# Intuitionistic Fuzzy Double Controlled Metric Spaces and Related Results

Authors: Misbah Farheen; Khalil Ahmed; Khalil Javed; Vahid Parvaneh; Fahim Ud Din; Umar Ishtiaq
Journal: Security and Communication Networks
 (2022)

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## Abstract

In this article, we introduce the concept of intuitionistic fuzzy double controlled metric spaces that generalizes the concept of intuitionistic fuzzy b-metric spaces. For this purpose, two noncomparable functions are used in triangle inequalities. We generalize the concepts of the Banach contraction principle and fuzzy contractive mappings by giving authentic proof of these mappings in the sense of intuitionistic fuzzy double controlled metric spaces. To validate the superiority of these results, examples are imparted. A possible application to solving integral equations is also set forth towards the end of this work to support the proposed results.


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## Body

## 1. Introduction

The concept of metric spaces and the Banach contraction principle are the backbone of the field of fixed point theory. Since the axiomatic interpretations of metric space, it has attracted researchers due to its spaciousness. So far, different developments in metric space have appeared in the literature, either by improving contraction conditions or by relaxing the axioms of metric space.Zadeh [1] was the first to put forward the concept of fuzzy sets and this idea has deeply influenced many scientific fields since its inception. Using the concepts of probabilistic metric space and fuzzy sets, fuzzy metric space was introduced in [2]. Afterward, the utility of FMS appeared in applied sciences such as fixed-point theory, image and signal processing, medical imaging, and decision-making. This concept succeeded in shifting a lot of mathematical structures within itself. In this continuation, Kramosil and Michalek [3] initiated the notion of fuzzy metric spaces. Khalil et al. [4] generalized this concept by introducing fuzzy b-metric-like spaces. Fuzzy metric space only discusses membership functions, so for dealing with membership and nonmembership functions, the notion of intuitionistic fuzzy metric spaces introduced by Park [5] and this concept was generalized into intuitionistic fuzzy b-metric spaces by Konwar [6]. In this connectedness, many important results appeared in the literature, such as fixed point theorems on intuitionistic fuzzy metric space [7], fixed point theorems for a generalized intuitionistic fuzzy contraction in intuitionistic fuzzy metric spaces [8], extension of fixed point results in intuitionistic fuzzy b-metric spaces [6], fixed points in intuitionistic fuzzy metric spaces [4], fuzzy fixed point [9], and some more work in generalized metric space in [10], ordered defined in fuzzy b-metric [11], partial metric defining the relation in [12], orthogonal neutrosophic metric space [13], and orthogonal partial metric space [14]. More details related to generalized metric spaces can be seen in [15].Recently, Saleem et al. [10] introduced the notion of fuzzy double controlled metric spaces and generalized the Banach contraction principle. This result was generalized in different spaces and different structures were attained using this topic. One may recall some basic results related to this topic, such as controlled fuzzy metric spaces and some related fixed point results in [16], controlled metric type spaces, and the related contraction principle in [17] and orthogonally controlled metric type spaces in [18], and very recently, the new aspects of metric spaces in [19].In this article, we aim to generalize the concept of intuitionistic fuzzy b-metric spaces and introduce the concept of intuitionistic fuzzy double controlled metric spaces. Some nontrivial examples are given and an application to solving integral equations is also imparted in this work. Table1 of abbreviations of notions will be used throughout this study.Table 1
Abbreviations.
AbbreviationsMeaningsFSsFuzzy setsFMSsFuzzy metric spacesIFBMSsIntuitionistic fuzzy b-metric spacesFDCMSsFuzzy double controlled metric spacesIFDCMSsIntuitionistic fuzzy double controlled metric spacesCTNContinuous triangle normCTCNContinuous triangle conorm


### 1.1. Preliminaries

First, we define some necessary definitions that are helpful for readers to understand the main results.Definition 1 (see [2]).
A binary operation∗: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTN if(1)
π∗µ=µ∗π,∀π,µ∈0,1;(2)
∗ is continuous;(3)
π∗1=π,∀π∈0,1;(4)
π∗µ∗ρ=π∗µ∗ρ,∀π,µ,ρ∈0,1;(5)
π≤ρ and µ≤d, with π,µ,ρ,d∈0,1, then π∗µ≤ρ∗d.Example 1 (see [20]).
Some fundamental examples of CTNs areπ∗µ=π·µ,π∗µ=minπ,µ and π∗µ=maxπ+µ−1,0.Definition 2 (see [2]).
A binary operation○: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTCN if it meets the following assertions:(1)
π○µ=µ○π,or allπ,µ∈0,1(2)
○ is continuous;(3)
π○0=0,for allπ∈0,1;(4)
π○µ○ρ=π○µ○ρ,for allπ,µ,ρ∈0,1;(5)
Ifπ≤ρ and µ≤d, with π,µ,ρ,d∈0,1, then π○µ≤ρ○d.Example 2 (see [2]).
π○µ=maxπ,µandπ○µ=minπ+µ,1 are examples of CTCNs.Definition 3 (see [21]).
Let functionsф,η:B×B⟶1,∞ be noncomparable. Let ∂:B×B⟶0,∞ be fulfilling:(a)
∂x,y=0ifx=y;(b)
∂x,y=∂y,x;(c)
∂x,y≤фx,z∂x,z+ηz,y∂z,y, for all x,y,z∈B. Then, ∂ is called a double controlled metric and B,∂ is called a double controlled metric space.Definition 4 (see [21]).
LetB≠∅ and ф,η:B×B⟶1,∞ be given noncomparable functions, and ∗ is a CTN and Ρ be a FS on B×B×0,∞ which is called fuzzy double controlled metric on B; if for all x,y,z∈B, the below circumstances are fulfilling:(I)
Ρx,y,0=0;(II)
Ρx,y,t=1for allt>0,if and only ifx=y;(III)
Ρx,y,t=Ρy,x,t;(IV)
Ρx,z,t+s≥Ρx,y,t/фx,y∗Ρy,z,s/ηy,z;(V)
Ρx,y,·:0,∞⟶0,1 is left continuous.
Then,B,Ρ,Q,∗ is named a FDCMS.Definition 5 (see [6]).
TakeB≠∅. Let ∗ be a CTN, ○ be a CTCN, b≥1, and Ρ,Q be FSs on B×B×0,∞, if the following for all x,y∈Bands,t>0:(I)
Ρx,y,t+Qx,y,t≤1;(II)
Ρx,y,t>0;(III)
Ρx,y,t=1⇔x=y;(IV)
Ρx,y,t=Ρy,x,t;(V)
Ρx,z,bt+s≥Ρx,y,t∗Ρy,z,s;(VI)
Ρx,y,· is a nondecreasing function of ℝ+andlimt⟶∞Ρx,y,t=1;(VII)
Qx,y,t>0;(VIII)
Qx,y,t=0⇔x=y;(IX)
Qx,y,t=Qy,x,t;(X)
Qx,z,bt+s≤Qx,y,t○Qy,z,s;(XI)
Qx,y,· is a nonincreasing function of ℝ+ and limt⟶∞Qx,y,t=0, then B,Ρ,Q,∗,○ is an IFBMS.


## 1.1. Preliminaries

First, we define some necessary definitions that are helpful for readers to understand the main results.Definition 1 (see [2]).
A binary operation∗: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTN if(1)
π∗µ=µ∗π,∀π,µ∈0,1;(2)
∗ is continuous;(3)
π∗1=π,∀π∈0,1;(4)
π∗µ∗ρ=π∗µ∗ρ,∀π,µ,ρ∈0,1;(5)
π≤ρ and µ≤d, with π,µ,ρ,d∈0,1, then π∗µ≤ρ∗d.Example 1 (see [20]).
Some fundamental examples of CTNs areπ∗µ=π·µ,π∗µ=minπ,µ and π∗µ=maxπ+µ−1,0.Definition 2 (see [2]).
A binary operation○: [0, 1] × [0, 1] ⟶ [0, 1] is called a CTCN if it meets the following assertions:(1)
π○µ=µ○π,or allπ,µ∈0,1(2)
○ is continuous;(3)
π○0=0,for allπ∈0,1;(4)
π○µ○ρ=π○µ○ρ,for allπ,µ,ρ∈0,1;(5)
Ifπ≤ρ and µ≤d, with π,µ,ρ,d∈0,1, then π○µ≤ρ○d.Example 2 (see [2]).
π○µ=maxπ,µandπ○µ=minπ+µ,1 are examples of CTCNs.Definition 3 (see [21]).
Let functionsф,η:B×B⟶1,∞ be noncomparable. Let ∂:B×B⟶0,∞ be fulfilling:(a)
∂x,y=0ifx=y;(b)
∂x,y=∂y,x;(c)
∂x,y≤фx,z∂x,z+ηz,y∂z,y, for all x,y,z∈B. Then, ∂ is called a double controlled metric and B,∂ is called a double controlled metric space.Definition 4 (see [21]).
LetB≠∅ and ф,η:B×B⟶1,∞ be given noncomparable functions, and ∗ is a CTN and Ρ be a FS on B×B×0,∞ which is called fuzzy double controlled metric on B; if for all x,y,z∈B, the below circumstances are fulfilling:(I)
Ρx,y,0=0;(II)
Ρx,y,t=1for allt>0,if and only ifx=y;(III)
Ρx,y,t=Ρy,x,t;(IV)
Ρx,z,t+s≥Ρx,y,t/фx,y∗Ρy,z,s/ηy,z;(V)
Ρx,y,·:0,∞⟶0,1 is left continuous.
Then,B,Ρ,Q,∗ is named a FDCMS.Definition 5 (see [6]).
TakeB≠∅. Let ∗ be a CTN, ○ be a CTCN, b≥1, and Ρ,Q be FSs on B×B×0,∞, if the following for all x,y∈Bands,t>0:(I)
Ρx,y,t+Qx,y,t≤1;(II)
Ρx,y,t>0;(III)
Ρx,y,t=1⇔x=y;(IV)
Ρx,y,t=Ρy,x,t;(V)
Ρx,z,bt+s≥Ρx,y,t∗Ρy,z,s;(VI)
Ρx,y,· is a nondecreasing function of ℝ+andlimt⟶∞Ρx,y,t=1;(VII)
Qx,y,t>0;(VIII)
Qx,y,t=0⇔x=y;(IX)
Qx,y,t=Qy,x,t;(X)
Qx,z,bt+s≤Qx,y,t○Qy,z,s;(XI)
Qx,y,· is a nonincreasing function of ℝ+ and limt⟶∞Qx,y,t=0, then B,Ρ,Q,∗,○ is an IFBMS.


## 2. Main Results

In this section, we introduce the concept of IFDCMSs and prove some fixed point (FP) results.Definition 6.
LetB≠∅ and ф,η:B×B⟶1,∞ be given noncomparable functions, and ∗ is a CTN and ○ is a CTCN. Ρ,Q are FSs on B×B×0,∞ which are named intuitionistic fuzzy double controlled metrics on B; if for all x,y,z∈B, the below circumstances are fulfilling:(I)
Px,y,t+Qx,y,t≤1(II)
Ρx,y,t>0;(III)
Ρx,y,t=1for allt>0,if and only ifx=y;(IV)
Ρx,y,t=Ρy,x,t;(V)
Ρx,z,t+s≥Ρx,y,t/фx,y∗Ρy,z,s/ηy,z;(VI)
Ρx,y,·:0,∞⟶0,1 is left continuous;(VII)
Qx,y,t<1;(VIII)
Qx,y,t=0for allt>0,if and only ifx=y;(IX)
Qx,y,t=Qy,x,t;(X)
Qx,z,t+s≤Qx,y,t/фx,y○Qy,z,s/ηy,z;(XI)
Qx,y,·:0,∞⟶0,1 is left continuous.
Then,B,Ρ,Q,∗,○ is called an IFDCMS.Remark 1.
If we takeфx,y=ηy,z=b≥1, then IFDCMS becomes an IFBMS.Example 3.
LetB=1,2,3andф,η:B×B⟶1,∞ be two noncomparable functions given by фx,y=x+y+1andηx,y=x2+y2−1. Define Ρ,Q:B×B×0,∞⟶0,1 as(1)Ρx,y,t=1,ifx=y,tt+maxx,y,ifotherwise.Qx,y,t=1,ifx=y,tt+maxx,y,ifotherwise.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.Proof.
Conditions (i)–(iv), (vi)–(ix), and (xi) are easy to examine Here, we prove (v) and (x).
Letx=1,y=2,andz=3. Then,(2)Ρ1,3,t+s=t+st+s+max1,3=t+st+s+3.
On the other hand,(3)Ρ1,2,tф1,2=t/ф1,2ф1,2+max1,2=t/4t/4+2=tt+8.Ρ2,3,sη2,3=s/η2,3s/η2,3+max2,3=s/12s/12+3=ss+36.
That is,(4)t+st+s+3≥tt+8·ss+36.
Then, it satisfies for allt,s>0. Hence,(5)Ρx,z,t+s≥Ρx,y,tфx,y∗Ρy,z,sфy,z.
Now,(6)Q1,3,t+s=max1,3t+s+max1,3=3t+s+3.
On the other hand,(7)Q1,2,tη1,2=max1,2t/η1,2+max1,2=2t/4+2=8t+8.Q2,3,sη2,3=max2,3s/η2,3+max2,3=3s/12+3=36s+36.
That is,(8)3t+s+3≤max8t+8,36s+36.
Then, it satisfies for allt,s>0. Hence,(9)Qx,z,t+s≤Qx,y,tфx,y○Qy,z,sфy,z.
On the same lines, one can examine all other cases. Hence,B,Ρ,Q,∗,○ is an IFDCMS.Remark 2.
The above example also satisfied for CTNπ∗µ=minπ,µ and CTCN π○µ=maxπ,µ.Example 4.
LetB=0,∞andф,η:B×B⟶1,∞ be two noncomparable functions given by фx,y=x+y+1andηx,y=x2+y2−1.
DefineΡ,Q:B×B×0,∞⟶0,1 as(10)Ρx,y,t=tt+x−y2,Qx,y,t=x−y2t+x−y2.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.Remark 3.
The above example also holds for(11)фx,y=1,ifx=y,1+maxx,yminx,y,ifx≠y,ηx,y=1,ifx=y,1+maxx2,y2minx2,y2,ifx≠y.Remark 4.
The above example is also satisfied for CTNπ∗µ=minπ,µ and CTCN π○µ=maxπ,µ.Example 5.
LetB=1,2,3andф,η:B×B⟶1,∞ be two noncomparable functions given by фx,y=x+y+1andηx,y=x2+y2−1. Define Ρ,Q:B×B×0,∞⟶0,1 as(12)Ρx,y,t=t+minx,yt+maxx,y,Qx,y,t=1−t+minx,yt+maxx,y.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.Proof.
It is easy to examine the line of the above example.Remark 5.
The above example is not IFDCMS if we take CTNπ∗µ=minπ,µ, CTCN π○µ=maxπ,µ, and x=1,y=2,z=3,t=0.02,s=0.03withфx,y=x+y+1andηx,y=x2+y2−1.Proposition 1.
LetB=0,1 and ф,η:B×B⟶0,1 be two noncomparable functions given by фx,y=2x+y+1andηx,y=2x2+y2+1. Define Q,Ρ as(13)Ρx,y,tn=e−x−y2/tn,Qx,y,tn=1−e−x−y2/tnfor allx,y∈B,t>0.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.Remark 6.
The above proposition is also satisfied for CTNπ∗µ=minπ,µ and CTCN π○µ=maxπ,µ.Proposition 2.
LetB=0,1 and ф,η:B×B⟶0,1 be two noncomparable functions given by фx,y=2x+y+1andηx,y=2x2+y2+1. Define Q,Ρ as(14)Ρx,y,tn=e−x−y2/tn−1,Qx,y,tn=1−e−x−y2/tn−1for allx,y∈B,t>0.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.Remark 7.
The above proposition is also satisfied for CTNπ∗µ=minπ,µ and CTCN π○µ=maxπ,µ.Definition 7.
LetB,Ρ,Q,∗,○ be an IFDCMS. Then, we define an open ball Bx,r,t with center x, radius r,0<r<1, and t>0 as follows:(15)Bx,r,t=y∈B:Ρx,y,t>1−r,Qx,y,t<r.Remark 8.
An IFDCMSB,Ρ,Q,∗,○ needs not to be a Hausdorff.Proof.
LetB=1,2,3andф,η:B×B⟶1,∞ be two noncomparable functions given by фx,y=x+y+1andηx,y=x2+y2−1. Define Ρ,Q:B×B×0,∞⟶0,1 as(16)Ρx,y,t=t+minx,yt+maxx,y,Qx,y,t=1−t+minx,yt+maxx,y.
Then,B,Ρ,Q,∗,○ is an IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ. Consider the open ball B1,0.4,6 centered at 1, with radius r=0.4 and t=6. Then,(17)B1,0.4,6=y∈B:Ρ1,y,6>0.6,Q1,y,6<0.4.
Now,(18)Ρ1,2,6=1+62+6=78=0.875,Q1,2,6=1−1+62+6=1−78=1−0.875=0.125,Ρ1,3,6=1+63+6=79=0.777,Q1,3,6=1−1+63+6=1−79=1−0.777=0.223.
Thus,B1,0.4,6=2,3. Now, consider the open ball B2,0.6,12 with radius r=0.6, centered at 2, and t=12. Then,(19)B2,0.6,12=y∈B:Ρ2,y,12>0.4,Q2,y,12<0.6.
Now,(20)Ρ2,1,12=1+122+12=1314=0.928,Q2,1,12=1−1+122+12=1−1314=1−0.928=0.071,Ρ2,3,12=2+123+12=1415=0.933,Q2,3,12=1−2+123+12=1−1415=1−0.933=0.066.
Thus,B2,0.6,12=1,3. Now,(21)B1,0.4,6∩B2,0.6,12=2,3∩1,3,=3≠.
Hence, an IFDCMS is not necessarily Hausdorff.Definition 8.
LetB,Ρ,Q,∗,○ be an IFDCMS and xn be a sequence in B. Then, xn is named to be(a)
Convergent, if there existsx∈B such that(22)limn⟶∞Ρxn,x,t=1,limn⟶∞Qxn,x,t=0,for allt>0.(b)
A Cauchy sequence (CS), if and only if for eachµ>0,t>0, there exists n0∈ℕ such that(23)Ρxn,xn+q,t≥1−µ,Qxn,xn+q,t≤µ,for alln,m≥n0.
If every Cauchy sequence is convergent inB, then B,Ρ,Q,∗,○ is called a complete IFDCMS.Lemma 1.
Letxn be a Cauchy sequence in IFDCMS B,Ρ,Q,∗,○ such that xn≠xm whenever m,n∈ℕ with n≠m. Then, the sequence xn can converge to at most one limit point.Proof.
Ccontrarily, assume thatxn⟶x and xn⟶y,forx≠y. Then, limn⟶∞Ρxn,x,t=1,limn⟶∞Qxn,x,t=0andlimn⟶∞Ρxn,y,t=1,limn⟶∞Qxn,y,t=0,for allt>0. Therefore,(24)Ρx,y,t≥Ρx,xn,t2фx,xn∗Ρxn,y,t2ηxn,y⟶1∗1,asn⟶∞,Qx,y,t≤Qx,xn,t2фx,xn○Qxn,y,t2ηxn,y⟶0○0,asn⟶∞.
That is,Ρx,y,t≥1∗1=1andQx,y,t≤0○0=0. Hence, x=y, that is, the sequence converges to at most one limit point.Lemma 2.
LetB,Ρ,Q,∗,○ be an IFDCMS. If for some 0<θ<1 and for any x,y∈B,t>0,(25)Ρx,y,t≥Ρx,y,tθ,Qx,y,t≤Qx,y,tθ.then x=y.Proof.
(1) implies that(26)Ρx,y,t≥Ρx,y,tθn,Qx,y,t≤Qx,y,tθn,n∈ℕ,t>0.
Now,(27)Ρx,y,t≥limn⟶∞Ρx,y,tθn=1,Qx,y,t≤limn⟶∞Qx,y,tθn=0,t>0.
By (iii) and (viii), thenx=y.
At this time, we prove the intuitionistic fuzzy controlled Banach contraction result.Theorem 1.
Suppose thatB,Ρ,Q,∗,○ is a complete IFDCMS in the company of ф,η:B×B⟶1,1/θwith0<θ<1 and suppose that(28)limt⟶∞Ρx,y,t=1andlimt⟶∞Qx,y,t=0.for all x,y∈B, and t>0. Let Ψ:B⟶B be a mapping satisfying for all x,y∈B and t>0. Then, Ψ has a unique FP.Proof.
Letx0 be a random integer of B and describe a sequence xn by xn=Ψnx0=Ψxn−1, n∈ℕ. By using 2 for all t>0, we have(29)Ρxn,xn+1,θt=ΡΨxn−1,Ψxn,θt≥Ρxn−1,xn,t≥Ρxn−2,xn−1,tθ≥Ρxn−3,xn−2,tθ2≥⋯≥Ρx0,x1,tθn−1,Qxn,xn+1,θt=QΨxn−1,Ψxn,θt≤Qxn−1,xn,t≤Qxn−2,xn−1,tθ,≤Qxn−3,xn−2,tθ2≤⋯≤Qx0,x1,tθn−1.
We obtain(30)Ρxn,xn+1,θt≥Ρx0,x1,tθn−1,Qxn,xn+1,θt≤Qx0,x1,tθn−1.
For anyq∈ℕ,usingvandx, we deduce(31)Ρxn,xn+q,t≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+q,t2ηxn+1,xn+q≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2∗Ρфxn+2,xn+q,t22ηxn+1,xn+qηxn+2,xn+q≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2∗Ρxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3∗Ρxn+3,xn+q,t23ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2∗Ρxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3∗Ρxn+3,xn+4,t24ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4∗⋯∗Ρxn+q−2,xn+q−1,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1∗Ρxn+q−1,xn+q,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯xn+q−1,xn+q,Qxn,xn+q,t≤Qxn,xn+1,t2фxn,xn+1Qxn+1,xn+q,t2ηxn+1,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,○Qxn+2,xn+q,t22ηxn+1,xn+qηxn+2,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,○Qxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,○Qxn+3,xn+q,t23ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,ΟQxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,○Qxn+3,xn+4,t24ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4○⋯○,Qxn+q−2,xn+q−1,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,○Qxn+q−1,xn+q,t2q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q.
Using (4) in the above inequalities, we deduce(32)≥Ρx0,x1,t2θn−1фxn,xn+1∗Ρx0,x1,t22θnηxn+1,xn+qфxn+1,xn+2,∗Ρx0,x1,t23θn+1ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,∗Ρx0,x1,t24θn+2ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4∗⋯∗,Ρx0,x1,t2q−1θn+q−2ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,∗Ρx0,x1,t2q−1θn+q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q,≤Qx0,x1,t2θn−1фxn,xn+1Qx0,x1,t22θnηxn+1,xn+qфxn+1,xn+2,○Qx0,x1,t23θn+1ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,○Qx0,x1,t24θn+2xn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4,○⋯○Qx0,x1,t2q−1θn+q−2ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,○Qx0,x1,t2q−1θn+q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q.
Using (2), ifn⟶∞, we deduce(33)limn⟶∞Ρxn,xn+q,t=1∗1∗⋯∗1=1,limn⟶∞Qxn,xn+q,t=0○0○⋯○0=0,i.e., xn is a CS. Since B,Ρ,Q,∗,○ is a complete IFDCMS, there exists(34)limn⟶∞xn=x.
Now, we investigate thatx is a FP of Ψ. Using v,x,and2, we obtain(35)Ρx,Ψx,t≥Ρx,xn+1,t2фx,xn+1∗Ρxn+1,Ψx,t2ηxn+1,Ψx≥Ρx,xn+1,t2фx,xn+1∗ΡΨxn,Ψx,t2ηxn+1,Ψx≥Ρx,xn+1,t2фx,xn+1∗Ρxn,x,t2θηxn+1,Ψx⟶1∗1=1,asn⟶∞, and(36)Qx,Ψx,t≤Qx,xn+1,t2фx,xn+1○Qxn+1,Ψx,t2ηxn+1,Ψx≤Qx,xn+1,t2фx,xn+1○QΨxn,Ψx,t2ηxn+1,Ψx≤Qx,xn+1,t2фx,xn+1○Qxn,x,t2θηxn+1,Ψx⟶0○0=0,asn⟶∞. This implies that Ψx=x, a FP. Now, we show the uniqueness. Suppose that Ψρ=ρ for some ρ∈B. Then,(37)1≥Ρρ,x,t=ΡΨρ,Ψx,t≥Ρρ,x,tθ=ΡΨρ,Ψx,tθ≥Ρρ,x,tθ2≥⋯≥Ρρ,x,tθn⟶1asn⟶∞0≤Qρ,x,t=QΨρ,Ψx,t≤Qρ,x,tθ=QΨρ,Ψx,tθ≤Qρ,x,tθ2≤⋯≤Qρ,x,tθn⟶0asn⟶∞.
By usingiiiandviii,thenx=ρ.Corollary 1.
Suppose thatB,Ρ,Q,∗,○ is a complete IFDCMS in the company of ф,η:B×B⟶1,1/θwith0<θ<1 and suppose that(38)limt⟶∞Ρx,y,t=1,limt⟶∞Qx,y,t=0,for all x,y∈B and t>0. Let Ψ:B⟶B be a mapping satisfying(39)ΡΨx,Ψy,θt≥minΡx,y,t,Ρx,Ψx,t,Ρy,Ψy,t,QΨx,Ψy,θt≤minQx,y,t,Qx,Ψx,t,Qy,Ψy,t,for all x,y∈B, and t>0. Then, Ψ has a unique FP.Proof.
It is easy to prove by using Theorem1 and Lemma 2.Definition 9.
LetB,Ρ,Q,∗,○ be an IFDCMS. A map Ψ:B⟶B is a D-controlled intuitionistic fuzzy contraction if there exists 0<θ<1, such that(40)1ΡфΨx,Ψy,t−1≤θ1Ρфx,y,t−1,QфΨx,Ψy,t≤θQфx,y,t,for all x,y∈Bandt>0.
Now, we prove a theorem for D-controlled intuitionistic fuzzy contractions.Theorem 2.
LetB,Ρ,Q,∗,○ be a complete IFDCMS with ф,η:B×B⟶1,∞ and suppose that(41)limt⟶∞Ρx,y,t=1,limt⟶∞Qx,y,t=0,for all x,y∈B and t>0. Let Ψ:B⟶B be a D-controlled intuitionistic fuzzy contraction. Further, suppose that for an arbitrary x0∈B,andn,q∈ℕ,xn=Ψnx0=Ψxn−1. Then, Ψ has a unique FP.Proof.
Letx0 be a random integer of B and describe a sequence xn by xn=Ψnx0=Ψxn−1, n∈ℕ. By using (5) and (6) for all t>0,n>q, we have(42)1Ρxn,xn+1,t−1=1ΡΨxn−1,xn,t−1≤θ1Ρxn−1,xn,t−1=θΡxn−1,xn,t−θ⇒1Ρxn,xn+1,t≤θΡxn−1,xn,t+1−θ≤θ2Ρxn−2,xn−1,t+θ1−θ+1−θ.
Continuing in this way, we get(43)1Ρxn,xn+1,t≤θnΡx0,x1,t+θn−11−θ+θn−21−θ+⋯+θ1−θ+1−θ≤θnΡx0,x1,t+θn−1+θn−2+⋯+11−θ≤θnΡx0,x1,t+1−θn.
We obtain(44)1θn/Ρx0,x1,t+1−θn≤Ρxn,xn+1,t,Qxn,xn+1,t=QΨxn−1,xn,t≤θQxn−1,xn,t=QΨxn−2,xn−1,t,≤θ2Qxn−2,xn−1,t≤⋯≤θnQx0,x1,t,for any q∈ℕUsingvandx, we deduce (45)Ρxn,xn+q,t≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+q,t2ηxn+1,xn+q,≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,∗Ρxn+2,xn+q,t22ηxn+1,xn+qηxn+2,xn+q,≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,∗Ρxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,∗Ρxn+3,xn+q,t23ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q,≥Ρxn,xn+1,t2фxn,xn+1∗Ρxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,∗Ρxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,∗Ρxn+3,xn+4,t24ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4∗⋯∗,Ρxn+q−2,xn+q−1,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,∗Ρxn+q−1,xn+q,t2q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q,Qxn,xn+q,t≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+q,t2ηxn+1,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,○Qxn+2,xn+q,t22ηxn+1,xn+qηxn+2,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,○Qxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,○Qxn+3,xn+q,t23ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q,≤Qxn,xn+1,t2фxn,xn+1○Qxn+1,xn+2,t22ηxn+1,xn+qфxn+1,xn+2,○Qxn+2,xn+3,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3,○Qxn+3,xn+4,t24ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+qфxn+3,xn+4○⋯○,Qxn+q−2,xn+q−1,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,○Qxn+q−1,xn+q,t2q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q,Ρxn,xn+q,t≥1θn/Ρx0,x1,t/2фxn,xn+1+1−θn∗1θn+1/Ρx0,x1,t/22ηxn+1,xn+qфxn+1,xn+2+1−θn+1,∗1θn+2/Ρx0,x1,t/23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3+1−θn+2∗⋯∗,1θn+q−2/Ρx0,x1,t/2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1+1−θn+q−2,∗1θn+q−1/Ρx0,x1,t/2q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q+1−θn+q−1,Qxn,xn+q,t≤θnQx0,x1,t2фxn,xn+1○θn+1Qx0,x1,t22ηxn+1,xn+qфxn+1,xn+2,○θn+2Qx0,x1,t23ηxn+1,xn+qηxn+2,xn+qфxn+2,xn+3○⋯○,θn+q−2Qx0,x1,t2q−1ηxn+1,xn+qηxn+2,xn+q⋯ηxn+q−2,xn+qфxn+q−2,xn+q−1,○θn+q−1Qx0,x1,t2q−1ηxn+1,xn+qηxn+2,xn+qηxn+3,xn+q⋯ηxn+q−1,xn+q.
Therefore,(46)limn⟶∞Ρxn,xn+q,t=1∗1∗⋯∗1=1,limn⟶∞Qxn,xn+q,t=0○0○⋯○0=0,i.e., xn is a CS. Since B,Ρ,Q,∗,○ is a complete IFDCMS, there exists(47)limn⟶∞xn=x.
Now, we investigate thatx is a FP of Ψ. Using vandx, we obtain(48)1ΡΨxn,Ψx,t−1≤θ1Ρxn,x,t−1=θΡxn,x,t−θ⇒1θ/Ρxn,x,t+1−θ≤ΡΨxn,Ψx,t.
Using the above inequality, we obtain(49)Ρx,Ψx,t≥Ρx,xn+1,t2фx,xn+1∗Ρxn+1,Ψx,t2ηxn+1,Ψx≥Ρx,xn+1,t2фx,xn+1∗ΡΨxn,Ψx,t2ηxn+1,Ψx≥Ρxn,xn+1,t2фx,xn+1∗1θ/Ρxn,x,t/2ηxn+1,Ψx+1−θ⟶1∗1=1,asn⟶∞, and(50)Qx,Ψx,t≤Qx,xn+1,t2фx,xn+1○Qxn+1,Ψx,t2ηxn+1,Ψx≤Qx,xn+1,t2фx,xn+1○QΨxn,Ψx,t2ηxn+1,Ψx≤Qxn,xn+1,t2фx,xn+1○θQxn,x,t2ηxn+1,Ψx⟶0○0=0asn⟶∞.
This implies thatΨx=x, a FP. Now, we show the uniqueness. Suppose that Ψρ=ρ for some ρ∈B. Then,(51)1Ρx,ρ,t−1=1ΡΨx,Ψρ,t−1≤θ1Ρx,ρ,t−1<1Ρx,ρ,t−1,a contradiction, and(52)Qx,ρ,t=QΨx,Ψρ,t≤θQx,ρ,t<Qx,ρ,t,a contradiction. Therefore, we must have Ρx,ρ,t=1andQx,ρ,t=0, hence x=ρ.Example 6.
LetB=0,1andф,η:B×B⟶1,∞ be two noncomparable functions given by(53)фx,y=1,ifx=y,1+maxx,yminx,y,ifx≠y≠0.ηx,y=1,ifx=y,1+maxx2,y2minx2,y2,ifx≠y.DefineΡ,Q:B×B×0,∞⟶0,1 as(54)Ρx,y,t=tt+x−y2,Qx,y,t=x−y2t+x−y2.Then,B,Ρ,Q,∗,○ is a complete IFDCMS with CTN π∗µ=πµ and CTCN π○µ=maxπ,µ.DefineΨ:B⟶BbyΨx=1−2−x/3 and take θ∈1/2,1. Then,(55)ΡΨx,Ψy,θt=Ρ1−2−x3,1−2−y3,θt=θtθt+1−2−x/3−1−2−y/32=θtθt+2−x−2−y2/9≥θtθt+x−y2/9=9θt9θt+x−y2≥tt+x−y2=Ρx,y,t,QΨx,Ψy,θt=Q1−2−x3,1−2−y3,θt=1−2−x/3−1−2−y/32θt+1−2−x/3−1−2−y/32=2−x−2−y2/9θt+2−x−2−y2/9=2−x−2−y29θt+2−x−2−y2≤x−y29θt+x−y2≤x−y2t+x−y2=Qx,y,t.Hence, all circumstances of Theorem1 are fulfilled and 0 is a unique fixed point for Ψ.


## 3. Application to Fuzzy Fredholm Integral Equation

LetB=Ce,g,ℝ be the set of the entire continuous functions so that their domain is real values and defined on e,g.Now, we consider the following fuzzy integral equation:(56)xl=fj+δ∫egFl,jxldj,forj∈e,g,where δ>0,fj is a fuzzy function of j:j∈e,g and F∈BDefineΡandQ by(57)Ρxl,yl,t=supl∈e,gtt+xl−yl2for allx,y∈Bandt>0,Ρxl,yl,t=1−supl∈e,gtt+xl−yl2for allx,y∈Bandt>0,with CTN and CTCN defined by π∗µ=π.µandπ○µ=maxπ,µ.Defineф,η:B×B⟶1,∞ as(58)фx,y=1,ifx=y,1+maxx,yminx,y,ifx≠y≠0.ηx,y=1,ifx=y,1+maxx2,y2minx2,y2,ifx≠y.Then,B,Ρ,Ρ,∗,○ is a complete IFDCMS.Assume that(59)Fl,jxl−Fl,jyl≤xl−yl,for x,y∈B, θ∈0,1 and for alll,j∈e,g.Also, considerδπ∫egdj2≤θ<1. Then, the fuzzy integral equation in equation (56) has a unique solution.Proof: .
DefineΨ:B⟶B by(60)Ψxl=fj+δ∫egFl,jeldj,for allj∈e,g.
Scrutinize that survival of an FP of the operatorΨ has come to the survival of solution of the fuzzy integral equation.
Now, for allx,y∈B, we obtain(61)ΡΨxl,Ψyl,θt=supl∈e,gθtθt+Ψxl−Ψyl2=supl∈e,gθtθt+fj+δ∫egFl,jeldj−fj−δ∫egFl,jeldj2=supl∈e,gθtθt+δ∫egFl,jeldj−δ∫egFl,jeldj2=supl∈e,gθtθt+Fl,jxl−Fl,jyl2δ∫egdj2≥supl∈e,gtt+xl−yl2≥Ρxl,yl,tQΨxl,Ψyl,θt=1−supl∈e,gθtθt+Ψxl−Ψyl2=1−supl∈e,gθtθt+fj+δ∫egFl,jeldj−fj−δ∫egFl,jeldj2=1−supl∈e,gθtθt+δ∫egFl,jeldj−δ∫egFl,jeldj2=1−supl∈e,gθtθt+Fl,jxl−Fl,jyl2δ∫egdj2≤1−supl∈e,gtt+xl−yl2≤Qxl,yl,t.
Therefore, all circumstances of Theorem1 are fulfilled. Hence, operator Ψ has a single FP. This implies that fuzzy integral (56) has a single solution.


## 4. Conclusion

Herein, we introduced the notion of intuitionistic fuzzy double controlled metric spaces and some new types of fixed point theorems in this new setting. Moreover, we provided a nontrivial example to demonstrate the viability of the proposed methods. We have supplemented this work with an application that demonstrates how the built method outperforms those found in the literature. Since our structure is more general than the class of fuzzy intuitionistic and double controlled metric spaces, our results and notions expand and generalize many previously published results [6, 22, 23].



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