# Fuzzy Triple Controlled Metric Spaces and Related Fixed Point Results

Authors: Salman Furqan; Hüseyin Işık; Naeem Saleem
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this study, we introduce fuzzy triple controlled metric space that generalizes certain fuzzy metric spaces, like fuzzy rectangular metric space, fuzzy rectangularb-metric space, fuzzy b-metric space, and extended fuzzy b-metric space. We use f,g,h, three noncomparable functions as follows: mqμ,η,t+s+w≥mqμ,ν,t/fμ,ν∗mqν,ξ,s/gν,ξ∗mqξ,η,w/hξ,η. We prove Banach fixed point theorem in the settings of fuzzy triple controlled metric space that generalizes Banach fixed point theorem for aforementioned spaces. An example is presented to support our main results. We also apply our technique to the uniqueness for the solution of an integral equation.


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## Body

## 1. Introduction and Preliminaries

Banach contraction mapping principle (BCMP) [1] has many applications in various scientific fields ([2–6]). BCMP was proved in 1922 and has been investigated by many researchers in different ways ([7–11]). In 1965, Zadeh [12] defined a fuzzy set that generalizes the definition of a crisp set by associating all elements with membership values between the interval 0,1. Since then, the fuzzy set theory has been used extensively in mathematics ([13–18]) and many other areas ([19–21]). The definition of fuzzy metric space was given by Kramosil and Michálek [22] in 1975. The fuzzy version of BCMP was proved by Grabiec [23]. He also extended the Edelstein theorem to fuzzy metric spaces. George and Veeramani [24] introduced the Hausdorff topology in fuzzy metric space and modified the definition of fuzzy metric space given in [22]. In 1989, Bakhtin [25] introduced the notion of a b-metric space that generalizes the definition of classical metric space. Branciari [26] introduced the rectangular metric space and proved some fixed point results. Roshan et al. [27] introduced the notion of b-rectangular metric space that generalizes the definition of a rectangular metric space.In [28], Nădăban utilized fuzzy sets in b-metric spaces and introduced the notion of a fuzzy b-metric space. He also discussed the topological properties of this new space. Kamran et al. [29] further generalized the definition of [25] by introducing the idea of extended b-metric space while in [30], Mehmood et al. applied the fuzzy sets to the definition in [29] by introducing the notion of an extended fuzzy b-metric space and proved BCMP on this space. In [31], authors defined fuzzy version of rectangular b-metric space while in [32], Asim et al. introduced the concept of extended rectangular b-metric space and proved related fixed point theorem. Recently, Saleem et al. [33] introduced the notion of fuzzy double controlled metric space and proved BCMP on such space. Abdeljawad et al. [34] modified the definition of controlled metric type space defined in [35] by giving the idea of a double controlled metric type space.Definition 1 ([36]).
Let∗:0,1×0,1⟶0,1 be a binary operation, then ∗ is said to be continuous triangular norm (in short, continuous t-norm), if for all x1,x2,x3,x4∈0,1, the following conditions are satisfied:
∗1∗x1,x2=∗x2,x1;
∗2∗x1,∗x2,x3=∗∗x1,x2,x3;
∗3∗ is continuous;
∗4∗x,1=x for every x∈0,1;
∗5∗x1,x2≤∗x3,x4 whenever x1≤x3,x2≤x4.Definition 2 ([28]).
LetF be a nonempty set, K≥1 a real number, ∗ a continuous t-norm, and m be a fuzzy set on F×F×0,∞. Then, m is said to be a fuzzy b-metric on F, if for all μ,ν,η∈F, m satisfies the following:
bm1mμ,ν,0=0 for t=0;
bm2mμ,ν,t=1 for all t>0, iff μ=ν;
bm3mμ,ν,t=mν,μ,t;
bm4mμ,η,Kt+s≥mμ,ν,t∗mν,η,s for all s,t>0;
bm5mμ,ν,·:0,∞⟶0,1 is left continuous and limt⟶∞mμ,ν,t=1.The quadrupleF,m,∗,K is called fuzzy b-metric space.Definition 3 ([30]).
LetF be a nonempty set,f:F×F⟶1,∞, ∗ a continuous t-norm, and mf is a fuzzy set on F×F×0,∞. Then, mf is extended fuzzy b-metric if for all μ,ν,η∈F, the following conditions are satisfied:
mf1mfμ,ν,0=0 for t=0;
mf2mfμ,ν,t=1 for all t>0, iff μ=ν;
mf3mfμ,ν,t=mfν,μ,t;
mf4mfμ,η,fμ,ηt+s≥mfμ,ν,t∗mfν,η,s for all s,t>0;
mf5mfμ,ν,·:0,∞⟶0,1 is left continuous.Then,F,mf,∗,f is known as an extended fuzzy b-metric space.Definition 4 ([37]).
LetF be a nonempty set, ∗ a continuous t-norm, and mr is a fuzzy set on F×F×0,∞. Then, mr is called a fuzzy rectangular metric if for any μ,ν∈F and all distinct points u,η∈F\μ,ν, the following conditions are satisfied:
mr1mrμ,ν,0=0 for t=0;
mr2mrμ,ν,t=1 for all t>0 iff μ=ν;
mr3mrμ,ν,t=mrν,μ,t;
mr4mrμ,ν,t+s+w≥mrμ,ξ,t∗mrξ,η,s∗mrη,ν,w for all t,s,w>0;
mr5mrμ,ν,·:0,∞⟶0,1 is left continuous, and limt⟶∞mrμ,ν,t=1.
Then,F,mr,∗ is called a fuzzy rectangular metric space.Definition 5 ([31]).
LetF be a nonempty set, b≥1 a real number, ∗ a continuous t-norm, and mrb be a fuzzy set on F×F×0,∞. Then, mrb is called fuzzy rectangular b-metric if for any μ,ν∈F and all distinct points ξ,η∈F\μ,ν, the following conditions are satisfied:
mrb1mrbμ,ν,0=0;
mrb2mrbμ,ν,t=1 for all t>0 iff μ=ν;
mrb3mrbμ,ν,t=mrbν,μ,t;
mrb4mrbμ,ν,bt+s+w≥mrbμ,ξ,t∗mrbξ,η,s∗mrbη,ν,w for all t,s,w>0;
mrb5mrbμ,ν,·:0,∞⟶0,1 is left continuous, and limt⟶∞mrbμ,ν,t=1.
Then,F,mrb,∗ is known as fuzzy rectangular b-metric space.Definition 6 ([34]).
Letf,g:F×F⟶1,∞ be two noncomparable functions, if for all μ,ν,η∈F,q:F×F⟶0,∞ satisfies the following:
qfg1qμ,ν=0 iff μ=ν;
qfg2qμ,ν=qν,μ;
qfg3qμ,ν≤fμ,ηqμ,η+gη,νqη,ν.
Then,F,q is known as double controlled metric type space.


## 2. Main Results

We first give the definition of a fuzzy triple controlled metric space as follows.Definition 7.
Letf,g,h:F×F⟶1,∞ be three noncomparable functions, ∗ a continuous t-norm, and mq is a fuzzy set on F×F×0,∞. Then, mq is called fuzzy triple controlled metric if for any μ,ν∈F and all distinct ξ,η∈F\μ,ν, the following conditions are satisfied:
mq1mqμ,ν,t>0;
mq2mqμ,ν,t=1 for all t>0 iff μ=ν;
mq3mqμ,ν,t=mqν,μ,t;
mq4mqμ,ν,t+s+w≥mq(μ,ξ,t/fμ,ξ∗mqξ,η,s/gξ,η∗mqη,ν,w/hη,ν, for all t,s,w>0;
mq5mqμ,ν,·:0,∞⟶0,1 is continuous.Then,F,mq,∗ is called a fuzzy triple controlled metric space.Remark 8.
(i)
Takingfμ,ξ=gξ,η=hη,ν=1 in (mq4), then we get the definition of fuzzy rectangular metric space [37].(ii)
Takingfμ,ξ=gξ,η=hη,ν=b≥1 in (mq4), then our definition reduces to the definition of fuzzy rectangular b-metric space [31].(iii)
Takingη=ν and s+w=t′ in (mq4), then our definition reduces to the definition of fuzzy double controlled metric space defined in [33].(iv)
Takingη=ν, fμ,ξ=gξ,η and s+w=t′ in (mq4), then our definition reduces to the definition of extended fuzzy b-metric space defined in [30].(v)
Takingη=ν, fμ,ξ=gξ,η=b≥1 and s+w=t′ in (mq4), then our definition reduces to the definition of fuzzy b-metric space defined in [28].The following example justifies Definition7.Example 1.
ConsiderF=1,2,3,4 and let f,g,h:F×F⟶1,∞ be three noncomparable functions defined as fμ,ν=1+μ+ν,gμ,ν=μ2+ν+1 and hμ,ν=μ2+ν2−1. Then, mq:F×F×0,∞⟶0,1 defined by
(1)mqμ,ν,t=minμ,ν+tmaxμ,ν+t,with product t-norm, that is, t1∗t2=t1t2, F,mq,∗ is a fuzzy triple controlled metric space.
Now,(2)f1,2=f2,1=4,f1,3=f3,1=5,f1,4=f4,1=6,f2,3=f3,2=6,f2,4=f4,2=7,f3,4=f4,3=8,f1,1=3,f2,2=5,f3,3=7,f4,4=9,g1,1=3,g1,2=4,g1,3=5,g1,4=6,g2,1=6,g2,2=7,g2,3=8,g2,4=9,g3,1=11,g3,2=12,g3,3=13,g3,4=14,g4,1=18,g4,2=19,g4,3=20,g4,4=21,h1,2=h2,1=4,h1,3=h3,1=9,h1,4=h4,1=16,h2,3=h3,2=12,h2,4=h4,2=19,h3,4=h4,3=24,h1,1=1,h2,2=7,h3,3=17,h4,4=31.
Axioms (mq1) to (mq3) and (mq5) can easily be verified, and we check only (mq4).
Letμ=1,η=4, then either ν=2 and ξ=3 or ν=3 and ξ=2. We prove for ν=2 and ξ=3. The proof for ν=3 and ξ=2 is similar.
(3)mq1,4,t+s+w=min1,4+t+s+wmax1,4+t+s+w=1+t+s+w4+t+s+w.
Now,(4)mq1,2,tf1,2=min1,2+t/f1,2max1,2+t/f1,2=1+t/42+t/4=4+t8+t,mq2,3,sg2,3=min2,3+s/g2,3max2,3+s/g2,3=2+s/83+s/8=16+s24+s,mq3,4,wh3,4=min3,4+w/h3,4max3,4+w/h3,4=3+w/244+w/24=72+w92+w.
Clearly,(5)mq1,4,t+s+w≥mq1,2,tf1,2∗mq2,3,sg2,3∗mq3,4,wh3,4.
Working like same steps, remaining cases can easily be proved. Hence,F,mq,∗ is a fuzzy triple controlled metric space.Remark 9.
(i) In Example1, F,mq,∗ is not a fuzzy triple controlled metric space, if we use minimum t-norm, i.e., t1∗t2=mint1,t2 instead of product t-norm.
Next, we define the convergence of a sequence as well as Cauchy sequence in the context of fuzzy triple controlled metric space.Definition 10.
Consider a fuzzy triple controlled metric spaceF,mq,∗. Then, a sequence μn in F is said to be
(1)
convergent, if for allt>0, there exists μ in F such that(6)limn⟶∞mqμn,μ,t=1,(2)
Cauchy iff for allt>0 and for each ε>0,t>0, there exists n0∈ℕ such that(7)mqμn,μm,t≥1−ε,forallm,n≥n0.
F,mq,∗ is called complete fuzzy triple controlled metric space, if every Cauchy sequence in F converges to some point μ in F.Definition 11.
LetF,mq,∗ be a fuzzy triple controlled metric space. Then, an open ball Bx,r,t, with center x, radius r,r∈0,1, and t>0, is given by
(8)Bx,r,t=y∈F:mqx,y,t>1−r,and the corresponding topology is defined as
(9)τmq=S⊂F:Bx,r,t⊂S.
Next example shows that a fuzzy triple controlled metric space is not Hausdorff.Example 2.
Consider the fuzzy triple controlled metric space as given in Example1. Then, the open ball centered at 1, radius 0.3, and t=5 is given by
(10)B1,0.3,5=y∈F:mq1,y,5>0.7.
Now,(11)mq1,2,5=1+52+5=67=0.8571,mq1,3,5=1+53+5=68=0.75,mq1,4,5=1+54+5=69=0.6666.
Thus,B1,0.3,5=2,3. Now, consider the open ball B2,0.2,7 with centered at 2, radius r=0.2, and t=7. Then,
(12)B2,0.2,7=y∈F:mq2,y,7>0.8,mq2,1,7=1+72+7=89=0.8888,mq2,3,7=2+73+7=910=0.9,mq2,4,7=2+74+7=911=0.8181.
Thus,B2,0.2,7=1,3,4 and so B1,0.3,5∩B2,0.2,7=2,3∩1,3,4=3≠∅. Hence, fuzzy triple controlled metric space F,mq,∗ is not Hausdorff.Theorem 12.
Letf,g,h:F×F⟶1,1/k be three noncomparable functions k∈0,1 and F,mq,∗ be a complete fuzzy triple controlled metric space such that
(13)limt⟶∞mqμ,ν,t=1.
LetT:F⟶F be a given mapping such that
(14)mqTμ,Tν,kt≥mqμ,ν,t,forallμ,ν∈F.
Then,T has a unique fixed point.Proof.
Choosea0 an arbitrary point in F. If Ta0=a0, then we are done. If Ta0≠a0 then Ta0=a1∈F. Continuing in this way, we have
(15)Ta1=TTa0=T2a0=a2,Ta2=TTa1=T2a1=T2Ta0=T3a0=a3,and so on,
(16)Tan=TTan−1=T2an−1=⋯=Tn+1a0=an+1.
So, we have iterative sequencean=Tan−1=Tna0. Applying (14) successively, we get
(17)mqan,an+1,t=mqTan−1,Tan,t≥mqan−1,an,tk=mqTan−2,Tan−1,tk≥mqan−2,an−1,tk2≥mqan−3,an−2,tk3≥⋯≥mqa0,a1,tkn.Now, writingt=t/3+t/3+t/3 and for any sequence an, using rectangular property, we have the following cases:Case 1.
Whenp=2m+1, i.e., p is odd, then
(18)mqan,an+2m+1,t≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+2m+1,t/3han+2,an+2m+1≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+3,t/32fan+2,an+3han+2,an+2m+1∗mqan+3,an+4,t/32gan+3,an+4han+2,an+2m+1∗mqan+4,an+2m+1,t/32han+4,an+2m+1han+2,an+2m+1≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+3,t/32fan+2,an+3han+2,an+2m+1∗mqan+3,an+4,t/32gan+3,an+4han+2,an+2m+1∗mqan+4,an+5,t/33fan+4,an+5han+2,an+2m+1han+4,an+2m+1∗mqan+5,an+6,t/33gan+5,an+6han+2,an+2m+1han+4,an+2m+1∗mqan+6,an+7,t/34fan+6,an+7han+2,an+2m+1han+4,an+2m+1han+6,an+2m+1∗mqan+7,an+8,t/34gan+7,an+8han+2,an+2m+1han+4,an+2m+1han+6,an+2m+1⋮∗mqan+2m−2,an+2m−1,t/3mfan+2m−2,an+2m−1han+2m−2,an+2m+1⋯han+2,an+2m+1∗mqan+2m−1,an+2m,t/3mgan+2m−1,an+2mhan+2m−2,an+2m+1⋯han+2,an+2m+1∗mqan+2m,an+2m+1,t/3mhan+2m,an+2m+1han+2m−2,an+2m+1⋯han+2,an+2m+1.
Applying (17) on right hand side, we deduce
(19)mqan,an+2m+1,t≥mqa0,a1,t/3fan,an+1kn∗mqa0,a1,t/3gan+1,an+2kn+1∗mqa0,a1,t/32fan+2,an+3han+2,an+2m+1kn+2∗mqa0,a1,t/32gan+3,an+4han+2,an+2m+1kn+3∗mqa0,a1,t/33fan+4,an+5han+2,an+2m+1han+4,an+2m+1kn+4∗mqa0,a1,t/33gan+5,an+6han+2,an+2m+1han+4,an+2m+1kn+5∗mqa0,a1,t/34fan+6,an+7han+2,an+2m+1han+4,an+2m+1han+6,an+2m+1kn+6∗mqa0,a1,t/34gan+7,an+8han+2,an+2m+1han+4,an+2m+1han+6,an+2m+1kn+7⋮∗mqa0,a1,t/3mfan+2m−2,an+2m−1han+2m−2,an+2m+1⋯han+2,an+2m+1kn+2m−2∗mqa0,a1,t/3mgan+2m−1,an+2mhan+2m−2,an+2m+1⋯han+2,an+2m+1k2m−1∗mqa0,a1,t/3mhan+2m,an+2m+1han+2m−2,an+2m+1⋯han+2,an+2m+1k2m.Case 2.
Whenp=2m, i.e., p is even, then
(20)mqan,an+2m,t≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+2m,t/3han+2,an+2m≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+3,t/32fan+2,an+3han+2,an+2m∗mqan+3,an+4,t/32gan+3,an+4han+2,an+2m∗mqan+4,an+2m,t/32han+4,an+2mhan+2,an+2m≥mqan,an+1,t/3fan,an+1∗mqan+1,an+2,t/3gan+1,an+2∗mqan+2,an+3,t/32fan+2,an+3han+2,an+2m∗mqan+3,an+4,t/32gan+3,an+4han+2,an+2m∗mqan+4,an+5,t/33fan+4,an+5han+2,an+2mhan+4,an+2m∗mqan+5,an+6,t/33gan+5,an+6han+2,an+2mhan+4,an+2m∗mqan+6,an+7,t/34fan+6,an+7han+2,an+2mhan+4,an+2mhan+6,an+2m∗mqan+7,an+8,t/34gan+7,an+8han+2,an+2mhan+4,an+2mhan+6,an+2m⋮∗mqan+2m−4,an+2m−3,t/3m−1fan+2m−4,an+2m−3han+2m−4,an+2m⋯han+2,an+2m∗mqan+2m−3,an+2m−2,t/3m−1gan+2m−3,an+2m−2han+2m−4,an+2m⋯han+2,an+2m∗mqan+2m−2,an+2m,t/3m−1han+2m−2,an+2mhan+2m−4,an+2m+1⋯han+2,an+2m.
Now applying (17), we deduce
(21)mqan,an+2m+1,t≥mqa0,a1,t/3fan,an+1kn∗mqa0,a1,t/3gan+1,an+2kn+1∗mqa0,a1,t/3/32fan+2,an+3han+2,an+2mkn+2∗mqa0,a1,t/3/32gan+3,an+4han+2,an+2mkn+3∗mqa0,a1,t/33fan+4,an+5han+2,an+2mhan+4,an+2m+1kn+4∗mqa0,a1,t/33gan+5,an+6han+2,an+2mhan+4,an+2m+1kn+5∗mqa0,a1,t/34fan+6,an+7han+2,an+2mhan+4,an+2m+1han+6,an+2mkn+6∗mqa0,a1,t/34gan+7,an+8han+2,an+2mhan+4,an+2mhan+6,an+2mkn+7⋮∗mqa0,a1,t/3m−1fan+2m−3,an+2m−2han+2m−4,an+2m⋯han+2,an+2mkn+2m−4∗mqa0,a1,t/3m−1gan+2m−1,an+2mhan+2m−2,an+2m+1⋯han+2,an+2m+1kn+2m−3∗mqa0,a1,t/3m−1han+2m−2,an+2mhan+2m−4,an+2m⋯han+2,an+2mkn+2m−2.
Using (13) for each case, we obtain
(22)limn⟶∞mqan,an+p,t=1∗1∗⋯∗1=1,which shows that an is a Cauchy sequence in F and converges to some a∈F (as F is complete), i.e.,
(23)limn⟶∞mqan,a,t=1.Now, we show thata is the fixed point of T. From (14),
(24)mqa,Ta,t≥mqa,an,t/3fa,an∗mqan,an+1,t/3gan,an+1∗mqan+1,Ta,t/3han+1,Ta≥mqa,an,t/3fa,an∗mqTan−1,Tan,t/3gan,an+1∗mqTan,Ta,t/3han+1,Ta≥mqa,an,t/3fa,an∗mqan−1,an,t/3gan,an+1k∗mqan,a,t/3han+1,Tak⟶1∗1∗1=1asn⟶∞,which shows that Ta=a, i.e., a is a fixed point of T. To prove a is unique, we assume a′ is also a fixed point of T, i.e., Ta′=a′. Then,
(25)mqa,a′,t=mqTa,Ta′,t≥mqa,a′,tk,which implies that a=a′, so the fixed point of T is unique.Example 3.
LetF=0,1 and mq:F×F×0,∞⟶0,1 be defined as mqμ,ν,t=e−∣μ−ν∣/t for all t>0. Let f,g,h:F×F⟶1,1/k be defined by fμ,ν=μ+ν+1,gν,ξ=ν2+ξ+1 and hξ,η=ξ2+η2+1. Then, F,mq,∗ is a complete fuzzy triple controlled metric space. Let T:F⟶F be given by Tμ=1−μ/4. Then,
(26)mqTμ,Tν,kt=mq1−μ4,1−ν4,kt=e−∣1−μ/4−1+ν/4∣/kt=e−μ/4−ν/4/kt≥e−μ−ν/4kt≥e−μ−ν/t=mqμ,ν,t,for all μ,ν∈F, where k∈13/50,1. Since all the conditions of Theorem 12 are satisfied, therefore T has a unique fixed point which is μ=4/5.Theorem12 generalizes Theorem 2.1 of [31] as follows.Corollary 13.
LetF,mq,∗ be a complete fuzzy rectangular b-metric space with b≥1 such that
(27)limt⟶∞mqμ,ν,t=1.
LetT:F⟶F be a mapping such that
(28)mqTμ,Tν,kt≥mqμ,ν,t,forallμ,ν∈F.
Then,T has a unique fixed point.Remark 14.
If we choosefμ,ν=gμ,ν=hμ,ν=1, then our main result reduces to the Banach contraction principle for the fuzzy rectangular metric space defined in [37].


## 3. Application

Consider the integral equation(29)μξ=gξ+∫0ξGξ,γ,μγdγ,where ξ∈I=0,1.Definefμ,ν=3μ+ν+1,gν,ξ=3μ2+ν2+1, and hξ,η=3μ2+ν2−1. Also, for all t>0, μ,ν∈CI,ℝ,(30)mqμ,ν,t=e−supξ∈Iμξ−νξ2/t.Then,CI,ℝ,mq is a complete fuzzy triple controlled metric space with product t-norm, where CI,ℝ is the space of all real valued continuous functions defined on I.Theorem 15.
Consider an integral operator defined onCI,ℝ as follows:
(31)Tμξ=gξ+∫0ξGξ,γ,μγdγ,g∈CI,ℝ,where G satisfies the following condition.
There existsf:I×I⟶0,∞ such that for all ξ,γ∈I,fξ,γ∈L1I,ℝ, and for all μ,ν∈CI,ℝ, we have
(32)Gξ,γ,μγ−Gξ,γ,νγ∣2≤f2ξ,γμγ−νγ2,where
(33)0<supξ∈I∫0ξf2ξ,γdγ≤k<1.
Then, the integral equation (29) has a unique solution.Proof.
Letμ,ν∈CI,ℝ. Note that
(34)mqTμ,Tν,kt=e−supξ∈ITμξ−Tνξ2/kt=e−supξ∈I∫0ξGξ,γ,μγ−Gξ,γ,νγdγ2/kt≥e−supξ∈I∫0ξGξ,γ,μγ−Gξ,γ,νγ2dγ/kt≥e−supξ∈I∫0ξf2ξ,γμγ−νγ2dγ/kt≥e−μγ−νγ2supξ∈I∫0ξf2ξ,γdγ/kt≥e−kμγ−νγ2/kt=e−μγ−νγ2/t≥e−supγ∈Iμγ−νγ2/t=mqμ,ν,t.
Thus,mqTμ,Tν,kt≥mqμ,ν,t for all μ,ν∈CI,ℝ and consequently, all the conditions of Theorem 12 are satisfied. Therefore, the integral equation (29) has a unique solution.


## 4. Conclusion

In this article, the concept of fuzzy triple controlled metric space is given which generalizes fuzzy rectangularb-metric space, rectangular fuzzy metric space. We have established Banach fixed point theorem in this space. An example is also presented that illustrates our main result. Our newly defined results can be used for further investigation in many existing results in the literature.



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*Source: 9936992-2021-05-20.xml*