# On Extendedb-Rectangular and Controlled Rectangular Fuzzy Metric-Like Spaces with Application

Authors: Naeem Saleem; Salman Furqan; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we introduce the notions of extendedb-rectangular and controlled rectangular fuzzy metric-like spaces that generalize many fuzzy metric spaces in the literature. We give examples to justify our newly defined fuzzy metric-like spaces and prove that these spaces are not Hausdorff. We use fuzzy contraction and prove Banach fixed point theorems in these spaces. As an application, we utilize our main results to solve the uniqueness of the solution of a differential equation occurring in the dynamic market equilibrium.


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## Body

## 1. Introduction and Preliminaries

In 1965, Zadeh [1] introduced the concept of a fuzzy set that generalizes the concept of an ordinary set or crisp set. Many authors have used fuzzy sets in different branches of mathematics extensively. For example, Kaleva [2] gave the idea of fuzzy differential equations, Buckley and Feuring [3] introduced the concept of fuzzy partial differential equations, and Puri and Ralescu [4] introduced the differentials of fuzzy functions. Fuzzy metric space is one of the most studied topic in fuzzy set theory. The definition of fuzzy metric space was given by Kramosil and Michálek [5] in 1975 which was later modified by George and Veeramani [6]. In 1983, Grabiec [7] established and proved the fuzzy version of the Banach contraction principle. Many researchers have used and extended this version in many fuzzy metric spaces (see [8–15] and references therein). Branciari [16] generalized the definition of classical metric space by introducing rectangular or Branciari metric space and proved some fixed point results. As a generalization of a Branciari metric space, authors in [17] gave the notion of b-rectangular metric space.Hitzler and Seda [18] introduced the idea of dislocated topology in which the self-distance between the points may not be zero. Amini-Harandi [19] introduced the definition of a metric-like space and proved related results. The notion of b-metric-like space was introduced by [20] as a generalization of metric-like space. Mlaiki et al. [21] generalized the definition of a rectangular metric space by defining rectangular metric-like space. The concept of a fuzzy metric-like space was introduced by Shukla and Abbas [14] as a generalization of [6] and proved related fixed point results. They generalized the definition of a fuzzy metric space in the sense that the self-distance may not be equal to one. The concept of fuzzy b-metric and fuzzy quasi-b-metric space was introduced by Nadaban [22]. The concepts of fuzzy double controlled metric space and fuzzy triple controlled metric space were given by Saleem et al. [13] and Furqan et al. [23], respectively. They also proved that these spaces are not Hausdorff.Definition 1 (see [19]).
LetϜ be a nonempty set, a mapping ρ:Ϝ×Ϝ⟶ℝ+∪0 is called metric-like if, for all ℏ1,ℏ2,ℏ3∈Ϝ,ρ satisfies the following:
(1)ML1ρℏ1,ℏ2=0⟹ℏ1=ℏ2,ML2ρℏ1,ℏ2=ρℏ2,ℏ1,ML3ρℏ1,ℏ3≤ρℏ1,ℏ2+ρℏ2,ℏ3.
The pairϜ,ρ is called a metric-like space.Example 1 (see [19]).
LetF=0,1, and ρ is given by ρℏ1,ℏ2=2 if ℏ1=ℏ2=0, and ρℏ1,ℏ2=1, otherwise. Then, Ϝ,ρ is a metric-like space.Definition 2 (see [20]).
LetϜ be a nonempty set and b≥1; a function ρ:Ϝ×Ϝ⟶ℝ+∪0 is called b-metric-like if ℏ1,ℏ2,ℏ3∈Ϝ,ρ satisfies the following:
(2)bML1ρℏ1,ℏ2=0⟹ℏ1=ℏ2,bML2ρℏ1,ℏ2=ρℏ2,ℏ1,bML3ρℏ1,ℏ3≤bρℏ1,ℏ2+ρℏ2,ℏ3.
The pairϜ,ρ is called b-metric-like space.Example 2 (see [20]).
Let Ϝ=0,∞ and ρ:0,∞×0,∞×0,∞ be defined as ρℏ1,ℏ2=ℏ1+ℏ22. Then, Ϝ,ρ is a b-metric-like space with b=2.Definition 3 (see [24]).
LetϜ be a nonempty set and ρ:Ϝ×Ϝ⟶0,∞ be a function; then, ρ is said to be a rectangular metric-like space if it satisfies the following:
(3)RML1ρℏ1,ℏ2=0⟹ℏ1=ℏ2,RML2ρℏ1,ℏ2=ρℏ2,ℏ1,RML3ρℏ1,ℏ4≤ρℏ1,ℏ2+ρℏ2,ℏ3+ρℏ3,ℏ4,foralldistinctℏ2,ℏ3∈Ϝ∖ℏ1,ℏ4.
The pairϜ,ρ is called rectangular metric-like space.Definition 4 (see [25]).
A binary operation∗ on 0,1, where ∗:0,1×0,1⟶0,1, is called a continuous triangular norm, if for all ℏ1,ℏ2,ℏ3,ℏ4∈0,1,the following conditions are satisfied:
∗1∗ℏ1,ℏ2=∗ℏ2,ℏ1∗2∗ℏ1,∗ℏ2,ℏ3=∗∗ℏ1,ℏ2,ℏ3∗3∗iscontinuous∗4∗ℏ,1=ℏfor every ℏ∈0,1,∗5∗ℏ1,ℏ2≤∗ℏ3,ℏ4wheneverℏ1≤ℏ3,ℏ2≤ℏ4.


## 2. Extended Fuzzyb-Rectangular Metric-Like Space

In this section, we introduce the definition of an extended fuzzyb-rectangular metric-like space.Definition 5.
Letf:Ϝ×Ϝ⟶1,∞ be a function; Mf is a fuzzy set on Ϝ×Ϝ×0,∞. Then, Mf is called an extended fuzzy b-rectangular metric-like with t-norm ∗, if for all ℏ1,ℏ4∈Ϝ and all distinct ℏ3,ℏ2∈Ϝ∖ℏ1,ℏ4; Mf satisfies the following:
(4)Mf1Mfℏ1,ℏ2,t>0,Mf2ifMfℏ1,ℏ2,t=1forallt>0,thenℏ1=ℏ2,Mf3Mfℏ1,ℏ2,t=Mfℏ2,ℏ1,t,Mf4Mfℏ1,ℏ4,t+s+w≥Mfℏ1,ℏ2,tfℏ1,ℏ4∗Mfℏ2,ℏ3,sfℏ1,ℏ4∗Mfℏ3,ℏ4,wfℏ1,ℏ4,forallt,s,w>0,Mf5Mfℏ1,ℏ2,·:0,∞⟶0,1iscontinuous.
Then,Ϝ,Mf,,f,∗ is called an extended fuzzy b-rectangular metric-like space.Remark 6.
(i)
By takingfx,y=b≥1, then extended fuzzy b-rectangular metric-like space reduces to fuzzy b-rectangular metric-like space.(ii)
By takingfx,y=1, then extended fuzzy b-rectangular metric-like space reduces to fuzzy rectangular metric-like space.The following example justifies Definition5.Example 3.
LetϜ=1,2,3,4. If we define ρ:Ϝ×Ϝ⟶0,∞ by ρℏ1,ℏ2=ℏ1+ℏ22 for all ℏ1,ℏ2 in Ϝ and f:Ϝ×Ϝ⟶1,∞ by fℏ1,ℏ2=ℏ12+ℏ22+1. Then, Mf:Ϝ×Ϝ×0,∞⟶0,1, given by
(5)Mfℏ1,ℏ2,t=tt+ρℏ1,ℏ2,forallt>0, is an extended fuzzy b-rectangular metric-like on Ϝ provided that ∗ is a minimum t-norm, that is, a∗b=mina,b for all a,b∈0,1.Clearly,(6)f1,2=f2,1=6,f1,3=f3,1=11,f1,4=f4,1=18,f2,3=f3,2=14,f2,4=f4,2=21,f3,4=f4,3=26,f1,1=3,f2,2=9,f3,3=19,f4,4=33,ρ1,2=ρ2,1=9,ρ2,3=ρ3,2=25,ρ3,4=ρ4,3=49,ρ1,3=ρ3,1=16,ρ2,4=ρ4,2=36,ρ1,4=ρ4,1=25,ρ1,1=4,ρ2,2=16,ρ3,3=36,ρ4,4=64,forallℏ1∈F.Note that the first three axioms (Mf1–Mf3) clearly hold. To prove the axiom Mf4, we discuss the following cases:Case 1.
Letℏ1=1,ℏ3=4. Then, we have
(7)Mf1,4,f1,4t+s+w=f1,4t+s+wf1,4t+s+w+ρ1,4=18t+s+w18t+s+w+25=1−2518t+s+w+225.
Also,(8)Mf1,2,t=tt+ρ1,2=tt+9=1−9t+9,Mf2,3,s=ss+ρ2,3=ss+25=1−25s+25,Mf3,4,w=ww+ρ3,4=ww+49=1−49w+49.
Note that(9)Mf1,4,f1,4t+s+w=1−2518t+s+w+25=1−225162t+162s+162w+225,Mf1,2,t=1−9t+9=1−22525t+225.
Clearly,(10)1−225162t+162s+162w+225>1−22525t+225.That is,
(11)Mf1,4,f1,4t+s+w>Mf1,2,t.Similarly, we have
(12)Mf1,4,f1,4t+s+w>Mf2,3,s,Mf1,4,f1,4t+s+w>Mf3,4,w.
Hence,(13)Mf1,4,f1,4t+s+w>Mf1,2,t∗Mf2,3,s∗Mf3,4,w.Case 2.
Letℏ1=2, and ℏ3=4. Then,
(14)Mf2,4,f2,4t+s+w=f2,4t+s+wf2,4t+s+w+ρ2,4=21t+s+w21t+s+w+36,=1−3621t+s+w+36,Mf2,3,t=tt+ρ2,3=tt+25=1−25t+25,Mf3,1,s=ss+ρ3,1=ss+16=1−16s+16,Mf1,4,w=ww+ρ1,4=ww+25=1−25w+25.
Now,(15)Mf2,4,f2,4t+s+w=1−3621t+s+w+36=1−900525t+525s+525w+900,Mf2,3,t=1−25t+25=1−90036t+900.
Clearly,(16)1−900525t+525s+525w+900>1−90036t+900implies that
(17)Mf1,4,f1,4t+s+w>Mf1,2,t.
Similarly, we obtain that(18)Mf2,4,f2,4t+s+w>Mf3,1,s,Mf2,4,f2,4t+s+w>Mf1,4,w.
Hence,(19)Mf2,4,f2,4t+s+w>Mf2,3,t∗Mf3,1,s∗Mf1,4,w.Case 3.
Ifℏ1=3 and ℏ3=4, then we have
(20)Mf3,4,f3,4t+s+w=f3,4t+s+wf3,4t+s+w+ρ3,4=26t+s+w26t+s+w+49,=1−4926t+s+w+49.
Also,(21)Mf3,1,t=tt+ρ3,1=tt+16=1−16t+16,Mf1,2,s=ss+ρ1,2=ss+9=1−9s+9,Mf2,4,w=ww+ρ2,4=ww+4=1−36w+36.
Now,(22)1−4926t+s+w+49=1−784416t+s+w+784,Mf3,1,t=1−16t+16=1−78449t+784.
Clearly,(23)1−784416t+416s+416w+784>1−44149t+784implies that
(24)Mf3,4,f1,4t+s+w>Mf3,1,t.
Similarly,(25)Mf3,4,f3,4t+s+w>Mf1,2,s,Mf3,4,f3,4t+s+w>Mf2,4,w.
Hence,(26)Mf3,4,f3,4t+s+w>Mf3,1,t∗Mf1,2,s∗Mf2,4,w.
Thus, in each case, we have(27)Mfℏ1,ℏ2,fℏ1,ℏ2t+s+w≥Mfℏ1,ℏ3,t∗Mfℏ3,ℏ4,s∗Mfℏ4,ℏ2,w,for all distinct ℏ1,ℏ2,ℏ3,ℏ4∈Ϝ and t,s,w>0. Hence, Ϝ,Mf,f,∗ is an extended fuzzy b-rectangular metric-like space.Definition 7.
LetϜ,Mf,f,∗ be an extended fuzzy b-rectangular metric-like space and ℏn be a sequence in Ϝ, then ℏn is
(1)
a convergent sequence, if there existsℏ in Ϝ such that(28)limn⟶∞Mfℏn,ℏ,t=Mfℏ,ℏ,t,forallt>0.(2)
a Cauchy sequence, if for allt>0 and for p≥1,(29)limn⟶∞Mfℏn+p,ℏn,texistsandisfinite.
An extended fuzzyb-rectangular metric-like space is said to be complete, if every Cauchy sequence converges to some h∈Ϝ.Definition 8.
LetϜ,Mf,f,∗ be an extended fuzzy b-rectangular metric metric-like space. A mapping T:Ϝ⟶Ϝ is said to be a fuzzy contractive mapping if
(30)1MfTℏ1,Tℏ2,t−1≤k1Mfℏ1,ℏ2,t−1forallℏ1,ℏ2∈F,t>0,andk∈0,1.Theorem 9.
LetϜ,Mf,f,∗ be a complete extended fuzzy b-rectangular metric-like space with f:Ϝ×Ϝ⟶1,1/k and T:Ϝ⟶Ϝ be a fuzzy contractive mapping such that
(31)limt⟶∞Mfℏ1,ℏ2,t=1.
Then,T has a unique fixed point.Proof.
Letℏ0 be an arbitrary point in Ϝ. If Tℏ0=ℏ0, then ℏ0 is the required fixed point. If Tℏ0≠ℏ0, then there exists ℏ1∈Ϝ such that Tℏ0=ℏ1. If Tℏ1=ℏ1, then ℏ1 is the required fixed point. Continuing in this way, we have the sequence ℏn in F such that Tℏn=Tn+1ℏ0=ℏn+1. Let t>0 and using inequality (30), we have
(32)1Mfℏn,ℏn+1,t−1=1MfTℏn−1,Tℏn,t−1≤k1Mfℏn−1,ℏn,t−1,so we have
(33)1Mfℏn,ℏn+1,t≤kMfℏn−1,ℏn,t+1−k.
Now,(34)kMfℏn−1,ℏn,t+1−k=kMfTℏn−2,Tℏn−1,t+1−k≤kkMfℏn−2,ℏn−1,t+1−k,so we have
(35)kMfℏn−1,ℏn,t≤k2Mfℏn−2,ℏn−1,t+k1−k+1−k.Continuing in this way, we have
(36)1Mfℏn,ℏn+1,t≤knMfℏ0,ℏ1,t+kn−11−k+kn−21−k+⋯+k1−k+1−k=knMfℏ0,ℏ1,t+kn−1+kn−2+⋯+k+11−k=knMfℏ0,ℏ1,t+1−kn,we have
(37)1kn/Mfℏ0,ℏ1,t+1−kn≤Mfℏn,ℏn+1,t.
Working in the same way, we also can prove that(38)1kn−2/Mfℏ0,ℏ2,t+1−kn−2≤Mfℏn−2,ℏn,t.Letℏn be a sequence in Ϝ; then, we have the following cases.Case 1.
Ifp is odd (say) p=2m+1 for m≥1, then
(39)Mfℏn,ℏn+2m+1,t≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m+1∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m+1∗Mfℏn+2,ℏn+2m+1,t/3fℏn,ℏn+2m+1≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m+1∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m+1∗Mfℏn+2,ℏn+3,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1∗Mfℏn+3,ℏn+4,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1∗Mfℏn+4,ℏn+2m+1,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m+1∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m+1∗Mfℏn+2,ℏn+3,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1∗Mfℏn+3,ℏn+4,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1∗Mfℏn+5,ℏn+6,t/33fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1∗Mfℏn+6,ℏn+7,t/34fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1∗Mfℏn+7,ℏn+8,t/34fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1⋯∗Mfℏn+2m,ℏn+2m+1,t/3mfℏn,ℏn+2m+1fℏn+2,ℏn+2m+1⋯fℏn+2m−2,ℏn+2m+1.⋮
Using inequality (37), we have
(40)Mfℏn,ℏn+2m+1,t≥1kn/Mfℏ0,ℏ1,t/3fℏn,ℏn+2m+1+1−kn∗1kn+1/Mfℏ0,ℏ1,t/3fℏn,ℏn+2m+1+1−kn+1∗1kn+2/Mfℏ0,ℏ1,t/32fℏn,ℏn+2m+1fℏn+2,ℏn+2m+1+1−kn+2∗⋯∗1kn+p−1/Mfℏ0,ℏ1,t/3mfℏn,ℏn+2m+1fℏn+2,ℏn+2m+1⋯fℏn+2m−2,ℏn+2m+1+1−kn+p−1.Taking limit n⟶∞, we have
(41)limn⟶∞Mfℏn,ℏn+2m+1,t≥1∗1∗⋯∗1=1.Case 2.
Ifp=2m,that is, p is even, then
(42)Mfℏn,ℏn+2m,t≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m∗Mfℏn+2,ℏn+2m,t/3fℏn,ℏn+2m,≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m∗Mfℏn+2,ℏn+3,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m∗Mfℏn+3,ℏn+4,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m∗Mfℏn+4,ℏn+2m,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m≥Mfℏn,ℏn+1,t/3fℏn,ℏn+2m∗Mfℏn+1,ℏn+2,t/3fℏn,ℏn+2m∗Mfℏn+2,ℏn+3,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m∗Mfℏn+3,ℏn+4,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m∗Mfℏn+4,ℏn+5,t/33fℏn,ℏn+2mfℏn+2,ℏn+2mfℏn+4,ℏn+2m∗Mfℏn+5,ℏn+6,t/33fℏn,ℏn+2mfℏn+2,ℏn+2mfℏn+4,ℏn+2m∗Mfℏn+6,ℏn+7,t/34fℏn,ℏn+2mfℏn+2,ℏn+2mfℏn+4,ℏn+2mfℏn+6,ℏn+2m⋯∗Mfℏn+2m−2,ℏn+2m,t/3m−1fℏn,ℏn+2mfℏn+2,ℏn+2m⋯fℏn+2m−4,ℏn+2m.
Using inequality (37) and (38), we have
(43)Mfℏn,ℏn+2m,t≥1kn/Mfℏ0,ℏ1,t/3fℏn,ℏn+2m+1−kn∗1kn+1/Mfℏ0,ℏ1,t/3fℏn,ℏn+2m+1−kn+1∗1kn+2/Mfℏ0,ℏ1,t/32fℏn,ℏn+2mfℏn+2,ℏn+2m+1−kn+2∗⋯∗1kn+p−1/Mfℏ0,ℏ2,t/3m−1fℏn,ℏn+2mfℏn+2,ℏn+2m⋯fℏn+2m−4,ℏn+2m+1−kn+p−1.Taking limit n⟶∞, we have
(44)limn⟶∞Mfℏn,ℏn+2m,t≥1∗1∗⋯∗1=1.
Thus,ℏn is a Cauchy sequence and converges to ℏ∈Ϝ (since Ϝ is complete). Now, we have to prove ℏ is the fixed point of T. Consider
(45)1MfTℏn,Tℏ,t−1≤k1Mfℏn,ℏ,t−1=kMfℏn,ℏ,t−k,1MfTℏn,Tℏ,t≤k1Mfℏn,ℏ,t−1+1=kMfℏn,ℏ,t+1−k,so,
(46)1k/Mfℏn,ℏ,t+1−k≤MfTℏn,Tℏ,t.
Now,(47)Mfℏ,Tℏ,t≥Mfℏ,ℏn,t/3fℏ,Tℏ∗Mfℏn,ℏn+1,t/3fℏ,Tℏ∗Mfℏn+1,Tℏ,t/3fℏ,Tℏ≥Mfℏ,ℏn,t/3fℏ,Tℏ∗MfTℏn−1,Tℏn,t/3fℏ,Tℏ∗MfTℏn,Tℏ,t/3fℏ,Tℏ≥Mfℏ,ℏn,t/3fℏ,Tℏ∗1k/Mfℏn,ℏ,t+1−k∗MfTℏn,Tℏ,t/3fℏ,Tℏ.
Applying limitn⟶∞ on the right-hand side, we have Mfℏ,Tℏ,t=1, which shows ℏ is the fixed point of T. For uniqueness, let ℏ′ be an other fixed point of T such that Tℏ′=ℏ′ and consider,
(48)1Mfℏ,ℏ′,t−1=1MfTℏ,Tℏ′,t−1≤k1Mfℏ,ℏ′,t−1<1Mfℏ,ℏ′,t−1,
which is a contradiction; thus,ℏ is the only fixed point of T.


## 3. Fuzzy Controlled Rectangular Metric-Like Space

In this section, we will give the definition of fuzzy controlled rectangular metric like space.Definition 10.
Letf:Ϝ×Ϝ⟶1,∞ be a function and Mc is a fuzzy set on Ϝ×Ϝ×0,∞. Then, Mc is called fuzzy controlled rectangular metric-like with ∗ as continuous t-norm; if for any ℏ1,ℏ2∈Ϝ and all distinct ℏ3,ℏ4∈Ϝ∖ℏ1,ℏ2, the following conditions are satisfied:
(49)Mc1Mcℏ1,ℏ2,t>0,Mc2ifMcℏ1,ℏ2,t=1forallt>0,thenℏ1=ℏ2,Mc3Mcℏ1,ℏ2,t=Mcℏ2,ℏ1,t,Mc4Mcℏ1,ℏ2,t+s+w≥Mcℏ1,ℏ3,tfℏ1,ℏ3∗Mcℏ3,ℏ4,sfℏ3,ℏ4∗Mcℏ4,ℏ2,wfℏ4,ℏ2,forallt,s,w>0,Mc5Mcℏ1,ℏ2,·:0,∞⟶0,1iscontinuous.
Then,Ϝ,Mc,∗ is called a fuzzy controlled rectangular metric-like space.Remark 11.
(i)
If we takefℏ1,ℏ3=fℏ3,ℏ4=fℏ4,ℏ2=fℏ1,ℏ2, then a fuzzy controlled rectangular metric-like space reduces to extended fuzzy b-rectangular metric-like space(ii)
If we takefℏ1,ℏ3=fℏ3,ℏ4=fℏ4,ℏ2=b≥1, then a fuzzy controlled rectangular metric-like space reduces to fuzzy b-rectangular metric-∗like space(iii)
If we takefℏ1,ℏ3=fℏ3,ℏ4=fℏ4,ℏ2=fℏ1,ℏ2=1, then a fuzzy controlled rectangular metric-like space reduces to fuzzy rectangular metric-like spaceFollowing example justifies the definition10.Example 4.
LetϜ=ℕ∪0, f:F×F⟶1,∞ and t1∗t2=t1·t2. Define Mc:Ϝ×Ϝ×0,∞⟶0,1 as
(50)Mcℏ1,ℏ2,t=exp−ℏ1+ℏ2t.
Then,Ϝ,Mc,∗ is a fuzzy controlled rectangular metric-like space.
We will proveMc4 as Mc1–Mc3 and Mc5 are easy to prove.
Now,(51)ℏ1+ℏ4≤ℏ1+ℏ2+ℏ2+ℏ3+ℏ3+ℏ4≤fℏ1,ℏ2t+s+wtℏ1+ℏ2+fℏ2,ℏ3t+s+wsℏ2+ℏ3+fℏ3,ℏ4t+s+wwℏ3+ℏ4,so we have
(52)ℏ1+ℏ4t+s+w≤ℏ1+ℏ2t/fℏ1,ℏ2+ℏ2+ℏ3s/fℏ2,ℏ3+ℏ3+ℏ4w/fℏ3,ℏ4−ℏ1+ℏ4t+s+w≥−ℏ1+ℏ2t/fℏ1,ℏ2+ℏ2+ℏ3s/fℏ2,ℏ3+ℏ3+ℏ4w/fℏ3,ℏ4exp−ℏ1+ℏ4t+s+w≥exp−ℏ1+ℏ2t/fℏ1,ℏ2·exp−ℏ2+ℏ3s/fℏ2,ℏ3·exp−ℏ3+ℏ4w/fℏ3,ℏ4.Thus
(53)Mcℏ1,ℏ4,t+s+w≥Mcℏ1,ℏ2,tfℏ1,ℏ2∗Mcℏ2,ℏ3,sfℏ2,ℏ3∗Mcℏ3,ℏ4,wfℏ3,ℏ4.
Hence,Ϝ,Mc,∗ is a fuzzy controlled rectangular metric-like space.Example 5.
ConsiderϜ=0,1,2,3 and let f:Ϝ×Ϝ⟶1,∞ be a function defined as fℏ1,ℏ2=1+ℏ1+ℏ2,fℏ2,ℏ3=ℏ22+ℏ3+2, and fℏ3,ℏ4=ℏ32+ℏ42+3. Define a rectangular metric-like space by ρℏ,ℏ′=maxℏ,ℏ′. Now define Mc:Ϝ×Ϝ×0,∞⟶0,1 as follows:
(54)Mcℏ1,ℏ2,t=tt+ρℏ1,ℏ2.
Then,Ϝ,Mc,∗ is a fuzzy controlled rectangular metric-like space with product t-norm.Definition 12.
LetϜ,Mc,∗ be a fuzzy controlled rectangular metric-like space and ℏn be a sequence in Ϝ; then, the sequence ℏn is called a convergent sequence if
(55)limn⟶∞Mcℏn,ℏ,t=Mcℏ,ℏ,t.Definition 13.
A sequenceℏn in Ϝ is called a Cauchy sequence if for t>0,p≥1(56)limn⟶∞Mcℏn,ℏn+p,texistsandisfinite.
If every Cauchy sequence converges inϜ, then Ϝ,Mc,∗ is known as a complete fuzzy controlled rectangular metric-like space.Definition 14.
LetϜ,Mc,∗ be a fuzzy controlled rectangular metric-like space. Then, an open ball Bℏ,r,t, with center ℏ and radius r, is given by
(57)Bℏ,r,t=ℏ′∈Ϝ:Mcℏ,ℏ′,t>1−r,and the corresponding topology is defined as
(58)τMc=C⊂Ϝ:Bℏ,r,t⊂C.
We show that a fuzzy controlled rectangular metric-like space need not be Hausdorff in the following example.Example 6.
Consider the fuzzy controlled rectangular metric-like space as in Example5. Now, define an open ball Bℏ1ℏ1,r1,t1 with center ℏ1=1, radius r1=0.12, and t1=10 as
(59)Bℏ11,0.12,10=ℏ′∈F:Mc1,ℏ′,10>0.88,
Let0∈F, then Mc1,0,10=10/10+max1,0=10/10+1=0.9, so 0∈Bℏ1ℏ1,r1,t.
Let1∈F, then Mc1,1,10=10/10+max1,1=10/10+1=0.9, so 1∈Bℏ1ℏ1,r1,t.
Let2∈F, then Mc1,2,10=10/10+max1,2=10/10+2=0.8333, so 2∉Bℏ1ℏ1,r1,t.
Let3∈F, then Mc1,3,10=10/10+max1,3=10/10+3=0.7692, so 3∉Bℏ1ℏ1,r1,t.
Hence,(60)Bℏ1ℏ1,r1,t1=0,1.
Now, define an open ballBℏ2ℏ2,r2,t2 with center ℏ2=2, radius r2=0.4, and t2=4 as
(61)Bℏ22,0.4,4=ℏ′∈Ϝ:Mc2,ℏ′,4>0.6.
Let0∈F, then Mc2,0,4=4/4+max2,0=4/4+2=0.6666, so 0∈Bℏ2ℏ2,r2,t.
Let1∈F, then Mc2,1,4=4/4+max2,1=4/4+2=0.6666, so 1∈Bℏ2ℏ2,r2,t.
Let2∈F, then Mc2,2,4=4/4+max2,2=4/4+2=0.6666, so 2∈Bℏ2ℏ2,r2,t.
Let3∈F, then Mc2,3,4=4/4+max2,3=4/4+3=0.5714, so 3∉Bℏ2ℏ2,r2,t.
Hence,(62)Bℏ2ℏ2,r2,t2=0,1,2.
Now,Bℏ1ℏ1,r1,t∩Bℏ2ℏ2,r2,t=0,1∩0,1,2=0,1≠∅. Thus, a fuzzy controlled rectangular metric-like space is not Hausdorff.Remark 15.
In light of Remark11, extended fuzzy rectangular b-metric-like, fuzzy rectangular b-metric-like, and a fuzzy rectangular metric-like spaces are also not Hausdorff.The following is the Banach fixed point theorem in the settings of a fuzzy controlled rectangular metric-like space.Theorem 16.
Letf:Ϝ×Ϝ⟶1,1/k, k∈0,1 be a function, and Ϝ,Mc,∗ be a complete fuzzy controlled rectangular metric-like space such that
(63)limt⟶∞Mcℏ,ℏ′,t=1.
Assume further thatT:Ϝ⟶Ϝ be self-mapping such that for all ℏ,ℏ′∈Ϝ,
(64)McTℏ,Tℏ′,kt≥Mcℏ,ℏ′,t.
Then,T has a unique fixed point.Proof.
Letℏ0 be an arbitrary point in Ϝ; then, we have the iterative sequence Tℏn=Tn+1ℏ0=ℏn+1. Now,
(65)Mcℏn,ℏn+1,t=McTℏn−1,Tℏn,t≥Mcℏn−1,ℏn,tk=McTℏn−2,Tℏn−1,tk≥Mcℏn−2,ℏn−1,tk2≥Mcℏn−3,ℏn−2,tk3⋯≥Mcℏ0,ℏ1,tkn,hence,
(66)Mcℏn,ℏn+1,t≥Mcℏ0,ℏ1,tkn.In the same way, we can prove that
(67)Mcℏn−2,ℏn,t≥Mcℏ0,ℏ2,tkn−2.Now writingt=t/3+t/3+t/3 and for any sequence ℏn, we have the cases below.Case 1.
Whenp is odd, p=2m+1(say), then
(68)Mcℏn,ℏn+2m+1,t≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+2m+1,t/3fℏn+2,ℏn+2m+1≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+3,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2m+1∗Mcℏn+3,ℏn+4,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2m+1∗Mcℏn+4,ℏn+2m+1,t/32fℏn+4,ℏn+2m+1fℏn+2,ℏn+2m+1≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+3,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2m+1∗Mcℏn+3,ℏn+4,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2m+1∗Mcℏn+4,ℏn+5,t/33fℏn+4,ℏn+5fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1∗Mcℏn+5,ℏn+6,t/33fℏn+5,ℏn+6fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1∗Mcℏn+6,ℏn+7,t/34fℏn+6,ℏn+7fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1∗Mcℏn+7,ℏn+8,t/34fℏn+7,ℏn+8fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1⋯∗Mcℏn+2m−2,ℏn+2m−1,t/3mfℏn+2m−2,ℏn+2m−1fℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1∗Mcℏn+2m−1,ℏn+2m,t/3mfℏn+2m−1,ℏn+2mfℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1∗Mcℏn+2m,ℏn+2m+1,t/3mfℏn+2m,ℏn+2m+1fℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1.
Applying (66), we deduce
(69)Mcℏn,ℏn+2m+1,t≥Mcℏ0,ℏ1,t/3fℏn,ℏn+1kn∗Mcℏ0,ℏ1,t/3fℏn+1,ℏn+2kn+1∗Mcℏ0,ℏ1,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2m+1kn+2∗Mcℏ0,ℏ1,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2m+1kn+3∗Mcℏ0,ℏ1,t/33fℏn+4,ℏn+5fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1kn+4∗Mcℏ0,ℏ1,t/33fℏn+5,ℏn+6fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1kn+5∗Mcℏ0,ℏ1,t/34fℏn+6,ℏn+7fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1kn+6∗Mcℏ0,ℏ1,t/34fℏn+7,ℏn+8fℏn+2,ℏn+2m+1fℏn+4,ℏn+2m+1fℏn+6,ℏn+2m+1kn+7⋯∗Mcℏ0,ℏ1,t/3mfℏn+2m−2,ℏn+2m−1fℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1kn+2m−2∗Mcℏ0,ℏ1,t/3mfℏn+2m−1,ℏn+2mfℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1k2m−1∗Mcℏ0,ℏ1,t/3mfℏn+2m,ℏn+2m+1fℏn+2m−2,ℏn+2m+1⋯fℏn+2,ℏn+2m+1k2m.Case 2.
Whenp is even, p=2m (say), then
(70)Mcℏn,ℏn+2m,t≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+2m,t/3fℏn+2,ℏn+2m≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+3,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2m∗Mcℏn+3,ℏn+4,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2m∗Mcℏn+4,ℏn+2m+1,t/32fℏn+4,ℏn+2m+1fℏn+2,ℏn+2m≥Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,ℏn+2,t/3fℏn+1,ℏn+2∗Mcℏn+2,ℏn+3,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2m∗Mcℏn+3,ℏn+4,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2m∗Mcℏn+4,ℏn+5,t/33fℏn+4,ℏn+5fℏn+2,ℏn+2mfℏn+4,ℏn+2m∗Mcℏn+5,ℏn+6,t/33fℏn+5,ℏn+6fℏn+2,ℏn+2mfℏn+4,ℏn+2m∗Mcℏn+6,ℏn+7,t/34fℏn+6,ℏn+7fℏn+2,ℏn+2mfℏn+4,ℏn+2mfℏn+6,ℏn+2m∗Mcℏn+7,ℏn+8,t/34fℏn+7,ℏn+8fℏn+2,ℏn+2mfℏn+4,ℏn+2mfℏn+6,ℏn+2m⋯∗Mcℏn+2m−4,ℏn+2m−3,t/3m−1fℏn+2m−4,ℏn+2m−3fℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2m∗Mcℏn+2m−3,ℏn+2m−2,t/3m−1fℏn+2m−3,ℏn+2m−2fℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2m∗Mcℏn+2m−2,ℏn+2m,t/3m−1fℏn+2m−2,ℏn+2mfℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2m.
Now applying (66) and (67) on the right-hand side, we deduce
(71)Mcℏn,ℏn+2m,t≥Mcℏ0,ℏ1,t/3fℏn,ℏn+1kn∗Mcℏ0,ℏ1,t/3fℏn+1,ℏn+2kn+1∗Mcℏ0,ℏ1,t/32fℏn+2,ℏn+3fℏn+2,ℏn+2mkn+2∗Mcℏ0,ℏ1,t/32fℏn+3,ℏn+4fℏn+2,ℏn+2mkn+3∗Mcℏ0,ℏ1,t/33fℏn+4,ℏn+5fℏn+2,ℏn+2mfℏn+4,ℏn+2mkn+4∗Mcℏ0,ℏ1,t/33fℏn+5,ℏn+6fℏn+2,ℏn+2mfℏn+4,ℏn+2mkn+5∗Mcℏ0,ℏ1,t/34fℏn+6,ℏn+7fℏn+2,ℏn+2mfℏn+4,ℏn+2mfℏn+6,ℏn+2mkn+6∗Mcℏ0,ℏ1,t/34fℏn+7,ℏn+8fℏn+2,ℏn+2mfℏn+4,ℏn+2mfℏn+6,ℏn+2mkn+7⋯∗Mcℏ0,ℏ1,t/3mfℏn+2m−4,ℏn+2m−3fℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2mkn+2m−2∗Mcℏ0,ℏ1,t/3mfℏn+2m−3,ℏn+2m−2fℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2mk2m−1∗Mcℏ0,ℏ1,t/3mfℏn+2m−2,ℏn+2mfℏn+2m−4,ℏn+2m⋯fℏn+2,ℏn+2mk2m−2.
Using (63) for each case, we obtain
(72)limn⟶∞Mcℏn,ℏn+p,t=1∗1∗⋯∗1=1,which shows ℏn is a Cauchy sequence in F, as F is complete, so there exists ℏ∈F such that
(73)limn⟶∞Mcℏn,ℏ,t=1.
Now, to proveℏ is a fixed point of T. From (64),
(74)Mcℏ,Tℏ,t≥Mcℏ,ℏn,t/3fℏ,ℏn∗Mcℏn,ℏn+1,t/3fℏn,ℏn+1∗Mcℏn+1,Tℏ,t/3fℏn+1,Tℏ≥Mcℏ,ℏn,t/3fℏ,ℏn∗McTℏn−1,Tℏn,t/3fℏn,ℏn+1∗McTℏn,Tℏ,t/3fℏn+1,Tℏ≥Mcℏ,ℏn,t/3fℏ,ℏn∗Mcℏn−1,ℏn,t/3fℏn,ℏn+1k∗Mcℏn,ℏ,t/3fℏn+1,Tℏk⟶1∗1∗1=1,
asn⟶∞ which shows ℏ is a fixed point of T. For uniqueness, we assume T has ℏ′ as an other fixed point, then
(75)Mcℏ,ℏ′,t=McTℏ,Tℏ′,t≥Mcℏ,ℏ′,tk,=McTℏ,Tℏ′,tk≥Mcℏ,ℏ′,tk2≥⋯≥Mcℏ,ℏ′,tkn,⟶1,
asn⟶∞. Hence ℏ=ℏ′, so T has a unique fixed point.Example 7.
LetI=0,1 and T:I⟶I a mapping given by
(76)Tℏ=kℏ2.
DefineMc:I×I×0,∞⟶0,1 by
(77)Mcℏ,ℏ′,t=e−ℏ+ℏ′2/2,forallt>0.
Note thatϜ,Mc,∗ is a complete fuzzy controlled rectangular metric-like space and
(78)McTℏ,Ty,kt=Mckℏ2,kℏ′2,kt=e−kℏ2+kℏ′22/kt=e−kℏ+ℏ′2/4kt=e−ℏ+ℏ′2/4t≥e−ℏ+ℏ′2/tforallk∈0,1=Mcℏ,ℏ′,t.
Thus, all the conditions of Theorem16 are satisfied. Moreover, ℏ=0 is a unique fixed point of T.


## 4. Application to Dynamic Market Equilibrium

Due to its vast applications in many fields, the fixed point theory has been used to prove the uniqueness of the solutions of many problems. This section is devoted to prove the unique solution of a differential equation appearing in dynamic market equilibrium.LetQs and Qd denote the supply and demand of a certain item, respectively, affected by price and trends. The economist wants to know the current price Pt of the item which is falling or rising at a decreasing or increasing rate. Let dPt/dt and d2Pt/dt2 denote, respectively, the first and second derivatives of the price Pt, and assume
(79)Qs=g1+u1Pt+e1dPtdt+c1d2Ptdt2,Qd=g2+u2Pt+e2dPtdt+c2d2Ptdt2,whereg1,g2,c1,c2,e1, and e2 are constants. In equilibrium, Qs=Qd; hence, we have
(80)g1+u1Pt+e1dPtdt+c1d2Ptdt2=g2+u2Pt+e2dPtdt+c2d2Ptdt2,which implies
(81)c1−c2d2Ptdt2+e1−e2dPtdt+u1−u2Pt=−g1−g2.Puttingc=c1−c2,e=e1−e2,u=u1−u2, and g=g1−g2 in (81), we have
(82)cd2Ptdt2+edPtdt+uPt=−g.On dividing byc, we have the following initial value problem:
(83)P′′+ecP′+ucPt=−gc,withP0=0,P′0=0. Note that equation (83) is equivalent to the following integral equation:
(84)Pt=∫0Tρt,rFt,r,Pr,whereρt,r is Green’s function defined by
(85)ρt,r=reμ/2t−r,if0≤r≤t≤T,teμ/2t−r,if0≤t≤r≤T.We will prove the uniqueness of the solution of the following integral equation:(86)Pt=∫0TGt,r,Prdr.DenoteF=C0,T, the space of all real-valued continuous functions defined over the interval 0,T. Now, define a complete fuzzy controlled metric-like space with product t-norm as
(87)Mcℏt,ℏ′t,s=e−supt∈0,Tℏt+ℏ′t2/s,with controlled functions fℏ1,ℏ2=1/3ℏ2+ℏ2+3,fℏ2,ℏ3=1/3ℏ22+ℏ32+4, and fℏ3,ℏ4=1/3ℏ3+ℏ4+5.Theorem 17.
Consider the operator(88)ΓPt=∫0TGt,r,Prdr.Assume the following conditions hold:
(89)G:0,T×0,T⟶ℝ+iscontinuous,Gt,r,Pr+Gt,r,Qr2≤f2t,rPr+Qr2,where
(90)supt∈0,T∫0Tf2t,rdr≤k<1.
Then, integral equation (86) has a unique solution.Proof.
LetP,Q∈Ϝ and s>0 and consider
(91)McΓPt,ΓQt,ks=e−supt∈0,TΓPt+ΓQt2/ks=e−supt∈0,T∫0TGt,r,Prdr+∫0TGt,r,Qrdr2/ks≥e−supt∈0,T∫0TGt,r,Prdr+Gt,r,Qr2dr/ks≥e−Pr+Qr2supt∈0,T∫0Tf2t,rdr/ks≥e−kPr+Qr2/ks=e−Pr+Qr2/s≥e−supr∈0,TPr+Qr2/s=McP,Q,s.By the application of Theorem 16, problem (86) has a unique solution.


## 5. Conclusion

In this article, the concepts of extendedb-rectangular and fuzzy controlled rectangular metric-like spaces are given that extend numerous fuzzy metric-like spaces. We proved that these classes of fuzzy metric-like spaces are not Hausdorff, and we have given examples to support our main results and definitions; also, we proved the Banach fixed point theorem in these spaces by using different types of contractions. We apply our results to prove the uniqueness of the solution of an integral equation appearing in dynamic market equilibrium.



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