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

The key objective of this research article includes the study of some rational type coincidence point and deriving common ﬁ xed 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 di ﬀ erential 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 di ﬀ erential equations in FM space will certainly play a vital role in the related ﬁ elds. It has the potential to be extended in any direction with di ﬀ erent types of weakly-compatible fuzzy-contraction conditions for self-mappings with di ﬀ erent types of di ﬀ erential equations.


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 singlevalued 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 intro-duced 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 fuzzycontraction 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 fuzzycontraction 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 selfmappings with different types of differential equations. This paper is organized as follows: Section 2 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 weaklycompatible fuzzy-contraction conditions for three selfmappings 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.

Preliminaries
In this section, we present some preliminaries related to our main work such as continuous t-norm, FM space, Cauchy sequence, complete FM space, fuzzy-contraction, fuzzycontractive sequence, and weakly-compatible self-mappings.
Definition 7 (see [15]). Let ðW, M r , * Þ be a FM space. A mapping A : W ⟶ W is said to be a fuzzy-contractive if ∃a ∈ ð0, 1Þ such that Definition 8 (see [31]). Let A 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]). Let A 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.

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 fuzzycontractive 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: Now we are in the position to present our first main result.
Next, we prove the uniqueness of a coincidence point in ðW, M r , * Þ for the mappings ℓ, A and B. 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, note that ða + bÞ < 1, where ða + b + 2cÞ < 1. Thus, we get that M r ðu, u * , tÞ = 1, that is, u = u * . By using the weak compatibility of the pair ðA, ℓÞ, ðB, ℓÞ 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 M r ðv, z, tÞ = 1 ⇒ v = z, for t > 0.
Corollary 11. Let a fuzzy metric M r is triangular in a complete FM space ðW, M r , * Þ and let A, B, ℓ : W ⟶ W be three self-mappings, satisfies for all w, x ∈ W, If we use identity map instead of ℓ, i.e., ℓ = I, in Theorem 10, we can get the following corollary: Corollary 12. Let a fuzzy metric M r is triangular in a complete FM space ðW, M r , * Þ and let A, B : W ⟶ W be two self-mappings, satisfies for all w, x ∈ W, for t > 0, 0 ≤ a, b, c < 1 with ða + b + 2cÞ < 1. Then, the mappings A and B have a unique CFP in W.
Remark 13. If we put the mappings A = B and ℓ = I (identity map) with constant c = 0 in Theorem 10, we obtained (Theorem 1 of [25]).

Example 14.
Let W = ½0, 1, * is a product continuous t -norm on W = ½0, 1 which is defined as ξ * ζ = ξ · ζ for all ξ, ζ ∈ W and a fuzzy metric M r : W 2 × ð0,∞Þ ⟶ ½0, 1 is defined by Then, it is easy to prove that M r is triangular and ðW, M r , * Þ is a complete FM space. The mappings A, B, ℓ : W ⟶ W be defined as Then, from (29), we have Hence, the self-mappings A, 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 Journal of Function Spaces Lastly, we simplify the third term of (6), then from (29), for t > 0, Hence, all the conditions of Theorem 10 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.