# On Fuzzy b-Metric-Like Spaces

Authors: Khalil Javed; Fahim Uddin; Hassen Aydi; Muhammad Arshad; Umar Ishtiaq; Habes Alsamir
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of this manuscript is to introduce the concept of fuzzy b-metric-like spaces and discuss some related fixed point results. Some examples are imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the obtained results, an application is provided to solve a first kind of Fredholm type integral equations.


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## Body

## 1. Introduction and Preliminaries

The concept of fuzzy sets was initiated by Zadeh [1], which gave a new aspect to research activity leading with the improvement of fuzzy systems. Afterwards, several researchers contributed towards some basic significant results in fuzzy sets.Kramosil and Michlek [2] introduced the concept of fuzzy metric spaces by generalizing the concepts of probabilistic metric spaces to fuzzy metric spaces. George and Veeramani [3] derived a Hausdorff topology which was initiated by a fuzzy metric to modify the concept of fuzzy metric spaces. Later on, the fixed point theory dealing with a fuzzy metric has been enriched with a number of different generalizations. Garbiec displayed the fuzzy version of Banach contraction principle in fuzzy metric spaces. For some necessary definitions, examples, and basic results, we refer to [4–6] and the references herein.As we know, fixed point theory plays a crucial role in proving the existence of solutions for different mathematical models and has a wide range of applications in different fields related to mathematics. This theory has intrigued many researchers. Recently, Harandi [7] initiated the concept of metric-like spaces, which generalizes the notion of metric spaces in a nice way. Alghamdi et al. [8] used the concept metric-like spaces to introduce the notion of b-metric-like spaces. Since then, a number of authors contributed in same. For more details, we refer to [7, 9–16]. In this sequel, Shukla and Abbas [8] generalized the concept of metric-like spaces and introduced fuzzy metric-like spaces. For more details on this topic, please see [17–29].In this article, our aim is to generalize the concept of b-metric-like spaces by introducing the concept of fuzzy b-metric-like spaces and prove some related fixed point results. We also support this work by some examples and an application to solve an integral equation.First, we write some notations used throughout this paper, as C-t-norm for a continuous triangular norm, b-metric-l for b-metric-like, F-metric-l for fuzzy metric-like, F-b-metric for fuzzy b-metric, F-b-metric-l for fuzzy b-metric-like, and s.t. for such that.Definition 1 [17].
A binary operation∗: 0,1×0,1⟶0,1 is called a C-t-norm if it satisfies the following assertions:
(1)
a∗b=b∗a,∀a,b∈0,1(2)
a∗1=a,∀a∈0,1(3)
a∗b∗c=a∗b∗c,∀a,b,c∈0,1(4)
Ifa≤c and b≤d, with a,b,c,d∈0,1, then a∗b≤c∗d
Some fundamental examples of a t-norm area∗b=a∙b,a∗b=mina,b and a∗b=maxa+b−1,0.Definition 2 [8].
A b-metric-l on a setѠ≠⏀ is a function μ:Ѡ×Ѡ⟶0,+∞ such that for allw,k,z∈Ѡ and b≥1, it satisfies the following conditions:
(1)
Ifμw,k=0⇒w=k(2)
μw,k=wk,w(3)
μw,k≤bμw,z+μz,kThe pairѠ,μ is called a b-metric-l space.Example 1 [8].
LetѠ=0,∞. Define the function μ:Ѡ×Ѡ⟶0,+∞ by
(1)μw,k=w+k2.Then,Ѡ,μ is called a b-metric-l space with b=2.Example 2 [8].
LetѠ=0,∞. Define the function μ:Ѡ×Ѡ⟶0,+∞ by.
(2)μw,k=maxw,k2.Then,Ѡ,μ is named as a b-metric-l space with b=2.Definition 3 [18].
A 3-tupleѠ,Ω,∗ is said an F-metric-l space if Ѡ≠⏀ is a random set, ∗ is a C-t-norm, and Ω is a fuzzy set on Ѡ×Ѡ×0,∞ meeting the conditions below for all w,k,z∈Ѡ,τ,s>0:
FL1)Ωw,k,τ>0
FL2) IfΩw,k,τ=1, then w=k
FL3)Ωw,k,τ=Ωk,w,τ
FL4)Ωw,z,τ+s≥Ωw,k,τ∗Ωk,z,s
FL5)Ωw,k,∙:0,∞⟶0,1 is continuous.Example 3 [18].
LetѠ=ℝ+, q∈ℝ+, and m>0. Define a t-norm by a∗b=ab and the fuzzy set Ω on Ѡ×Ѡ×0,∞ by
(3)Ωw,k,τ=qτqτ+mmaxw,k,∀w,k∈Ѡ,τ>0.Then,Ѡ,Ω,∗ is an F-metric-l space.Definition 4 [5].
A 3-tupleѠ,Ω,∗ is said an F-b-metric space if Ѡ is a random (nonempty) set, ∗ is a C-t-norm, and Ω is a fuzzy set on Ѡ×Ѡ×0,∞ meeting the conditions below for all w,k,z∈Ѡ,τ,s>0 and a provided real numberb≥1:
FB1)Ωw,k,τ>0
FB2)Ωw,k,τ=1 iff w=k
FB3)Ωw,k,τ=Ωk,w,τ
FB4)Ωw,z,τ+s≥Ωw,k,τ/b∗Ωk,z,s/b
FB5)Ωw,k,∙:0,∞⟶0,1 is continuous.Example 4 [9].
LetΩw,k,τ=e−w−kp/τ, where p>1 is a real number. It is then simple to show that Ω is an F-b-metric with b=2p−1.


## 2. Main Results

We start this section with the introduction of F-b-metric-l spaces and prove some related fixed point results.Definition 5.
A 4-tupleѠ,Ω,∗,b is named an F-b-metric-l space if Ѡ≠⏀ is a random set, ∗ is a C-t-norm, and Ω is a fuzzy set on Ѡ×Ѡ×0,∞ meeting the following conditions below for all w,k,z∈Ѡ,τ,s>0:
B1)Ωw,k,τ>0
B2) IfΩw,k,τ=1, then w=k
B3)Ωw,k,τ=Ωk,w,τ
B4)Ωw,z,bτ+s≥Ωw,k,τ∗Ωk,z,s, for b∈ℕ
B5)Ωw,k,∙:0,∞⟶0,1 is continuous.Remark 6.
In the above definition, a setѠ is endowed by an F-b-metric-l with a t-norm (∗). An F-b-metric-l space does not satisfy the (FB2) condition of F-b-metric spaces, that is, the self-distance may not be equal to 1, i.e., Ωw,w,τ≠1 for all τ>0,for some or may be for all w∈Ѡ. But all other conditions are the same. Hence, the F-b-metric-l may not be a F-b-metric, but the converse is true.Definition 7.
LetѠ,Ω,∗,b be an F-b-metric-l space. For x∈X,Ѳ∈0,1,τ>0, we define the open ball as
(4)Bx,Ѳ,τ=k∈Ѡ:Ωw,k,τ>1−Ѳ.
Then,(5)TM=T⊂Ѡ:w∈T⟺thereexistτ>0andѲ∈0,1;Bx,Ѳ,τ⊆T,is a topology on Ѡ.The following simplest example shows that an F-b-metric-l need not to be an F-b-metric.Example 5.
TakeѠ=0,∞. Consider the t-norm defined by a∗b=ab, then
(6)Ωw,k,τ=ew+k2/τ−1,∀w,k∈Ѡ,τ>0.
is an F-b-metric-l. But it is not an F-b-metric.Proof.
(B1), (B2), (B3), and (B5) are obvious. Here, we prove (B4). We have(7)w+z≤bτ+sτw+k+bτ+ssk+z.
Here,b is an arbitrary integer. We have
(8)∣w+z∣bτ+s≤w+kτ+k+zs.
Hence,(9)e∣w+z∣/bτ+s≤ew+k/τ+ek+z/s.
Sinceew is an increasing function for w>0, one writes
(10)ew+z2/bτ+s−1≥ew+k2/τ−1+ek+z2/s−1.
That is,(11)Ωw,z,bτ+s≥Ωw,k,τ∗Ωk,z,s,∀w,k,z∈Ѡ,∀τ,s>0
Hence, (B4) satisfied.
Now, we have to prove thatѠ,Ω,∗,b is not an F-b-metric space. For this purpose, we investigate the self-distance. Indeed,
(12)Ωw,w,τ=ew+w2/τ−1=1ew+w2/τ≠1,∀τ>0,∀w∈Ѡ.
Hence,Ѡ,Ω,∗,b is not an F-b-metric space.The following example shows that an F-b-metric-l space need not be continuous.Example 6.
LetѠ=0,∞,Ωw,k,τ=e−dw,k/τ,∀w,k∈Ѡ,τ>0 and
(13)dw,k=0,ifw=k,2k+w2,ifk,w∈0,1,12k+w2,otherwise.
If we consider the t-norm defined bya∗b=ab, then Ѡ,Ω,∗,b is an F-b-metric-l space with a coefficient b=4. To illustrate the discontinuity, we have
(14)limn⟶∞Ω0,1−1n,τ=limn⟶∞e−21−1/n2=e−2=Ω0,1,τ.
However, since(15)limn⟶∞Ω1,1−1n,τ=limn⟶∞e−22−1/n2=e−8≠1=Ω1,1,τ.
One can assert thatΩw,k,τ is not continuous.Proposition 8.
LetѠ,μ be a b-metric-l space. Then, Ѡ,Ω,∗,b is an F-b-metric-l space defined as
(16)Ωw,k,τ=e−μw,k/τn,∀τ>0,∀w,k∈Ѡ,n∈ℕ.Proof.
(B1), (B2), (B3), and (B5) are obvious. Here, we prove (B4). We have(17)μw,z≤bμw,k+μk,z.
Therefore,(18)μw,zτ+sn≤bμw,k+μk,zτ+sn.
That is,(19)μw,zbτ+sn≤μw,kτn+μk,zsn.
Hence,(20)eμw,z/bτ+sn≥eμw,k/τn+eμk,z/sn.
That is,Ωw,z,bτ+s≥Ωw,k,τ∗Ωk,z,s. Hence, (B4) is satisfied and Ѡ,Ω,∗,b is an F-b-metric-l space.Definition 9.
A sequencewn in a F-b-metric-l space Ѡ,Ω,∗,b is said to be convergent to w∈Ѡ, if
(21)limn⟶∞Ωwn,w,τ=Ωw,w,τ,∀τ>0.Definition 10.
A sequencewn in an F-b-metric-l space Ѡ,Ω,∗,b is said to be Cauchy if
(22)limn⟶∞Ωwn,wn+p,τ,∀τ>0,p≥1,exists and is finite.Definition 11.
An F-b-metric-l spaceѠ,Ω,∗,b is said to be complete if every Cauchy sequence wn in Ѡ converges to some w∈Ѡ such that
(23)limn⟶∞Ωwn,w,τ=Ωw,w,τ=limn⟶∞Ωwn,wn+p,τ,∀τ>0,p≥1.Definition 12.
LetѠ,Ω,∗,b be an F-b-metric-l space. A mapping F:Ѡ⟶Ѡ is said to be fuzzy contractive if there exists q∈0,1 such that
(24)1ΩFw,Fk,τ−1≤q1Ωw,k,τ−1,for all w,k∈Ѡ and τ>0. Here, q is called the fuzzy contractive constant ofF.Theorem 13.
LetѠ,Ω,∗,b be a complete F-b-metric-l space and F:Ѡ⟶Ѡ be a fuzzy contractive mapping with a fuzzy contractive constantq, then F has a unique fixed point υ∈Ѡ so that Ωυ,υ,τ=1 for all τ>0.Proof.
For an arbitraryw0∈Ѡ, define a sequence wn in Ѡ by
(25)w1=Fw0,w2=F2w0=Fw1,⋯,wn=Fnw0=Fwn−1foralln∈ℕ.
Ifwn=wn−1 for some n∈ℕ, then wn is a fixed point ofF. We assume that wn≠wn−1 for all n∈ℕ. For τ>0 and n∈ℕ, we get from (24)
(26)1Ωwn,wn+1,τ−1=1ΩFwn−1,Fwn,τ−1≤q1Ωwn−1,wn,τ−1.
We have(27)1Ωwn,wn+1,τ≤qΩwn−1,wn,τ+1−q,∀τ>0.
Therefore,(28)qΩFwn−2,Fwn−1,τ+1−q≤q2Ωwn−2,wn−1,τ+q1−q+1−q.
Continuing in this way, we get(29)1Ωwn,wn+1,τ≤qnΩw0,w1,τ+qn−11−q+qn−21−q+⋯+q1−q+1−q≤qnΩw0,w1,τ+qn−1+qn−2+⋯+11−q≤qnΩw0,w1,τ+1−qn.
We have(30)1qn/Ωw0,w1,τ+1−qn≤Ωwn,wn+1,τ,∀τ>0,n∈ℕ.
Now, forp≥1 and n∈ℕ, we have
(31)Ωwn,wn+p,τ≥Ωwn,wn+1,τb∗Ωwn+1,wn+p,τb≥Ωwn,wn+1,τb∗Ωwn+1,wn+2,τb2∗Ωwn+2,wn+p,τb2.
Continuing in this way, we get(32)Ωwn,wn+p,τ≥Ωwn,wn+1,τb∗Ωwn+1,wn+2,τb2∗⋯∗Ωwn+p−1,wn+p,τbp−1.
By using (30) in the above inequality, we have
(33)Ωwn,wn+p,τ≥1qn/Ωw0,w1,τ/b+1−qn∗1qn+1/Ωw0,w1,τ/b2+1−qn+1∗⋯∗1qn+p−1/Ωw0,w1,τ/bp−1+1−qn+p−1≥1qn/Ωw0,w1,τ/b+1∗1qn+1/Ωw0,w1,τ/b2+1∗⋯∗1qn+p−1/Ωw0,w1,τ/bp−1+1.
Here,b is an arbitrary positive integer, and as q∈0,1, we deduce from the above expression that
(34)limn⟶∞Ωwn,wn+p,τ=1forallτ>0,p≥1.
Therefore,wn is a Cauchy sequence in Ѡ,Ω,∗,b. By the completeness of Ѡ,Ω,∗,b,
there isυ∈Ѡ such that
(35)limn⟶∞Ωwn,υ,τ=limn⟶∞Ωwn,wn+p,τ=limn⟶∞Ωυ,υ,τ=1,∀τ>0,p≥1.
Now, we prove thatυ is a fixed point for F. For this, we obtain from (24) that
(36)1ΩFwn,Fυ,τ−1≤q1Ωwn,υ,τ−1=qΩwn,υ,τ−q,1q/Ωwn,υ,τ+1−q≤ΩFwn,Fυ,τ
Using the above inequality, we obtain(37)Ωυ,Fυ,τ≥Ωυ,wn+1,τ2b∗Ωwn+1,Fυ,τ2b=Ωυ,wn+1,τ2b∗ΩFwn,Fυ,τ2b≥Ωυ,wn+1,τ2b∗1q/Ωwn,υ,τ/2b+1−q.
Taking limit asn⟶∞ and using (35) in the above expression, we get that Ωυ,Fυ,τ=1, that is, Fυ=υ. Therefore, υ is a fixed point of F and Ωυ,υ,τ=1 for all τ>0.
Now, we investigate the uniqueness of the fixed pointυ of F. Let v be another fixed point of F, s.t. Ωυ,v,t<1 for some τ>0. It follows from (24) that
(38)1Ωυ,v,τ−1=1ΩFυ,Fv,τ−1≤q1Ωυ,v,τ−1<1Ωυ,v,τ−1,
a contradiction. Therefore, we must haveΩυ,v,τ=1, for all τ>0, and hence υ=v.Corollary 14.
LetѠ,Ω,∗,b be a complete F-b-metric-l space and F:Ѡ⟶Ѡ be a mapping satisfying
(39)1ΩFnw,Fnk,τ−1≤q1Ωw,k,τ−1,
for somen∈ℕ,∀w,k∈Ѡ,τ>0, where 0<q<1. Then, F has a unique fixed point υ∈Ѡ and Ωυ,υ,τ=1,∀τ>0.Proof.
υ∈Ѡ is the unique fixed point of Fn by using Theorem 13, and Ωυ,υ,τ=1,∀τ>0.Fυ is also a fixed point of Fn as FnFυ=Fυ and from Theorem 13, Fυ=υ, υ is the unique fixed point, since the unique fixed point of F is also the unique fixed point of Fn.Example 7.
LetѠ=0,2 and the t-norm be defined asa∗b=ab. Given Ω as
(40)Ωw,k,τ=e−maxw,k2/τ,forallw,k∈Ѡandτ>0.
(From Example2 and Proposition 8, Ω is an F-b-metric-l).Then,Ѡ,Ω,∗,b is a complete F-b-metric-l space. Define F:Ѡ⟶Ѡ as
(41)Fw=0,w=1,w2,w∈0,1,w4,w∈1,2.Then, we have 8 cases:Case 1.
Ifw=k=1, then Fw=Fk=0.Case 2.
Ifw=1 and k∈0,1, then Fw=0 and Fk=k/2.Case 3.
Ifw=1 and k∈1,2, then Fw=0 and Fk=k/4.Case 4.
Ifw∈0,1 and k∈1,2, then Fw=w/2 and Fk=k/4.Case 5.
Ifw∈0,1 and k∈0,1, then Fw=w/2 and Fk=k/2.Case 6.
Ifw∈0,1 and k=1, then Fw=w/2 and Fk=0.Case 7.
Ifw∈1,2 and k=1, then Fw=w/4 and Fk=0.Case 8.
Ifw∈1,2 and k∈1,2, then Fw=w/4 and Fk=k/4.All above cases satisfy the fuzzy contraction:(42)1ΩFw,Fk,τ−1≤q1Ωw,k,τ−1,with q∈1/2,1thefuzzycontractiveconstant. Hence F is a fuzzy contractive mapping with q∈1/2,1. All conditions of Theorem 13 are satisfied. Also, 0 is the unique fixed point of F and Ω0,0,τ=1,∀τ>0.Theorem 15.
LetѠ,Ω,∗,b be a complete F-b-metric-l space such that
(43)limτ⟶∞Ωw,k,τ=1,for all w,k∈Ѡ,τ>0, and F:Ѡ⟶Ѡ be a mapping satisfying the condition
(44)ΩFw,Fk,ατ≥Ωw,k,τ,for all w,k∈Ѡ,τ>0, where α∈0,1. Then F has a unique fixed point υ∈Ѡ and
(45)Ωυ,υ,τ=1,∀τ>0.Proof.
For an arbitraryw0∈Ѡ, define a sequence wn in Ѡ by
(46)w1=Fw0,w2=F2w0=Fw1,⋯,wn=Fnw0=Fwn−1foralln∈ℕ.
Ifwn=wn−1 for some n∈ℕ, then wn is a fixed point ofF. We assume that wn≠wn−1 for all n∈ℕ. For τ>0 and n∈ℕ, we get from (44) that
(47)Ωwn,wn+1,τ≥Ωwn+1,wn,ατ=ΩFwn,Fwn−1,ατ≥Ωwn,wn−1,τ,for all n∈ℕ and τ>0. Therefore, by applying the above expression, we can deduce that
(48)Ωwn+1,wn,τ≥Ωwn+1,wn,ατ=ΩFwn,Fwn−1,ατ≥Ωwn,wn−1,τ=ΩFwn−1,Fwn−2,τ≥Ωwn−1,wn−2,τα≥⋯≥Ωw1,w0,ταn,for all n∈ℕ, p≥1, and τ >0. Thus, we have
(49)Ωwn,wn+p,τ≥Ωwn,wn+1,τb∗Ωwn+1,wn+p,τb.
Continuing in this way, we get(50)Ωwn,wn+p,τ≥Ωwn,wn+1,τb∗Ωwn+1,wn+2,τb2∗⋯∗Ωwn+p−1,wn+p,τbp−1.
Using (48) in the above inequality, we deduce
(51)Ωwn,wn+p,τ≥Ωw0,w1,τbαn∗Ωw0,w1,τb2αn+1∗⋯∗Ωw0,w1,τbp−1αn+p−1.
Here,b is an arbitrary positive integer. We know that limn⟶∞Ωw,k,τ=1,∀w,k∈Ѡ and τ>0,α∈0,1. So, from (51), we deduce that
(52)limn⟶∞Ωwn,wn+p,τ=1∗1∗⋯∗1=1,∀τ>0,p≥1.
Hence,wn is a Cauchy sequence. The hypothesis of completeness of the F-b-metric-l space Ѡ,Ω,∗,b ensures that there exists υ∈Ѡ such that
(53)limn⟶∞Ωwn,υ,τ=limn⟶∞Ωwn,wn+p,τ=Ωυ,υ,τ=1,∀τ>0,p≥1.
Now, we derive thatυ∈Ѡ is a fixed point of F. We have
(54)Ωυ,Fυ,τ≥Ωυ,wn+1,τ2b∗Ωwn+1,Fυ,τ2b,∀τ>0=Ωυ,wn+1,τ2b∗ΩFwn,Fυ,τ2b≥Ωυ,wn+1,τ2b∗Ωwn,υ,τ2bα.
Taking limit asn⟶+∞, and by (53), we get
(55)Ωυ,Fυ,τ=1∗1=1.
Therefore,υ is a fixed point of F and Fυ,υ,τ=1,∀τ>0.
Now, we investigate the uniqueness of fixed point. For this, assume thatv and υ are two fixed points of F. Then, by (44), we have
(56)Ωυ,v,τ=ΩFυ,Fv,τ≥Ωυ,v,τα,Ωυ,v,τ≥Ωυ,v,τα,∀τ>0
We obtain(57)Ωυ,v,τ≥Ωυ,v,ταn,∀n∈ℕ.
Taking limit asn⟶+∞ and using the fact limτ⟶∞Ωw,k,τ=1, so υ=v; hence the fixed point is unique.Example 8.
LetѠ=0,1 and the t-norm be defined as a∗b=ab. Also, Ω is defined as
(58)Ωw,k,τ=e−w+k2/τ,∀w,k∈Ѡ,τ>0.
Then,Ѡ,Ω,∗,b is a complete F-b-metric-l space. Define F:Ѡ⟶Ѡ by
(59)Fw=0,w∈0,12,w6,w∈12,1.Now,limτ⟶∞Ωw,k,τ=limτ⟶∞e−w+k2/τ=1. Forα∈1/2,1, we have four cases:Case 1.
Ifw,k∈0,1/2, then Fw=Fk=0. Here,
(60)ΩFw,Fk,ατ=Ω0,0,ατ=e0≥e−w+k2/τ=Ωw,k,τ.Case 2.
Ifw∈0,1/2 and k∈1/2,1, then Fw=0 and Fk=k/6. We have
(61)ΩFw,Fk,ατ=Ω0,w6,ατ=e−0+k/62/ατ=e−k2/36ατ≥e−w+k2/τ=Ωw,k,τ.Case 3.
Ifw,k∈1/2,1, then Fw=w/6 and Fk=k/6. Here,
(62)ΩFw,Fk,ατ=Ωw6,k6,ατ=e−w+k2/36ατ≥e−w+k2/τ=Ωw,k,τ.Case 4.
Ifw∈1/2,1 and k∈0,1/2, then Fw=w/6 and Fk=0. Then,
(63)ΩFw,Fk,ατ=Ωw6,0,ατ=e−w2/36ατ≥e−w+k2/τ=Ωw,k,τ.From all 4 cases, we obtain that(64)ΩFw,Fk,ατ≥Ωw,k,τ.Hence, all conditions of Theorem13 are satisfied, and 0 is the unique fixed point ofF. Also,
(65)Ωυ,υ,τ=Ω0,0,τ=e0=1,∀τ>0.


## 3. An Application to an Integral Equation

Consider the following integral equation:(66)wl=∫0J∂l,r,wrdr,where J>0 and ∂:0,J×0,J×ℝ+⟶ℝ+.LetѠ=C0,J be the set of all continuous real valued functions defined on0,J. Consider the b-metric-l given as
(67)μw,k=∣w∣+∣k∣2.Then, by Proposition8,
(68)Ωwl,kl,τ=maxl∈0,Je−wl+kl2/τ,∀w,k∈Ѡ.Clearly,Ѡ,Ω,∗,b is a complete F-b-metric-l space. Let
(69)Fwl=∫0J∂l,r,wrdr,∀w∈Ѡ,forl∈0,J. Observe that the existence of a solution of (66) is equivalent to the existence of a fixed point ofF.Theorem 16.
Assume that the following hypotheses hold:(1)
∂:0,J×0,J×ℝ+⟶ℝ+ is continuous(2)
∀l,r∈0,J, there is a continuous function β:0,J×0,J⟶ℝ+ such that(70)∣∂l,r,wr∣+∣∂l,r,kr∣≤α1/2βl,rwl+kl,where α∈0,1 and b>1.(3)
limτ⟶∞Ωwl,kl,τ=1andSupl∈0,J∫0Jβl,rdr≤1
Then, the integral Equation (66) has a unique solution.Proof.
For alll∈0,J, we have
(71)ΩFwl,Fkl,ατ=e−Fwl+Fkl2/ατ=e−∣∫0J∂l,r,wrdr∣+∣∫0J∂l,r,krdr∣2/ατ≥e−∫0J∂l,r,wrdr+∫0J∂l,r,krdr2/ατ=e−∫0J∂l,r,wr+∂l,r,krdr2/ατ≥e−∫0Jα1/2βl,rwl+kldr2/ατ=e−∫0Jα1/2βl,rwl+kl21/2dr2/ατ=Supl∈0,Je−α∣wl∣+∣kl∣2∫0Jβl,rdr2/ατ≥e−wl+kl2/τ=Ωwl,kl,τ.
Therefore,ΩFwl,Fkl,ατ≥Ωwl,kl,τ. Also, observe that all conditions of Theorem 15 are satisfied. Hence, the operator F has a unique fixed point. This means that the integral Equation (66) has a unique solution.Example 9.
Consider the nonlinear integral equation below(72)wl=18∫01rwrdr.
Then, it has a solution inѠ.Proof.
LetF:Ѡ⟶Ѡ be defined by Fwl=1/8∫01rwrdr. Set ∂l,r,wr=1/8rwr in Theorem 16, we get
(i)
∂:0,J×0,J×ℝ+⟶ℝ+ is continuous(ii)
∀l,r∈0,J, there is a continuous function βl,r=r such that(73)∣∂l,r,wr∣+∣∂l,r,kr∣=14×r2wr+kr≤14rwr+kr=1161/2rwr+kr=α1/2βl,rwl+kl,where α=1/16∈0,1/b(iii)
limτ⟶∞Ωwl,kl,τ=1andSupl∈0,J∫01βl,r=Supl∈0,J∫01rdr=1/2<1
Hence, all hypotheses of Theorem16 are fulfilled. Therefore, the problem (72) has a solution on Ѡ.


## 4. Conclusion

Fixed point techniques are used to solve many mathematical problems, as differential and integral equations, integro-differential equations, game theory, and economics. The intent of this manuscript is to present a new space, so-called a fuzzy b-metric-like space. Topological properties and related examples are addressed, so our fixed point results are new. Ultimately, to illustrate the practical side of the theoretical results, a solution of a nonlinear integral equation is given.



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*Source: 6615976-2021-03-15.xml*