# Ordered-Theoretic Fixed Point Results in Fuzzyb-Metric Spaces with an Application

Authors: Khalil Javed; Fahim Uddin; Hassen Aydi; Aiman Mukheimer; Muhammad Arshad
Journal: Journal of Mathematics
 (2021)

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## Abstract

The aim of this manuscript is to initiate the study of the Banach contraction inR-fuzzy b-metric spaces and discuss some related fixed point results to ensure the existence and uniqueness of a fixed point. A nontrivial example is imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the provided results, an application is presented to solve the first kind of a Fredholm-type integral equation.


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## Body

## 1. Introduction and Preliminaries

Since the axiomatic interpretation of metric spaces and the inception of the Banach contraction principle, many authors have studied fixed point theory vividly. A number of results have been introduced, and metric fixed point has been generalized in different directions. In this connectedness, Bakhtin [1] and Czerwik [2] gave a generalization of a metric space and named it as a b-metric space. Zadeh [3] introduced the concept of fuzzy sets and generalized the concept of metric spaces and fuzzy sets and named them as fuzzy metric spaces, which became a point of interest for many authors [2, 4]. Nădăban [5] extended the concept of a fuzzy metric and introduced the notion of fuzzy b-metric spaces. For related works in this setting, refer to [6–9].Recently, Baghani and Ramezani [10] tossed the concept of orthogonal sets and gave an extension of the Banach contraction principle. For more details, refer to [10–24].In this article, we further aim to establish fixed point results in the setting ofR-complete fuzzy b-metric spaces. We provide an example dealing with an R-fuzzy b-metric space, but it is not a fuzzy b-metric space. The presented results improve and generalize many results in the literature.First, we recall some basic definitions and notions, which are essential for this work.Definition 1 (see [11]).
A binary operation∗: [0, 1] × [0, 1] ⟶ [0, 1] is referred to as a continuous t-norm if the following assumptions hold:(1)
e∗f=f∗e,∀e,f∈0,1(2)
e∗1=e,∀e∈0,1(3)
e∗f∗s=e∗f∗s,∀e,f,s∈0,1(4)
Ife≤s and f≤u, with e,f,s,u∈0,1, then e∗f≤s∗u
Some fundamental examples of at-norm are e∗f=e⋅f,e∗f=mine,f, and e∗f=maxe+f−1,0.Definition 2 (see [12, 13]).
A 3-tupleH,M,∗ is said to be a fuzzy metric space if H≠M is an arbitrary set, ∗ is a continuous t-norm, and M is a fuzzy set on H×H×0,∞ meeting the following conditions for all σ,H,z∈M,τ,s>0:(B1)
Mσ,M,τ>0(B2)
Mσ,M,τ=1 iff σ=M(B3)
Mσ,M,τ=MM,σ,τ(B4)
Mσ,z,τ+s≥Mσ,M,τ∗MM,z,s(B5)
Mσ,M,M:0,∞⟶0,1 is continuousExample 1 (see [12]).
LetH,d be a metric space with a continuous t-norm ɑ∗M=ɑ⋅M, and let M be a fuzzy set defined on H×H×0,∞ by(1)Mσ,M,τ=ττ+dσ,M.
Then,H,M,∗ is called a standard fuzzy metric space.Definition 3 (see [6]).
A 4-tupleH,M,∗,u is said to be a fuzzy b-metric space if H≠M is an arbitrary set, ∗ is a continuous t-norm, and M is a fuzzy set on H×H×0,∞ meeting the following conditions for all σ,M,z∈H,τ,s>0 and for a given real number u≥1:(B1)
Mσ,M,τ>0(B2)
Mσ,M,τ=1 iff σ=M(B3)
Mσ,M,τ=MM,σ,τ(B4)
Mσ,z,τ+s≥Mσ,M,τ/u∗MM,z,s/u(B5)
Mσ,M,M:0,∞⟶0,1 is continuousExample 2 (see [7]).
LetMσ,M,τ=e−σ−Mp/τ, where p>1 represents a real number. It is then simple to prove that M is a fuzzy b-metric with u=2p−1. It should be noted that, for p = 2, H,M,∗ is not a fuzzy metric space.Definition 4.
AssumeH≠M and R∈H×H is a binary relation. Suppose there exists σ0∈M such that σ0Rσ or σRσ0 for all σ∈H. Then, we say that H is an R-set.Example 3.
(i)
LetH=0,∞ and define σRM if σM=minσ,M; then, by putting σ0=1, H,R is an R-set.(ii)
SupposeM is a set of scalar matrices of order 2×2 with entries from natural numbers (i.e., M=ɑ00ɑ, for all ɑ∈N). Define the relation R by(2)ARBifdetA≤detB.
Then, by takingA=I, M,R is an R-set.Definition 5 (see [10]).
Suppose thatH,R is an R-set. A sequence σn for all n∈ℕ is said to be an R-sequence if ∀n;σnRσn+1 or ∀n;σn+1Rσn.Definition 6.
(see [14]).(a)
A metric spaceH,d is an R-metric space if H,R is an R-set.(b)
A mappingƑ:H⟶H is R-continuous at σ∈H if for each R-sequence σn for all n∈ℕ in H if limn⟶∞dσn,σ=0, then limn⟶∞dƑσn,Ƒσ=0. Furthermore, Ƒ is R-continuous if Ƒ is R-continuous at each σ∈H.(c)
A mappingƑ:H⟶H is called R-preserving if σRƑ, then ƑσRƑM for allσ,M∈H.(d)
AnR-sequence σn in H is said to be an R-Cauchy sequence if for every ε > 0, there exists an integer N such that d(σn, σm) < ε for all n ≥ ℕ and m ≥ ℕ. It is clear that σnRσm or σmRσn.(e)
H is R-complete if every R-Cauchy sequence is convergent.


## 2. Main Results

We start this section with the introduction ofR-fuzzy b-metric spaces.Definition 7.
LetH≠M and R be a reflexive binary relation on H. Let ∗ be a continuous t-norm and H be a fuzzy set on H×H×0,∞. Suppose that, for all τ,s>0 and for all σ,M,z∈H, with either (σRz or zRσ), either (σRM or MRσ), and either (MRz or zRM), the following conditions hold:(1)
Mσ,M,τ>0(2)
Mσ,M,τ=1 if and only if σ=M(3)
Mσ,M,τ=MM,σ,τ(4)
Mσ,z,τ+s≥Mσ,M,τ/u∗MM,z,s/u, where u≥1(5)
Mσ,M,M:0,∞⟶0,1 is continuous
Then,H,M,∗,u,R is called an R-fuzzy b-metric space with the coefficient u≥1.Remark 1.
In the above definition, the setH is endowed with a reflexive binary relation R, and M is a fuzzy set on H×H×0,∞ satisfying (1)–(5) for those comparable elements with respect to the reflexive binary relation R. An R-fuzzy b-metric may not be a fuzzy b-metric.
The following simplest example shows that theR-fuzzy b-metric with u=4 does not need to be a fuzzy b-metric with u=4.Example 4.
LetH=−1,1 and Mσ,M,τ=e−σ−M3/τ. Define a binary relation such that σRM iff σ≥M. It is clear that Mσ,M,τ is an R-fuzzy b-metric on H with u=4.
Note that forσ=0.1,M=0.5, and z=0.8, the following condition does not hold:(3)Mσ,z,τ+s≥Mσ,M,τu∗MM,z,su.
So,Mσ,M,τ is not a fuzzy b-metric.Definition 8.
LetH,M,∗,u,R represent an R-fuzzy b-metric space.(a)
A sequenceσn for all n∈ℕ is said to be an R-sequence if ∀n;σnRσn+1 or ∀n;σn+1Rσn.(b)
A Cauchy sequenceσn is said to be an R-Cauchy sequence if ∀n;σnRσn+1 or ∀n;σn+1Rσn.(c)
A mappingƑ:H⟶H is R-continuous at σ∈M if for each R-sequence σn for all n∈ℕ in M with limn⟶∞Mσn,σ,τ=1 for all τ>0, then limn⟶∞MƑσn,Ƒσ,τ=1 for all τ>0. Furthermore, Ƒ is R-continuous if Ƒ is R-continuous at each σ∈H.(d)
A mappingƑ:H⟶H is called R-preserving if σRM, then ƑσRƑM for allσ,M∈H.(e)
If eachR-Cauchy sequence is convergent, then M is R-complete.
Motivated by the work of Baghani and Ramezani [10] and Hezarjaribi et al. [14], we introduce the concept of Banach contraction principle in the setting of R-fuzzy b-metric spaces.Definition 9.
LetH,M,∗,u,R be an R-fuzzy b-metric space. A map Ƒ:H⟶H is an R-contraction if there exists q∈0,1 such that, for every τ>0 and σ,M∈H with σRM, we have(4)MƑσ,ƑM,qτ≥Mσ,M,τ.Theorem 1.
Assume thatH,M,∗,u,R is an R-complete fuzzy b-metric space. Let Ƒ:H⟶H be an R-continuous, R-contraction, and R-preserving mapping. Thus, Ƒ has a unique fixed point σ∗∈H. Furthermore,(5)limn⟶∞MƑnσ,σ∗,τ=1,for allσ∈Handτ>0.Proof.
SinceH,M,∗,u,R is an R-complete fuzzy b-metric space, there exists σ0∈H such that(6)σ0RM,for allM∈H.
This yields thatσ0RƑσ0. Assume that(7)σ1=Ƒσ0,σ2=Ƒ2σ0=Ƒσ1,…,σn=Ƒnσ0=Ƒσn−1,for alln∈N.
SinceƑ is R-preserving, σn is an R-sequence and Ƒ is an R-contraction. Thus,(8)Mσn+1,σn,qτ=MƑσn,Ƒσn−1,qτ≥Mσn,σn−1,τ,for all n∈N and τ>0. Therefore, by applying the above expression, we deduce(9)Mσn+1,σn,τ≥Mσn+1,σn,qτ=MƑσn,Ƒσn−1,qτ≥Mσn,σn−1,τ=MƑσn−1,Ƒσn−2,τ≥Mσn−1,σn−2,τq≥…≥Mσ1,σ0,τqn,for all n∈N and τ > 0. Thus, from (9) and (B4), we have(10)Mσn,σn+p,τ≥Mσn,σn+1,τu∗Mσn+1,σn+p,τu≥Mσn,σn+1,τu∗Mσn+1,σn+2,τu2∗Mσn+2,σn+3,τu3∗…∗Mσn+p−1,σn+p,τun+p≥Mσ1,σ0,τuqn∗Mσ1,σ0,τu2qn∗…∗Mσ1,σ0,τun+pqn.
Here,u is an arbitrary positive integer. We know that limn⟶∞Mσ,M,τ=1 for all σ,M∈H and τ>0. From (10), we get(11)limn⟶∞Mσn,σn+p,τ≥1∗1∗…∗1=1.
Then,σn is an R-Cauchy sequence. The hypothesis of R-completeness of the fuzzy b-metric space H,M,∗,u,R ensures that there exists σ∗∈H such that Mσn,σ∗,τ⟶1 as n⟶+∞ for all τ>0. Since Ƒ is an R-continuous mapping, one writes Mσn+1,Ƒσ∗,τ=MƑσn,Ƒσ∗,τ⟶1 as n⟶+∞. Hence,(12)Mσ∗,Ƒσ∗,τ≥Mσ∗,σn+1,τ2u∗Mσn+1,Ƒσ∗,τ2u.
Asn⟶+∞, we get Ƒσ∗,Ƒσ∗,τ=1∗1=1; hence, Ƒσ∗=σ∗.
To show the uniqueness of the fixed point for the mappingƑ, assume that σ∗ and M∗ are two fixed points of Ƒ such that σ∗≠M∗. We have(13)σ0Rσ∗andσ0RM∗.
SinceM is R-preserving, we can write(14)Ƒnσ0RƑnσ∗andƑnσ0RƑnƑυ∗,for all n∈N. Using (4), we have(15)MƑnσ0,Ƒnσ∗,τ≥MƑnσ0,Ƒnσ∗,qτ≥Mσ0,σ∗,τqn,MƑnσ0,Ƒnυ∗,τ≥MƑnσ0,Ƒnυ∗,qτ≥Mσ0,υ∗,τqn.
Hence,(16)Mσ∗,υ∗,τ=MƑnσ∗,Ƒnυ∗,τ≥MƑnσ0,Ƒnσ∗,τ2u∗MƑnσ0,Ƒnυ∗,τ2u≥Mσ0,σ∗,τ2uqn∗Mσ0,υ∗,τ2uqn⟶1asn⟶∞.
So,σ∗=M∗; hence, σ∗ is the unique fixed point.Corollary 1.
LetH,M,∗,u,R be an R-complete fuzzy b-metric space. Let H:H⟶H be an R-contraction and R-preserving. Also, if σn is an R-sequence with σn⟶σ∈Ƒ, then σRσn for all n∈ℕ. Therefore, M has a unique fixed point σ∗∈H. Furthermore, limn⟶∞MƑnσ,σ&lowast;,τ=1, for all σ∈H and τ>0.Proof.
The proof of this result moves along the same lines as in Theorem1, that is, σn is an R-Cauchy sequence and converges to σ∗∈H. Hence, σ∗Rσn for all n∈ℕ. From (4), we have(17)MƑσ∗,σn+1,τ=MƑσ∗,Ƒσn,τ≥MƑσ∗,Ƒσn,τq≥Mσ∗,σn,τ.
Also,(18)limn⟶∞MƑσ∗,σn+1,τ=1.
Hence,(19)Mσ∗,Ƒσ∗,τ≥Mσ∗,σn+1,τ2u∗Mσn+1,Ƒσ∗,τ2u.
Asn⟶+∞, we get Mσ∗,Ƒσ∗,τ=1∗1=1, and so, Ƒσ∗=σ∗. The rest of the proof is the same as in Theorem 1.Corollary 2.
LetƑ,M,∗,u,R be an R-complete fuzzy b-metric space and Ƒ:H⟶H be an R-continuous and R-preserving mapping. Suppose that there exist q∈0,1/2 and τ>0 such that(20)MƑσ,Ƒυ,qτ≥MƑσ,σ,τ2+MƑυ,υ,τ2.
Then,M has a unique fixed point.Corollary 3.
LetH,M,∗,u,R be an R-complete fuzzy b-metric space and Ƒ:H⟶H be an R-continuous and R-preserving mapping. Assume that there exist q∈0,1/u and τ>0 such that(21)MƑσ,Ƒυ,qτ≥minMƑσ,σ,τ,MƑυ,υ,τ.
Then,Ƒ has a unique fixed point.Proof.
The proof is a part of the next corollary.Corollary 4.
LetƑ,M,∗,u,R be an R-complete fuzzy b-metric space and Ƒ:H⟶H be an R-continuous and R-preserving mapping. Assume that there exist q∈0,1/u and τ>0 such that(22)MƑσ,Ƒυ,qτ≥minMƑσ,σ,τ,MƑυ,υ,τ,Mσ,υ,τ.
Then,Ƒ has a unique fixed point.Proof.
This corollary is a generalization of Theorem 2.5 in [8]. It is easy to prove this result by the help of Theorem 1 of this article and Theorem 2.5 of [8].Example 5.
LetH=−1,1. The relation on H is defined as σRMMσ≥M. Define the R-fuzzy b-metric given as in Example 4:(23)Mσ,M,τ=e−σ−M3/τ,ifτ>0,0,ifτ≤0,with the t-norm ɑ∗M=ɑMM. Let σn be an R-sequence in H such that σn=1. Hence, σnconverges to 1. Therefore, H,M,∗,u,R is an R-complete fuzzy b-metric space with u=4.
DefineƑ:H⟶H by(24)Ƒσ=σ4ifσ∈0,1,0ifσ∈−1,0.
Note the following:(1)
Ifσ∈0,1 and υ∈0,1, then Ƒσ=σ/4 andƑυ=υ/4(2)
Ifσ∈0,1 and υ∈−1,0, then Ƒσ=σ/4 and Ƒυ=0(3)
Ifσ∈−1,0 and σ∈−1,0, then Ƒσ=0 and Ƒυ=0
In all cases, we haveƑσ≥ƑM. Thus, Ƒ is an R-preserving map.
Letσn be an arbitrary R-sequence in H so that σn converges to σ∈H. Now,(25)limn⟶∞Mσn,σ,τ=limn⟶∞e−σn−σ3/τ.
Asσn converges to σ∈H, we have e−03/τ=e0=1.
Now, we need to show thatlimn⟶∞MƑσn,Ƒσ,qτ=1. For this purpose, there are some cases.(1)
Takeσn,σ∈−1,0; then,(26)limn⟶∞MƑσn,Ƒσ,qτ=limn⟶∞M0,0,qτ=limn⟶∞e0=1.(2)
Takeσn,σ∈0,1; then,(27)limn⟶∞MƑσn,Ƒσ,qτ=limn⟶∞Mσn4,σ4,qτ=limn⟶∞e−σn−σ3/64qτ.Asσn converges to σ∈H, we have e−03/64τ=e0=1.(3)
Now, takeσn∈0,1 and σ∈−1,0; then,(28)limn⟶∞MƑσn,Ƒσ,qτ=limn⟶∞Mσn4,0,qτ=limn⟶∞e−σn3/64qτ.
Asn⟶∞, we can easily see limn⟶∞e−σn3/64qτ=e0=1.
Hence,Ƒ is R-continuous.
For eachσ,M∈H with σRM, we have the following.Case (a) Forσ,H∈0,1, we have(29)MƑσ,Ƒυ,qτ=Mσ4,υ4,qτ=e−σ−M3/64qτ≥e−σ−M3/τ=Mσ,υ,τ.Case (b) Forσ, υ∈−1,0, we have(30)MƑσ,Ƒυ,qτ=M0,0,qτ=e0≥e−σ−M3/τ=Mσ,υ,τ.
Hence,Ƒ is an R-contraction. Hence, by Theorem 1, Ƒ has a unique fixed point.


## 3. An Application to an Integral Equation

Within this part, we apply Theorem1.Letƕ=Cɑ,M,ℝ be the set of all continuous real-valued functions defined on ɑ,M.Now, we consider the following Fredholm-type integral equation of first kind:(31)σl=∫ɑMFl,τσldτ,forl,τ∈ɑ,M,where F∈H. Define M as in Example 4, that is,(32)Mσl,υl,τ=supl∈ɑ,Me−σl−Ml3/τ,for allσ,υ∈ƕandτ>0.Then,ƕ,M,∗,u,R is an R-complete fuzzy b-metric space.Theorem 2.
Assume thatFl,τσl−Fl,τMl≤qσl−Ml for σ,M∈H, q∈0,1, and ∀l,τ∈ɑ,M. Also, consider ∫ɑMdτ=M−ɑ=1. Let Ƒ:H⟶H be(i)
R-preserving(ii)
R-contraction(iii)
R-continuous
Then, the Fredholm-type integral equation of first kind in equation (31) has a unique solution.Proof.
DefineƑ:ƕ⟶ƕ by(33)Ƒσl=∫ɑMFl,τσldτ,for alll,τ∈ɑ,M.(i)
DefineR as σlRMlMσl≥Ml. We have σlRMl, and we want to show that ƑσlRƑMl. We see that σl and Ƒσl belong to H because Ƒ is a self-map. So, observe that if σlRMl, then it must be ƑσlRƑMl. Hence, Ƒ is R-preserving.(ii)
Observe that the existence of a fixed point ofƑ is equivalent to the existence of a solution of Fredholm-type integral equation (31). Now, for all σ,M∈M, we have(34)MMσl,ƑMl,qτ=e−Ƒσl−ƑMl3/qτ=e−∫ɑMFl,τσldτ–∫ɑMFl,τMldτ3/qτ=e−∫ɑMFl,τσl–Fl,τMldτ/qτ≥e−∫ɑMqσl–Mldτ/qτ≥supl∈ɑ,Me−qσl−Ml3∫ɑMdτ/qτ=supl∈ɑ,Me−σl−Ml3/τ=Mσl,Ml,τ.Hence,Ƒ is an R-contraction.(iii)
Supposepn is an R-sequence in H such that pn converges to p∈H. Because Ƒ is R-preserving, Ƒpn is an R-sequence for each n∈ℕ. From (ii), we have(35)MƑpnl,Ƒpl,qτ≥Ƒpnl,pl,τ.
Aslimn⟶∞Mpnl,pl,τ=1, for all τ>0, it is clear that(36)limn⟶∞ƑMpnl,Ƒpl,qτ=1.
Hence,Ƒ is R-continuous.
Now, assume thatσ and M are two fixed points of ℱ; then, we have(37)Mσl,Ml,τ≥Mσl,Ml,τq.
Thus, for alln∈ℕ,(38)Mσl,Ml,τ≥Mσl,Ml,τqn=e−σl−Ml3/τ/qn.
Taking the limit asn⟶∞ and using the fact limn⟶∞Mσl,Ml,τ=1, we get σl=Ml.
Consequently, all the conditions of Theorem1 hold. The operator Ƒ therefore has a unique fixed point.


## 4. Conclusion

Herein, we introduced the notion ofR-fuzzy b-metric spaces and we proved some related fixed point results. Moreover, we presented some examples to illustrate the feasibility of the proposed methods and obtained results. We have also enriched this work with an application. Since our framework is more general than the class of fuzzy and fuzzy b-metric spaces, our results extend and generalize several existing ones in the literature.



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