# A New Fixed Point Theorem on Generalized Quasimetric Spaces

Authors: Luljeta Kikina; Kristaq Kikina; Kristaq Gjino
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

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.


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## Body

## 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:(1.1)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space (gqms) on the lines of quasimetric space, where the tetrahedral inequalityd(x,y)≤d(x,z)+d(z,w)+d(w,y) has been replaced by quasitetrahedral inequalityd(x,y)≤k[d(x,z)+d(z,w)+d(w,y)].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 (fork=1). Also, every qms is a gqms, while the converse is not true.In gqms, contrary to a metric space, the “open” ballsB(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]).
LetX 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 (a)
d(x,y)=0 if and only if x=y,(b)
d(x,y)=d(y,x),(c)
d(x,y)≤d(x,z)+d(z,w)+d(w,y)  (tetrahedral inequality).Then dis called a generalized metric and (X,d) is a generalized metric space (or shortly gms).Definition 1.2 (see [14]).
LetX 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:
(i)
d(x,y)=0⇔x=y;(ii)
d(x,y)=d(y,x);(iii)
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 setB(a,r)={x∈X:d(x,a)<r} is called “open” ball with center a∈X and radius r>0.
The familyτ={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:
(1.2)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:(1.3)d(xn,xm)=d(1-1n,1-1m)=1,∀n,m∈N.SinceB(1,r)∩B(2,r)≠ϕ for all r>0, the (X,d) is non-Hausdorff generalized metric space.The functiond 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.(1)
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.(2)
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 mapT:X→X is called contraction if there exists 0<c<1/k such that(1.4)d(Tx,Ty)≤cd(x,y)∀x,y∈X.Definition 1.9.
LetT:X→X be a mapping where X is a gqms. For each x∈X, let
(1.5)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 variablesf:R+3→R satisfying the following properties: (a)
f is nondecreasing in respect to each variable,(b)
f(t,t,t)≤t, t∈R+will be noted by 𝔽3 and every such function will be called an 𝔽3-function.
Some examples of𝔽3-function are as follows:
(1)
f(t1,t2,t3)=max{t1,t2,t3},(2)
f(t1,t2,t3)=[max{t1t2,t2t3,t3t1}]1/2,(3)
f(t1,t2,t3)=[max{t1p,t2p,t3p}]1/p,p>0,(4)
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.
Ifm>2 is odd, then writing m=2p+1, p≥1, by quasitetrahedral inequality, we can easily show that
(2.1)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.
Ifm>2 is even, then writing m=2p, p≥2 and using the same arguments as before we can get
(2.2)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:
(2.3)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.
Letx0 be an arbitrary point in X. Define the sequences (xn) and (yn) inductively as follows:
(2.4)xn=Syn=(ST)nx0,y1=Tx0,yn+1=Txn=(TS)ny1,n≥1.
Denote
(2.5)dn=d(xn,xn+1),ρn=ρ(yn,yn+1),n=1,2,….
Using the inequality (2) we get
(2.6)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
(2.7)ρn≤cdn-1.
Using the inequality (2.3) we have
(2.8)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
(2.9)dn≤c2dn-1≤cdn-1.
Using the mathematical induction, by the inequalities (2.7) and (2.9), we get
(2.10)dn≤cnd(x0,x1),ρn≤cnd(x0,x1).
So
(2.11)lim⁡n→∞d(xn,xn+1)=lim⁡n→∞ρ(yn,yn+1)=0.
Applying the inequality (2), we get
(2.12)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
(2.13)ρ(yn,yn+2)≤max⁡{cnd(x0,x1),cd(xn-1,xn+1)}.
Similarly, using (2.3), we obtain
(2.14)d(xn,xn+2)≤max⁡{cnd(x0,x1),cd(xn-1,xn+1)}.
Using the mathematical induction, we get
(2.15)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
(2.16)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.
Supposexp=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
(2.17)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 equalityxp=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 thatxn≠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 limitsα 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 obtain(2.18)d(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 thatα 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 obtain(2.19)ρ(β,Tα)≤k[ρ(β,ynq-1)+ρ(ynq-1,ynq)+ρ(ynq,Tα)].
Then if q→∞, we get
(2.20)ρ(β,Tα)≤klim⁡q→∞___ρ(ynq,Tα).
Using the inequality (2), for x=α and y=yn-1 we obtain
(2.21)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
(2.22)lim⁡n→∞___ρ(Tα,yn)≤cf2(lim⁡n→∞___ρ(yn-1,Tα),0,0).
And so,
(2.23)lim⁡n→∞___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
(2.24)Tα=β.
It follows similarly that
(2.25)Sβ=α.
By (2.24) and (2.25) we obtain
(2.26)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 Cases1 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 (2.27)d(α,α′)=d(Sβ,STα′)≤cf1(d(α′,α),0,ρ(Tα,Tα′)).
And so,we have
(2.28)d(α,α′)≤cρ(Tα,Tα′).
If  Tα≠Tα′, in similar way by (2) for x=STα and y=Tα′, we have
(2.29)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

(1)
Ifk1=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 off1 and f2 in Theorem 2.2 we get different theorems.(2)
Forf1=f2=f, where  f(t1,t2,t3)=max{t1,t2,t3} we have an extension of Fisher’s theorem [15] in generalized quasimetric spaces.(3)
Forf1=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.(4)
Forf1(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 differentf.



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