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.
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:
To introduce the notion of intuitionistic extended fuzzy b-metric-like space.
To enhance the literature of intuitionistic fuzzy fixed point theory.
To plot some graphical structure of obtained result.
To prove the existence and uniqueness of established results via integral equations.
To provide an application dynamic market equilibrium.
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:
ν∗ω=ω∗ν,∀ν,ω∈[0,1];
∗ is continuous;
ν∗1=ν,∀ν∈[0,1];
(ν∗ω)∗κ=ν∗(ω∗κ),∀ν,ω,κ∈[0,1];
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:
ν∘ω=ω∘ν,∀ν,ω∈[0,1];
∘ is continuous;
ν∘0=0,(∀)ν∈[0,1];
(ν∘ω)∘κ=ν∘(ω∘κ),∀ν,ω,κ∈[0,1];
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:
P(ϑ,δ)=0impliesϑ=δ;
P(ϑ,δ)=P(δ,ϑ);
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:
Qb(ϑ,δ,T)>0;
Qb(ϑ,δ,T)=1 impliesϑ=δ;
Qb(ϑ,δ,T)=Qb(δ,ϑ,T);
Qb(ϑ,β,(T+S))≥Qb(ϑ,δ,T)∗Qb(δ,β,S);
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:
Δα(ϑ,δ,0)=0;
Δα(ϑ,δ,T)=1⇔ϑ=δ;
Δα(ϑ,δ,T)=Δα(δ,ϑ,T);
Δα(ϑ,β,α(ϑ,β)(T+S))≥Δα(ϑ,δ,T)∗Δα(δ,β,S);
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:
Mb(ϑ,δ,T)+Nb(ϑ,δ,T)≤1;
Mb(ϑ,δ,T)>0;
Mb(ϑ,δ,T)=1⇔ϑ=δ;
Mb(ϑ,δ,T)=Mb(δ,ϑ,T);
Mb(ϑ,β,b(T+S))≥Mb(ϑ,δ,T)∗Mb(δ,β,S);
Mb(ϑ,δ,⋅) is a non-decreasing (ND) function of R+and limT→∞Mb(ϑ,δ,T)=1;
Nb(ϑ,δ,T)>0;
Nb(ϑ,δ,T)=0⇔ϑ=δ;
Nb(ϑ,δ,T)=Nb(δ,ϑ,T);
Nb(ϑ,β,b(T+S))≤Nb(ϑ,δ,T)∘Nb(δ,β,S);
Nb(ϑ,δ,⋅) is a non-increasing (NI) function of R+ and limT→∞Nb(ϑ,δ,T)=0,
then (D,Mb,Nb,∗,∘) is an IFBMS.
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:
Mϕ(ϑ,δ,T)+Nϕ(ϑ,δ,T)≤1;
Mϕ(ϑ,δ,T)>0;
Mϕ(ϑ,δ,T)=1impliesϑ=δ;
Mϕ(ϑ,δ,T)=Mϕ(δ,ϑ,T);
Mϕ(ϑ,β,ϕ(ϑ,β)(T+S))≥Mϕ(ϑ,δ,T)∗Mϕ(δ,β,S);
Mϕ(ϑ,δ,⋅) is a ND function of R+ and limT→∞Mϕ(ϑ,δ,T)=1;
Nϕ(ϑ,δ,T)>0;
Nϕ(ϑ,δ,T)=0implies ϑ=δ;
Nϕ(ϑ,δ,T)=Nϕ(δ,ϑ,T);
Nϕ(ϑ,β,ϕ(ϑ,β)(T+S))≤Nϕ(ϑ,δ,T)∘Nϕ(δ,β,S);
Nϕ(ϑ,δ,⋅) is a NI function of R+ and limT→∞Nϕ(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}2for 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 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.
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).
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)=0for 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)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)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∗⋯∗=1andlimn→∞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
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)=1−kTkT+max{ϑ6,δ6}=1−TT+max{ϑk6,δk6}≤1−TT+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.
Variation of L.H.S. = <inline-formula id="ieqn-315"><mml:math id="mml-ieqn-315"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> with R.H.S. = <inline-formula id="ieqn-316"><mml:math id="mml-ieqn-316"><mml:mspace width="thinmathspace" /><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> of an Example 3.15 for <inline-formula id="ieqn-317"><mml:math id="mml-ieqn-317"><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> and <inline-formula id="ieqn-318"><mml:math id="mml-ieqn-318"><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></inline-formula>
The matrix of values of L.H.S. = <inline-formula id="ieqn-324"><mml:math id="mml-ieqn-324"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula> in which first row represents the values of <inline-formula id="ieqn-325"><mml:math id="mml-ieqn-325"><mml:mi>δ</mml:mi></mml:math></inline-formula> and first column represents the values of <inline-formula id="ieqn-326"><mml:math id="mml-ieqn-326"><mml:mi>ϑ</mml:mi></mml:math></inline-formula>
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
0.9677
0.9375
0.9090
0.8823
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.1
0.9677
0.9677
0.9375
0.9090
0.8823
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.2
0.9375
0.9375
0.9375
0.9090
0.8823
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.3
0.9090
0.9090
0.9090
0.9090
0.8823
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.4
0.8823
0.8823
0.8823
0.8823
0.8823
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.5
0.8571
0.8571
0.8571
0.8571
0.8571
0.8571
0.8333
0.8108
0.7894
0.7692
0.7500
0.6
0.8333
0.8333
0.8333
0.8333
0.8333
0.8333
0.8333
0.8108
0.7894
0.7692
0.7500
0.7
0.8108
0.8108
0.8108
0.8108
0.8108
0.8108
0.8108
0.8108
0.7894
0.7692
0.7500
0.8
0.7894
0.7894
0.7894
0.7894
0.7894
0.7894
0.7894
0.7894
0.7894
0.7692
0.7500
0.9
0.7692
0.7692
0.7692
0.7692
0.7692
0.7692
0.7692
0.7692
0.7692
0.7692
0.7500
1
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
0.7500
The matrix of values of R.H.S. = <inline-formula id="ieqn-327"><mml:math id="mml-ieqn-327"><mml:mspace width="thinmathspace" /><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula> in which first row represents the values of <inline-formula id="ieqn-328"><mml:math id="mml-ieqn-328"><mml:mi>δ</mml:mi></mml:math></inline-formula> and first column represents the values of <inline-formula id="ieqn-329"><mml:math id="mml-ieqn-329"><mml:mi>ϑ</mml:mi></mml:math></inline-formula>
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
0.9090
0.8333
0.7692
0.7142
0.6666
0.6250
0.5882
0.5555
0.5263
0.5000
0.1
0.9090
0.9090
0.8333
0.7692
0.7142
0.6666
0.6250
0.5882
0.5555
0.5263
0.5000
0.2
0.8333
0.8333
0.8333
0.7692
0.7142
0.6666
0.6250
0.5882
0.5555
0.5263
0.5000
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
Variation of L.H.S. = <inline-formula id="ieqn-319"><mml:math id="mml-ieqn-319"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> with R.H.S. = <inline-formula id="ieqn-320"><mml:math id="mml-ieqn-320"><mml:mspace width="thinmathspace" /><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> of an Example 3.15 for <inline-formula id="ieqn-321"><mml:math id="mml-ieqn-321"><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> and <inline-formula id="ieqn-322"><mml:math id="mml-ieqn-322"><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></inline-formula>
The matrix of values of R.H.S. = <inline-formula id="ieqn-330"><mml:math id="mml-ieqn-330"><mml:mspace width="thinmathspace" /><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula> in which first row represents the values of <inline-formula id="ieqn-331"><mml:math id="mml-ieqn-331"><mml:mi>δ</mml:mi></mml:math></inline-formula> and first column represents the values of <inline-formula id="ieqn-332"><mml:math id="mml-ieqn-332"><mml:mi>ϑ</mml:mi></mml:math></inline-formula>
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.0909
0.1666
0.2307
0.2857
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.1
0.0909
0.0909
0.1666
0.2307
0.2857
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.2
0.1666
0.1666
0.1666
0.2307
0.2857
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.3
0.2307
0.2307
0.2307
0.2307
0.2857
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.4
0.2857
0.2857
0.2857
0.2857
0.2857
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.5
0.3333
0.3333
0.3333
0.3333
0.3333
0.3333
0.3750
0.4117
0.4444
0.4736
0.5000
0.6
0.3750
0.3750
0.3750
0.3750
0.3750
0.3750
0.3750
0.4117
0.4444
0.4736
0.5000
0.7
0.4117
0.4117
0.4117
0.4117
0.4117
0.4117
0.4117
0.4117
0.4444
0.4736
0.5000
0.8
0.4444
0.4444
0.4444
0.4444
0.4444
0.4444
0.4444
0.4444
0.4444
0.4736
0.5000
0.9
0.4736
0.4736
0.4736
0.4736
0.4736
0.4736
0.4736
0.4736
0.4736
0.4736
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
The matrix of values of L.H.S. = <inline-formula id="ieqn-333"><mml:math id="mml-ieqn-333"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mi>ϑ</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mi>δ</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula> in which first row represents the values of <inline-formula id="ieqn-334"><mml:math id="mml-ieqn-334"><mml:mi>δ</mml:mi></mml:math></inline-formula> and first column represents the values of <inline-formula id="ieqn-335"><mml:math id="mml-ieqn-335"><mml:mi>ϑ</mml:mi></mml:math></inline-formula>
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.0322
0.0625
0.0909
0.1176
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.1
0.0322
0.0322
0.0625
0.0909
0.1176
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.2
0.0625
0.0625
0.0625
0.0909
0.1176
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.3
0.0909
0.0909
0.0909
0.0909
0.1176
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.4
0.1176
0.1176
0.1176
0.1176
0.1176
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.5
0.1428
0.1428
0.1428
0.1428
0.1428
0.1428
0.1666
0.1891
0.2105
0.2307
0.2500
0.6
0.1666
0.1666
0.1666
0.1666
0.1666
0.1666
0.1666
0.1891
0.2105
0.2307
0.2500
0.7
0.1891
0.1891
0.1891
0.1891
0.1891
0.1891
0.1891
0.1891
0.2105
0.2307
0.2500
0.8
0.2105
0.2105
0.2105
0.2105
0.2105
0.2105
0.2105
0.2105
0.2105
0.2307
0.2500
0.9
0.2307
0.2307
0.2307
0.2307
0.2307
0.2307
0.2307
0.2307
0.2307
0.2307
0.2500
1
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
0.2500
Graph of <inline-formula id="ieqn-323"><mml:math id="mml-ieqn-323"><mml:mi>f</mml:mi><mml:mi>ϑ</mml:mi><mml:mo>=</mml:mo><mml:mi>ϑ</mml:mi></mml:math></inline-formula>, 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:D→D is an intuitionistic extended fuzzy b-like contraction mapping if there exists 0<k<1 such that
1Mϕ(fϑ,fδ,T)−1≤k[1Mϕ(ϑ,δ,T)−1]and
Nϕ(fϑ,fδ,T)≤kNϕ(ϑ,δ,T),for all ϑ,δ∈D 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
limT→∞Mϕ(ϑ,δ,T)=1andlimT→∞Nϕ(ϑ,δ,T)=0for all ϑ,δ∈D and T>0. Let f:D→D be an intuitionistic extended fuzzy b-like contraction mapping. Further, suppose that for an arbitrary ϑ0∈D, andn,q∈N, we have ϑn=fnϑ0=fνn−1. Then f has a unique FP.
Proof: Let ϑ0 be in D . Take ϑn=fnϑ0=fνn−1, n∈N. By using (4) and (5) for all T>0,n>q, we have
1Mϕ(ϑn,ϑn+1,T)−1=1Mϕ(fϑn−1,ϑn,T)−1≤k[1Mϕ(ϑn−1,ϑn,T)−1]=kMϕ(ϑn−1,ϑn,T)−k.
That is
1Mϕ(ϑn,ϑn+1,T)≤kMϕ(ϑn−1,ϑn,T)+(1−k)≤k2Mϕ(ϑn−2,ϑn−1,T)+k(1−k)+(1−k).
Continuing in this way, we get
1Mϕ(ϑn,ϑn+1,T)≤knMϕ(ϑ0,ϑ1,T)+kn−1(1−k)+kn−2(1−k)+⋯+k(1−k)+(1−k)≤knMϕ(ϑ0,ϑ1,T)+(kn−1+kn−2+⋯+1)(1−k)≤knMϕ(ϑ0,ϑ1,T)+(1−kn).
We obtain
1knMϕ(ϑ0,ϑ1,T)+(1−kn)≤Mϕ(ϑn,ϑn+1,T)and
Nϕ(ϑn,ϑn+1,T)=Nϕ(fϑn−1,ϑn,T)≤kNϕ(ϑn−1,ϑn,T)=Nϕ(fϑn−2,ϑn−1,T)≤k2Nϕ(ϑn−2,ϑn−1,T)≤⋯≤knNϕ(ϑ0,ϑ1,T)for all q∈N, 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+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), we deduce
Mϕ(ϑn,ϑn+q,T)≥1knMϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)))+(1−kn)∗1kn+1Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)))+(1−kn+1)∗1kn+2Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)))+(1−kn+2)∗⋯∗1kn+q−1Mϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)⋯ϕ(ϑn+q−1,ϑn+q)))+(1−kn+q−1)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+q−1Nϕ(ϑ0,ϑ1,Tq(ϕ(ϑn,ϑn+q)ϕ(ϑn+1,ϑn+q)ϕ(ϑn+2,ϑn+q)⋯ϕ(ϑn+q−1,ϑn+q))).
Consequently, for all n,q∈N, we obtain ϕ(ϑn,ϑn+q)k<1,where0<k<1. Therefore, from (vi),(xi),(1)and forn→∞,limn→∞Mϕ(ϑn,ϑn+q,T)=1∗1∗⋯∗=1andlimn→∞Nϕ(ϑn,ϑn+q,T)=0∘0∘⋯∘0=0,i.e., {ϑn} is a GCS. Since (D,Mϕ,Nϕ,∗,∘) is a G-complete IEFBMLS, there exists
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 investigate that ϑ is a FP of f. Using (v),(x)and(1), we obtain
1Mϕ(fϑn,fϑ,T)−1≤k[1Mϕ(ϑn,ϑ,T)−1]=kMϕ(ϑn,ϑ,T)−k.
That is,
1kMϕ(ϑn,ϑ,T)+(1−k)≤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))+(1−k)→1∗1=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))→0∘0=0asn→∞.
This yields that fϑ=ϑ, a FP. Now, we show the uniqueness. Suppose fc=c for some c∈D, then
1Mϕ(ϑ,c,T)−1=1Mϕ(fϑ,fc,T)−1≤k[1Mϕ(ϑ,c,T)−1]<1Mϕ(ϑ,c,T)−1which is a contradiction. Also,
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}ifϑ≠δ
Also take
Mϕ(ϑ,δ,T)=e−max{ϑ,δ}2TandNϕ(ϑ,δ,T)=1−e−max{ϑ,δ}2Twith ν∗ω=ν.ωandν∘ω=max{ν,ω}. Then (D,Mϕ,Nϕ,∗,∘) is a G-complete IEFBMLS. Define f:D→D by
f(ϑ)={0ifϑ∈[0,12],ϑ9ifϑ∈(12,1]
Then we have four cases:
If ϑ,δ∈[0,12],thenfϑ=fδ=0;
If ϑ∈[0,12]andδ∈(12,1],thenfϑ=0andfδ=δ9;
If δ∈[0,12]andϑ∈(12,1],thenfδ=0andfϑ=δ9;
If ϑ,δ∈(12,1],thenfϑ=δ9andfδ=δ9;
In all 1−4 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)−1≤k[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=δkifϑ≠δ,Nϕ(ϑ,δ,T)={0ifϑ=δ1−kifϑ≠δfor all k∈D. Then (D,Mϕ,Nϕ,∗,∘) is an G-complete IEFBMLS with CTN a∗b=ab and CTCN a∘b=max{a,b}. Define f:D→D by
f(ϑ)=1−2−ϑ3.
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)−1≤k[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.
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)+β∫egF(l,j)ϑ(l)djforl,j∈[e,g]where β>0,f(j) is a fuzzy function of j∈[e,g] and F∈D. Define MϕandNϕ by
Mϕ(ϑ(l),δ(l), T)=supl∈[e,g]TT+max{ϑ(l),δ(l)}2and
Nϕ(ϑ(l),δ(l), T)=1−supl∈[e,g]TT+max{ϑ(l),δ(l)}2for all ϑ,δ∈D and T>0, with the CTN and CTCN defined by ν∗ω=ν.ωandν∘ω=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,δ∈D, k∈(0,1) and ∀l,j∈[e,g]. Also consider (β∫egdj)2≤k<1. then fuzzy integral equation in Eq. (9) has a unique solution.
Proof: Define f:D→D by
fϑ(l)=f(j)+β∫egF(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,δ∈D, 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)+β∫egF(l,j)e(l)dj,f(j)+β∫egF(l,j)e(l)dj}2=supl∈[e,g]kTkT+max{β∫egF(l,j)e(l)dj,β∫egF(l,j)e(l)dj}2=supl∈[e,g]kTkT+max{F(l,j)ϑ(l),F(l,j)δ(l)}2(β∫egdj)2≥supl∈[e,g]TT+max{ϑ(l),δ(l)}2≥Mϕ(ϑ(l),δ(l), T).Nϕ(fϑ(l),fδ(l),kT)=1−supl∈[e,g]kTkT+max{fϑ(l),fδ(l)}2=1−supl∈[e,g]kTkT+max{f(j)+β∫egF(l,j)e(l)dj,f(j)+β∫egF(l,j)e(l)dj}2=1−supl∈[e,g]kTkT+max{β∫egF(l,j)e(l)dj,β∫egF(l,j)e(l)dj}2=1−supl∈[e,g]kTkT+max{F(l,j)ϑ(l),F(l,j)δ(l)}2(β∫egdj)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:D→D be
fϑ(l)=f(j)+β∫egF(l,j)e(l)dj for all l,j∈[e,g].
Suppose the below conditions meet:
max{F(l,j)ϑ(l),F(l,j)δ(l)}≤max{ϑ(l),δ(l)} for ϑ,δ∈D, k∈(0,1) and ∀l,j∈[e,g],
(β∫egdj)2≤k<1
Then integral Eq. (9) has a solution.
We can prove easily by follow the above proof.
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)dT2Qd=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
(ω1−ω2)d2P(T)dT2+(α1−α2)ddP(T)dT+(υ1−υ2)P(T)=−(σ1−σ2)
Letting ω=ω1−ω2,α=α1−α2,υ=υ1−υ2andσ=σ1−σ2 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,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(T−r)if0≤r≤T≤TTαμ2(r−T)if0≤T≤r≤T≤T.
We will show the existence of a solution to the integral equation
P(T)=∫0TG(T,r,P(r))dr.
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}+Tand
Nϕ(ϑ,ω,T)=1−supTϵ[0,T]min{ϑ,ω}+Tmax{ϑ,ω}+Tfor all ϑ,ωϵD with the CTN ′∗′ such that a∗b=a⋅b and CTCN ∘ such that a∘b=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:D→D defined by
fP(T)=∫0TG(T,r,P(r)dr.
Theorem 4.1 Consider Eq. (11) and suppose that
G:[0,T]×[0,T]→R+ is continuous function,
There exist a continuous function ξ:[0,T]×[0,T]→R+ such that supT∈[0,T]∫0Tξ(T,r)dr≥1;
Then, the integral Eq. (11) has a unique solution.
Proof: For ϑ,ω∈D, 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+kT∫0Tmax{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT≥supTϵ[0,T]∫0Tξ(T,r)min{ϑ(r),ω(r)}dr+kT∫0Tξ(T,r)max{ϑ(r),ω(r)}dr+kT≥supTϵ[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)=1−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=1−supTϵ[0,T]∫0Tmin{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT∫0Tmax{G(T,r,ϑ(r)),G(T,r,ω(r))}dr+kT≤1−supTϵ[0,T]∫0Tξ(T,r)min{ϑ(r),ω(r)}dr+kT∫0Tξ(T,r)max{ϑ(r),ω(r)}dr+kT≤1−supTϵ[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,ω∈D, and all conditions of Theorem 3.14 are satisfied. Therefore, Eq. (11) has a unique fixed point.
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.
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.
ReferencesZadeh, L. A. (1965). Fuzzy sets. Information and Control,8(3),338–353. DOI 10.1016/S0019-9958(65)90241-X.Schweizer, B., Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics,10(1),314–334. DOI 10.2140/pjm.Kramosil, I., Michlek, J. (1975). Fuzzy metric and statistical metric spaces. Kybernetika,11(5),336–344.George, A., Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems,64(3),395–399. DOI 10.1016/0165-0114(94)90162-7.Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems,27(3),385–389. DOI 10.1016/0165-0114(88)90064-4.Dey, D., Saha, M. (2014). An extension of banach fixed point theorem in fuzzy metric space. Boletim da Sociedade Paranaense de Matematica,32(1),299–304. DOI 10.5269/bspm.v32i1.17260.Nadaban, N. (2016). Fuzzy b-metric spaces. International Journal of Computers Communications and Control,11(2),273–281. DOI 10.15837/ijccc.2016.2.2443.Gregory, V., Sapena, A. (2002). On fixed point theorems in fuzzy metric spaces. Fuzzy set and Systems,125(2),245–253. DOI 10.1016/S0165-0114(00)00088-9.Bashir, S., Saleem, N., Husnine, S. M. (2021). Fixed point results of a generalized reversed F-contraction mapping and its application. Aims Mathematics,6(8),8728–8741. DOI 10.3934/math.2021507.Harandi, A. (2012). Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications,2012(1),1–10.Alghamdi, M. A., Hussain, N., Salimi, P. (2013). Fixed point and coupled fixed point theorems on b-metric-like spaces. Journal of Inequalities and Applications,2013(1),1–25. DOI 10.1186/1029-242X-2013-402.Shukla, S., Abbas, M. (2014). Fixed point results in fuzzy metric-like spaces. Iranian Journal of Fuzzy Systems,11(5),81–92.Javed, K., Uddin, F., Aydi, H., Arshad, M., Ishtiaq, U.et al. (2021). On fuzzy b-metric-like spaces. Journal of Function Spaces,2021,1–10. DOI 10.1155/2021/6615976.Mehmood, F., Ali, R., Ionescu, C., Kamran, T. (2017). Extended fuzzy b-metric spaces. Journal of Mathematical Analysis,6(8),124–131.Park, J. H. (2004). Intuitionistic fuzzy metric spaces. chaos. Solitons and Fractals,22(5),1039–1046. DOI 10.1016/j.chaos.2004.02.051.Rafi, M., Noorani, M. S. M. (2006). Fixed theorems on intuitionistic fuzzy metric space. Iranian Journal of Fuzzy Systems,3(1),23–29.Alaca, C., Turkoglu, D., Yildiz, C. (2006). Fixed points in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals,29(5),1073–1078. DOI 10.1016/j.chaos.2005.08.066.Mohamad, A. (2007). Fixed-point theorems in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals,34(5),1689–1695. DOI 10.1016/j.chaos.2006.05.024.Saleem, N., Iqbal, I., Iqbal, B., Radenovic, S. (2020). Coincidence and fixed points of multivalued F-contractions in generalized metric space with application. Journal of Fixed Point Theory and Applications,22(4),1–24. DOI 10.1007/s11784-020-00815-3.Saleem, N., Ali, B., Raza, Z., Abbas, M. (2019). Fixed points of suzuki-type generalized multivalued (f, θ, L) almost contractions with applications. Filomat,33(2),499–518. DOI 10.2298/FIL1902499S.Saleem, N., Abbas, M., de la sen, M. (2019). Optimal approximate solution of coincidence points equations in fuzzy metric spaces. Mathematics,7(4),1–13. DOI 10.3390/math7040327.Saleem, N., Abbas, M., Raza, Z. (2017). Fixed fuzzy point results of generalized suzuki type F-contraction mappings in ordered metric spaces. Georgian Journal of Mathematics,24(4),1–14.de la sen, M., Abbas, M., Saleem, N. (2017). On optimal fuzzy best proximity coincidence points of proximal contractions involving cyclic mappings in non-archimedean fuzzy metric spaces. Mathematics,5(2),1–20.Saleem, N., Abbas, M., Raza, Z. (2016). Optimal coincidence best approximation solution in non-archimedean fuzzy metric spaces. Iranian Journal of Fuzzy Systems,13(3),113–124.Abbas, M., Saleem, N., de la sen, M. (2016). Optimal coincidence point results in partially ordered non-archimedean fuzzy metric spaces. Fixed Point Theory and Applications,2016(1),1–18. DOI 10.1186/1687-1812-2012-1.Saleem, N., Ali, B., Abbas, M., Raza, Z. (2015). Fixed points of suzuki type generalized multivalued mappings in fuzzy metric spaces with applications. Fixed Point Theory and Applications,2015(1),1–18. DOI 10.1186/s13663-015-0284-7.Shukla, S., Gopal, D., Roldán-López-deHierro, A. F. (2016). Some fixed point theorems in 1-M-complete fuzzy metric-like spaces. International Journal of General Systems,45(8),815–829. DOI 10.1080/03081079.2016.1153084.Abbas, M., Lael, F., Saleem, N. (2020). Fuzzy b-metric spaces: Fixed point results for contraction correspondences and their application. Axioms,9(2),1–12. DOI 10.3390/axioms9020036.Sintunavarat, W., Kumam, P. (2012). Fixed theorems for a generalized intuitionistic fuzzy contraction in intuitionistic fuzzy metric spaces. Thai Journal of Mathematics,10(1),123–135.Saadati, R., Park, J. H. (2006). On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals,27(2),331–344. DOI 10.1016/j.chaos.2005.03.019.Konwar, K. (2020). Extension of fixed results in intuitionistic fuzzy b-metric spaces. Journal of Intelligent & Fuzzy Systems,39(5),7831–7841. DOI 10.3233/JIFS-201233.Mahmood, T., Ali, W., Ali, Z., Chinram, R. (2021). Power aggregation operators and similarity measures based on improved intuitionistic hesitant fuzzy sets and their applications to multiple attribute decision making. Computer Modeling in Engineering & Sciences,126(3),1165–1187. DOI 10.32604/cmes.2021.014393.