On nabla conformable fractional Hardy-type inequalities on arbitrary time scales

*Correspondence: ahmedeldeeb@azhar.edu.eg 1Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt Full list of author information is available at the end of the article Abstract The main aim of the present article is to introduce some new ∇-conformable dynamic inequalities of Hardy type on time scales. We present and prove several results using chain rule and Fubini’s theorem on time scales. Our results generalize, complement, and extend existing results in the literature. Many special cases of the proposed results, such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities, are obtained and analyzed.


Introduction
Fractional calculus theory has an important role in the mathematical analysis and applications. Fractional calculus (FC), the theory of integrals and derivatives of noninteger order, is a field of research with a history dating back to Abel, Riemann, and Liouville (see [33] for a historical summary). Indeed, the most famous and extensively studied formulation, is called the Riemann-Liouville fractional integral in their honor. The corresponding fractional derivative is obtained by a composition of fractional integral with integer order derivative.
The definitions of fractional integrals and derivatives are not unique, and many definitions of fractional derivative operators have been introduced and successfully applied to solve complex systems in science and engineering (see [17,32,37]). Recently, study on fractional dynamic equations is very widespread around the world and is useful in pure and applied mathematics, physics, engineering, biology, economics, etc. They use an integral in its formulation, especially Cauchy's integral formula with some modifications. Therefore, they sometimes require a difficult calculation to obtain. Riemann-Liouville and Caputo fractional derivatives do not satisfy the nonlinear derivative rules as product, quotient, and chain rules. The mean value theorem and Rolle's theorem are not formulated using the definitions of Riemann-Liouville and Caputo fractional derivatives.
Recently, depending just on the basic limit definition of the derivative, Khalil et al. [31] proposed a new simple definition of the fractional derivative called conformable derivative T α f (t) (α ∈ (0, 1]) of a function f : for all t > 0, α ∈ (0, 1], this definition found wide resonance in the scientific community interested in fractional calculus, see [29,30,48]. Therefore, calculating the derivative by this definition is easily compared to the definitions that are based on integration. The researchers in [31] also suggested a definition for the α-conformable integral of a function η as follows: After that, Abdeljawad [4] made an extensive research of the newly introduced conformable calculus. In his work, he generalized the definition of conformable derivative T a α f (t) for t > a ∈ R + as follows: where f : R + → R. Benkhettou et al. [38] introduced a conformable calculus on an arbitrary time scale, which is a natural extension of the conformable calculus.
In the last few decades many authors pointed out that derivatives and integrals of noninteger order are very suitable for the description of properties of various real materials, e.g., polymers. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. These are some of the advantages of fractional derivatives in comparison with classical integer-order models.
Time scales theory, which has become a trend, began with S. Hilger. In his PhD thesis, this concept was initiated in order to get the continuous theorem and the discrete theorem in one theorem [27]. In [12,13], Bohner and Peterson introduced the most basic concepts and definitions related to the theory of time scales. Next, some basic definitions and concepts about the fractional analysis, which are used in this manuscript, were given and adapted from [12,13,34,38]. Any nonempty arbitrary closed subset of the real numbers is called a time scale T. We assume that T has the standard topology on the real numbers R. Now, let σ : T → T be the forward jump operator defined by and ρ : T :→ T be the backward jump operator defined by and left-scattered if ρ(t) < t. Points that are simultaneously right-dense and left-dense are called dense points, and points that are simultaneously right-scattered and left-scattered are called isolated points. The forward and backward graininess functions μ and ν, for a time scale T, are defined by μ(t) := σ (t)t and ν(t) := tρ(t), respectively. In [11], the authors studied a version of the nabla conformable fractional derivative on arbitrary time scales. Namely, for a function η : T → R, the nabla conformable fractional derivative, T ∇,α η(t) ∈ R of order α ∈ (0, 1] at t ∈T κ and t > 0 was defined as: Given any > 0, there is a δ-neighborhood U t ⊂ T of t, δ > 0 such that for all s ∈ U t . The nabla conformable fractional integral is defined by Rahmat et al. [38] presented a new type of conformable nabla derivative and integral which involves the time scale power function G n (t, s) for s, t ∈ T and also generalizes the definition of the nabla conformable fractional derivative and integral on time scales in [11]. The time scale power function takes the form (ta) η for T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. [31]. T × T − → R + for n ∈ N 0 is defined by and its inverse function G -n : T × T − → R + is then given by (1.4) We use the convention G 0 (t, s) = 1 for all s, t ∈ T.
, n ∈ N, Remark 1.3 Regarding the generalization of the power function G α (t, s) to real values of α ≥ 0 (instead of integers), we recall a broadly accepted extension of its particular cases (1.5) and (1.7) in the form (see [14]) (1pq).

Definition 1.4 (Conformable nabla derivative)
Given a function f : T − → R and a ∈ T, f is (γ , a)-nabla differentiable at t > a, if it is nabla differentiable at t, and its (γ , a)-nabla derivative is defined by where the function G 1-γ (t, a) is as defined in (1.3). If ∇ γ a [f (t)] exists in some interval (a, a + ) T , > 0, then we define Next, we provide the (γ , a)-nabla derivatives of sums, products, and quotients of (γ , a)nabla differentiable functions. . (1.10) exists and is finite.
We need the relations between different types calculus on time scales T and continuous calculus, discrete calculus, and quantum calculus as follows. Note that: For the case T = R, we have the classical conformable integral as defined in [4], namely (1.14) Theorem 1.9 Let γ ∈ (0, 1] and a ∈ T. Then, for any ld-continuous function f : The function F is called a (γ , a)-nabla antiderivative of f .
The study of Hardy-type inequalities attracted and still attracts the attention of many researchers. Over several decades many generalizations, extensions, and refinements have been made to the above inequalities, we refer the interested reader to the papers [1,8,9,18,19,23,24,35,36,41,44], see also [2,21,22,24] and the references cited therein.

Theorem 1.12 Suppose that the continuous function
The constant ( p p-1 ) p in (1.16) is sharp. Copson [16] obtained another classical discrete inequality of Hardy type.
Renaud [40] proved the following two results which are the reverse discrete and continuous versions of inequality (1.17).
ı=1 is a sequence of nonnegative and nonincreasing real numbers. (1.18)

Theorem 1.15 Suppose that η is a nonnegative and nonincreasing function on the interval
Renaud [40] proved the following result.
Very recently, Zakarya et al. [49] gave an α-conformable version of Theorem 1.20 on time scales as follows.

Theorem 1.21
Assume that T is a time scale with ω ∈ (0, ∞) T . If k ≤ 0 < h < 1 and α ∈ (0, 1], define In this paper, motivated by the results in [7,23,43], we introduce a new nabla version of Hardy-type dynamic inequalities via conformable fractional ∇-integral of order γ ∈ (0, 1] on time scales. These inequalities have a completely new form. Therefore, as special case, we obtain some new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities.

Main results
Now, we are ready to state and prove our main results. Throughout this section, any time scale T is unbounded above, and we will assume that the right-hand sides of the inequalities converge if the left-hand sides converge.

Theorem 2.1
Assume that T is a time scale with 0 ≤ r ∈ T. Moreover, suppose that f and λ are nonnegative Id-continuous functions on [r, ∞) T with f nondecreasing. If p ≥ 1 and β ≤ 0, then for t ≥ a ∈ T and γ ∈ (0, 1] we have that Applying the chain rule (1.9) and using ∇ γ ,x a denotes the (γ , a)-nabla derivative with respect to x, we get and so (note that x ≥ t ≥ r and hence, because is nondecreasing, Integrating both sides with respect to x over [t, ∞) T gives Integrating both sides again, but this time with respect to t over [r, ∞) T , produces Using Fubini's theorem on time scales, inequality (2.4) can be rewritten as Now, from the chain rule (1.10), there exists c ∈ [ρ(t), t] such that (here ∇ γ ,t a denotes the (γ , a)-nabla derivative with respect to t) This shows the validity of inequality (2.1). Now, as special cases of our results, we give the continuous, discrete, and quantum αconformable inequalities. Namely, in cases of time scales T = R, T = hZ, T = Z, and T = q N 0 .

Corollary 2.4
For T = Z, we simply take h = 1 in Corollary 2.3. In this case, inequality It is interesting to discuss inequality (2.1) after changing the limit of integral t r λ(s)∇ γ a s to be from t to ∞. Let us do that in the following theorem.

Theorem 2.6 Under the same hypotheses of Theorem 2.1 with β > 1, then we have that
By utilizing the chain rule (1.9) and using ∇ γ From (2.8) and (2.9) we get Thus, Therefore, upon integrating both sides with respect to x over [r, t] T , Since p-γ +1 (r) = 0, we have Then, by integrating both sides with respect to t over [r, ∞) T , we get (2.10) With the help of Fubini's theorem on time scales, inequality (2.10) can be rewritten as Combining (2.12) and (2.11) yields from which inequality (2.7) follows.
We are ready to present several special cases of our results to continuous, discrete, and quantum α-conformable inequalities. Namely, in cases of time scales T = R, T = hZ, T = Z, and T = q N 0 .

Corollary 2.9
For T = Z, we simply take h = 1 in Corollary 2.8. In this case, inequality  Proof As f is nondecreasing, we have for x ≥ r

Theorem 2.11 Under the same hypotheses of Theorem
(2.14) Using the chain rule (1.9) and the fact that Combining (2.14) with (2.15) gives and thus Therefore, Again using the chain rule (1.10), there is c ∈ [ρ(t), t] such that This completes the proof.
Again, we present some special cases of our results to the continuous, discrete, and quantum α-conformable inequalities. Namely, in cases of time scales T = R, T = hZ, T = Z, and T = q N 0 .

Corollary 2.12 If
We next discuss inequality (2.13) for the case when the limit of integral t r λ(s)∇ γ a s is changed from t to ∞.

Theorem 2.16
Under the same hypotheses of Theorem 2.1 with β ≥ 1, then we have that Employing the chain rule (1.9) and using ∇ Thus, which is our desired inequality (2.19).

Conclusion
In this important work, we discussed some new dynamic inequalities of Hardy type using nabla integral on time scales. By employing the conformable fractional ∇-conformableintegral on time scales, several ∇-conformable Hardy-type inequalities on time scales have been proved. Our proposed results show the potential for producing some original continuous, discrete, and quantum inequalities. We further presented some relevant inequalities as special cases: discrete inequalities and integral inequalities. These results may be used to obtain more generalized results of several obtained inequalities before.