# Double Controlled Quasi-Metric Type Spaces and Some Results

Authors: Abdullah Shoaib; Sabeena Kazi; Asifa Tassaddiq; Shaif S. Alshoraify; Tahair Rasham
Journal: Complexity
 (2020)

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## Abstract

Abdeljawad et al. (2018) introduced a new concept, named double controlled metric type spaces, as a generalization of the notion of extendedb-metric spaces. In this paper, we extend their concept and introduce the concept of double controlled quasi-metric type spaces with two incomparable functions and prove some unique fixed point results involving new types of contraction conditions. Also, we introduce the concept of α−μ−k double controlled contraction and prove some related fixed point results. We give several examples to show that our results are the proper generalization of the existing works.


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## Body

## 1. Introduction and Preliminaries

The theory of fixed points takes an important place in the transition from classical analysis to modern analysis. One of the most remarkable work on fixed point theory was done by Banach [1]. Various generalizations of Banach fixed point theorem were made by numerous mathematicians, see [2–4]. One of the abstraction of the metric spaces is the quasi-metric space that was introduced by Wilson [5]. The commutativity condition does not hold in general in quasi-metric spaces. Several authors used this concept to prove some fixed point results, see [6–18]. On the other hand, Bakhtin [19] and Czerwik [20] established the idea of b-metric spaces. Lateral, many authors got several fixed point results, for instance, see [21–25]. Kamran et al. [26] introduced a new idea to generalized b-metric spaces, named as extended b-metric spaces, see also [27–30]. They replaced the parameter b≥1 in the triangle inequality by the control function θ:G×G⟶1,∞. Nurwahyu [31] introduced dislocated quasi-extended b-metric space and obtained several fixed point results. Mlaiki et al. [32] generalized the triangle inequality in b-metric spaces by using control function in a different style and introduced controlled metric type spaces. Recently, Abdeljawad et al. [33] generalized the idea of extended b-metric spaces as well as controlled metric type spaces and introduced double controlled metric type spaces. They replaced the control function θ in triangle inequality by two control functions α and μ. Now, we recall some basic definitions and examples that will be used in this paper.Definition 1 (see [33]).
Given noncomparable functionsα,μ:G×G⟶[1,+∞. If f:G×G⟶0,+∞ satisfies(q1)
fυ,γ=0 if and only if υ=γ(q2)
fυ,γ=fγ,υ(q3)
fυ,γ≤αυ,efυ,e+μe,γfe,γ, for all υ,γ,e∈G
Then,f is called a double controlled metric type with the functions α, μ and the pair G,f is called double controlled metric type space with the functions α,μ.Theorem 1 (see [33]).
LetG,f be a complete double controlled metric type space with the functions α,μ:G×G⟶1,+∞ and let T:G⟶G be a given mapping. Suppose that the following conditions are satisfied.There existsk∈0,1 such that(1)fTc˙,Ty≤kfc˙,y,for allc˙,y∈G.Forυ0∈G, choose υp=Tpυ0. Assume that(2)supm≥1limi⟶+∞αυi+1,υi+2αυi,υi+1μυi+1,υm<1k.In addition, for everyυ∈G, we have(3)limp⟶+∞αυ,υpandlimp⟶+∞μυp,υexist and are finite.Then,T has a unique fixed point υ∗∈G.Definition 2.
Given noncomparable functionsα,μ:G×G⟶1,+∞. If f:G×G⟶0,+∞ satisfies(Q1)fυ,γ=0 if and only if υ=γ(Q2)fυ,γ≤αυ,efυ,e+μe,γfe,γ, for all υ,γ,e∈G
Then,f is called a double controlled quasi-metric type with the functions α and μ and G,f is called a double controlled quasi-metric type space. If μe,γ=αe,γ, then G,f is called a controlled quasi-metric type space.Remark 1.
Any quasi-metric space, double controlled metric type space, and controlled quasi-metric type space are also double controlled quasi-metric type space, but the converse is always not true (see Examples1–3).Example 1.
LetG=0,1,2. Define f:G×G⟶0,+∞ by f0,1=4, f0,2=1, f1,0=3=f1,2, f2,0=0, f2,1=2, and f0,0=f1,1=f2,2=0.
Defineα,μ:G×G⟶1,+∞ as α0,1=α1,0=α1,2=1, α0,2=5/4, α2,0=10/9, α2,1=20/19, α0,0=α1,1=α2,2=1, μ0,1=μ1,0=μ0,2=μ1,2=1, μ2,0=3/2, μ2,1=11/8, and μ0,0=μ1,1=μ2,2=1.
To show that the usual triangle inequality in quasi-metric is not satisfied. Letυ=0, e=2, and γ=1, then we have(4)f0,1=4>3=f0,2+f2,1,this shows that f is a double controlled quasi-metric type for all υ,γ,e∈G, but it is not a controlled quasi-metric type. Indeed,(5)f0,1=4>25576=α0,2f0,2+α2,1f2,1.
Also, it is not a double controlled metric type space because we have(6)f0,1=4=α0,2f0,2+μ2,1f2,1≠f1,0.Definition 3.
LetG,f be a double controlled quasi-metric type space with two functions. A sequence υp is convergent to some υ in G if and only if limp⟶+∞fυp,υ=limp⟶+∞fυ,υp=0.Definition 4.
LetG,f be a double controlled quasi-metric type space with two functions:(i)
The sequenceυp is a left Cauchy if and only if for every ε>0, we obtain a positive integer pε such that fυm,υp<ε, for all p>m>pε or limp,m⟶+∞fυm,υp=0(ii)
The sequenceυp is a right Cauchy if and only if for every ε>0, we obtain a positive integer pε such that fυm,υp<ε, for all m>p>pε or limp,m⟶+∞fυm,υp=0(iii)
The sequenceυp is a dual Cauchy if and only if it is left Cauchy as well as right CauchyDefinition 5.
LetG,f be a double controlled quasi-metric type space. Then, G,fG,f is(i)
Left complete if and only if each left Cauchy sequence inG is convergent(ii)
Right complete if and only if each right Cauchy sequence inG is convergent(iii)
Dual complete if and only if every left Cauchy as well as right Cauchy sequence inG is convergent
Note that each dual complete double controlled quasi-metric type space is left complete as well as right complete, but converse is not true in general.


## 2. Main Results

In this section, we generalize the definition of the fixed point for double controlled quasi-metric type spaces with two incomparable functionsα and μ which are given as follows.Theorem 2.
LetG,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G⟶1,+∞ and let T:G⟶G be a given mapping. Suppose that the following conditions are satisfied.There existsk∈0,1 such that(7)fTc˙,Ty≤kfc˙,y,for allc˙,y∈G.Forυ0∈G, choose υp=Tpυ0. Assume that(8)limi,m⟶+∞αυi+1,υi+2αυi,υi+1μυi+1,υm<1k.In addition, for everyυ∈G, we have(9)limp⟶+∞αυ,υpandlimp⟶+∞μυp,υexist and are finite.Then,T has a unique fixed point υ∗∈G.Proof.
Letυ0∈G be an arbitrary element and υp be the sequence defined as above. If υ0=Tυ0, then υ0 be a fixed point of T. By (7), we have(10)fυp,υp+1≤kpfυ0,υ1,p∈ℕ.
For all natural numbersp<m, we have(11)fυp,υm≤αυp,υp+1fυp,υp+1+μυp+1,υmfυp+1,υm≤αυp,υp+1fυp,υp+1+μυp+1,υmαυp+1,υp+2fυp+1,υp+2+μυp+1,υmμυp+2,υmfυp+2,υm≤αυp,υp+1fυp,υp+1+μυp+1,υmαυp+1,υp+2fυp+1,υp+2+μυp+1,υmμυp+2,υmαυp+2,υp+3fυp+2,υp+3+μυp+1,υmμυp+2,υmμυp+3,υmfυp+3,υm≤⋯≤αυp,υp+1fυp,υp+1+∑i=p+1m−2∏j=p+1iμυj,υmαυi,υi+1fυi,υi+1+∏k=p+1m−1μυk,υmfυm−1,υm≤αυp,υp+1kpfυ0,υ1+∑i=p+1m−2∏j=p+1iμυj,υmαυi,υi+1kifυ0,υ1+∏i=p+1m−1μυi,υmkm−1fυ0,υ1≤αυp,υp+1kpfυ0,υ1+∑i=p+1m−1∏j=p+1iμυj,υmαυi,υi+1kifυ0,υ1≤αυp,υp+1kpfυ0,υ1+∑i=p+1m−1∏j=0iμυj,υmαυi,υi+1kifυ0,υ1.
Let(12)Sp=∑i=0p∏j=0iμυj,υmαυi,υi+1ki.
Hence, we have(13)fυp,υm≤fυ0,υ1kpαυp,υp+1+Sm−1−Sp.
Letai=∏j=0iμυj,υmαυi,υi+1ki. By using (8), we have limi⟶+∞ai+1/ai<1. By the ratio test, the infinite series ∑i=1∞∏j=0iμυj,υmαυi,υi+1ki is convergent, and let p,m tend to infinity in (13), which implies that(14)limp,m⟶+∞fυp,υm=0.
SinceG,f is a left complete double controlled quasi-metric type space, there exists some υ∗∈G such that(15)limp⟶+∞fυp,υ∗=limp⟶+∞fυ∗,υp=0.
By using (Q2) and (7), we have(16)fυ∗,Tυ∗≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tυ∗fυp+1,Tυ∗≤αυ∗,υp+1fυ∗,υp+1+kμυp+1,Tυ∗fυp,υ∗fυ∗,Tυ∗.
By taking limitp tends to infinity together with (9) and (15), we get fυ∗,Tυ∗=0, that is, Tυ∗=υ∗. Now, we have to show that the fixed point of T is unique for this; let ξ∈G be such that Tξ=ξ and υ∗≠ξ, so we have(17)fυ∗,ξ=fTυ∗,Tξ≤kfυ∗,ξ.
So,υ∗=ξ. Hence, υ∗ is a unique fixed point of T.Example 2.
LetG=0,1,2. Define f:G×G⟶0,+∞ by(18)fc˙,y012003418125045215140.
Givenα,μ:G×G⟶1,+∞ as(19)αυ,y0120121202132112111,μυ,y012011110531111092121.
It is easy to see thatG,f is a double controlled quasi-metric type and the given function f is not a controlled metric type for the function α. Indeed,(20)f1,2=45>6380=α1,0f0,2+α0,1f0,1.
TakeT0=T2=0, T1=2, and k=1/2. We observe the following cases:(i)
Ifc˙=0 and y=1, we have(21)fTc˙,Ty=18<12×34=kfc˙,y.So, inequality (7) holds. Also, it holds if c˙=1 and.(ii)
Inequality (7) holds trivially in the cases when c˙=0 and y=2 and if c˙=2 and y=0.(iii)
Ifc˙=1 and y=2, we get(22)fTc˙,Ty=15<12×45=kfc˙,y.
Similarly, in the case whenc˙=2 and y=1, we have fTc˙,Ty<kfc˙,y. So, (7) holds for all cases. Now, let υ0=2, so we have υ1=Tυ0=T2=0 and υ2=0, υ3=0,⋯(23)limi,m⟶+∞αυi+1,υi+2αυi,υi+1μυi+1,υm=1<2=1k.
That is, (8) holds. In addition, for each υ∈G, we have(24)limp⟶+∞αυ,υp<∞andlimp⟶+∞μυp,υ<∞.
That is, (9) holds. Hence, all conditions of Theorem 2 are satisfied, and υ=0 is a unique fixed point.


## 3. Further Results

In this section, we introduce the concept ofα−μ−k double controlled contraction and prove related fixed point results with some examples.Definition 6.
LetG be a nonempty set, G,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G⟶1,+∞, and Tˇ:G⟶G be a given mapping. Assume there exists k∈0,1/2 such that(25)h1=supkαc˙,y,c˙,y∈G<12,h2=supkμc˙,y,c˙,y∈G<12.
Suppose that the following conditions are satisfied:(26)fTˇc˙,Tˇy≤kfc˙,Tˇy+fy,Tˇc˙,for allc˙,y∈G.
Forυ0∈G and υp=Tˇpυ0, we have(27)limi,m⟶+∞αυi+1,υi+2αυi,υi+1μυi+1,υm<1−hh,where h=maxh1,h2. Also, for each υ∈G, we have(28)limp⟶+∞αυp,υ<∞,limp⟶+∞αυ,υp<∞,limp⟶+∞μυp,υ<∞.
Then,Tˇ is called α−μ−k double controlled contraction.Theorem 3.
LetG,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G⟶1,+∞ and let Tˇ:G⟶G be α−μ−k double controlled contraction. Then, Tˇ has a unique fixed point υ∗∈G.Proof.
Letυ0∈G be an arbitrary element and υp be the sequence defined as above. If υ0=Tˇυ0, then υ0 is a fixed point of Tˇ. By (26), we have(29)fυp,υp+1=fTˇυp−1,Tˇυp≤kfυp−1,Tˇυp+fυp,Tˇυp−1≤kfυp−1,υp+1+fυp,υp≤kαυp−1,υpfυp−1,υp+kμυp,υp+1fυp,υp+1fυp,υp+1≤h1fυp−1,υp+h2fυp,υp+1,by Definition6≤hfυp−1,υp+hfυp,υp+11−hfυp,υp+1≤hfυp−1,υp,(30)fυp,υp+1≤h1−hfυp−1,υp.
Now,(31)fυp−1,υp=fTˇυp−2,Tˇυp−1≤maxfTˇυp−2,Tˇυp−1,fTˇυp−1,Tˇυp−2,≤kfυp−2,υp+fυp−1,υp−1,≤kαυp−2,υp−1fυp−2,υp−1+kμυp−1,υpfυp−1,υp,≤h1fυp−2,υp−1+h2fυp−1,υp,by Definition61−hfυp−1,υp≤hfυp−2,υp−1,fυp−1,υp≤h1−hfυp−2,υp−1.
Combining (30) and the above inequality, we get(32)fυp,υp+1≤h1−h2fυp−2,υp−1.
Continuing in this way, we obtain(33)fυp,υp+1≤h1−hpfυ0,υ1.
Now, to prove thatυp is a Cauchy sequence, for all natural numbers p<m, we have(34)fυp,υm≤αυp,υp+1fυp,υp+1+∑i=p+1m−2∏j=p+1iμυj,υmαυi,υi+1fυi,υi+1+∏k=p+1m−1μυk,υmfυm−1,υm.
Using (33), we get(35)fυp,υm≤αυp,υp+1h1−hpfυ0,υ1+∑i=p+1m−2∏j=p+1iμυj,υmαυi,υi+1h1−hifυ0,υ1+∏i=p+1m−1μυi,υmh1−hm−1fυ0,υ1≤αυp,υp+1h1−hpfυ0,υ1+∑i=p+1m−1∏j=0iμυj,υmαυi,υi+1h1−hifυ0,υ1,Sp=∑i=0p∏j=0iμυj,υmαυi,υi+1h1−hi.
Hence, we have(36)fυp,υm≤fυ0,υ1h1−hpαυp,υp+1+Sm−1−Sp.
Letai=∏j=0iμυj,υmαυi,υi+1h/1−hi. By using (27), we have limi⟶+∞ai+1/ai<1. By the ratio test, the infinite series ∑i=1∞∏j=0iμυj,υmαυi,υi+1h/1−hi is convergent, and let p,m tend to infinity in (36), which yield(37)limp,m⟶+∞fυp,υm=0.
So, the sequenceυp is a left Cauchy. Since G,f is a left complete double controlled quasi-metric type space, there must be exist some υ∗∈G such that(38)limp⟶+∞fυp,υ∗=0=limp⟶+∞fυ∗,υp.
We claim thatTˇυ∗=υ∗. By (26), we have(39)fυ∗,Tˇυ∗≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗fυp+1,Tˇυ∗≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗kfυp,Tˇυ∗+fυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2fυp−1,Tˇυ∗+fυ∗,υp+μυp+1,Tˇυ∗kfυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2αυp−1,υ∗fυp−1,υ∗+μυ∗,Tˇυ∗fυ∗,Tˇυ∗+fυ∗,υp+μυp+1,Tˇυ∗kfυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2αυp−1,υ∗fυp−1,υ∗≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗kfυp,Tˇυ∗+fυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2fυp−1,Tˇυ∗+fυ∗,υp+μυp+1,Tˇυ∗kfυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2αυp−1,υ∗fυp−1,υ∗+μυ∗,Tˇυ∗fυ∗,Tˇυ∗+fυ∗,υp+μυp+1,Tˇυ∗kfυ∗,υp+1≤αυ∗,υp+1fυ∗,υp+1+μυp+1,Tˇυ∗k2αυp−1,υ∗fυp−1,υ∗+h22fυ∗,Tˇυ∗+μυp+1,Tˇυ∗k2fυ∗,υp+μυp+1,Tˇυ∗kfυ∗,υp+1.
By taking limit asp tend to infinity together with (38), we get(40)1−h22fυ∗,Tˇυ∗≤0.
Hence,υ∗=Tˇυ∗, which is a contradiction. Now, we have to show that the fixed point of Tˇ is unique for this let υ∗∗∈G such that Tˇυ∗∗=υ∗∗, so we have(41)fυ∗,υ∗∗=fTˇυ∗,Tˇυ∗∗≤kfυ∗,Tˇυ∗∗+fυ∗∗,Tˇυ∗≤kfυ∗,υ∗∗+fυ∗∗,υ∗≤k1−kfυ∗∗,υ∗≤k1−k2fυ∗,υ∗∗⋮≤k1−k2nfυ∗,υ∗∗.
By taking limit asn tend to infinity, we have υ∗=υ∗∗. Hence, υ∗ is a unique fixed point of Tˇ.Example 3.
TakeG=0,1,2. Define f:G×G⟶0,+∞ by(42)fc˙,y01200251419200452157100.
Givenα,μ:G×G⟶1,+∞ as αυ,y01201102/100116/511211/1011 and μυ,y012016/511/1011112111. It is easy that G,f is a double controlled quasi-metric type space. The given f is not a controlled metric for the function α. Indeed,(43)f2,1=710>307500=α2,0f2,0+α0,1f0,1.
TakeTˇ0=Tˇ2=2, Tˇ1=0, and k=2/5, and we observe the follows cases:(i)
Ifc˙=0 and y=1, we have(44)fTˇc˙,Tˇy=15≤825=250+45=kfc˙,Tˇy+fy,Tˇc˙.Ifc˙=1 and y=0, we get(45)fTˇc˙,Tˇy=14≤825=kfc˙,Tˇy+fy,Tˇc˙.(ii)
It is straightforward in the case when we takec˙=0 and y=2.(iii)
Ifc˙=1 and y=2, we get(46)14≤25=2545+15=kfc˙,Tˇy+fy,Tˇc˙.
Similarly, in the case when we takec˙=2 and y=1, that is, inequality (26) holds, we have(47)h1=supkαc˙,y,c˙,y∈G<12,h2=supkμc˙,y,c˙,y∈G<12,and h=maxh1,h2=12/25. Now, let υ0=1, and we have υ1=Tˇυ0=Tˇ1=0, υ2=Tˇυ1=Tˇ0=2, υ3=Tˇ2=2, υ3=2,…(48)limi,m⟶+∞αυi+1,υi+2αυi,υi+1μυi+1,υm=1<1−hh,which shows that (27) holds. In addition, for each υ∈G, we have(49)limp⟶+∞αυp,υ<∞,limp⟶+∞αυ,υp<∞,limp⟶+∞μυp,υ<∞.
That is, (28) holds. All conditions of Theorem 3 are proved, and v=2 is the unique fixed point.Definition 7.
LetG,f be a complete quasi-b-metric space. Tˇ:G⟶G is called Chatterjee-type b-contraction if the following conditions are satisfied:(50)fTˇc˙,Tˇy≤kfc˙,Tˇy+fy,Tˇc˙,for all c˙,y∈G, k∈0,1/2, and(51)b<1−kbkb.Theorem 4.
LetG,f be a complete quasi-b-metric space and Tˇ:G⟶G be Chatterjee-type b-contraction. Then, Tˇ has a unique fixed point.Remark 2.
In the Example3, f is a quasi-b-metric with b≥16/13, but we cannot apply Theorem 4 because Tˇ is not a Chatterjee-type b-contraction. Indeed, b≰1−kb/kb, for all b≥16/13.


## 4. Conclusion

In the present paper, we have obtained sufficient conditions to ensure the existence of the fixed point for different types of contractive mappings in the setting of double controlled quasi-metric type spaces. Examples are given to demonstrate the variety of our results. New results in quasi-b-metric spaces, extended b-metric spaces, extended quasi-b-metric spaces, controlled metric spaces, and controlled quasi-metric spaces can be obtained as corollaries of our results. Also, results in right complete and dual complete double controlled quasi-metric type spaces can be obtained in a similar way. It is natural to ask, Are there other multivalued contraction mappings which can be applied to obtain more results in the double controlled quasi-metric type spaces? Is there interest to find serious applications to integral equations and dynamical systems?



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