IN AN UNBOUNDED D -METRIC SPACE
BIJENDRA SINGH, SHISHIR JAIN, AND SHOBHA JAIN Received 16 March 2004 and in revised form 24 August 2004
Rhoades (1996) proved a fixed point theorem in a boundedD-metric space for a con- tractive self-map with applications. Here we establish a more general fixed point theorem in an unboundedD-metric space, for two self-maps satisfying a general contractive con- dition with a restricted domain ofx and y. This has been done by using the notion of semicompatible maps inD-metric space. These results generalize and improve the results of Rhoades (1996), Dhage et al. (2000), and Veerapandi and Rao (1996). These results also underline the necessity and importance of semicompatibility in fixed point theory of D-metric spaces. All the results of this paper are new.
1. Introduction
There have been a number of generalizations of metric spaces. One such generalization is generalized metric space orD-metric space initiated by Dhage [1] in 1992. He proved some results on fixed points for a self-map satisfying a contraction for complete and boundedD-metric spaces. Rhoades [4] generalized Dhage’s contractive condition by in- creasing the number of factors and proved the existence of unique fixed point of a self- map inD-metric space. Recently, motivated by the concept of compatibility for metric space, Singh and Sharma [5] introduced the concept ofD-compatibility of maps inD- metric space and proved some fixed point theorems using a contractive condition.
In [4], the following theorem has been established.
Theorem1.1 (Rhoades [4]). LetX be a complete and boundedD-metric space, and let T be a self-map ofX satisfyingD(Tx,T y,Tz)≤kMax{D(x,y,z),D(x,Tx,z),D(y,T y,z), D(x,T y,z),D(y,Tx,z)}, for allx,y,zinX, 0≤k <1. ThenThas a unique fixed pointpin XandTis continuous atp.
The object of this paper is to generalize this contraction by increasing the number of factors in it from five to ten. We involve two semicompatible self-maps in it, one of which is continuous. Further, the domain ofxand y from the whole space has been reduced to some orbits only. The first result gives a sufficient condition for existence of a unique fixed point withT is continuous and (S,T) semicompatible and the second one gives
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:5 (2005) 789–801 DOI:10.1155/IJMMS.2005.789
the same result ifScontinuous. For the sake of completeness, following Dhage [1,2], we recall some definitions and known results inD-metric space.
2. Preliminaries
In what follows (X,D) will denote aD-metric space,Nthe set of all natural numbers, and R+the set of all positive real numbers.
Definition 2.1. LetX be a nonempty set. A generalized metric (orD-metric) onXis a function:D:X×X×X→R+that satisfies the following conditions.
(D-1)D(x,y,z)=0 iffx=y=z(sufficiency).
(D-2)D(x,y,z)=D(p{x,y,z}), (symmetry) wherepis a permutation function.
(D-3)D(x,y,z)≤D(x,y,a) +D(x,a,z) +D(a,y,z), for allx,y,z, andainX(tetrahedral inequality).
The pair (X,D) is called a generalized metric (orD-metric) space. Geometrically aD- metricD(x,y,z) represents the perimeter of a triangle whose vertices arex,y, andz. Im- mediate examples of such a function are
(a)D(x,y,z)=Max{d(x,y),d(y,z),d(x,z)}, (b)D(x,y,z)=d(x,y) +d(y,z) +d(x,z).
Here,dis the ordinary metric onX.
Definition 2.2[1]. A sequence{xn}in aD-metric space is said toD-converge to a pointx if for anyε >0, there exists a positive integern0such thatD(xn,xm,x)< ε, for alln,m > n0. A sequence{xn}is said to be aD-Cauchy sequence if for eachε >0 there existsn0∈N such thatD(xn,xn+p,xn+p+t)< ε, for alln > n0, for allp,t∈N.Xis said to be complete if each Cauchy sequence of it converges to some point ofX.
In aD-metric space, ifDis continous in two variables, then the limit of sequence is unique, if it exists. Throughout this paper theD-metric is assumed to be continous in two variables.
Definition 2.3. Let (X,D) be aD-metric space andSbe a nonempty subset ofX. We define the diameter ofSasδd(S)=Sup{D(x,y,z)|x,y,z∈S}.
Definition 2.4[6]. LetTbe a multi-valued map onD-metric space (X,D). Letx0∈X. A sequence{xn}inXis said to be an orbit ofTatx0denoted byO(T,x0) ifxn−1∈Tn−1(x0), that is,xn∈Txn−1, for alln∈N. The set of all orbits ofTatx0is denoted byF(T,x0). An orbitO(T,x0) is said to be bounded if its diameter is finite.
Definition 2.5 [5]. Self mapsS and T on a D-metric space (X,D) are said to be D- compatible if limn→∞D(STxn,TSxn,z)=0, wherez=STxn orTSxn, when ever{xn}is a sequence inXsuch that limn→∞Txn=limn→∞Sxn=x∈X.
Definition 2.6. A pair (S,T) of self-mappings of aD-metric space is said to be semicom- patible if limn→∞STxn=Tx, when ever{xn} is a sequence inX such that limn→∞Txn
=limn→∞Sxn=x∈X.
It follows that (S,T) is semicompatible andSy=T ythenST y=TSy.
Propostion2.7. SandTareD-compatible self-maps on aD-metric space(X,D)andTis continuous then the pair(S,T)is semicompatible.
Proof. Let{Sxn} →u,{Txn} →ufor someu∈X. To show this,STxn→Tu. AsTis con- tinuousTSxn→Tu.
Now, as (S,T) is D-compatible we have limn→∞D(STxn,STxn,TSxn)=0. That is, limn→∞D(STxn,STxn+p,TSxn)=0. That is, limn→∞D(STxn,STxn+p,Tu)=0. That is, limn→∞STxn=Tu. Hence (S,T) is semicompatible.
Propostion2.8. SandTare semicompatible self-maps on aD-metric space(X,D)andT is continuous then(S,T)isD-compatible.
Proof. Let{Sxn} →u,{Txn} →uand asT is continuousTSxn→Tu. The semicompati- bility of (S,T) gives STxn → Tu. Now, limn→∞D(STxn,STxn,TSxn) =D(Tu,Tu,
Tu)=0. Hence (S,T) isD-compatible.
The following is an example of a pair of self-maps (S,T) which is semicompatible but not compatible. Further, it is shown that the semicompatibility of the pair (S,T) need not imply the semicompatibility of (T,S).
Example 2.9. LetX=[0, 1] and consider theD-metric space (X,D), whereDis defined by D(x,y,z)=Max{|x−y|,|y−z|,|z−x|}, for allx,y,z∈X. Define a self-map as follows:
Sx=x if 0≤x <1 2, Sx=1 ifx≥1
2.
(2.1)
LetIbe the identity map onXandxn=1/2−1/n. Then{Ixn} = {xn} →1/2 and{Sxn} → 1/2. Again,{ISxn} = {Sxn} →1/2=S(1/2). Thus (I,S) is not semicompatible though it is compatible. Also for any sequence{xn}inXsuch that{xn} →xand{Sxn} →xwe have {SIxn} = {Sxn} →x=Ix. Thus (S,I) is always semicompatible.
Remark 2.10. The above example gives an important aspect of semicompatibility inD- metric space as the pair (I,S) is commuting, weakly commuting, compatible, and weak compatible, still it is not semicompatible.
The following is an example of a pair of maps which is semicompatible but not com- patible.
Example 2.11. Let X=[0, 2], defineD(x,y,z)=Max{|x−y|,|y−z|,|z−x|}, for all x,y,z∈X. Define self-mapsAandSonXas follows:
Sx=1, x∈[0, 1),
2, x=1,
x+ 3
5 , x∈(1, 2], Ax=2, x∈[0, 1],
x
2, x∈(1, 2].
(2.2)
Takexn=2−1/(2n) then we haveS(1)=A(1)=2 andS(2)=A(2)=1. AlsoSA(1)= AS(1)=1 andSA(2)=AS(2)=2. HenceAxn→1 andSxn→1,ASxn→2, andSAxn→1.
Now,
nlim→∞DASxn,ASxm,Sy=D(2, 2, 2)=0,
nlim→∞MASxn,SAxn,ASxn=D(2, 1, 2)=1=0. (2.3)
Hence (A,S) is semicompatible but it is not compatible.
Propostion2.12. LetSandTbe two self-maps of aD-metric space(X,D)such thatS(X)⊆ T(X). For somex0∈Xdefine sequences{xn}and{yn}inX bySxn−1=Txn=yn, for all n∈N. Then
(i)O(T−1S,x0)= {x0,x1,x2,x3,...,xn,...}, (ii)O(ST−1,Sx0)= {y1,y2,y3,...,yn,...}.
Proof. AsSx0=Tx1we havex1∈T−1Sx0andSx1=Tx2givesx2∈T−1Sx1=(T−1S)2x0. Similarly,Sxn−1=Txngivesxn∈T−1Sxn−1=(T−1S)nx0. Again,
y1=Sx0, y2=Sx1∈ST−1Sx0=
ST−1Sx0, y3=Sx2∈ST−1ST−1Sx0=
ST−12Sx0. (2.4)
Similarly,yn∈(ST−1)n−1Sx0.
Letϕdenote the class of functionsφ:R+→R+which are upper semicontinuous non- decreasing andφ(t)< t, fort >0. If the orbit{yn}is bounded, defineγi=δd{yi,yi+1, yi+2,...},i=1, 2,.... Thenγnis finite for allnand also{γn}is a nonincreasing sequence andγn≥0, for alln. Henceγn→γ(γ≥0) asn→ ∞.
Lemma2.13. LetS,T,xn,yn,x0be as above. If (i)some orbit{yn} =O(ST−1,Sx0)is bounded, (ii)for allx,y,zinO(T−1S,x0), for someφ∈ϕ,
D(Sx,Sy,Sz)≤φMaxD(Tx,T y,Tz),D(Sx,Tx,Tz),D(Sy,T y,Tz),D(Sx,T y,Tz), D(Sy,Tx,Tz),D(Tx,T y,Sz),D(Sx,Tx,Sz),
D(Sy,T y,Sz),D(Sx,T y,Sz),D(Sy,Tx,Sz),
(2.5)
thenlimn→∞γn=0and{yn}is aD-Cauchy sequence inO(ST−1,Sx0).
Proof. We have from condition (ii) Dyn,yn+p,yn+p+t
=DSxn−1,Sxn+p−1,Sxn+p+t−1
≤φMaxDyn−1,yn+p−1,yn+p+t−1
,Dyn−1,yn,yn+p+t−1
, Dyn+p−1,yn+p,yn+p+t−1
,Dyn,yn+p−1,yn+p+t−1
, Dyn−1,yn+p,yn+p+t−1
,Dyn−1,yn+p−1,yn+p+t , Dyn−1,yn,yn+p+t
,Dyn+p−1,yn+p,yn+p+t , Dyn,yn+p−1,yn+p+t
,Dyn−1,yn+p,yn+p+t
≤φγn−1
.
(2.6)
Taking sup overpandtwe haveγn≤φ(γn−1) and lettingn→ ∞we getγ≤φ(γ)< γif γ >0, which is a contradiction. Henceγ=0, that is,γn→0, asn→ ∞.
Thus,D(yn,yn+p,yn+p+t)≤φ(γn−1)→0 asn→ ∞. Hence{yn}is aD-Cauchy sequence.
3. Main results
Theorem3.1. LetSandTbe self-maps of aD-metric space(X,D)satisfying the following.
(1)S(X)⊆T(X).
(2)The pair(S,T)is semicompatible andTis continuous.
(3)For somex0∈X, some orbit{yn} =O(ST−1,Sx0)is bounded and complete.
(4)For allx,y∈O(T−1S,x0)∪O(ST−1,Sx0)and for allz∈Xand for someφ∈ϕ, D(Sx,Sy,Sz)≤φMaxD(Tx,T y,Tz),D(Sx,Tx,Tz), D(Sy,T y,Tz),
D(Sx,T y,Tz), D(Sy,Tx,Tz),D(Tx,T y,Sz), D(Sx,Tx,Sz), D(Sy,T y,Sz), D(Sx,T y,Sz), D(Sy,Tx,Sz).
(3.1)
ThenSandThave a unique common fixed point.
Proof. Forx0∈X, construct sequences{xn}and{yn}inXasSxn−1=Txn=yn, for all n∈N. Then byLemma 2.13,{yn}is a Cauchy sequence inO(ST−1,Sx0) which is com- plete. Hence,
yn=Txn=Sxn−1−→u∈X. (3.2)
AsTis continuous and (S,T) is semicompatible we get
T2xn−→Tu, STxn−→Tu. (3.3)
Step 1. Puttingx=Txn,y=Txn,z=xnin condition (4) we get
DSTxn,STxn,Sxn≤φMax{DTTxn,TTxn,Txn,DSTxn,TTxn,Txn, DSTxn,TTxn,Txn,DSTxn,TTxn,Txn, DSTxn,TTxn,Txn,DTTxn,TTxn,Sxn, DSTxn,TTxn,Sxn,DSTxn,TTxn,Sxn, DSTxn,TTxn,Sxn
,DSTxn,TTxn,Sxn .
(3.4)
Taking limit asn→ ∞and using (3.2) and (3.3) we get
D(Tu,Tu,u)≤φD(Tu,Tu,u)=⇒Tu=u. (3.5) Step 2. Puttingx=xn,y=xn, andz=uin condition (4) we get
DSxn,Sxn,Su≤φMaxDTxn,Txn,Tu,DSxn,Txn,Tu,DSxn,Txn,Tu, DSxn,Txn,Tu,DSxn,Txn,Tu,DTxn,Txn,Su, DSxn,Txn,Su,DSxn,Txn,Su,DSxn,Txn,Su, DSxn,Txn,Su.
(3.6)
Taking limit asn→ ∞using (3.2) and (3.5) we get
D(u,u,Su)≤φD(u,u,Su)< D(u,u,Su) ifD(u,u,Su)>0, (3.7) which is a contradiction, henceD(u,u,Su)=0, that is,u=Su. Henceu=Su=Tu. That is,uis a common fixed point ofSandT.
Step 3(Uniqueness). Letwbe another common fixed point ofSandT, then
w=Sw=Tw. (3.8)
Puttingx=xn,y=xn, andz=win condition (4) we get
D(Sxn,Sxn,Sw)≤φMaxDTxn,Txn,Tw,DSxn,Txn,Tw,DSxn,Txn,Tw, DSxn,Txn,Tw,DSxn,Txn,Tw,DTxn,Txn,Sw, DSxn,Txn,Sw,DSxn,Txn,Sw,DSxn,Txn,Sw, DSxn,Txn,Sw.
(3.9)
Taking limit asn→ ∞using (3.2) to (3.8) we getD(u,u,w)≤φD(u,u,w)< D(u,u,w) ifD(u,u,w)>0 which is a contradiction, henceD(u,u,w)=0, that is,u=w. Hence,uis a unique common fixed point ofSandT.
Remark 3.2. It is clear from the above proof that the satisfaction of condition (4) ofTheorem 3.1for allx,y∈O(T−1S,x0)∪O(ST−1,Sx0) and for allz∈O(ST−1,Sx0) en- sures the existence of a common fixed point ofSandT.
Remark 3.3. In view ofProposition 2.7, in condition (2) ofTheorem 3.1the semicom- patibility of the maps can be replaced by compatibility.
The following theorem is a counterpart ofTheorem 3.1and establishes the existence of a unique common fixed point of a pair of semicompatible maps (S,T) whenSis con- tinuous.
Theorem3.4. Let SandT be self-maps of aD-metric space(X,D)satisfying conditions (1), (3) ofTheorem 3.1and
(1)the pair(S,T)is semicompatible and S is continuous,
(2)for allx,y∈O(T−1S,x0)and for allz∈X, and for someφ∈ϕ, D(Sx,Sy,Sz)≤φMaxD(Tx,T y,Tz),D(Sx,Tx,Tz),D(Sy,T y,Tz),
D(Sx,T y,Tz),D(Sy,Tx,Tz),D(Tx,T y,Sz),D(Sx,Tx,Sz), D(Sy,T y,Sz),D(Sx,T y,Sz),D(Sy,Tx,Sz).
(3.10) ThenSandThave a unique common fixed point.
Proof. Forx0∈X, construct sequences{xn}and{yn}inXas in the proof ofTheorem 3.1 thenSxn→u,Txn→u. AsSis continuous we getSTxn→Suand as (S,T) is semicompat- ible we getSTxn→Tu. As the limit of the sequence is unique we get
Su=Tu. (3.11)
Step 4. Puttingx=xn,y=xn, andz=uin condition (2) we get DSxn,Sxn,Su≤φMaxDTxn,Txn,Tu,DSxn,Txn,Tu,
DSxn,Txn,Tu,DSxn,Txn,Tu,DSxn,Txn,Tu, DTxn,Txn,Su,DSxn,Txn,Su,DSxn,Txn,Su, DSxn,Txn,Su,DSxn,Txn,Su.
(3.12)
Taking limit asn→ ∞using (3.2) and (3.11) we get
D(u,u,Su)≤φD(u,u,Su)< D(u,u,Su) ifD(u,u,Su)>0, (3.13) which is a contradiction. Hence u=Su and we get u the common fixed point of S andT.
The uniqueness of fixed point follows fromStep 3ofTheorem 3.1.
Remark 3.5. In view of Propositions2.7and2.8, condition (1) of the above theorem is not in the close reach of compatibility, weak commutativity. Thus it highlights the necessity and importance of semicompatibility in the fixed point theory ofD-metric spaces.
InTheorem 3.1if we takeφ(t)=λt, for allt∈R+,λ <1. Asλ <1,φ∈ϕand we get the following.
Corollary3.6. LetSandTbe self-maps of aD-metric space(X,D)satisfying conditions (1), (3), (2), ofTheorem 3.1or (1) ofTheorem 3.4and
(1)for allx,y∈O(T−1S,x0)∪O(ST−1,Sx0),z∈X, there existsλ∈[0, 1)such that D(Sx,Sy,Sz)≤λMaxD(Tx,T y,Tz),D(Sx,Tx,Tz),D(Sy,T y,Tz),
D(Sx,T y,Tz),D(Sy,Tx,Tz),D(Tx,T y,Sz), D(Sx,Tx,Sz),D(Sy,T y,Sz),D(Sx,T y,Sz), D(Sy,Tx,Sz).
(3.14)
ThenSandThave a unique common fixed point.
In the above corollary if we takeT=I, the identity map onX, then conditions (1), (2) ofTheorem 3.1are satisfied trivially and we get the following.
Corollary3.7. LetSbe a self-map on aD-metric space(X,D)such that forx0∈X the orbitO(S,x0)is bounded and complete and for allx,y∈O(S,x0), for allz∈X, there exists λ∈[0, 1)such that
D(Sx,Sy,Sz)≤λMaxD(x,y,z),D(Sx,x,z),D(Sy,y,z),D(Sx,y,z), D(Sy,x,z),D(x,y,Sz),D(Sx,x,Sz),D(Sy,y,Sz), D(Sx,y,Sz),D(Sy,x,Sz).
(3.15)
ThenShas a unique fixed point.
If we restrict the contractive condition of the above corollary to the maximum of those factors which containzin the third place of functionD, we get the following.
Corollary3.8. LetSbe a self-map on aD-metric space(X,D)such that for somex0∈X the orbitO(S,x0)is bounded and complete and for allx, y∈O(S,x0), for allz∈X, there existsλ∈[0, 1)such that
(A)D(Sx,Sy,Sz)≤λMax{D(x,y,z),D(Sx,x,z),D(Sy,y,z),D(x,Sy,z),D(y,Sx,z)}. ThenShas a unique fixed pointuinXandSis continuous atu.
Proof. The existence of the unique fixed pointufollows fromCorollary 3.8. To prove the continuity, let{zn} ⊂Xsuch thatzn→u. Puttingx=xm,y=xm, andz=znin condition (A) we have
DSxm,Sxm,Szn
≤λMaxDxm,xm,zn
,DSxm,xm,zn
,DSxm,xm,zn
, Dxm,Sxm,zn
,Dxm,Sxm,zn
. (3.16)
Taking limit asm→ ∞we get Du,u,Szn
≤λMaxDu,u,zn
,Du,u,zn
,Du,u,zn
,Du,u,zn
,Du,u,zn . (3.17) Thus, D(u,u,Szn)≤λD(u,u,zn) and hence limn→∞D(u,u,Szn)=0. Now, D(Szn+p, Szn,u)≤D(Szn+p,u,u) +D(u,Szn,u) implies that limn→∞D(Szn+p,Szn,u)=0. Thus,{Szn} D-converges tou. Therefore,Sis continuous atu.
Remark 3.9. The above corollary improves and generalizes [4, Theorem 1] in which the required domain ofxandyin the contractive condition is needed to be the whole space Xwhile in this corollary the required domain ofx,yis just an orbitO(S,x0). Further, in the above corollary boundedness of just an orbit is taken where as in [4] the boundedness of the whole space was assumed.
Example 3.10 (of Corollary 3.8). Let X = {2n|n∈Z} ∪ {0} = {1, 1/2, 1/22,..., 0, 2, 22,...},d(x,y)= |x−y|, for allx,y∈X. Consider
D(x,y,z)=Max|x−y|,|y−z|,|z−x|
∀x,y,z∈X. (3.18) LetSbe a self-map onXgiven bySx=x/2, for allx∈X.
ThenO(S, 1)= {1, 1/2, 1/22,...}andO(S, 1)=O(S, 1)∪ {0}. Thus the orbitO(S, 1) is bounded and complete, howeverXis not bounded. To show the existence ofλof condi- tion (A), letM(x,y,z)=Max{D(x,y,z), D(Sx,x,z),D(Sy,y,z),D(x,Sy,z),D(y,Sx,z)}. Case 1. x,y, andz∈O(S, 1), takingx,y, andzin descending order. We takex=1/2n−1, y=1/2n+p−1,z=1/2n+p+t−1thenSx=1/2n,Sy=1/2n+p,Sz=1/2n+p+t. Now,
D(Sx,Sy,Sz)= 1 2n−
1 2n+p+t, M(x,y,z)=Max
1 2n−1−
1
2n+p+t−1, 1 2n+p−1−
1 2n+p+t−1, 1
2n− 1 2n+p+t−1
= 1 2n−1−
1 2n+p+t−1 =2
1 2n−
1 2n+p+t
.
(3.19)
Thus,D(Sx,Sy,Sz)=(1/2)M(x,y,z) in this case.
Note. In casex,z, andyare in descending order or elsez,x,yare in descending order, the computation can be made similarly and forλ=1/2 condition (A) is satisfied.
Case 2. x,y∈O(S, 1) andz=0. We takex=1/2n−1,y=1/2n+p−1,z=0 thenSx=1/2n, Sy=1/2n+p,Sz=0. Now,
D(Sx,Sy,Sz)= 1
2n, M(x,y,z)=Max 1
2n−1, 1 2n+p−1, 1
2n
= 1
2n−1. (3.20) Thus,D(Sx,Sy,Sz)=(1/2)M(x,y,z) in this case too.
Case 3. x,y∈O(S, 1),z=X−O(S, 1), andz=0. We takex=1/2n−1,y=1/2n+p−1,z= 2m+1, thenSx=1/2n,Sy=1/2n+p,Sz=2m. Now,
D(Sx,Sy,Sz)=2m− 1 2n+p, M(x,y,z)=Max
2m+1− 1
2n+p−1, 2m+1− 1
2n, 2m+1− 1 2n+p
, M(x,y,z)=2m+1− 1
2n+p =2
2m− 1 2n+p+1
>2
2m− 1 2n+p
=2D(Sx,Sy,Sz).
(3.21)
Thus for allx,y∈O(S, 1), for allz∈X, condition (A) is satisfied. Hence all the conditions ofCorollary 3.8are satisfied and “0” is the unique common fixed point ofS.
Remark 3.11. The above example shows that the assumption of boundedness of the whole spaceXin [4] can be removed.
In [3], Dhage et al. established the following result.
Theorem3.12 (Dhage et al. [3]). Let(X,D)be aD-metric space,f a self-map onX. Sup- pose that there existsx0inXsuch thatO(x0)is bounded andf-orbitally complete. Suppose also that f satisfies
(A)D(f x,f y,f z)≤λMax{D(x,y,z),D(x,f x,z)}, forx,y,z∈O(x0) for some0≤λ <1. Then f has a unique fixed point inX.
InCorollary 3.8, if we restrict the maximum to only the first two factors in the con- tractive condition, we get the following.
Corollary3.13. LetSbe a self-map on aD-metric space(X,D)such that for somex0∈X the orbitO(S,x0)is bounded and complete and for someλ∈[0, 1)
D(Sx,Sy,Sz)≤λMaxD(x,y,z),D(x,Sx,z) ∀x,y∈OS,x0
,z∈X. (3.22)
ThenShas a unique fixed point andSis continuous at it.
Remark 3.14. This corollary improvesTheorem 3.1of Dhage et al. [3] in the sense that in the contractive condition the domain ofx,yis just an orbitO(S,x0) and not its closure.
Also this result corrects the above said result in the sense that the domain ofz in the condition must be the whole spaceXand not just the closure of the orbitO(S,x0), for otherwise the uniqueness of the fixed point does not follow. The following is an example of it.
Example 3.15. Let X= {2n|n∈Z} ∪ {0} = {1, 1/2, 1/22,..., 0, 2, 22,...} and d(x,y)=
|x−y|, for allx,y∈X. Consider
D(x,y,z)=Max|x−y|,|y−z|,|z−x|
∀x,y,z∈X. (3.23)
Let a self-mapSon theD-metric space (X,D) be given by
Sx=
x
2 ifx∈
1,1 2, 1
22,...,0
, 22
x ifx∈ {2, 22,...}.
(3.24)
Then O(S, 1)= {1, 1/2, 1/22,...} and O(S, 1)=O(S, 1)∪ {0}. Thus the orbit O(S, 1) is bounded and complete. To show the existence ofλof condition (A) see the following.
Case 4. x,y,z∈O(S, 1). We take x=1/2n−1, y=1/2n+p−1,z=1/2n+p+t−1. As seen in Case 1ofExample 3.10, forλ=1/2, condition (A) is satisfied.
Case 5. x,y∈O(S, 1) andz=0. As seen inCase 2ofExample 3.10, forλ=1/2, condition (A) is satisfied.
Case 6. x=0 andy,z∈O(S, 1). We takex=0, y=1/2n−1,z=1/2n+p−1 thenSx=0, Sy=1/2n,Sz=1/2n+p. Now,
D(Sx,Sy,Sz)= 1
2n, MaxD(x,y,z),D(x,Sx,z)=Max 1
2n−1, 1 2n+p−1
= 1 2n−1.
(3.25)
Thus, forλ=1/2, condition (A) is satisfied.
The cases{x=0,y=0,z=0}and{x=0,y=0,z=1/2n}trivially satisfy (A) forλ= 1/2. Thus for allx,y,z∈O(S, 1), condition (A) is satisfied. Hence all the conditions of [3, Theorem 3.1] are satisfied, stillShas two fixed points which are “0” and “2.”
Remark 3.16. In the above example, takex=1/2,y=1/22, andz=2 then we haveSx= 1/22,Sy=1/23, andSz=2. Now,
D(Sx,Sy,Sz)=2−1 8=
15 8 , D(x,y,z)=2− 1
22 = 7
4, D(Sx,x,z)=2− 1 22 =
7 4.
(3.26)
Thus the contractive condition ofCorollary 3.13 is not satisfied for all x, y∈O(S, 1) andz∈X. Hence,Corollary 3.13too cannot assure the uniqueness of a fixed point in Example 3.15.
It is to be noted thatExample 3.10is also an example ofCorollary 3.13.
Corollary3.17. LetSandTbe self-maps of aD-metric space(X,D)satisfying conditions (1), (2), (3) ofTheorem 3.1and
(1)letai,i=1to10be constants such thatai≥0for alli,ai<1and for allx,y∈ O(T−1S,x0)∪O(ST−1,Sx0),z∈X
D(Sx,Sy,Sz)≤a1D(Tx,T y,Tz) +a2D(Sx,Tx,Tz) +a3D(Sy,T y,Tz) +a4D(Sx,T y,Tz) +a5D(Sy,Tx,Tz) +a6D(Tx,T y,Sz) +a7D(Sx,Tx,Sz) +a8D(Sy,T y,Sz) +a9D(Sx,T y,Sz) +a10D(Sy,Tx,Sz).
(3.27)
ThenSandThave a unique common fixed point.
Proof. Letai=λ(<1). Let
M(x,y,z)=MaxD(Tx,T y,Tz),D(Sx,Tx,Tz),D(Sy,T y,Tz), D(Sx,T y,Tz),D(Sy,Tx,Tz),D(Tx,T y,Sz), D(Sx,Tx,Sz),D(Sy,T y,Sz),D(Sx,T y,Sz), D(Sy,Tx,Sz).
(3.28)
Then from condition (1) ofCorollary 3.17we have D(Sx,Sy,Sz)≤
10 i=1
aiM(x,y,z)
=M(x,y,z) 10 i=1
ai
=λM(x,y,z) (λ <1).
(3.29)
The rest of the proof follows fromCorollary 3.6.
In Corollary 3.17 if we take T =I, the identity map on X, then conditions (1), (2) ofTheorem 3.1 are trivially satisfied and if we take a3=a4=a5=a6=a7=a9= a10=0, we get the following.
Corollary3.18. LetSbe a self-map on aD-metric space(X,D)such that for somex0∈X, the orbitO(S,x0)is bounded and complete and
D(Sx,Sy,Sz)≤a1D(x,y,z) +a2D(x,Sx,z) +a8D(Sy,y,Sz) ∀x,y∈OS,x0
,z∈X. (3.30)
ThenShas a unique fixed point.
Remark 3.19. The above corollary generalizes and improves [7, Theorem 1] in which the required domain ofxand ywas needed to be the whole spaceX. In this corollary the required domain ofxandyis just an orbitO(S,x0) and onlyzvaries over the whole space X.
References
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84(1992), no. 4, 329–336.
[2] ,A common fixed point principle inD-metric spaces, Bull. Calcutta Math. Soc.91(1999), no. 6, 475–480.
[3] B. C. Dhage, A. M. Pathan, and B. E. Rhoades,A general existence principle for fixed point theo- rems inD-metric spaces, Int. J. Math. Math. Sci.23(2000), no. 7, 441–448.
[4] B. E. Rhoades,A fixed point theorem for generalized metric spaces, Int. J. Math. Math. Sci.19 (1996), no. 3, 457–460.
[5] B. Singh and R. K. Sharma,Common fixed points via compatible maps inD-metric spaces, Rad.
Mat.11(2002), no. 1, 145–153.
[6] T. Veerapandi and K. Chandrasekhara Rao,Fixed point theorems of some multivalued mappings in aD-metric space, Bull. Calcutta Math. Soc.87(1995), no. 6, 549–556.
[7] ,Fixed points in Dhage metric spaces, Pure Appl. Math. Sci.43(1996), no. 1-2, 9–14.
Bijendra Singh: School of Studies in Mathematics, Vikram University, Ujjain 456010, Madhya Pradesh, India
E-mail address:[email protected]
Shishir Jain: Shri Vaishnav Institute of Technology & Science, Gram Baroli, Alwasa, Indore 453331, Madhya Pradesh, India
E-mail address:jainshishir11@rediffmail.com
Shobha Jain: MB Khalsa College, Raj Mohalla, Indore 452002, Madhya Pradesh, India E-mail address:[email protected]
