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

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 b≥1 with a function α: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-metric-like space (IEFBMLS). In which, we generalize the concept of IFBMS by replacing the coefficient b≥1 with a function α:D×D→[1,∞) in both triangular inequalities and we replace condition (III) of IFBMS, Mb(ϑ,δ,T)=1⇔ϑ=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.    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.    If ν≤κ and ω≤d, with ν,ω,κ,d∈[0,1], then ν∗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.    If ν≤κ and ω≤d, with ν,ω,κ,d∈[0,1], then ν∘ω≤κ∘d.

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,β∈D. 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,β∈DandT,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,β∈DandT,S>0:

Δ1)   Δα(ϑ,δ,0)=0;

Δ2)   Δα(ϑ,δ,T)=1⇔ϑ=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, b≥1 and Mb,Nb be FSs on D×D×(0,∞). If (D,Mb,Nb,∗,∘) verifies the following for all ϑ,δ∈DandS,T>0:

(I) Mb(ϑ,δ,T)+Nb(ϑ,δ,T)≤1;

(II) Mb(ϑ,δ,T)>0;

(III) Mb(ϑ,δ,T)=1⇔ϑ=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  limT→∞⁡Mb(ϑ,δ,T)=1;

(VII) Nb(ϑ,δ,T)>0;

(VIII) Nb(ϑ,δ,T)=0⇔ϑ=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 limT→∞⁡Nb(ϑ,δ,T)=0,

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,δ∈DandS,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 limT→∞⁡Mϕ(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 limT→∞⁡Nϕ(ϑ,δ,T)=0,

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,δ∈DandT>0. Define the CTN by: ν∗ω=ν⋅ω and CTCN ′′∘′′ by ν∘ω=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 a∗b=ab and CTCN a∘b=maxa,b.

Remark 3.5 Above example also satisfied for CTN a∗b=min{a,b} and CTCN a∘b=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)=1−T+min{ϑ,δ}T+max{ϑ,δ}.

Then (D,Mϕ,Nϕ,∗,∘) is an IEFBMLS with CTN a∗b=ab and CTCN a∘b=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)=e−max{ϑ,δ}Tn,Nϕ(ϑ,δ,T)=1−e−max{ϑ,δ}Tnfor  alln∈N,  ϑ,δ∈D,T>0.

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

Remark 3.8 The above proposition also satisfied for CTN a∗b=min{a,b} and CTCN a∘b=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  alln∈N,  ϑ,δ∈D,T>0.

Then (D,Mϕ,Nϕ,∗,∘) is an IEFBMLS with CTN a∗b=ab and CTCN a∘b=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,δ∈D  and  T>0, define CTN ∗ by ν∗ω=ν⋅ω and CTCN ∘ by ν∘ω=max{ν,ω} and define ϕ by

ϕ(ϑ,δ)={1ifϑ=δ,1+max{ϑ,δ}min{ϑ,δ}ifϑ≠δ

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,limn→∞⁡Mϕ(ϑn,ϑn+q,T)andlimn→∞⁡Nϕ(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,

limn→∞⁡Mϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T)andlimn→∞⁡Nϕ(ϑn,ϑ,T)=Nϕt(ϑ,ϑ,T).

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

limn→∞⁡Mϕ(ϑn,ϑn+q,T)=limn→∞⁡Mϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T),

limn→∞⁡Nϕ(ϑn,ϑn+q,T)=limn→∞⁡Nϕ(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

limT→∞⁡Mϕ(ϑ,δ,T)=1andlimT→∞⁡Nϕ(ϑ,δ,T)=0(1)

for all ϑ,δ∈D and T>0. Let f:D→D be a mapping satisfying

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

for all ϑ,δ∈D and T>0 with 0<k<1. Further, suppose that for an arbitrary ϑ0∈D    andn,q∈N, we have

ϕ(ϑn,ϑn+q)<1k,

where ϑn=fnϑ0=fνn−1. Then f has a unique FP.

Proof: Let ϑ0 be a random element in D and consider ϑn=fnϑ0=fνn−1, n∈N. By using (2) for all T>0, we have

Mϕ(ϑn,ϑn+1,kT)=Mϕ(fϑn−1,fϑn,kT)≥Mϕ(ϑn−1,ϑn,T)≥Mϕ(ϑn−2,ϑn−1,Tk)≥Mϕ(ϑn−3,ϑn−2,Tk2)≥⋯≥Mϕ(ϑ0,ϑ1,Tkn)

and

Nϕ(ϑn,ϑn+1,kT)=Nϕ(fϑn−1,fϑn,kT)≤Nϕ(ϑn−1,ϑn,T)≤Nϕ(ϑn−2,ϑn−1,Tk)≤Nϕ(ϑn−3,ϑn−2,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 q∈N,  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+q−1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)⋯ϕ(ϑn+q−1,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+q−1,ϑn+q,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)⋯ϕ(ϑn+q−1,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+q−1,ϑn+q))kn+q−1)

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+q−1,ϑn+q))kn+q−1).

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

limn→∞⁡Mϕ(ϑn,ϑn+q,T)=1∗1∗⋯∗=1

and

limn→∞⁡Nϕ(ϑn,ϑn+q,T)=0∘0∘⋯∘0=0.

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

limn→∞⁡Mϕ(ϑn,ϑn+q,T)=limn→∞⁡Mϕ(ϑn,ϑ,T)=Mϕ(ϑ,ϑ,T),

limn→∞⁡Nϕ(ϑn,ϑn+q,T)=limn→∞⁡Nϕ(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)))→1∗1=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)))→0∘0=0asn→∞.

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

1≥Mϕ(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

0≤Nϕ(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)=1−TT+max{ϑ,δ}

with CTN ∗ such that a∗b=a⋅b and CTCN ∘ such that a∘b=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×D→D as