# Double Controlled Dislocated Quasi-Metric Type Spaces and Some Results

Authors: Abdullah Shoaib; Shaif S. Alshoraify; Muhammad Arshad
Journal: Journal of Mathematics
 (2020)

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## Abstract

The aim of this work is to introduce double controlled dislocated quasi-metric type spaces and to obtain fixed point results for a pair of multivalued mappings satisfying Kannan-type double controlled contraction in such spaces. An example has been built and a remark has been given which shows that how our result can be used when a corresponding new result in dislocated quasib-metric type spaces cannot be used. Our results generalize and extend many existing results in the literature.


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## Body

## 1. Introduction and Preliminaries

The field of fixed point theory covers both pure and applied mathematics. Fixed point theory is a special branch of functional analysis and its results are used to show the solution existence of different mathematical models. A multivalued mappingB from C to the subsets of C has a fixed point y∈C, if y∈By. If we take elements of C instead of subsets of C, then B represents a single valued mapping from C to C. A single valued mapping B:C⟶C has a fixed point y∈C if y=By. Fixed point results for multivalued mappings generalize the results for single-valued mappings. Fixed point results of multivalued mappings have applications in engineering, economics, Nash equilibria, and game theory [1–4]. Due to its importance, many interesting results have been proved in the setting of multivalued mappings, for example, see [5–14].By excluding one and a half condition, out of three conditions of a metric space, we obtain dislocated quasi-metric space [11]. Complete dislocated quasi-metric space is a generalization of 0-complete and complete quasi-partial metric space [15–17]. Dislocated quasi-metric also generalizes dislocated metric, partial metric, and quasi-metric. Fixed point results in dislocated quasi-metric space can be seen in [18–22].Ran and Reurings [23] gave a fixed point result with an order and obtained solution to matrix equations as an application. Nieto et al. [24] gave an extension to the result in [23] for ordered mappings and used it to give a unique solution for ODE with periodic boundary conditions. Altun et al. [25] introduced a new approach to common fixed point of mappings, satisfying a generalized contraction with a new restriction in a complete ordered metric space.On the contrary, Kamran et al. [26] introduced a new concept of generalized b metric space, named as extended b-metric spaces, see also [27]. They replaced the parameter b≥1 in triangle inequality by the control function θ:X×X⟶1,∞. Recently, Mlaiki et al. [27] generalized the triangle inequality in b-metric space by using control function in a different style and introduced controlled metric type spaces. Very recently, Abdeljawad et al. [28] generalized the concept of controlled metric type space by introducing two control functions αr,y and μr,y and establishing double controlled metric type space. In this paper, we extend the result of Altun et al. [25] in four different ways by using(i)
Multivalued mappings instead of single-valued mappings(ii)
Kannan-type contraction instead of Banach-type contraction(iii)
LeftK-sequentially complete double controlled dislocated quasi-metric type space instead of complete metric space(iv)
A function which generalized the partial order relationOur results unify, extend, and generalize several comparable results in the existing literature. We give the following definitions and results which will be useful to understand the paper.Definition 1.
(see [28]). Given noncomparable functions α,μ:ℑ×ℑ⟶1,∞. Suppose that q:ℑ×ℑ⟶0,∞ satisfies the following:(q1) qr,y=0 if and only if r=y(q2) qr,y=qy,r(q3) qr,y≤αr,zqr,z+μz,yqz,y, for all r,z,y∈ℑ
Then,q is called double controlled metric type with the functions α and μ, and the pair ℑ,q is called double controlled metric type space with the functions αandμ.Theorem 1 (see [28]).
Letℑ,q be a complete double controlled metric type space with the functions α,μ:ℑ×ℑ⟶1,∞ and let T:ℑ⟶ℑ be a given mapping. Suppose that the following conditions are satisfied.
There existsk∈0,1 such that(1)qTx,Ty≤kqx,y,for allx,y∈ℑ.Forυ0∈ℑ, choose υn=Tnυ0. Assume that(2)supm≥1limi⟶∞αυi+1,υi+2αυi,υi+1μυi+1,υm<1k.In addition, for everyυ∈ℑ, we have(3)limi⟶∞αυ,υnlimi⟶∞μυn,υexist and are finite.Then,T has a unique fixed point υ∗∈ℑ.Definition 2.
Given noncomparable functionsα,μ:ℑ×ℑ⟶1,∞. If ρq:ℑ×ℑ⟶0,∞ satisfies(4)ρq1qω,v=ρqv,ω=0,thenω=v,ρq2qω,v≤αω,eqω,e+μe,vqe,v,for all ω,v,e∈ℑ, then ρq is called a double controlled dislocated quasi-metric type with the functions α and μ and ℑ,ρq is called a double controlled dislocated quasi-metric type space. If μe,v=αe,v then ℑ,ρq is called a controlled quasi-metric type space.Remark 1.
Any dislocated quasi-metric space or double controlled metric-type space is double controlled dislocated quasi-metric type space, but the converse is not true in general. Also, a controlled dislocated quasi-metric type space is also double controlled quasi-metric type space. The converse is not true in general (see Examples1 and 2).Example 1.
Letℑ=0,1,2,3. Define ρq:ℑ×ℑ⟶0,∞ by ρq0,1=0, ρq0,2=1, ρq0,3=1/4,ρq1,0=1/2,ρq1,2=2, ρq1,3=1/3,ρq2,0=1/2, ρq2,1=1, ρq2,3=1/3,ρq3,0=3/2,ρq3,1=2,ρq3,2=1/4,ρq0,0=1/2, ρq1,1=0, ρq2,2=2, and ρq3,3=0.
Defineα,μ:ℑ×ℑ⟶1,∞ as α0,1=α1,2=α0,2=1, α1,0=4/3, α2,0=2, α3,1=4/3, α2,3=3, α0,3=4/3, α1,3=3,α2,1=1, α3,0=4/3,α3,2=2,α0,0=α1,1=α2,2=α3,3=1,
μ1,2=μ2,1=3/2, μ2,0=2, μ3,0=μ0,3=μ1,0=μ0,1=μ1,3=μ3,1=1,μ3,2=4,μ2,3=1,μ0,2=4/3, and μ0,0=μ1,1=μ2,2=μ3,3=1. It is obvious that ρq is double controlled dislocated quasi-metric type for all ω,v,e∈ℑ. It is clear that ρq is not double controlled metric-type space. Also, it is not controlled dislocated quasi-metric type. Indeed,(5)ρq1,2=2>32=α1,3ρq1,3+α3,2ρq3,2.Definition 3.
Letℑ,ρq be a double controlled dislocated quasi-metric type space:(i)
A sequencern in ℑ,ρq is called left K-Cauchy if ∀ε>0, ∃n0∈N such that ρqrm,rn<ε and ∀n>m>n0.(ii)
A sequencern is double controlled dislocated quasi and converges to r if limn⟶∞ρqrn,r=limn⟶∞ρqr,rn=0 or for any ε>0, there exists n0∈N, such that, for all n>n0,ρqr,rn<ε and ρqrn,r<ε. In this case, r is called a ρq-limit of rn.(iii)
ℑ,ρq is called left K-sequentially complete if every left K-Cauchy sequence in ℑ,ρq converges to a point r∈ℑ such that ρqr,r=0.Definition 4.
Letℑ,ρq be a double controlled dislocated quasi-metric type space. Let K be a nonempty subset of ℑ and let r∈ℑ. An element y0∈K is called a best approximation in K if(6)ρqr,K=ρqr,y0,whereρqr,K=infy∈Kρqr,y,ρqK,r=ρqy0,r,whereρqK,r=infy∈Kρqy,r.
If eachr∈ℑ has at least one best approximation in K, then K is called a proximinal set. We denote the set of all proximinal subsets of ℑ by Pℑ.Definition 5.
(see [11]). The function Hρq:Pℑ×Pℑ⟶0,∞, defined by(7)HρqA,B=maxsupa∈Aρqa,B,supb∈BρqA,b,is called double controlled dislocated quasi-Hausdorff metric type on Pℑ. Also, Pℑ,Hρqis known as double controlled dislocated quasi-Hausdorff metric type space.
Following similar arguments of Lemma 1.7 given by Shoaib [11], we obtain the following lemma.Lemma 1.
Letℑ,ρq be a double controlled dislocated quasi-metric type space. Let Pℑ,Hρq be a double controlled dislocated quasi-Hausdorff metric type space on Pℑ. Then, for all A,B∈Pℑ and for each a∈A, there exists ba∈B, such that HρqA,B≥ρqa,ba and HρqB,A≥ρqba,a.


## 2. Main Result

Letℑ,ρq be a double controlled complete dislocated quasi-metric type space and r0∈ℑ and S,T:ℑ⟶Pℑ be multifunctions on ℑ. Let r1∈Sr0 be an element such that ρqr0,Sr0=ρqr0,r1 and ρqSr0,r0=ρqr1,r0. Let r2∈Tr1 be such that ρqr1,Tr1=ρqr1,r2 and ρqTr1,r1=ρqr2,r1. Let r3∈Sr2 be such that ρqr2,Sr2=ρqr2,r3 and so on. Thus, we construct a sequence rnof points in ℑ such that r2n+1∈Sr2n and r2n+2∈Tr2n+1, with ρqr2n,Sr2n=ρqr2n,r2n+1, ρqSr2n,r2n=ρqr2n+1,r2n, ρqr2n+1,Tr2n+1=ρqr2n+1,r2n+2, and ρqTr2n+1,r2n+1=ρqr2n+2,r2n+1, where n=0,1,2,…. We denote this iterative sequence by TSrn. We say that TSrn is a sequence in ℑ generated by r0. If T=S, then we say that Trn is a sequence in ℑ generated by r0. Let M⊆ℑ, define β∗r,M=infβr,a,a∈M and β∗M,y=infβb,y,b∈M.Definition 6.
Letℑ be a nonempty set, β:ℑ×ℑ⟶0,+∞ be a mapping, and S,T:ℑ⟶Pℑ be the multivalued mappings; then, the pair S,T is said to be a pair of β∗-Alt multivalued mapping, if(8)aβ∗r,Sr≥1,ρqr,Sr=ρqr,y,ρqSr,r=ρqy,rimplyβ∗Sy,y≥1,bβ∗Sr,r≥1,rhoqr,Tr=ρqr,y,ρqTr,r=ρqy,rimplyβ∗y,Sy≥1.Definition 7.
Letℑ,ρq be a complete double controlled dislocated quasi-metric type space and S,T be a pair of β∗-Alt multivalued mapping. Then, S,T is called β∗ Kannan-type double controlled contraction, if for every two consecutive points x,y belonging to the range of an iterative sequence TSrn with β∗Sy,y≥1 and β∗x,Sx≥1, and we have(9)maxHρqSx,Ty,HρqTy,Sx≤tρqy,Ty+ρqx,Sx,where t∈0,1/2. Also, the terms of the sequence TSrn satisfy the following:(10)supm≥1limi⟶∞αri+1,ri+2αri,ri+1μri+1,rm<1−tt.Theorem 2.
Letℑ,ρq be a left K-sequentially complete double controlled dislocated quasi-metric type space.S,T be the pair of β∗ Kannan-type double controlled contraction. Assume that(i)
The setGS=r:β∗r,Sr≥1 contains r0 and is closed.(ii)
For everyr∈ℑ, we have(11)limn⟶∞αr,rnlimn⟶∞μrn,r<1t.Then,TSrn⟶r∈ℑ. Also, if (9) holds for each r,y∈r∗, then S and T have a common fixed point r∗ in ℑ and ρqr∗,r∗=0.Proof.
Asr0 be an arbitrary element of GS, from condition (i) β∗r0,Sr0≥1. Let TSrn be the iterative sequence in ℑ generated by a point r0∈ℑ. Since β∗r0,Sr0≥1, ρqr0,Sr0=ρqr0,r1 and ρqSr0,r0=ρqr1,r0. As S,T is β∗-Alt multivalued mapping, so β∗Sr1,r1≥1. Now, β∗Sr1,r1≥1, ρqr1,Tr1=ρqr1,r2, and ρqTr1,r1=ρqr2,r1 imply that β∗r2,Sr2≥1. By induction, we deduce that β∗r2p,Sr2p≥1 and β∗Sr2p+1,r2p+1≥1, for all p=0,1,2,⋯. Now, by Lemma 1, we have(12)ρqr2p,r2p+1≤HρqTr2p−1,Sr2p,(13)ρqr2p+1,r2p+2≤HρqSr2p,Tr2p+1.
Asr2p,r2p+1∈TSrn, β∗r2p,Sr2p≥1, and β∗Sr2p+1,r2p+1≥1, then by using condition (9) in inequality (13), we have(14)ρqr2p+1,r2p+2≤tρqr2p,Sr2p+ρqr2p+1,Tr2p+1≤tρqr2p,r2p+1+ρqr2p+1,r2p+2≤t1−tρqr2p,r2p+1,ρqr2p+1,r2p+2≤c´ρqr2p,r2p+1,wherec´=t1−t.
Asr2p,r2p−1∈TSrn, β∗r2p,Sr2p≥1, and β∗Sr2p−1,r2p−1≥1, then by using condition (9) in inequality (12), we obtain(15)ρqr2p,r2p+1≤tρqr2p−1,Tr2p−1+ρqr2p,Sr2p≤tρqr2p−1,r2p+ρqr2p,r2p+1≤t1−tρqr2p−1,r2p,ρqr2p,r2p+1≤c´ρqr2p−1,r2p.
Asr2p−2,r2p−1∈TSrn, β∗r2p−2,Sr2p−2≥1, and β∗Sr2p−1,r2p−1≥1, then by using condition (9), we obtain(16)ρqr2p−1,r2p≤tρqr2p−2,Sr2p−2+ρqr2p−1,Tr2p−1≤tρqr2p−2,r2p−1+ρqr2p−1,r2p≤t1−tρqr2p−2,r2p−1≤c´ρqr2p−2,r2p−1.
Using (16) in (15), we have(17)ρqr2p,r2p+1≤c´2ρqr2p−2,r2p−1.
Asr2p−2,r2p−1∈TSrn, β∗Sr2p−1,r2p−1≥1, and β∗r2p−2,Sr2p−2≥1, then by using condition (9), we obtain(18)ρqr2p−2,r2p−1≤tρqr2p−3,r2p−2+ρqr2p−2,r2p−1≤ć3ρqr2p−3,r2p−2.
From (17) and (18), we have(19)ć2ρqr2p−2,r2p−1≤ć3ρqr2p−3,r2p−2.
Using (19) in (15), we have(20)ρqr2p,r2p+1≤ć3ρqr2p−3,r2p−2.
Continuing in this way, we obtain(21)ρqr2p,r2p+1≤ć2pρqr0,r1.
Similarly, by using (14)–(16) and continuing in this way, we obtain(22)ρqr2p+1,r2p+2≤ć2p+1ρqr0,r1.
Combining inequalities (21) and (22), we have(23)ρqrn,rn+1≤ćnρqr0,r1.
Now, to prove thatrn is a left Cauchy sequence, for all natural numbers n<m, we have(24)ρqrn,rm≤αrn,rn+1ρqrn,rn+1+μrn+1,rmρqrn+1,rm≤αrn,rn+1ρqrn,rn+1+μrn+1,rmαrn+1,rn+2ρqrn+1,rn+2+μrn+1,rmμrn+2,rmρqrn+2,rm≤αrn,rn+1ρqrn,rn+1+μrn+1,rmαrn+1,rn+2ρqrn+1,rn+2+μrn+1,rmμrn+2,rmαrn+2,rn+3ρqrn+2,rn+3+μrn+1,rmμrn+2,rmμrn+3,rmρqrn+3,rm≤αrn,rn+1ρqrn,rn+1+∑i=n+1m−2∏j=n+1iμrj,rmαri,ri+1ρqri,ri+1+∏k=n+1m−1μrk,rmρqrm−1,rm,≤αrn,rn+1t1−tnρqr0,r1+∑i=n+1m−1∏j=0iμrj,rmαri,ri+1t1−tiρqr0,r1.
Hence, we have(25)ρqrn,rm≤ρqr0,r1t1−tnαrn,rn+1+Sm−1−Sn,where(26)Sn=∑i=1n∏j=0iμrj,rmαri,ri+1t1−ti.
By using ratio test and inequality (10), it can be proved easily that the infinite series ∑i=1∞∏j=0iμrj,rmαri,ri+1t/1−ti is convergent. Letting n,m⟶∞ in (25), yields(27)limn,m⟶∞ρqrn,rm=0.
So, the sequenceTSrn is a left Cauchy sequence. Since ℑ,ρq is a left K-sequentially double controlled complete dislocated quasi-metric type space, there exists r∗∈ℑ such that TSrn⟶r∗∈ℑ and(28)limn⟶∞ρqrn,r∗=limn⟶∞ρqr∗,rn=0.
Sinceℑ,ρq is a left K-sequentially complete and GS is closed subset of ℑ, so GS,ρq is also left K -sequentially complete. As β∗r2p,,Sr2p≥1 for all p∈ℕ. So, r2n is a subsequence of TSrn contained in GS. By completeness of GS,ρq and uniqueness of limit, r2n⟶r∗∈GS, that is,(29)β∗r∗,Sr∗≥1.
Now, we suppose thatρqr∗,Sr∗≠0. By Lemma 1, we have(30)ρqr2n,Sr∗≤HρqTr2n−1,Sr∗.
Asβ∗r∗,Sr∗≥1 and β∗Sr2n−1,r2n−1≥1, by using (9), we have(31)ρqr2n,Sr∗≤tρqr2n−1,r2n+ρqr∗,Sr∗.
Takinglimn⟶∞ on both sides of inequality (31), we obtain(32)limn⟶∞ρqr2n,Sr∗≤limn⟶∞tρqr2n−1,r2n+ρqr∗,Sr∗,limn⟶∞ρqr2n,Sr∗≤tρqr∗,Sr∗.
Now,(33)ρqr∗,Sr∗≤αr∗,r2nρqr∗,r2n+μr2n,Sr∗ρqr2n,Sr∗.
Takinglimn⟶∞ of inequality (33) and using inequality (11) and (28), we obtain(34)ρqr∗,Sr∗≤limn⟶∞μr2n,Sr∗ρqr2n,Sr∗.
Using inequalities (11) and (32) in inequality (34), we obtain(35)ρqr∗,Sr∗<ρqr∗,Sr∗.
It is a contradiction; therefore,(36)ρqr∗,Sr∗=0.
Now, suppose thatρqSr∗,r∗≠0. By Lemma 1, we have(37)ρqSr∗,r2n≤HρqSr∗,Tr2n−1.
Now,β∗r∗,Sr∗≥1 and β∗Sr2n−1,r2n−1≥1, so by (9), we have(38)ρqSr∗,r2n≤tρqr∗,Sr∗+ρqr2n−1,r2n.
Takinglimn⟶∞ on both sides of inequality (38), we obtain(39)limn⟶∞ρqSr∗,r2n≤tρqr∗,Sr∗.
Now,(40)ρqSr∗,r∗≤αSr∗,r2nρqSr∗,r2n+μr2n,r∗ρqr2n,r∗.
Takinglimn⟶∞ of inequality (40) and using inequalities (11) and (28), we obtain(41)ρqSr∗,r∗<1tlimn⟶∞ρqSr∗,r2n.
By using inequality (39) in inequality (41), we obtain(42)ρqSr∗,r∗<ρqr∗,Sr∗=0,by.
It is a contradiction. Hence,ρqr∗,Sr∗=ρqSr∗,r∗=0, so r∗∈Sr∗. Now,(43)ρqr∗,r∗≤αr∗,r2nρqr∗,r2n+μr2n,r∗ρqr2n,r∗.
This impliesρqr∗,r∗=0 as n⟶∞. As β∗r∗,Sr∗≥1 and ρqr∗,Sr∗=ρqSr∗,r∗=0=ρqr∗,r∗. So, Definition 6 implies(44)β∗Sr∗,r∗≥1.
Now, by Lemma1, we have(45)ρqr2n+1,Tr∗≤HρqSr2n,Tr∗.
As,β∗r2n,Sr2n≥1 and β∗Sr∗,r∗≥1, so by (9), we have(46)ρqr2n+1,Tr∗≤tρqr2n,r2n+1+ρqr∗,Tr∗.
Takinglimn⟶∞ on both sides of inequality (46), we obtain(47)limn⟶∞ρqr2n+1,Tr∗≤limn⟶∞tρqr2n,r2n+1+ρqr∗,Tr∗.
Lettingn⟶∞ and using inequality (27), we have(48)limn⟶∞ρqr2n+1,Tr∗≤tρqr∗,Tr∗.
Since,(49)ρqr∗,Tr∗≤αr∗,r2nρqr∗,r2n+μr2n+1,Tr∗ρqr2n+1,Tr∗,by taking limn⟶∞ and using inequality (11) and (46), we obtain(50)ρqr∗,Tr∗<1tlimn⟶∞ρqr2n+1,Tr∗.
By using inequalities (11) and (48), we obtain(51)ρqr∗,Tr∗<ρqr∗,Tr∗.
It is a contradiction. Thus,ρqr∗,Tr∗=0. Similar to the arguments above, we obtain(52)ρqTr∗,r∗=0.
Hence,r∗∈Tr∗. Hence, r∗ is a common fixed point for S and T.Example 2.
Letℑ=0,∞∩ℚ+. Define ρqr,y=r+2y2 if r≠y and ρqr,y=0 if r=y. Then, ℑ,ρq is a complete double controlled dislocated quasi-metric type space with(53)αr,y=2,ifr,y≥1,r+22,otherwise,μr,y=1ifr,y≥1,y+22,otherwise.
Let,(54)βr,y=1,ifr∈Ay∈B,14,otherwise.
Now,(55)A=r:β∗r,Sr≥1=0,1,164,14096,…,B=y:β∗Sy,y≥1=0,18,1512,….
Define the mappingsS,T:ℑ⟶Pℑ as follows:(56)Ty=y8,y4∩ℚ+,for ally∈0,1,18,164,1512,14096,…,y+2,2y+1∩ℚ+,if otherwise,Sy=18y,for ally∈0,1,18,164,1512,14096,…,y+1,y+3∩ℚ+,if otherwise.
The givenρq is not a controlled dislocated quasi-metric type space for the function α. Indeed,(57)ρq1,3=49>37.5=α1,0ρq1,0+α0,3ρq0,3.
Now,ρqr0,Sr0=ρq1,S1=ρq1,1/8=1+2/82=5/42, we define the sequence TSrn=1,1/8,1/64,1/512,1/4096,… in ℑ generated by r0=1.
Note thatβ∗r,Sr≥1, ρqr,Sr=ρqr,y, and ρqSr,r=ρqy,r imply β∗Sy,y≥1. Also, β∗Sr,r≥1, ρqr,Tr=ρqr,y, and ρqTr,r=ρqy,r implyβ∗y,Sy≥1. So, the pair S,T is β∗-Alt multivalued mapping on TSrn.
Now, for allr,y∈ℑ∩TSrn with β∗Sy,y≥1, β∗r,Sr≥1, and t=2/5, we observe that all cases are satisfied, that is,(58)maxHρqSr,Ty,HρqTr,Sy≤tρqr,Sr+ρqy,Ty.
Now,r0=1, and we have r1=1/8, r2=1/64, r3=1/512,….(59)supm≥1limi⟶∞αri+1,ri+2αri,ri+1μri+1,rm≈0.8<1−tt=32.
That is, the pairS,T is β∗ Kannan-type double controlled contraction. Let r0=1, and we have(60)GS=r:β∗r,Sr≥1r∈TSrn=0,1,164,14096,….
That is (i) holds. Finally, for everyr∈TSrn, we have(61)limn⟶∞αr,rn<52,limn⟶∞rn,r<52.
Thus, all the hypothesis of Theorem2 are satisfied. In fact, 0 is a common fixed point of S and T.Definition 8.
Letℑ,ρq be a complete dislocated quasi b-metric type space and S,T be a pair of β∗-Alt multivalued mapping. Then, the pair S,T is called β∗ Kannan-type b-contraction, if for every two consecutive points randy belonging to the range of an iterative sequence TSrn with β∗Sy,y≥1 and β∗r,Sr≥1, and we have(62)maxHpqSy,Tr,HρqTr,Sy≤tρqr,Tr+ρqy,Sy,whenever, t∈0,1/2 and(63)b<1−tt.Theorem 3.
Letℑ,ρq be a left K-sequentially complete dislocated quasi b-metric space and a pair S,T be a β∗ Kannan type b-contraction. Assume the following.The setGS=r:β∗r,Sr≥1 is closed and contained r0. Then, TSrn⟶r∗∈ℑ. Also, if (62) holds for each r,y∈r∗, then S and T have a common fixed point r∗ in ℑ and ρqr∗,r∗=0.Remark 2.
In Example 2,ρqr,y=r+2y2 is a dislocated quasi b-metric with b=2, but we cannot apply Theorem 3 because the pair S,T is not β∗ Kannan type b-contraction, indeed b≮1−t/t=3/2.


## 3. Conclusion

The aim of this work is to introduce double controlled dislocated quasi-metric type spaces and related definitions as a generalization of double controlled metric-type spaces. We have removed one and a half restriction out of three restrictions of double controlled metric-type spaces. This new space is a generalization of metric space, quasi-metric space, dislocated metric space, dislocated quasi-metric space, partial metric space, quasi-partial metric space,b-metric space, quasi-b-metric space, dislocated b-metric space, dislocated quasi-b-metric space, extended b-metric space, dislocated extended b-metric space, quasi-extended b-metric space, dislocated quasi-extended b-metric space, and double controlled metric space. We establish a new generalized contraction on a sequence and obtain fixed point results for a pair of multivalued mappings in left k-sequentially complete double controlled dislocated quasi-metric type spaces. New results in ordered spaces and new results for graphic contractions can be obtained as corollaries of our results. An example has been built to demonstrate the novelty of results. [29]



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