφ0 0.
On the studying of fixed points, various conditions ofφhave been studied by many different authors3,5,6.
Remark 3.10. The functions mix, y in Theorems 3.4 and 3.5 can be generalized to the compositionsφmix, yfori1,2.
Example 3.11. Let X, d be the symmetric space and A, B, S,and T be the functions as in Example 3.8. Recall thatX, dsatisfiesW3but does not satisfy bothH.EandC.C. Let φt 2/3t, t∈Randk3/2.Then, for anyx, y∈X,dAx, By≤φmix, yfori1,2.
Note that the pairsA, SandB, Tsatisfy propertyE-A, andAX⊂TX,BX⊂SX,andSX ared-closedτd-closed.
Therefore,A, B, S,andT satisfy all conditions of5, Theorem 2.4and satisfy also all conditions of5, Theorem 2.5. But the pairsA, SandB, Thave no points of coincidence, and also the pairsA, SandB, Thave no common fixed points.
Acknowledgments
The authors are very grateful to the referees for their helpful suggestions. The first author was supported by Hanseo University, 2007.
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