Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales

This paper mainly deals with introducing and studying the properties of generalized nabla differentiability for fuzzy functions on time scales via Hukuhara difference. Further, we obtain embedding results on En for generalized nabla differentiable fuzzy functions. Finally, we prove a fundamental theorem of a nabla integral calculus for fuzzy functions on time scales under generalized nabla differentiability. The obtained results are illustrated with suitable examples.


Introduction
The theory of dynamic equations on time scales is a genuinely new subject and the research related to this area is developing rapidly. Time scale theory has been developed to unify continuous and discrete structures, and it allows solutions for both differential and difference equations at a time and extends those results to dynamic equations. Basic results in time scales and dynamic equations on time scales are found in [1][2][3][4][5][6]. In [7], the author illustrated an example where delta derivative needs more assumptions than nabla derivative. Some recent studies in economics [8], production, inventory models [9], adaptive control [10], neural networks [11], and neural cellular networks [12] suggest nabla derivative is also preferable and it has fewer restrictions than delta derivative on time scales.
On the other hand, when we expect to investigate a real world phenomenon absolutely, it is important to think about a number of unsure factors too. To specify these vague or imprecise notions, Zadeh [13] established fuzzy set theory. The theory of fuzzy differential equations (FDEs) and its applications was developed and studied by Kaleva [14], Lakshmikantham and Mohapatra [15]. The concept based on Hukuhara differentiability has a shortcoming that the solution to a FDEs exists only for increasing length of support. To overcome this shortcoming, Bede and Gal [16] studied generalized Hukuhara differentiability for fuzzy functions. In light of this preferred advantage, many authors [17][18][19] tend their enthusiasm to the generalized Hukuhara differentiability for fuzzy set valued functions.
The calculus of fuzzy functions on time scales was studied by Fard and Bidgoli [20]. Vasavi et al. [21][22][23][24] introduced Hukuhara delta derivative, second-type Hukuhara delta derivative, and generalized Hukuhara delta derivatives by using Hukuhara difference, and they studied fuzzy dynamic equations on time scales. Wang et al. [25] introduced and studied almost periodic fuzzy vector-valued functions on time scales. Deng et al. [26] studied fractional nabla-Hukuhara derivative on time scales. Recently, Leelavathi et al. [27] introduced and studied properties of nabla Hukuhara derivative for fuzzy functions on time scales. However, this derivative has the disadvantage that it exists only for the fuzzy functions on time scales which have a diameter with an increasing length. For the fuzzy functions with decreasing length of diameter on time scales, Leelavathi et al. [28] introduced the second-type nabla Hukuhara derivative and studied its properties. Later, they continued to study fuzzy nabla dynamic equations under the first and second-type nabla Hukuhara derivatives in [29] under generalized differentiability by using generalized Hukuhara difference in [30]. Consider a simple fuzzy function F(s) = s c, s ∈ T ∩ [−2, 2], where c = (1, 2, 3) is a triangular fuzzy number. Clearly, F(s) has decreasing length of diameter in T ∩ [−2, 0] and increasing length of diameter in T ∩ [0, 2]. Therefore, the fuzzy function F(s) is neither a nabla Hukuhara differentiable (as defined in [27]) nor a second-type nabla Hukuhara differentiable (as defined in [28]) on T ∩ [−2, 2]. In this context, it is required to define a nabla Hukuhara derivative for a fuzzy function which may have both increasing and decreasing length of diameter on a time scale. To address this issue, in the present work, we define a new derivative called generalized nabla derivative for fuzzy functions on time scales via Hukuhara difference and study their properties. In [31], the authors introduced a nabla integral for fuzzy functions on time scales and obtained fundamental properties. In the present work, we continue to study nabla integral for fuzzy functions on time scales and prove a fundamental theorem of nabla integral calculus for generalized nabla differentiable functions.
The rest of this paper is arranged as follows. In Section 2, we present some basic definitions, properties, and results relating to the calculus of fuzzy functions on time scales. In Section 3, we establish the nabla Hukuhara generalized derivative for fuzzy functions on time scales and obtain its fundamental properties. The results are highlighted with suitable examples. In Section 4, we prove an embedding theorem on E n and obtain the results connecting to generalized nabla differentiability on time scales. Using these results, we finally prove the fundamental theorem of nabla integral calculus for fuzzy functions on time scales under generalized nabla differentiability and a numerical example is provided to verify the validity of the theorem.

Preliminaries
Let k ( n ) be the family of all nonempty convex compact subsets of n . Define the set addition and scalar multiplication in k ( n ) as usual. Then, by [14], k ( n ) is a commutative semi-group under addition with cancellation laws. Further, if β, γ ∈ and U, V ∈ k ( n ), then Let P and Q be two bounded nonempty subsets of n . By using the Pampeiu-Hausdorff metric, we define the distance between P and Q as follows: where . is the Euclidean norm in n . Then, ( k ( n ), d H ) becomes a separable and complete metric space [14]. Define: If there exists a t ∈ n such that u(t) = 1, then u is said to be normal.
For 0 ≤ λ ≤ 1, denote [u] λ = {t ∈ n : u(t) ≥ λ}; then, from the above conditions, we have that the λ-level set [u] λ ∈ k ( n ). By Zadeh's extension principle, a mapping h : R n × R n → R n can br extended to g : E n × E n → E n by , for all p, q ∈ E n and h is continuous. The scalar multiplication and addition ⊕ of p, q ∈ E n is defined as where d H is the Pampeiu-Hausdorff metric defined in k ( n ). Then, (E n , D H ) is a complete metric space [14]. The following theorem extends the properties of addition and scalar multiplication of fuzzy number valued functions ( F = E 1 ) to E n [14].
The properties of addition and scalar multiplication of fuzzy number valued functions ( F = E 1 ) are easily extended to E n .

Definition 1 ([14]
). Let K, L ∈ E n . If there exists M ∈ E n such that K = L ⊕ M, then we say that M is the Hukuhara difference of K and L and is denoted by K h L.
For any K, L, M, N ∈ E n and β ∈ , the following hold: provided the Hukuhara differences exists. A triangular fuzzy number is denoted by three points as t = (t 1 , t 2 , t 3 ). This representation is denoted as membership function In addition, λ-level sets of triangular fuzzy number t is an interval defined by λ-cut operation, 1]. Clearly, the triangular fuzzy number is in E 1 .

Remark 2 ([14]
). A mapping Φ : I → E n is said to be integrably bounded if there exists an integrable function h such that x ≤ h(s), for all x ∈ Φ 0 (s).

Definition 3 ([14]
). Let Φ : I → E n . The integral of Φ over I is denoted by I Φ(s)ds or y x Φ(s)ds, where g is a level wise selection of measurable functions of Φ λ for 0 < λ ≤ 1.
A mapping Φ : I → E n is said to be integrable over I if Φ is integrably bounded and strongly measurable function and also I Φ(s)ds ∈ E n .

Definition 4 ([18]).
A fuzzy function Φ : I → E n is said to be differentiable from left at s 0 if for δ > 0, there exists P ∈ E n , such that the following holds: Here, P is the derivative of Φ from left at s 0 and is denoted as Φ − (s 0 ).

Definition 5 ([18]).
A fuzzy function Φ : I → E n is said to be differentiable from right at s 0 if, for δ > 0, there exists P ∈ E n , such that the following holds: Here, P is the derivative of Φ from right at s 0 and is denoted as Φ + (s 0 ). The limits are taken over (E n , D H ).

Definition 6 ([18]
). If Φ is both left-differentiable and right-differentiable at s 0 , then Φ is said to be differentiable at s 0 and Φ − (s 0 ) = Φ + (s 0 ) = P . Here, P is called the derivative of Φ at s 0 and we consider one-sided derivative at the end points of I.

Generalized Nabla Hukuhara Differentiability on Time Scales
This section is concerned with defining and studying the properties of ∇ g derivative for fuzzy functions on time scales. In addition, we illustrate the results with suitable examples.

Definition 7 ([21]
). For any given > 0, there exists a δ > 0, such that the fuzzy function Φ : Here, T-limit denotes the limit on time scale in the metric space (E n , D H ).

Remark 4. From the above definition, we have
where the zero element in E n is given by0.
, then, for every > 0, there exists a δ > 0, such that k , if there exists an element Φ ∇ g + (s) ∈ E n with the property that, for every given > 0, there exists a neighborhood N T [a,b] of s for some δ > 0 and 0 ≤h ≤ δ, .
exist as a finite number and holds any one of the following: For 0 ≤h < 1 and for allh ≥ 0, to each Similarly, we can prove Φ is continuous at s, if ∇ g is right-differentiable at s.
Since is arbitrary, we get any one of (i)-(iv).
The converse proposition of Theorem 3(a) may not be true. That is a fuzzy function which is continuous may not be differentiable.
and Φ is continuous at left scattered point s. Then, from (GH3), we have and Φ is continuous at left scattered point s. Then, from (GH4), we have

Remark 9. A fuzzy function
(a) If Φ is ∇ g differentiable as in (GH1) at ld-point s or ∇ g differentiable as (GH4) at left scattered point s, If Φ is ∇ g differentiable as (GH2) at ld-point s or ∇ g differentiable as (GH3) at left scattered point s, (1) If Φ and Ψ are both ∇ g differentiable of same kind, then: → E n is also ∇ g differentiable of same kind at s with (2) If Φ and Ψ are different kinds of ∇ g differentiable at s, and (Φ h Ψ) exists for s ∈ T k .
Multiplying the above equation with −1 and it follows that The case when Φ and Ψ are ∇ g differentiable as in (GH4) is similar to the previous one. k , similar to 1(a), we have Φ( (s)) = Φ(s) ⊕ u(s) and Ψ( (s)) = Ψ(s) ⊕ v(s). Consider Multiplying the above equation with −1 ν(s) , we get the desired result. In a similar way, we can easily prove the other case.

Integration of Fuzzy Functions on Time Scales
In this section, we prove fundamental theorem of nabla integral calculus for fuzzy functions on time scales under generalized fuzzy nabla differentiable functions on time scales.
First, we prove an embedding theorem on E n and obtain some results which are useful to prove the main theorem. To prove the these results, we make use of Definitions 1-3 and Theorem 4 in [31].
Proof. First, we show that X = C[0, 1] × C[0, 1] is a Banach space. Consider a cauchy sequence l n 0 = ( f n 0 , g n 0 ) and for * > 0, there exists N > 0, n 0 > N such that n 0 , m 0 > N implies l m 0 − l n 0 < * , that is which yields the result that f n 0 (λ) → f and g n 0 (λ) → g as 1] isometrically and isomorphically, we need to prove the following: , for any u, v ∈ E n and p, q ≥ 0; and Let i(u) = (u − , u + ). The λ-level set of u ∈ E n can be written as Therefore, Thus, (a) is proved. Now, consider We make use the Proposition 3.1 and Remark 3.4 in [18] to prove the following results.
and s is left dense; then, the proof is similar to the proof of Theorem 8 [16]. Now, for s being left scattered, we have Again, in the same way, However, Finally, (i • Φ) ∇ (s) = i(Φ ∇ g (s)) = −i * (Φ ∇ g (s)).

Conclusions
This paper is concerned with investigating a new derivative called generalized nabla derivative for fuzzy functions on time scales and studies some basic properties of ∇ g derivative. In addition, we prove a fundamental theorem of nabla integral calculus for fuzzy functions on time scales under generalized differentiability on time scales. The advantage of ∇ g derivative is that it is exists even for a fuzzy function having increasing and decreasing length of diameter on a time scale. The results obtained in this paper include results of Leelavathi et al. [27], when the function having only increasing length of diameter, and the results of Leelavathi et al. [28], when the function having only decreasing length of diameter. The obtained results are illustrated with numerical examples. In the future, we propose to study fuzzy nabla dynamic equations on time scales under generalized nabla derivative and their applications.