Some New Coupled Fixed-Point Findings Depending on Another Function in Fuzzy Cone Metric Spaces with Application

In this paper, we introduce the new concept of coupled ﬁxed-point (FP) results depending on another function in fuzzy cone metric spaces (FCM-spaces) and prove some unique coupled FP theorems under the modiﬁed contractive type conditions by using “the triangular property of fuzzy cone metric.” Another function is self-mapping continuous, one-one, and subsequently convergent in FCM-spaces. In support of our results, we present illustrative examples. Moreover, as an application, we ensure the existence of a common solution of the two Volterra integral equations to uplift our work.


Introduction
Fixed-point theory is one of the most interesting areas of research. In 1922, Banach [1] proved a "Banach contraction principle" stated as follows: "a single-valued contractive type mapping in a complete metric space has a unique FP." After the publication of this principle, many researchers have contributed their ideas to the problems on fixed points in the context of metric spaces for single-valued and multivalued mappings with different types of applications. Kannan [2] and Chatterjea [3] proved some fixed-point theorems, while Reich [4,5] presented some remarks concerning contractive type mappings in complete metric spaces. Covitz and Nadler [6] and Daffer and Kaneko [7] proved some multivalued fixedpoint theorems, while Kaewkhao and Neammanee [8] established fixed-point theorems for multivalued Zamfirescu mapping in complete metric spaces. In 2007, Huang and Zhang [9] introduced the notion of cone metric space in which they extended and modified the concept of metric spaces. ey proved the convergence properties and some fixed-point results by using the concept of the underlying cone are normal.
Meanwhile, in 2008, Rezapour and Hamlbarani [10] proved fixed-point theorems without the assumption of normality of cone. After that, many others contributed their ideas to the problems on fixed-point results in cone metric spaces. Some of their contributions to the problems on cone metric spaces for fixed points can be found in [11][12][13][14].
Initially, the concept of fuzzy set theory was given by Zadeh [15]. Recently, the fuzzy set theory has been investigated, applied, and modified in many directions, in which the one direction of this theory is fuzzy logic, which has a wide range of applications again in many directions such as in engineering fields, business, and education. In education, fuzzy logic is used for the student results evaluation, which can be directly monitored by the teacher. Some of the references related to an education system based on fuzzy logic can be found in [16][17][18][19]. e other direction of the fuzzy set is the fuzzy metric theory. e notion of FM-space was introduced by Kramosil and Michalek [20]; they used the concept of a fuzzy set on metric space and proved some basic properties of the FM-space. After that, the stronger form of the metric fuzziness was given by George and Veeramani [21]. Later on, Gregori and Sapena [22] proved some contractive type FP theorems in FM-spaces. Recently, in 2020, Li et al. [23] proved some strongly coupled FP theorems by using cyclic contractive type mappings in complete FM-spaces. Meanwhile Rehman et al. [24] presented the concept of rational type contraction mappings and proved some FP theorems in complete FMspaces with an application.
In 2015, Oner et al. [25] introduced the concept of fuzzy cone metric spaces (FCM-spaces) and proved some basic properties and "a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy." Later on, Rehman and Li [26] established some generalized fuzzy cone-contractive type results for FP without the assumption that "all the sequences are Cauchy." After that, Jabeen et al. [27] proved common FP theorems for quasi-contraction by using the concept of compatible and weakly compatible for three self-mappings with an integral type application. In 2020, Chen et al. [28] introduced the concept of coupled contractive type mappings in FCM-spaces and proved some coupled FP results with application to nonlinear integral type application. Recently, in 2021, Rehman and Aydi [29] proved some rational type common FP theorems in FCM-spaces with an application.
In [30], Guo and Lakshmikantham introduced the coupled FP results for the nonlinear operator with applications. After that, some coupled FP theorems in partially ordered metric spaces were proved by Bhaskar and Lakshmikantham [31] and Lakshmikantham and Ciric [32]. In 2010, Sedghi et al. [33] proved common coupled FP theorems for commuting mappings in FM-spaces. Meanwhile Moradi [34] presented some results on "Kannan FP on complete and generalized metric spaces which depends on another function" by using the concept of subsequence convergence and continuity.
In this paper, we use the above concepts together and prove some unique coupled FP theorems depending on another function in FCM-spaces. Moreover, we present an application of the two Volterra integral equations for a common solution to support our results. is new concept will play an important role in the theory of fixed point to prove more coupled FP and strongly coupled FP results in complete FCM-spaces with the application of different types of differential equations. is paper is organized as follows: Section 2 gives preliminary concepts. In Section 3, we use the concepts of Guo and Lakshmikantham [30], Moradi [34], Chen et al. [28], and Jabeen et al. [27] all together and establish some unique coupled FP results depending on another continuous function which is one-one and subsequently convergent in FCM-spaces. In Section 4, we present an application of the two Volterra integral equations for the existence of a common solution to support our main work. In the last section (Section 5), we present the conclusion of our work.

Preliminaries
Definition 1. Let G be any set. A fuzzy set A in G is a function whose domain is G and the range is [0, 1].
Definition 2 (see [35] (i) * is associative and commutative (ii) * is continuous Definition 3 (see [9]). Let E be a real Banach space, and P is a subset of E. en, P is called a cone if (i) P is closed and nonempty and P ≠ 0 { } (ii) If α, β ∈ R, α, β ≥ 0 and g, h ∈ P, then αg + βh ∈ P (iii) If both g ∈ P and − g ∈ P, then g � 0 A partial ordering on a given cone P ⊂ E is defined by g ⪯ h ⇔ h − g ∈ P. g ⪯ h stands for g ⪯ h and g ≠ h, while g ≪ h stands for h − g ∈ int(P). In this paper, all cones have a nonempty interior.
Definition 4 (see [21]). A 3-tuple (G, M c , * ) is said to be an FM-space if G is any set, * is continuous t-norm, and M c is a fuzzy set on G 2 × (0, ∞) satisfying Definition 5 (see [25]). A 3-tuple (G, M c , * ) is said to be an FCM-space if P is a cone of E, G is an arbitrary set, * is continuous t-norm, and M c is a fuzzy set on G 2 × int(P) satisfying 1] is continuous, for g, h, k ∈ G, and t, s ≫ 0 Definition 6 (see [25]). Let a 3-tuple (G, M c , * ) be an FCM- Lemma 1 (see [25]). Let (G, M c , * ) be an FCM-space and let a sequence Definition 7 (see [26]). Let (G, M c , * ) be an FCM-space. e fuzzy cone metric M c is triangular if Definition 8 (see [25]). Let (G, M c , * ) be an FCM-space and Definition 9 (see [31]). An element Now, in the following main results, we shall prove some unique coupled FP theorems depending on another function which is continuous, one-one, and subsequently convergent in FCM-spaces. We present some illustrative examples in support of our results. As a further study, we shall present two Volterra integral equations to ensure the existence of common solution to support our work.

Main Results
Now, we are in the position to present our first main result.
A is a continuous, one-one, and subsequently convergent selfmapping on G, that is, for all g, h, ξ, η ∈ G, t ≫ 0, and α, β, c ∈ [0, 1] with α + 2β + 2c < 1. en € B has a unique coupled FP. Also, if _ A converges sequently, then for every g 0 ∈ G the iterative sequence € B ℓ g o converges to this coupled FP.
Proof. Consider any g 0 , h 0 ∈ G; we define sequences g ℓ and h ℓ in G such that Now, from (5) for t ≫ 0, we have After simplification, we get that where θ is the same as in (8). Now, from (8) and (9) and by induction, for t ≫ 0, we have that It is shown that _ Ag ℓ is a fuzzy cone contractive sequence; therefore, lim Hence, proving that _ Ag ℓ is a Cauchy sequence, we have that Now, for sequence h ℓ ℓ ≥ 0 and from (5), for t ≫ 0, we have Mathematical Problems in Engineering where θ is the same as in (8). Similarly, from (5) for t ≫ 0, we have where θ is the same as in (8). Now, from (15) and (16) and by induction, for t ≫ 0, we have that It is shown that _ Ah ℓ is a fuzzy cone contractive sequence.
Now, for j > ℓ and for t ≫ 0, we have Hence, proving that _ Ah ℓ ℓ ≥ 0 is a Cauchy sequence, we have that Since G is complete, _ Ag ℓ and _ Ah ℓ are Cauchy sequences in G; therefore, _ Ag ℓ ⟶ g ∈ G and _ Ah ℓ ⟶ h ∈ G as ℓ ⟶ ∞; that is, lim ℓ⟶∞ _ Ag ℓ � g and lim ℓ⟶∞ _ Ah ℓ � h. Since _ A is subsequently convergent, g ℓ has a convergent subsequence. So there exist g ∈ G and g ℓ(k) in G such that lim k⟶∞ g ℓ(k) � g. Since _ A is continuous, lim k⟶∞ g ℓ(k) � g, and lim k⟶∞ _ Ag n(k) � _ Ag. Now, from (5), for t ≫ 0, we have After simplification, for t ≫ 0, we have Hence, we get that we have to prove that € B(h, g) � h. en, from (5), for t ≫ 0, we have

Mathematical Problems in Engineering
After simplification, for t ≫ 0, we have Hence, we get that For uniqueness, suppose that (g 1 , h 1 ) and (h 1 ,

Mathematical Problems in Engineering
Hence, we get that M c ( _ Ag, _ Ag 1 , t) � 1 for t ≫ 0; this implies that g � g 1 . Similarly, again from (5), for t ≫ 0, we have Hence, we get that M c ( _ Ah, _ Ah 1 , t) � 1 for t ≫ 0, and this implies that h � h 1 . A is a continuous, one-one, and subsequently convergent selfmapping on G, that is, for all g, h, ξ, η ∈ G, t ≫ 0, and α, β ∈ [0, 1] with α + 2β < 1. en € B has a unique coupled FP. Also, if _ A converges sequently, then, for every g 0 ∈ G, the iterative sequence € B ℓ g o converges to this coupled FP.

Corollary 2. Let € B: G × G ⟶ G be a mapping in a complete FCM-space (G, M c , * ) in which M c is triangular and _
A is a continuous, one-one, and subsequently convergent selfmapping on G, that is, for all g, h, ξ, η ∈ G, t ≫ 0, and α, c ∈ [0, 1] with α + 2c < 1. en € B has a unique coupled FP. Also, if _ A converges sequently, then, for every g 0 ∈ G, the iterative sequence € B ℓ g o converges to this coupled FP.
If we use _ A as an identity self-mapping, that is, _ A � I in eorem 1, then we get the following corollary.

Theorem 2. Let € B: G × G ⟶ G be a mapping in a complete FCM-space (G, M c , * ) in which M c is triangular and _
A is a continuous, one-one, and subsequently convergent selfmapping on G, that is, Proof. Let any g 0 , h 0 ∈ G, and we define sequence g ℓ by Now, from (34), for t ≫ 0, we have Mathematical Problems in Engineering where η � (α + β)/(1 − β − c) < 1. Similarly, again from (34), for t ≫ 0, we have where η is the same as in (37). Now, from (37) and (38) and by induction, for t ≫ 0, we have Hence, we get that _ Ag ℓ ℓ≥0 is a fuzzy cone contractive sequence; therefore, Now, for j > ℓ and for t ≫ 0, we have Hence, proving that _ Ag ℓ is a Cauchy sequence, we have that lim ℓ,j⟶∞ M c _ Ag ℓ , _ Ag j , t � 1, t ≫ 0.
Now, again from (34), for t ≫ 0, we have After simplification, we get that for t ≫ 0, We have