# Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations

Authors: Saif Ur Rehman; Hassen Aydi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations:μτ=∫0τΓτ,v,μvdv and ντ=∫0τΓτ,v,νvdv, for all μ∈U, v∈0,η, and 0<η∈ℝ, where U=C0,η,ℝ is the space of all ℝ-valued continuous functions on the interval 0,η and Γ:0,η×0,η×ℝ⟶ℝ.


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## Body

## 1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle,” which is stated as follows: “A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point.” This principle plays a very important role in the fixed point theory. A number of researches have generalized it in many directions for single-valued and multivalued mappings in the context of metric spaces. Some of the findings can be found in [2–13] and the references therein. Currently, the fixed point theory is one of the most interested research areas in the field of mathematics. In the last decades, it has been investigated in many fields, such as game theory, graph theory, economics, computer sciences, and engineering.The theory of fuzzy sets was introduced by Zadeh [14], while the concept of a fuzzy metric space (FM space) was given by Kramosil and Michalek [15]. After that, the stronger form of the metric fuzziness was presented by George and Veeramani in [16]. Later on, in [17], Gregori and Sapena proved some contractive-type fixed point results in complete FM spaces. Some more fixed point results in FM spaces can be found in [18–27] and the references therein.Initially, in 2007, the concept of a cone metric space was reintroduced by Huang and Zhang [28]. They proved some nonlinear contractive-type fixed point results in cone metric spaces. After the publication of this article, a number of researchers have contributed their ideas in cone metric spaces. Some of such works can be found in [29–34] and the references therein.In 2015, the basic concept of a fuzzy cone metric space (FCM space) was given by Öner et al. [35]. They presented some key attributes and a “fuzzy cone Banach contraction theorem” in FCM spaces. Later, Rehman and Li [36] extended and improved a “fuzzy cone Banach contraction theorem” and proved some generalized fixed point theorems in FCM spaces. Some more properties and related fixed point results can be found in [37–47].The aim of this research work is to establish some rational-type fuzzy cone-contraction results in FCM spaces. We use the concept of [36, 39] and prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces. Some illustrative examples are presented. In the last section, we give an application of two Fredholm integral equations (FIEs).


## 2. Preliminaries

Definition 1 [47].
An operation∗:0,12⟶0,1 is called a continuous t-norm if
(i)
∗ is commutative, associative, and continuous(ii)
1∗η1=η1 and η1∗η2≤η3∗η4, whenever η1≤η3 and η2≤η4, for all η1,η2,η3,η4∈0,1
The basict-norms: the minimum, the product, and the Lukasiewicz continuous t-norms are defined by [47]
(1)η1∗η2=minη1,η2η1∗η2=η1η2,η1∗η2=maxη1+η2−1,0.Definition 2 [35].
A 3-tupleU,Mr,∗ is said to be a FCM space if P is a cone of E, U is an arbitrary set, ∗ is a continuous t-norm, and Mr is a fuzzy set on U2×intP satisfying the following conditions:
(1)
∀ν1,ν2∈U;Mrν1,ν2,t>0 and Mrν1,ν2,t=1⇔ν1=ν2(2)
∀ν1,ν2∈U;Mrν1,ν2,t=Mrν2,ν1,t(3)
∀ν1,ν2,ν3∈U;Mrν1,ν2,t∗Mrν2,ν3,s≤Mrν1,ν3,t+s(4)
∀ν1,ν2∈U;Mrν1,ν2,.:intP⟶0,1 is continuous
for allt,s∈sinintP.Definition 3 [35].
LetU,Mr,∗ be a FCM space and ν1∈U and νj be a sequence in U.
(i)
νj converges to ν1 if for c∈0,1 and t≫θ there is j1∈N such that Mrνj,ν1,t>1−c, for j≥j1. We may write this limj⟶∞νj=ν1 or νj⟶ν1 as j⟶∞(ii)
νj is Cauchy if for c∈0,1 and t≫θ there is j1∈N such that Mrνj,νk,t>1−c, for j,k≥j1(iii)
U,Mr,∗ is complete if every Cauchy sequence is convergent in U(iv)
νj is fuzzy cone contractive if there is a∈0,1 so that(2)1Mrνj,νj+1,t−1≤a1Mrνj−1,νj,t−1,fort≫θ,j≥1.Lemma 4 [35].
LetU,Mr,∗ be a FCM space and let νj be sequence in U converging to a point ν1∈U iff Mrνj,ν1,t⟶1 as j⟶∞ for each t≫θ.Definition 5 [36].
LetU,Mr,∗ be a FCM space. The fuzzy cone metric Mr is triangular if
(3)1Mrν1,ν3,t−1≤1Mrν1,ν2,t−1+1Mrν2,ν3,t−1,∀ν1,ν2,ν3∈U,t≫θ.Definition 6 [35].
LetU,Mr,∗ be a FCM space and ℓ:U⟶U. Then, ℓ is said to be fuzzy cone contractive if there is a∈0,1 such that
(4)1Mrℓν1,ℓν2,t−1≤a1Mrν1,ν2,t−1,∀ν1,ν2∈U,t≫θ.
A “fuzzy cone Banach contraction theorem” [35] is stated as follows: “Let U,Mr,∗ be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and ℓ:U⟶U be a fuzzy cone contractive mapping. Then, ℓ has a unique fixed point.”In this paper, we present some rational-type fuzzy cone-contraction theorems in FCM spaces by using the concept of [36, 39]. Namely, we prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces without the assumption that the fuzzy cone contractive sequences are Cauchy. We use “the triangular property of the fuzzy cone metric.” We also present some illustrative examples to support our work. In the last section, an application of Fredholm integral equations is provided.


## 3. Main Results

In this section, we prove some common fixed point theorems via generalized rational-type fuzzy cone-contraction conditions in FCM spaces.Theorem 7.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(5)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+bMrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1+cMrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1,
for allμ,ν∈U, t≫θ, a∈0,1, and b,c,d≥0 with a+b+c+2d<1. Then, ℓ and ℏ have a common fixed point in U.Proof.
Fixμ0∈U and construct a sequence of points in U such that
(6)μ2j+1=ℓμ2j,μ2j+2=ℏμ2j+1,j≥0.
Then, by (5), for t≫θ,
(7)1Mrμ2j+1,μ2j+2,t−1=1Mrℓμ2j,ℏμ2j+1,t−1≤a1Mrμ2j,μ2j+1,t−1+bMrμ2j,μ2j+1,tMrμ2j,ℏμ2j+1,2t∗Mrμ2j+1,ℓμ2j,2t−1+cMrμ2j,ℓμ2j,t∗Mrμ2j+1,ℏμ2j+1,tMrμ2j,μ2j+1,t∗Mrμ2j,ℏμ2j+1,2t∗Mrμ2j+1,ℓμ2j,2t−1+d1Mrμ2j,ℓμ2j,t−1+1Mrμ2j+1,ℏμ2j+1,t−1=a1Mrμ2j,μ2j+1,t−1+bMrμ2j,μ2j+1,tMrμ2j,μ2j+2,2t∗Mrx2j+1,x2j+1,2t−1+cMrμ2j,μ2j+1,t∗Mrμ2j+1,μ2j+2,tMrμ2j,μ2j+1,t∗Mrμ2j,μ2j+2,2t∗Mrμ2j+1,μ2j+1,2t−1+d1Mrμ2j,μ2j+1,t−1+1Mrμ2j+1,μ2j+2,t−1=a1Mrμ2j,μ2j+1,t−1+bMrμ2j,μ2j+1,tMrμ2j,μ2j+2,2t−1+cMrμ2j+1,μ2j+2,tMrμ2j,μ2j+2,2t−1+d1Mrμ2j,μ2j+1,t−1+1Mrμ2j+1,μ2j+2,t−1.By Definition2 (3), Mrμ2j,μ2j+2,2t≥Mrμ2j,μ2j+1,t∗Mrμ2j+1,μ2j+2,t, for t≫θ. One writes
(8)1Mrμ2j+1,μ2j+2,t−1≤a1Mrμ2j,μ2j+1,t−1+bMrμ2j,μ2j+1,tMrμ2j,μ2j+1,t∗Mrμ2j+1,μ2j+2,t−1+cMrμ2j+1,μ2j+2,tMrμ2j,μ2j+1,t∗Mrμ2j+1,μ2j+2,2t−1+d1Mrμ2j,μ2j+1,t−1+1Mrμ2j+1,μ2j+2,t−1.After simplification, we get that(9)1Mrμ2j+1,μ2j+2,t−1≤γ1Mrμ2j,μ2j+1,t−1,fort≫θ,whereγ=a+c+d/1−b−d<1 since a+b+c+2d<1. Similarly,
(10)1Mrμ2j+2,μ2j+3,t−1=1Mrℓμ2j+2,ℏμ2j+1,t−1≤a1Mrμ2j+1,μ2j+2,t−1+bMrμ2j+1,μ2j+2,tMrμ2j+2,ℏμ2j+1,2t∗Mμ2j+1,ℓμ2j+2,2t−1+cMrμ2j+2,ℓμ2j+2,t∗Mrμ2j+1,ℏμ2j+1,tMrμ2j+1,μ2j+2,t∗Mrμ2j+2,ℏμ2j+1,2t∗Mrμ2j+1,ℓμ2j+2,2t−1+d1Mrμ2j+2,ℓμ2j+2,t−1+1Mrμ2j+1,ℏμ2j+1,t−1=a1Mrμ2j+1,μ2j+2,t−1+bMrμ2j+1,μ2j+2,tMrμ2j+2,μ2j+2,2t∗Mrx2j+1,x2j+3,2t−1+cMrμ2j+2,μ2j+3,t∗Mrμ2j+1,μ2j+2,tMrμ2j+1,μ2j+2,t∗Mrμ2j+2,μ2j+2,2t∗Mrμ2j+1,μ2j+3,2t−1+d1Mrμ2j+2,μ2j+3,t−1+1Mrμ2j+1,μ2j+2,t−1=a1Mrμ2j+1,μ2j+2,t−1+bMrμ2j+1,μ2j+2,tMrμ2j+1,μ2j+3,2t−1+cMrμ2j+2,μ2j+3,tMrμ2j+1,μ2j+3,2t−1+d1Mrμ2j+2,μ2j+3,t−1+1Mrμ2j+1,μ2j+2,t−1.Again, by Definition2 (3), Mrμ2j+1,μ2j+3,2t≥Mrμ2j+1,μ2j+2,t∗Mrμ2j+2,μ2j+3,t, for t≫θ. We have
(11)1Mrμ2j+2,μ2j+3,t−1≤a1Mrμ2j+1,μ2j+2,t−1+bMrμ2j+1,μ2j+2,tMrμ2j+1,μ2j+2,t∗Mrμ2j+2,μ2j+3,t−1+cMrμ2j+2,μ2j+3,tMrμ2j+1,μ2j+2,t∗Mrμ2j+2,μ2j+3,2t−1+d1Mrμ2j+1,μ2j+2,t−1+1Mrμ2j+2,μ2j+3,t−1.After simplification, we have(12)1Mrμ2j+2,μ2j+3,t−1≤γ1Mrμ2j+1,μ2j+2,t−1,fort≫θ,where the value ofγ is the same as in (9). Now, from (9) and (12) and by induction, we have
(13)1Mrμ2j+2,μ2j+3,t−1≤γ1Mrμ2j+1,μ2j+2,t−1≤γ21Mrμ2j+1,μ2j+2,t−1≤⋯≤γ2j+21Mrμ0,μ1,t−1⟶0,asj⟶∞,which shows thatμj is a fuzzy cone-contractive sequence in U, and we get that
(14)limj⟶∞Mrμ2j+1,μ2j+2,t=1,fort≫θ.Note thatMr is triangular; then, for all k>j≥j0,
(15)1Mrμj,μk,t−1≤1Mrμj,μj+1,t−1+1Mrμj+1,μj+2,t−1+⋯+1Mrμk−1,μk,t−1≤γj+γj+1+⋯+γk−11Mrμ0,μ1,t−1≤γj1−γ1Mrμ0,μ1,t−1⟶0,asj⟶∞,which yields thatμj is a Cauchy sequence in U. Since U,Mr,∗ is complete, there is v1∈U such that
(16)limj⟶∞Mrμ2j+1,v1,t=1,fort≫θ.Now, we prove thatℏv1=v1. Since Mr is triangular,
(17)1Mrv1,ℏv1,t−1≤1Mrv1,μ2j+1,t−1+1Mrμ2j+1,ℏv1,t−1,fort≫θ.By (5), (14), and (16), for t≫θ,
(18)1Mrμ2j+1,ℏv1,t−1=1Mrℓμ2j,ℏv1,t−1≤a1Mrμ2j,v1,t−1+bMrμ2j,v1,tMrμ2j,ℏv1,2t∗Mrv1,ℓμ2j,2t−1+cMrμ2j,ℓμ2j,t∗Mrv1,ℏv1,tMrμ2j,v,t∗Mrμ2j,ℏv1,2t∗Mrv1,ℓμ2j,2t−1+d1Mrμ2j,ℓμ2j,t−1+1Mrν1,ℏν1,t−1=a1Mrμ2j,v1,t−1+bMrμ2j,v1,tMrμ2j,ℏv1,2t∗Mrv1,μ2j+1,2t−1+cMrμ2j,μ2j+1,t∗Mrv1,ℏv1,tMrμ2j,v1,t∗Mrμ2j,ℏv1,2t∗Mrv1,μ2j+1,2t−1+d1Mrμ2j,μ2j+1,t−1+1Mrν1,ℏν1,t−1.Again, by Definition2 (3), Mrμ2j,ℏv1,2t≥Mrμ2j,v1,t∗Mrv1,ℏv1,t, for t≫θ. It follows that
(19)1Mrμ2j+1,ℏv1,t−1≤a1Mrμ2j,v1,t−1+bMrμ2j,v1,tMrμ2j,v1,t∗Mrv1,ℏv1,t∗Mrv1,μ2j+1,2t−1+cMrμ2j,μ2j+1,t∗Mrv1,ℏv1,tMrμ2j,v1,t∗Mrμ2j,v1,t∗Mrv1,ℏv1,t∗Mrv1,μ2j+1,2t−1+d1Mrμ2j,μ2j+1,t−1+1Mrν1,ℏν1,t−1⟶b+d1Mrv1,ℏv1,t−1,asj⟶∞.Then,(20)limsupj⟶∞1Mrμ2j+1,ℏv1,t−1≤b+d1Mrv1,ℏv1,t−1,fort≫θ.This together with (17) and (16) implies
(21)1Mrv1,ℏv1,t−1≤b+d1Mrv1,ℏv1,t−1,fort≫θ.Note thatb+d<1 because a+b+c+2d<1. Then, Mrv1,ℏv1,t=1, that is, ℏv1=v1. Similarly, we can show that ℓv1=v1 because Mr is triangular. Therefore,
(22)1Mrv1,ℓv1,t−1≤1Mrv1,μ2j+2,t−1+1Mrμ2j+2,ℓv1,t−1,fort≫θ.Now, again by (5), (14), and (16), one writes for t≫θ(23)1Mrμ2j+2,ℓv1,t−1=1Mrℓv1,ℏμ2j+1,t−1≤a1Mrv1,μ2j+1,t−1+bMrv1,μ2j+1,tMrv1,ℏμ2j+1,2t∗Mrμ2j+1,ℓv1,2t−1+cMrv1,ℓv1,t∗Mrμ2j+1,ℏμ2j+1,tMrv1,μ2j+1,t∗Mrv1,ℏμ2j+1,2t∗Mrμ2j+1,ℓv1,2t−1+d1Mrν1,ℓν1,t−1+1Mrμ2j+1,ℏμ2j+1,t−1=a1Mrv1,μ2j+1,t−1+bMrv1,μ2j+1,tMrv1,μ2j+2,2t∗Mrμ2j+1,ℓv1,2t−1+cMrv1,ℓv1,t∗Mrμ2j+1,μ2j+2,tMrv1,μ2j+1,t∗Mrv1,μ2j+2,2t∗Mrμ2j+1,ℓv1,2t−1+d1Mrν1,ℓν1,t−1+1Mrμ2j+1,μ2j+2,t−1.Again, by Definition2 (3), Mrμ2j+1,ℓv1,2t≥Mrμ2j+1,v1,t∗Mrv1,ℓv1,t, for t≫θ. It follows that
(24)1Mrμ2j+1,ℓv1,t−1≤a1Mrv1,μ2j+1,t−1+bMrv1,μ2j+1,tMrv1,μ2j+2,2t∗Mrμ2j+1,v1,t∗Mrv1,ℓv1,t−1+cMrv1,ℓv1,t∗Mrμ2j+1,μ2j+2,tMrv1,μ2j+1,t∗Mrv1,μ2j+2,2t∗Mrμ2j+1,v1,t∗Mrv1,ℓv1,t−1+d1Mrν1,ℓν1,t−1+1Mrμ2j+1,μ2j+2,t−1⟶b+d1Mrv1,ℓv1,t−1,asj⟶∞.Then,(25)limsupj⟶∞1Mrμ2j+2,ℓv1,t−1≤b+d1Mrv1,ℓv1,t−1,fort≫θ.This together with (22) and (16) implies
(26)1Mrv1,ℓv1,t−1≤b+d1Mrv1,ℓv1,t−1,fort≫θ.Note thatb+d<1 since a+b+c+2d<1. Then, Mrv1,ℓv1,t=1, that is, ℓv1=v1.Hence,v1 is a common fixed point of ℓ and ℏ.Example 1.
LetU=0,∞, ∗ be a continuous t-norm and Mr:U2×0,∞⟶0,1 be written as
(27)Mrμ,ν,t=tt+μ−ν,∀μ,ν∈U,t≫θ.
Then, easily one can verify thatMr is triangular and U,Mr,∗ is a complete FCM space. Now, we define ℓ,ℏ:U⟶U by
(28)ℓμ=ℏμ=3μ8,ifμ∈0,1,4μ5+75,ifμ∈1,∞.
Then, fort≫θ, we have
(29)1Mrℓμ,ℏν,t−1=ℓμ−ℏνt=381Mrμ,ν,t−1.
Hence, the pair of self-mappingℓ,ℏ is a fuzzy cone-contraction. Now, from Definition 2 (3), Mrμ,ℏν,2t≥Mrμ,ν,t∗Mrν,ℏν,t and Mrν,ℓμ,2t≥Mrν,μ,t∗Mrμ,ℓμ,t, for t≫θ. We get the following:
(30)Mrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤1Mrμ,ν,t∗Mrμ,ℓμ,t∗Mrν,ℏν,t−1=1t/t+μ−νt/t+μ−ℓμt/t+ν−ℏν−1=t+μ−νt+μ−ℓμt+ν−ℏνt3−1=t+μ−ν5t/8μ+ν+25/64μν+t2μ−νt3=5t+μ−ν8tμ+ν+5μν64t3+μ−νt,Mrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤1Mrμ,ν,t3−1=1t/t+μ−ν3−1=t+μ−ν3t3−1=μ−ν3+3tμ−νt+μ−νt3,1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1=1t/t+μ−ℓμ−1+1t/t+ν−ℏν−1=5μ+ν8t.
Hence, from the above, we conclude that all the conditions of Theorem7 are satisfied with a=3/8, b=c=1/6, and d=1/8. The mappings ℓ and ℏ have a common fixed point, i.e., ℓ7=ℏ7=7∈0,∞.Puttingb=0 in Theorem 7, we get the following corollary.Corollary 8.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(31)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+cMrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1,for all μ,ν∈U, t≫θ, a∈0,1, and c,d≥0 with a+c+2d<1. Then, ℓ and ℏ have a common fixed point in U.In the following corollary, we prove that the mappingsℓ and ℏ have a unique common fixed point in U by using the constant c=0 in Theorem 7.Corollary 9.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(32)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+bMrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1,for all μ,ν∈U, t≫θ, a∈0,1, and b,d≥0 with a+b+2d<1. Hence, ℓ and ℏ have a unique common fixed point in U.Proof.
It follows from the proof of Theorem7 that v1 is a common fixed point of ℓ and ℏ in U. For uniqueness, let u1 be another common fixed point of ℓ and ℏ in U such that ℓu1=ℏu1=u1 and ℓv1=ℏv1=v1. Then, by view of (32),
(33)1Mru1,v1,t−1=1Mrℓu1,ℏv1,t−1≤a1Mru1,v1,t−1+bMru1,v1,tMru1,ℏv1,2t∗Mrv1,ℓu1,2t−1+d1Mrμ1,ℓμ1,t−1+1Mrν1,ℏν1,t−1.
By Definition2 (3),
(34)Mrv1,ℓu1,2t≥Mrv1,u1,t∗Mru1,ℓu1,t=Mrv1,u1,t∗1=Mrv1,u1,t,fort≫θ,Mru1,ℏv1,2t≥Mru1,v1,t∗Mrv1,ℏv1,t=Mru1,v1,t∗1=Mru1,v1,t,fort≫θ.
It follows that(35)1Mru1,v1,t−1≤a1Mru1,v1,t−1+bMru1,v1,tMru1,v1,t∗Mrv1,u1,t−1+d1Mrμ1,μ1,t−1+1Mrν1,ν1,t−1=a+b1Mru1,v1,t−1=a+b1Mrℓu1,ℏv1,t−1≤a+b21Mru1,v1,t−1≤⋯≤a+bj1Mru1,v1,t−1⟶0,asj⟶∞.
Sincea+b<1, one writes Mru1,v1,t=1, i.e., u1=v1 for t≫θ.Corollary 10.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(36)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+d1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1,
for allμ,ν∈U, t≫θ, a∈0,1, and d≥0 with a+2d<1. Then, ℓ and ℏ have a unique common fixed point in U.Example 2.
As in Example1, let Mr:U2×0,∞⟶0,1 be defined by
(37)Mrμ,ν,t=tt+μ−ν/2,∀μ,ν∈U,t≫θ.
Then, easily one can verify thatMr is triangular and U,Mr,∗ is a complete FCM space. Now, we define self-mappings ℓ,ℏ:U⟶U by
(38)lμ=2μ3+13,μ∈0,1,4μ5+85,μ∈1,∞,ℏν=2ν3+13,ν∈0,1,5ν6+43,ν∈1,∞.
Then, from (36), for t≫θ, we have
(39)1Mrℓμ,ℏν,t−1=μ−ν3t≤23μ−ν2t+μ+ν−242t≤23μ−ν2t+17μ+ν−26t=a1Mrμ,ν,t−1+d1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1.
Hence, all the conditions of Corollary10 are satisfied with a=2/3 and d=1/7. The mappings ℓ and ℏ have a common fixed point, i.e., ℓ8=ℏ8=8∈0,∞.Theorem 11.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(40)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+bMrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1+cMrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrν,ℓμ,t−1+1Mrμ,ℏν,t−1,
for allμ,ν∈U, t≫θ, a∈0,1, and b,c,d≥0 with a+b+c+2d<1. Then, ℓ and ℏ have a common fixed point in U.Proof.
The proof is similar as the proof of Theorem7.Corollary 12.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(41)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+cMrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrν,ℓμ,t−1+1Mrμ,ℏν,t−1,
for allμ,ν∈U, t≫θ, a∈0,1, and c,d≥0 with a+c+2d<1. Then, ℓ and ℏ have a common fixed point in U.Corollary 13.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(42)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+bMrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1+d1Mrν,ℓμ,t−1+1Mrμ,ℏν,t−1,
∀μ,ν∈U, t≫θ, a∈0,1, and b,d≥0 with a+b+2d<1. Then, ℓ and ℏ have a unique common fixed point in U.Proof.
It is as theproof of Theorem 7. Let v1 be a common fixed point of ℓ and ℏ in U. Let u1 be another common fixed point of ℓ and ℏ in U such that ℓu1=ℏu1=u1 and ℓv1=ℏv1=v1. Then, by view of (42),
(43)1Mru1,v1,t−1=1Mrℓu1,ℏv1,t−1≤a1Mru1,v1,t−1+bMru1,v1,tMru1,ℏv1,2t∗Mrv1,ℓu1,2t−1+d1Mrv1,ℓu1,t−1+1Mru1,ℏv1,t−1.By Definition 2 (3),
(44)Mrv1,ℓu1,2t≥Mrv1,u1,t∗Mru1,ℓu1,t=Mrv1,u1,t∗1=Mrv1,u1,t,fort≫θ,Mru1,ℏv1,2t≥Mru1,v1,t∗Mrv1,ℏv1,t=Mru1,v1,t∗1=Mru1,v1,t,fort≫θ.
It follows that(45)1Mru1,v1,t−1≤a1Mru1,v1,t−1+bMru1,v1,tMru1,v1,t∗Mrv1,u1,t−1+d1Mrv1,u1,t−1+1Mru1,v1,t−1=a+b+2d1Mru1,v1,t−1,fort≫θ.
Since0<a+b+2d<1, Mru1,v1,t=1, i.e., u1=v1.Corollary 14.
LetU,Mr,∗ be a complete FCM space in which Mr is triangular. Let ℓ,ℏ:U⟶U be a pair of self-mappings so that
(46)1Mrℓμ,ℏν,t−1≤a1Mrμ,ν,t−1+d1Mrν,ℓμ,t−1+1Mrμ,ℏν,t−1,
for allμ,ν∈U, t≫θ, a∈0,1, and d≥0 with a+2d<1. Then, ℓ and ℏ have a unique common fixed point in U.Example 3.
LetU=0,1. As in Example 2, we define self-mappings ℓ,ℏ:U⟶U by
(47)ℓμ=2μ5+17,μ∈0,12,3μ4+316,μ∈12,1,ℏν=2ν5+17,ν∈0,12,2ν3+14,ν∈12,1.
Now, from (46), for t≫θ, we have
(48)1Mrℓμ,ℏν,t−1=ℓμ−ℏν2t=μ−ν5t≤25μ−ν2t+2721μ+ν−1070t≤25μ−ν2t+27ν−2μ/5−1/72t+μ−2ν/5−1/72t=a1Mrν,μ,t−1+d1Mrν,ℓμ,t−1+1Mrμ,ℏν,t−1.
Hence, all the conditions of Corollary14 are satisfied with a=2/5 and d=2/7. The mappings ℓ and ℏ have a common fixed point, i.e., ℓ3/4=ℏ3/4=3/4∈0,1.


## 4. Application

In this section, we present an application on Fredholm integral equations. LetU=C0,η,ℝ be the space of all ℝ-valued continuous functions on the interval 0,η, where 0<η∈ℝ. The Fredholm integral equations are
(49)μτ=∫0ηK1τ,v,μvdv,ντ=∫0ηK2τ,v,νvdv,∀μ,ν∈U,whereτ∈0,η and K1,K2:0,η×0,η×ℝ⟶ℝ. The induced metric d:U2⟶ℝ be defined as
(50)dμ,ν=supτ∈0,ημτ−ντ=μ−ν,whereμ,ν∈C0,η,ℝ=U.The binary operation∗ is defined by α∗λ=αλ for all α,λ∈0,η. A standard fuzzy metric Mr:U2×0,∞⟶0,1 is given as
(51)Mrμ,ν,t=tt+dμ,ν,fort>0,∀μ,ν∈U.Then, easily one can verify thatMr is triangular and U,Mr,∗ is a complete FCM space.Theorem 15.
The two FIEs are(52)μτ=∫0ηK1τ,v,xvdv,ντ=∫0ηK2τ,v,νvdv,
whereτ∈0,1 and μ,ν∈U. Assume that K1,K2:0,1×0,1×ℝ⟶ℝ are such that Aμ,Bν∈E for every μ,ν∈E, where
(53)Aμτ=∫0ηK1τ,v,μvdv,Bντ=∫0ηK2τ,v,νvdv.
If there existsβ∈0,1 such that for all μ,ν∈U,
(54)Aμ−Bν≤βNℓ,ℏ,μ,ν,where
(55)Nℓ,ℏ,μ,ν=maxμ−ν,μ−Aμ+ν−Bν,1t23tμ−ν2+μ−ν3,1t2t+μ−νtμ−Aμ+tν−Bν+μ−Aμ·ν−Bν.
Then, the two FIEs defined in (49) have a common solution in U.Proof.
Define the mappingsℓ,ℏ:E⟶E by
(56)ℓμ=Aμ,ℏν=Bν.
The FIEs in (49) have a common solution if and only if ℓ and ℏ have a common fixed point in U. Now, we have to show that Theorem 7 is applied to the integral operators ℓ and ℏ. Then, for all μ,ν∈U, we have the following four cases.
(a)
IfNℓ,ℏ,μ,ν=∥μ−ν∥ in (55), then from (51) and (54), we have(57)1Mrℓμ,ℏν,t−1=dℓμ,ℏνt≤βNℓ,ℏ,μ,νt=βμ−νt=β1Mrμ,ν,t−1.
This implies that(58)1Mrℓμ,ℏν,t−1≤β1Mrμ,ν,t−1,fort≫θ,
for allμ,ν∈U such that ℓμ≠ℏν. It is obvious that the inequality (58) holds if ℓμ=ℏν. Thus, the integral operators ℓ and ℏ satisfy all the conditions of Theorem 7 with β=a and b=c=d=0 in (5). The integral operators ℓ and ℏ have a common fixed point, i.e., (49) has a common solution in U.
(b)
IfNℓ,ℏ,μ,ν=1/t2t+μ−νtμ−Aμ+tν−Bν+μ−Aμ·ν−Bν in (55), then from (51) and (54), we have(59)1Mrℓμ,ℏν,t−1=dℓμ,ℏνt≤βNℓ,ℏ,μ,νt=β1t3t+μ−νtμ−Aμ+tν−Bν+μ−Aμ·ν−Bν.
It yields that(60)1Mrℓμ,ℏν,t−1≤β1t3t+μ−νtμ−Aμ+tν−Bν+μ−Aμ·ν−Bν,
for allμ,ν∈U and for t≫θ. Here, we simplify the term Mrμ,ν,t/Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1 by using Definition 2 (3) and (51), for t≫θ, we have
(61)Mrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤Mrμ,ν,tMrμ,ν,t∗Mrν,ℏν,t∗Mrν,μ,t∗Mrμ,ℓμ,t−1=1Mrμ,ν,t∗Mrν,ℏν,t∗Mrμ,ℓμ,t−1=t+μ−νt+ν−ℏνt+μ−ℓμt3−1=1t3t2μ−ν+t2ν−ℏν+μ−ℓμ+tμ−νν−ℏν+μ−ℓμtμ−ℓμ·ν−ℏν+μ−ν·μ−ℓμ·ν−ℏν=1t3t2μ−ν+t+μ−νtν−ℏν+tμ−ℓμ+μ−ℓμ·ν−ℏν.
This implies that(62)Mrμ,ν,tMrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤1t3t2μ−ν+t+μ−νtν−ℏν+tμ−ℓμ+μ−ℓμ·ν−ℏν,
for allμ,ν∈U and for t≫θ. Now, from (60) and (62), we have
(63)1Mrℓμ,ℏν,t−1≤βMrμ,ν,tMrμ,ℓμ,t∗Mrν,ℏν,2t−1,fort≫θ,
for allμ,ν∈U such that ℓμ≠ℏν. It is obvious that the inequality (63) holds if ℓμ=ℏν. Thus, the integral operators ℓ and ℏ satisfy all the conditions of Theorem 7 with β=b and a=c=d=0 in (5). The integral operators ℓ and ℏ have a common fixed point, i.e., (49) has a common solution in U.
(c)
IfNℓ,ℏ,μ,ν=1/t23t∥μ−ν∥2+∥μ−ν∥3 in (55), then from (51) and (54), we have(64)1Mrℓμ,ℏν,t−1=dℓμ,ℏνt≤βNℓ,ℏ,μ,νt=β3tμ−ν2+μ−ν3t3.This implies
(65)1Mrℓμ,ℏν,t−1≤β3tμ−ν2+μ−ν3t3,
for allμ,ν∈U and for t≫θ. Here, we simplify the term Mrμ,ℓμ,t∗Mrν,ℏν,t/Mrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1, by Definition 2 (3). For t≫θ, we have
(66)Mrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤Mrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ν,t∗Mrν,ℏν,t∗Mrν,μ,t∗Mrμ,ℓμ,t−1=1Mrμ,ν,t∗Mrμ,ν,t∗Mrν,μ,t−1.
In view of (51) and after routine calculation, we get
(67)Mrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1≤3t2∥μ−ν∥+3t∥μ−ν∥2+∥μ−ν∥3t2,
fort≫θ. Now, from (65) and (67), we have
(68)1Mrℓμ,ℏν,t−1≤βMrμ,ℓμ,t∗Mrν,ℏν,tMrμ,ν,t∗Mrμ,ℏν,2t∗Mrν,ℓμ,2t−1,fort≫θ,
for allμ,ν∈U such that ℓμ≠ℏν. It is obvious that the inequality (68) holds if ℓμ=ℏν. Thus, the integral operators ℓ and ℏ satisfy all the conditions of Theorem 7 with β=c and a=b=d=0 in (5). The integral operators ℓ and ℏ have a common fixed point, i.e., (49) has a common solution in U.
(d)
IfNℓ,ℏ,μ,ν=μ−Aμ+ν−Bν in (55), then from (51) and (54), we have(69)1Mrℓμ,ℏν,t−1=dℓμ,ℏνt≤βNℓ,ℏ,μ,νt=βμ−Aμ+ν−Bνt=β1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1.
This implies that(70)1Mrℓμ,ℏν,t−1≤β1Mrμ,ℓμ,t−1+1Mrν,ℏν,t−1,fort≫θ,
for allμ,ν∈U such that ℓμ≠ℏν. It is obvious that the inequality (70) holds if ℓμ=ℏν. Thus, the integral operators ℓ and ℏ satisfy all the conditions of Theorem 7 with β=d and a=b=c=0 in (5). The integral operators ℓ and ℏ have a common fixed point, i.e., (49) has a common solution in U.


## 5. Conclusion

In this paper, we presented the concept of rational-type fuzzy cone contractions in FCM spaces and some common fixed point results under generalized rational-type fuzzy cone-contraction conditions in complete FCM spaces by using the “triangular property of fuzzy cone metric” as a basic tool. Moreover, we resolved some Fredholm integral equations as an application. So, one can use this concept to prove more rational-type fuzzy cone-contraction results in complete FCM spaces with different types of applications.



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