# Rational Type Fuzzy-Contraction Results in Fuzzy Metric Spaces with an Application

Authors: Saif Ur Rehman; Ronnason Chinram; Chawalit Boonpok
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper aims to introduce the new concept of rational type fuzzy-contraction mappings in fuzzy metric spaces. We prove some fixed point results under the rational type fuzzy-contraction conditions in fuzzy metric spaces with illustrative examples to support our results. This new concept will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an application of a nonlinear integral type equation to get the existing result for a unique solution to support our work.


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## Body

## 1. Introduction

The theory of fixed point is one of the most interesting areas of research in mathematics. In the last decades, a lot of work was dedicated to the theory of fixed point. A pointμ belonging to a nonempty set U is called a fixed point of a mapping ℓ:U⟶U if and only if ℓμ=μ. In 1922, Stefan Banach, a well-known mathematician, proved a Banach contraction principle in [1], which is stated as “A self-mapping in a complete metric space satisfying the contraction condition has a unique fixed point.” After the publication of this principle, many researchers contributed their ideas to the theory of fixed point and proved different contractive type mapping results for single and multivalued mappings in the context of metric spaces for fixed point, coincidence point, and common fixed point. Some of these results can be found in [2–13].In 1965, the theory of fuzzy set was introduced by Zadeh [14]. Recently, this theory is used, investigated, and applied in many directions. One direction is the evaluation of test results which is the application of fuzzy logic in the processing of students evaluation; moreover, the application is expected to represent the mechanisms of human thought processes capable of resolving the problem of evaluation of students, which can be directly monitored by the teacher (for example, see [15–19]). Many researchers have extensively developed the theory of fuzzy sets and their applications in different fields. Some of their results can be found in [20–29] the references therein.The other direction is the generalization of metric spaces to fuzzy metric spaces. In [30], Kramosil and Michalek introduced the concept of fuzzy metric spaces (FM-space) and some more notions. Later on, the stronger form of the metric fuzziness was given by George and Veeramani [31]. In 2002, Gregory and Sapena [32] proved some contractive type fixed point theorems in FM-spaces. Some more fixed point results in the said space can be found in [33–41].This research work aims to present the new concept of rational type fuzzy-contraction mappings inG-complete FM-spaces. We use the concept of Gregory and Sapena [32] and the “triangular property of fuzzy metric” presented by Bari and Vetro [33] and prove some unique fixed point theorems under the rational type fuzzy-contraction conditions in G-complete FM-spaces with some illustrative examples. This new theory will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an integral type application in the sense of Jabeen et al. [42] to prove a result for a unique solution to support our work. The application section of the paper is more important; one can use this concept and present different types of nonlinear integral type equations for the existence of unique solutions for their results. Some integral type application results in the theory of fixed point can be found in [43–46].


## 2. Preliminaries

Definition 1.
(see [47]). An operation ∗:0,12⟶0,1 is called a continuous t-norm, if(i)
∗ is commutative, associative, and continuous.(ii)
1∗ξ1=ξ1 and ξ1∗ξ2≤ξ3∗ξ4, whenever ξ1≤ξ3 and ξ2≤ξ4, ∀ξ1,ξ2,ξ3,ξ4∈0,1.
The basict-norms, the minimum, the product, and the Lukasiewicz continuous t-norms are defined as follows (see [47]):(1)ξ1∗ξ2=minξ1,ξ2,ξ1∗ξ2=ξ1ξ2,ξ1∗ξ2=maxξ1+ξ2−1,0.Definition 2.
(see [31]). A 3-tuple U,Mr,∗ is said to be a FM-space if U is an arbitrary set, ∗ is a continuous t-norm, and Mr is a fuzzy set on U2×0,∞ satisfying the following conditions:(i)
Mrμ1,μ∗,t>0 and Mrμ1,μ∗,t=1⟺μ1=μ∗(ii)
Mrμ1,μ∗,t=Mrμ∗,μ1,t(iii)
Mrμ1,μ,t∗Mrμ,μ∗,s≤Mrμ1,μ∗,t+s(iv)
Mrμ1,μ∗,.:0,∞⟶0,1 is continuous,∀μ,μ1,μ∗∈U and t,s∈0,∞.Lemma 1 (see [31]).
Mrμ1,μ∗,∗ is nondecreasing ∀μ1,μ∗∈U.Definition 3.
(see [31]). Let U,Mr,∗ be a FM-space, v1∈U, and a sequence μj in U is(i)
Converges tov1 if ξ∈0,1 and t>0∃j1∈ℕ, such that Mrμj,v1,t>1−ξ, ∀j≥j1. We may write this limj⟶∞μj=v1 or μj⟶μ1 as j⟶∞.(ii)
Cauchy sequence ifξ∈0,1 and t>0∃j1∈ℕ such that Mrμj,μk,t>1−ξ, ∀j,k≥j1.(iii)
U,Mr,∗ is complete if every Cauchy sequence is convergent in U.(iv)
[32] fuzzy-contractive if ∃a∈0,1 such that(2)1Mrμj,μj+1,t−1≤a1Mrμj−1,μj,t−1,fort>0,j≥1.
In the sense of Gregori and Sapena [32], a sequence μj in a FM-space is said to be G-Cauchy if limjMrμj,μj+p,t=1, for t>0 and p>0. A FM-space U,Mr,∗ is called G-complete if every G-Cauchy sequence is convergent.
Throughout this paper,ℕ represents the set of natural numbers.Lemma 2 (see [31]).
LetU,Mr,∗ be a FM-space and let a sequence μj in U converge to a point v1∈U iff Mrμj,v1,t⟶1, as j⟶∞, for t>0.Definition 4.
(see [33]). Let U,Mr,∗ be a FM-space. The fuzzy metric Mr is triangular, if(3)1Mrμ1,μ∗,t−1≤1Mrμ1,μ,t−1+1Mrμ,μ∗,t−1,∀μ,μ1,μ∗∈U,t>0.Definition 5 (see [32]).
LetU,Mr,∗ be a FM-space and ℓ:U⟶U. Then, ℓ is said to be fuzzy-contractive if ∃a∈0,1 such that(4)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1,∀μ1,μ∗,∈U,t>0.
In the following, we present some rational type fixed point results under the rational type fuzzy-contraction conditions inG-complete FM-spaces by using the “triangular property of fuzzy metric.” We present illustrative examples to support our results. In the last section of this paper, we present an integral type application for a unique solution to support our work.


## 3. Main Result

In this section, we define rational type fuzzy-contraction maps and prove some unique fixed point theorems under the rational type fuzzy-contraction mappings inG-complete FM-spaces.Definition 6.
LetU,Mr,∗ be a FM-space; a mapping ℓ:U⟶U is called a rational type fuzzy-contraction if ∃a,b∈0,1 such that(5)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1+bMrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1,∀μ1,μ∗,∈U, t>0.Theorem 1.
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U is a rational type fuzzy-contraction satisfying (5) with a+b<1. Then, ℓ has a unique fixed point in U.Proof.
Fixμ0∈U and μj+1=ℓμj,j≥0. Then, by (5), for t>0,j≥1,(6)1Mrμj,μj+1,t−1=1Mrℓμj−1,ℓμj,t−1≤a1Mrμj−1,μj,t−1+bMrμj−1,μj,tMrμj−1,ℓμj−1,t∗Mrμj,ℓμj−1,2t−1=a1Mrμj−1,μj,t−1+bMrμj−1,μj,tMrμj−1,μj,t∗Mrμj,μj,2t−1,and after simplification,(7)1Mrμj,μj+1,t−1≤a1Mrμj−1,μj,t−1,fort>0.
Similarly,(8)1Mrμj−1,μj,t−1≤a1Mrμj−2,μj−1,t−1,fort>0.
Now, from (7) and (8) and by induction, for t>0, we have that(9)1Mrμj,μj+1,t−1≤a1Mrμj−1,μj,t−1≤a21Mrμj−2,μj−1,t−1≤⋯≤aj1Mrμ0,μ1,t−1⟶0,asj⟶∞.
Hence,μj is a fuzzy-contractive sequence in U,Mr,∗; therefore,(10)limj⟶∞Mrμj,μj−1,t=1,fort>0.
Now, we show thatμj is a G-Cauchy sequence; let j∈ℕ, and there is a fixed q∈ℕ such that(11)Mrμj,μj+q,t=Mrμj,μj+q,1q+1q+⋯+1q︸q−timest≥Mrμj,μj+1,tq∗Mrμj+1,μj+2,tq∗⋯∗Mrμj+q−1,μj+q,tq⟶1∗1∗⋯∗1︸q−times=1,asj⟶∞.
Hence, it is proved thatμj is a G-Cauchy sequence. Since U,Mr,∗ is G-complete, ∃v1∈U such that μj⟶v1, as j⟶∞, i.e.,(12)limj⟶∞Mrμj,v1,t=1,fort>0.
SinceMr is triangular, from (5), (10), and (12), for t>0, we have(13)1Mrv1,ℓv1,t−1≤1Mrv1,μj+1,t−1+1Mrℓμj,ℓv1,t−1≤1Mrv1,μj+1,t−1+a1Mrμj,v1,t−1+bMrμj,v1,tMrμj,ℓμj,t∗Mrv1,ℓμj,2t−1=1Mv1,μj+1,t−1+a1Mrμj,v1,t−1+bMrμj,v1,tMrμj,μj+1,t∗Mrv1,μj+1,2t−1⟶0,asj⟶∞.
Hence,Mrv1,ℓv1,t=1⇒ℓv1=u1, for t>0.
Uniqueness. Let ∃z1∈U such that ℓz1=z1 and ℓv1=v1; then, from (5) and by using Definition 2 (iii), for t>0, we have(14)1Mrv1,z1,t−1=1Mrℓv1,ℓz1,t−1≤a1Mrv1,z1,t−1+bMrv1,z1,tMrv1,ℓv1,t∗Mrz1,ℓv1,2t−1≤a1Mrv1,z1,t−1+bMrv1,z1,tMrz1,v1,t∗Mrv1,v1,t−1=a1Mrv1,z1,t−1=a1Mrℓv1,ℓz1,t−1≤a21Mrv1,z1,t−1≤⋯≤aj1Mrv1,z1,t−1⟶0,asj⟶∞.
Hence, it is proved thatMrv1,z1,t=1, and this implies that v1=z1.Corollary 1 (fuzzy Banach contraction principle).
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U is a fuzzy-contraction satisfying (4) with a∈0,1. Then, ℓ has a unique fixed point in U.Example 1.
LetU=0,∞, ∗ be a continuous t-norm, and Mr:U2×0,∞⟶0,1 be defined as(15)Mrμ1,μ∗,t=tt+4μ1−4μ∗/5,∀μ1,μ∗∈U,t>0.
Then, one can easily verify thatMr is triangular and U,Mr,∗ is a G-complete FM-space. Now we define a mapping ℓ:U⟶U as(16)ℓμ1=3μ14,ifμ1∈0,1,2μ13+8,ifμ1∈1,∞.
Then, we have(17)1Mrℓμ1,ℓμ∗,t−1=341Mrμ1,μ∗,t−1,∀μ1,μ∗∈U,t>0.
Hence, a mappingℓ is a fuzzy contraction. Now, from Definition 2 (iii), for t>0,(18)Mrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1≤Mrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,μ1,t∗Mrμ1,ℓμ1,t−1=1Mrμ1,ℓμ1,t∗Mrμ1,ℓμ1,t−1=1Mrμ1,ℓμ1,t2−1=2μ15t2μ15+t.
Hence, all the conditions of Theorem1 are satisfied with a=3/4 and b=2/9. A mapping ℓ has a fixed point, i.e., ℓ24=24∈0,∞.
Next, we present a generalized rational type fuzzy-contraction theorem.Theorem 2.
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U satisfies(19)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1+bMrμ1,μ∗,t∗Mrμ∗,ℓμ∗,tMrμ1,ℓμ1,t∗Mrμ1,ℓμ∗,2t−1+cMrμ1,ℓμ1,tMrμ1,ℓμ∗,2t−1+Mrμ∗,ℓμ∗,tMrμ1,ℓμ∗,2t−1+d1Mrμ1,ℓμ1,t−1+1Mrμ∗,ℓμ∗,t−1,∀μ1,μ∗∈U,t>0,a,b,c,d≥0 with a+b+2c+2d<1. Then, ℓ has a unique fixed point.Proof.
Fixμ0∈U and μj+1=ℓμj,j≥0. Then, by (19), for t>0,j≥1,(20)1Mrμj,μj+1,t−1=1Mrℓμj−1,ℓμj,t−1≤a1Mrμj−1,μj,t−1+bMrμj−1,μj,t∗Mrμj,ℓμj,tMrμj−1,ℓμj−1,t∗Mrμj−1,ℓμj,2t−1+cMrμj−1,ℓμj−1,tMrμj−1,ℓμj,2t−1+Mrμj,ℓμj,tMrμj−1,ℓμj,2t−1+d1Mrμj−1,ℓμj−1,t−1+1Mrμj,ℓμj,t−1=a1Mrμj−1,μj,t−1+bMrμj−1,μj,t∗Mrμj,μj+1,tMrμj−1,μj,t∗Mrμj−1,μj+1,2t−1+cMrμj−1,μj,tMrμj−1,μj+1,2t−1+Mrμj,μj+1,tMrμj−1,μj+1,2t−1+d1Mrμj−1,μj,t−1+1Mrμj,μj+1,t−1.
From Definition2 (iii), Mrμj−1,μj+1,2t≥Mrμj−1,μj,t∗Mrμj,μj+1,t,fort>0, and after simplification, we have(21)1Mrμj,μj+1,t−1≤β1Mrμj−1,μj,t−1,whereβ=a+b+c+d1−c−d<1.
Similarly, fort>0, we have(22)1Mrμj−1,μj,t−1≤β1Mrμj−2,μj−1,t−1,whereβ=a+b+c+d1−c−d<1.
Now, from (21) and (22) and by induction, for t>0, we have(23)1Mrμj,μj+1,t−1≤β1Mrμj−1,μj,t−1≤β21Mrμj−2,μj−1,t−1≤⋯≤βj1Mrμ0,μ1,t−1⟶0,asj⟶∞.
Hence,μj is a rational type fuzzy-contractive sequence in U such that(24)limj⟶∞Mrμj,μj+1,t=1,fort>0.
Now we have to show thatμj is a G-Cauchy sequence; let j∈ℕ, and there is a fixed q∈ℕ such that(25)Mrμj,μj+q,t=Mrμj,μj+q,1q+1q+⋯+1q︸q−timest≥Mrμj,μj+1,tq∗Mrμj+1,μj+2,tq∗⋯∗Mrμj+q−1,μj+q,tq⟶1∗1∗⋯∗1︸q−times=1,asj⟶∞.
Hence, it is proved thatμj is a G-Cauchy sequence. Since U,Mr,∗ is G-complete, then ∃v1∈U such that μj⟶v1, as j⟶∞, i.e.,(26)limj⟶∞Mrμj,v1,t=1,fort>0.
SinceMr is triangular,(27)1Mrv1,ℓv1,t−1≤1Mrv1,μj+1,t−1+1Mrμj+1,ℓv1,t−1,fort>0.
Now from (19), (24), and (26), for t>0, we have(28)1Mrμj+1,ℓv1,t−1=1Mrℓμj,ℓv1,t−1≤a1Mrμj,v1,t−1+bMrμj,v1,t∗Mrv1,ℓv1,tMrμj,ℓμj,t∗Mrμj,ℓv1,2t−1+cMrμj,ℓμj,tMrμj,ℓv1,2t−1+Mrv1,ℓv1,tMrμj,ℓv1,2t−1+d1Mrμj,ℓμj,t−1+1Mrv1,ℓv1,t−1=a1Mrμj,v1,t−1+bMrμj,v1,t∗Mrv1,ℓv1,tMrμj,μj+1,t∗Mrμj,ℓv1,2t−1+cMrμj,μj+1,tMrμj,ℓv1,2t−1+Mrv1,ℓv1,tMrμj,ℓv1,2t−1+d1Mrμj,μj+1,t−1+1Mrv1,ℓv1,t−1.
From Definition2 (iii), Mrμj,ℓv1,2t≥Mrμj,v1,t∗Mrv1,ℓv1,t,fort>0, and we have(29)1Mrμj+1,ℓv1,t−1≤a1Mrμj,v1,t−1+bMrμj,v1,t∗Mrv1,ℓv1,tMrμj,μj+1,t∗Mrμj,v1,t∗Mrv1,ℓv1,t−1+cMrμj,μj+1,tMrμj,v1,t∗Mrv1,ℓv1,t−1+Mrv1,ℓv1,tMrμj,v1,t∗Mrv1,ℓv1,t−1+d1Mrμj,μj+1,t−1+1Mrv1,ℓv1,t−1⟶c+d1Mrv1,ℓv1,t−1,asj⟶∞.
Then,(30)limsupj⟶∞1Mrμj+1,ℓv1,t−1≤c+d1Mrv1,ℓv1,t−1,fort>0.
Now, from (26), (27), and (30), as j⟶∞, we get that(31)1Mrv1,ℓv1,t−1≤c+d1Mrv1,ℓv1,t−1,fort>0,and c+d<1 where a+b+2c+2d<1, and hence Mrv1,ℓv1,t=1, i.e., ℓv1=v1, for t>0.
Uniqueness. Let ∃z1∈U such that ℓz1=z1 and ℓv1=v1. Then, from (19) and from Definition 2 (iii), for t>0, we have(32)1Mrv1,z1,t−1=1Mrℓv1,ℓz1,t−1≤a1Mrv1,z1,t−1+bMrv1,z1,t∗Mrz1,ℓz1,tMrv1,ℓv1,t∗Mrv1,ℓz1,2t−1+cMrv1,ℓv1,tMrv1,ℓz1,2t−1+Mrz1,ℓz1,tMrv1,ℓz1,2t−1+d1Mrv1,ℓv1,t−1+1Mrz1,ℓz1,t−1=a1Mrv1,z1,t−1+bMrv1,z1,tMrv1,z1,2t−1+c1Mrv1,z1,2t−1+1Mrv1,z1,2t−1=a1Mrv1,z1,t−1+bMrv1,z1,tMrv1,z1,t∗Mrz1,z1,t−1+c1Mrv1,z1,t∗M−rz1,z1,t−1+1Mrv1,z1,t∗Mrz1,z1,t−1=a+2c1Mrv1,z1,t−1=a+2c1Mrℓv1,ℓz1,t−1≤a+2c21Mrv1,z1,t−1≤⋯≤a+2cj1Mrv1,z1,t−1⟶0,asj⟶∞,wherea+2c<1.
Hence,Mrv1,z1,t=1, and this implies that v1=z1, for t>0.Corollary 2.
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U satisfies(33)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1+bMrμ1,μ∗,t∗Mrμ∗,ℓμ∗,tMrμ1,ℓμ1,t∗Mrμ1,ℓμ∗,2t−1+d1Mrμ1,ℓμ1,t−1+1Mrμ∗,ℓμ∗,t−1,∀μ1,μ∗∈U, t>0, a,b,d≥0 with a+b+2d<1. Then, ℓ has a unique fixed point.Corollary 3.
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U satisfies(34)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1+cMrμ1,ℓμ1,tMrμ1,ℓμ∗,2t−1+Mrμ∗,ℓμ∗,tMrμ1,ℓμ∗,2t−1+d1Mrμ1,ℓμ1,t−1+1Mrμ∗,ℓμ∗,t−1,∀μ1,μ∗∈U, t>0, a,c,d≥0 with a+2c+2d<1. Then, ℓ has a unique fixed point.Corollary 4.
LetU,Mr,∗ be a G-complete FM-space in which Mr is triangular and a mapping ℓ:U⟶U satisfies(35)1Mrℓμ1,ℓμ∗,t−1≤a1Mrμ1,μ∗,t−1+d1Mrμ1,ℓμ1,t−1+1Mrμ∗,ℓμ∗,t−1,∀μ1,μ∗∈U, t>0, a,d≥0 with a+2d<1. Then, ℓ has a unique fixed point.Example 2.
From Example1, we define Mr as(36)Mrμ1,μ∗,t=tt+μ1−μ∗/2,∀μ1,μ∗∈U,t>0.
Then, one can easily show thatMr is triangular and U,Mr,∗ is G-complete FM-space. Now we define a mapping ℓ:U⟶U as(37)ℓμ1=3μ17,ifμ1∈0,1,3μ14+1,ifμ1∈1,∞.
Then, we have(38)1Mrℓμ1,ℓμ∗,t−1=371Mrμ1,μ∗,t−1,∀μ1,μ∗∈U,t>0.
A mappingℓ satisfies (4), and hence ℓ is a fuzzy contraction. Now, from Definition 2 (iii), Mrμ1,ℓμ∗,2t≥Mrμ1,μ∗,t∗Mrμ∗,ℓμ∗,t for t>0, and after simplification, we get the following:(39)Mrμ1,μ∗,t∗Mrμ∗,ℓμ∗,tMrμ1,ℓμ1,t∗Mrμ1,ℓμ∗,2t−1≤1Mrμ1,ℓμ1,t−1=2μ17t,Mrμ1,ℓμ1,tMrμ1,ℓμ∗,2t−1+Mrμ∗,ℓμ∗,tMrμ1,ℓμ∗,2t−1≤1071Mrμ1,μ∗,t−1=5μ1−μ∗7t,1Mrμ1,ℓμ1,t−1+1Mrμ∗,ℓμ∗,t−1=2μ1+μ∗7t.
Hence, all the conditions of Theorem2 are satisfied with a=3/7, b=c=1/9, and d=1/12, and ℓ has a unique fixed point, i.e., ℓ4=4∈0,∞.


## 4. Application

In this section, we present an integral type application to support our work. LetU=C0,η,ℝ be the space of all ℝ-valued continuous functions on the interval 0,η, where 0<η∈ℝ. The nonlinear integral equation is(40)μ1τ=∫0τΓτ,v,μ1vdv,∀μ1∈U,where τ,v∈0,η and Γ:0,η×0,η×ℝ⟶ℝ. The induced metric m:U2⟶ℝ can be defined as(41)mμ1,μ∗=supτ∈0,ημ1τ−μ∗τ=μ1−μ∗,whereμ1,μ∗∈C0,η,ℝ=U.The binary operation∗ is defined by α∗λ=αλ, ∀α,λ∈0,η. A standard fuzzy metric Mr:U2×0,∞⟶0,1 can be defined as(42)Mrμ1,μ∗,t=tt+mμ1,μ∗,fort>0,∀μ1,μ∗∈U.Then, one can easily verify thatMr is triangular and U,Mr,∗ is a G-complete FM-space.Theorem 3.
Let the integral equation be defined in (40), and there exists β∈0,1, satisfying(43)mℓμ1,ℓμ∗≤βNℓ,μ1,μ∗∀μ1,μ∗∈U,where(44)Nℓ,μ1,μ∗=maxμ1−μ∗,2μ1−ℓμ1,∀μ1,μ∗∈U.
Then, the integral equation in (40) has a unique solution in U.Proof.
Define the integral operatorℓ:U⟶U by(45)ℓμ1τ=∫0τΓτ,v,μ1vdv,∀μ1∈U.
Notice thatℓ is well defined and (40) has a unique solution if and only if ℓ has a unique fixed point in U. Now we have to show that Theorem 1 applies to the integral operator ℓ. Then, ∀μ1,μ∗∈U, we have the following two cases:(a)
IfNℓ,μ1,μ∗=μ1−μ∗ in (44), then, from (42) and (43), we have(46)1Mrℓμ1,ℓμ∗,t−1=mℓμ1,ℓμ∗t≤βNℓ,μ1,μ∗t=βμ1−μ∗t=β1Mrμ1,μ∗,t−1,and this implies that(47)1Mrℓμ1,ℓμ∗,t−1≤β1Mrμ1,μ∗,t−1,fort>0,∀μ1,μ∗∈U such that ℓμ1≠ℓμ∗. Inequality (47) holds if ℓμ1=ℓμ∗. Thus, the integral operator ℓ satisfies all the conditions of Theorem 1 with β=a and b=0 in (5). The integral operator ℓ has a unique fixed point, i.e., (40) has a solution in U.(b)
IfNℓ,μ1,μ∗=μ1−ℓμ1 in (44), then, from (42) and (43), we have(48)1Mrℓμ1,ℓμ∗,t−1=mℓμ1,ℓμ∗t≤βNℓ,μ1,μ∗t=βμ1−ℓμ1t≤2βμ1−ℓμ1t,and this implies that(49)1Mrℓμ1,ℓμ∗,t−1≤2βμ1−ℓμ1t,fort>0.
Here, we simplify the termMrμ1,μ∗,t/Mrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1, and by using Definition 2 (iii) and (42), for t>0, we have(50)Mrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1≤Mrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,μ1,t∗Mrμ1,ℓμ1,t−1=1Mrμ1,ℓμ1,t2−1=t+mμ1,ℓμ12−t2t2=2mμ1,ℓμ1t+mμ1,ℓμ1t2=2μ1−ℓμ1t+μ1−ℓμ1t2,and this implies that(51)Mrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1≤2μ1−ℓμ1t+μ1−ℓμ1t2,fort>0.
Now from (49) and (51), we have(52)1Mrℓμ1,ℓμ∗,t−1≤βMrμ1,μ∗,tMrμ1,ℓμ1,t∗Mrμ∗,ℓμ1,2t−1,fort>0,∀μ1,μ∗∈U such that ℓμ1≠ℓμ∗. Inequality (52) holds if ℓμ1=ℓμ∗. Thus, the integral operator ℓ satisfies all the conditions of Theorem 1 with β=b and a=0 in (5). The integral operator ℓ has a unique fixed point, i.e., (40) has a solution in U.


## 5. Conclusion

In this paper, we have presented the concept of rational type fuzzy-contraction maps inFM-spaces and proved some rational type fixed point theorems in G-complete FM-spaces under the rational type fuzzy-contraction conditions by using the “triangular property of fuzzy metric.” In the last section, we presented an integral type application for rational type fuzzy-contraction maps and proved a result of a unique solution for an integral operator in FM-space. In this direction, one can prove more rational type fuzzy-contraction results in G-complete FM-spaces with different types of applications.



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