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A new conformable nabla derivative and its application on arbitrary time scales
Advances in Difference Equations volume 2021, Article number: 238 (2021)
Abstract
In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function \(\widehat{\mathcal{G}}_{\eta }(t, a)\) for \(t,a\in \mathbb{T}\). The time scale power function takes the form \((t-a)^{\eta }\) for \(\mathbb{T}=\mathbb{R}\) which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form \((t-a)^{(\eta )}\) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form \(\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))\), and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.
1 Introduction
Fractional calculus becomes an important area of research in mathematical analysis and applications. Various definition of fractional derivative operators such as Riemann–Liouville, Caputo, Grünwald–Letnikov, and Erdélyi–Kober to mention few, were introduced and successfully applied to solve complex systems in science and engineering (see [1–3]). However, the semigroup properties of these fractional operators behave well in some cases only.
In 2014, Khalil et al. [4] introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative \(T_{\alpha }f(t)\) (\(\alpha \in (0,1]\)) of a function \(f:\mathbb{R}^{+}\to \mathbb{R}\) at \(t>0\) which depends on the basic limit definition of the classical derivative. This new fractional derivative definition has attracted much attention in recent years. In 2015, Abdeljawad [5] made an extensive research of the newly introduced conformable fractional calculus. In his work, he generalizes the definition of conformable fractional derivative \(T_{\alpha }^{a} f(t)\) for \(t>a\in \mathbb{R}^{+}\) as
In this case, if the function \(f(t)\) is differentiable, then
It is clear that if \(\alpha =1\), we have \(T^{a}_{1} f(t)=\lim_{\epsilon \to 0} \frac{f(t+\epsilon )-f(t)}{\epsilon }=f'(t)\).
The conformable fractional integral of a function \(f(t)\) from \(a\in \mathbb{R}^{+}\) is defined by
Note that for \(a=0\), we have \(T^{0}_{\alpha }f(t)=T_{\alpha }f(t)\) and \(I_{\alpha }^{0} f(t)=I_{\alpha }f(t)\), as suggested in [4].
An important reason for its introduction is specified by the fact that this derivative has many well-known properties of integer-order derivatives, among which we can highlight Rolle’s theorem, mean value theorem, product rule, quotient rule, chain rule, fractional power series expansion, and fractional Laplace transform (see [4–7]), which the classical fractional derivatives do not obey. To further justify the introduction of the conformable fractional derivative and its consequences, we refer to [8].
Since then, the theory of conformable fractional calculus and its applications have been studied by many authors (see, for instance, [9–15]). Chung [16] used the conformable fractional derivative and integral to discuss fractional Newtonian mechanics and Rezazadeh et al. [17] investigated the stability of linear conformable fractional systems from the point view of control theory (see also [18]). It is worth noting that the conformable fractional derivative does not have a physical meaning as the Riemann–Liouville or Caputo derivatives.
In [19] Anderson and Ulness made a remark that the discussed derivative is local in nature and hence the correct name must be “conformable derivative” instead as introduced “conformable fractional derivative.” It is worth mentioning here the criticism made about the definition of conformable fractional derivative by the authors of [20, 21].
On the other hand, Benkhettou et al. [22] extended the definition of (conformable) fractional derivative to arbitrary time scales. Meanwhile, in [23], the authors have studied a version of the nabla conformable fractional derivative on arbitrary time scales. Namely, for a function \(f:\mathbb{T}\to \mathbb{R}\), the nabla conformable fractional derivative, \(T_{\nabla ,\alpha } f(t)\in \mathbb{R}\) of order \(\alpha \in (0,1]\) at \(t\in \mathbb{T}_{\kappa }\) and \(t>0\), was defined as follows.
For any \(\epsilon >0\), there exists a neighborhood \(U=(t-\delta , t+\delta )\cap \mathbb{T}\) for some \(\delta >0\) such that
The nabla conformable fractional integral is defined by
Further development on conformable fractional derivative and its applications on arbitrary time scales can be seen through the articles [24–32].
Motivated by the results in [5], we generalize the definition of the nabla conformable fractional derivative and integral on time scales in [23] by replacing \((t-a)^{1-\alpha }\) with a generalized time scale power function \(\widehat{\mathcal{G}}_{1-\gamma }(t,a)\). With the new definition of the nabla conformable derivative, we are able to define the conformable exponential function and study its properties. We also prove an analogue of Gronwall’s inequality which will be useful in establishing the stability of the nabla conformable fractional dynamical systems on time scales.
2 Preliminaries
In this section, we review some basic concepts and notations of calculus of time scales used in this article. The reader interested in the subject of time scales is referred to [33–35].
A time scale \(\mathbb{T}\) is an arbitrary closed nonempty subset of \(\mathbb{R}\). The forward and backward jump operators \(\sigma , \rho :\mathbb{T}\to \mathbb{T}\) are defined by
respectively. The forward and backward graininess functions \(\mu , \nu :\mathbb{T}\to [0,\infty )\) are defined by
respectively. We say that \(t\in \mathbb{T}\) is right-scattered (resp., left-scattered) if \(\sigma (t)>t\) (resp., \(\rho (t)< t\)). Points that are right- and left-scattered at the same time are called isolated. Also, if \(t<\sup \mathbb{T}\) and \(\sigma (t)=t\), then t is called right-dense, and if \(t>\inf \mathbb{T}\) and \(\rho (t)=t\), then t is called left-dense. If \(\mathbb{T}\) has a right-scattered minimum m, then \(\mathbb{T}_{\kappa }=\mathbb{T}-\{m\}\). Otherwise, \(\mathbb{T}_{\kappa }=\mathbb{T}\).
Throughout this paper we assume that \(a,b\in \mathbb{T}\) and \(a< b\). We then define the interval \([a,b]\) in \(\mathbb{T}\) by \([a,b]_{\mathbb{T}}:=\{t\in \mathbb{T}\mid a\leq t\leq b\}\). Open intervals and half-open intervals are defined analogously.
We assume throughout that a time scale \(\mathbb{T}\) has the topology that it inherits from the real numbers with the standard topology. For \(f:\mathbb{T}\to \mathbb{R}\) and \(t\in \mathbb{T}_{\kappa }\), define the nabla derivative of \(f(t)\), denoted \(f^{\nabla }(t)\), to be the number (provided it exists) with the property that given any \(\epsilon >0\), there is a neighborhood \(\mathcal{U}\) of t (i.e., \(\mathcal{U}_{t}=(t-\delta , t+\delta )\cap \mathbb{T}\) for some \(\delta >0\)) such that
for all \(s\in U\). For \(\mathbb{T}=\mathbb{R}\), we have \(f^{\nabla }=f'\), the usual derivative, and for \(\mathbb{T}=\mathbb{Z}\) we have the backward difference operator, \(f^{\nabla }(t)=\nabla f(t):=f(t)-f(t-1)\).
A function \(f:\mathbb{T}\to \mathbb{R}\) is left-dense continuous (ld-continuous) provided it is continuous at left-dense points in \(\mathbb{T}\) and its right-sided limits exist (finite) at right-dense point in \(\mathbb{T}\). The set of all ld-continuous functions \(f:\mathbb{T}\to \mathbb{R}\) will be denoted by \(C_{\mathrm{ld}}(\mathbb{T})\). Similarly, the set of all ld-continuous and nabla differentiable functions will be denoted by \(C_{\mathrm{ld}}^{\nabla }(\mathbb{T})\).
Theorem 2.1
([34])
Assume \(f:\mathbb{T}\to \mathbb{R}\) is a function and let \(t\in \mathbb{T}_{\kappa }\). Then we have:
-
(i)
If f is nabla differentiable at t, the f is continuous at t.
-
(ii)
If f is continuous at a left-scattered t, the f is nabla differentiable at t with
$$ f^{\nabla }(t)=\frac{f(t)-f(\rho (t))}{\nu (t)}. $$ -
(iii)
If f is left-dense, then f is nabla differentiable at t if and only if the limit
$$ \frac{f(t)-f(s)}{t-s} $$exists as a finite number. In this case
$$ f^{\nabla }(t)=\frac{f(t)-f(s)}{t-s}. $$
A function \(F:\mathbb{T}\to \mathbb{R}\) is called a nabla antiderivative of \(f:\mathbb{T}\to \mathbb{R}\), provided that \(F^{\nabla }(t)=f(t)\) holds for all \(t\in \mathbb{T}_{\kappa }\), then the nabla integral of f is defined by
It follows that, if \(f\in C_{\mathrm{ld}}(\mathbb{T})\) and \(t\in \mathbb{T}_{\kappa }\), then
Note that if \(\vert f(t) \vert \leq g(t)\) on \([a,b)\), then
Let \(f:\mathbb{T}\times \mathbb{T}\to \mathbb{R}\) be a function and denote the nabla derivative of \(f(t,s)\) with respect to t (for fixed s) by \(f^{\nabla }(t,s)\). If f and \(f^{\nabla }\) are continuous, then
Now, we remind further aspects of time scale calculus, which will be needed later; see, e.g., [34].
Definition 2.2
Let \(\mathbb{T}\) be a time scale. A function \(f:\mathbb{T}\times \mathbb{R}\to \mathbb{R}\) is called
-
(i)
ld-continuous, if g defined by \(g(t):=f(t,y(t))\) is ld-continuous for any ld-function \(y:\mathbb{T}\to \mathbb{R}\).
-
(ii)
bounded on a set \(S\subset \mathbb{T}\times \mathbb{R}\), if there exists a constant \(M>0\) such that
$$ \bigl\vert f(t,y(t) \bigr\vert \leq M\quad \text{for all }(t,y)\in S. $$ -
(iii)
Lipschitz continuous on a set \(S\subset \mathbb{T}\times \mathbb{R}\), if there exists a constant \(L>0\) such that
$$ \bigl\vert f(t, y_{1})-f(t,y_{2}) \bigr\vert \leq L \vert y_{1}-y_{2} \vert \quad \text{for all } (t,y_{1}), (t,y_{2})\in S. $$
The following lemma is a nabla version of the Proposition 2.6 in [36].
Lemma 2.3
Suppose \(\mathbb{T}\) is a time scale and f is an increasing ld-continuous function on the interval \([a, b]_{\mathbb{T}}\). If f̃ is the extension of f to the real interval \([a, b]\) given by
then
Proof
Let \(r\in [a, b]_{\mathbb{T}}\) be a left-scattered point. Then we have
Since f is an increasing function, its extension f̃ will be an increasing continuous function. Thus, the mean value theorem for integrals implies
and
Hence, we have
Suppose, if \([a, b]\) has only one left-scattered point s, then we have
If we repeat the above steps for n left-scattered points in \([a, b]\), we obtain
This completes the proof. □
Next, we introduce the generalized time scale power function \(\mathcal{G}_{n}(t,s)\in \mathbb{R}^{+}\) for \(n\in \mathbb{N}_{0}\) and \(s,t\in \mathbb{T}\).
Definition 2.4
Let \([s,t]\subset \mathbb{T}\) and \(s< t\). The generalized time scale power function \(\mathcal{G}_{n}:\mathbb{T}\times \mathbb{T}\to R^{+}\) for \(n\in \mathbb{N}_{0}\) is defined by
and its inverse function \(\mathcal{G}_{-n}:\mathbb{T}\times \mathbb{T}\to R^{+}\) is then given by
We use the convention \(\widehat{\mathcal{G}}_{0}(t,s)=1\) for all \(s,t\in \mathbb{T}\).
Notice that
Corollary 2.5
For \(h>0\), \(\mathbb{T}=h\mathbb{Z}=\{hk: k\in \mathbb{Z}\}\), we have \(\rho ^{k}(s)=s-kh\). Then
and
where
For \(\mathbb{T}=q^{\mathbb{N}_{0}}\), we have \(\rho ^{k}(s)=sq^{-k}\). Then we write
Remark 2.6
Regarding the generalization of the power function, \(\widehat{\mathcal{G}}_{\alpha }(t,s)\) to real values of \(\alpha \geq 0\) (instead of integers n), we recall broadly accepted extension of its particular cases (2.5) and (2.7) in the form (see [37])
where \((p,\tilde{q})_{\infty }=\prod_{j=0}^{\infty }(1-p\tilde{q}^{j})\).
Remark 2.7
The integer iteration of the functions \(\sigma ^{n}\), \(\rho ^{n}\) follows the standard composition rule. The problem of noninteger iteration of the functions \(\sigma ^{\gamma }\), \(\rho ^{\gamma }\) for \(\gamma \in \mathbb{R}\) on discrete time scales has been discussed in [37]. In particular, for \(\mathbb{T}=hZ\), \(h>0\), we have
For \(\mathbb{T}=\overline{q^{\mathbb{Z}}}\), \(q>1\), we have
where \([\gamma ]_{q}=\frac{q^{\gamma }-1}{q-1}\).
We also have
which implies \(\sigma =\rho ^{-1}\) and \(\sigma ^{-1} =\rho \), respectively.
3 Conformable nabla derivative
Several variants of conformable (fractional) derivative have been defined on time scales (see [22, 28, 30]). In this section we give the definition of conformable nabla derivative (strictly following [20]) depending on the function \(\widehat{\mathcal{G}}_{1-\gamma }(t,a)\) (as defined in Sect. 2 above).
Throughout the paper, the operator \(\bigtriangledown _{a}^{\gamma }\) is referred to as the conformable fractional nabla derivative of order \(\gamma \in (0,1]\) on \(\mathbb{T}\) at a point \(t>a\in \mathbb{T}\).
Definition 3.1
Given a function \(f:\mathbb{T}\to \mathbb{R}\) and \(a\in \mathbb{T}\), f is \((\gamma ,a)\)-nabla differentiable at \(t > a\), if it is nabla differentiable at t, and its \((\gamma ,a)\)-nabla derivative is defined by
where the function \(\widehat{\mathcal{G}}_{1-\gamma }(t,a)\) as defined in (2.2).
If \(\bigtriangledown _{a}^{\gamma }[f(t)]\) exists in some interval \((a,a+\epsilon )_{\mathbb{T}}\), \(\epsilon >0\), then we define
if the limit \(\lim_{t\to a^{+}}\bigtriangledown _{a}^{\gamma }[f(t)]\) exists.
Moreover, we call f is \((\gamma ,a)\)-nabla differentiable on \(\mathbb{T}_{\kappa }\) (\(a\in \mathbb{T}_{\kappa }\)) provided \(\bigtriangledown _{a}^{\gamma }[f(t)]\) exists for all \(t\in \mathbb{T}_{\kappa }\). The function \(\bigtriangledown _{a}^{\gamma }:\mathbb{T}_{\kappa }\to \mathbb{R}\) is then called the \((\gamma ,a)\)-nabla derivative of f on \(\mathbb{T}_{\kappa }\).
Some useful properties concerning the \((\gamma ,a)\)-nabla derivative are given next. The proof of this theorem is similar to that in [23], hence omitted.
Theorem 3.2
If \(f:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable at \(t\in \mathbb{T}_{\kappa }\), where \(t>a\) and \(\gamma \in (0,1]\), then f is continuous at t.
Remark 3.3
For \(\mathbb{T}=\mathbb{R}\) and \(t>a\in \mathbb{R}\), then we have
which is the conformable fractional derivative defined in [5].
One the other hand, if \(t\in \mathbb{Z}\) and \(t>a\), then we have the \((\gamma ,a)\)-(backward) difference
a new definition in discrete setting.
Example 3.4
-
(i)
Let \(a\in \mathbb{T}\). If \(f:\mathbb{T}\to \mathbb{R}\) is defined by \(f(t)=\alpha \) for all \(t\in \mathbb{T}\), \(t>a\), where \(\alpha \in \mathbb{R}\) a constant, then we have \(\bigtriangledown _{a}^{\gamma }[\alpha ]=0\).
-
(ii)
Let \(a\in \mathbb{T}\). If \(f:\mathbb{T}\to \mathbb{R}\) is defined by \(f(t)=t-a\) for all \(t\in \mathbb{T}\), \(t>a\), then
$$ \bigtriangledown _{a}^{\gamma }[t-a]=\widehat{ \mathcal{G}}_{1- \gamma }(t,a)\quad \text{for all } t\in \mathbb{T}. $$ -
(iii)
If \(f:\mathbb{T}\to \mathbb{R}\) is defined by \(f(t)=t^{2}\) for all \(t\in [a,b]_{\mathbb{T}}\), then
$$ \bigtriangledown _{a}^{\gamma } \bigl[t^{2} \bigr]= \bigl( \rho (t)+t \bigr) \widehat{\mathcal{G}}_{1-\gamma }(t,a)\quad \text{for all } t \in [a,b]_{\mathbb{T}}. $$
Next, we provide the \((\gamma ,a)\)-nabla derivatives of sums, products, and quotients of \((\gamma ,a)\)-nabla differentiable functions. The proof is omitted since it is similar to that in [23].
Theorem 3.5
Assume \(f,g:\mathbb{T}\to \mathbb{R}\) are \((\gamma ,a)\)-nabla differentiable at \(t\in \mathbb{T}_{\kappa }\), \(t>a\). Then:
-
(i)
The sum \(f+g:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable at t with
$$ \bigtriangledown _{a}^{\gamma }(rf+sg) (t)=r\bigtriangledown _{a}^{ \gamma }f(t)+s\bigtriangledown _{a}^{\gamma }g(t). $$ -
(ii)
For any \(k\in \mathbb{R}\), \(kf:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable at t with
$$ \bigtriangledown _{a}^{\gamma }(kf) (t)=k\bigtriangledown _{a}^{ \gamma }f(t). $$ -
(iii)
The product \(fg:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable at t with
$$ \bigtriangledown _{a}^{\gamma }(fg) (t)= \bigl[\bigtriangledown _{a}^{ \gamma }f(t) \bigr]g(t)+f \bigl(\rho (t) \bigr) \bigtriangledown _{a}^{\gamma }g(t)=f(t) \bigtriangledown _{a}^{\gamma }g(t)+ \bigtriangledown _{a}^{\gamma }f(t) \bigl[g \bigl( \rho (t) \bigr) \bigr]. $$ -
(iv)
If \(g(t)g(\rho (t))\neq 0\), then \(\frac{f}{g}\) is \((\gamma ,a)\)-nabla differentiable at t with
$$ \bigtriangledown _{a}^{\gamma } \biggl(\frac{f}{g} \biggr) (t)=- \frac{[\bigtriangledown _{a}^{\gamma }f(t)]g(t)-f(t)\bigtriangledown _{a}^{\gamma }g(t)}{g(t)g(\rho (t))}. $$In the case \(f\equiv 1\), we have
$$ \bigtriangledown _{a}^{\gamma } \biggl(\frac{1}{g} \biggr) (t)=- \frac{\bigtriangledown _{a}^{\gamma }g(t)}{g(t)g(\rho (t))}. $$
Lemma 3.6
(Chain rule)
Let \(g\in C_{\mathrm{ld}}^{\nabla }(\mathbb{T})\) and assume that \(f:\mathbb{R}\to \mathbb{R}\) is continuously differentiable. Then \(f\circ g:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable and satisfies
Proof
The nabla version of the chain rule is given by
This can be obtained by following the proof of the chain rule in [34] and applying the ordinary substitution rule from calculus to find
Hence, we have
This completes the proof. □
Lemma 3.7
Let \(m\in \mathbb{N}\), \(a\in \mathbb{T}\) and suppose \(f:\mathbb{T}\to \mathbb{R}\) is \((\gamma ,a)\)-nabla differentiable at \(t\in \mathbb{T}\).
-
(i)
For f defined by \(f(t)=\widehat{\mathcal{G}}_{m}(t,s)\), \(s\in \mathbb{T}\), we have
$$ \bigtriangledown _{a}^{\gamma }f(t)=\widehat{\mathcal{G}}_{1- \gamma }(t,a) \sum_{k=0}^{m-1}\widehat{ \mathcal{G}}_{k} \bigl(\rho (t),s \bigr) \widehat{\mathcal{G}}_{m-1-k}(t,s). $$ -
(ii)
For g defined by \(g(t)=\widehat{\mathcal{G}}_{-m}(\sigma ^{m}(t),s)\), \(s\in \mathbb{T}\), we have
$$ \bigtriangledown _{a}^{\gamma }g(t)=-\widehat{ \mathcal{G}}_{1- \gamma }(t,a)\sum_{k=0}^{m-1} \frac{1}{\widehat{\mathcal{G}}_{m-k}(\sigma ^{k-1}(t),s)\widehat{\mathcal{G}}_{k+1}(\sigma ^{m-1-k}(t),s)}. $$
Proof
It is sufficient to prove the results for a discrete time scale \(\mathbb{T}\). We will prove by induction. If \(m=1\), then \(f(t)=\widehat{\mathcal{G}}_{1}(t,s)=t-s\) and, clearly, \(\bigtriangledown _{a}^{\gamma }(t)=\widehat{\mathcal{G}}_{1- \gamma }(t,a)\) holds by Example 3.4(ii). Now we assume that
holds for \(f(t)=\widehat{\mathcal{G}}_{m}(t,s)\). Let
By the product rule (Theorem 3.5(iii)), we obtain
Hence, by mathematical induction, part (i) holds.
Next, for \(g(t)=\widehat{\mathcal{G}}_{-m}(\sigma ^{m}(t),s)=\frac{1}{f(t)}\) we apply Theorem 3.5(iv) to obtain
provided \(\widehat{\mathcal{G}}_{m-k}(\sigma (t),s)\widehat{\mathcal{G}}_{k+1}( \sigma (t),s)\neq 0\). □
Example 3.8
Consider the time scale
We have \(\sigma (t)=\sqrt{t^{2}+1}\) and \(\rho (t)=\sqrt{t^{2}-1}\) for \(t\in \mathbb{T}\). Let \(\gamma =\frac{1}{2}\), \(a=s=0\in \mathbb{T}\).
-
(i)
Let \(f(t)=\widehat{\mathcal{G}}_{2}(t,0)=t^{(2)}\) for \(t\in \mathbb{T}\) (\(t>1\)), then we have
$$\begin{aligned} \bigtriangledown _{0}^{\frac{1}{2}} \bigl[t^{(2)} \bigr]&=t^{( \frac{1}{2})}\sum_{k=0}^{1} \widehat{\mathcal{G}}_{k} \bigl(\sqrt{t^{2}-1},0 \bigr) \widehat{\mathcal{G}}_{1-k}(t,0) \\ &=t^{(\frac{1}{2})} \bigl[t^{(1)}+ \bigl(\sqrt{t^{2}-1} \bigr)^{(1)} \bigr] \\ &= \bigl[t+\sqrt{t^{2}-1} \bigr]t^{(\frac{1}{2})}. \end{aligned}$$ -
(ii)
Let \(g(t)=\widehat{\mathcal{G}}_{-2}(\sigma ^{2}(t),0)=\frac{1}{t^{(2)}}\) for \(t\in \mathbb{T}\) (\(t>1\)), then
$$\begin{aligned} \bigtriangledown _{0}^{\frac{1}{2}} \biggl[\frac{1}{t^{(2)}} \biggr] &=-t^{( \frac{1}{2})}\sum_{k=0}^{1} \frac{1}{\widehat{\mathcal{G}}_{2-k}(\sqrt{t^{2}+k-1},0)\widehat{\mathcal{G}}_{k+1}(\sqrt{t^{2}+1-k},0)} \\ &=- \biggl[\frac{1}{(\sqrt{t^{2}-1})^{(2)}(\sqrt{t^{2}+1})}+ \frac{1}{ t^{(2)}(t)} \biggr]t^{(\frac{1}{2})}. \end{aligned}$$
4 Conformable ∇-integrals
In this section, we give the definition of conformable fractional ∇-integral of order \(\gamma \in (0,1]\) starting from \(a\in \mathbb{T}\) (or simply γ-nabla integral from a) on a time scale \(\mathbb{T}\).
Definition 4.1
Assume that \(\gamma \in (0,1]\), \(a, t_{1}, t_{2}\in \mathbb{T}\), \(a\leq t_{1}\leq t_{2}\) and \(f\in C_{\mathrm{ld}}(\mathbb{T})\), then we say that f is \((\gamma ,a)\)-nabla integrable on the interval \([t_{1}, t_{2}]_{\mathbb{T}}\) if the following integral:
exists and is finite.
Remark 4.2
If \(t_{1}\in \mathbb{T}\) and \(\sup \mathbb{T}=\infty \), and \(f\in C_{\mathrm{ld}}(\mathbb{T})\), then the improper nabla integral has the form
Example 4.3
For the case \(\mathbb{T}=\mathbb{R}\), we have the classical conformable integral as defined in [5], namely
For \(\mathbb{T}=h\mathbb{Z}\) \(h>0\), we have a new conformable fractional h-sum given by
For \(\mathbb{T}=q^{\mathbb{N}_{0}}\), we have a new conformable fractional q-sum given by
Example 4.4
Let \(\mathbb{T}=\mathbb{Z}\), \(a=0\) and \(\gamma =1/2\). Then
The proof of the following theorem is similar to that of Theorem 8 in [23], hence omitted.
Theorem 4.5
Let \(\gamma \in (0, 1]\) and \(a\in \mathbb{T}\). Then, for any ld-continuous function \(f:\mathbb{T}\to \mathbb{R}\), there exists a function \(F:\mathbb{T}\to \mathbb{R}\) such that
The function F is called an \((\gamma ,a)\)-nabla antiderivative of f.
Lemma 4.6
Assume \(a, t_{1}, t_{2}\in \mathbb{T}\), \(a\leq t_{1}\leq t_{2}\) and \(f\in C_{\mathrm{ld}}(\mathbb{T})\). Let the function \(F:\mathbb{T}\to \mathbb{R}\) be \((\gamma ,a)\)-nabla differentiable on \([t_{1}, t_{2}]\) such that
Then
Proof
Since \(f\in C_{\mathrm{ld}}(\mathbb{T})\), by Theorem 4.5, we have
□
Similar to the proof of Theorem 9 in [23] (see also [34]), one can prove the following theorem.
Theorem 4.7
Let \(\gamma \in (0,1]\). If \(a, t_{1}, t_{2}, t_{3}\in \mathbb{T}\), \(a\leq t_{1}\leq t_{2}\leq t_{3}\), \(\alpha \in \mathbb{R}\), and \(f,g\in C_{\mathrm{ld}}(\mathbb{T})\), then
-
(i)
\(\int _{t_{1}}^{t_{2}}[f(t)+ g(t)]\bigtriangledown _{a}^{\gamma }t= \int _{t_{1}}^{t_{2}}f(t)\bigtriangledown _{a}^{\gamma }t+\int _{t_{1}}^{t_{2}}g(t) \bigtriangledown _{a}^{\gamma }t\);
-
(ii)
\(\int _{t_{1}}^{t_{2}}\alpha f(t)\bigtriangledown _{a}^{\gamma }t= \alpha \int _{t_{1}}^{t_{2}} f(t)\bigtriangledown _{a}^{\gamma }t\);
-
(iii)
\(\int _{t_{1}}^{t_{2}} f(t)\bigtriangledown _{a}^{\gamma }t=-\int _{t_{2}}^{t_{1}} f(t)\bigtriangledown _{a}^{\gamma }t\);
-
(iii)
\(\int _{t_{1}}^{t_{3}}f(t)\bigtriangledown _{a}^{\gamma }t=\int _{t_{1}}^{t_{2}}f(t) \bigtriangledown _{a}^{\gamma }t+ \int _{t_{2}}^{t_{3}}f(t) \bigtriangledown _{a}^{\gamma }t\);
-
(iv)
\(\int _{t_{1}}^{t_{2}} [f(t)(\bigtriangledown _{a}^{\gamma }g)(t)] \bigtriangledown _{a}^{\gamma }t=[f(t)g(t)]_{t_{1}}^{t_{2}}-\int _{t_{1}}^{t_{2}} \bigtriangledown _{a}^{\gamma }(t) g(\rho (t))\bigtriangledown _{a}^{ \gamma }t\);
-
(v)
\(\int _{t_{1}}^{t_{1}} f(t)\bigtriangledown _{a}^{\gamma }t=0\);
-
(vi)
If \(\vert f(t) \vert \leq g(t)\) on \([t_{1},t_{2})\), then
$$ \biggl\vert \int _{t_{1}}^{t_{2}} f(t)\bigtriangledown _{a}^{\gamma }t \biggr\vert \leq \int _{t_{1}}^{t_{2}} g(t)\bigtriangledown _{a}^{\gamma }t. $$
Theorem 4.8
Assume \(\gamma \in (0,1]\) and \(a\in \mathbb{T}\). If \(f\in C_{\mathrm{ld}}(\mathbb{T})\) and \(t\in \mathbb{T}_{\kappa }\) with \(\rho (t)>a\), then
Proof
Since \(f\in C_{\mathrm{ld}}(\mathbb{T})\), by Lemma 4.6, we have
□
Theorem 4.9
Let \(\gamma \in (0,1]\), \(a,t\in \mathbb{T}\), \(t\geq a\) and \(f\in C_{\mathrm{ld}}(\mathbb{T})\). Then, for all \(t\in \mathbb{T}\), we have
Proof
Since \(f\in C_{\mathrm{ld}}(\mathbb{T})\) and \(\bigtriangledown _{a}^{-\gamma }f(t)\) is \((\gamma ,a)\)-nabla integrable, we have
□
Theorem 4.10
Let \(f:[a,b]_{\mathbb{T}}\to \mathbb{R}\) be \((\eta ,a)\)-differentiable for \(\eta \in (0,1]\). Then, for all \(t>a\), we have
Proof
Since f is differentiable, by Definition 4.1, we have
□
5 Conformable exponential function
In this section we define the conformable fractional nabla exponential function on time scales.
Definition 5.1
The function \(p:\mathbb{T}\to \mathbb{R}\) is said to be ν-regressive if
holds. Define the ν-regressive class of ld-functions on \(\mathbb{T}_{\kappa }\) to be
and define the set \(\mathcal{R}_{\nu }^{+}\) of all positive regressive elements of \(\mathcal{R}_{\nu }\) by
Definition 5.2
For \(p\in \mathcal{R}_{\nu }\), define circle minus p by
Definition 5.3
For \(h>0\), let
Define the ν-cylinder transformation \(\widehat{\xi }_{h}(z):\mathbb{C}_{h}\to \mathbb{Z}_{h}\) by
where Log is the principle logarithm function.
The inverse transform \(\widehat{\xi }^{-1}_{h}\) is then given by
The following lemma is an easy exercise.
Lemma 5.4
It holds that
Definition 5.5
Let \(\alpha \in (0,1]\), \(a,s,t\in \mathbb{T}\), \(a\leq s\leq t\) and \(p\in \mathcal{R}_{\nu }\). Then we define the (conformable) fractional nabla exponential function of order γ from a by
Remark 5.6
In the case \(s=a\), we write \(\widehat{e}^{\gamma }_{p}(t,a;a)=\widehat{e}^{\gamma }_{p}(t,a)\) and, if \(\gamma =1\), we have the usual nabla exponential function as defined in [35].
Example 5.7
Let \(\mathbb{T}=\mathbb{R}\) and \(\lambda \in \mathcal{R}_{\nu }\) a constant. Then
Notice that, if \(\gamma =1\), we have \(\widehat{e}_{\lambda }^{\gamma }(t,a)=\widehat{e}_{\lambda }(t,a)=e^{ \lambda (t-a)}\), which coincides with the classical exponential function.
Example 5.8
Let \(t,a\in \mathbb{T}=\mathbb{Z}\) and let \(\lambda \in \mathcal{R}_{\nu }\) be a constant. Then
Indeed, by (5.1) and (5.3), we have
Since \(\nabla t^{(\gamma )}=\gamma t^{(\gamma -1)}\), we obtain
It follows that
Note that, for \(\gamma =1\), we have
which coincides with the nabla exponential function given in [35].
Theorem 5.9
Let \(p,q\in \mathcal{R}_{\nu }\) and \(a,s,t\in \mathbb{T}\) (\(a< s\leq t\)). Then:
-
(i)
\(\widehat{e}_{0}^{\gamma }(t,a)=1\), \(\widehat{e}^{\gamma }_{p}(a,a)=1\);
-
(ii)
\(\widehat{e}^{\gamma }_{p}(\rho (t),a)= (1-\nu (t)p(t) )^{ \widehat{\mathcal{G}}_{\gamma -1}(\sigma ^{\gamma -1}(t),a)} \widehat{e}^{\gamma }_{p}(t,a)\);
-
(iii)
\(\frac{1}{\widehat{e}^{\gamma }_{p}(t,a)}=\widehat{e}^{\gamma }_{ \ominus _{\nu }p}(t,a)\);
-
(iv)
\(\widehat{e}^{\gamma }_{p}(t,a)= \frac{1}{\widehat{e}^{\gamma }_{p}(a,t)}=\widehat{e}^{\gamma }_{ \ominus _{\nu }p}(a,t)\);
-
(v)
\(\widehat{e}^{\gamma }_{p}(t,a)\widehat{e}^{\gamma }_{\ominus _{\nu }p}(s,a)= \widehat{e}^{\gamma }_{p}(t,s;a)\);
-
(vi)
\(\widehat{e}^{\gamma }_{p}(t,a)\widehat{e}_{q}^{\gamma }(t,a)= \widehat{e}_{p \oplus _{\nu }q}^{\gamma }(t,a)\);
-
(vii)
\(\frac{\widehat{e}^{\gamma }_{p}(t,a)}{\widehat{e}_{q}^{\gamma }(t,a)}= \widehat{e}_{p \ominus _{\nu }q}^{\gamma }(t,a)\).
Proof
(i) Clear by definition.
(ii) It is sufficient to prove the property for left-scattered points (i.e., \(\rho (t)< t\)),
By Theorem 4.8,
and substituting into (5.4), we get the required property.
(iii) Follows directly from Lemma 5.4. Indeed,
hence the property (iii).
(iv) The result follows directly from (5.3) and part (iii) of this theorem.
(v) By Lemma 5.4 and Definition 5.5, we have
(vi)
(vi) The result follows easily using parts (iii) and (v). □
Lemma 5.10
Let \(g\in C_{\mathrm{ld}}^{\nabla }(\mathbb{T})\), \(g(t)>0\) and \(\frac{g^{\nabla }(t)}{g(t)}\in \mathcal{R}_{\nu }\). Then
where Log is the principle logarithm function.
Proof
Let \(f(x)=\operatorname{Log} [x]\). Clearly, \(f:\mathbb{R}^{+}\to \mathbb{R}\) is continuous on \(\mathbb{R}^{+}\). By using Lemma 3.6, we have
□
6 Conformal dynamic equations and inequalities
In this section, we consider solutions to the conformable dynamic equations and prove several related inequalities.
Let \(f\in C_{\mathrm{ld}}([a,b]_{\mathbb{T}}\times \mathbb{R})\) and \(x\in C_{\mathrm{ld}}^{\nabla }([a,b]_{\mathbb{T}})\). Consider the following conformable fractional dynamic equation:
However, one can also consider the integral form of the equation:
The integral form is useful for proving the existence and uniqueness of solutions or for studying analytical properties of solutions.
Lemma 6.1
Let \(\gamma \in (0,1]\) and \(f:[a,b]_{\mathbb{T}}\times \mathbb{R}\to \mathbb{R}\). Function \(x\in C_{\mathrm{ld}}([a,b]_{\mathbb{T}})\) is a solution of problem (6.1) if and only if x is a solution of (6.2).
Proof
\((\Rightarrow )\) By Theorem 4.10, we have
\((\Leftarrow )\) By taking \((\gamma ,a)\)-nabla derivative of both sides of (6.2), we have
These completes the proof. □
Let \(\mathbb{X}\) be the Banach space of all \(x\in C_{\mathrm{ld}}([a,b]_{\mathbb{T}})\) with the norm
In the following theorem, we examine the solution to problem (6.1) on Banach space \(\mathbb{X}\).
Theorem 6.2
Let the function \(f:\mathbb{X}\times \mathbb{R}\to \mathbb{R}\) be Lipschitz continuous with Lipschitz constant \(L>0\). If
then problem (6.1) has a unique solution in \(\mathbb{X}\).
Proof
Define the operator \(T:\mathbb{X}\to \mathbb{X}\) by
We show that T has a fixed point, which is a unique solution of (6.1) on \([a,b]_{\mathbb{T}}\). For that, we show that T is a contraction mapping on \(\mathbb{X}\).
Now, for \(x, y\in \mathbb{X}\), we have
Since for \(\gamma \in (0,1)\), \(\gamma -1<0\), By Lemma 2.3 and Remark 2.6, we obtain that
holds for all \(t\in [a,b]_{\mathbb{T}}\).
It follows that
Thus, for \(\frac{L(t-a)^{\gamma }}{\gamma } <1\), T is a contraction mapping on \(\mathbb{X}\). Hence, by Banach fixed point theorem, T has a unique fixed point x in \(\mathbb{X}\). This completes the proof. □
Considering the existence and uniqueness of the solution of the equation (6.1) above, by letting \(f(t,y(t))=\Psi _{\nu }^{\gamma }(p(t)y(t))\), we are able to prove the following theorems.
Theorem 6.3
For \(\gamma \in (0,1]\), \(a\in \mathbb{T}\) and \(\Psi _{\nu }^{\gamma }(p)\in \mathcal{R}_{\nu }^{+}\), the exponential function \(\widehat{e}^{\gamma }_{p}(t,a)\), defined by (5.3), is a unique solution of the following Cauchy problem:
where
Proof
It is sufficient to prove the property for left-scattered points (i.e., \(\rho (t)< t\)). By the definition of conformable nabla derivative and part (ii) of Theorem 5.9, we have
where
and
□
Remark 6.4
Note that, if \(\gamma =1\), we obtain the usual ∇-derivative of the exponential function \(\widehat{e}^{\gamma }_{p}(t,a)\), i.e.,
For \(\mathbb{T}=\mathbb{R}\), we have \(\nu (t)=0\) and \(\Psi _{0}^{\gamma }(p(t))=p(t)\). Hence
The following lemma is useful.
Lemma 6.5
Let \(\Psi _{\nu }^{\gamma }(p(t))\) be defined as in Theorem 6.3above. Then
Proof
□
Theorem 6.6
Suppose \(p\in \mathcal{R}_{\nu }^{+}\). Let \(\gamma \in (0,1]\), \(a\in \mathbb{T}\) and \(y_{0}\in \mathbb{R}\). The unique solution of the initial value problem
is given by
Proof
Using Theorem 6.3 and Lemma 6.5, we get
Since \(y(a)=\widehat{e}^{\gamma }_{\ominus _{\nu }p}(a,a)y_{0}=y_{0}\), we have the desired result. □
Theorem 6.7
If \(p\in \mathcal{R}_{\nu }^{+}\) and \(a,b,c\in \mathbb{T}\) are such that \(a< b\leq c\), then
and
Proof
We use Lemma 6.5 and the properties in Theorem 5.9 to find
It follows that
which proves the desired identity. □
For the introduction to the Gronwall’s inequality, we refer to [34, 38]. The following dynamic inequality is useful in proving the Gronwall’s inequality in this new setting.
Theorem 6.8
Let \(y, f\in C_{\mathrm{ld}}(\mathbb{T})\), \(p\in \mathcal{R}_{\nu }^{+}\) and \(a\in \mathbb{T}\). Then
implies
Proof
Using the product rule in Theorem 3.5 and Lemma 6.5, we get
Note that, from Theorem 5.9(ii), we have
which implies that
It follows that
Now, taking \((\eta ,a)\)-integral on both sides of the latter inequality yields
Using parts (iii) and (v) of Theorem 5.9 yields
This completes the proof. □
Next, we prove Gronwall’s inequality, which will be useful in establishing the stability of conformable dynamical systems on time scales.
Theorem 6.9
(Gronwall’s inequality)
Let \(y, f\in C_{\mathrm{ld}}(\mathbb{T})\), \(p\in \mathcal{R}_{\nu }^{+}\), and \(a\in \mathbb{T}\). Then
implies
Proof
Define \(g(t)=\int _{a}^{t} \Psi _{\nu }^{\gamma }(p(\tau ))y(\tau ) \bigtriangledown _{a}^{\gamma }\tau \). Then \(g(a)=0\) and
By Theorem 6.8,
Since \(y(t)\leq f(t)+g(t)\), the claim follows. □
If we take \(\mathbb{T}=h\mathbb{N}_{0}\) and \(a=0\) in Gronwall’s inequality, we obtain the following example.
Example 6.10
Let \(\mathbb{T}=h\mathbb{Z}\cap [0,\infty )\). If \(y,f:\mathbb{T}\to \mathbb{R}\) and \(\eta >0\) is a constant such that
then
Corollary 6.11
Let \(y\in C_{\mathrm{ld}}(\mathbb{T})\), \(p\in \mathcal{R}_{\nu }^{+}\), \(\lambda \in \mathbb{R}\), \(a\in \mathbb{T}\) and \(y\geq 0\). Then
implies
Proof
By letting \(f(t)=\lambda \) in Theorem 6.9, we have
where we use Theorem 6.7 and \(\widehat{e}^{\gamma }_{p}(t,a;a)=\widehat{e}^{\gamma }_{p}(t,a)\). This completes the proof. □
Finally, we present a new version of Gronwall’s inequality in this new setting (see [38]). The proof of the following theorem is similar to that in [38].
Theorem 6.12
Let \(y\in C_{\mathrm{ld}}(\mathbb{T})\), \(-p(t)\in \mathcal{R}_{\nu }^{+}\), \(\lambda \in \mathbb{R}\), \(a\in \mathbb{T}\), and \(y\geq 0\). Then
implies
Proof
The proof is given in two cases.
Case I. For \(t\in [a,\infty )_{\mathbb{T}}\), we have
Then, by Corollary 6.11, we have
Case II. For \(t\in (-\infty , a]_{\mathbb{T}}\), let
For any \(s\in [t,a]_{\mathbb{T}}\), we have
Since, \(y\geq 0\), \(p\geq 0\), \(\lambda >0\), we have \(\lambda -z(s)>0\). It follows that
Multiplying by \(-\Psi _{\nu }^{\gamma }(p(s))\) both sides of the above inequality, it follows that
Notice that
It follows that
Since \(-p\in \mathcal{R}_{\nu }^{+}\), we have
Integrating the above inequality over the interval \([t,a]_{\mathbb{T}}\) and using Lemma 5.10 leads to
Therefore,
This completes the proof. □
7 Concluding remarks
In this paper we propose a new conformable nabla derivative and integral on arbitrary time scales which generalize the conformable fractional derivative and integral introduced in [5]. With this new definition of conformable derivative, we are able to define the conformable exponential function which is the solution to the linear conformable dynamic equation on time scales. Several useful results pertaining this exponential function are obtained. As an application, in the last section of this paper, we study the conformable dynamic equations and inequalities. We prove the Gronwall’s inequality which is useful in establishing the stability of nabla conformable dynamical systems on time scales.
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Acknowledgements
We thank Professor Sayed Khalil Mohamed Marzuki Elagan for his support and advise provided. The authors also would like to thank the reviewers and the editors for their valuable comments and suggestions. The authors would like to acknowledge the grant from Ministry of Higher Education, Malaysia (grant FRGS/1/2019/STG06/UKM/01/3) for financial support.
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The main idea of this paper was proposed by SR and MN. SR prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Segi Rahmat, M.R., Noorani, M.S.M. A new conformable nabla derivative and its application on arbitrary time scales. Adv Differ Equ 2021, 238 (2021). https://doi.org/10.1186/s13662-021-03385-x
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DOI: https://doi.org/10.1186/s13662-021-03385-x