In this manuscript, the concept of a cyclic tripled type fuzzy cone contraction mapping in the setting of fuzzy cone metric spaces is introduced. Also, some theoretical results concerned with tripled fixed points are given without a mixed monotone property in the mentioned space. Moreover, under this concept, some strong tripled fixed point results are obtained. Ultimately, to support the theoretical results non-trivial examples are listed and the existence of a unique solution to a system of integral equations is presented as an application.
After the emergence of Banach’s theorem [1], fixed-point technology has a special resonance not only in the field of mathematical analysis but also in many vital disciplines such as topology, dynamical system, control and economic theory, biology, global analysis and variant branches of engineering [2–6]. Because of the importance of this approach, it became the main controller in the study of the existence and uniqueness of the solution to many differentials, integral and functional equations [7–16].
Mixed monotone mappings and coupled fixed points were first presented by Bhaskar and Lakshmikantham [17]. They introduced some theoretical coupled fixed point results and applied this contribution to discuss the existence of a solution to a periodic boundary value problem in the framework of partially ordered metric spaces (POMSs). The researchers then generalized this point, including those who were interested in generalizing the space and others who worked to generalize the contractive condition. Because there is so much backlog in this direction, the reader can refer to [18–22].
It’s not over, coupled fixed points have been circulated to tripled fixed points (TFPs) by Berinde and Borcut [23, 24]. Some new theoretical and practical results on this point have been summarized by the same authors in POMSs, for more details, see [25–30].
The concept of cone metric spaces was presented by Huang and Zhang [31]. They replaced the set of real numbers with an ordered Banach space. Also, some results concerned with fixed points and their applications are given by the same authors. After this article came out, it became very easy to convert all properties of normal metric spaces into cone metric spaces, with a slight change in the contractive stipulation, or to use extended contractive stipulations for single and multi-valued mappings [32–34].
The fuzzy set theory was presented by Zadeh [35], while fuzzy metric spaces and some important concepts were shown by Kramosi et al. [36]. In some cases, they compared the concept of fuzzy metric with the statistical metric spaces, and they argued that both concepts were equivalent. Because of the many applications of this trend, many authors worked in this direction and obtained some pivotal results, for example, see, [37–45].
Continuing on the previous approach, in this article, the notions of a tripled and cyclic tripled type fuzzy cone contraction (FCC) mapping are presented in the framework of fuzzy cone metric spaces (FCMSs). Also, some TFP results without the mixed monotone property (MMP) in POMSs and some supportive examples are derived. Finally, the theoretical results are involved to study the existence and uniqueness of the solution to a system of nonlinear integral equations.
Preliminaries
Definition 2.1. [46] An operation ∗ : [0, 1] × [0, 1] → [0, 1] is known as a continuous ν-norm if it verifies the two assertions below:
The elementary concepts of continuous ν-norm such as the product, the Lukasiewicz ν-norm and the minimum are presented in [46], respectively as follows:
Here, refers to the set of natural numbers, ϝ is the real Banach space and ϑ represents a zero element in ϝ.
Definition 2.2. [31] A subset ϒ∈ ϝ is described by a cone if:
A partial ordering on a given cone ϒ⊂ ϝ is given by θ ≤ ρ ⇔ ρ - θ ∈ ϒ. θ < ρ symbolize θ ≤ ρ and θ ≠ ρ, while θ ⪡ ρ symbolize ρ - θ ∈ ϒ0, where ϒ0 stands for the interior of ϒ, it should be noted that all cones have non-empty interior.
Definition 2.3. [38] A trio (Ω, Θϖ, ∗) is called a FCMS if a cone ϒ ∈ ϝ , Ω is an arbitrary set, ∗ is a continuous ν-norm, and Θϖ is a fuzzy set on Ω2 × ϒ0 so that the assertions below hold:
Θϖ (θ, ρ, ν) > ϑ and Θϖ (θ, ρ, ν) =1 iff θ = ρ ;
Θϖ (θ, ρ, ν) = Θϖ (ρ, θ, ν) ;
Θϖ (θ, ρ, ν)∗ Θϖ (ρ, δ, μ) ≤ Θϖ (θ, δ, ν + μ) ;
Θϖ (θ, ρ, ·) : ϒ0 → [0, 1] is continuous,
for all θ, ρ, δ ∈ Ω and ν, μ ∈ ϒ0 .
Definition 2.4. [38] Assume that (Ω, Θϖ, ∗) is a FCMS, θ ∈ Ω and (θi) is a sequence in Ω . Then
(θi) is called converge to θ if, for ν ⪢ ϑ and 0 < u < 1, there exists so that Θϖ (θi, θ, ν) >1 - u, for all i > i1, and we can write or θi → θ as i → ∞ .
(θi) is called a Cauchy sequence if, for ν ⪢ ϑ and 0 < u < 1, there exists so that
if every Cauchy sequence is convergent in Ω, then we say a trio (Ω, Θϖ, ∗) is complete.
(θi) is called a FCC if there exists β ∈ (0, 1) so that
Definition 2.5. [39] Assume that (Ω, Θϖ, ∗) is an FCMS, the fuzzy cone metric Θϖ is triangular if
Lemma 2.6.[38] Assume that (Ω, Θϖ, ∗) is a FCMS, θ ∈ D and (θi) is a sequence in Ω, then
Definition 2.7. [38] Assume that (Ω, Θϖ, ∗) is an FCMS and Ξ : Ω → Ω . Then Ξ is said to be a FCC if there exists g ∈ (0, 1) so that
Tripled fixed point results
Definition 3.1. [23] Let Ω be a non-empty set. We say that a trio (θ, ρ, δ) ∈ Ω3 is a TFP of the mapping Ξ : Ω3 → Ω if θ = Ξ (θ, ρ, δ) , ρ = Ξ (ρ, δ, θ) and δ = Ξ (δ, θ, ρ) .
Example 3.2. Let Ω = [0, ∞) and Ξ : Ω3 → Ω be a mapping given by
Then there is a unique TFP of Ξ, whenever θ = ρ = δ.
Definition 3.3. Let Ξ : Ω3 → Ω be a mapping in a metric space (Ω, ϖ) and D be a non-empty subset of Ω6. Then we say that D is Ξ-invariant subset of Ω6 iff for all θ, ρ, δ, s, q, z ∈ Ω, the assumptions below hold:
(θ, ρ, δ, s, q, z)∈ D ⇔ (s, q, z, θ, ρ, δ) ∈ D ;
(θ, ρ, δ, s, q, z)∈ D ⇔
Definition 3.4. Suppose that (Ω, ϖ) is a metric space and D ∈ Ω6 which verifies the transitive property iff for all θ, ρ, δ, s, q, z, h, p, a ∈ Ω so that (θ, ρ, δ, s, q, z) , (s, q, z, h, p, a) ∈ D ⇒ (θ, ρ, δ, h, p, a) ∈ D .
The following examples illustrate the above two definitions.
Example 3.5. Let Ω ={ 0, 1, 2, 3, 4, 5 } and Ξ : Ω3 → Ω be a mapping defined by
It clear that D = { 1, 2, 3 } 6 ⊆ Ω6 is Ξ-invariant, which verifies the transitive property.
Example 3.6. Let and Ξ : Ω3 → Ω be a mapping defined by
Then D = [(- ∞ , -1) ∪ (1, ∞)] 6 ⊆ Ω6 is Ξ-invariant, which verifies the transitive property.
Now, we state and proof our first main results.
Theorem 3.7.Suppose that D is a non-empty subset of a complete fuzzy cone metric spaces (CFCMS) (Ω, Θϖ, ∗) in which Θϖ is triangular and φ : [0, ∞) → [0, ∞) is a function with
ϑ = φ (ϑ) < φ (η) < η and φ (u) < η for all η > ϑ. Let Ξ : Ω3 → Ω be a mapping so thatfor all (θ, ρ, δ, h, p, a) ∈ D . Suppose either
Ξis continuous, or
If for any three sequences (θi) , (ρi) and (δi) with {θi+1, ρi+1, δi+1, θi, ρi, δi} ∈ D such that (θi) → θ, (ρi) → ρ and (δi) → δ, for all i ≥ 1, then (θ, ρ, δ, θi+1, ρi+1, δi+1) ∈ D, for all i ≥ 1 . Moreover, if there is (θ0, ρ0, δ0) ∈ Ω3 so thatand Ξ is Ξ-invariant set which verifies the transitive property,
then there is a TFP of Ξ, i.e., θ = Ξ (θ, ρ, δ) , ρ = Ξ (ρ, δ, θ) and δ = Ξ (δ, θ, ρ) .
Proof. Assume that (θi) , (ρi) and (δi) are three sequences in Ω and Ξ (Ω3) ⊆ Ω so that
for all If ∃ such that θi∗ = θi∗-1, ρi∗ = ρi∗-1 and δi∗ = δi∗-1 then, we have
Thus (θi∗-1, ρi∗-1, δi∗-1) is a TFP of Ξ and this finishes the proof.
So let θi ≠ θi-1 or ρi ≠ ρi-1 or δi ≠ δi-1 for all . Since
and D is Ξ-invariant set, we get
Analogously, by the fact that D is Ξ-invariant set, we can get
Continuing with the same scenario, we find that
Consider
that is
Now, we shall show that
Since (θi, ρi, δi, θi-1, ρi-1, δi-1) ∈ D, for all , and by (1), for ν ⪢ ϑ, we can write
Since D is Ξ-invariant set and (θi, ρi, δi, θi-1, ρi-1, δi-1) ∈ D, , we get (ρi-1, δi-1, θi-1, ρi, δi, θi) ∈ D .
Now, again from (1), for ν ⪢ ϑ and (ρi-1, δi-1, θi-1, ρi, δi, θi) ∈ D, , we have
Similarly, as D is Ξ-invariant set and (θi, ρi, δi, θi-1, ρi-1, δi-1) ∈ D, , we get (δi-1, θi-1, ρi-1, δi, θi, ρi) ∈ D .
Now, again from (1), for ν ⪢ ϑ and (δi-1, θi-1, ρi-1, δi, θi, ρi) ∈ D, for all , one can write
Adding (2)-(4), we have
Since for all η > ϑ, φ (η) < η, then from (5), one sees that
This implies that (ℶ i) is a monotone decreasing sequence, hence for ℶ ≥ ϑ .
Next, we claim that ℶ = ϑ . By a contradiction, suppose that ℶ > ϑ, letting i→ ∞ on the both sides of (5), for all η > ϑ, we can write
a contradiction. Hence ℶ = ϑ, and for ν ⪢ ϑ, we have
Next, we prove that (θi), (ρi) and (δi) are Cauchy sequences in (Ω, Θϖ, ∗).
By supposition, consider at least one, (θi), (ρi) or (δi) is not a Cauchy sequence, then there is ɛ > ϑ and two subsequences ik and jk with ik > jk ≥ k, so that
for all . Let ik be the smallest integer so that ik > jk ≥ k . Then for ν ⪢ ϑ, and using (6), we can write
Since Θϖ is a triangle inequality and from (7) and (8), for ν ⪢ ϑ, we get
Now, setting k→ ∞ and from (6), we get Since ik > jk and D verifies the transitive property, then we get
It follows from (1) and (9) that, for ν ⪢ ϑ,
Similarly, for ν ⪢ ϑ, one can write
and
Adding (10)–(12), we have
Letting k→ ∞ on both sides of (13), for all η > ϑ, we get
a contradiction. Hence (θi), (ρi) and (δi) are Cauchy sequences in (Ω, Θϖ, ∗) . Since (Ω, Θϖ, ∗) is complete then there are θ, ρ and δ in Ω so that
Ultimately, we shall show that Ξ (θ, ρ, δ) = ρ, Ξ (ρ, δ, θ) = ρ and Ξ (δ, θ, ρ) = δ . If hypothesis (i) holds, then we obtain
similarly
and
Hence, θ = Ξ (θ, ρ, δ) , ρ = Ξ (ρ, δ, θ) and δ = Ξ (δ, θ, ρ), i.e., Ξ has a TFP in Ω .
If the hypothesis (ii) holds, then we have three sequences (θi) , (ρi) and (δi) converging to θ, ρ and δ, respectively for some θ, ρ, δ ∈ Ω. Then we get (θ, ρ, δ, θi, ρi, δi) ∈ D, for all
Since Θϖ is triangular and by (1), for ν ⪢ ϑ, we obtain
This leads to Θϖ (Ξ (θ, ρ, δ) , θ, ν) =1, this implies that θ = Ξ (θ, ρ, δ). Similarly, one can reach that ρ = Ξ (ρ, δ, θ) and δ = Ξ (δ, θ, ρ). Thus Ξ has a TFP in Ω.□
Example 3.8. Suppose that Ω = [0, ∞) , ∗ is a continuous ν-norm and Ξ : Ω × Ω × (0, ∞) → [0, 1] is described by
where ϖ (θ, ρ) = |θ - ρ| is a usual metric ∀θ, ρ ∈ Ω, and ν ⪢ ϑ . Define the mapping Ξ : Ω3 → Ω by
It is clear that if we set ρ1 = 6 and ρ2 = 7, ρ1 ≤ ρ2 and Ξ (θ, ρ1, δ) ≤ Ξ (θ, ρ2, δ) so the MMP is not satisfied here. Moreover, define the mapping by ∀η > ϑ .
By routine calculations, for all θ, ρ, δ, h, p, a ∈ Ω, we can write
Hence all requirements of Theorem 3.7 are fulfilled with D = Ω6 and Ξ has a unique TFP, i.e., T (3, 3, 3) =3 .
Put ψ (η) = γη, where γ > [0, 1) in Theorem 3.7, we have the result below.
Corollary 3.9.Suppose that D is a non-empty subset of a CFCMS (Ω, Θϖ, ∗) in which Θϖ is triangular. Let Ξ : Ω3 → Ω be a given mapping and there is γ ∈ [0, 1) so thatfor all (θ, ρ, δ, h, p, a) ∈ D . Suppose either
Ξis continuous, or
If for any three sequences (θi) , (ρi) and (δi) with {θi+1, ρi+1, δi+1, θi, ρi, δi} ∈ Dsuch that (θi) → θ, (ρi) → ρand (δi) → δ, for alli ≥ 1, then (θ, ρ, δ, θi+1, ρi+1, δi+1) ∈ D, for alli ≥ 1 . Moreover, if there is (θ0, ρ0, δ0) ∈ Ω3so thatand Ξ is Ξ-invariant set which verifies the transitive property,
then there is a TFP ofΞ.
To discuss the uniqueness of a TFP, we present the following theorem:
Theorem 3.10.In addition to hypotheses of Theorem 3.7, let for all (θ, ρ, δ) , (h, p, a) ∈ Ω3, there is (s, q, z) ∈ Ω3 so that (θ, ρ, δ, h, p, a) ∈ D and (h, p, a, s, q, z) ∈ D . Then Ξ has a unique TFP in Ω.
Proof. Based on the proof of Theorem 3.7, the mapping Ξ has a TFP in Ω. Suppose that (θ, ρ, δ) and (h, p, a) are TFPs of Ξ, i.e., θ = Ξ (θ, ρ, δ) , ρ = Ξ (ρ, δ, θ) , δ = Ξ (δ, θ, ρ) , h = Ξ (h, p, a) , p = Ξ (p, a, h) and a = Ξ (a, h, p) .
We prove that θ = h, ρ = p and δ = a . From our assumptions, there is (s, q, z) ∈ Ω3 so that (θ, ρ, δ, h, p, a) ∈ D and (h, p, a, s, q, z) ∈ D .
Let s = s0, q = q0 and z = z0 and define three sequences (si) , (qi) and (zi) so that si = Ξ (si-1, qi-1, zi-1) , qi = Ξ (qi-1, zi-1, si-1) and zi = Ξ (zi-1, si-1, qi-1) , for all Since D is Ξ-invariant and
then
this implies that (θ, ρ, δ, s1, q1, z1) ∈ D . By the property of Ξ-invariant, we get
this implies that (θ, ρ, δ, s2, q2, z2) ∈ D . Repeating this argument, we get (θ, ρ, δ, si, qi, zi) ∈ D, for all Now, in view of (1), for ν ⪢ ϑ, one can write
Since D is Ξ-invariant and (θ, ρ, δ, si, qi, zi) ∈ D, for all we get (qi, zi, si, ρ, δ, θ) ∈ D, for all Again by (1), for ν ⪢ ϑ, one can write
Similarly
Adding (14)-(16) for ν ⪢ ϑ, we get
Repeating the same argument, for ν ⪢ ϑ, we have
Now, from φ (η) < η and it follows that for all η > ϑ, hence, from (17) for ν ⪢ ϑ, we have
Similarly, for ν ⪢ ϑ, we conclude that
Since Θϖ is triangular and by (18) and (19), we get
this leads to
Thus, and Hence θ = h, ρ = p and δ = a respectively. Therefore Ξ has a unique TFP. □
In the following part, we shall prove some TFP results by POMSs in FCMSs.
Let (Ω, ≤) be a partially ordered set (POS). A mapping Ξ : Ω → Ω is defined as non-decreasing (resp. non-increasing) if for each θ, ρ ∈ Ω so that θ ≤ ρ ⇒ Ξ (θ) ≤ Ξ (ρ) (resp. Ξ (ρ) ≤ Ξ (θ)).
Definition 3.11. [17] Let (Ω, ≤) be a POS. We say that the mapping Ξ : Ω3 → Ω have a MMP if Ξ is a monotone non-decreasing in the first and the third argument and it is a monotone non-increasing in the second argument, i.e., for all θ, ρ, δ ∈Ω :
θ, θ∗ ∈ Ω, θ ≤ θ∗ ⇒ Ξ (θ, ρ, δ) ≤ Ξ (θ∗, ρ, δ) .
ρ, ρ∗ ∈ Ω, ρ ≤ ρ∗ ⇒ Ξ (θ, ρ, δ) ≥ Ξ (θ, ρ∗, δ) .
δ, δ∗ ∈ Ω, δ ≤ δ∗ ⇒ Ξ (θ, ρ, δ) ≤ Ξ (θ, ρ, δ∗).
Example 3.12. Suppose that (Ω, ϖ) is a metric space with partial ordered " ≤ " and (Ω, Θϖ, ∗) is an FCMS equipped with
for all θ, ρ ∈ Ω and ν ⪢ ϑ, where ϖ (θ, ρ) = |θ - ρ| .
The mapping Ξ : Ω3 → Ω satisfies the MMP, for all θ, ρ, δ ∈Ω because
and
Consider D = {(θ, ρ, δ, θ∗, ρ∗, δ∗) ∈ Ω6 : θ ≥ θ∗, ρ ≤ ρ∗, δ ≥ δ∗)} , where D ⊆ Ω6 . Then D is Ξ-invariant subset of Ω6 which verifies the transitive property.
Theorem 3.13.Let (Ω, ≤) be a POS and (Ω, Θϖ, ∗) be a CFCMS in which Θϖ is triangular. Let the mapping with φ (ϑ) = ϑ < φ (η) < η, and for all η > ϑ. Assume that the mapping Ξ : Ω3 → Ω has a MMP and verifiesfor all θ, ρ, δ, h, p, a ∈ Ω for which θ ≥ h, ρ ≤ p and δ ≥ a . Suppose either
Ξis continuous, or
Ωhas the two assertions below:
if a non-decreasing sequences (θi) and (δi) in Ω where θi → θ and δi → δ, then θi ≤ θ and δi ≤ δ, for all ,
if a non-increasing sequence (ρi) in Ω where ρi → ρ, then ρi ≥ ρ, for all .
If there exist θ0, ρ0, δ0 ∈ Ω so that θ0 ≤ Ξ (θ0, ρ0, δ0) , ρ0 ≥ Ξ (ρ0, δ0, θ0) and δ0 ≤ Ξ (δ0, θ0, ρ0) with θ = Ξ (θ, ρ, δ) , ρ = Ξ (ρ, δ, θ) and δ = Ξ (δ, θ, ρ) , then Ξ has a TFP.
Proof. Suppose that a subset D ⊆ Ω6 is defined by D = {(θ, ρ, δ, θ∗, ρ∗, δ∗) ∈ Ω6 : θ ≥ θ∗, ρ ≤ ρ∗, δ ≥ δ∗} .
It follows from Example 3.12 that D is Ξ-invariant subset of Ω6 which verifies the transitive property. From (20), for all θ, ρ, δ, h, p, a ∈ Ω we get (θ, ρ, δ, h, p, a) ∈ D . Since θ0, ρ0, δ0 ∈ Ω so that
then, we get (Ξ (θ0, ρ0, δ0) , Ξ (ρ0, δ0, θ0) , Ξ (δ0, θ0, ρ0) , θ0, ρ0, δ0) ∈ D .
Now, if the assumption (2) holds and (θi) , (ρi) and (δi) are any three sequences in Ω so that (θi) and (δi) are non-decreasing sequences where θi → θ and δi → δ and (ρi) is a non-increasing sequence with ρi → ρ . Then, we have
Therefore (θ, ρ, δ, θi, ρi, δi) ∈ D, Thus the hypothesis (ii) of Theorem 3.7 is satisfied. Hence all requirements of Theorem 3.7 are fulfilled. Then Ξ has a TFP.□
Corollary 3.14.In addition to the hypotheses of Theorem 3.13, suppose that for all (θ, ρ, δ) , (s, q, z) ∈ Ω3, there exist (h, p, a) ∈ Ω3 so that θ ≥ h, ρ ≤ p, δ ≥ a and s ≥ h, q ≤ p, z ≥ a . Then Ξ has a unique TFP.
Proof. Suppose that a subset D ⊆ Ω6 is defined by D = {(θ, ρ, δ, θ∗, ρ∗, δ∗) ∈ Ω6 : θ ≥ θ∗, ρ ≤ ρ∗, δ ≥ δ∗} . Then by Example 3.12, we get D is Ξ-invariant subset of Ω6 which verifies the transitive property. Hence according to the proof of Theorem 3.13, we obtain the existence of the TFP.
For the uniqueness, assume that for all (θ, ρ, δ) and (s, q, z) ∈ Ω3, there exist (h, p, a) ∈ Ω3 so that θ ≥ h, ρ ≤ p, δ ≥ a and s ≥ h, q ≤ p, z ≥ a . This implies that (θ, ρ, δ, h, p, a) ∈ D and (h, p, a, s, q, z) ∈ D . Hence all assumptions of Theorem 3.10 are satisfied and Ξ has a unique TFP. □
Strong TFP results in FCMSs
We start this part with introducing the definitions below.
Definition 4.1. Assume that D, G and H are three non-empty subsets of a given set Ω . A mapping Ξ : Ω3 → Ω, so that Ξ (θ, ρ, δ) ∈ D if θ ∈ G, ρ ∈ H and δ ∈ D, Ξ (θ, ρ, δ) ∈ G if θ ∈ H, ρ ∈ D and δ ∈ G and Ξ (θ, ρ, δ) ∈ H if θ ∈ D, ρ ∈ G, and δ ∈ H is said to be a cyclic map with respect with (w.r.t.) D, G and H.
Definition 4.2. Let Ω be a non-empty set. A trio (θ, ρ, δ) ∈ Ω3 is called a TFP of the mapping Ξ : Ω3 → Ω if Ξ (θ, ρ, δ) = θ, Ξ (ρ, δ, θ) = ρ and Ξ (δ, θ, ρ) = δ and it is said to be a strong TFP if θ = ρ = δ, that is Ξ (θ, θ, θ) = θ.
Definition 4.3. Assume that D, G and H are three non-empty subsets of a metric space (Ω, ϖ). We say that a mapping Ξ : Ω3 → Ω is a cyclic tripled Kannan type contraction w.r.t. D, G and H if Ξ is cyclic w.r.t. D, G and H verifying the inequality
where θ, p ∈ D, ρ, a ∈ G, δ, h ∈ H, and
Definition 4.3 can be generalized in FCMS as follows:
Definition 4.4. Assume that D, G and H are three non-empty closed subsets of an FCMS (Ω, Θϖ, ∗) . We say that a mapping Ξ : Ω3 → Ω is a cyclic tripled Kannan type FCC w.r.t. D, G and H if Ξ is cyclic w.r.t. D, G and H verifying the inequality
where θ, p ∈ D, ρ, a ∈ G, δ, h ∈ H, and
Definition 4.5. A mapping Ξ : Ω3 → Ω is called a generalized cyclic tripled type FCC condition in FCMSs if Ξ verifies
where θ, p ∈ D, ρ, a ∈ G and δ, h ∈ H, for ν ⪢ ϑ, and It should be noted that the inequality (22) is reduced to (21) if we take and d = e = 0 .
Now, we present our first result of this section.
Theorem 4.6.Assume that D, G and H are three non-empty closed subsets of a CFCMS (Ω, Θϖ, ∗) , where Θϖ is triangular and the mapping Ξ : Ω3 → Ω is a generalized cyclic tripled type FCC w.r.t. D, G and H . Assume that Ξ verifies (22) with b + c + 2d + 2e < 1 . Then D∩ G ∩ H = ∅ and Ξ has a strong TFP in D ∩ G ∩ H .
Proof. Define θ0 ∈ D, ρ0 ∈ G and δ0 ∈ H . Suppose that (θi) , (ρi) and (δi) are three sequences described as follows:
for all i ≥ ϑ . Then (θi) ⊂ D, (ρi) ⊂ G and (δi) ⊂ H since Ξ is a cyclic mapping w.r.t. D, G and H . Let us denote
Then g ∈ (0, 1) for b + c + 2d + 2e < 1 . We show that, for ν ⪢ ϑ and i ≥ ϑ,
Clearly, (24) holds for i = ϑ . Let (24) holds for i = k, ν ⪢ ϑ, then from (22), one can write
which implies that
Similarly, based on (22), we can write
which leads to
Again using (22), one can write
which implies that
Then, by mathematical induction inequality (24) is fulfilled. Based on (24) with i ≥ ϑ for ν ⪢ ϑ, we obtain
Hence (24) is fulfilled for i = k + 1 . Thus (24) holds, for all i ≥ ϑ. Also, by (22) for i ≥ ϑ, we get
We assume that ς = max {d, e} and σ = max {b, c} , then, we get
This together with (24) verifies that
for ν ⪢ ϑ. Thus, for i, j ≥ ϑ, without loss of generally, let i ≤ j,
This proves that (θi) is a Cauchy sequence and convergent in Ω . Because D, G and H are non-empty closed subsets of Ω, one can write
Analogously,
and
Then, from (26)-(28), one sees that
Since Θϖ is a triangular, by (24) and (25), we obtain that
Thus, Θϖ (θ, ρ, ν) =1 . Similarly, we conclude that Θϖ (ρ, δ, ν) =1 and Θϖ (δ, θ, ν) =1, for ν ⪢ ϑ . This conclude that θ = ρ = δ ∈ D ∩ G ∩ H .
Now, we shall prove that (θ, ρ, δ) is a strong TFP of Ξ . According to the Θϖ triangular property, we can write
for ν ⪢ ϑ . By (22) and (26)-(28), we get
Then,
Hence, this together with (29) leads to
Since (c + e) <1, then Θϖ (θ, Ξ (θ, ρ, δ) , ν) =1, this leads to Ξ (θ, ρ, δ) = θ = ρ = δ . Therefore a trio (θ, ρ, δ) is a strong TFP of Ξ . □
If we put e = 0 and d = 0 in the above theorem, we have Corollary 4.7 and Corollary 4.8, respectively.
Corollary 4.7.Suppose that D, G and H are three non-empty closed subsets of a CFCMS (Ω, Θϖ, ∗) , where Θϖ is triangular and the mapping Ξ : Ω3 → Ω is a cyclic tripled type FCC w.r.t. D, G and H . Let Ξ verifieswhereθ, p ∈ D, ρ, a ∈ Gandδ, h ∈ H, forν ⪢ ϑ, andwithb + c + 2d < 1. ThenD∩ G ∩ H = ∅ andΞhas a strong TFP inD ∩ G ∩ H.
Corollary 4.8.Assume that D, G and H are three non-empty closed subsets of a CFCMS (Ω, Θϖ, ∗) , where Θϖ is triangular and the mapping Ξ : Ω3 → Ω is a cyclic tripled type FCC w.r.t. D, G and H . Let Ξ verifieswhere θ, p ∈ D, ρ, a ∈ G and δ, h ∈ H, for ν ⪢ ϑ and b, c, with b + c + 2e < 1 . Then D∩ G ∩ H ≠ ∅ and Ξ has a strong TFP in D ∩ G ∩ H .
The result of Kannan type for a cyclic TFP in FCMSs can be obtained If we set b = c and d = e = 0 in (22) as follows:
Corollary 4.9.Suppose that D, G and H are three non-empty closed subsets of a CFCMS (Ω, Θϖ, ∗) , where Θϖ is triangular and the mapping Θϖ : Ω3 → Ω is a cyclic tripled Kannan type FCC w.r.t. D, G and H verifying (21), for some Then D∩ G ∩ H ≠ ∅ and Ξ has a strong TFP in D ∩ G ∩ H .
To support the result of Theorem 4.6, we present the following example.
Example 4.10. Suppose that Ω = [0, ∞) , ∗ is a continuous ν-norm and Ξ : Ω × Ω × (0, ∞) → [0, 1] is defined by
where ϖ (θ, ρ) = |θ - ρ| is a usual metric, for all θ, ρ ∈ Ω, and ν > ϑ . Then easily one can prove that Θϖ is triangular and (Ω, Θϖ, ∗) is a CFCMS. Suppose that D = [0, 1] , and are three non-empty closed subsets of Ω with ϖ (D, G) =0, ϖ (G, H) =0 and ϖ (H, D) =0 . Define a continuous mapping Ξ : Ω3 → Ω by
Then the mapping Ξ is a cyclic mapping w.r.t. D, G and H. Indeed, if we choose and then we have
Now, for ν ⪢ ϑ, one can get
Therefore, Inequality (22) is fulfilled with and for ν ⪢ ϑ . Hence all requirements of Theorem 4.6 are justified with D = Ω6 and Ξ has a strong TFP, i.e., T (3, 3, 3) =3 ∈ (1, ∞) .
The second theorem of this part is presented as the following:
Theorem 4.11.Suppose that D, G and H are three non-empty closed subsets of a CFCMS (Ω, Θϖ, ∗) , where Θϖ is triangular and the mapping Ξ : Ω3 → Ω is a cyclic tripled contractive type mapping w.r.t. D, G and H for some verifyingwhere θ, p ∈ D, ρ, a ∈ G and δ, h ∈ H, for ν ⪢ 0 . Then D∩ G ∩ H ≠ ∅ and Ξ has a strong TFP in D ∩ G ∩ H .
Proof. Define θ0 ∈ D, ρ0 ∈ G and δ0 ∈ H . Suppose that (θi) , (ρi) and (δi) are three sequences defined by (23).
Now, we shall show that (θi) is a Cauchy sequence. From (31), we get
If Θϖ (θi+1, ρi+2, ν) is minimum,
then is the maximum in (32) which is not true. So, we have
Analogously
and
Adding (33)-(35), for ν ⪢ ϑ, one can write
Again by (32) and by the same manner, one can get
Analogously
and
Adding (37)-(39), and substituting in (36), for ν ⪢ ϑ, we get
By continuing with the same approach, for ν ⪢ ϑ, we obtain that
Then (40) is true for all i ≥ ϑ . Now, for an integer k,
Then, we have the two cases below:
(i) If Θϖ (ρk, θk+1, ν) is minimum,
then is the maximum in (41) so that
(ii) If Θϖ (θk, ρk+1, ν) is minimum,
then is the maximum in (41) so that
Adding (42) and (43), one can write
By the same method, one can obtain
and
where Applying (40) in (44)-(46) for ν ⪢ ϑ,
for k ≥ ϑ . Because Θϖ is triangular, then by (40) and (47), one sees that
for i ≥ ϑ . Now, for m, i ≥ ϑ, without loss of generally, we can suppose that m > i and
This shows that (θi) is a Cauchy sequence and convergent in Ω . Since D, G and H are non-empty closed subsets of Ω, then
Analogously
and
It follows from (48)-(50) that
Since Θϖ is a triangular, by (40) and (48), we can write
Hence, Θϖ (θ, ρ, ν) =1, similarly, one can obtain that Θϖ (ρ, δ, ν) =1 and Θϖ (δ, θ, ν) =1, for ν ⪢ ϑ . This implies that θ = ρ = δ ∈ D ∩ G ∩ H .
Finally, we show that (θ, ρ, δ) is a strong TFP of Ξ . Since Θϖ is a triangular, then we have
for ν ⪢ ϑ . By (31) and (48)-(50), we obtain
If 1 is the minimum of {1, Θϖ (θ, Ξ (θ, ρ, δ) , ν)}, then it follows from (51) that Θϖ (θ, Ξ (θ, ρ, δ) , ν) =1, as i → ∞ , for ν ⪢ ϑ . Thus, θ = ρ = δ ∈ D ∩ G ∩ H .
Otherwise, if Θϖ (θ, Ξ (θ, ρ, δ) , ν) is the minimum of {1, Θϖ (θ, Ξ (θ, ρ, δ) , ν)}, then we get
Now, from (51), we have
this leads to
a contradiction. Therefore, Θϖ (θ, Ξ (θ, ρ, δ) , ν) =1, for ν ⪢ ϑ, this implies that Ξ (θ, ρ, δ) = θ = ρ = δ, i.e., a trio (θ, ρ, δ) is a strong TFP of Ξ.□
Example 4.12. Assume that all data of Example 4.10 hold. Then it follows from (31) that
Therefore, all requirements of Theorem 4.11 are fulfilled with for ν ⪢ ϑ, with D = Ω6 and Ξ has a strong TFP, i.e., T (3, 3, 3) =3 ∈ (1, ∞).
Supportive application
In this part, we will use the results of Theorem 3.7 to discuss the existence and the uniqueness of the solutions of the following nonlinear integral system:
where η ∈ [0, ℓ] and is a continuous function. Assume that Ω = C ([0, ℓ]) is the space of all real continuous functions on [0, ℓ] , with The induced metric ϖ : Ω2 → Ω is given by
where .
Define a binary relation ∗ and fuzzy metric Θϖ : Ω2 × (0, ∞) → Ω is defined by x ∗ y = xy for all x, y ∈ [0, ℓ] and
for ν > ϑ and respectively. It is clear that Θϖ is triangular and (Ω, Θϖ, ∗) is a CFCMS.
We postulate that the following assumptions hold:
is continuous function.
for all η ∈ [0, ℓ] and for with θ ≥ h, ρ ≤ p and δ ≥ a, we get
φ : [0, ∞) → [0, ∞) is non-decreasing, continuous and verifies 0 = φ (0) < φ (η) < η and for all η > 0 .
Now, we shall state and prove our main theorem of this section.
Theorem 5.1.Under assertions (a1) and (a2), System (52) has a unique solution (where ).
Proof. Define the mapping by
Assume that D = {(θ, ρ, δ, h, p, a) ∈ Ω3 × Ω3 : θ (η) ≥ h (η) , ρ (η) ≤ p (η) and δ (η) ≥ a (η) , for all η ∈ [0, ℓ]}. It is clear that a subset D of Ω6 is Ξ-invariant which verifies the transitive property. Clearly, stipulation (a1) of Theorem 3.7 is fulfilled.
Now, we shall prove that is a TFP of the mapping Ξ.
Assume that (θ, ρ, δ, h, p, a) ∈ D, by hypothesis (a1) , for all η ∈ [0, ℓ], one can write
This leads to
Then, we have
for all (θ, ρ, δ, h, p, a) ∈ D . Hence the contractive stipulation 1 is fulfilled. Furthermore, it is easily to illustrate that so that . Therefore all hypotheses of Theorem 3.7 are fulfilled. Then the mapping Ξ has a unique TFP, which is a unique solution to System (52).□
Conclusion
Fuzzy set theory has been considered, utilized, and modified in various trends, in which the one direction of this theory is fuzzy logic, which has a lot of vital applications such as engineering fields, business, education, etc. In this manuscript, we presented the notion of a cyclic tripled type FCC mapping in the setting of FCMSs. In addition, some theoretical results concerned with TFPs are introduced without MMP in POMSs. Moreover, some strong TFP theorems are obtained under this notion in the mentioned space. Finally, to strengthen the theoretical results, non-trivial examples are listed and the existence and uniqueness of the solution of the nonlinear integral equations are obtained. In lieu of the current nonlinear integral equation, the authors can use various types of applications such as Riemann integral equations, Lebesgue integral equations, and integro-differential equations to support their findings.
Availability of data and material
The data used to support the findings of this study are available from the corresponding author upon request.
Competing interests
The authors declare that they have no competing interests concerning the publication of this article.
Funding
Not applicable.
Author’s contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Footnotes
Acknowledgments
The authors are very grateful and thank the editor-in-chief, associate editor and each of the reviewers for their good comments to improve the presentation of the manuscript.
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