We obtain a new fixed point theorem in generalized quasimetric spaces. This result generalizes, unify, enrich, and extend some theorems of well-known authors from metric spaces to generalized quasimetric spaces.
1. Introduction and Preliminaries
The concept of metric space, as an ambient space in fixed point theory, has been generalized in several directions. Some of such generalizations are quasimetric spaces, generalized metric spaces, and generalized quasimetric spaces.
The concept of quasimetric spaces is treated differently by many authors. In this paper our concept is in line with this treated in [1–6], and so forth and the triangular inequality d(x,y)≤d(x,z)+d(z,y) is replaced by quasi-triangular inequality:d(x,y)≤k[d(x,z)+d(z,y)],k≥1.
In 2000 Branciari [7] introduced the concept of generalized metric spaces (gms) (the triangular inequality d(x,y)≤d(x,z)+d(z,y) is replaced by tetrahedral inequalityd(x,y)≤d(x,z)+d(z,w)+d(w,y)). Starting with the paper of Branciari, some classical metric fixed point theorems have been transferred to gms (see [8–13]).
Recently L. kikina and k. kikina [14] introduced the concept of generalized quasimetric space (gqms) on the lines of quasimetric space, where the tetrahedral inequalityd(x,y)≤d(x,z)+d(z,w)+d(w,y) has been replaced by quasitetrahedral inequalityd(x,y)≤k[d(x,z)+d(z,w)+d(w,y)]. The well-known fixed point theorems of Banach and of Kannan have been transferred to such a space.
The metric spaces are a special case of generalized metric spaces and generalized metric spaces are a special case of generalized quasimetric spaces (for k=1). Also, every qms is a gqms, while the converse is not true.
In gqms, contrary to a metric space, the “open” balls B(a,r)={x∈X:d(x,a)<r} are not always open sets, and consequently, a generalized quasidistance is not always continuous of its variables. The gqms is not always a Hausdorff space and the convergent sequence (xn) in gqms is not always a Cauchy sequence (see Example 1.3).
Under this situation, it is reasonable to consider if some well-known fixed point theorems can be obtained in generalized quasimetric space.
The aim of this paper is to generalize, unify, and extend some theorems of well-known authors such as of Fisher and Popa, from metric spaces to generalized quasimetric spaces.
Let us start with the main definitions.
Definition 1.1 (see [7]).
Let X be a set and d:X2→R+ a mapping such that for all x,y∈X and for all distinct points z,w∈X, each of them different from x and y, one has
Then dis called a generalized metric and (X,d) is a generalized metric space (or shortly gms).
Definition 1.2 (see [14]).
Let X be a set. A nonnegative symmetric function ddefined on X·X is called a generalized quasidistance on X if and only if there exists a constant k≥1 such that for all x,y∈X and for all distinct points z,w∈X, each of them different from x and y, the following conditions hold:
d(x,y)=0⇔x=y;
d(x,y)=d(y,x);
d(x,y)≤k[d(x,z)+d(z,w)+d(w,y)].
Inequality (2.7) is often called quasitetrahedral inequality and k is often called the coefficient of d. A pair (X,d) is called a generalized quasimetric space if X is a set and d is a generalized quasidistance on X.
The set B(a,r)={x∈X:d(x,a)<r} is called “open” ball with center a∈X and radius r>0.
The family τ={Q⊂X:∀a∈Q,∃r>0,B(a,r)⊂Q} is a topology on X and it is called induced topology by the generalized quasidistance d.
The following example illustrates the existence of the generalized quasimetric space for an arbitrary constantk≥1.
Example 1.3 (see [14]).
LetX={1-(1/n):n=1,2,…}∪{1,2}. Define d:X·X→R as follows:
d(x,y)={0,forx=y,1n,forx∈{1,2},y=1-1nory∈{1,2},x=1-1n,x≠y,3k,forx,y∈{1,2},x≠y,1,otherwise.
Then it is easy to see that (X,d) is a generalized quasimetric space and is not a generalized metric space (for k>1).
Note that the sequence (xn)=(1-(1/n)) converges to both 1 and 2 and it is not a Cauchy sequence:d(xn,xm)=d(1-1n,1-1m)=1,∀n,m∈N.
Since B(1,r)∩B(2,r)≠ϕ for all r>0, the (X,d) is non-Hausdorff generalized metric space.
The function d is not continuous: 1=limn→∞d(1-(1/n),1/2)≠d(1,1/2)=1/2.
In [14] the following is proved.
Proposition 1.4.
If (X,d) is a quasimetric space, then (X,d) is a generalized quasimetric space. The converse proposition does not hold true.
Definition 1.5.
A sequence {xn} in a generalized quasimetric space (X,d) is called Cauchy sequence if limn,m→∞d(xn,xm)=0.
Definition 1.6.
Let (X,d) be a generalized quasimetric space. Then one has the following.
A sequence {xn} in X is said to be convergent to a point x∈X (denoted by limn→∞xn=x) if limn→∞d(xn,x)=0.
It is called compact if every sequence contains a convergent subsequence.
Definition 1.7.
A generalized quasimetric space (X,d) is called complete, if every Cauchy sequence is convergent.
Definition 1.8.
Let (X,d) be a gqms and the coefficient of d is k.
A map T:X→X is called contraction if there exists 0<c<1/k such thatd(Tx,Ty)≤cd(x,y)∀x,y∈X.
Definition 1.9.
Let T:X→X be a mapping where X is a gqms. For each x∈X, let
O(x)={x,Tx,T2x,…},
which will be called the orbit of T at x. The space X is said to be T-orbitally complete if and only if every Cauchy sequence which is contained in O(x) converges to a point in X.
Definition 1.10.
The set of all upper semicontinuous functions with 3 variables f:R+3→R satisfying the following properties:
f is nondecreasing in respect to each variable,
f(t,t,t)≤t, t∈R+
will be noted by 𝔽3 and every such function will be called an 𝔽3-function.
Some examples of 𝔽3-function are as follows:
f(t1,t2,t3)=max{t1,t2,t3},
f(t1,t2,t3)=[max{t1t2,t2t3,t3t1}]1/2,
f(t1,t2,t3)=[max{t1p,t2p,t3p}]1/p,p>0,
f(t1,t2,t3)=(at1t2+bt2t3+ct3t1)1/2, where a,b,c≥0 and a+b+c<1.
2. Main Result
We state the following lemma which we will use for the proof of the main theorem.
Lemma 2.1.
Let (X,d) be a generalized quasimetric space and {xn} is a sequence of distinct point (xn≠xm for all n≠m) in X. If d(xn,xn+1)≤cnl, 0≤c<1/k<1, for all n∈N and limn→∞d(xn,xn+2)=0, then {xn} is a Cauchy sequence.
Proof.
If m>2 is odd, then writing m=2p+1, p≥1, by quasitetrahedral inequality, we can easily show that
d(xn,xn+m)≤k[d(xn,xn+1)+d(xn+1,xn+2)+d(xn+2,xn+m)]≤kd(xn,xn+1)+k2d(xn+1,xn+2)+k2d(xn+2,xn+m)≤kcnl+k2cn+1l+k2d(xn+2,xn+m)≤⋯≤kcnl+k2cn+1l+k3cn+2l+⋯+km-1cn+m-2l+km-1cn+m-1l≤kcnl+k2cn+1l+k3cn+2l+⋯+km-1cn+m-2l+kmcn+m-1l≤kcnl[1+kc+⋯+(kc)m-1]=kcnl1-(kc)m1-kc<kcnl11-kc.
Therefore, limn→∞d(xn,xn+m)=0.
If m>2 is even, then writing m=2p, p≥2 and using the same arguments as before we can get
d(xn,xn+m)≤k[d(xn,xn+2)+d(xn+2,xn+3)+d(xn+3,xn+m)]≤kd(xn,xn+2)+kcn+2l+kd(xn+3,xn+m)≤kd(xn,xn+2)+kcn+2l+k2[d(xn+3,xn+4)+d(xn+4,xn+5)+d(xn+5,xn+m)]≤⋯≤kd(xn,xn+2)+kcn+2l+k2cn+3l+⋯+km-2cn+m-1l=kd(xn,xn+2)+kcn+2l[1+kc+⋯+(kc)m-3]=kd(xn,xn+2)+kcn+2l1-(kc)m-21-kc<kd(xn,xn+2)+kcn+2l11-kc.
And so limn→∞d(xn,xn+m)=0. It implies that {xn} is a Cauchy sequence in X. This completes the proof of the lemma.
We state the following theorem.
Theorem 2.2.
Let (X,d) and (Y,ρ) be two generalized quasimetric spaces with coefficients k1 and k2, respectively. Let T be a mapping of X into Y and S a mapping of Y into X satisfying the following inequalities:
d(Sy,STx)≤cf1{d(x,Sy),d(x,STx),ρ(y,Tx)},ρ(Tx,TSy)≤cf2{ρ(y,Tx),ρ(y,TSy),d(x,Sy)},
for all x∈X and y∈Y, where 0<c<1/k≤1, k=max{k1,k2}, f1,f2∈𝔽3. If there exists x0∈X such that O(x0) is ST-orbitally complete in X and O(Tx0) is TS-orbitally complete in Y, then ST has a unique fixed point α in X and TS has a unique fixed point β in Y. Further, Tα=β and Sβ=α.
Proof.
Let x0 be an arbitrary point in X. Define the sequences (xn) and (yn) inductively as follows:
xn=Syn=(ST)nx0,y1=Tx0,yn+1=Txn=(TS)ny1,n≥1.
Denote
dn=d(xn,xn+1),ρn=ρ(yn,yn+1),n=1,2,….
Using the inequality (2) we get
ρn=ρ(yn,yn+1)=ρ(Txn-1,TSyn)≤cf2(ρ(yn,yn),ρ(yn,yn+1),d(xn-1,xn))=cf2(0,ρn,dn-1).
By this inequality and properties of f2, it follows that
ρn≤cdn-1.
Using the inequality (2.3) we have
dn=d(xn,xn+1)=d(Syn,STxn)≤cf1(d(xn,xn),d(xn,xn+1),ρ(yn,yn+1))=cf1(0,dn,ρn),
and so dn≤cρn. By this inequality and (2.7) we obtain
dn≤c2dn-1≤cdn-1.
Using the mathematical induction, by the inequalities (2.7) and (2.9), we get
dn≤cnd(x0,x1),ρn≤cnd(x0,x1).
So
limn→∞d(xn,xn+1)=limn→∞ρ(yn,yn+1)=0.
Applying the inequality (2), we get
ρ(yn,yn+2)=ρ(Txn-1,TSyn+1)≤cf2(ρ(yn+1,yn),ρ(yn+1,yn+2),d(xn-1,xn+1))=cf2(ρn,ρn+1,d(xn-1,xn+1))≤cmax{cnd(x0,x1),d(xn-1,xn+1)},
and so
ρ(yn,yn+2)≤max{cnd(x0,x1),cd(xn-1,xn+1)}.
Similarly, using (2.3), we obtain
d(xn,xn+2)≤max{cnd(x0,x1),cd(xn-1,xn+1)}.
Using the mathematical induction, we get
d(xn,xn+2)≤max{cnd(x0,x1),cd(xn-1,xn+1)}≤max{cnd(x0,x1),c2d(xn-2,xn)}≤⋯≤max{cnd(x0,x1),cnd(x0,x2)}=cnmax{d(x0,x1),d(x0,x2)}=cnl,
and so
d(xn,xn+2)≤cnl,similarlyρ(yn,yn+2)≤cnl,
where l=max{d(x0,x1),d(x0,x2)}.
We divide the proof into two cases.
Case 1.
Suppose xp=xq for some p,q∈N, p≠q. Let p>q. Then (ST)px0=(ST)p-q(ST)qx0=(ST)qx0; that is, (ST)nα=α where n=p-q and (ST)qx0=α. Now if n>1, by (2.10), we have
d(α,STα)=d[(ST)nα,(ST)n+1α]≤cnd(α,STα).
Since 0<c<1, d(α,STα)=0. So STα=α and hence α is a fixed point of ST.
By the equality xp=xq it follows that yp+1=yq+1. We take β=(TS)qTx0 and, in similar way, we prove that β is a fixed point of TS.
Case 2.
Assume that xn≠xm for all n≠m. Then, from (2.10), (2.16), and Lemma 2.1 is derived that {xn} is a Cauchy sequence in X. Since O(x0) is ST-orbitally complete, there exists α∈X such that limn→∞xn=α. In the same way, we show that the sequence (yn) is a Cauchy sequence and there exists a β∈Y such that limn→∞yn=β.
We now prove that the limits α and β are unique. Suppose, to the contrary, that α′≠α is also limn→∞xn. Since xn≠xm for all n≠m, there exists a subsequence (xnk) of (xn) such that xnk≠α and xnk≠α′ for all k∈N. Without loss of generality, assume that (xn) is this subsequence. Then by Tetrahedral property of Definition 1.1 we obtaind(α,α′)≤k[d(α,xn)+d(xn,xn+1)+d(xn+1,α′)].
Letting n tend to infinity we get d(α,α′)=0 and so α=α′, in the same way for β.
Let us prove now that α is a fixed point of ST. First we prove that β=Tα. In contrary, if β≠Tα, the sequence (yn) does not converge to Tα and there exists a subsequence (ynq) of (yn) such that ynq≠Tα for all q∈N. Then by Tetrahedral property of Definition 1.1 we obtainρ(β,Tα)≤k[ρ(β,ynq-1)+ρ(ynq-1,ynq)+ρ(ynq,Tα)].
Then if q→∞, we get
ρ(β,Tα)≤klimq→∞___ρ(ynq,Tα).
Using the inequality (2), for x=α and y=yn-1 we obtain
ρ(Tα,yn)=ρ(Tα,TSyn-1)≤cf2(ρ(yn-1,Tα),ρ(yn-1,TSyn-1),d(α,Syn-1))=cf2(ρ(yn-1,Tα),ρ(yn-1,yn),d(α,xn-1)).
Letting n tend to infinity we get
limn→∞___ρ(Tα,yn)≤cf2(limn→∞___ρ(yn-1,Tα),0,0).
And so,
limn→∞___ρ(Tα,yn)=0.
Since limq→∞___ρ(ynq,Tα)≤limn→∞___ρ(Tα,yn), by (2.23) and (2.20), we have ρ(β,Tα)=0 and so
Tα=β.
It follows similarly that
Sβ=α.
By (2.24) and (2.25) we obtain
STα=Sβ=α,TSβ=Tα=β.
Thus, we proved that the points α and β are fixed points of ST and TS, respectively.
Let us prove now the uniqueness (for Cases 1 and 2 in the same time). Assume that α′≠α is also a fixed point of ST. By (2.3) for x=α′ and y=β we get d(α,α′)=d(Sβ,STα′)≤cf1(d(α′,α),0,ρ(Tα,Tα′)).
And so,we have
d(α,α′)≤cρ(Tα,Tα′).
If Tα≠Tα′, in similar way by (2) for x=STα and y=Tα′, we have
ρ(Tα,Tα′)≤cd(α,α′).
By (2.28) and (2.29) we get d(α,α′)=0. Thus, we have again α=α′. The uniqueness of β follows similarly. This completes the proof of the theorem.
3. Corollaries
If k1=k2=1, then by Theorem 2.2 we obtain [12, Theorem 2.1], that generalize and extend the well-known Fisher fixed point theorem [15] from metric space to generalized metric spaces.
For different expressions of f1 and f2 in Theorem 2.2 we get different theorems.
For f1=f2=f, where f(t1,t2,t3)=max{t1,t2,t3} we have an extension of Fisher’s theorem [15] in generalized quasimetric spaces.
For f1=f2=f, where f(t1,t2,t3)=[max{t1t2,t2t3,t3t1}]1/2, we have an extension of Popa’s theorem [13] in generalized quasimetric spaces.
For f1(t1,t2,t3)=(a1t1t2+b1t2t3+c1t3t1)1/2 and f2(t1,t2,t3)=(a2t1t2+b2t2t3+c2t3t1)1/2 we obtain an extension of Popa’s Corollary [13] in generalized quasimetric spaces.
Remark 3.1.
We can obtain many other similar results for different f.
BramantiM.BrandoliniL.Schauder estimates for parabolic nondivergence operators of Hörmander type20072341177245229897010.1016/j.jde.2006.07.015ZBL1140.35361KikinaL.gjonileta@yahoo.comKikinaK.kristaqkikina@yahoo.comA related fixed point theorem for m mappings on m complete quasi-metric spacesMathematica Cluj. In pressKikinaL.gjonileta@yahoo.comKikinaK.kristaqkikina@yahoo.comGeneralized fixed point theorem for three mappings on three quasi-metric spacesJournal of Computational Analysis and Applications. In pressPepoB.Fixed points for contractive mappings of third order in pseudo-quasimetric spaces199014473481110609410.1016/0019-3577(90)90015-FZBL0723.54040PeppoC.Fixed point theorems for (ϕ,k,i,j)-mappings201072256257010.1016/j.na.2009.06.0972579324XiaQ.The geodesic problem in quasimetric spaces2009192452479248197010.1007/s12220-008-9065-4ZBL1200.54013BranciariA.A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces2000571-231371771669ZBL0963.54031AzamA.ArshadM.Kannan fixed point theorem on generalized metric spaces20081145482475781ZBL1161.54022DasP.A fixed point theorem on a class of generalized metric spaces200212933DasP.DeyL. K.A fixed point theorem in a generalized metric space200733133392294745ZBL1137.54024KikinaL.gjonileta@yahoo.comKikinaK.kristaqkikina@yahoo.comA fixed point theorem in generalized metric spacesDemonstratio Mathematica. In pressKikinaL.gjonileta@yahoo.comKikinaK.kristaqkikina@yahoo.comFixed points on two generalized metric spaces2011529-3214591467SarmaI. R.RaoJ. M.RaoS. S.Contractions over generalized metric spaces2009231801822521195ZBL1173.54311KikinaL.gjonileta@yahoo.comKikinaK.kristaqkikina@yahoo.comTwo fixed point theorems on a class of generalized quasi-metric spacesJournal of Computational Analysis and Applications. In pressFisherB.Fixed points on two metric spaces19811636333337653063ZBL0472.54031