# A Proposal to the Study of Contractions in Quasi-Metric Spaces

Authors: Hamed H. Alsulami; Erdal Karapınar; Farshid Khojasteh; Antonio-Francisco Roldán-López-de-Hierro
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

We investigate the existence and uniqueness of a fixed point of an operator via simultaneous functions in the setting of complete quasi-metric spaces. Our results generalize and improve several recent results in literature.


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## Body

## 1. Introduction and Preliminaries

One of the attractive research subjects in the fixed point theory is the investigation of the existence and uniqueness of (common) fixed point of various operators in the setting of quasi-metric space. Very recently, Jleli and Samet [1] and Samet et al. [2] reported that G-metrics, introduced by Mustafa and Sims [3], can be deduced from quasi-metrics by taking q(x,y)=G(x,y,y). Consequently, the authors in [1, 2] proved that several fixed point results in the setting of G-metric spaces can be deduced from the corresponding theorems in the context of quasi-metric spaces. The importance of these results follows from the simplicity of construction of quasi-metric despite the notion of G-metric.In this paper, we investigate the existence and uniqueness of a fixed point of operators via simultaneous functions, defined by Khojasteh et al. [4], in the setting of complete quasi-metric spaces. We also observed that several existing results can be concluded from our main results. We also show that some result in the context of G-metric spaces can be deduced from the corresponding theorems in the framework of quasi-metric spaces.For the sake of completeness, we recollect basic notions, definitions, and fundamental results. LetA,B⊆X be two nonempty subsets of a set X and let T:A→B be a mapping. A point x∈X is called afixed point of the mapping T if Tx=x.Definition 1.
LetX be a nonempty set and let q:X×X→[0,+∞) be a given function which satisfies(1)
q(x,y)=0 if and only if x=y;(2)
q(x,y)≤q(x,z)+q(z,y) for any points x,y,z∈X. Then q is called a quasi-metric and the pair (X,q) is called a quasi-metric space.It is evident that any metric space is a quasi-metric space, but the converse is not true in general. Now, we recall convergence and completeness on quasi-metric spaces.Definition 2.
Let(X,q) be a quasi-metric space and let {xn} be a sequence in X and x∈X. The sequence {xn} converges to x if
(1)lim⁡n→∞q(xn,x)=lim⁡n→∞q(x,xn)=0.Remark 3.
A convergent sequence in a quasi-metric space has a unique limit.Remark 4.
If{xn} converges to x in a quasi-metric space (X,q), then
(2)lim⁡n→∞q(xn,y)=q(x,y),∀y∈X.
In other words, q is a continuous mapping on its first argument. This property follows from q(xn,y)≤q(xn,x)+q(x,y) and q(x,y)≤q(x,xn)+q(xn,y). Therefore,
(3)q(x,y)-q(x,xn)≤q(xn,y)≤q(xn,x)+q(x,y)1111111111∀n∈N.Definition 5 (see, e.g., [1, 2]).
Let(X,q) be a quasi-metric space and let {xn} be a sequence in X. We say that {xn} is left-Cauchy if, for every ε>0, there exists a positive integer N=N(ε) such that q(xn,xm)<ε for all n≥m>N.Definition 6 (see, e.g., [1, 2]).
Let(X,q) be a quasi-metric space and let {xn} be a sequence in X. We say that {xn} is right-Cauchy if, for every ε>0, there exists a positive integer N=N(ε) such that q(xn,xm)<ε for all m≥n>N.Definition 7 (see, e.g., [1, 2]).
Let(X,q) be a quasi-metric space and let {xn} be a sequence in X. We say that {xn} is Cauchy if, for every ε>0, there exists a positive integer N=N(ε) such that q(xn,xm)<ε for all m,n>N.Remark 8.
A sequence{xn} in a quasi-metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy.Definition 9 (see, e.g., [1, 2]).
Let(X,q) be a quasi-metric space. We say that(1)
(X,q) is left-complete if each left-Cauchy sequence in X is convergent;(2)
(X,q) is right-complete if each right-Cauchy sequence in X is convergent;(3)
(X,q) is complete if each Cauchy sequence in X is convergent.


## 2. Simulation Functions

The notion ofsimulation function was introduced by Khojasteh et al. in [4].Definition 10 (see [4]).
A simulation function is a mappingζ:[0,∞)×[0,∞)→R satisfying the following conditions:(ζ1)
ζ(0,0)=0;(ζ2)
ζ(t,s)<s-t for all t,s>0;(ζ3)
if{tn}, {sn} are sequences in (0,∞) such that lim⁡n→∞tn=lim⁡n→∞sn>0 then
(4)limsup⁡n→∞ζ(tn,sn)<0.
LetZ be the family of all simulation functions ζ:[0,∞)×[0,∞)→R.Before presenting our main fixed point results using simulation functions, we show a wide range of examples to highlight their potential applicability to the field of fixed point theory. In the following results, the mappingζ is defined from [0,∞)×[0,∞) into R.Definition 11 (Khan et al. [5]).
An altering distance function is a continuous, nondecreasing mappingϕ:[0,∞)→[0,∞) such that ϕ-1({0})={0}.Example 12.
Letϕ and ψ be two altering distance functions such that ψ(t)<t≤ϕ(t) for all t>0. Then the mapping
(5)ζ1(t,s)=ψ(s)-ϕ(t),∀t,s∈[0,∞)
is a simulation function.If, in the previous example,ϕ(t)=t and ψ(t)=λt for all t≥0, where λ∈[0,1), then we obtain the following particular case of simulation function:
(6)ζB(t,s)=λs-t,∀t,s∈[0,∞).Example 13.
Ifφ:[0,∞)→[0,∞) is a lower semicontinuous function such that φ-1(0)={0} and we define ζR:[0,∞)×[0,∞)→R by
(7)ζR(t,s)=s-φ(s)-t,∀s,t∈[0,∞),
then ζR is a simulation function.If, in the previous example,φ is continuous, we deduce the following case.Example 14.
Ifφ:[0,∞)→[0,∞) is a continuous function such that φ(t)=0⇔t=0 and we define
(8)ζR(t,s)=s-φ(s)-t,∀s,t∈[0,∞),
then ζR is a simulation function.Example 15.
Letf,g:[0,∞)→(0,∞) be two continuous functions with respect to each variable such that f(t,s)>g(t,s) for all t,s>0 and define
(9)ζ(t,s)=s-f(t,s)g(t,s)t,∀t,s∈[0,∞).
Then ζ is a simulation function.Example 16.
Ifφ:[0,∞)→[0,1) is a function such that limsup⁡t→r+φ(t)<1 for all r>0 and we define
(10)ζT(t,s)=sφ(s)-t,∀s,t∈[0,∞),
then ζT is a simulation function.Example 17.
Ifη:[0,∞)→[0,∞) is an upper semicontinuous mapping such that η(t)<t for all t>0 and η(0)=0 and we define
(11)ζBW(t,s)=η(s)-t,∀s,t∈[0,∞),
then ζBW is a simulation function.Example 18.
Ifϕ:[0,∞)→[0,∞) is a function such that ∫0ε‍ϕ(u)du exists and ∫0ε‍ϕ(u)du>ε, for each ε>0, and we define
(12)ζK(t,s)=s-∫0t‍ϕ(u)du,∀s,t∈[0,∞),
then ζK is a simulation function.Example 19.
Leth:[0,∞)×[0,∞)→[0,∞) be a function such that h(t,s)<1 for all t,s>0 and limsup⁡n→∞h(tn,sn)<1 provided that {tn} and {sn}⊂(0,+∞) are two sequences such that lim⁡n→∞tn=lim⁡n→∞sn>0, and we define
(13)ζU(t,s)=sh(t,s)-t,∀s,t∈[0,∞),
and then ζU is a simulation function.The following results are more theoretical.Proposition 20.
Letη:[0,∞)×[0,∞)→R be a function such that η(0,0)=0 and there exists ζ∈Z verifying that η(t,s)≤ζ(t,s) for all s,t≥0. Then η∈Z.Proof.
For allt,s>0, η(t,s)≤ζ(t,s)<s-t. If {tn} and {sn} are sequences in (0,∞) such that lim⁡n→∞tn=lim⁡n→∞sn=δ>0, then limsup⁡n→∞η(tn,sn)≤limsup⁡n→∞ζ(tn,sn)<0.Proposition 21.
Let{ηi}i∈N⊂Z. Then the following statements hold.(a)
For eachk∈N, the function η(k)min⁡:R×R→R defined by
(14)η(k)min⁡(t,s)=min⁡{η1(t,s),η2(t,s),…,ηk(t,s)}∀t,s≥0is a simulation function (i.e.,η(k)min⁡∈Z for any k∈N).(b)
For eachk∈N, the function η(k)¯:R×R→R defined by
(15)η(k)¯(t,s)=1k∑i=1k‍ηi(t,s),∀t,s≥0is a simulation function (i.e.,η(k)¯∈ for any k∈N).Proof.
Sinceη(k)min⁡(t,s)≤η1(t,s) for all t,s>0, the conclusion (a) is a direct consequence of Proposition 20. Next, we prove the conclusion (b). Let k∈N be given. It is obvious that η(k)¯(t,s)<s-t for all s,t>0 because
(16)η(k)¯(t,s)=1k∑i=1k‍ηi(t,s)<1k∑i=1k‍(s-t)=s-t.
Let {tn},{sn}⊂(0,+∞) be two sequences such that lim⁡n→∞tn=lim⁡n→∞sn=δ>0. For any n∈N, we have
(17)limsup⁡n→∞η(k)¯(tn,sn)=1k∑i=1k‍limsup⁡n→∞ηi(tn,sn)<0.


## 3. Main Results

In this section we use simulation functions to present a very general kind of contractions on quasi-metric spaces, and we prove related existence and uniqueness fixed point theorems.Definition 22.
Let(X,q) be a quasi-metric space. We will say that a self-mapping T:X→X is a Z-contraction if there exists ζ∈Z such that
(18)ζ(q(Tx,Ty),q(x,y))≥0,∀x,y∈X.For clarity, we will use the termZq-contraction when we want to highlight that T is a Z-contraction on a quasi-metric space involving the quasi-metric q. In such a case, we will say that T is a Zq-contraction with respect toζ.Next, we observe some useful properties ofZq-contractions in the context of quasi-metric spaces.Remark 23.
By axiom(ζ3), it is clear that a simulation function must verify ζ(r,r)<0 for all r>0. Consequently, if T is a Zq-contraction with respect to ζ∈Zq, then
(19)q(Tx,Ty)≠q(x,y),∀distinctx,y∈X.
In other words, if T is a Zq-contraction, then it cannot be an isometry.We will prove that if aZq-contraction has a fixed point, then it is unique.Lemma 24.
If aZq-contraction in a quasi-metric space has a fixed point, then it is unique.Proof.
Let(X,q) be a quasi-metric space and let T:X→X be a Zq-contraction with respect to ζ∈Z. We are reasoning by contradiction. Suppose that there are two distinct fixed points u,v∈X of the mapping T. Then q(u,v)>0. By (18), we have
(20)0≤ζ(q(Tu,Tv),q(u,v))=ζ(q(u,v),q(u,v)),
which is a contradiction due to Remark 23.Inspired by Browder and Petryshyn’s paper [6], we will characterize the notions of asymptotically right-regularity and asymptotically left-regularity for a self-mapping T in the context of quasi-metric space (X,q).Definition 25.
We will say that a self-mappingT:X→X on a quasi-metric space (X,q) is(i)
asymptotically right-regular at a pointx∈X if lim⁡n→∞q(Tnx,Tn+1x)=0;(ii)
asymptotically left-regular at a pointx∈X if lim⁡n→∞q(Tn+1x,Tnx)=0;(iii)
asymptotically regular if it is both asymptotically right-regular and asymptotically left-regular.Now, we show that aZq-contraction is asymptotically regular at every point of X.Lemma 26.
EveryZq-contraction on a quasi-metric space is asymptotically regular.Proof.
Letx be an arbitrary point of a quasi-metric space (X,q) and let T:X→X be a Zq-contraction with respect to ζ∈Z. If there exists some p∈N such that Tpx=Tp-1x, then y=Tp-1x is a fixed point of T; that is, Ty=y. Consequently, we have that Tny=y for all n∈N, so
(21)q(Tnx,Tn+1x)=q(Tn-p+1Tp-1x,Tn-p+2Tp-1x)=q(Tn-p+1y,Tn-p+2y)=q(y,y)=0,
for sufficient large n∈N. Thus, we conclude that
(22)lim⁡n→∞q(Tnx,Tn+1x)=0.
Similarly, lim⁡n→∞q(Tn+1x,Tnx)=0, so T is asymptotically regular at x. On the contrary, suppose that Tnx≠Tn-1x for all n∈N; that is,
(23)q(Tnx,Tn-1x)>0,∀n∈N.
On what follows, from (18) and (ζ2), we have that, for all n∈N,
(24)0≤ζ(q(Tn+1x,Tnx),q(Tnx,Tn-1x))<q(Tnx,Tn-1x)-q(Tn+1x,Tnx).
In particular,(25)q(Tn+1x,Tnx)<q(Tnx,Tn-1x),∀n∈N.
The above inequality yields that {q(Tnx,Tn-1x)} is a monotonically decreasing sequence of nonnegative real numbers. Thus, there exists r∈[0,∞) such that lim⁡n→∞q(Tnx,Tn+1x)=r≥0. We will prove that r=0. Suppose, on the contrary, that r>0. Since T is Zq-contraction with respect to ζ∈Zq, by (ζ3), we have
(26)0≤limsup⁡n→∞ζ(q(Tn+1x,Tnx),q(Tnx,Tn-1x))<0,
which is a contradiction. Thus, r=0 and this proves that lim⁡n→∞q(Tnx,Tn+1x)=0. Hence, T is an asymptotically right-regular mapping at x. Similarly, it can be demonstrated that T is asymptotically left-regular at x.Given a self-mappingT:X→X, a sequence {xn}⊆X is called aPicard sequence ofT (orgenerated byT) if xn=Txn-1 for all n∈N.Remark 27.
In the proof of the previous result we have proved that ifT:X→X is a Zq-contraction on a quasi-metric space (X,q) and {xn=Tn-1x1} is a Picard sequence of T, then either there exists n0∈N such that xn0 is a fixed point of T (i.e., xn0+1=Txn0=xn0) or
(27)0<q(Tn+1x,Tnx)<q(Tnx,Tn-1x),0<q(Tnx,Tn+1x)<q(Tn-1x,Tnx),11111111∀n∈N.Now, we show that every Picard sequence{xn} generated by a Zq-contraction is always bounded.Lemma 28.
Let(X,q) be a quasi-metric space and let T:X→X be a Zq-contraction with respect to ζ. If {xn} is a Picard sequence generated by T, then {q(xn,xm):n,m∈N} is bounded.Proof.
Letx0∈X be arbitrary and let {xn} be defined iteratively by xn+1=Txn for all n≥0. If there exists some n≥0 and p≥1 such that xn+p=xn, then the set {xn:n∈N} is finite, so it is bounded. Hence, assume that xn+p≠xn for all n≥0 and p≥1. In this case, by Remark 27, we have that
(28)0<q(xn+1,xn)<q(xn,xn-1),0<q(xn,xn+1)<q(xn-1,xn),111111∀n∈N.
Notice that by Lemma 26,
(29)lim⁡n→∞q(xn+1,xn)=lim⁡n→∞q(xn,xn+1)=0.
In particular, there exists n0∈N such that
(30)q(xn+1,xn)<1,q(xn,xn+1)<1,∀n≥n0.
We will prove that {xn:n∈N} is bounded reasoning by contradiction. We distinguish between right and left boundedness. Suppose that the set
(31)D={q(xm,xn):m>n}
is not bounded. Then we can find n1>n0 such that q(xn1,xn0)>1. If n1 is the smallest natural number, greater than n0, verifying this property, then we can suppose that
(32)q(xp,xn0)≤1,∀p∈{n0,n0+1,…,n1-1}.
Again, as D is not bounded, there exists n2>n1 such that
(33)q(xn2,xn1)>1,q(xp,xn1)≤1,111111∀p∈{n1,n1+1,…,n2-1}.
Repeating this process, there exists a partial subsequence {xnk} of {xn} such that, for all k≥1,
(34)q(xnk+1,xnk)>1,q(xp,xnk)≤1,111111∀p∈{nk,nk+1,…,nk+1-1}.
Therefore, by the triangular inequality, we have that, for all k,
(35)1<q(xnk+1,xnk)≤q(xnk+1,xnk+1-1)+q(xnk+1-1,xnk)≤q(xnk+1,xnk+1-1)+1.
Letting k→∞ in (35) and using (29) we obtain
(36)lim⁡k→∞q(xnk+1,xnk)=1.
By (28), we have q(xnk+1,xnk)≤q(xnk+1-1,xnk-1). Therefore using the triangular inequality we obtain
(37)1<q(xnk+1,xnk)≤q(xnk+1-1,xnk-1)≤q(xnk+1-1,xnk)+q(xnk,xnk-1)≤1+q(xnk,xnk-1).
Letting k→∞ and using (29) we obtain
(38)lim⁡k→∞q(xnk+1-1,xnk-1)=1.
Owing to the fact that T is a Zq-contraction with respect to ζ∈Zq, we deduce from (ζ3) that, for all k,
(39)0≤limsup⁡k→∞ζ(q(Txnk+1-1,Txnk-1),q(xnk+1-1,xnk-1))=limsup⁡k→∞ζ(q(xnk+1,xnk),q(xnk+1-1,xnk-1))<0,
which is a contradiction. This proves that D={q(xm,xn):m>n} is bounded. Similarly, it can be proved that D′={q(xm,xn):m<n} is also bounded. Therefore, the set {q(xm,xn):m,n∈N} is bounded.In the next theorem we prove the existence of fixed point of aZq-contraction.Theorem 29.
EveryZ-contraction on a complete quasi-metric space has a unique fixed point. In fact, every Picard sequence converges to its unique fixed point.Proof.
Let(X,q) be a complete quasi-metric space and let T:X→X be a Zq-contraction with respect to ζ. Take x0∈X and consider the Picard sequence {xn=Tnx0}n≥0. If {xn} contains a fixed point of T, the proof is finished. In other case, Lemma 26 and Remark 27 guarantee that
(40)0<q(xn+1,xn)<q(xn,xn-1),0<q(xn,xn+1)<q(xn-1,xn),111111111111111111∀n∈N.(41)lim⁡n→∞q(xn+1,xn)=lim⁡n→∞q(xn,xn+1)=0.
We are going to show that {xn} is a left Cauchy sequence. For this purpose, taking into account that Lemma 28 guarantees that {q(xm,xn):m,n∈N} is bounded, we can consider the sequence {Cn}⊂[0,∞) given by
(42)Cn=sup⁡({q(xi,xj):i≥j≥n}),∀n∈N.
It is clear that the sequence {Cn} is a monotonically nonincreasing sequence of nonnegative real numbers. Therefore, it is convergent; that is, there exists C≥0 such that lim⁡n→∞Cn=C. Let us show that C=0 reasoning by contradiction. If C>0 then, by definition of Cn, for every k∈N there exists nk,mk∈N such that mk>nk≥k and
(43)Ck-1k<q(xmk,xnk)≤Ck.
Hence,
(44)lim⁡k→∞q(xmk,xnk)=C.
By using (40) and the triangular inequality, we have, for all k,
(45)q(xmk,xnk)≤q(xmk-1,xnk-1)≤q(xmk-1,xmk)+q(xmk,xnk)+q(xnk,xnk-1).
Letting k→∞ in the above inequality and using (41) and (44), we derive that
(46)lim⁡k→∞q(xmk-1,xnk-1)=C.
Due to fact that T is a Zq-contraction with respect to ζ∈Zq and by using (ζ3), (18), (44), and (46), we have
(47)0≤limsup⁡k→∞ζ(q(Txmk,Txnk),q(xmk,xnk))=limsup⁡k→∞ζ(q(xmk-1,xnk-1),q(xmk,xnk))<0,
which is a contradiction. This contradiction concludes that C=0 and, hence, {xn} is a left Cauchy sequence. Similarly, it can be proved that {xn} is a right Cauchy sequence. Therefore, {xn} is a Cauchy sequence. Since (X,q) is a complete quasi-metric space, there exists u∈X such that lim⁡n→∞xn=u.
We will show that the pointu is a fixed point of T reasoning by contradiction. Suppose that Tu≠u; that is, q(u,Tu)>0. By Remark 4,
(48)lim⁡n→∞q(Txn,Tu)=lim⁡n→∞q(xn+1,Tu)=q(u,Tu)>0.
Therefore, there is n0∈N such that
(49)q(Txn,Tu)>0,∀n≥n0.
In particular, Txn≠Tu. This also means that xn≠u for all n≥n0. As q(Txn,Tu)>0 and q(xn,u)>0, axiom (ζ2) and property (18) imply that, for all n≥n0,
(50)0≤ζ(q(Txn,Tu),q(xn,u))<q(xn,u)-q(Txn,Tu).
In particular, 0≤q(Txn,Tu)≤q(xn,u) for all n≥n0, which means that
(51)lim⁡n→∞q(xn+1,Tu)=lim⁡n→∞q(Txn,Tu)=0.
Similarly, it can be proved that lim⁡n→∞q(Tu,xn+1)=0. Therefore, {xn} converges, at the same time, to u and to Tu. By the unicity of the limit, u=Tu, which contradicts Tu≠u. As a consequence, u is a fixed point of T. Notice that the uniqueness of the fixed point follows from Lemma 24.Next, we show a variety of cases in which Theorem29 can be applied. Firstly, we mention the analog of the celebrated Banach contraction principle [7] in quasi-metric spaces.Corollary 30 (see, e.g., [1]).
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping such that
(52)q(Tx,Ty)≤λq(x,y),∀x,y∈X,
where λ∈[0,1). Then T has a unique fixed point in X.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζB∈Z, where ζB is defined by ζB(t,s)=λs-t for all s,t∈[0,∞) (see (6)).The following example shows that the above theorem is a proper generalization of the analog of Banach contraction principle.Example 31.
Letα,β,k∈(0,1) be such that α≤k. Let X=[0,1] and q:X×X→[0,∞) be a function defined by
(53)q(x,y)={x-y,ifx≥y,β(y-x),ifx<y.
Then (X,q) is a complete quasi-metric space (but it is not a metric space). Consider the mapping T:X→X defined as Tx=αx for all x∈X. It is clear that it is a Zq-contraction with respect to ζ∈Z, where
(54)ζ(t,s)=ks-t,∀t,s∈[0,∞).
Indeed, if x≥y, then Tx≥Ty. Hence, we get that
(55)ζ(q(Tx,Ty),q(x,y))=ζ(q(αx,αy),q(x,y))=ζ(α(x-y),x-y)=k(x-y)-α(x-y)=(k-α)(x-y)≥0.
If x<y then Tx<Ty. Hence, we get that
(56)ζ(q(Tx,Ty),q(x,y))=ζ(q(αx,αy),q(x,y))=ζ(β(αy-αx),β(y-x))=kβ(y-x)-β(αy-αx)=β(k-α)(y-x)≥0.
Notice that all conditions in Theorem 29 are satisfied and T has a unique fixed point, which is x=0.In the following corollaries we obtain some known and some new results in fixed point theory via simulation functions.Corollary 32 (Rhoades type).
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping satisfying the following condition:
(57)q(Tx,Ty)≤q(x,y)-φ(q(x,y))∀x,y∈X,
where φ:[0,∞)→[0,∞) is a lower semicontinuous function and φ-1(0)={0}. Then T has a unique fixed point in X.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζR∈Z, where ζR is defined by ζR(t,s)=s-φ(s)-t for all s,t∈[0,∞) (see Example 13).Remark 33.
Note that Rhoades assumed in [8] that the function φ was continuous and nondecreasing and it verified lim⁡t→∞φ(t)=∞. In Corollary 32, we replace these conditions by the lower semicontinuity of φ, which is a weaker condition. Therefore, our result is stronger than Rhoades’ original version.Corollary 34.
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping. Suppose that for every x,y∈X,
(58)q(Tx,Ty)≤φ(q(x,y))q(x,y)
for all x,y∈X, where φ:[0,+∞)→[0,1) is a function such that limsup⁡t→r+φ(t)<1 for all r>0. Then T has a unique fixed point.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζT∈Z, where ζT is defined by ζT(t,s)=sφ(s)-t for all s,t∈[0,∞) (see Example 16).Corollary 35.
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping. Suppose that, for every x,y∈X,
(59)q(Tx,Ty)≤η(q(x,y))
for all x,y∈X, where η:[0,+∞)→[0,+∞) is an upper semicontinuous mapping such that η(t)<t for all t>0 and η(0)=0. Then T has a unique fixed point.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζBW∈Z, where ζBW is defined by ζBW(t,s)=η(s)-t for all s,t∈[0,∞) (see Example 17).Corollary 36.
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping satisfying the following condition:
(60)∫0q(Tx,Ty)‍ϕ(t)dt≤q(x,y)∀x,y∈X,
where ϕ:[0,∞)→[0,∞) is a function such that ∫0ϵ‍ϕ(t)qt exists and ∫0ϵ‍ϕ(t)dt>ϵ, for each ϵ>0. Then T has a unique fixed point in X.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζK∈Z, where ζK is defined by
(61)ζK(t,s)=s-∫0t‍ϕ(u)qu∀s,t∈[0,∞)
(see Example 18).Corollary 37.
Let(X,q) be a complete quasi-metric space and let T:X→X be a mapping satisfying the following condition:
(62)q(Tx,Ty)≤h(q(Tx,Ty),q(x,y))q(x,y)∀x,y∈X,
where h:[0,∞)×[0,∞)→[0,∞) is a function such that h(t,s)<1 and limsup⁡n→∞h(tn,sn)<1 provided that {tn} and {sn}⊂(0,+∞) are two sequences such that lim⁡n→∞tn=lim⁡n→∞sn. Then T has a unique fixed point in X.Proof.
The result follows from Theorem29 taking into account that T is a Zq-contraction with respect to ζU∈Z, where ζU is defined by ζU(t,s)=sh(t,s)-t for all s,t∈[0,∞) (see Example 19).Example 38.
The following example is inspired by Remark3 in Boyd and Wong [9]. Let X=[0,1]∪{2,3,4,…} and let us define
(63)q(x,y)={0,ifx=y,x-y,ifx,y∈[0,1],x>y,x+y,otherwise.
It is apparent that (X,q) is a complete quasi-metric space, but it is not a metric space (for instance, q(1,2)≠q(2,1)). Let us consider the mappings T:X→X, η:[0,∞)→R, and ζ:[0,∞)×[0,∞)→R defined by
(64)Tx={x-12x2,ifx∈[0,1],x-1,ifx∈{2,3,4,…},η(t)={t-14t2,ift∈[0,2],t-12,ift>2,ζ(t,s)=η(s)-t,∀t,s≥0.
Although η is not an upper semicontinuous mapping, it is easy to show that ζ is a simulation function (if {tn}→δ>0 and {sn}→δ, then lim⁡sup⁡n→∞ζ(tn,sn)≤max⁡(-δ2/4,-1/2)<0). Furthermore, it can be proved that
(65)ζ(q(Tx,Ty),q(x,y))={0,ifx=yor{x,y}={0,2},(x+3y)(x-y)4,ifx,y∈[0,1],x>y,(x-y)24,ifx,y∈[0,1],x<y,1+y(4-y)2,ifx=2,0<y≤1,1+x22,ify=2,0<x≤1,1+y22,ifx∈{3,4,5,…},y∈[0,1],1+x22,ify∈{3,4,5,…},x∈[0,1],32,ifx,y∈{2,3,4,…},x≠y.
Therefore, T is a Zq-contraction with respect to ζ. Using Theorem 29, T has a unique fixed point, which is x=0.
As Boyd and Wong pointed out in [9], as
(66)lim⁡n→∞q(Tn,0)q(n,0)=lim⁡n→∞Tnn=lim⁡n→∞n-1n=1,
there can be no decreasing function φ with φ(t)<1 for t>0 and for which (58) holds. Furthermore, since
(67)lim⁡x→0+q(Tx,0)q(x,0)=lim⁡x→0+Txx=lim⁡x→0+x-x2/2x=lim⁡x→0+(1-x2)=1,
there is no increasing function φ with φ(t)<1 for t>0 and for which (58) holds.Example 39.
LetX=[0,∞) (it is also possible to consider X=[0,A], where A>0) and let us define
(68)q(x,y)={x-y,ifx≥y,y-x2,ifx<y.
It is clear that (X,q) is a complete quasi-metric space, but it is not a metric space since q(1,2)≠q(2,1). Let us define T:X→X and ζ:[0,∞)×[0,∞)→R by
(69)Tx=log⁡(x+1),∀x∈X;ζ(t,s)=log⁡(s+1)-t,∀t,s≥0.
Then ζ∈Z and T is a Zq-contraction with respect to ζ. Therefore, T has a unique fixed point, which is x=0.


## 4. Consequences: Fixed Point Results in the Context ofG-Metric Spaces

In this section, we show the applicability of our main results to the framework ofG-metric spaces, and we indicate that some existing fixed point results in that setting can be easily derived from our main theorems. First, we recall some basic definitions and fundamental results on this topic which can be found in the literature.Definition 40 (Mustafa and Sims [3]).
A generalized metric (or aG-metric) on a nonempty set X is a mapping G:X×X×X→[0,∞) satisfying the following properties for all x,y,z,a∈X:(G1)
G(x,y,z)=0  if  x=y=z;(G2)
0<G(x,x,y)  for all  x,y∈X  with  x≠y;(G3)
G(x,x,y)≤G(x,y,z)  for all  x,y,z∈X  with  y≠z;(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯ (symmetry in all three variables);(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z)  (rectangle inequality).
In such a case, the pair(X,G) is called a G-metric space.The following result gives some examples of well-knownG-metrics.Lemma 41.
If(X,d) is a metric space and we define Gmax⁡,Gsum:X×X×X→[0,+∞), for all x,y,z∈X, by
(70)Gmax⁡(x,y,z)=max⁡{d(x,y),d(y,z),d(z,x)},Gsum(x,y,z)=d(x,y)+d(y,z)+d(z,x),
then Gmax⁡ and Gsum are G-metrics on X.Example 42.
LetX=[0,∞). The function G:X×X×X→[0,+∞), defined by
(71)G(x,y,z)=|x-y|+|y-z|+|z-x|,
for all x,y,z∈X, is a G-metric on X.Conversely, aG-metric always induces quasi-metrics and also metrics.Lemma 43.
Let(X,G) be a G-metric space and let us define qG,qG′,dGm,dGs:X×X→[0,∞), for all x,y∈X, by
(72)qG(x,y)=G(x,x,y),qG′(x,y)=G(x,y,y),dGm(x,y)=max⁡{G(x,x,y),G(x,y,y)},dGs(x,y)=G(x,x,y)+G(x,y,y).
Then qG and qG′ are quasi-metrics on X and dGm and dGs are metrics on X.The notions of convergence, Cauchy sequence, and completeness in aG-metric space are as follows.Definition 44.
Let(X,G) be a G-metric space, and let {xn} be a sequence of points of X. We say that {xn} is G-convergent to x∈X if
(73)lim⁡n,m→∞G(x,xn,xm)=0;
that is, for any ε>0, there exists N∈N such that G(x,xn,xm)<ε, for all n,m≥N. We call x the limit of the sequence and write {xn}→x or lim⁡n→∞xn=x.Proposition 45.
If(X,G) is a G-metric space, then the following statements are equivalent:(1)
{xn} is G-convergent to x;(2)
{G(xn,xn,x)}→0 as n→∞;(3)
{G(xn,x,x)}→0 as n→∞.Definition 46.
Let(X,G) be a G-metric space. A sequence {xn} is called a G-Cauchy sequence if, for any ε>0, there exists N∈N such that G(xn,xm,xl)<ε for all m,n,l≥N; that is, {G(xn,xm,xl)}→0 as n,m,l→+∞.Proposition 47.
Let(X,G) be a G-metric space. Then the following are equivalent:(1)
the sequence{xn} is G-Cauchy,(2)
for anyε>0, there exists N∈N such that G(xn,xm,xm)<ε, for all m,n≥N.Definition 48.
AG-metric space (X,G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).For more details onG-metric space, we refer, for example, to [3, 10, 11].Lemma 49 (Agarwal et al. [12]).
Let(X,G) be a G-metric space and let us consider the quasi-metrics qG and qG′ as in Lemma 43. Then the following statements hold.(1)
qG(x,y)≤2qG′(x,y)≤4qG(x,y) for all x,y∈X.(2)
In(X,qG) and in (X,qG′), a sequence is right-convergent (resp., left-convergent) if and only if it is convergent. In such a case, its right-limit, its left-limit, and its limit coincide.(3)
In(X,qG) and in (X,qG′), a sequence is right-Cauchy (resp., left-Cauchy) if and only if it is Cauchy.(4)
In(X,qG) and in (X,qG′), every right-convergent (resp., left-convergent) sequence has a unique right-limit (resp., left-limit).(5)
If{xn}⊆X and x∈X, then {xn}→Gx⇔{xn}→qGx⇔{xn}→qG′x.(6)
If{xn}⊆X, then {xn} is G-Cauchy ⇔{xn} is qG-Cauchy ⇔{xn} is qG′-Cauchy.(7)
(X,G) is complete ⇔(X,qG) is complete ⇔(X,qG′) is complete.We present the following version of Theorem29 in the context of G-metric spaces, using the quasi-metric qG defined in Lemma 43.Corollary 50.
Let(X,G) be a complete G-metric space and let T:X→X be a mapping such that there exists ζ∈Z verifying
(74)ζ(G(Tx,Ty,Ty),G(x,y,y))≥0,∀x,y∈X.
Then T has a unique fixed point in X. Furthermore, every Picard sequence generated by T converges to the unique fixed point of T.Proof.
Since(X,G) is complete, then item 7 of Lemma 49 guarantees that (X,qG) is a complete quasi-metric space, and T is a ZqG-contraction in (X,qG) with respect to ζ.The following results are consequence of Corollaries30–37 applied to the quasi-metric qG(x,y)=G(x,x,y) for all x,y∈X (generated by a G-metric).Corollary 51.
Let(X,G) be a complete G-metric space and let T:X→X be a mapping satisfying the following condition:
(75)G(Tx,Ty,Ty)≤λG(x,y,y),∀x,y∈X,
where λ∈[0,1). Then T has a unique fixed point in X.Corollary 52 (see, e.g., [13]).
Let(X,G) be a complete G-metric space and let T:X→X be a mapping satisfying the following condition:
(76)G(Tx,Ty,Ty)≤G(x,y,y)-φ(G(x,y,y)),∀x,y∈X,
where φ:[0,∞)→[0,∞) is lower semicontinuous function and φ-1(0)={0}. Then T has a unique fixed point in X.Corollary 53 (see, e.g., [14]).
Let(X,G) be a complete G-metric space and let T:X→X be a mapping. Suppose that, for every x,y∈X,
(77)G(Tx,Ty,Ty)≤φ(G(x,y,y))G(x,y,y)
for all x,y∈X, where φ:[0,+∞)→[0,1) is a mapping such that limsup⁡t→r+φ(t)<1, for all r>0. Then T has a unique fixed point.Corollary 54 (cf. [15]).
Let(X,G) be a complete G-metric space and let T:X→X be a mapping. Suppose that, for every x,y∈X,
(78)G(Tx,Ty,Ty)≤η(G(x,y,y))
for all x,y∈X, where η:[0,+∞)→[0,+∞) is an upper semicontinuous mapping such that η(t)<t for all t>0 and η(0)=0. Then T has a unique fixed point.Corollary 55.
Let(X,G) be a complete G-metric space and let T:X→X be a mapping satisfying the following condition:
(79)∫0G(Tx,Ty,Ty)‍ϕ(t)qt≤G(x,y,y),∀x,y∈X,
where φ:[0,∞)→[0,∞) is a function such that ∫0ϵ‍ϕ(t)qt exists and ∫0ϵ‍ϕ(t)qt>ϵ, for each ϵ>0. Then T has a unique fixed point in X.Finally, we point out that, obviously, if we replaceG(Tx,Ty,Ty) and G(x,y,y) in Corollaries 50–55 by the expressions G(Tx,Ty,Tz) and G(x,y,z), respectively, then the conclusion is still valid (because the contractive conditions are stronger).



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