# Fixed Point Theorems for Ćirić TypeZ-Contractions in Generating Spaces of Quasi-Metric Family

Authors: Seong-Hoon Cho
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, the notions of Ćirić type (I)Z-contractions and Ćirić type (II) Z-contractions in generating spaces of quasi-metric family are introduced and new fixed point theorems for such two contractions are established. We give examples to illustrate main results.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle forms the basis of metric fixed point theory. Because its importance, many authors generalized this contraction principle by generalizing the certain contraction conditions.Especially, Ćirić [1] proved a result on nonunique fixed points as follows:If a mapT:T:X,d→X,d, where X,d is a metric space, satisfies(1)mindTx,Ty,dx,Tx,dy,Ty−mindx,Ty,dy,Tx≤kdx,y,for allx,y∈X, where k∈0,1, then T has a fixed point whenever X,d is T-orbitally complete.Also, Ćirić [2] obtained the following result:If a mapT:X,d→X,d satisfies the following condition(2)dTx,Ty≤kmaxdx,y,dx,Tx,dy,Ty,12dx,Ty+dy,Tx,for allx,y∈X, where k∈0,1, then T has a unique fixed point provided that X,d is T-orbitally complete.Recently, Khojasteh et al. [3] presented the notion of Z-contraction by using a simulation function and obtained the following result:If a mapT:X,d→X,d is Z-contraction with respect to a simulation funtion ζ, that is,(3)ζdTx,Ty,dx,y≥0,for allx,y∈X, then T has a unique fixed point when X,d is complete.They unified the some existing metric fixed point results. Afterwards, many authors (for example, [4–8]) obtained generalizations of the result of [3].Also, a lot of authors generalized the Banach contraction principle by introducing the concepts of generalized metrics, for example, Branciari metric, b-metric, quasi-metric, semi-metric, G-metric, cone metric, fuzzy metric, Menger Probabilistic metric.In particular, Chang et al. [9, 10] introduced the concept of a generating space of a quasi-metric family and gave some examples and properties of generating space of a quasi-metric family, and they obtained some fixed point and minimization results in such spaces.In this paper, we introduce the concepts of Ćirić type (I)Z-contraction maps and Ćirić type (II) Z-contraction maps in generating spaces of quasi-metric family, and we establish new fixed point theorems for such two contraction maps.LetX be a nonempty set and {da:a∈0,1} be a family of mappings da;X×X→0,∞∀a∈0,1.ThenX,{da:a∈0,1} is called a generating space of quasi-metric family [9, 10] if the following are satisfied:(QM1)∀a∈0,1,dax,y=0 if and only if x=y;(QM2)∀a∈0,1,dax,y=day,x;(QM3)∀a∈0,1,∃b∈0,a such that dax,z≤dbx,y+dby,z;(QM4)∀x,y∈X,dax,y is nonincreasing and left continuous in a.It follows from (QM3) and (QM4) that∀a∈0,1,∃b∈0,a such that(4)dax,z≤dcx,y+dcy,z∀c∈0,b.Example 1.
LetX,d be a metric space, and let dax,y=dx,y∀a∈0,1,∀x,y∈X.
ThenX,d is a generating space of quasi-metric family.Example 2.
LetX=R, and let dax,y=1/a|x−y|∀a∈0,1,∀x,y∈X.
ThenX,{da:a∈0,1} is a generating space of quasi-metric family.Example 3.
LetX={a,b,c,d}, and let d:X×X→0,∞ be a map such that(5)da,b=db,a=3,db,c=dc,b=da,c=dc,a=1,da,d=dd,a=db,d=dd,b=dc,d=dd,c=4,dx,x=0∀x∈X.
Letdax,y=1/adx,y∀a∈0,1,∀x,y∈X.
ThenX,{da:a∈0,1} is a generating space of quasi-metric family.
Note thatd is not a metric on X. In fact, the following inequality is not satisfied:(6)da,b≤da,c+dc,b.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, {xn}⊂X be a sequence, and x∈X.
Then we say that(1)
{xn} converges to x (denoted by limn→∞xn=x) if and only if limn→∞daxn,x=0∀a∈0,1;(2)
{xn} is a Cauchy sequence if and only if ∀ϵ>0,∃n0∈N such that(7)∀n,m>n0,daxn,xm<ϵ∀a∈0,1;(3)
X,{da:a∈0,1} is complete if and only if every Cauchy sequence in X is convergent;(4)
T:X→X is continuous at x if and only if ∀ϵ>0,∃δ>0 such that ∀a∈0,1,daTx,Ty<ϵ, whenever dax,y<δ;(5)
T:X→X is continuous if and only if it is continuous at each point in X.Remark 4.
Every generating spaceX,{da:a∈0,1} of quasi-metric family is a Hausdorff space in the topology τ{da} induced by the family of neighborhoods:(8)Vxϵ,a:x∈X,ϵ>0,a∈0,1,
whereVxϵ,a={y∈X:dax,y<ϵ} (see [11]).
Thus, every convergent sequence in generating spaces of quasi-metric family has a unique limit.
Note that every convergent sequence is a Cauchy sequence.
Also, note thatT:X→X is continuous at x if and only if(9)∀a∈0,1,limn→∞daTxn,Tx=0wheneverlimn→∞daxn,x=0.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping.
Then we say that
(1)
T  is T-orbitally continuous if and only if(10)∀a∈0,1,limi→∞daTnix,x=0implieslimi→∞daTTnix,Tx=0,∀x∈X.
(2)
X
,
{
d
a
:
a
∈
0
,
1
} is T-orbitally complete if and only if every Cauchy sequence of the form {xni=Tnix:i=1,2,3,⋯},x∈X is convergent in X.Remark 5.
LetX,{da:a∈0,1}  be a generating space of quasi-metric family, and let T:X→X be a mapping.(1)
ifT:X→X is continuous, then it is T-orbitally continuous;(2)
ifX,{da:a∈0,1} complete, then it is T-orbitally complete.Remark 6.
A generating space of quasi-metric family induce a fuzzy metric space and Menger probabilistic metric space (see [12] and [13], respectively).
LetZ be the family of all mappings ζ:0,∞×0,∞→R such that(ζ1) ζ0,0=0;(ζ2) ζt,s<s−t∀s,t>0;(ζ3) for any sequence {tn},{sn}⊂0,∞(11)limn→∞supζtn,sn<0,wheneverlimn→∞tn=limn→∞sn>0.
We say thatζ∈Z is a simulation function [9].
Note that for allt>0, ζt,t<0.Example 7.
Letζi:0,∞→0,∞,i=1,2,3,4,5,6,7,8, be functions defined as follows:(1)
ζ1t,s=ϕ1s−ϕ2t∀t,s≥0, where ϕ1,ϕ2:0,∞→0,∞ are continuous functions such that(12)ϕ1−10=0,ϕ2−10=0andϕ2t<t≤ϕ1t∀t>0;(2)
ζ2t,s=s−ft,s/gt,st∀t,s≥0, where f,g:0,∞×0,∞→0,∞ are continuous finctions with respect to each variables such that(13)ft,s>gt,s∀t,s>0;(3)
ζ3t,s=s−ϕs−t∀t,s≥0, where ϕ:0,∞→0,∞ is a lower semi-continuous function with ϕ−1{0}=0;(4)
ζ4t,s=sφs−t∀t,s≥0, where φ:0,∞→0,1 is a function such that(14)limt→r+supφt<1∀r>0;(5)
ζ5t,s=ψs−t∀t,s≥0, where ψ:0,∞→0,∞ is an upper semicontinuous function such that(15)ψt<t∀t>0andψ0=0;(6)
ζ6t,s=ks−t∀t,s≥0, where k∈0,1;(7)
ζ7t,s=s−∫0tνudu∀s,t≥0, where ν:0,∞→0,∞ is a function such that(16)∫0ϵνudu>ϵ∀ϵ>0;(8)
ζ8t,s=s1+s−t∀s,t≥0.
Thenζi∈Z,∀i=1,2,3,4,5,6,7,8.
For more examples of simulation functions, we can find in [8, 9, 11, 12].Lemma 8.
LetX,{da:a∈0,1} be a generating space of quasi-metric family. Suppose that {xn}⊂X is not a Cauchy sequence.
Then there exists anϵ>0 for which we can find two subsequences {xmk} and {xnk} of {xn} such that mk is the smallest index with mk>nk>k(17)daxmk,xnk≥ϵ,wherea∈0,1anddsxmk−1,xnk<ϵ∀s∈0,1.
Further if(18)limn→∞dsxn−1,xn=0∀s∈0,1,
then we have(1)
limn→∞dcxmk,xnk=ϵ∀c∈0,a;(2)
limn→∞dbxmk+1,xnk=ϵ,whereb∈0,a;(3)
limn→∞dbxmk,xnk+1=ϵ,whereb∈0,a;(4)
limn→∞dbxmk+1,xnk+1=ϵ,whereb∈0,a;Proof.
If{xn}⊂X is not a Cauchy sequence, then from definition (17) holds.
Suppose that (18) is satisfied.
Then from (17) we have that for each c∈0,a(19)ϵ≤daxmk,xnk≤dcxmk,xnk≤dbxmk,xmk−1+dbxmk−1,xnk<dbxmk,xmk−1+ϵ,forsomeb∈0,c.
By takingk→∞ in above inequality(20)limk→∞dcxmk,xnk=ϵ∀c∈0,a.
Hence (1) is proved.
We show that (2) holds.
It follows from (QM3) and (QM4) that fora∈0,1,∃b∈0,a such that ∀c∈0,b,(21)daxmk,xnk≤dcxmk,xmk+1+dcxmk+1,xnk,(22)dcxmk+1,xnk≤dexmk+1,xmk+dexmk,xnkforsomee∈0,c,
Lettingk→∞ in Equations (21 and 22), and using Equations (18 and 19) we obtain(23)limk→∞dcxmk+1,xnk=ϵ∀c∈0,b.
In particular, we have(24)limk→∞dbxmk+1,xnk=ϵ,whereb∈0,a.
Thus proof of Equation (2) is complete.
In similar with the proof of Equation (2), we have(25)limk→∞dcxmk,xnk+1=ϵ∀c∈0,b,
and so(26)limk→∞dbxmk,xnk+1=ϵ,whereb∈0,a.
We now show that Equation (4) holds.
Fora,∃b∈0,a such that ∀c∈0,b,(27)daxmk+1,xnk≤dcxmk+1,xnk+1+dcxnk+1,xnk,
and(28)dcxmk+1,xnk+1≤dexmk+1,xmk+dexmk,xnk+1∀e∈0,c.
By takingk→∞ in Equations (17) and (29), and using Equations (18), (23) and (25) we obtain(29)limk→∞dcxmk+1,xnk+1=ϵ∀c∈0,b,
and so(30)limk→∞dbxmk+1,xnk+1=ϵ,whereb∈0,a.
Hence (4) is proved.Lemma 9.
LetX,{da:a∈0,1} be a generating space of quasi-metric family. Suppose that (31)limn→∞daxn,x=0andlimn→∞dayn,y=0∀a∈0,1.
Then we have(32)limn→∞daxn,yn=dax,y∀a∈0,1.Proof.
Since(33)daxn,yn≤dbxn,x+dbx,yn,b∈0,a,(34)dax,y≤dcx,yn+dcyn,y,c∈0,a,
we have(35)daxn,yn≤dbxn,x+dbx,yn≤dbxn,x+dbcx,yn,(36)dax,y≤dcx,yn+dcyn,y≤dbcx,yn+dcyn,y,
wheredbcyn,y=max{dbyn,y,dcyn,y} and dbcx,yn=max{dbx,yn,dcx,yn}.
Hence we have(37)0≤daxn,yn−dax,y=daxn,yn−dbcx,yn+dbcx,yn−dax,y≤daxn,yn−dbcx,yn+dbcx,yn−dax,y≤dbxn,x+dcyn,y≤dexn,x+deyn,ywhere0<e<minb,c.
By takingn→∞, we obtain(38)limn→∞daxn,yn=dax,y∀a∈0,1.


## 2. Fixed Point Theorems

### 2.1. Nonunique Fixed Point Results

LetX,{da:a∈0,1} be a generating space of quasi-metric family.A mappingT:X→X is called Ćirić type (I) Z -contraction with respect to ζ∈Z if(39)ζma,x,y,dax,y≥0,∀a∈0,1,for allx,y∈X, where ma,x,y=daTx,Ty(40)−mindax,Ty,day,Tx,dax,Tx,day,Ty.Now, we prove our first fixed point result.Theorem 10.
LetX,{da:a∈0,1}be a generating space of quasi-metric family, and let T:X→X be a Ćirić type (I)Z-contraction mapping with respect to ζ. IfX is T-orbitally complete, then  T has a fixed point.Proof.
Letx0∈X be any fixed point. Define a sequence {xn}⊂X by xn=Txn−1=Tnx0∀n=1,2,3⋯.
Ifxn0=xn0+1 for some n0∈N, then xn0 is a fixed point of T, and the proof is finished.
Assume thatxn−1≠xn∀n=1,2,3⋯.
We infer that∀n=1,2,3,⋯(41)ma,xn−1,xn=daTxn−1,Txn−mindaxn−1,Txn,daxn,Txn−1,daxn−1,Txn−1,daxn,Txn=daxn,xn+1−mindaxn−1,xn+1,daxn,xn,daxn−1,xn,daxn,xn+1=daxn,xn+1.
It follows from (41) and (42) that ∀a∈0,1,∀n=1,2,3,⋯(42)0≤ζma,xn−1,xn,daxn−1,xn=ζdaxn,xn+1,daxn−1,xn<daxn−1,xn−daxn,xn+1.
Consequently, we obtain that∀a∈0,1,∀n=1,2,3,⋯(43)daxn,xn+1<daxn−1,xn.
Sincedaxn−1,xn is a decreasing sequence bounded from below by 0, there exists r≥0 such that(44)limn→∞daxn−1,xn=r,∀a∈0,1.
We now show thatlimn→∞daxn−1,xn=0,∀a∈0,1.
On the contrary, assume that(45)limn→∞daxn−1,xn=r>0,wherea∈0,1.
Lettn:=ma,xn−1,xn and sn:=daxn−1,xn,a∈0,1,∀n=1,2,3,⋯.
Then(46)limn→∞sn=limn→∞tn=r>0.
From conditionζ3 we have(47)0≤limsupn→∞ζtn,sn<0,
which is a contradiction. Hence we haver=0, and hence(48)limn→∞daxn,xn−1=0,∀a∈0,1.
We now show that{xn} is a Cauchy sequence.
Suppose that{xn} is not a Cauchy sequence.
By Lemma8, there exist ϵ>0 and two subsequences {xnk},{xnk} of {xn} satisfying (17).
Since (49) holds, from Lemma 8 we have that(49)limn→∞dbxmk,xnk=ε,andlimn→∞dbxmk+1,xnk=limn→∞dbxmk,xnk+1=limn→∞dbxmk+1,xnk+1=εwhereb∈0,a
Let(50)tk=mb,xmk,xnk=dbxmk+1,xnk+1−mindbxmk,xnk+1,dbxmk+1,xnk,dbxmk,xmk+1,dbxnk,xnk+1,sk=dbxmk,xnk.
Then(51)limk→∞tk=limk→∞sk=ε>0.
It follows fromξ3 that(52)0≤limk→∞supζtk,sk<0,
which is a contradiction.
Thus{xn=Tnx0} is a Cauchy sequence. It follows from T-orbitally completness of X that there exists x∗∈X such that(53)limn→∞daxn,x∗=0,∀a∈0,1.
From Lemma9(54)limn→∞daxn,Tx∗=dax∗,Tx∗,∀a∈0,1.
Thus it follows from (40) that(55)0≤ζma,xn,x∗,daxn,x∗<daxn,x∗−ma,xn,x∗,
where(56)ma,xn,x∗=daTxn,Tx∗−mindaxn,Tx∗,dax∗,Txn,daxn,Txn,dax∗,Tx∗=daxn+1,Tx∗−mindaxn,Tx∗,dax∗,xn+1,daxn,xn+1,dax∗,Tx∗.
Hence(57)ma,xn,x∗<daxn,x∗.
Lettingn→∞ in above inequality, and using Equations (49), (54) and (55), we have(58)dax∗,Tx∗=0∀a∈0,1.
Thusx∗=Tx∗.Example 11.
LetX=0,∞ with dax,y=1/a|x−y|∀a∈0,1,∀x,y∈X.
ThenX,{da:a∈0,1} is a generating space of quasi-metric family.
Define a mappingT:X→X as follows:(59)Tx=18x.
Letζt,s=ϕs−ψt, where(60)ψt=2tandϕt=14t∀t≥0.
Thenζ∈Z.
We have that(61)ζma,x,y,dax,y=ϕdax,y−ψma,x,y=ϕ1ax−y−ψ181ax−y−mindax,Tx,day,Ty,dax,Ty,day,Tx=2mindax,Tx,day,Ty,dax,Ty,day,Tx≥0.
ThusT is a Ćirić type (I) Z-contraction with respect to ζ.
All conditions of Theorem10 are satisfied. Hence, from Theorem 10 T has a fixed point, x∗=0.Corollary 12.
LetX,d be a metric space, and let T:X→X be a mapping such that(62)ζmx,y,dx,y≥0.
for allx,y∈X, where mx,y=dTx,Ty(63)−mindx,Ty,dy,Tx,dx,Tx,dy,Ty.
IfX is T-orbitally complete, then T has a fixed point.


### 2.2. Unique Fixed Point Results

LetX,{da:a∈0,1} be a generating space of quasi-metric family.A mappingT:X→X is called Ćirić type (II) Z-contraction with respect to ζ∈Z if(64)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,na,x,y≥0,wherena,x,y=max{dax,y,dax,Tx,day,Ty,1/2dax,Ty+day,Tx} and ζ∈Z is nondecreasing with respect to the second variable.Theorem 13.
LetX,{da:a∈0,1}be a generating space of quasi-metric family, and letT:X→X be a Ćirić type (II) Z-contraction with respect to ζ∈Z. If X is T-orbitally complete, then T has a unique fixed point.Proof.
Firstly, we show the uniqueness of fixed point whenever it exists.
Suppose thatT has two fixed points, say u,v∈X such that u≠v.
Then from (54) we have(65)0≤ζdaTu,Tv,na,u,v=ζdau,v,dau,v<dau,v−dau,v=0,
which is a contradiction. ThusT has a unique fixed point if it exists.
Secondly, we show the existence of fixed point.
As in proof of Theorem10, let us define a sequence {xn}⊂X by xn=Txn−1=Tnx0∀n=1,2,3⋯, where x0∈X is any fixed point, such that(66)xn−1≠xn∀n=1,2,3⋯.
We infer that(67)na,xn−1,xn=maxdaxn−1,xn,daxn−1,Txn−1,daxn,Txn,12daxn−1,Txn+daxn,Txn−1=maxdaxn−1,xn,daxn−1,xn,daxn,xn+1,12daxn−1,xn+1+daxn,xn≤maxdaxn−1,xn,daxn−1,xn,daxn,xn+1,12dbxn−1,xn+dbxn,xn+1≤maxdbxn−1,xn,dbxn−1,xn,dbxn,xn+1,12dbxn−1,xn+dbxn,xn+1=maxdbxn−1,xn,dbxn,xn+1whereb∈0,a.
Hence(68)nb,xn−1,xn≤maxdbxn−1,xn,dbxn,xn+1.It follows from 65 that(69)0≤ζdbTxn−1,Txn,nb,xn,xn+1=ζdbxn,xn+1,nb,xn,xn+1<nb,xn,xn+1−dbxn,xn+1≤maxdbxn−1,xn,dbxn,xn+1−dbxn,xn+1,
which implies(70)dbxn,xn+1<dbxn−1,xn∀n=1,2,3,⋯.
Thus there existsr≥0 such that(71)limn→∞dbxn−1,xn=r.
We show thatr=0.
Assume thatr≠0.
Let(72)sn=maxdbxn−1,xn,dbxn,xn+1andtn=dbxn,xn+1∀n=1,2,3,⋯.
Thenlimn→∞sn=limn→∞tn=r>0.
It follows from (54 and 55) that(73)0≤limsupζdbTxn−1,Txn,nb,xn−1,xn≤limsupζtn,sn<0,
which is a contradiction. Hencer=0, and hence(74)limn→∞dbxn−1,xn=0,∀b∈0,a.
We now show that{xn} is a Cauchy sequence.
Suppose that{xn} is not a Cauchy sequence.
By Lemma1.1, there exist ϵ>0 and two subsequences {xnk},{xnk} of {xn} satisfying (17).
Since (73) holds, it follows from Lemma 1.1 that(75)limn→∞dcxmk,xnk=ε,andlimn→∞dcxmk+1,xnk=limn→∞dcxmk,xnk+1=limn→∞dcxmk+1,xnk+1=εforsomec∈0,b.
Letsk=nc,xmk,xnk and tn=dcxmk+1,xnk+1.
Then(76)limk→∞nc,xmk,xnk=limk→∞dcxmk,xnk=ϵ.
Thus from (ζ3) we have(77)0≤limsupζtk,sk<0,
which is a contradiction. Thus{xn} is a Cauchy sequence.
Hence there existsx∗∈X such that(78)limn→∞daxn,x∗=0∀a∈0,1.
Letsn=na,xn,x∗ and tn=daxn,Tx∗,a∈0,1,∀n=1,2,3,⋯.
IfdaTx∗,x∗>0,wherea∈0,1, then from Lemma 9 we have(79)limn→∞sn=limn→∞tn=dax∗,Tx∗>0.
Hence(80)0≤limsupζtn,sn<0,
which is a contradiction.
Thus(81)daTx∗,x∗=0∀a∈0,1.
Hencex∗=Tx∗.Corollary 14.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(82)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,la,x,y≥0,
wherela,x,y=max{dax,y,1/2dax,Tx+day,Ty,1/2dax,Ty+day,Tx} and ζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.Corollary 15.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(83)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,dax,y≥0,
whereζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.
By takingζ=ζ6 in Corollary 2.5, we have the following result.Corollary 16.
(Banach) LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(84)∀a∈0,1,∀x,y∈X,daTx,Ty≤kdax,y,wherek∈0,1.
IfX is T-orbitally complete, then T has a unique fixed point.Corollary 17.
LetX,d be a metric space, and let T:X→X be a mapping such that(85)∀x,y∈X,ζdTx,Ty,nx,y≥0,
wherenx,y=max{dx,y,dx,Tx,dy,Ty,1/2dx,Ty+dy,Tx} and ζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.Example 18.
LetX={2/n:n=2,3,4,⋯}∪{0}, and let(86)dax,y=1ax−y∀a∈0,1.LetT:X→X be a map defined as(87)Tx=2n+1,x=2n,n=2,3,4,⋯,x=0,0.Obviously,X,{da:a∈0,1} is complete, and hence it is T-orbitally complete.We now show thatT is a Ćirić type (II) Z-contraction with respect to ζ7t,s=s−∫0tνudu with νu=1+2u,∀u≥0.Case 1: Letx=2/n,y=0.Then we have(88)ζ7daT2n,T0,na,2n,0=ζ71a⋅2n+1,1a⋅2n=1a⋅2n−∫01/a⋅2/n+11+2udu0=1a⋅2n−1a⋅2n+1+1a⋅2n+12=1a⋅4nn+13>0.Case 2: Letx=2/n,y=2/nm<n.(89)ζ7daT2n,T2m,na,2n,2m=ζ71a⋅2n+1−2m+1,1a⋅2n−2m=1a⋅2n−2m−∫01/a⋅2/n+1−2/m+11+2udu=1a⋅2n−2m−1a⋅2n+1−2m+1+1a⋅2n+1−2m+12=1a⋅2m−nn+1m+1n+1m+1−mn+4m−n2mnn+12m+12=1a⋅2m−nn+1m+1−mn+4m−n2mnn+1m+1>0.ThusT is a Ćrić type (II) Z-contraction with respect to ζ6. All conditions of Theorem 13 are satisfied, and T has a unique fixed point theorem x∗=0.However, Bananch contraction principle in the setting of generating spaces of quasi-metric family, i.e. Corollary16 is not satisfied.In fact, if forx=2/n,y=2/n+1∀n=2,3,4,⋯(90)∀a∈0,1,daTx,Ty≤kdax,y,wherek∈0,1,then we have that(91)1a2n+1−2n+2≤k1a2n−2n+1,which implies(92)k≥nn+1n+1n+2∀n=2,3,4,⋯.By takingn→∞, we have k≥1 which is a contradiction.Thus Theorem13 is a generalization of the Banach contraction principle in generating space of quasi-metric family.


## 2.1. Nonunique Fixed Point Results

LetX,{da:a∈0,1} be a generating space of quasi-metric family.A mappingT:X→X is called Ćirić type (I) Z -contraction with respect to ζ∈Z if(39)ζma,x,y,dax,y≥0,∀a∈0,1,for allx,y∈X, where ma,x,y=daTx,Ty(40)−mindax,Ty,day,Tx,dax,Tx,day,Ty.Now, we prove our first fixed point result.Theorem 10.
LetX,{da:a∈0,1}be a generating space of quasi-metric family, and let T:X→X be a Ćirić type (I)Z-contraction mapping with respect to ζ. IfX is T-orbitally complete, then  T has a fixed point.Proof.
Letx0∈X be any fixed point. Define a sequence {xn}⊂X by xn=Txn−1=Tnx0∀n=1,2,3⋯.
Ifxn0=xn0+1 for some n0∈N, then xn0 is a fixed point of T, and the proof is finished.
Assume thatxn−1≠xn∀n=1,2,3⋯.
We infer that∀n=1,2,3,⋯(41)ma,xn−1,xn=daTxn−1,Txn−mindaxn−1,Txn,daxn,Txn−1,daxn−1,Txn−1,daxn,Txn=daxn,xn+1−mindaxn−1,xn+1,daxn,xn,daxn−1,xn,daxn,xn+1=daxn,xn+1.
It follows from (41) and (42) that ∀a∈0,1,∀n=1,2,3,⋯(42)0≤ζma,xn−1,xn,daxn−1,xn=ζdaxn,xn+1,daxn−1,xn<daxn−1,xn−daxn,xn+1.
Consequently, we obtain that∀a∈0,1,∀n=1,2,3,⋯(43)daxn,xn+1<daxn−1,xn.
Sincedaxn−1,xn is a decreasing sequence bounded from below by 0, there exists r≥0 such that(44)limn→∞daxn−1,xn=r,∀a∈0,1.
We now show thatlimn→∞daxn−1,xn=0,∀a∈0,1.
On the contrary, assume that(45)limn→∞daxn−1,xn=r>0,wherea∈0,1.
Lettn:=ma,xn−1,xn and sn:=daxn−1,xn,a∈0,1,∀n=1,2,3,⋯.
Then(46)limn→∞sn=limn→∞tn=r>0.
From conditionζ3 we have(47)0≤limsupn→∞ζtn,sn<0,
which is a contradiction. Hence we haver=0, and hence(48)limn→∞daxn,xn−1=0,∀a∈0,1.
We now show that{xn} is a Cauchy sequence.
Suppose that{xn} is not a Cauchy sequence.
By Lemma8, there exist ϵ>0 and two subsequences {xnk},{xnk} of {xn} satisfying (17).
Since (49) holds, from Lemma 8 we have that(49)limn→∞dbxmk,xnk=ε,andlimn→∞dbxmk+1,xnk=limn→∞dbxmk,xnk+1=limn→∞dbxmk+1,xnk+1=εwhereb∈0,a
Let(50)tk=mb,xmk,xnk=dbxmk+1,xnk+1−mindbxmk,xnk+1,dbxmk+1,xnk,dbxmk,xmk+1,dbxnk,xnk+1,sk=dbxmk,xnk.
Then(51)limk→∞tk=limk→∞sk=ε>0.
It follows fromξ3 that(52)0≤limk→∞supζtk,sk<0,
which is a contradiction.
Thus{xn=Tnx0} is a Cauchy sequence. It follows from T-orbitally completness of X that there exists x∗∈X such that(53)limn→∞daxn,x∗=0,∀a∈0,1.
From Lemma9(54)limn→∞daxn,Tx∗=dax∗,Tx∗,∀a∈0,1.
Thus it follows from (40) that(55)0≤ζma,xn,x∗,daxn,x∗<daxn,x∗−ma,xn,x∗,
where(56)ma,xn,x∗=daTxn,Tx∗−mindaxn,Tx∗,dax∗,Txn,daxn,Txn,dax∗,Tx∗=daxn+1,Tx∗−mindaxn,Tx∗,dax∗,xn+1,daxn,xn+1,dax∗,Tx∗.
Hence(57)ma,xn,x∗<daxn,x∗.
Lettingn→∞ in above inequality, and using Equations (49), (54) and (55), we have(58)dax∗,Tx∗=0∀a∈0,1.
Thusx∗=Tx∗.Example 11.
LetX=0,∞ with dax,y=1/a|x−y|∀a∈0,1,∀x,y∈X.
ThenX,{da:a∈0,1} is a generating space of quasi-metric family.
Define a mappingT:X→X as follows:(59)Tx=18x.
Letζt,s=ϕs−ψt, where(60)ψt=2tandϕt=14t∀t≥0.
Thenζ∈Z.
We have that(61)ζma,x,y,dax,y=ϕdax,y−ψma,x,y=ϕ1ax−y−ψ181ax−y−mindax,Tx,day,Ty,dax,Ty,day,Tx=2mindax,Tx,day,Ty,dax,Ty,day,Tx≥0.
ThusT is a Ćirić type (I) Z-contraction with respect to ζ.
All conditions of Theorem10 are satisfied. Hence, from Theorem 10 T has a fixed point, x∗=0.Corollary 12.
LetX,d be a metric space, and let T:X→X be a mapping such that(62)ζmx,y,dx,y≥0.
for allx,y∈X, where mx,y=dTx,Ty(63)−mindx,Ty,dy,Tx,dx,Tx,dy,Ty.
IfX is T-orbitally complete, then T has a fixed point.


## 2.2. Unique Fixed Point Results

LetX,{da:a∈0,1} be a generating space of quasi-metric family.A mappingT:X→X is called Ćirić type (II) Z-contraction with respect to ζ∈Z if(64)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,na,x,y≥0,wherena,x,y=max{dax,y,dax,Tx,day,Ty,1/2dax,Ty+day,Tx} and ζ∈Z is nondecreasing with respect to the second variable.Theorem 13.
LetX,{da:a∈0,1}be a generating space of quasi-metric family, and letT:X→X be a Ćirić type (II) Z-contraction with respect to ζ∈Z. If X is T-orbitally complete, then T has a unique fixed point.Proof.
Firstly, we show the uniqueness of fixed point whenever it exists.
Suppose thatT has two fixed points, say u,v∈X such that u≠v.
Then from (54) we have(65)0≤ζdaTu,Tv,na,u,v=ζdau,v,dau,v<dau,v−dau,v=0,
which is a contradiction. ThusT has a unique fixed point if it exists.
Secondly, we show the existence of fixed point.
As in proof of Theorem10, let us define a sequence {xn}⊂X by xn=Txn−1=Tnx0∀n=1,2,3⋯, where x0∈X is any fixed point, such that(66)xn−1≠xn∀n=1,2,3⋯.
We infer that(67)na,xn−1,xn=maxdaxn−1,xn,daxn−1,Txn−1,daxn,Txn,12daxn−1,Txn+daxn,Txn−1=maxdaxn−1,xn,daxn−1,xn,daxn,xn+1,12daxn−1,xn+1+daxn,xn≤maxdaxn−1,xn,daxn−1,xn,daxn,xn+1,12dbxn−1,xn+dbxn,xn+1≤maxdbxn−1,xn,dbxn−1,xn,dbxn,xn+1,12dbxn−1,xn+dbxn,xn+1=maxdbxn−1,xn,dbxn,xn+1whereb∈0,a.
Hence(68)nb,xn−1,xn≤maxdbxn−1,xn,dbxn,xn+1.It follows from 65 that(69)0≤ζdbTxn−1,Txn,nb,xn,xn+1=ζdbxn,xn+1,nb,xn,xn+1<nb,xn,xn+1−dbxn,xn+1≤maxdbxn−1,xn,dbxn,xn+1−dbxn,xn+1,
which implies(70)dbxn,xn+1<dbxn−1,xn∀n=1,2,3,⋯.
Thus there existsr≥0 such that(71)limn→∞dbxn−1,xn=r.
We show thatr=0.
Assume thatr≠0.
Let(72)sn=maxdbxn−1,xn,dbxn,xn+1andtn=dbxn,xn+1∀n=1,2,3,⋯.
Thenlimn→∞sn=limn→∞tn=r>0.
It follows from (54 and 55) that(73)0≤limsupζdbTxn−1,Txn,nb,xn−1,xn≤limsupζtn,sn<0,
which is a contradiction. Hencer=0, and hence(74)limn→∞dbxn−1,xn=0,∀b∈0,a.
We now show that{xn} is a Cauchy sequence.
Suppose that{xn} is not a Cauchy sequence.
By Lemma1.1, there exist ϵ>0 and two subsequences {xnk},{xnk} of {xn} satisfying (17).
Since (73) holds, it follows from Lemma 1.1 that(75)limn→∞dcxmk,xnk=ε,andlimn→∞dcxmk+1,xnk=limn→∞dcxmk,xnk+1=limn→∞dcxmk+1,xnk+1=εforsomec∈0,b.
Letsk=nc,xmk,xnk and tn=dcxmk+1,xnk+1.
Then(76)limk→∞nc,xmk,xnk=limk→∞dcxmk,xnk=ϵ.
Thus from (ζ3) we have(77)0≤limsupζtk,sk<0,
which is a contradiction. Thus{xn} is a Cauchy sequence.
Hence there existsx∗∈X such that(78)limn→∞daxn,x∗=0∀a∈0,1.
Letsn=na,xn,x∗ and tn=daxn,Tx∗,a∈0,1,∀n=1,2,3,⋯.
IfdaTx∗,x∗>0,wherea∈0,1, then from Lemma 9 we have(79)limn→∞sn=limn→∞tn=dax∗,Tx∗>0.
Hence(80)0≤limsupζtn,sn<0,
which is a contradiction.
Thus(81)daTx∗,x∗=0∀a∈0,1.
Hencex∗=Tx∗.Corollary 14.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(82)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,la,x,y≥0,
wherela,x,y=max{dax,y,1/2dax,Tx+day,Ty,1/2dax,Ty+day,Tx} and ζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.Corollary 15.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(83)∀a∈0,1,∀x,y∈X,ζdaTx,Ty,dax,y≥0,
whereζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.
By takingζ=ζ6 in Corollary 2.5, we have the following result.Corollary 16.
(Banach) LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(84)∀a∈0,1,∀x,y∈X,daTx,Ty≤kdax,y,wherek∈0,1.
IfX is T-orbitally complete, then T has a unique fixed point.Corollary 17.
LetX,d be a metric space, and let T:X→X be a mapping such that(85)∀x,y∈X,ζdTx,Ty,nx,y≥0,
wherenx,y=max{dx,y,dx,Tx,dy,Ty,1/2dx,Ty+dy,Tx} and ζ∈Z is nondecreasing with respect to the second variable.
IfX is T-orbitally complete, then T has a unique fixed point.Example 18.
LetX={2/n:n=2,3,4,⋯}∪{0}, and let(86)dax,y=1ax−y∀a∈0,1.LetT:X→X be a map defined as(87)Tx=2n+1,x=2n,n=2,3,4,⋯,x=0,0.Obviously,X,{da:a∈0,1} is complete, and hence it is T-orbitally complete.We now show thatT is a Ćirić type (II) Z-contraction with respect to ζ7t,s=s−∫0tνudu with νu=1+2u,∀u≥0.Case 1: Letx=2/n,y=0.Then we have(88)ζ7daT2n,T0,na,2n,0=ζ71a⋅2n+1,1a⋅2n=1a⋅2n−∫01/a⋅2/n+11+2udu0=1a⋅2n−1a⋅2n+1+1a⋅2n+12=1a⋅4nn+13>0.Case 2: Letx=2/n,y=2/nm<n.(89)ζ7daT2n,T2m,na,2n,2m=ζ71a⋅2n+1−2m+1,1a⋅2n−2m=1a⋅2n−2m−∫01/a⋅2/n+1−2/m+11+2udu=1a⋅2n−2m−1a⋅2n+1−2m+1+1a⋅2n+1−2m+12=1a⋅2m−nn+1m+1n+1m+1−mn+4m−n2mnn+12m+12=1a⋅2m−nn+1m+1−mn+4m−n2mnn+1m+1>0.ThusT is a Ćrić type (II) Z-contraction with respect to ζ6. All conditions of Theorem 13 are satisfied, and T has a unique fixed point theorem x∗=0.However, Bananch contraction principle in the setting of generating spaces of quasi-metric family, i.e. Corollary16 is not satisfied.In fact, if forx=2/n,y=2/n+1∀n=2,3,4,⋯(90)∀a∈0,1,daTx,Ty≤kdax,y,wherek∈0,1,then we have that(91)1a2n+1−2n+2≤k1a2n−2n+1,which implies(92)k≥nn+1n+1n+2∀n=2,3,4,⋯.By takingn→∞, we have k≥1 which is a contradiction.Thus Theorem13 is a generalization of the Banach contraction principle in generating space of quasi-metric family.


## 3. Consequence

By applying simulation functions of Example3 to Theorem 10 and Theorem 13, we have some fixed point results.Especially, takingζ8 for ζ in Theorem 10 and Theorem 13, we have the following results.Corollary 19.
LetX,{da:a∈0,1} be a generating space of quasi-metric family, and let T:X→X be a mapping such that(93)∀a∈0,1,∀x,y∈X,ma,x,y≤dax,y1+dax,y.
IfX is T-orbitally complete, then T has a fixed point.Corollary 20.
LetX,dbe a metric space, and let T:X→Xbe a mapping such that(94)∀x,y∈X,mx,y≤dx,y1+dx,y.
IfX is T-orbitally complete, then T has a fixed point.Corollary 21.
LetX,{da:a∈0,1} be a gessnerating space of quasi-metric family, and let T:X→X be a mapping such that(95)∀a∈0,1,∀x,y∈X,daTx,Ty≤na,x,y1+na,x,y.
IfX is T-orbitally complete, then T has a unique fixed point.Corollary 22.
LetX,d be a metric space, and let T:X→X be a mapping such that(96)∀x,y∈X,dTx,Ty≤nx,y1+nx,y.
If X is T-orbitally complete, then T has a unique fixed point.



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