Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

Naeem Saleem,Khalil Javed,Fahim Uddin,Umar Ishtiaq,Khalil Ahmed,Thabet Abdeljawadand Manar A.Alqudah

1Department of Mathematics,University of Management and Technology,Lahore,54770,Pakistan

2Department of Math&Stats,International Islamic University Islamabad,Islamabad,44000,Pakistan

3Abdus Salam School of Mathematical Sciences,Government College University,Lahore,54600,Pakistan

4Office of Research,Innovation and Commercialization,University of Management and Technology,Lahore,54770,Pakistan

5Department of Mathematics and Sciences,Prince Sultan University,Riyadh,11586,Saudi Arabia

6Department of Medical Research,China Medical University,Taichung,40402,Taiwan

7Department of Mathematical Sciences,Faculty of Sciences,Princess Nourah Bint Abdulrahman University,Riyadh,11671,Saudi Arabia

ABSTRACT In this manuscript,our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces.We establish some fixed point theorems in this setting.Also,we plot some graphs of an example of obtained result for better understanding.We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space.Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions.Triangular conorms are known as dual operations of triangular norms.The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.

KEYWORDS Fixed point;extended fuzzy b-metric like space;intuitionistic extended fuzzy b-metric-like space;integral equation

1 Introduction

After being given the notion of fuzzy sets (FSs) by Zadeh [1], many researchers provided many generalizations.Schweizer et al.[2]introduced the notion of continuous t-norms.In this continuation,Kramosil et al.[3]introduced the approach of fuzzy metric spaces,while George et al.[4]introduced the concept of fuzzy metric spaces.Garbiec[5]gave the fuzzy interpretation of the Banach contraction principle in fuzzy metric spaces.Dey et al.[6]established an extension of Banach fixed point theorem in fuzzy metric space.Nadaban[7]introduced the notion of fuzzy b-metric spaces.Gregory et al.[8]proved various fixed point theorems in fuzzy metric spaces.Bashir et al.[9]established several fixed point results of a generalized reversed F-contraction mapping and its application.

Recently,Harandi[10]initiated the concept of metric-like spaces,which generalized the notion of metric spaces in a nice way.Alghamdi et al.[11]used the concept of metric-like spaces and introduced the notion of b-metric-like spaces.In this sequel,Shukla et al.[12]generalized the concept of metriclike spaces and introduced fuzzy metric-like spaces and Javed et al.[13]introduced the concept of fuzzy b-metric-like spaces and prove some fixed point results.

Mehmood et al.[14] presented the notion of fuzzy extended b-metric spaces (FEBMSs) by replacing the coefficientb≥1 with a functionα: D×D →[1,∞).The approach of intuitionistic fuzzy metric spaces was tossed by Park et al.[15–18], Saleem et al.[19–28] proved several fixed theorems on intuitionistic fuzzy metric space.Sintunavarat et al.[29]established various fixed theorems for a generalized intuitionistic fuzzy contraction in intuitionistic fuzzy metric spaces.Saadati et al.[30]did amazing work in the sense of intuitionistic fuzzy topological spaces.Later,Konwar[31]presented the concept of an intuitionistic fuzzy b metric space (IFBMS).Mahmood et al.[32] did power aggregation operators and similarity measures based on improved intuitionistic hesitant fuzzy sets and their applications to multiple attribute decision making.

In this manuscript, we aim to introduce the concept of intuitionistic extended fuzzy b-metriclike space (IEFBMLS).In which, we generalize the concept of IFBMS by replacing the coefficientb≥1 with a functionα: D×D →[1,∞)in both triangular inequalities and we replace condition(III) of IFBMS,Mb(?,δ,T)= 1 ??=δbyMb(?,δ,T)= 1 implies?=δand similarly, we replace ‘’?by implies’’in condition (VIII) of IFBMS.So, presented results in this manuuscript are more generalized in the existing literature.Also,we provide some fixed point(FP)results,non-trivial examples,an application to integral equations and application dynamic market equilibrium.

Main objectives of this manuscript are:

(a) To introduce the notion of intuitionistic extended fuzzy b-metric-like space.

(b) To enhance the literature of intuitionistic fuzzy fixed point theory.

(c) To plot some graphical structure of obtained result.

(d) To prove the existence and uniqueness of established results via integral equations.

(e) To provide an application dynamic market equilibrium.

2 Preliminaries

The following definitions are helpful in the sequel.

Definition 2.1[15] A binary operation ?: [0, 1] × [0, 1] →[0, 1] is called a continuous triangle norm(briefly CTN)if:

1.ν*ω=ω*ν, ?ν,ω∈[0, 1];

2.*is continuous;

3.ν*1=ν, ?ν∈[0, 1];

4.(ν*ω)*κ=ν*(ω*κ), ?ν,ω,κ∈[0, 1];

5.Ifν≤κandω≤d,withν,ω,κ,d∈[0, 1],thenν*ω≤κ*d.

Definition 2.2[15]A binary operation °:[0,1] ×[0,1]→[0,1]is called a continuous triangle conorm(briefly CTCN)if it meets the below assertions:

1.ν°ω=ω°ν, ?ν,ω∈[0, 1];

2.°is continuous;

3.ν°0=0,(?) ν∈[0, 1];

4.(ν°ω)°κ=ν°(ω°κ), ?ν,ω,κ∈[0, 1];

5.Ifν≤κandω≤d,withν,ω,κ,d∈[0, 1],thenν°ω≤κ°d.

Definition 2.3[10] A mappingP: D × D → [1,∞), where D?, fulfilling the below circumstances:

a.P(?,δ)=0 implies?=δ;

b.P(?,δ)=P(δ,?);

c.P(?,δ)≤P(?,β)+P(β,δ);

for all?,δ,β∈D.ThenPis called a metric-like and(D,P)is named metric-like space.

Definition 2.4[12]Take D?.Let*be a CTN andQbbe a FS on D×D×(0,∞).A three tuple(D,Qb,*)is called fuzzy metric like space,if it verifies the following for all?,δ,β∈D andT,S ＞0:(F1)Qb(?,δ,T)＞0;

(F2)Qb(?,δ,T)=1 implies?=δ;

(F3)Qb(?,δ,T)=Qb(δ,?,T);

(F4)Qb(?,β,(T+S))≥Qb(?,δ,T)*Qb(δ,β,S);

(F5)Qb(?,δ, ·):(0,∞)→[0, 1]is continuous.

Definition 2.5[14]A 4-tuple(D,Δα,*,α)is called an FEBMS if D is a non-empty set,α:D×D →[1,∞),*is a CTN andΔαis a FS on D×D×(0,∞),so that for all?,δ,β∈D andT,S ＞0:

Δ1)Δα(?,δ,0)=0;

Δ2)Δα(?,δ,T)=1 ??=δ;

Δ3)Δα(?,δ,T)=Δα(δ,?,T);

Δ4)Δα(?,β,α(?,β)(T+S))≥Δα(?,δ,T)*Δα(δ,β,S);

Δ5)Δα(?,δ, ·):(0,∞)→[0,1]is continuous.

Definition 2.6[31] Take D?.Let * be a CTN, ° be a CTCN,b≥1 andMb,Nbbe FSs on D×D×(0,∞).If(D,Mb,Nb,*, °)verifies the following for all?,δ∈D andS,T ＞0:

(I)Mb(?,δ,T)+Nb(?,δ,T)≤1;

(II)Mb(?,δ,T)＞0;

(III)Mb(?,δ,T)=1 ??=δ;

(IV)Mb(?,δ,T)=Mb(δ,?,T);

(V)Mb(?,β,b(T+S))≥Mb(?,δ,T)*Mb(δ,β,S);

(VI)Mb(?,δ, ·)is a non-decreasing(ND)function of R+andMb(?,δ,T)=1;

(VII)Nb(?,δ,T)＞0;

(VIII)Nb(?,δ,T)=0 ??=δ;

(IX)Nb(?,δ,T)=Nb(δ,?,T);

(X)Nb(?,β,b(T+S))≤Nb(?,δ,T)°Nb(δ,β,S);

(XI)Nb(?,δ, ·)is a non-increasing(NI)function of R+andNb(?,δ,T)=0,then(D,Mb,Nb,*, °)is an IFBMS.

3 Main Result

In this section,we introduce the notion of an IEFBMLS and prove some related FP results.

Definition 3.1Let D?,*be a CTN,°be a CTCN,φ: D×D →[1,∞)be a mapping andMφ,Nφbe FSs on D×D×(0,∞).If(D,Mφ,Nφ,*, °)is such that for?,δ∈D andS,T ＞0:

(i)Mφ(?,δ,T)+Nφ(?,δ,T)≤1;

(ii)Mφ(?,δ,T)＞0;

(iii)Mφ(?,δ,T)=1 implies?=δ;

(iv)Mφ(?,δ,T)=Mφ(δ,?,T);

(v)Mφ(?,β,φ(?,β)(T+S))≥Mφ(?,δ,T)*Mφ(δ,β,S);

(vi)Mφ(?,δ, ·)is a ND function of R+and=1;

(vii)Nφ(?,δ,T)＞0;

(viii)Nφ(?,δ,T)=0 implies?=δ;

(ix)Nφ(?,δ,T)=Nφ(δ,?,T);

(x)Nφ(?,β,φ(?,β)(T+S))≤Nφ(?,δ,T)°Nφ(δ,β,S);

(xi)Nφ(?,δ, ·)is a NI function of R+and=0,then(D,Mφ,Nφ,*, °)is an IEFBMLS.

Remark 3.2In the above definition,the self distance in condition(iii)may not be equal to 1 and in condition(viii)the self distance may not be equal to 0.In triangular inequalities,we useφ:D×D →[1,∞).So,this is cleared that IEFBMLS may not be an IFBMS but converse is true.

Example 3.3Let D=(0,∞), defineMφ,Nφ:D×D×(0,∞)→[0,1]by

for all?,δ∈D andT ＞0.Define the CTN by:ν*ω=ν·ωand CTCN′′°′′byν°ω= max{ν,ω}and define′′φ′′by

Example 3.4Let D=(0,∞)andα:D×D →[1,∞)be a function given byφ(?,δ)=?+δ+1.DefineMφ,Nφ:D×D×(0,∞)→[0,1]as

and

Remark 3.5Above example also satisfied for CTNa*b=min{a,b}and CTCNa°b=max{a,b}.

Example 3.6Let D=(0,∞)andφ:D×D →[1,∞)be a function given byφ(?,δ)=?+δ+1.DefineMφ,Nφ:D×D×(0,∞)→[0,1]as

and

Proposition 3.7Let D =(0,∞)andφ: D × D →[1,∞)be a function given byφ(?,δ)=2(?+δ+1)DefineN,Mas

Remark 3.8The above proposition also satisfied for CTNa*b= min{a,b}and CTCNa°b=max{a,b}.

Proposition 3.9Let D = [0,1] andφ: D × D →[1,∞)be a function given byφ(?,δ)=2(?+δ+1).DefineMφ,Nφas

Example 3.10Let D=(0,∞), defineMφ,Nφ:D×D×(0,∞)→[0,1]by

for all?,δ∈D andT ＞0,define CTN*byν*ω=ν·ωand CTCN°byν°ω= max{ν,ω}and defineφby

Remark 3.11In the above all examples self distance may not be equal to 1 and 0.In particular,assume an example 3.9,take?=δ,then

Remark 3.12In the above Examples 3.3, 3.4, 3.6, 3.7, 3.10 and Proposition 3.9, it is easy to see that self-distance is not equal to 1 as in condition (iii) and the self-distance is not equal to 0 as in condition (viii) in Definition 3.1.So, the Examples 3.3, 3.4, 3.6, 3.7, 3.10 and Proposition 3.9 are becomes IEFBMLSs but not becomes IFBMSs.

Definition 3.13Letbe an IEFBMLS.Then

Now,we consider intuitionistic extended fuzzy like contractions.

Theorem 3.14Letbe a G-complete IEFBMLS(with the functionφ: D×D →[1,∞))and suppose that

for all?,δ∈D andT ＞0.Let f :D →D be a mapping satisfying

for all?,δ∈D andT ＞0 with 0＜k ＜1.Further,suppose that for an arbitrary?0∈D andn,q∈N,we have

where?n=f n?0=f νn－1.Thenfhas a unique FP.

Proof:Let?0be a random element in D and consider?n=f n?0=f νn－1,n∈N.By using(2)for allT ＞0,we have

and

We obtain

for anyq∈N,and using(v)and(x),we deduce

and

That is,{?n}is a GCS.Since(D,Mφ,Nφ,*,°)is a G-complete IEFBMLS,there ? in D so that

Now,using(v), (x) and(1),we obtain

and

This implies that f ? =?.To prove the uniqueness,suppose that fc=c for some c ∈D,then

and

By using(iii) and (viii), we get ? =c.

Example 3.15Let D=[0,1]and define Mφ,Nφ:D×D×(0,∞)→[0,1]as

and

with CTN*such thata*b=a·band CTCN°such thata°b=max{a,b}.Defineφ:D×D →[1,∞)as

Then clearly(D,Mφ,Nφ,*, °)is a complete IEFBMLS.Now,define f :D×D →D as

Letk∈then we have the following:

and

Observe that all the conditions of Theorem 3.14 are satisfied and 0 is a unique fixed point, i.e.,f (0)=0.

Now,we use the Example 3.15 to show the graphical view of contraction mapping and a unique fixed point.Below,in Fig.1,we show the graphical view of Mφ(f?,fδ,kT)=Mφ(?,δ,T).Table 1 shows the values of Mφ(f?,fδ,kT)and Table 2 shows the values of Mφ(?,δ,T).In Fig.2,we show the graphical view of Nφ(f?,fδ,kT)=Nφ(?,δ,T).Table 3 shows the values of Nφ(f?,fδ,kT)and Table 4 shows the values of Nφ(?,δ,T).In Fig.3,we show the view of unique fixed point.

Figure 1:Variation of L.H.S.=Mφ(f ?,f δ,kT)with R.H.S.= Mφ(?,δ,T)of an Example 3.15 for T =1 and

Table 1:The matrix of values of L.H.S.=Mφ(f ?,f δ,kT),in which first row represents the values of δ and first column represents the values of ?

Table 2:The matrix of values of R.H.S.= Mφ(?,δ,T),in which first row represents the values of δ and first column represents the values of ?

Table 2 (continued)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.7692 0.7692 0.7692 0.7692 0.7142 0.6666 0.6250 0.5882 0.5555 0.5263 0.5000 0.4 0.7142 0.7142 0.7142 0.7142 0.7142 0.6666 0.6250 0.5882 0.5555 0.5263 0.5000 0.5 0.6666 0.6666 0.6666 0.6666 0.6666 0.6666 0.6250 0.5882 0.5555 0.5263 0.5000 0.6 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.5882 0.5555 0.5263 0.5000 0.7 0.5882 0.5882 0.5882 0.5882 0.5882 0.5882 0.5882 0.5882 0.5555 0.5263 0.5000 0.8 0.5555 0.5555 0.5555 0.5555 0.5555 0.5555 0.5555 0.5555 0.5555 0.5263 0.5000 0.9 0.5263 0.5263 0.5263 0.5263 0.5263 0.5263 0.5263 0.5263 0.5263 0.5263 0.5000 1 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000

Figure 2: Variation of L.H.S.= Nφ(f ?,f δ,kT) with R.H.S.= Nφ(?,δ,T) of an Example 3.15 for T =1 and k

Table 3:The matrix of values of R.H.S.= Nφ(?,δ,T),in which first row represents the values of δ and first column represents the values of ?

?

Table 4:The matrix of values of L.H.S.=Nφ(f ?,f δ,kT),in which first row represents the values of δ and first column represents the values of ?

Figure 3:Graph of f ? =?,we can see that both lines intersect each other at 0.This shows that 0 is a unique fixed point

Definition 3.16Letbe an IEFBMLS.A map f : D →D is an intuitionistic extended fuzzy b-like contraction mapping if there exists 0＜k ＜1 such that

and

for all?,δ∈D andT ＞0.

Now,we prove the following theorem related to above contraction mapping.

Theorem 3.17Letbe a G-complete IEFBMLS withφ: D × D →[1,∞).Suppose that

for all?,δ∈D andT ＞0.Let f : D →D be an intuitionistic extended fuzzy b-like contraction mapping.Further,suppose that for an arbitrary?0∈D, andn,q∈N,we have?n=f n?0=f νn－1.Thenfhas a unique FP.

Proof:Let?0be in D.Take?n=f n?0=f νn－1,n∈N.By using(4)and(5)for allT ＞0,n ＞q,we have

Continuing in this way,we get

We obtain

and

for allq∈N,Using(v)and(x),we deduce

and

Using(3),(v)and(x),we deduce

and

Consequently,for alln,q∈N,we obtainwhere 0＜k ＜1.Therefore,from(vi),(xi), (1)and forn→∞,

and

i.e.,{?n}is a GCS.SinceD,Mφ,Nφ,*,°is a G-complete IEFBMLS,there exists

Now,we investigate that?is a FP off.Using(v),(x)and (1),we obtain

That is,

It implies that

and

This yields thatf ?=?, a FP.Now, we show the uniqueness.Supposefc=cfor somec∈D,then

which is a contradiction.Also,

Again, it is a contradiction.Therefore, we must haveMφ(?,c,T)= 1 andNφ(?,c,T)= 0,hence?=c.

Example 3.18Let D=[0,1].Defineφby

Also take

withν*ω=ν.ωandν°ω=max{ν,ω}.Thenis a G-complete IEFBMLS.Define f :D →D by

Then we have four cases:

In all 1－4 cases,

are satisfied fork∈and also

Observe that all circumstances of Theorems 3.14 and 3.17 are fulfilled,and 0 is a unique FP off.

Example 3.19Let D=[0,1] andφ:D×D →[1,∞)be a function given byφ(?,δ)=?+δ+1.DefineMφ,Nφ:D×D×(0,∞)→[0,1]as

for allk∈D.Thenis an G-complete IEFBMLS with CTNa*b=aband CTCNa°b=max{a,b}.Define f :D →D by

Then,

and

Nφ(f ?,f δ,T)≤kNφ(?,δ,T)

Observe that all circumstances of Theorems 3.14 and 3.17 are fulfilled,and 0 is a unique FP off.

4 Application to Fuzzy Fredholm Integral Equations

Let D =C([e,g], R)be the set of all continuous real valued functions defined on the interval[e,g].

Now,we let the fuzzy integral equation

whereβ ＞0,f (j)is a fuzzy function ofj∈[e,g]andF∈D.DefineMφandNφby

and

for all?,δ∈D andT ＞0, with the CTN and CTCN defined byν*ω=ν.ωandν°ω=maxν,ω.Defineφ:D×D →[1,∞)by

Assume that

max{F(l,j)?(l),F(l,j)δ(l)}≤max{?(l),δ(l)} for?,δ∈D,k∈(0, 1)and ?l,j∈[e,g].Also consider≤k ＜1.then fuzzy integral equation in Eq.(9)has a unique solution.

Proof:Define f :D →D by

Scrutinize that survival of an FP of the operatorfis come to the survival of solution of the fuzzy integral equation.

Now for all?,δ∈D,we obtain

Therefore,all the conditions of Theorem 3.11 are fulfilled.Hence operatorfhas a unique FP.This implies that fuzzy integral Eq.(9)has a unique solution.

Corollary 3.1Letbe a G-complete IFBMS.Define f :D →D be

Suppose the below conditions meet:

Then integral Eq.(9)has a solution.

We can prove easily by follow the above proof.

5 Application to Dynamic Market Equilibrium

In real business cycle models, economy is always in its long run equilibrium but in Keynesian business cycle theory the economy could be above or below the long-term potential,full employment GDP.While the real business cycle model seeks to overcome the distinction between the long run growth model and the real business cycle.Now we show how our established result can be used to find the unique solution to an integral equation in dynamic market equilibrium economics.

Let us denote the supplyQβand demandQd,in many markets,current prices and pricing trends(whether prices are rising or dropping and whether they are rising or falling at an increasing or decreasing rate) have an impact.The economist, therefore, wants to know what the current price isP(T),by using the first derivative,and the second derivative.Assume that,

σ1,σ2,υ1,υ2,α1andα2are constants.If pricing clears the market at each point in time,comment on the dynamic stability of the market.In equilibrium,Qβ=Qd.So,

since

Lettingω=ω1－ω2,α=α1－α2,υ=υ1－υ2andσ=σ1－σ2in above,we have

Dividing through byω,P (T)is governed by the following initial value problem:

whereξ(T,r)is Green′s function given by

We will show the existence of a solution to the integral equation

Let D=C ([0,T])set of real continuous functions defined on [0,T] forT ＞0,we define

and

for all?,ω ∈D with the CTN′*′such thata*b=a·band CTCN°such thata°b= max{a,b}.Defineφ:D×D →[1, ∞).As

Q(ω,d)=1+ω+d.

Theorem 4.1Consider Eq.(11)and suppose that

Then,the integral Eq.(11)has a unique solution.

Proof:For?,ω∈D,by using of assumptions(i)to(iii),we have

and

ThusMφ(f ?,f ω,kT)≥Mφ(?,ω,T)andNφ(f ?,f ω,kT)≤Nφ(?,ω,T)for all?,ω∈D,and all conditions of Theorem 3.14 are satisfied.Therefore,Eq.(11)has a unique fixed point.

6 Conclusion

Herein,we introduced the notion of intuitionistic extended fuzzy b-metric-like spaces and some new types of fixed point theorems in this new setting.Moreover, we provided non-trivial examples and plotted some graphs to demonstrate the viability of the proposed methods.We provided an application of the obtained results in a dynamic equilibrium market.We have supplemented this work with applications demonstrating how the built method outperforms those found in the literature.Since our structure is more general than the class of fuzzy b-metric like space and intuitionistic fuzzy b-metric space, our results and notions expand and generalize several previously published results.This work can easily extend to the structure of neutrosophic extended b-metric-like spaces,controlled intuitionistic fuzzy b-metric-like spaces,and many other structures.

Acknowledgement:The authors are grateful to their universities for their support.

Author’s Contributions:All authors contributed equally in writing this article.All authors read and approved the final manuscript.

Availability of Data and Materials:Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Funding Statement:Princess Nourah bint Abdulrahman University Researchers Supporting Project No.(PNURSP2022R14),Princess Nourah bint Abdulrahman University,Riyadh,Saudi Arabia.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.