# A New Approach to Fuzzy Differential Equations Using Weakly-Compatible Self-Mappings in Fuzzy Metric Spaces

Authors: Iqra Shamas; Saif Ur Rehman; Naeem Jan; Abdu Gumaei; Mabrook Al-Rakhami
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The key objective of this research article includes the study of some rational type coincidence point and deriving common fixed point (CFP) results for rational type weakly-compatible three self-mappings in fuzzy metric (FM) space. The “triangular property of FM” is used as a fundamental tool. Moreover, some unique coincidence points and CFP theorems were presented for three self-mappings in an FM space under the conditions of rational type weakly-compatible fuzzy-contraction. In addition, some suitable examples are also given. Furthermore, an application of fuzzy differential equations is provided in the aid of the proposed work. Hence, the innovative direction of rational type weakly-compatible fuzzy-contraction with the application of fuzzy differential equations in FM space will certainly play a vital role in the related fields. It has the potential to be extended in any direction with different types of weakly-compatible fuzzy-contraction conditions for self-mappings with different types of differential equations.


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## Body

## 1. Introduction

In 1922, Banach [1] proved a “Banach contraction principle for fixed point (FP)” which is stated as “A self-mapping in a complete metric space satisfy the contraction condition has a unique FP.” After the publication of the “Banach contraction principle,” many researchers contributed in flourishing the FP theory. Thus, different types of contractive results were established for FP and CFP for different types of mappings in the context of metric spaces. Kannan [2], Chatterjea [3, 4], Ali et al. [5], Covitz and Nadler [6], Altun et al. [7], Khan [8], Rehman et al. [9], and Sahin [10] proved some single-valued and multivalued contractive type FP, CFP, and best proximity point results in different types of spaces.In 1965, the concept of fuzzy sets was given by Zadeh [11], and this concept is investigated, used, and applied in many directions. In [12], Kramosil and Michalek used the concept of fuzzy sets together with metric space and introduced the notion of FM space and some more notions. In 1988, Grabiec [13] used the concept of Kramosil and Michalek [12] and proved two FP theorems of “Banach and Edelstein contraction mapping theorems on complete and compact fuzzy metric spaces, respectively.” Later on, George and Veeramani [14] presented the stronger form of metric fuzziness. In 2002, Gregory and Sapena [15] proved some contractive type FP theorems in complete FM spaces in the sense of [12, 14]. In [16], Sadeghi et al. extended and improved the result of Gregory and Sapena [15] and proved some FP and coincidence point theorems by using an implicit relation for set-valued-mappings on complete partially ordered FM spaces. Bari and Vetro [17] established some FP results by using attractors and weak-fuzzy contractive mappings in FM space. While Hadzic and Pap [18] proved “a FP theorem for multi-valued mappings in probabilistic metric spaces with an application in FM spaces.” Imdad and Ali [19], Shamas et al. [20], Som [21], and Pant and Chauhan [22] proved FP and CFP theorems for different contractive type mappings in FM spaces. Saleh et al. [23, 24] established different contractive type FP results in FM spaces. Recently, Rehman et al. [25] introduced the rational type fuzzy-contraction condition in FM spaces and proved some FP theorems with an application. In 2015, Oner et al. [26] introduced the notion of fuzzy cone metric (FCM) space. They proved some basic properties and a “fuzzy cone Banach contraction theorem” for FP with the assumption that “the fuzzy cone contractive sequences are Cauchy.” Later on, Rehman and Li [27] established some FP theorems in FCM space without the assumption that the “fuzzy cone contractive sequences are Cauchy.” Jabeen et al. [28] proved some common FP results in FCM spaces by using the contractive type weakly-compatible self-mappings with an application. By using the concept of [27], Rehman and Aydi [29] proved some rational type CFP theorems in FCM spaces with an application to Fredholm integral equations.The chief purpose behind this article is the introduction of a new concept of rational type weakly-compatible fuzzy-contraction maps for three self-mappings in FM spaces. We used the “triangular property of fuzzy metric” as an elementary tool and proved some unique coincidence points and CFP theorems under the rational type weakly compatible fuzzy-contraction conditions for three self-mappings in FM spaces with some suitable examples. Moreover, we applied the fuzzy differential equations for a unique solution in order to support our study. As a result, the current novel direction of rational type weakly-compatible fuzzy-contraction with the application of fuzzy differential equations in FM space will play a vital role in the related fields. It has the potential to be extended in any direction with different types of weakly-contraction conditions for self-mappings with different types of differential equations.This paper is organized as follows: Section2 named as preliminaries consists of the basic concepts related to our main work. In Section 3, we established some coincidence points and CFP theorems under the rational type weakly-compatible fuzzy-contraction conditions for three self-mappings in FM spaces with examples to verify the validity of our work. Section 4 deals with the application of the fuzzy differential equations to support our work. Finally, the last section, Section 5 concludes this article.


## 2. Preliminaries

In this section, we present some preliminaries related to our main work such as continuoust-norm, FM space, Cauchy sequence, complete FM space, fuzzy-contraction, fuzzy-contractive sequence, and weakly-compatible self-mappings.Definition 1 (see [30]).
An operation∗:0,12⟶0,1 is called a continuous t-norm, if
(i)
∗ continuous(ii)
∗ is commutative and associative(iii)
1∗ζ1=ζ1 and ζ1∗ζ2≤ζ3∗ζ4, whenever ζ1≤ζ3 and ζ2≤ζ4, ∀ζ1,ζ2,ζ3,ζ4∈0,1The basic continuoust-norms of minimum, product, and Lukasiewicz are defined by Schweizer and Sklar [30], respectively, as follows
(i)
The minimumt-norm is ζ1∗ζ2=minζ1,ζ2(ii)
The productt-norm is ζ1∗ζ2=ζ1ζ2(iii)
The Lukasiewiczt-norm is ζ1∗ζ2=maxζ1+ζ2−1,0Definition 2 (see [14]).
A 3-tupleW,Mr,∗ is said to be a FM space if W is an arbitrary set, ∗ is a continuous t-norm and Mr is a fuzzy set on W2×0,∞ satisfying the following conditions:
(i)
Mrw,x,t>0(ii)
Mrw,x,t=1⇔w=x(iii)
Mrw,x,t=Mrx,w,t(iv)
Mrw,ρ,t∗Mrρ,x,s≤Mrw,x,t+s(v)
Mrw,x,.:0,∞⟶0,1 is continuous
(1)∀w,x,ρ∈Wandt,s∈0,∞.Lemma 3 (see [14]).
Mrw,x,∗ is nondecreasing ∀w,x∈W.Definition 4 (see [14, 15]).
LetW,Mr,∗ be a FM space, w∈W and wj is a sequence in W. Then,
(i)
wj converges to w if ρ∈0,1 and t>0∃j1∈ℕ, such that Mrwj,w,t>1−ρ, ∀j≥j1. We may write this limj⟶∞wj=w or wj⟶w as j⟶∞(ii)
wj is a Cauchy sequence if ρ∈0,1 and t>0∃j1∈ℕ such that Mrwj,wk,t>1−ρ, ∀j,k≥j1(iii)
W,Mr,∗ is complete if every Cauchy sequence is convergent in W(iv)
Fuzzy-contractive if∃a∈0,1 and satisfying(2)1Mrwj,wj+1,t−1≤a1Mrwj−1,wj,t−1fort>0,j≥1.
Throughout this paper,ℕ represents the set of natural numbers.Lemma 5 (see [14]).
LetW,Mr,∗ be a FM space. A sequence wj in W converges to w∈W if and only if Mrwj,w,t⟶1, as j⟶∞, for t>0.Definition 6 (see [17]).
LetW,Mr,∗ be a FM space. The fuzzy metric Mr is triangular, if
(3)1Mrw,x,t−1≤1Mrw,ρ,t−1+1Mrρ,x,t−1∀w,ρ,x∈W,t>0.Definition 7 (see [15]).
LetW,Mr,∗ be a FM space. A mapping A:W⟶W is said to be a fuzzy-contractive if ∃a∈0,1 such that
(4)1MrAw,Ax,t−1≤a1Mrw,x,t−1∀w,x∈W,t>0.Definition 8 (see [31]).
LetA and ℓ be two self-mappings on a nonempty set W (i.e., A,ℓ:W⟶W). If there exists u∈W and u=Av=ℓv for some v∈W. Then, v is called a coincidence point of A and ℓ, and u is called a point of coincidence of the mappings A and ℓ. The mappings A and ℓ are said to be weakly-compatible if they commute at their coincidence point, i.e., Av=ℓv for some v∈W, then Aℓv=ℓAv.Proposition 9 (see [31]).
LetA and ℓ be weakly-compatible self-mappings on a nonempty set W. If A and ℓ have a unique point of coincidence such that u=Av=ℓv, then, u is known as the unique common FP of A and ℓ.


## 3. Main Result

This section deals with the main results of our paper, here, we establish some coincidence point and CFP theorems under the rational type weakly-compatible fuzzy-contractive for three self-mappings in FM spaces with some suitable examples. Throughout main results, we use the concept of a binary operation∗ is a continuous product t-norm which is defined as:
(5)ξ∗ζ=ξ·ζforallξ,ζ∈0,1.Now we are in the position to present our first main result.Theorem 10.
Let a fuzzy metricMr is triangular in a complete FM space W,Mr,∗ and let A,B,ℓ:W⟶W be three self-mappings, satisfies for all w,x∈W,
(6)1MrAw,Bx,t−1≤a1Mrℓw,ℓx,t−1+bMrℓw,ℓx,tMrℓx,Aw,2t·Mrℓw,Bx,2t−1+c1Mrℓw,Aw,t−1+1Mrℓx,Bx,t−1,for t>0 and 0≤a,b,c<1 with a+b+2c<1. If AW∪BW⊂ℓW, where ℓW is a complete subspace of W. Then A,B, and ℓ have a unique point of coincidence. Moreover, if the pairs A,ℓ and B,ℓ are weakly compatible. Then, A,B, and ℓ have a unique CFP in W.Proof.
Letw0 be the arbitrary point of W. Using the condition AW∪BW⊂ℓW choose a sequence wi in W such that
(7)ℓw2i+1=Aw2iandℓw2i+2=Bw2i+1,foralli≥0.
Now, by (6), for t>0,
(8)1Mrℓw2i+1,ℓw2i+2,t−1=1MrAw2i,Bw2i+1,t−1≤a1Mrℓw2i,ℓw2i+1,t−1+bMrℓw2i,ℓw2i+1,tMrℓw2i+1,Aw2i,2t·Mrℓw2i,Bw2i+1,2t−1+c1Mrℓw2i,Aw2i,t−1+1Mrℓw2i+1,Bw2i+1,t−1=a1Mrℓw2i,ℓw2i+1,t−1+bMrℓw2i,ℓw2i+1,tMrℓw2i+1,ℓw2i+1,2t·Mrℓw2i,ℓw2i+2,2t−1+c1Mrℓw2i,ℓw2i+1,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a1Mrℓw2i,ℓw2i+1,t−1+bMrℓw2i,ℓw2i+1,tMrℓw2i,ℓw2i+2,2t−1+c1Mrℓw2i,ℓw2i+1,t−1+1Mrℓw2i+1,ℓw2i+2,t−1.
Now by using Definition2 (iv), Mrℓw2i,ℓw2i+2,2t≥Mrℓw2i,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,t for t>0,
(9)1Mrℓw2i+1,ℓw2i+2,t−1≤a1Mrℓw2i,ℓw2i+1,t−1+bMrℓw2i,ℓw2i+1,tMrℓw2i,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,t−1+c1Mrℓw2i,ℓw2i+1,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a1Mrℓw2i,ℓw2i+1,t−1+b1Mrℓw2i+1,ℓw2i+2,t−1+c1Mrℓw2i,ℓw2i+1,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a+c1Mrℓw2i,ℓw2i+1,t−1+b+c1Mrℓw2i+1,ℓw2i+2,t−1.
After simplification, we obtain(10)1Mrℓw2i+1,ℓw2i+2,t−1≤Y1Mrℓw2i,ℓw2i+1,t−1fort>0,where Y=a+c/1−b−c<1. Similarly, again by the view of (6), for t>0,
(11)1Mrℓw2i+2,ℓw2i+3,t−1=1MrAw2i+2,Bw2i+1,t−1≤a1Mrℓw2i+2,ℓw2i+1,t−1+bMrℓw2i+2,ℓw2i+1,tMrℓw2i+1,Aw2i+2,2t·Mrℓw2i+2,Bw2i+1,2t−1+c1Mrℓw2i+2,Aw2i+2,t−1+1Mrℓw2i+1,Bw2i+1,t−1=a1Mrℓw2i+2,ℓw2i+1,t−1+bMrℓw2i+2,ℓw2i+1,tMrℓw2i+1,ℓw2i+3,2t·Mrℓw2i+2,ℓw2i+2,2t−1+c1Mrℓw2i+2,ℓw2i+3,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a1Mrℓw2i+2,ℓw2i+1,t−1+bMrℓw2i+2,ℓw2i+1,tMrℓw2i+1,ℓw2i+3,2t−1+c1Mrℓw2i+2,ℓw2i+3,t−1+1Mrℓw2i+1,ℓw2i+2,t−1.
Now by using Definition2 (iv), Mrℓw2i+1,ℓw2i+3,2t≥Mrℓw2i+1,ℓw2i+2,t·Mrℓw2i+2,ℓw2i+3,t for t>0,
(12)1Mrℓw2i+2,ℓw2i+3,t−1≤a1Mrℓw2i+2,ℓw2i+1,t−1+bMrℓw2i+2,ℓw2i+1,tMrℓw2i+1,ℓw2i+2,t·Mrℓw2i+2,ℓw2i+3,t−1+c1Mrℓw2i+2,ℓw2i+3,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a1Mrℓw2i+2,ℓw2i+1,t−1+b1Mrℓw2i+2,ℓw2i+3,t−1+c1Mrℓw2i+2,ℓw2i+3,t−1+1Mrℓw2i+1,ℓw2i+2,t−1=a+c1Mrℓw2i+1,ℓw2i+2,t−1+b+c1Mrℓw2i+2,ℓw2i+3,t−1.
After simplification, we obtain(13)1Mrℓw2i+2,ℓw2i+3,t−1≤Y1Mrℓw2i+1,ℓw2i+2,t−1fort>0,where Y value is same as in (10). Now from (10), (13), and by induction,
(14)1Mrℓw2i+2,ℓw2i+3,t−1≤Y1Mrℓw2i+1,ℓw2i+2,t−1≤Y21Mrℓw2i,ℓw2i+1,t−1≤⋯≤Y2i+21Mrℓw0,ℓw1,t−1⟶0,asi⟶∞.
Hence,ℓwii≥0 is a fuzzy-contractive sequence in W,Mr,∗, therefore,
(15)limi⟶∞Mrℓwi,ℓwi−1,t=1fort>0.
SinceMr is triangular, j>i>i0,
(16)1Mrℓwi,ℓwj,t−1≤1Mrℓwi,ℓwi+1,t−1+1Mrℓwi+1,ℓwi+2,t−1+⋯+1Mrℓwi−1,ℓwj,t−1≤Yi1Mrℓwi,ℓwi+1,t−1+Yi+11Mrℓwi+1,ℓwi+2,t−1+⋯+Yj−11Mrℓwi−1,ℓwj,t−1≤Yi+Yi+1+⋯+Yj−11Mrℓw0,ℓw1,t−1≤Yi1−Y1Mrℓw0,ℓw1,t−1⟶0,asi⟶∞.
This shows thatℓwi is a Cauchy sequence, and ℓW is a complete subspace of W. Hence, ∃u,v∈W such that ℓwi⟶u=ℓv as i⟶∞, i.e.,
(17)limi⟶∞Mru,ℓwi,t=Mru,ℓv,t=1fort>0.
SinceMr is triangular,
(18)1Mrℓv,Av,t−1≤1Mrℓv,ℓw2i+2,t−1+1Mrℓw2i+2,Av,t−1fort>0.
Now from (6), (15), (17), and by using Definition 2 (iv), for t>0,
(19)1Mrℓw2i+2,Av,t−1=1MrAv,Bw2i+1,t−1≤a1Mrℓv,ℓw2i+1,t−1+bMrℓv,ℓw2i+1,tMrℓw2i+1,Av,2t·Mrℓv,Bw2i+1,2t−1+c1Mrℓv,Av,t−1+1Mrℓw2i+1,Bw2i+1,t−1≤a1Mrℓv,ℓw2i+1,t−1+bMrℓv,ℓw2i+1,tMrℓw2i+1,ℓv,t·Mr,ℓv,Av,t·Mrℓv,ℓw2i+2,2t−1+c1Mrℓv,Av,t−1+1Mrℓw2i+1,ℓw2i+2,t−1⟶b+c1Mrℓv,Av,t−1asi⟶∞.
Then,(20)limsupi⟶∞1Mrℓw2i+2,Av,t−1≤b+c1Mrℓv,Av,t−1fort>0.
Now, from (17), (18), and (20), we obtain
(21)1Mrℓv,Av,t−1≤b+c1Mrℓv,Av,t−1fort>0.
Notice thatb+c<1, where a+b+2c<1, therefore, Mrℓv,Av,t=Mru,Av,t=1⇒u=ℓv=Av for t>0. Next, we have to prove that u=ℓv=Bv. Since, Mr is triangular,
(22)1Mrℓv,Bv,t−1≤1Mrℓv,ℓw2i+1,t−1+1Mrℓw2i+1,Bv,t−1fort>0.
Now, again from (6), (15), (17), and by using Definition 2 (iv), for t>0,
(23)1Mrℓw2i+1,Bv,t−1=1MrAw2i,Bv,t−1≤a1Mrℓw2i,ℓv,t−1+bMrℓw2i,ℓv,tMrℓv,Aw2i,2t·Mrℓw2i,Bv,2t−1+c1Mrℓw2i,Aw2i,t−1+1Mrℓv,Bv,t−1≤a1Mrℓw2i,ℓv,t−1+bMrℓw2i,ℓv,tMrℓv,ℓw2i+1,2t·Mrℓw2i,ℓv,t·Mrℓv,Bv,t−1+c1Mrℓw2i,ℓw2i+1,t−1+1Mrℓv,Bv,t−1.
Then,(24)limsupi⟶∞1Mrℓw2i+1,Bv,t−1≤b+c1Mrℓv,Bv,t−1fort>0.
Now, from (17), (22), and (24), we obtain
(25)1Mrℓv,Bv,t−1≤b+c1Mrℓv,Bv,t−1fort>0.
Note thatb+c<1, where a+b+2c<1, therefore, Mrℓv,Bv,t=Mru,Bv,t=1⇒u=ℓv=Av for t>0. Hence, u is a common coincidence point of the mappings ℓ,AandBinW such that u=ℓv=Av=Bv.
Next, we prove the uniqueness of a coincidence point inW,Mr,∗ for the mappings ℓ,AandB. Let u∗ be the other common coincidence point in W such that u∗=ℓv∗=Av∗=Bv∗ for some v∗∈W. Then, from (6) and by using Definition 2 (iv), for t>0,
(26)1Mru,u∗,t−1=1Mrℓv,ℓv∗,t−1=1MrAv,Bv∗,t−1≤a1Mrℓv,ℓv∗,t−1+bMrℓv,ℓv∗,tMrℓv∗,Av,2t·Mrℓv,Bv∗,2t−1+c1Mrℓv,Av,t−1+1Mrℓv∗,Bv∗,t−1=a1Mru,u∗,t−1+bMru,u∗,tMru∗,u,2t·Mru,u∗,2t−1+c1Mru,u,t−1+1Mru∗,u∗,t−1≤a+b1Mru,u∗,t−1,note that a+b<1, where a+b+2c<1. Thus, we get that Mru,u∗,t=1, that is, u=u∗. By using the weak compatibility of the pair A,ℓ,B,ℓ and by using Preposition 9, we can get a unique CFP of the mappings A,B, and ℓ. Let ∃z∈W such that, ℓz=Az=Bz=z. Hence, we get that Mrv,z,t=1⇒v=z, for t>0.Corollary 11.
Let a fuzzy metricMr is triangular in a complete FM space W,Mr,∗ and let A,B,ℓ:W⟶W be three self-mappings, satisfies for all w,x∈W,
(27)1MrAw,Bx,t−1≤a1Mrℓw,ℓx,t−1+bMrℓw,ℓx,tMrℓx,Aw,2t·Mrℓw,Bx,2t−1,for t>0 and 0≤a,b<1 with a+b<1. If AW∪BW⊂ℓW, where ℓW is a complete subspace of W. Then, A,B, and ℓ have a unique point of coincidence. Moreover, if the pairs A,ℓ and B,ℓ are weakly compatible. Then, A,B, and ℓ have a unique CFP in W.If we use identity map instead ofℓ, i.e., ℓ=I, in Theorem 10, we can get the following corollary:Corollary 12.
Let a fuzzy metricMr is triangular in a complete FM space W,Mr,∗ and let A,B:W⟶W be two self-mappings, satisfies for all w,x∈W,
(28)1MrAw,Bx,t−1≤a1Mrw,x,t−1+bMrw,x,tMrx,Aw,2t·Mrw,Bx,2t−1+c1Mrw,Aw,t−1+1Mrx,Bx,t−1,for t>0, 0≤a,b,c<1 with a+b+2c<1. Then, the mappings A and B have a unique CFP in W.Remark 13.
If we put the mappingsA=B and ℓ=I (identity map) with constant c=0 in Theorem 10, we obtained (Theorem 1 of [25]).Example 14.
LetW=0,1, ∗ is a product continuous t-norm on W=0,1 which is defined as ξ∗ζ=ξ·ζ for all ξ,ζ∈W and a fuzzy metric Mr:W2×0,∞⟶0,1 is defined by
(29)Mrw,x,t=tt+w−x,∀w,x∈W,andt>0.
Then, it is easy to prove thatMr is triangular and W,Mr,∗ is a complete FM space. The mappings A,B,ℓ:W⟶W be defined as
(30)Aw=Bw=4w3w+9andℓw=2w3∀w∈W.
Then, from (29), we have
(31)1MrAw,Bx,t−1=1tAw−Bx=1t4w3w+9−4x3x+9=36w−36x3w+93x+9≤36w−36x81=23ℓw−ℓxt=231Mrℓw,ℓx,t−1fort>0.
Hence, the self-mappingsA,B, and ℓ are satisfied the weakly-compatible fuzzy-contraction condition in FM spaces. Next, we simplify the second term of (6), then, by using Definition 2 (iv) and from (29), for t>0, we have
(32)Mrℓw,ℓx,tMrℓx,Aw,2t·Mrℓw,Bx,2t−1≤Mrℓw,ℓx,tMrℓx,ℓw,t·Mrℓw,Aw,t·Mrℓw,ℓx,t·Mrℓx,Bx,t−1=1Mrℓx,ℓw,t·Mrℓw,Aw,t·Mrℓx,Bx,t−1=t+ℓw−ℓx·t+ℓw−Aw·t+ℓx−Bxt3−1=t+2w/3−2x/3·t+2w/3−4w/3w+9·t+2x/3−4w/3w+9t3−1=t+2w/3−2x/3·t+2w2+w/3w+9·t+2x2+x/3x+9t3−1=t+2w/3−2x/3·t2+2tw2+w/3w+9+x2+x/3x+9+4w2+w·x2+x/3w+9·3x+9t3−1≤t+2/3w−x·t2+2t/813w2x+3wx2+9w2+x2+6wx+9w+x+4/81w2x2+w2x+wx2+wxt3−1=t+2/3w−x·t2+2/819tw2+x2+2w2x2+wx6t+2+wx3t+2+9w+xt3−1=1t32t23w−x+281t+23w−x9tw2+x2+2w2x2+wx6t+2+wx3t+2+9w+x.
Lastly, we simplify the third term of (6), then from (29), for t>0,
(33)1Mrw,Aw,t−1+1Mrx,Bx,t−1=1tℓw−Aw+ℓx−Bx=1t2w3−4w3w+9+2x3−4x3x+9=1t2w2+w3w+9+2x2+x3x+9≤281t3wx+9w+x+9w2+x2+6wx.
Hence, all the conditions of Theorem10 are satisfied with a=2/3,b=1/9, and c=1/5. The mappings A,B, and ℓ have a unique CFP, that is, 0.Theorem 15.
Let a fuzzy metricMr is triangular in a complete FM space W,Mr,∗ and let A,B,ℓ:W⟶W be three self-mappings, satisfies for all w,x∈W,
(34)1MrAw,Bx,t−1≤a1Mrℓw,ℓx,t−1+b1UA,B,ℓ,w,x,t−1+cMrℓw,Aw,t·Mrℓx,Bx,tMrℓw,Bx,2t·Mrℓw,ℓx,t·Mrℓx,Aw,2t−1,where
(35)UA,B,ℓ,w,x,t=maxMrℓw,ℓx,t,Mrℓw,Aw,t,Mrℓx,Bx,t,Mrℓx,Aw,t,Mrℓw,Bx,t,for t>0 and 0≤a,b,c<1 with a+b+c<1. If AW∪BW⊂ℓW, where ℓW is a complete subspace of W. Then A,B, and ℓ have a coincidence point in W.Proof.
Letw0 be the arbitrary point of W. Using the condition AW∪BW⊂ℓW choose a sequence wi in W such that
(36)ℓw2i+1=Aw2iandℓw2i+2=Bw2i+1,foralli≥0.
Now, by (34), for t>0,
(37)1Mrℓw2i+1,ℓw2i+2,t−1=1MrAw2i,Bw2i+1,t−1≤a1Mrℓw2i,ℓw2i+1,t−1+b1UA,B,ℓ,w2i,w2i+1,t−1+cMrℓw2i,Aw2i,t·Mrℓw2i+1,Bw2i+1,tMrℓw2i,Bw2i+1,2t·Mrℓw2i,ℓw2i+1,t·Mrℓw2i+1,Aw2i,2t−1=a1Mrℓw2i,ℓw2i+1,t−1+b1UA,B,ℓ,w2i,w2i+1,t−1+cMrℓw2i,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,tMrℓw2i,ℓw2i+2,2t·Mrℓw2i,ℓw2i+1,t−1,where
(38)UA,B,ℓ,w2i,w2i+1,t=maxMrℓw2i,ℓw2i+1,t,Mrℓw2i,Aw2i,t,Mrℓw2i+1,Bw2i+1,t,Mrℓw2i+1,Aw2i,t,Mrℓw2i,Bw2i+1,t=maxMrℓw2i,ℓw2i+1,t,Mrℓw2i,ℓw2i+1,t,Mrℓw2i+1,ℓw2i+2,t,Mrℓw2i+1,ℓw2i+1,t,Mrℓw2i,ℓw2i+2,t=maxMrℓw2i,ℓw2i+1,t,Mrℓw2i+1,ℓw2i+2,t,1,Mrℓw2i,ℓw2i+2,t=1.
Now from (37), (38), and by using Definition 2 (iv), for t>0, we obtain
(39)1Mrℓw2i+1,ℓw2i+2,t−1≤a1Mrℓw2i,ℓw2i+1,t−1+cMrℓw2i,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,tMrℓw2i,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,t·Mrℓw2i,ℓw2i+1,t−1.
After simplification, fort>0,
(40)1Mrℓw2i+1,ℓw2i+2,t−1≤ζ1Mrℓw2i,ℓw2i+1,t−1whereζ=a+c<1.
whereζ=a+c<1. Similarly, again by view of (34), for t>0,
(41)1Mrℓw2i+2,ℓw2i+3,t−1=1MrAw2i+2,Bw2i+1,t−1≤a1Mrℓw2i+2,ℓw2i+1,t−1+b1UA,B,ℓ,w2i+2,w2i+1,t−1+cMrℓw2i+2,Aw2i+2,t·Mrℓw2i+1,Bw2i+1,tMrℓw2i+2,Bw2i+1,2t·Mrℓw2i+2,ℓw2i+1,t·Mrℓw2i+1,Aw2i+2,2t−1=a1Mrℓw2i+2,ℓw2i+1,t−1+b1UA,B,ℓ,w2i+2,w2i+1,t−1+cMrℓw2i+2,ℓw2i+3,t·Mrℓw2i+1,ℓw2i+2,tMrℓw2i+2,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+3,2t−1,where
(42)UA,B,ℓ,w2i+2,w2i+1,t=maxMrℓw2i+2,ℓw2i+1,t,Mrℓw2i+2,Aw2i+2,t,Mrℓw2i+1,Bw2i+1,t,Mrℓw2i+1,Aw2i+2,t,Mrℓw2i+2,Bw2i+1,t=maxMrℓw2i+2,ℓw2i+1,t,Mrℓw2i+2,ℓw2i+3,t,Mrℓw2i+1,ℓw2i+2,t,Mrℓw2i+2,ℓw2i+3,t,Mrℓw2i+2,ℓw2i+2,t=maxMrℓw2i+2,ℓw2i+1,t,Mrℓw2i+2,ℓw2i+3,t,Mrℓw2i+1,ℓw2i+3,t,1=1.
Now from (41), (42), and by using Definition 2 (iv),
(43)1Mrℓw2i+2,ℓw2i+3,t−1≤a1Mrℓw2i+2,ℓw2i+1,t−1+cMrℓw2i+2,ℓw2i+3,t·Mrℓw2i+1,ℓw2i+2,tMrℓw2i+2,ℓw2i+1,t·Mrℓw2i+1,ℓw2i+2,t·Mrℓw2i+2,ℓw2i+3,t−1.
Then after simplification, fort>0,
(44)1Mrℓw2i+2,ℓw2i+3,t−1≤ζ1Mrℓw2i+1,ℓw2i+2,t−1whereζ=a+c<1.
Now, from (40), (44), and by induction,
(45)1Mrℓw2i+2,ℓw2i+3,t−1≤ζ1Mrℓw2i+1,ℓw2i+2,t−1≤ζ21Mrℓw2i,ℓw2i+1,t−1≤⋯≤ζ2i+21Mrℓw0,ℓw1,t−1⟶0asi⟶∞.
Hence,ℓwi is a fuzzy-contractive sequence in W,Mr,∗, therefore
(46)limi⟶∞Mrℓwi,ℓwi−1,t=1fort>0.
SinceMr is triangular, j>i>i0,
(47)1Mrℓwi,ℓwj,t−1≤1Mrℓwi,ℓwi+1,t−1+1Mrℓwi+1,ℓwi+2,t−1+⋯+1Mrℓwi−1,ℓwj,t−1≤ζi1Mrℓwi,ℓwi+1,t−1+ζi+11Mrℓwi+1,ℓwi+2,t−1+⋯+ζj−11Mrℓwi−1,ℓwj,t−1≤ζi+ζi+1+⋯+ζj−11Mrℓw0,ℓw1,t−1≤ζi1−ζ1Mrℓw0,ℓw1,t−1⟶0asi⟶∞.
This shows thatℓwi is a Cauchy sequence and ℓW is a complete subspace of W. Hence, ∃u,v∈W such that ℓwi⟶u=ℓv as i⟶∞, i.e.,
(48)limi⟶∞Mru,ℓwi,t=Mru,ℓv,t=1fort>0.
SinceMr is triangular,
(49)1Mrℓv,Av,t−1≤1Mrℓv,ℓw2i+2,t−1+1Mrℓw2i+2,Av,t−1fort>0.
Now, from (34), (46), (48), and by using Definition 2 (iv), for t>0, we have that
(50)1Mrℓw2i+2,Av,t−1=1MrAv,Bw2i+1,t−1≤a1Mrℓv,ℓw2i+1,t−1+b1UA,B,ℓ,v,w2i+1,t−1+cMrℓv,Av,t·Mrℓw2i+1,Bw2i+1,tMrℓv,Bw2i+1,2t·Mrℓv,ℓw2i+1,t·Mrℓw2i+1,Av,2t−1≤a1Mrℓv,ℓw2i+1,t−1+b1UA,B,ℓ,v,w2i+1,t−1+cMrℓv,Av,t·Mrℓw2i+1,ℓw2i+2,tMrℓv,ℓw2i+2,2t·Mrℓv,ℓw2i+1,t·Mrℓw2i+1,ℓv,t·Mrℓv,Av,t−1⟶b1UA,B,ℓ,v,w2i+1,t−1asi⟶∞,where
(51)UA,B,ℓ,v,w2i+1,t=maxMrℓv,ℓw2i+1,t,Mrℓv,Av,t,Mrℓw2i+1,Bw2i+1,t,Mrℓw2i+1,Av,t,Mrℓv,Bw2i+1,t=maxMrℓv,ℓw2i+1,t,Mrℓv,Av,t,Mrℓw2i+1,ℓw2i+2,t,Mrℓw2i+1,Av,t,Mrℓv,ℓw2i+2,t⟶max1,Mrℓv,Av,t=1asi⟶∞.
Now from (50) and (51), for t>0, we have
(52)limi⟶∞sup1Mrℓw2i+2,Av,t−1=0fort>0.
By using the value (48) and (52) in (49) with limit i⟶∞, we get that Mrℓv,Av,t=1⇒u=ℓv=Av for t>0. Next, we have to prove that u=ℓv=Bv. Since, Mr is triangular,
(53)1Mrℓv,Bv,t−1≤1Mrℓv,ℓw2i+1,t−1+1Mrℓw2i+1,Bv,t−1fort>0.
Now, from (34), (46), (48), and by using Definition 2 (iv), for t>0, we have that
(54)1Mrℓw2i+1,Bv,t−1=1MrAw2i,Bv,t−1≤a1Mrℓw2i,ℓv,t−1+b1UA,B,ℓ,w2i,v,t−1+cMrℓw2i,Aw2i,t·Mrℓv,Bv,tMrℓw2i,Bv,2t·Mrℓw2i,ℓw2i,t·Mrℓv,Aw2i,2t−1≤a1Mrℓw2i,ℓv,t−1+b1UA,B,ℓ,w2i,v,t−1+cMrℓw2i,ℓw2i+1,t·Mrℓv,Bv,tMrℓw2i,ℓv,t·Mrℓv,Bv,t·Mrℓw2i,ℓv,t·Mrℓv,ℓw2i+1,t−1⟶b1UA,B,ℓ,v,w2i,t−1asi⟶∞,where
(55)UA,B,ℓ,w2i,v,t=maxMrℓw2i,ℓv,t,Mrℓw2i,Aw2i,t,Mrℓv,Bv,t,Mrℓv,Aw2i,t,Mrℓw2i,Bv,t=maxMrℓw2i,ℓv,t,Mrℓw2i,ℓw2i+1,t,Mrℓv,Bv,t,Mrℓv,ℓw2i+1,t,Mrℓw2i,Bv,t⟶max1,Mrℓv,Bv,t=1asi⟶∞.
Now from (54) and (55), for t>0,
(56)limi⟶∞sup1Mrℓw2i+1,Bv,t−1=0fort>0.
By using the value (48) and (56) in (53) with limit i⟶∞, we obtain Mrℓv,Bv,t=1⇒u=ℓv=Bv for t>0. Hence, we obtain that u is a common coincidence point of the mappings ℓ,A, and B in W such that u=ℓv=Av=Bv.Corollary 16.
Let a fuzzy metricMr is triangular in a complete FM space W,Mr,∗ and let A,B,ℓ:W⟶W be three self-mappings, satisfies for all w,x∈W,
(57)1MrAw,Bx,t−1≤a1Mrℓw,ℓx,t−1+b1UA,B,ℓ,w,x,t−1,where
(58)UA,B,ℓ,w,x,t=maxMrℓw,ℓx,t,Mrℓw,Aw,t,Mrℓx,Bx,t,Mrℓx,Aw,t,Mrℓw,Bx,t,for t>0 and 0≤a,b<1 with a+b<1. If AW∪BW⊂ℓW, where ℓW is a complete subspace of W. Then A,B, and ℓ have a coincidence point in W. Moreover, if the pairs A,ℓ and B,ℓ are weakly compatible. Then, A,B, and ℓ have a unique CFP in W.Proof.
From the proof of Theorem15, u is a common coincidence point of the mappings ℓ,A,andBinW such that u=ℓv=Av=Bv for some v∈W. Now we prove the uniqueness of the coincidence point, let ∃u∗∈W is another common coincidence point of ℓ,A, and B in W such that u∗=ℓv∗=Av∗=Bv∗ for some v∗∈W. Then from (57), for t>0,
(59)1Mru,u∗,t−1=1Mrℓv,ℓv∗,t−1=1MrAv,Bv∗,t−1≤a1Mrℓv,ℓv∗,t−1+b1UA,B,ℓ,v,v∗,t−1,where
(60)UA,B,ℓ,v,v∗,t=maxMrℓv,ℓv∗,t,Mrℓv,Av,t,Mrℓv∗,Bv∗,t,Mrℓv∗,Av,t,Mrℓv,Bv∗,t=maxMrℓv,ℓv∗,t,1=1.
Thus,(61)1Mru,u∗,t−1≤a1Mrℓv,ℓv∗,t−1=a1Mru,u∗,t−1fort>0.
Noticing that1−a≠0 where a+b<1, therefore, Mru,u∗,t=1⇒u=u∗ for t>0. By using the weak compatibility of the pair A,ℓ,B,ℓ and by Preposition 9, we can get a unique CFP of the mappings A,B, and ℓ. Let ∃z∈W such that, ℓz=Az=Bz=z. Hence, we get that Mrv,z,t=1⇒v=z, for t>0.Example 17.
From Example14, it is proved that the three self-mappings A,B, and ℓ are weakly compatible fuzzy-contractive in FM-spaces, that is,
(62)1MrAw,Bx,t−1=231Mrℓw,ℓx,t−1fort>0.
Next, we calculate the value of the second term, present in (57). Then, we have the following cases:
(1)
If the maximum value ofUA,B,ℓ,w,x,t=Mrℓw,ℓx,t, for t>0. Then, by using (29)
(63)1UA,B,ℓ,w,x,t−1=1Mrℓw,ℓx,t−1=1tℓw−ℓx=23tw−xfort>0.(2)
If the maximum value ofUA,B,ℓ,w,x,t=Mrℓw,Aw,t, for t>0. Then, by using (29)
(64)1UA,B,ℓ,w,x,t−1=1Mrℓw,Aw,t−1=1tℓw−Aw=1t2w3−4w3w+9=1t2w2+w3w+9≤29tw2+wfort>0.(3)
If the maximum value ofUA,B,ℓ,w,x,t=Mrℓx,Bx,t, for t>0. Then, by using (29)
(65)1UA,B,ℓ,w,x,t−1=1Mrℓx,Bx,t−1=1tℓx−Bx=1t2x3−4x3x+9=1t2x2+x3x+9≤29tx2+xfort>0.(4)
If the maximum value ofUA,B,ℓ,w,x,t=Mrℓx,Aw,t, for t>0. Then, by using (29)
(66)1UA,B,ℓ,w,x,t−1=1Mrℓx,Aw,t−1=1tℓx−Aw=1t2x3−4w3w+9=1t2wx+3x−2w3x+9≤29twx+3x−2wfort>0.(5)
If the maximum value ofUA,B,ℓ,w,x,t=Mrℓw,Bx,t, for t>0. Then, by using (29)
(67)1UA,B,ℓ,w,x,t−1=1Mrℓw,Bx,t−1=1tℓw−Bx=1t2w3−4x3x+9=1t2wx+3w−2x3w+9≤29twx+3w−2xfort>0.
Hence, all the cases together with (62) and with contacts a=2/3 and b=2/7, we have
(68)1MrAw,Bx,t−1≤231Mrℓw,ℓx,t−1+271UA,B,ℓ,w,x,t−1,where
(69)UA,B,ℓ,w,x,t=maxMrℓw,ℓx,t,Mrℓw,Aw,t,Mrℓx,Bx,t,Mrℓx,Aw,t,Mrℓw,Bx,tfort>0.
Thus, all the hypotheses of Corollary16 are satisfied with a=2/3, b=2/7, and the mappings A,B, and ℓ have a unique CFP, namly, 0.


## 4. Application to the Fuzzy Differential Equations

In this section, we present an application of the fuzzy differential equations (FDEs) to support our main work. From the book of Lakshmikantham and Mohapatra [32], we have the following FDEs.LetF be the space of all fuzzy subsets w of R where w:R⟶0,1.(70)w′′v=hv,wv,w′v,v∈J=a,b,wv1=w1,wv2=w2,v1,v2∈J=a,b,where h:J×F×F⟶F is a continuous function. This problem is equivalent to the integral equation
(71)wv=∫v1v2Qv,ξhξ,wξ,w′ξdξ+Bv,where Green’s function Q is given by
(72)Qv,ξ=v2−vξ−v1v2−v1,v1≤ξ≤v≤v2,v2−ξv−v1v2−v1,v1≤v≤ξ≤v2.AndBv satisfies B′′=0,Bv1=w1,Bv2=w2. Here, we recall some properties of Qv,ξ, that is,
(73)∫v1v2Qv,ξdξ≤v2−v128,∫v1v2Qvv,ξdξ≤v2−v12.LetC=C1J,F, ∗ is a continuous t-norm, and a fuzzy metric Mr:C×C×0,∞⟶0,1 be defined as
(74)Mrw,x,t=tt+Dw,xwhereDw,x=w−x,∀w,x∈C, and t>0. Then, it is easy to prove that Mr is triangular and C,Mr,∗ is a complete FM-space.Now, we prove the existing result for the above boundary value problem by using Corollary16.Theorem 18.
Assume thath1,h2:J×F×F⟶F and let there exist α,β∈0,1 with α≤β such that for all w,x∈C1J,F, satisfies
(75)h1v,wv,w′v−h2v,xv,x′v≤αwv−xv+βw′v−x′v.
Let there existsη∈0,1 such that
(76)Dwv,xv=Dw,x≤ηMA,B,ℓ,w,x,where
(77)MA,B,ℓ,w,x=maxℓw−ℓx,ℓw−Aw,ℓx−Bx,ℓx−Aw,ℓw−Bx.
Then the integral equations(78)wv=∫v1v2Qv,ξh1ξ,wξ,w′ξdξ+Bv,v∈J,xv=∫v1v2Qv,ξh2ξ,xξ,x′ξdξ+Bv,v∈J,have a unique common solution in C1v1,v2,F.Proof.
Suppose thatC=v1,v2,F with metric
(79)Dw,x=maxv1≤v≤v2α∣wv−xv∣+βw′v−x′v.
The spaceC,D is a complete metric space. Now, we define the operators A,B,ℓ:C⟶C as
(80)Aw=Gw+B,Bx=Hx+B,ℓw=w,andℓx=x,where
(81)Gwv=∫v1v2Qv,ξh1ξ,wξ,w′ξdξ,v∈J,Hxv=∫v1v2Qv,ξh2ξ,xξ,x′ξdξ,v∈J,where h1,h2∈CJ×F×F,F,w,x∈C1J,F, and B∈CJ,F. Now by the properties of Qv,ξ, and from (79), (80) and by using the hypothesis, we have
(82)Awv−Bxv≤∫v1v2Qv,ξh1ξ,wξ,w′ξ−h2ξ,xξ,x′ξdξ≤Dw,x∫v1v2Qv,ξdξ≤v2−v128Dw,x≤Dw,x8,Aw′v−Bx′v≤∫v1v2Qvv,ξh1ξ,wξ,w′ξ−h2ξ,xξ,x′ξdξ≤Dw,x∫v1v2Qvv,ξdξ≤v2−v12Dw,x≤Dw,x2.
Now, from the above and by view of (75), and (79), we have that
(83)DAw,Bx=maxv1≤v≤v2αAwv−Bxv+βAw′v−Bx′v≤αDw,x8+βDw,x2≤58βDw,x.
Now, from (76), we have that
(84)DAw,Bx≤58βDw,x≤λMA,B,ℓ,w,x,where λ=5/8βη<1. Now we are in the position to apply Corollary 16 to get that the mappings A,B, and ℓ have a unique CFP w∗∈C, i.e., w∗ is a solution of the BVP. We have the following cases.
(1)
Ifℓw−ℓx is the maximum term in (77), then MA,B,ℓ,w,x=ℓw−ℓx. Now from (74) and (84), we have
(85)1MrAw,Bx,t−1=DAw,Bxt≤λMA,B,ℓ,w,xt=λℓw−ℓxt=λ1Mrℓw,ℓx,t−1.
This implies that(86)1MrAw,Bx,t−1≤λ1Mrℓw,ℓx,t−1fort>0,for all w,x∈C. Thus, the operators A,B, and ℓ satisfy all the conditions of Corollary 16 with λ=a+b in (57). Then, the operators A,B, and ℓ have a unique CFP w∗∈C, i.e., w∗ is a solution of the BVP (4.1).
(2)
Ifℓw−Aw is the maximum term in (77), then MA,B,ℓ,w,x=ℓw−Aw. Now from (74) and (84), we have
(87)1MrAw,Bx,t−1=DAw,Bxt≤λMA,B,ℓ,w,xt=λℓw−Awt=λ1Mrℓw,Aw,t−1.
This implies that(88)1MrAw,Bx,t−1≤λ1Mrℓw,Aw,t−1∀w,x∈Candt>0.
Similarly, if∣ℓx−Bx∣ is the maximum term in (77), then, MA,B,ℓ,w,x=∣ℓx−Bx∣. Now, by using (74) and (84), we get that
(89)1MrAw,Bx,t−1≤λ1Mrℓx,Bx,t−1∀w,x∈Candt>0.
Again, if∣ℓx−Aw∣ is the maximum term in (77), then, MA,B,ℓ,w,x=∣ℓx−Aw∣. Now, by using (74) and (84), we get that
(90)1MrAw,Bx,t−1≤λ1Mrℓx,Aw,t−1∀w,x∈Candt>0.
Next, if∣ℓw−Bx∣ is the maximum term in (77), then, MA,B,ℓ,w,x=∣ℓw−Bx∣. Now by using (74) and (84), we get that
(91)1MrAw,Bx,t−1≤λ1Mrℓw,Bx,t−1∀w,x∈Candt>0.
Hence, from (88), (89), (90), and (91), the operators A,B, and ℓ satisfy all the conditions of Corollary 16 with λ=b and a=0 in (57). Thus, the operators A,B, and ℓ have a unique CFP w∗∈C, i.e., w∗ is a solution of the BVP (70).


## 5. Conclusion

In this paper, we established the concept of rational type weakly-compatible fuzzy-contraction for three self-mappings in complete FM spaces and proved some coincidence point and CFP results under the rational type weakly-compatible fuzzy-contraction conditions by using the “triangular property of fuzzy metric” in complete FM spaces with examples. Further, as an application, we presented fuzzy differential equations and proved that the solution of the FDEs has a unique CFP of the integral operators, that is,A,B, and ℓ. This new direction of rational type weakly-compatible fuzzy-contraction with the application of FDEs in FM space will play a very important role. This idea can be extended and generalized in different directions with different types of weakly-compatible fuzzy-contraction conditions for self-mappings with different types of differential equations in the context of FM spaces.



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