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ARTICLE

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

Naeem Saleem1, Khalil Javed2, Fahim Uddin3, Umar Ishtiaq4, Khalil Ahmed2, Thabet Abdeljawad5,6,*, Manar A. Alqudah7

1 Department of Mathematics, University of Management and Technology, Lahore, 54770, Pakistan
2 Department of Math & Stats, International Islamic University Islamabad, Islamabad, 44000, Pakistan
3 Abdus Salam School of Mathematical Sciences, Government College University, Lahore, 54600, Pakistan
4 Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, 54770, Pakistan
5 Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
6 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
7 Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, 11671, Saudi Arabia

* Corresponding Author: Thabet Abdeljawad. Email:

Computer Modeling in Engineering & Sciences 2023, 135(1), 109-131. https://doi.org/10.32604/cmes.2022.021031

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

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 metric-like 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 coefficient b1 with a function α:D×D[1,). The approach of intuitionistic fuzzy metric spaces was tossed by Park et al. [1518], Saleem et al. [1928] 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-metric-like space (IEFBMLS). In which, we generalize the concept of IFBMS by replacing the coefficient b1 with a function α:D×D[1,) in both triangular inequalities and we replace condition (III) of IFBMS, Mb(ϑ,δ,T)=1ϑ=δ by Mb(ϑ,δ,T)=1impliesϑ=δ and 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.    νω=ων,ν,ω[0,1];

2.     is continuous;

3.    ν1=ν,ν[0,1];

4.    (νω)κ=ν(ωκ),ν,ω,κ[0,1];

5.    If νκ and ωd, with ν,ω,κ,d[0,1], then νωκd.

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.    νω=ων,ν,ω[0,1];

2.     is continuous;

3.    ν0=0,()ν[0,1];

4.    (νω)κ=ν(ωκ),ν,ω,κ[0,1];

5.    If νκ and ωd, with ν,ω,κ,d[0,1], then νωκd.

Definition 2.3 [10] A mapping P:D×D[1,), where D, fulfilling the below circumstances:

a. P(ϑ,δ)=0impliesϑ=δ;

b. P(ϑ,δ)=P(δ,ϑ);

c. P(ϑ,δ)P(ϑ,β)+P(β,δ);

for all ϑ,δ,βD. Then P is called a metric-like and (D,P) is named metric-like space.

Definition 2.4 [12] Take D. Let be a CTN and Qb be a FS on D×D×(0,). A three tuple (D,Qb,) is called fuzzy metric like space, if it verifies the following for all ϑ,δ,βDandT,S>0:

(F1)   Qb(ϑ,δ,T)>0;

(F2)   Qb(ϑ,δ,T)=1  impliesϑ=δ;

(F3)   Qb(ϑ,δ,T)=Qb(δ,ϑ,T);

(F4)   Qb(ϑ,β,(T+S))Qb(ϑ,δ,T)Qb(δ,β,S);

(F5)   Qb(ϑ,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 and Δα is a FS on D×D×(0,), so that for all ϑ,δ,βDandT,S>0:

Δ1)   Δα(ϑ,δ,0)=0;

Δ2)   Δα(ϑ,δ,T)=1ϑ=δ;

Δ3)   Δα(ϑ,δ,T)=Δα(δ,ϑ,T);

Δ4)   Δα(ϑ,β,α(ϑ,β)(T+S))Δα(ϑ,δ,T)Δα(δ,β,S);

Δ5)   Δα(ϑ,δ,):(0,)[0,1] is continuous.

Definition 2.6 [31] Take D. Let be a CTN, be a CTCN, b1 and Mb,Nb be FSs on D×D×(0,). If (D,Mb,Nb,,) verifies the following for all ϑ,δDandS,T>0:

(I) Mb(ϑ,δ,T)+Nb(ϑ,δ,T)1;

(II) Mb(ϑ,δ,T)>0;

(III) Mb(ϑ,δ,T)=1ϑ=δ;

(IV) Mb(ϑ,δ,T)=Mb(δ,ϑ,T);

(V) Mb(ϑ,β,b(T+S))Mb(ϑ,δ,T)Mb(δ,β,S);

(VI) Mb(ϑ,δ,) is a non-decreasing (ND) function of R+and  limTMb(ϑ,δ,T)=1;

(VII) Nb(ϑ,δ,T)>0;

(VIII) Nb(ϑ,δ,T)=0ϑ=δ;

(IX) Nb(ϑ,δ,T)=Nb(δ,ϑ,T);

(X) Nb(ϑ,β,b(T+S))Nb(ϑ,δ,T)Nb(δ,β,S);

(XI) Nb(ϑ,δ,) is a non-increasing (NI) function of R+ and limTNb(ϑ,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 3.1 Let D, be a CTN, be a CTCN, ϕ:D×D[1,) be a mapping and Mϕ,Nϕ be FSs on D×D×(0,). If (D,Mϕ,Nϕ,,) is such that for ϑ,δDandS,T>0:

(i) Mϕ(ϑ,δ,T)+Nϕ(ϑ,δ,T)1;

(ii) Mϕ(ϑ,δ,T)>0;

(iii) Mϕ(ϑ,δ,T)=1impliesϑ=δ;

(iv) Mϕ(ϑ,δ,T)=Mϕ(δ,ϑ,T);

(v) Mϕ(ϑ,β,ϕ(ϑ,β)(T+S))Mϕ(ϑ,δ,T)Mϕ(δ,β,S);

(vi) Mϕ(ϑ,δ,) is a ND function of R+ and limTMϕ(ϑ,δ,T)=1;

(vii) Nϕ(ϑ,δ,T)>0;

(viii) Nϕ(ϑ,δ,T)=0implies  ϑ=δ;

(ix) Nϕ(ϑ,δ,T)=Nϕ(δ,ϑ,T);

(x) Nϕ(ϑ,β,ϕ(ϑ,β)(T+S))Nϕ(ϑ,δ,T)Nϕ(δ,β,S);

(xi) Nϕ(ϑ,δ,) is a NI function of R+ and limTNϕ(ϑ,δ,T)=0,

then (D,Mϕ,Nϕ,,) is an IEFBMLS.

Remark 3.2 In 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 use ϕ:D×D[1,). So, this is cleared that IEFBMLS may not be an IFBMS but converse is true.

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

Mϕ(ϑ,δ,T)=TT+max{ϑ,δ}2,Nϕ(ϑ,δ,T)=max{ϑ,δ}2T+max{ϑ,δ}2

for all ϑ,δDandT>0. Define the CTN by: νω=νω and CTCN by νω=max{ν,ω} and define ϕ by

ϕ(ϑ,δ)={1ifϑ=δorϑ(0,1),1+max{ϑ,δ}ifotherwise.

Then (D,Mϕ,Nϕ,,) is an IEFBMLS.

Example 3.4 Let D=(0,)andα:D×D[1,) be a function given by ϕ(ϑ,δ)=ϑ+δ+1. Define Mϕ,Nϕ:D×D×(0,)[0,1] as

Mϕ(ϑ,δ,T)=TT+max{ϑ,δ}

and

Nϕ(ϑ,δ,T)=max{ϑ,δ}T+max{ϑ,δ}.

Then (D,Mϕ,Nϕ,,) is an IEFBMLS with CTN ab=ab and CTCN ab=maxa,b.

Remark 3.5 Above example also satisfied for CTN ab=min{a,b} and CTCN ab=max{a,b}.

Example 3.6 Let D=(0,)andϕ:D×D[1,) be a function given by ϕ(ϑ,δ)=ϑ+δ+1. Define Mϕ,Nϕ:D×D×(0,)[0,1] as

Mϕ(ϑ,δ,T)=T+min{ϑ,δ}T+max{ϑ,δ}

and

Nϕ(ϑ,δ,T)=1T+min{ϑ,δ}T+max{ϑ,δ}.

Then (D,Mϕ,Nϕ,,) is an IEFBMLS with CTN ab=ab and CTCN ab=maxa,b.

Proposition 3.7 Let D=(0,) and ϕ:D×D[1,) be a function given by ϕ(ϑ,δ)=2(ϑ+δ+1) Define N,M as

Mϕ(ϑ,δ,T)=emax{ϑ,δ}Tn,Nϕ(ϑ,δ,T)=1emax{ϑ,δ}Tnfor  allnN,  ϑ,δD,T>0.

Then (D,Mϕ,Nϕ,,) is an IEFBMLS with CTN ab=ab and CTCN ab=maxa,b.

Remark 3.8 The above proposition also satisfied for CTN ab=min{a,b} and CTCN ab=max{a,b}.

Proposition 3.9 Let D=[0,1] and ϕ:D×D[1,) be a function given by ϕ(ϑ,δ)=2(ϑ+δ+1). Define Mϕ,Nϕ as

Mϕ(ϑ,δ,T)=[emax{ϑ,δ}Tn]1,Nϕ(ϑ,δ,T)=1[emax{ϑ,δ}Tn]1for  allnN,  ϑ,δD,T>0.

Then (D,Mϕ,Nϕ,,) is an IEFBMLS with CTN ab=ab and CTCN ab=maxa,b.

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

Mϕ(ϑ,δ,T)=TT+max{ϑ,δ},Nϕ(ϑ,δ,T)=max{ϑ,δ}T+max{ϑ,δ}

for all ϑ,δD  and  T>0, define CTN by νω=νω and CTCN by νω=max{ν,ω} and define ϕ by

ϕ(ϑ,δ)={1ifϑ=δ,1+max{ϑ,δ}min{ϑ,δ}ifϑδ

Then (D,Mϕ,Nϕ,,) be an IEFBMLS.

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

Mϕ(ϑ,ϑ,T)=TT+max{ϑ,ϑ}1,

Nϕ(ϑ,ϑ,T)=max{ϑ,ϑ}T+max{ϑ,ϑ}0.

Remark 3.12 In 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 3.13 Let (D,Mϕ,Nϕ,,) be an IEFBMLS. Then

(a)   A sequence {ϑn} in   D is named to be a G-Cauchy sequence (GCS) if and only if for all q>0and  T>0,limnMϕ(ϑn,ϑn+q,T)andlimnNϕ(ϑn,ϑn+q,T) exists and finite.

(b)   A sequence {ϑn} in   D is named to be G-convergent (GC) to ϑ in D, if and only if for all T>0,

limnMϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T)andlimnNϕ(ϑn,ϑ,T)=Nϕt(ϑ,ϑ,T).

(c)   An IEFBMS is named to be complete iff each GCS is convergent, i.e.,

limnMϕ(ϑn,ϑn+q,T)=limnMϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T),

limnNϕ(ϑn,ϑn+q,T)=limnNϕ(ϑn,ϑ,T)=Nϕ(ϑ,ϑ,T)

Now, we consider intuitionistic extended fuzzy like contractions.

Theorem 3.14 Let (D,Mϕ,Nϕ,,) be a G-complete IEFBMLS (with the function ϕ:D×D[1,)) and suppose that

limTMϕ(ϑ,δ,T)=1andlimTNϕ(ϑ,δ,T)=0(1)

for all ϑ,δD and T>0. Let f:DD be a mapping satisfying

Mϕ(fϑ,fδ,kT)Mϕ(ϑ,δ,T)andNϕ(fϑ,fδ,kT)Nϕ(ϑ,δ,T)(2)

for all ϑ,δD and T>0 with 0<k<1. Further, suppose that for an arbitrary ϑ0D    andn,qN, we have

ϕ(ϑn,ϑn+q)<1k,

where ϑn=fnϑ0=fνn1. Then f has a unique FP.

Proof: Let ϑ0 be a random element in D and consider ϑn=fnϑ0=fνn1, nN. By using (2) for all T>0, we have

Mϕ(ϑn,ϑn+1,kT)=Mϕ(fϑn1,fϑn,kT)Mϕ(ϑn1,ϑn,T)Mϕ(ϑn2,ϑn1,Tk)Mϕ(ϑn3,ϑn2,Tk2)Mϕ(ϑ0,ϑ1,Tkn)

and

Nϕ(ϑn,ϑn+1,kT)=Nϕ(fϑn1,fϑn,kT)Nϕ(ϑn1,ϑn,T)Nϕ(ϑn2,ϑn1,Tk)Nϕ(ϑn3,ϑn2,Tk2)Nϕ(ϑ0,ϑ1,Tkn).

We obtain

Mϕ(ϑn,ϑn+1,kT)Mϕ(ϑ0,ϑ1,Tkn)andNϕ(ϑn,ϑn+1,kT)Nϕ(ϑ0,ϑ1,Tkn)(3)

for any qN,  T=qTT=Tq+Tq++Tqand  using(v)  and  (x), we deduce

Mϕ(ϑn,ϑn+q,T)Mϕ(ϑn,ϑn+1,Tq(ϕ(ϑn,ϑn+q)))Mϕ(ϑn+1,ϑn+2,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))Mϕ(ϑn+2,ϑn+3,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))Mϕ(ϑn+q1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q)))

and

Nϕ(ϑn,ϑn+q,T)Nϕ(ϑn,ϑn+1,Tq(ϕ(ϑn,ϑn+q)))Nϕ(ϑn+1,ϑn+2,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))Nϕ(ϑn+2,ϑn+3,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))Nϕ(ϑn+q1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q))).

Using (3) (v)  and  (x), one writes

Mϕ(ϑn,ϑn+q,T)Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q))kn)Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q))kn+1)Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q))kn+2)Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q))kn+q1)

and

Nϕ(ϑn,ϑn+q,T)Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q))kn)Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q))kn+2)Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q))kn+3)Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q))kn+q1).

We obtain for all n,qN,ϕ(ϑn,ϑn+q)k<1, where 0<k<1. Therefore, from (vi),  (xi), (1) and  lettingn,

limnMϕ(ϑn,ϑn+q,T)=11=1

and

limnNϕ(ϑn,ϑn+q,T)=000=0.

That is, {ϑn} is a GCS. Since (D,Mϕ,Nϕ,,) is a G-complete IEFBMLS, there ϑ in D so that

limnMϕ(ϑn,ϑn+q,T)=limnMϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T),

limnNϕ(ϑn,ϑn+q,T)=limnNϕ(ϑn,ϑ,T)=Nϕ(ϑ,ϑ,T)

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

Mϕ(fϑ,ϑ,T)Mϕ(fν,fϑn,T2(ϕ(fϑ,ϑ)))Mϕ(fϑn,ν,T2(ϕ(fϑ,ϑ)))Mϕ(ν,ϑn,T2(ϕ(fϑ,ϑ))k)Mϕ(ϑn+1,ϑn,T2(ϕ(fϑ,ϑ)))11=1asn,

and

Nϕ(fϑ,ϑ,T)Nϕ(fν,fϑn,T2(ϕ(fϑ,ϑ)))Nϕ(fϑn,ν,T2(ϕ(fϑ,ϑ)))Nϕ(ν,ϑn,T2(ϕ(fϑ,ϑ))k)Nϕ(ϑn+1,ϑn,T2(ϕ(fϑ,ϑ)))00=0asn.

This implies that fϑ=ϑ. To prove the uniqueness, suppose that fc=c for some cD, then

1Mϕ(c,ϑ,T)=Mϕ(fc,fϑ,T)Mϕ(c,ϑ,Tk)=Mϕ(fc,fϑ,Tk)

Mϕ(c,ϑ,Tk2)Mϕ(c,ϑ,Tkn)1asn,

and

0Nϕ(c,ϑ,T)=Nϕ(fc,fϑ,T)Nϕ(c,ϑ,Tk)=Nϕ(fc,fϑ,Tk)Nϕ(c,ϑ,Tk2)Nϕ(c,ϑ,Tkn)0asn.

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

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

Mϕ(ϑ,δ,T)=TT+max{ϑ,δ}

and

Nϕ(ϑ,δ,T)=1TT+max{ϑ,δ}

with CTN such that ab=ab and CTCN such that ab=max{a,b}. Define ϕ:D×D[1,) as

ϕ(ϑ,δ)={1+ϑifϑ>δ;1+δifδ>ϑ;1+ϑ+δifotherwise.

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

f(ϑ)=ϑ6.

Let k[12,1), then we have the following:

Mϕ(fϑ,fδ,kT)=Mϕ(ϑ6,δ6,kT)=kTkT+max{ϑ6,δ6}=TT+max{ϑk6,δk6}TT+max{ϑ,δ}=Mϕ(ϑ,δ,T)

and

Nϕ(fϑ,fδ,kT)=Nϕ(ϑ6,δ6,kT)=1kTkT+max{ϑ6,δ6}=1TT+max{ϑk6,δk6}1TT+max{ϑ,δ}=Nϕ(ϑ,δ,T).

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 ϑ, f δ, kT) = Mφ (ϑ, δ, T ). Table 1 shows the values of Mφ (f ϑ, f δ, kT) and Table 2 shows the values of Mφ (ϑ, δ, T). In Fig. 2, we show the graphical view of Nφ (f ϑ, f δ, kT) = Nφ (ϑ, δ, T). Table 3 shows the values of Nφ (f ϑ, f δ, kT ) and Table 4 shows the values of Nφ (ϑ, δ, T). In Fig. 3, we show the view of unique fixed point.

Figure 1: Variation of L.H.S. = Mϕ(fϑ,fδ,kT) with R.H.S. = Mϕ(ϑ,δ,T) of an Example 3.15 for T=1 and k[12,1).

Figure 2: Variation of L.H.S. = Nϕ(fϑ,fδ,kT) with R.H.S. = Nϕ(ϑ,δ,T) of an Example 3.15 for T=1 and k[12,1).

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

Definition 3.16 Let (D,Mϕ,Nϕ,,) be an IEFBMLS. A map f:DD is an intuitionistic extended fuzzy b-like contraction mapping if there exists 0<k<1 such that

1Mϕ(fϑ,fδ,T)1k[1Mϕ(ϑ,δ,T)1](4)

and

Nϕ(fϑ,fδ,T)kNϕ(ϑ,δ,T),(5)

for all ϑ,δD  and  T>0.

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

Theorem 3.17 Let (D,Mϕ,Nϕ,,) be a G-complete IEFBMLS with ϕ:D×D[1,). Suppose that

limTMϕ(ϑ,δ,T)=1andlimTNϕ(ϑ,δ,T)=0(6)

for all ϑ,δD and T>0. Let f:DD be an intuitionistic extended fuzzy b-like contraction mapping. Further, suppose that for an arbitrary ϑ0D,  andn,qN, we have ϑn=fnϑ0=fνn1. Then f has a unique FP.

Proof: Let ϑ0 be in D . Take ϑn=fnϑ0=fνn1, nN. By using (4) and (5) for all T>0,n>q, we have

1Mϕ(ϑn,ϑn+1,T)1=1Mϕ(fϑn1,ϑn,T)1

k[1Mϕ(ϑn1,ϑn,T)1]=kMϕ(ϑn1,ϑn,T)k.

That is

1Mϕ(ϑn,ϑn+1,T)kMϕ(ϑn1,ϑn,T)+(1k)k2Mϕ(ϑn2,ϑn1,T)+k(1k)+(1k).

Continuing in this way, we get

1Mϕ(ϑn,ϑn+1,T)knMϕ(ϑ0,ϑ1,T)+kn1(1k)+kn2(1k)++k(1k)+(1k)

knMϕ(ϑ0,ϑ1,T)+(kn1+kn2++1)(1k)knMϕ(ϑ0,ϑ1,T)+(1kn).

We obtain

1knMϕ(ϑ0,ϑ1,T)+(1kn)Mϕ(ϑn,ϑn+1,T)(7)

and

Nϕ(ϑn,ϑn+1,T)=Nϕ(fϑn1,ϑn,T)kNϕ(ϑn1,ϑn,T)=Nϕ(fϑn2,ϑn1,T)k2Nϕ(ϑn2,ϑn1,T)knNϕ(ϑ0,ϑ1,T)(8)

for all qN,  T=qTT=Tq+Tq++Tq. Using (v) and (x), we deduce

Mϕ(ϑn,ϑn+q,T)Mϕ(ϑn,ϑn+1,Tq(ϕ(ϑn,ϑn+q)))Mϕ(ϑn+1,ϑn+2,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))

Mϕ(ϑn+2,ϑn+3,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))

Mϕ(ϑn+q1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q)))

and

Nϕ(ϑn,ϑn+q,T)Nϕ(ϑn,ϑn+1,Tq(ϕ(ϑn,ϑn+q)))Nϕ(ϑn+1,ϑn+2,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))Nϕ(ϑn+2,ϑn+3,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))Nϕ(ϑn+q1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q))).

Using (3), (v)  and  (x), we deduce

Mϕ(ϑn,ϑn+q,T)1knMϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)))+(1kn)1kn+1Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))+(1kn+1)1kn+2Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))+(1kn+2)1kn+q1Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q)))+(1kn+q1)

and

Nϕ(ϑn,ϑn+q,T)knNϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)))kn+1Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))kn+2Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))kn+q1Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)ϕ(ϑn+q1,ϑn+q))).

Consequently, for all n,qN, we obtain ϕ(ϑn,ϑn+q)k<1,where0<k<1. Therefore, from (vi),  (xi), (1) and  forn,

limnMϕ(ϑn,ϑn+q,T)=11=1

and

limnNϕ(ϑn,ϑn+q,T)=000=0,

i.e., {ϑn} is a GCS. Since (D,Mϕ,Nϕ,,) is a G-complete IEFBMLS, there exists

limnMϕ(ϑn,ϑn+q,T)=limnMϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T),

limnNϕ(ϑn,ϑn+q,T)=limnNϕ(ϑn,ϑ,T)=Nϕ(ϑ,ϑ,T).

Now, we investigate that ϑ is a FP of f. Using (v),  (x)and (1), we obtain

1Mϕ(fϑn,fϑ,T)1k[1Mϕ(ϑn,ϑ,T)1]=kMϕ(ϑn,ϑ,T)k.

That is,

1kMϕ(ϑn,ϑ,T)+(1k)Mϕ(fϑn,fϑ,T).

It implies that

Mϕ(ϑ,fϑ,T)Mϕ(ϑ,ϑn+1,T2ϕ(ϑ,fϑ))Mϕ(fϑn,fϑ,T2ϕ(ϑ,fϑ))Mϕ(ϑn,ϑn+1,T2ϕ(ϑ,fϑ))1kMϕ(ϑn,ϑ,T2ϕ(ϑ,fϑ))+(1k)11=1asn

and

Nϕ(ϑ,fϑ,T)Mϕ(ϑ,ϑn+1,T2ϕ(ϑ,fϑ))Nϕ(fϑn,fϑ,T2ϕ(ϑ,fϑ))Nϕ(ϑn,ϑn+1,T2ϕ(ϑ,fϑ))kNϕ(ϑn,ϑ,T2ϕ(ϑ,fϑ))00=0asn.

This yields that fϑ=ϑ, a FP. Now, we show the uniqueness. Suppose fc=c for some cD, then

1Mϕ(ϑ,c,T)1=1Mϕ(fϑ,fc,T)1

k[1Mϕ(ϑ,c,T)1]<1Mϕ(ϑ,c,T)1

Nϕ(ϑ,c,T)=Nϕ(fϑ,fc,T)kNϕ(ϑ,c,T)<Nϕ(ϑ,c,T).

Again, it is a contradiction. Therefore, we must have Mϕ(ϑ,c,T)=1andNϕ(ϑ,c,T)=0, hence ϑ=c.

Example 3.18 Let   D=[0,1]. Define ϕ by

ϕ(ϑ,δ)={1ifϑ=δ,1+max{ϑ,δ}ifϑδ

Also take

Mϕ(ϑ,δ,T)=emax{ϑ,δ}2TandNϕ(ϑ,δ,T)=1emax{ϑ,δ}2T

with νω=ν.ωandνω=max{ν,ω}. Then (D,Mϕ,Nϕ,,) is a G-complete IEFBMLS. Define f:DD by

f(ϑ)={0ifϑ[0,12],ϑ9ifϑ(12,1]

Then we have four cases:

1.    If ϑ,δ[0,12],thenfϑ=fδ=0;

2.    If ϑ[0,12]andδ(12,1],thenfϑ=0andfδ=δ9;

3.    If δ[0,12]andϑ(12,1],thenfδ=0andfϑ=δ9;

4.    If ϑ,δ(12,1],thenfϑ=δ9andfδ=δ9;

In all 14 cases,

Mϕ(fϑ,fδ,kT)Mϕ(ϑ,δ,T)andNϕ(fϑ,fδ,kT)Nϕ(ϑ,δ,T)

are satisfied for k[12,1), and also

1Mϕ(fϑ,fδ,T)1k[1Mϕ(ϑ,δ,T)1]andNϕ(fϑ,fδ,T)kNϕ(ϑ,δ,T)

Satisfied for k[12,1).

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

Example 3.19 Let D=[0,1]andϕ:D×D[1,) be a function given by ϕ(ϑ,δ)=ϑ+δ+1. Define Mϕ,Nϕ:D×D×(0,)[0,1] as

Mϕ(ϑ,δ,T)={1ifϑ=δkifϑδ,

Nϕ(ϑ,δ,T)={0ifϑ=δ1kifϑδ

for all kD. Then (D,Mϕ,Nϕ,,) is an G-complete IEFBMLS with CTN ab=ab and CTCN ab=max{a,b}. Define f:DD by

f(ϑ)=12ϑ3.

Then,

Mϕ(fϑ,fδ,kT)Mϕ(ϑ,δ,T)andNϕ(fϑ,fδ,kT)Nϕ(ϑ,δ,T)

are satisfied for k[12,1), and also

1Mϕ(fϑ,fδ,T)1k[1Mϕ(ϑ,δ,T)1]

and

Nϕ(fϑ,fδ,T)kNϕ(ϑ,δ,T)

satisfied for k[12,1).

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

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

ϑ(l)=f(j)+βegF(l,j)ϑ(l)djforl,j[e,g](9)

where β>0,f(j) is a fuzzy function of j[e,g] and FD. Define MϕandNϕ by

Mϕ(ϑ(l),δ(l),  T)=supl[e,g]TT+max{ϑ(l),δ(l)}2

and

Nϕ(ϑ(l),δ(l),  T)=1supl[e,g]TT+max{ϑ(l),δ(l)}2

for all ϑ,δD and T>0, with the CTN and CTCN defined by νω=ν.ωandνω=maxν,ω. Define ϕ:D×D[1,) by

ϕ(ϑ,δ)={1ifϑ=δ;1+max{ϑ,δ}ifotherwise.

Then (D,Mϕ,Nϕ,,) be a G-complete IEFBMLS.

Assume that

max{F(l,j)ϑ(l),F(l,j)δ(l)}max{ϑ(l),δ(l)} for ϑ,δD, k(0,1) and l,j[e,g]. Also consider (βegdj)2k<1. then fuzzy integral equation in Eq. (9) has a unique solution.

Proof: Define f:DD by

fϑ(l)=f(j)+βegF(l,j)e(l)djfor alll,j[e,g]

Scrutinize that survival of an FP of the operator f is come to the survival of solution of the fuzzy integral equation.

Now for all ϑ,δD, we obtain

Mϕ(fϑ(l),fδ(l),kT)=supl[e,g]kTkT+max{fϑ(l),fδ(l)}2

=supl[e,g]kTkT+max{f(j)+βegF(l,j)e(l)dj,f(j)+βegF(l,j)e(l)dj}2

=supl[e,g]kTkT+max{βegF(l,j)e(l)dj,βegF(l,j)e(l)dj}2

=supl[e,g]kTkT+max{F(l,j)ϑ(l),F(l,j)δ(l)}2(βegdj)2

supl[e,g]TT+max{ϑ(l),δ(l)}2

Mϕ(ϑ(l),δ(l),  T).

Nϕ(fϑ(l),fδ(l),kT)=1supl[e,g]kTkT+max{fϑ(l),fδ(l)}2

=1supl[e,g]kTkT+max{f(j)+βegF(l,j)e(l)dj,f(j)+βegF(l,j)e(l)dj}2

=1supl[e,g]kTkT+max{βegF(l,j)e(l)dj,βegF(l,j)e(l)dj}2

=1supl[e,g]kTkT+max{F(l,j)ϑ(l),F(l,j)δ(l)}2(βegdj)2

supl[e,g]TT+max{ϑ(l),δ(l)}2

Nϕ(ϑ(l),δ(l),  T).

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

Corollary 3.1 Let (D,Mϕ,Nϕ,,) be a G-complete IFBMS. Define f:DD be

fϑ(l)=f(j)+βegF(l,j)e(l)dj for all l,j[e,g].

Suppose the below conditions meet:

I. max{F(l,j)ϑ(l),F(l,j)δ(l)}max{ϑ(l),δ(l)} for ϑ,δD, k(0,1) and l,j[e,g],

II. (βegdj)2k<1

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 supply Qβ and demand Qd, 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 is P(T), by using the first derivative dP(T)dT, and the second derivative d2P(T)dT2. Assume that,

Qβ=σ1+υ1P(T)+α1dP(T)dT+ω1d2P(T)dT2

Qd=σ2+υ2P(T)+α2dP(T)dT+ω2d2P(T)dT2.

σ1,σ2,υ1,υ2,α1andα2 are constants. If pricing clears the market at each point in time, comment on the dynamic stability of the market. In equilibrium,Qβ=Qd. So,

σ1+υ1P(T)+α1dP(T)dT+ω1d2P(T)dT2=σ2+υ2P(T)+α2dP(T)dT+ω2d2P(T)dT2.

since

(ω1ω2)d2P(T)dT2+(α1α2)ddP(T)dT+(υ1υ2)P(T)=(σ1σ2)

Letting ω=ω1ω2,α=α1α2,υ=υ1υ2andσ=σ1σ2 in above, we have

ωd2P(T)dT2+αdP(T)dT+υP(T)=σ.

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

{P+αωP+υωP(T)=σωP(0)=0P(0)=0,(10)

where α2ω=4υωandυα=μ is a continuous function. It is easy to show that the problem (10) is equivalent to the integral equation

P(T)=0Tξ(T,r)f(T,r,P(r))dr.

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

ξ(T,r)={rαμ2(Tr)if0rTTTαμ2(rT)if0TrTT.

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

P(T)=0TG(T,r,P(r))dr.(11)

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

Mϕ(ϑ,ω,T)=supTϵ[0,T]min{ϑ,ω}+Tmax{ϑ,ω}+T

and

Nϕ(ϑ,ω,T)=1supTϵ[0,T]min{ϑ,ω}+Tmax{ϑ,ω}+T

for all ϑ,ωϵD with the CTN such that ab=ab and CTCN such that ab=max{a,b}. Define ϕ:D×D[1,). As

Q(ω,d)=1+ω+d.

It is easy to show that (D,Mϕ,Nϕ,,  )  is  a  complete  IEFBMLS and f:DD defined by

fP(T)=0TG(T,r,P(r)dr.

Theorem 4.1 Consider Eq. (11) and suppose that

(i)    G:[0,T]×[0,T]R+ is continuous function,

(ii)    There exist a continuous function ξ:[0,T]×[0,T]R+ such that supT[0,T]0Tξ(T,r)dr1;

(iii)

max{G(T,r,ϑ(r))G(T,r,ω(r))}ξ(T,r)max{ϑ(r),ω(r)}andmin  {G(T,r,ϑ(r))G(T,r,ω(r))}ξ(T,r)minϑ(r),ω(r).

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

Proof: For ϑ,ωD, by using of assumptions (i) to (iii), we have

Mϕ(fϑ,fω,kT)=supTϵ[0,T]min{0TG(T,r,ϑ(r))dr,0TG(T,r,ω(r))dr}+kTmax{0TG(T,r,ϑ(r))dr,0TG(T,r,ω(r))dr}+kT=supTϵ[0,T]0Tmin{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT0Tmax{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kTsupTϵ[0,T]0Tξ(T,r)min{ϑ(r),ω(r)}dr+kT0Tξ(T,r)max{ϑ(r),ω(r)}dr+kTsupTϵ[0,T]min{ϑ(r),ω(r)}0Tξ(T,r)dr+kTmax{ϑ(r),ω(r)}0Tξ(T,r)dr+kT(min{ϑ(r),ω(r)}+Tmax{ϑ(r),ω(r)}+T)=Mϕ(ϑ,ω,T)

and

Nϕ(fϑ,fω,kT)=1supTϵ[0,T]min{0TG(T,r,ϑ(r))dr,0TG(T,r,ω(r))dr}+kTmax{0TG(T,r,ϑ(r))dr,0TG(T,r,ω(r))dr}+kT

=1supTϵ[0,T]0Tmin{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT0Tmax{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT

1supTϵ[0,T]0Tξ(T,r)min{ϑ(r),ω(r)}dr+kT0Tξ(T,r)max{ϑ(r),ω(r)}dr+kT

1supTϵ[0,T]min{ϑ(r),ω(r)}0Tξ(T,r)dr+kTmax{ϑ(r),ω(r)}0Tξ(T,r)dr+kT

1(min{ϑ(r),ω(r)}+Tmax{ϑ(r),ω(r)}+T)=Nϕ(ϑ,ω,T).

Thus Mϕ(fϑ,fω,kT)Mϕ(ϑ,ω,T)and  Nϕ(fϑ,fω,kT)Nϕ(ϑ,ω,T)  for  allϑ,ωD, 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.

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APA Style
Saleem, N., Javed, K., Uddin, F., Ishtiaq, U., Ahmed, K. et al. (2023). Unique solution of integral equations via intuitionistic extended fuzzy b-metric-like spaces. Computer Modeling in Engineering & Sciences, 135(1), 109-131. https://doi.org/10.32604/cmes.2022.021031
Vancouver Style
Saleem N, Javed K, Uddin F, Ishtiaq U, Ahmed K, Abdeljawad T, et al. Unique solution of integral equations via intuitionistic extended fuzzy b-metric-like spaces. Comput Model Eng Sci. 2023;135(1):109-131 https://doi.org/10.32604/cmes.2022.021031
IEEE Style
N. Saleem et al., "Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces," Comput. Model. Eng. Sci., vol. 135, no. 1, pp. 109-131. 2023. https://doi.org/10.32604/cmes.2022.021031

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