Some Rational Coupled Fuzzy Cone Contraction Theorems in Fuzzy Cone Metric Spaces with an Application

In this paper, we establish the new concept of rational coupled fuzzy cone contraction mapping in fuzzy cone metric spaces and prove some unique rational-type coupled ﬁxed-point theorems in the framework of fuzzy cone metric spaces by using “the triangular property of fuzzy cone metric.” To ensure the existence of our results, we present some illustrative unique coupled ﬁxed-point examples. Furthermore, we present an application of a Lebesgue integral-type contraction mapping in fuzzy cone metric spaces and to prove a unique coupled ﬁxed-point theorem.


Introduction
In 1965, the theory of fuzzy sets was introduced by Zadeh [1]. Kramosil and Michalek [2] introduced the notion of FMS by using continuous t-norm with fuzzy sets. Afterward, Grabiec [3] established the completeness property of the FMS and proved a "Fuzzy Banach Contraction Principle for a unique fixed point (FP) in complete FMS." Since then, many contributed to this theory concerning FP results (e.g., see [4][5][6]). Later on, in 1994, George and Veeramani [7] modified the concept of FMS introduced by Kramosil and Michalek [2], and they presented the topological properties and proved Baire's theorem on complete FMS. In 2002, some contractive-type FP theorems were proved by Gregory and Sapena [8] on complete FMS by using the concept of [2,7]. Some related FP concepts in FMSs can be found in [9][10][11][12]. Recently, the rational-type fuzzy contraction concept in FMS is given by Rehman et al. [13], and they proved some FP results with an application.
Jaggi [14] proved the rational-type FP result for a contractive condition. However, Harjani et al. [15] modified the concept of Jaggi [14] and proved a generalized result in "partially ordered metric space." In 2011, Luong and uan [16] proved generalized rational weak contraction results in "partially ordered metric space," which is a generalization of the result of [14]. In [17], Guo and Lakshmikantham presented the concept of coupled FP with applications by using the nonlinear operator. Later on, Bhaskar [18] and Lakshmikantham [19] proved coupled FP results in "partially ordered metric space." In [20], Sedghi et al. used commuting mappings and established some common coupled FP theorems in FMSs.
In 2007, the notion of cone metric space (CMS) was introduced by Huang and Zhang [21]. ey proved some basic convergence properties and FP theorems on CMS. In 2008, Abbas et al. [22] proved some common FP theorems without continuity for noncommuting mappings on CMS. After that, many others contributed their ideas to the problem of FP results in CMS. Some of their FP contributions can be found in [23][24][25].
Oner et al. [26] introduced the concept of fuzzy cone metric space (FCMS) and proved a "fuzzy cone Banach contraction theorem" for FP in complete FCMSs in which they assumed that the "fuzzy cone contractive (fc − contractive) sequences are Cauchy." In [27], Rehman and Li proved some FP theorems in FCMSs without the assumption of "fc − contractive sequences are Cauchy" by using the "triangular property of FCM." Some more FP findings in the said space can be found in [28][29][30][31]. Recently, Chen et al. [32] and Rehman and Aydi [33] established some coupled FP and common FP results, respectively, in FCMs with integral types of applications. Waheed et al. [34] proved some coupled FP theorems in FCMSs depending on another function with an application to Volterra integral equations.
In this paper, we prove some rational-type unique coupled FP theorems in FCMSs under the rational type fc − contractive conditions with supportive examples. In addition, to verify the validity of our work, we present an application of a Lebesgue integral-type contraction condition theorem to support our work. e layout of this paper is as follows: Section 2 consists of some basic preliminary concepts. In Section 3, we define the rational coupled fc − contractive mapping in FCMS and prove some unique rational coupled FP results in complete FCMSs with suitable examples. Section 4 deals with the application of Lebesgue integral-type contraction mapping to get the existence result of unique coupled FP theorems in complete FCMSs.

Preliminaries
In this section, we recall some basic definitions and lemmas related to our main results. roughout the complete paper, N represents a set of natural numbers and ⊺-norm represents a continuous t-norm as defined in [35]. (1) * is associative, commutative, and continuous Definition 2. Let E be a real Banach space, 0 ∈ E. en, a subset C ⊂ E is called a cone: (1) If C ≠ ∅, closed, and C ≠ 0 { } (2) If κ 1 , κ 2 ≥ 0 and g, z ∈ C, then κ 1 g + κ 2 z ∈ C (3) If −z, z ∈ C, then z � 0.
A partial ordering is defined on a given cone C ⊂ E by g ≤ z⇔z − g ∈ C. g≺z stands for g ≤ z and g ≠ z, while g ≪ z stands for z − g ∈ int(C). In this paper, all cones have a nonempty interior. Definition 3. A 3-tuple (G, M r , * ) is said to be a FMS if G is any set, * is a ⊺-norm, and M r is a fuzzy set on G × G × (0, ∞) which satisfies the following: is continuous; ∀g 1 , g 2 , g 3 ∈ G and s, ⊺ > 0 Definition 4. A 3-tuple (G, M r , * ) is said to be a FCMS if C is a cone of E, G is an arbitrary set, * is a ⊺-norm, and M r is a fuzzy set on G × G × int(C) which satisfies the following: is continuous; ∀g 1 , g 2 , g 3 ∈ G, and s, ⊺ ≫ 0 Definition 5. Let a 3-tuple (G, M r , * ) be a FCMS and 9 ∈ G and a sequence g J in G (1) Converges to 9 if c ∈ (0, 1) and ⊺ ≫ 0 and there is We may write this lim J⟶∞ g J � 9 or g J ⟶ 9 as J ⟶ ∞.
en, Γ is known as a fc − contractive if ∃η ∈ (0, 1) such that (1) Journal of Mathematics Γ(g, h) � g, and Γ(h, g) � h. (2) Furthermore, we shall study some unique coupled FP results in FCMSs under the rational coupled fc − contraction conditions with examples. Also, we present an application of the Lebesgue integral-type rational coupled fc − contraction mapping to get a unique rational coupled FP result in FCMSs.

Main Results
In this section, we shall present our main results with illustrative examples.
Definition 9. Let (G, M r , * ) be a FCMS. A mapping Γ: G × G ⟶ G is called a rational coupled fc − contraction if ∃η 1 ∈ (0, 1) and η 2 ≥ 0 such tha Theorem 1. Assume that (G, M r , * ) be a complete FCMS in which M r is triangular and a mapping Γ: G × G ⟶ G is a rational coupled fc − contraction satisfying (3). en, Γ has a unique coupled FP in G.
Proof. Let any g 0 , h 0 ∈ G; we define sequences g J and h J in G such that Now, from (3) and (4), for ⊺ ≫ 0, is implies that Similarly, Now, from (6) and (7) and by induction, for ⊺ ≫ 0, we have that Hence, g J is a Cauchy sequence. Since, by the completeness of (G, M r , * ), ∃g ∈ G, so that Now, for sequence h J and from (3) and (4), for ⊺ ≫ 0, is implies that Similarly, Now, from (13) and (14) and by induction, for ⊺ ≫ 0, we have that is shows h J is a fc − contractive sequence; therefore, 4 Journal of Mathematics Now, for ℓ > J and for ⊺ ≫ 0, we have Hence, h J is a Cauchy sequence. Since, by the completeness of (G, M r , * ), ∃g ∈ G so that Now, we shall prove that Γ(g, h) � g. Since M r is triangular and by the view of (3), (9), and (11), for ⊺ ≫ 0, for ⊺ ≫ 0. Next, we have to prove that h � Γ(h, g); therefore, by the triangular property of M r and by the view of (3), (16), and Journal of Mathematics (3) and by using Definition 4 (4), for ⊺ ≫ 0, Hence, we get that M r (g, g 1 , ⊺) � 1⇒g � g 1 for ⊺ ≫ 0. Similarly, again from (3) and by using Definition 4 (4), for Hence, we get that ∀g, h, ξ, κ ∈ G, ⊺ ≫ 0, and η 1 ∈ (0, 1). en, Γ has a unique coupled FP in G.
Proof. Let any g 0 , h 0 ∈ G; we define sequences g J and h J in G such that Now, from (28) and (29), for ⊺ ≫ 0,
Proof. Let any g o , h o ∈ G; we define sequences g J and h J in G such that Now, from (67) and from the proof of eorem 1, for Similarly, again by using the arguments, we have Now, from (70) and (71) and by induction, for ⊺ ≫ 0, we have is shows that g J is a fc − contractive sequence, and therefore, Hence, we get that lim J⟶∞ M r g J , g J+1 , ⊺ � 1, for ⊺ ≫ 0.

Conclusion
We established the new concept of rational coupled fc-contraction mapping in FCMSs and proved some unique rational coupled FP theorems in FCMSs under the rational coupled fc-contraction conditions by using the "triangular property of fuzzy cone metric" with the help of some suitable examples to unify our work. In the last section, we presented an application of the Lebesgue integral-type coupled contraction theorem for unique rational coupled FP in complete FCMSs. By using this concept, one can prove more rational coupled-type fc-contraction results in complete FCMSs with different integral types of application to prove unique coupled FP results.

Data Availability
Data sharing does not apply to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.