Coupled fixed point analysis in fuzzy cone metric spaces with an application to nonlinear integral equations

In this paper, we introduce the concept of coupled type and cyclic coupled type fuzzy cone contraction mappings in fuzzy cone metric spaces. We establish some coupled fixed point results without the mixed monotone property, and also present some coupled fixed results using the partial order metric in the said space. We present some strong coupled fixed point theorems using cyclic coupled type fuzzy cone contraction mappings in fuzzy cone metric spaces. Moreover, we present an application of nonlinear integral equations for the existence of a unique solution to support our work.


Introduction
Initially, Kirk et al. [1] introduced a cyclic contractive type mapping which ensures the existence of the best proximity points in complete metric spaces. Several cyclic type mapping results can be found (see, e.g., [2][3][4]). Later on, Lakshmikantham and Circ [5] presented the concept of coupled fixed point in partially ordered metric spaces which has a wide range of applications in partial differential equations and boundary value problems. In 2014, Choudhury and Maity [6] proved the result on cyclic coupled Kannan type contraction for a strong coupled fixed point. For more coupled fixed point results, see [7,8]. More related works and references are in [9][10][11][12].
Huang and Zhang [13] presented an idea of a cone metric space by using an ordered Banach space instead of real numbers. They proved some nonlinear contractive type fixed point results in cone metric spaces. After this article, several authors have contributed their ideas in the field of cone metric spaces. They established different types of contractive results for fixed point, coincidence point, and a common fixed point in cone metric spaces (see [14][15][16][17][18][19][20][21][22][23] and the references therein).
The fuzzy set theory was initiated by Zadeh [24], while Kramosil et al. [25] introduced fuzzy metric spaces and some more notions. They compared the notion of fuzzy metric with the statistical metric spaces and proved that both the conceptions are equivalent in some cases. Later, George et al. [26] presented the stronger form of the metric fuzziness. Some fixed point and common fixed point results in fuzzy metric spaces can be found in, e.g., [27][28][29][30].
Oner et al. [31] introduced the fuzzy cone metric space or shortly (FCM-space) and proved a fuzzy cone Banach contraction theorem for a fixed point in FCM-spaces with the assumption of Cauchy sequences. Some more topological properties, fixed point and common fixed point results can be found in, e.g., [32][33][34][35][36].
In this paper, we present a new concept of coupled type and cyclic coupled type fuzzy cone contraction mappings in FCM-spaces. The rest of the paper is organized as follows. Section 2 consists of preliminary concepts. In Sect. 3, we prove some coupled fixed point results without the mixed monotone property in the sense of Sintunavarat et al. [8], and we prove some coupled fixed point theorems via partial ordered metric FCM-spaces. In Sect. 4, we establish some strong coupled fixed point results for the generalized cyclic type fuzzy cone contraction mapping in FCM-space in the sense of Choudhury et al. [6]. In Sect. 5, we present an application of nonlinear integral equations for the existence of a unique solution to support our work. Finally, the conclusion is discussed in Sect. 6, and some illustrative examples are presented in the paper to support our work. (i) * is associative, commutative, and continuous.
Throughout this paper E represents the real Banach space and θ is the zero of E, while N represents the set of natural numbers.
A partial ordering on a given cone P ⊂ E is defined by μ ν ⇔ νμ ∈ P. μ ≺ ν stands for μ ν and μ = ν, while μ ν stands for νμ ∈ int(P). In this paper, all cones have nonempty interior.

Definition 2.3 ([31])
A three-tuple (U, F m , * ) is said to be a FCM-space if a cone P ⊂ E, U is an arbitrary set, * is a continuous t-norm, and F m is a fuzzy set on U 2 × int(P) satisfying the following: for all μ, ν, ω ∈ U and t, s ∈ int(P).

Coupled fixed point results in FCM-spaces
Definition 3.1 ([38]) Let U be a nonempty set, and an ordered pair (μ, ν) ∈ U × U is called a coupled fixed point of the mapping T : Then T has a unique coupled fixed point in U for all μ = ν.
for all (μ, ν, x, y) ∈ A. Assume that either: and A is a T-invariant set which satisfies the transitive property, then T has a coupled fixed point such that μ = T(μ, ν) and ν = T(ν, μ).
Repeating the same argument, we get Now we have to show that Adding (3.3) and (3.4), we have Since ∀ τ > 0, ψ(τ ) < τ , then from (3.5) we have Hence, (δ i ) is a monotone decreasing sequence, therefore lim i→∞ δ i = δ for δ ≥ 0. Next, we have to show that δ = 0. By the contrary case, let δ > 0, taking the limit i → ∞ on both sides of (3.5), i.e., lim r→τ + ψ(r) < τ , ∀ τ > 0, then which is a contradiction to the fact that δ > 0. Hence δ = 0, therefore for t θ we have Next, we have to show that (μ i ) and (ν i ) are Cauchy sequences in (U, F m , * ). By supposition, let at least one, (μ i ) or (ν i ), be not a Cauchy sequence. Then ∃ ε > 0 and the two subsequences of integers i k and j k with i k > j k ≥ k such that Further, we choose i k is the smallest integer such that i k > j k ≥ k and (3.6) holds, for t θ , we have By using the F m triangle inequality and from (3.7) and (3.8), for t θ , we have Now taking limit k → ∞, and from (3.6), we have lim k→∞ r k = ε > 0. Since i k > j k and A satisfies the transitive property, we get Now, in the view of (3.1) and (3.10), for t θ , we get Similarly, for t θ , Adding (3.11) and (3.12), we have Now, using the limit k → ∞ on both sides of (3.13), i.e., lim r→τ Now, finally we have to show that T(μ, ν) = μ and T(ν, μ) = ν. If assertion (1) holds, then we have Hence, μ = T(μ, ν) and ν = T(ν, μ), i.e., T has a coupled fixed point in U. Suppose that, if assertion (2) holds. We obtain two sequences (μ i ) and (ν i ) converging to μ and ν respectively for some μ, ν ∈ U. Then, by supposition, we have (μ, ν, Since F m is triangular and by (3.1), for t θ , we have Hence we get that F m (T(μ, ν), μ, t) = 1, this implies that μ = T(μ, ν). Similarly, we can prove that ν = T(ν, μ). Thus, T has a coupled fixed point in U.

Corollary 3.8 Let A be a nonempty subset of a complete FCM-space
for all (μ, ν, x, y) ∈ A. Assume that either: , then A is a T-invariant set which satisfies the transitive property. Then T has a coupled fixed point.
In the following theorem we prove the uniqueness of a coupled fixed point of a mapping T on U. Theorem 3.9 By addition to the hypotheses of Theorem 3.6, assume that ∀ (μ, ν), (g, h) ∈ X 2 , ∃ (x, y) ∈ U 2 such that (μ, ν, μ, ν) ∈ A and (g, h, x, y) ∈ A. Then T has a unique coupled fixed point in U.
Proof From the proof of Theorem 3.6, the mapping T has a coupled fixed point in U. Assume that (μ, ν) and (g, h) are coupled fixed points of T, i.e., T(μ, ν) = μ, T(ν, μ) = ν, T(g, h) = g, and T(h, g) = h. We have to show that μ = g and ν = h.
Again, by the property of T-invariance, we have Repeating this argument, we get (μ, ν, Adding (3.16) and (3.17), for t θ , we have Repeating the same argument, for t θ and ∀ i ∈ N, we have Now from mapping ψ, ψ(τ ) < τ and lim r→τ + ψ(r) < τ , it follows that lim i→∞ ψ i (τ ) = 0, ∀ τ > 0. Hence, from (3.19), for t θ , we get Similarly, for t θ , we can prove that Since F m is triangular and from (3.20) and (3.21) ∀ i ∈ N, and t θ , we have Hence T has a unique coupled fixed point. This completes the proof.
In the following we prove some coupled fixed point results by partial ordered metric space in FCM-spaces.
Let C(U) denote the collection of all subsets of a set U, and a pair (U, ) denotes the partially ordered set with partially ordered . A mapping F : U → U is known as nondecreasing (resp. nonincreasing) if ∀ a, b ∈ U such that a b ⇒ F(a) F(b) (resp. F(b) F(a)).

Definition 3.10 ([38]
) Let a pair (U, ) be a partially ordered set, and a mapping T : U × U → U is known to have a mixed monotone property if T : U × U → U is monotone nondecreasing in the first argument, and T is monotone nonincreasing in the second argument ∀ μ, ν ∈ U: Example 3.11 Let (U, m) be a metric space with partial ordered " ", let (U, F m , * ) be an FCM-space defined ∀ μ, ν ∈ U and t θ as follows: We define a subset A ⊆ U 4 by A = {(μ, μ * , ν, ν * ) ∈ U 4 : μ ν, μ * ν * }. Then A is a Tinvariant subset of U 4 which satisfies the transitive property.

22)
for all (μ, ν, x, y) ∈ U for which μ x and ν y. Assume that either: (1) T is continuous, or (2) U has the following properties: μ), then T has a coupled fixed point.
Therefore, (μ, ν, μ i , ν i ) ∈ A, i ∈ N, hence assertion (2) of Theorem 3.6 is satisfied. Since all the hypotheses of Theorem 3.6 are satisfied, F has a coupled fixed point.

Corollary 3.13
In addition to the hypotheses of Theorem 3.12, assume that ∀ (μ, ν), (g, h) ∈ U 2 , ∃ (x, y) ∈ X 2 such that μ x, ν y and g x, h y. Then T has a unique coupled fixed point.
Proof Let a subset A ⊆ X 4 be defined as A = {(μ, μ * , ν, ν * ) ∈ X 4 : μ ν and μ * ν * }. Now, from Example 3.11, we conclude that A is a T-invariant set which satisfies the transitive property. Then, through the proof of Theorem 3.12, easily from simple calculation we can get the existence of a coupled fixed point.
Uniqueness: Now we have to show the unique coupled fixed point of the mapping T. Since ∀ (μ, ν), (g, h) ∈ U 2 , ∃ (x, y) ∈ U 2 such that μ x, ν y and g x, h y, we conclude that (μ, ν, x, y) ∈ A and (g, h, x, y) ∈ A. Thus all the hypotheses of Theorem 3.9 hold and T has a unique coupled fixed point.   where μ, y ∈ A, ν, x ∈ B, and a ∈ (0, 1 2 ).
In the following, we shall study a more generalized cyclic coupled type fuzzy cone contraction condition in (U, F m , * ) and prove some strong coupled fixed point results in FCMspaces. A mapping T : U × U → U is known as a generalized cyclic coupled type fuzzy cone contraction condition in FCM-spaces if T satisfies the inequality where μ, y ∈ A, ν, x ∈ B, t θ , and a, b, c, d ∈ [0, ∞). We note that (4.3) is the same as (4.2) if a = b ∈ (0, 1 2 ) and c = d = 0. Also, we illustrate some examples to support our results.

Theorem 4.5 Assume that A and B are two nonempty closed subsets of a complete FCMspace (U, F m , * ) in which F m is triangular and T : U ×U → U is a generalized cyclic coupled type fuzzy cone contraction w.r.t A and B. Suppose that T satisfies (4.3) with (a + b + 2c + 2d) < 1. Then A ∩ B = ∅ and T has a strong coupled fixed in A ∩ B.
Proof Fix μ 0 ∈ A and ν 0 ∈ B. Let (μ i ) and (ν i ) be two sequences defined as Then (μ i ) ⊂ A and (ν i ) ⊂ B since T is a cyclic mapping w.r.t A and B. We denote the following: .
Then h ∈ (0, 1) for a + b + 2c + 2d < 1. We claim that, for t θ and i ≥ 0, It is clear that (4.5) holds for i = 0. Suppose that (4.5) holds for i = k for t θ , then by (4.3) we have Similarly, in view of (4.3), Thus, by the induction hypothesis, i.e., (4.5) with i = k for t θ , we have That is, (4.5) holds for i = k + 1. Therefore, we have proved that (4.5) holds for all i ≥ 0 by induction. Meanwhile, by (4.3), for i ≥ 0, Here, we suppose that α = max{a, b} and β = max{c, d}, then we have This together with (4.5) implies that Then, for i, j ≥ 0, without loss of generality we assume that i ≤ j, This implies that (μ i ) is a Cauchy sequence and hence convergent in X. Since A is a nonempty closed subset of U, therefore Similarly, So, from (4.7) and (4.8), we have Since F m is triangular, by (4.5) and (4.6), Therefore, F m (μ, ν, t) = 1, for t θ and hence μ = ν ∈ A ∩ B. Now we have to prove that μ is a strong coupled fixed point of T by using the F m triangularity condition, we have for t θ . In view of (4.3), (4.7), and (4.8), T(μ, ν), t) -1 , as i → ∞.

Corollary 4.6 Assume that A and B are two nonempty subsets of a complete FCM-space (U, F m , * ) in which F m is triangular and T : U × U → U is a cyclic coupled type fuzzy cone contraction w.r.t. A and B. Suppose that T satisfies the inequality
where μ, y ∈ A, ν, x ∈ B, and t θ for a, b, c ∈ [0, ∞) with (a + b + 2c) < 1. Then A ∩ B = ∅ and T has a strong coupled fixed point in A ∩ B. T(x, y), t) -1 ,

Corollary 4.7 Assume that A and B are two nonempty closed subsets of a complete FCMspace (U, F m , * ) in which F m is triangular and T : U × U → U is a cyclic coupled type fuzzy cone contraction w.r.t. A and B. Suppose that T satisfies the inequality
where μ, y ∈ A, ν, x ∈ B, and t θ for some a, b, d ∈ [0, ∞) with (a + b + 2d) < 1. Then A ∩ B = ∅, and T has a strong coupled fixed point in A ∩ B.
If a = b and c = d = 0 in (4.3), then we may get the following corollary of K annan type for a cyclic coupled fixed point in FCM-spaces.

Corollary 4.8 Assume that A and B are two nonempty closed subsets of a complete FCMspace (U, F m , * ) in which F m is triangular and T : U × U → U is a cyclic coupled K annan type fuzzy cone contraction w.r.t A and B satisfying (4.2)
for some a ∈ (0, 1 2 ). Then A ∩ B = ∅, and T has a strong coupled fixed in A ∩ B.

Theorem 4.11
Assume that A and B are two nonempty closed subsets of a complete FCMspace (U, F m , * ) in which F m is triangular and T : U ×U → U is a cyclic coupled contractive type mapping w.r.t. A and B for some a ∈ [0, 1) satisfying where μ, y ∈ A, ν, x ∈ B, and t θ . Then A ∩ B = ∅ and T has a strong coupled fixed point in A ∩ B.
Proof Let μ o ∈ A and ν o ∈ B be two fixed elements, and let (μ i ) and (ν i ) be any two sequences in A and B, respectively, which are defined as follows: (4.14) Now, we have to show that (μ i ) is a Cauchy sequence. Then from (4.13) we have is the maximum in (4.15), which is not possible. Therefore, Similarly, Adding (4.16) and (4.17), for t θ , we have Now, again by (4.13) and similar as above, we get (4.20) Adding (4.19) and (4.20), and substituting in (4.18) for t θ , we get Continuing this process, for t θ , we have (4.21) Thus (4.21) is true ∀ i ≥ 0. Now, for an integer k, Then, again we may have the following two cases: Adding (4.23) and (4.24), where a * = a 2 , and by (4.21) for t θ , we have Since F m is triangular, and by (4.21) and (4.25), Now, for m, i ≥ 0, without loss of generality we may assume that m > i, This shows that (μ i ) is a Cauchy sequence and hence convergent in X. Since A = ∅ a closed subset of U, therefore Similarly, Hence, from (4.26) and (4.27), we have Since F m is triangular, by (4.21) and (4.26), we get Now we show that μ is a strong coupled fixed point of T. Since F m is triangular, for t θ , we have Then, in view of (4.13), (4.26), and (4.27), If 1 is the minimum of {1, F m (μ, T(μ, ν), t)}, then directly from (4.28) we may get that F m (μ, T(μ, ν), t) = 1 as i → ∞, which implies that T(μ, ν) = μ = ν. Secondly, if F m (μ, T(μ, ν), t) is the minimum of {1, F m (μ, T(μ, ν), t)}, then we have Now, from (4.28), which is a contradiction. Hence F m (μ, T(μ, ν), t) = 1 ⇒ T(μ, ν) = μ = ν is a strong coupled fixed point of T.
Example 4.12 As from Example 4.10, and in view of (4.13), we have that 19x -4y 20 T(μ, ν) , m x, T(x, y) Hence all the conditions of Theorem 4.11 are satisfied with a = 6 7 for t θ , and 5 is the strong coupled fixed point of T, that is, T(5, 5) = 5.

An application of nonlinear integral equations
In this section, we present an application of nonlinear integral equations to support our results. Let

Conclusion
In this paper, we have introduced the concept of coupled type and cyclic coupled type fuzzy cone contraction mappings in fuzzy cone metric spaces. We have established some coupled fixed point results without the mixed monotone property and also we have presented some more coupled fixed results via partial order metric in fuzzy cone metric spaces. We have proved some strong coupled fixed point theorems for cyclic type fuzzy cone contraction mappings. As a consequence, the main results of this paper extend and unify several results given in the literature of coupled fixed points. Moreover, we presented an integral type application for the existence of unique solution in fuzzy cone metric spaces to support our work.