COMPLEXITYComplexity1099-05261076-2787Hindawi10.1155/2020/34609383460938Research ArticleDouble Controlled Quasi-Metric Type Spaces and Some ResultsShoaibAbdullah1KaziSabeena2https://orcid.org/0000-0002-6165-8055TassaddiqAsifa2AlshoraifyShaif S.3https://orcid.org/0000-0003-3712-072XRashamTahair3ZargarzadehHassan1Department of Mathematics and StatisticsRiphah International UniversityIslamabadPakistanriphah.edu.pk2Department of Basic Sciences and HumanitiesCollege of Computer and Information SciencesMajmaah UniversityAl-Majmaah 11952Saudi Arabiamu.edu.sa3Department of Mathematics and StatisticsInternational Islamic UniversityH-10Islamabad 44000Pakistaniiu.edu.pk20202072020202023032020310520201906202020720202020Copyright © 2020 Abdullah Shoaib et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abdeljawad et al. (2018) introduced a new concept, named double controlled metric type spaces, as a generalization of the notion of extended b-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 αμk 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.

Majmaah UniversityR-1441-137
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 . Various generalizations of Banach fixed point theorem were made by numerous mathematicians, see . One of the abstraction of the metric spaces is the quasi-metric space that was introduced by Wilson . The commutativity condition does not hold in general in quasi-metric spaces. Several authors used this concept to prove some fixed point results, see . On the other hand, Bakhtin  and Czerwik  established the idea of b-metric spaces. Lateral, many authors got several fixed point results, for instance, see . Kamran et al.  introduced a new idea to generalized b-metric spaces, named as extended b-metric spaces, see also . They replaced the parameter b1 in the triangle inequality by the control function θ:G×G1,. Nurwahyu  introduced dislocated quasi-extended b-metric space and obtained several fixed point results. Mlaiki et al.  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.  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 [<xref ref-type="bibr" rid="B33">33</xref>]).

Given noncomparable functions α,μ:G×G[1,+. If f:G×G0,+ satisfies

fυ,γ=0 if and only if υ=γ

fυ,γ=fγ,υ

fυ,γαυ,efυ,e+μe,γfe,γ, for all υ,γ,eG

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 [<xref ref-type="bibr" rid="B33">33</xref>]).

Let G,f be a complete double controlled metric type space with the functions α,μ:G×G1,+ and let T:GG be a given mapping. Suppose that the following conditions are satisfied.

There exists k0,1 such that(1)fTc˙,Tykfc˙,y,for all c˙,yG.

For υ0G, choose υp=Tpυ0. Assume that(2)supm1limi+αυi+1,υi+2αυi,υi+1μυi+1,υm<1k.

In addition, for every υG, we have(3)limp+αυ,υp and limp+μυp,υ exist and are finite.

Then, T has a unique fixed point υG.

Definition 2.

Given noncomparable functions α,μ:G×G1,+. If f:G×G0,+ satisfies

(Q1) fυ,γ=0 if and only if υ=γ

(Q2) fυ,γαυ,efυ,e+μe,γfe,γ, for all υ,γ,eG

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 Examples 13).

Example 1.

Let G=0,1,2. Define f:G×G0,+ 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.

Define α,μ:G×G1,+ 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. Let υ=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,eG, 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+μ2,1f2,1f1,0.

Definition 3.

Let G,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.

Let G,f be a double controlled quasi-metric type space with two functions:

The sequence υp 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

The sequence υp 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

The sequence υp is a dual Cauchy if and only if it is left Cauchy as well as right Cauchy

Definition 5.

Let G,f be a double controlled quasi-metric type space. Then, G,fG,f is

Left complete if and only if each left Cauchy sequence in G is convergent

Right complete if and only if each right Cauchy sequence in G is convergent

Dual complete if and only if every left Cauchy as well as right Cauchy sequence in G 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 functions α and μ which are given as follows.

Theorem 2.

Let G,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G1,+ and let T:GG be a given mapping. Suppose that the following conditions are satisfied.

There exists k0,1 such that(7)fTc˙,Tykfc˙,y,for all c˙,yG.

For υ0G, choose υp=Tpυ0. Assume that(8)limi,m+αυi+1,υi+2αυi,υi+1μυi+1,υm<1k.

In addition, for every υG, we have(9)limp+αυ,υp and limp+μυp,υ exist and are finite.

Then, T has a unique fixed point υG.

Proof.

Let υ0G 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+1kpfυ0,υ1,p.

For all natural numbers p<m, we have(11)fυp,υmαυp,υp+1fυp,υp+1+μυp+1,υmfυp+1,υmαυp,υp+1fυp,υp+1+μυp+1,υmαυp+1,υp+2fυp+1,υp+2+μυp+1,υmμυp+2,υmfυp+2,υmαυp,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,υmαυp,υp+1fυp,υp+1+i=p+1m2j=p+1iμυj,υmαυi,υi+1fυi,υi+1+k=p+1m1μυk,υmfυm1,υmαυp,υp+1kpfυ0,υ1+i=p+1m2j=p+1iμυj,υmαυi,υi+1kifυ0,υ1+i=p+1m1μυi,υmkm1fυ0,υ1αυp,υp+1kpfυ0,υ1+i=p+1m1j=p+1iμυj,υmαυi,υi+1kifυ0,υ1αυp,υp+1kpfυ0,υ1+i=p+1m1j=0iμυj,υmαυi,υi+1kifυ0,υ1.

Let(12)Sp=i=0pj=0iμυj,υmαυi,υi+1ki.

Hence, we have(13)fυp,υmfυ0,υ1kpαυp,υp+1+Sm1Sp.

Let ai=j=0iμυj,υmαυi,υi+1ki. By using (8), we have limi+ai+1/ai<1. By the ratio test, the infinite series i=1j=0iμυj,υmαυi,υi+1ki is convergent, and let p,m tend to infinity in (13), which implies that(14)limp,m+fυp,υm=0.

Since G,f is a left complete double controlled quasi-metric type space, there exists some υG such that(15)limp+fυp,υ=limp+fυ,υp=0.

By using (Q2) and (7), we have(16)fυ,Tυαυ,υp+1fυ,υp+1+μυp+1,Tυfυp+1,Tυαυ,υp+1fυ,υp+1+kμυp+1,Tυfυp,υfυ,Tυ.

By taking limit p 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 ξG be such that Tξ=ξ and υξ, so we have(17)fυ,ξ=fTυ,Tξkfυ,ξ.

So, υ=ξ. Hence, υ is a unique fixed point of T.

Example 2.

Let G=0,1,2. Define f:G×G0,+ by(18)fc˙,y012003418125045215140.

Given α,μ:G×G1,+ as(19)αυ,y0120121202132112111,μυ,y012011110531111092121.

It is easy to see that G,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.

Take T0=T2=0, T1=2, and k=1/2. We observe the following cases:

If c˙=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.

Inequality (7) holds trivially in the cases when c˙=0 and y=2 and if c˙=2 and y=0.

If c˙=1 and y=2, we get

(22)fTc˙,Ty=15<12×45=kfc˙,y.

Similarly, in the case when c˙=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 υG, we have(24)limp+αυ,υp< and limp+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 of αμk double controlled contraction and prove related fixed point results with some examples.

Definition 6.

Let G be a nonempty set, G,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G1,+, and Tˇ:GG be a given mapping. Assume there exists k0,1/2 such that(25)h1=supkαc˙,y,c˙,yG<12,h2=supkμc˙,y,c˙,yG<12.

Suppose that the following conditions are satisfied:(26)fTˇc˙,Tˇykfc˙,Tˇy+fy,Tˇc˙,for all c˙,yG.

For υ0G and υp=Tˇpυ0, we have(27)limi,m+αυi+1,υi+2αυi,υi+1μυi+1,υm<1hh,where h=maxh1,h2. Also, for each υG, we have(28)limp+αυp,υ<,limp+αυ,υp<,limp+μυp,υ<.

Then, Tˇ is called αμk double controlled contraction.

Theorem 3.

Let G,f be a left complete double controlled quasi-metric type space with the functions α,μ:G×G1,+ and let Tˇ:GG be αμk double controlled contraction. Then, Tˇ has a unique fixed point υG.

Proof.

Let υ0G 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=fTˇυp1,Tˇυpkfυp1,Tˇυp+fυp,Tˇυp1kfυp1,υp+1+fυp,υpkαυp1,υpfυp1,υp+kμυp,υp+1fυp,υp+1fυp,υp+1h1fυp1,υp+h2fυp,υp+1,by Definition 6hfυp1,υp+hfυp,υp+11hfυp,υp+1hfυp1,υp,(30)fυp,υp+1h1hfυp1,υp.

Now,(31)fυp1,υp=fTˇυp2,Tˇυp1maxfTˇυp2,Tˇυp1,fTˇυp1,Tˇυp2,kfυp2,υp+fυp1,υp1,kαυp2,υp1fυp2,υp1+kμυp1,υpfυp1,υp,h1fυp2,υp1+h2fυp1,υp,by Definition 61hfυp1,υphfυp2,υp1,fυp1,υph1hfυp2,υp1.

Combining (30) and the above inequality, we get(32)fυp,υp+1h1h2fυp2,υp1.

Continuing in this way, we obtain(33)fυp,υp+1h1hpfυ0,υ1.

Now, to prove that υp is a Cauchy sequence, for all natural numbers p<m, we have(34)fυp,υmαυp,υp+1fυp,υp+1+i=p+1m2j=p+1iμυj,υmαυi,υi+1fυi,υi+1+k=p+1m1μυk,υmfυm1,υm.

Using (33), we get(35)fυp,υmαυp,υp+1h1hpfυ0,υ1+i=p+1m2j=p+1iμυj,υmαυi,υi+1h1hifυ0,υ1+i=p+1m1μυi,υmh1hm1fυ0,υ1αυp,υp+1h1hpfυ0,υ1+i=p+1m1j=0iμυj,υmαυi,υi+1h1hifυ0,υ1,Sp=i=0pj=0iμυj,υmαυi,υi+1h1hi.

Hence, we have(36)fυp,υmfυ0,υ1h1hpαυp,υp+1+Sm1Sp.

Let ai=j=0iμυj,υmαυi,υi+1h/1hi. By using (27), we have limi+ai+1/ai<1. By the ratio test, the infinite series i=1j=0iμυj,υmαυi,υi+1h/1hi is convergent, and let p,m tend to infinity in (36), which yield(37)limp,m+fυp,υm=0.

So, the sequence υp is a left Cauchy. Since G,f is a left complete double controlled quasi-metric type space, there must be exist some υG such that(38)limp+fυp,υ=0=limp+fυ,υp.

We claim that Tˇυ=υ. By (26), we have(39)fυ,Tˇυαυ,υp+1fυ,υp+1+μυp+1,Tˇυfυp+1,Tˇυαυ,υp+1fυ,υp+1+μυp+1,Tˇυkfυp,Tˇυ+fυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2fυp1,Tˇυ+fυ,υp+μυp+1,Tˇυkfυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2αυp1,υfυp1,υ+μυ,Tˇυfυ,Tˇυ+fυ,υp+μυp+1,Tˇυkfυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2αυp1,υfυp1,υαυ,υp+1fυ,υp+1+μυp+1,Tˇυkfυp,Tˇυ+fυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2fυp1,Tˇυ+fυ,υp+μυp+1,Tˇυkfυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2αυp1,υfυp1,υ+μυ,Tˇυfυ,Tˇυ+fυ,υp+μυp+1,Tˇυkfυ,υp+1αυ,υp+1fυ,υp+1+μυp+1,Tˇυk2αυp1,υfυp1,υ+h22fυ,Tˇυ+μυp+1,Tˇυk2fυ,υp+μυp+1,Tˇυkfυ,υp+1.

By taking limit as p tend to infinity together with (38), we get(40)1h22fυ,Tˇυ0.

Hence, υ=Tˇυ, which is a contradiction. Now, we have to show that the fixed point of Tˇ is unique for this let υG such that Tˇυ=υ, so we have(41)fυ,υ=fTˇυ,Tˇυkfυ,Tˇυ+fυ,Tˇυkfυ,υ+fυ,υk1kfυ,υk1k2fυ,υk1k2nfυ,υ.

By taking limit as n tend to infinity, we have υ=υ. Hence, υ is a unique fixed point of Tˇ.

Example 3.

Take G=0,1,2. Define f:G×G0,+ by(42)fc˙,y01200251419200452157100.

Given α,μ:G×G1,+ 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.

Take Tˇ0=Tˇ2=2, Tˇ1=0, and k=2/5, and we observe the follows cases:

If c˙=0 and y=1, we have

(44)fTˇc˙,Tˇy=15825=250+45=kfc˙,Tˇy+fy,Tˇc˙.

If c˙=1 and y=0, we get

(45)fTˇc˙,Tˇy=14825=kfc˙,Tˇy+fy,Tˇc˙.

It is straightforward in the case when we take c˙=0 and y=2.

If c˙=1 and y=2, we get

(46)1425=2545+15=kfc˙,Tˇy+fy,Tˇc˙.

Similarly, in the case when we take c˙=2 and y=1, that is, inequality (26) holds, we have(47)h1=supkαc˙,y,c˙,yG<12,h2=supkμc˙,y,c˙,yG<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<1hh,which shows that (27) holds. In addition, for each υG, 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.

Let G,f be a complete quasi-b-metric space. Tˇ:GG is called Chatterjee-type b-contraction if the following conditions are satisfied:(50)fTˇc˙,Tˇykfc˙,Tˇy+fy,Tˇc˙,for all c˙,yG, k0,1/2, and(51)b<1kbkb.

Theorem 4.

Let G,f be a complete quasi-b-metric space and Tˇ:GG be Chatterjee-type b-contraction. Then, Tˇ has a unique fixed point.

Remark 2.

In the Example 3, f is a quasi-b-metric with b16/13, but we cannot apply Theorem 4 because Tˇ is not a Chatterjee-type b-contraction. Indeed, b1kb/kb, for all b16/13.

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?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Each author equally contributed to this paper, read, and approved the final manuscript.

Acknowledgments

Sabeena Kazi would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under the Project no. R-1441-137.