Applications of Structural Nabla Derivatives on Time Scales to Dynamic Equations

Abstract

We present here more general concepts of Hausdorff derivatives (structural Nabla derivatives) on a timescale. We examine structural Nabla integration on temporal scales. Using the fixed-point theorem, we establish adequate criteria for the question of existence and uniqueness of the solution to an initial value problem characterized by structural Nabla derivatives on timescales. Furthermore, some features of the new operator are proven and illustrated by using concrete examples.

1. Introduction, Basics and Preliminaries

Fractal calculus and fractional calculus have recently become the most popular topics in both mathematics and engineering for non-differential solutions [1,2,3,4,5,6,7,8,9,10]. As a result, the use of the fractional derivative has seen significant progress and attention in many disciplines of science [6,7,8,9]. Dynamic equations on timescales have been successfully used to unify differential and difference equations [11,12,13,14]. Recently, several results have presented the generalization of fraction derivatives on timescales. Bastos et al. [12] defined a new definition of fractional-order derivative on timescales. In [15], a novel technique and very interesting results with more generalization were obtained (please see [16,17]). In [17], a new useful definition for local fractional derivatives was given, called the conformal fractional derivative, which was subsequently developed by many researchers (see [18,19,20,21]).

The Hausdorff derivative, like the Tsallis derivative and the Jackson derivative, can be considered as a generalization of the well-known Leibniz derivative. Note that the Hausdorff derivative (structural Nabla derivatives) has been associated with non-extensivity in systems presenting fractal aspects. The fractal derivative (Hausdorff derivative) on fractal time is defined by

$$\frac{\partial h}{{\partial }^{\lambda }t}=\underset{t\to s}{lim}\frac{h\left(t\right)-h\left(s\right)}{t^{\lambda }-s^{\lambda }},$$

where $\lambda$ is the fractal dimensions of time. A more general definition is given, as in [6,7,8,9], as follows:

$$\frac{{\partial }^{\delta }h}{{\partial }^{\lambda }t}=\underset{t\to s}{lim}\frac{h^{\delta }\left(t\right)-h^{\delta }\left(s\right)}{t^{\lambda }-s^{\delta }},$$

where $\delta$ represents the fractal dimensions of space.

This article is organized as follows. We define the structural Nabla derivatives in Definition 4 and structural Nabla integration on timescales in Definition 5, and we study their important properties. This is the first step in this direction. Next, we provide the initial value problem and establish certain minimal requirements to obtain the unique solution for structural Nabla derivative initial value problems on an arbitrary timescale (see Theorems 9 and 10). We end our studies by introducing illustrative examples.

We now state and recall certain basics, tools, and results, which will be helpful subsequently. For this end, let $\mathbb{T}$ be a closed set in $\mathbb{R}$, representing a timescale. We establish the forward- and backward-leap operators,

$$\upsilon ,\tau :\mathbb{T}\to \mathbb{T},$$

by

$$\upsilon \left(t\right)=inf\left(\left]t,+\infty \right[\cap \mathbb{T}\right)\ge t,$$

and

$$\tau \left(t\right)=sup\left(\left]-\infty ,t\right[\cap \mathbb{T}\right)\le t,$$

respectively. To make it simpler, let us denote

$${\mathcal{R}}_s=\left\{t\in \mathbb{T}:\upsilon \left(t\right)>t\right\},$$

and

$${\mathcal{R}}_s=\left\{t\in \mathbb{T}:\upsilon \left(t\right)>t\right\},$$

and

$${\mathcal{L}}_s=\left\{t\in \mathbb{T}:\tau \left(t\right)<t\right\}.$$

and

$${\mathcal{R}}_d=\left\{t\in \mathbb{T}:\upsilon \left(t\right)=t<sup\mathbb{T}\right\},$$

and

$${\mathcal{L}}_d=\left\{t\in \mathbb{T}:\tau \left(t\right)=t<inf\mathbb{T}\right\}.$$

Definition 1.

Let $t\in \mathbb{T}$, as we say the following:

• If $t\in {\mathcal{R}}_s$ (or $t\in {\mathcal{L}}_s$), then t is right-scattered (or left-scattered) and isolated if it is both;
• If $t\in {\mathcal{R}}_d$ (or $t\in {\mathcal{L}}_d$), then t is right-dense (or left-dense);
• It $t\in {\mathcal{R}}_d\cap {\mathcal{L}}_d$, then t is dense.

The graininess and backward-graininess functions $\nu ,\mu :\mathbb{T}\to {\mathbb{R}}^{+}$ can be expressed as $\mu \left(t\right)=\upsilon \left(t\right)-t$ and $\nu \left(t\right)=t-\tau \left(t\right)$. We note

$${\mathbb{T}}^{\kappa }=\overline{\mathbb{T}\setminus \left\{\tau \left(sup\mathbb{T}\right)\right\}}\mathrm{and}{\mathbb{T}}_{\kappa }=\overline{\mathbb{T}\setminus \left\{\upsilon \left(inf\mathbb{T}\right)\right\}}$$

If $\mathbb{T}$ is bounded, then we have ${\mathbb{T}}_0\subseteq {\mathbb{T}}_{\kappa }$, with ${\mathbb{T}}_0=\mathbb{T}\setminus \{min\mathbb{T}\}$. For $\alpha ,\beta \in \mathbb{T}$, we construct the closed interval,

$${[\alpha ,\beta ]}_{\mathbb{T}}=\left\{t\in \mathbb{T}:t\in [\alpha ,\beta ]\right\}.$$

Definition 2

([13]). A function $h:\mathbb{T}\to \mathbb{R}$ is called $ld$-continuous if it is continuous at left-dense points in $\mathbb{T}$ and has right-sided limits at right-dense points in $\mathbb{T}$; it is denoted as $h\in {\mathcal{C}}_{ld}(\mathbb{T},\mathbb{R})$.

Definition 3

([14]). Let $h:\mathbb{T}\to \mathbb{R}$ and $t\in {\mathbb{T}}_{\kappa }$; the ∇-derivative of h at t, expressed as $h^{\nabla }\left(t\right)$, is given to the number (provided that it exists) with the following criterion: if any $\epsilon >0$ there exists a neighborhood $\mathcal{U}$ of t where

$$\left|h^{\tau }\left(t\right)-h\left(s\right)-h^{\nabla }\left(t\right)\left(\tau \left(t\right)-s\right)\right|\le \epsilon \left|\tau \left(t\right)-s\right|,\forall s\in \mathcal{U}.$$

We claim that h is considered ∇-differentiable if $h^{\nabla }\left(t\right)$ exists $\forall t\in {\mathbb{T}}_{\kappa }$. The function $h^{\nabla }:\mathbb{T}\to \mathbb{R}$ is referred to as the ∇-derivative of h on ${\mathbb{T}}_{\kappa }$. We note

$${\mathcal{C}}_{ld}^1\left(\mathbb{T},\mathbb{R}\right)=\left\{h:\mathbb{T}\to \mathbb{R}:h^{\nabla }\in {\mathcal{C}}_{ld}\left(\mathbb{T},\mathbb{R}\right)\right\}.$$

Definition 4

(The timescale structural derivative [1]). Suppose that $t\in {\mathbb{T}}^{\kappa }$, $h,p:\mathbb{T}\to \mathbb{R}$, and $\varrho >0$. The number $h^{{\Delta }_p^{\varrho }}\left(t\right)$ is defined as follows, granted that it exists: for any $\epsilon >0$, there exists a neighborhood $\mathcal{U}$ of t in which

$$\left|[h^{\varrho }\left(\upsilon \left(t\right)\right)-h^{\varrho }\left(s\right)]-h^{{\Delta }_p^{\varrho }}\left(t\right)[p\left(\upsilon \left(t\right)\right)-p\left(s\right)]\right|\le \epsilon \left|p\left(\upsilon \right(t\left)\right)-p\left(s\right)\right|,\forall s\in \mathcal{U}.$$

The structural derivative of h at t (in relation to ϱ and $p(·)$) is signified with the expression $h^{{\Delta }_p^{\varrho }}\left(t\right)$. Moreover, $h^{{\Delta }_p^{\varrho }}\left(t\right)$) develops for any $t\in {\mathbb{T}}^{\kappa }$, and we argue that h is structurally differentiable on ${\mathbb{T}}^{\kappa }$ (or ${\Delta }_p^{\varrho }$–differentiable).

For instance, in practical applications to handle coarse-grained fractal spaces (fractal porosity), which are discontinuous in compact Euclidean space, and models of continuous hydrodynamic flows (see [18]), we give an example specifying that structural derivatives on timescales make sense. In particular, we treat a function as structurally differentiable on a broad timescale, even when it is not classically differentiable. It is crucial to be able to differentiate between non-smooth functions.

Example 1.

Assume that $p,f:\mathbb{T}\to \mathbb{R}$, $\lambda >0$ and $t_0$ is left-dense in $\mathbb{T}$. If $f\left(t\right)={\left(t-t_0\right)}^{\alpha }$, $p\left(t\right)={\left(t-t_0\right)}^{\beta }$, where $\frac{\beta }{\lambda }\le \alpha <1$, then

$$f^{{\nabla }_p^{\lambda }}\left(t_0\right)=\underset{s\to t_0}{lim}{\left(s-t_0\right)}^{\lambda \alpha -\beta }=\left\{\begin{array}{cc}0,& \mathrm{if}\alpha \lambda >\beta ,\\ 1,& \mathrm{if}\alpha \lambda >\beta .\end{array}\right.$$

But, $f^{\nabla }\left(t_0\right)={lim}_{s\to t_0}{\left(s-t_0\right)}^{\alpha -1}$ does not exist.

Theorem 1

([1]). Let $h,p:\mathbb{T}\to \mathbb{R}$, $t\in {\mathbb{T}}^{\kappa }$ and $\varrho >0$. Then, the resulting characteristics apply:

(1) 

If h is continuous at $t\in {\mathcal{R}}_s$, then h is ${\Delta }_p^{\varrho }$-differentiable at t with

$$h^{{\Delta }_p^{\varrho }}\left(t\right)=\frac{h^{\varrho }\left(\upsilon \left(t\right)\right)-h^{\varrho }\left(t\right)}{p\left(\upsilon \right(t\left)\right)-p\left(t\right)}.$$

(2) 

If $t\in {\mathcal{R}}_d$, then h is structural differentiable at t if and only if the limit

$$\underset{s\to t}{lim}\frac{h^{\varrho }\left(t\right)-h^{\varrho }\left(s\right)}{p\left(t\right)-p\left(s\right)},$$

exists and is finite. Thus,

$$h^{{\Delta }_p^{\varrho }}\left(t\right)=\underset{s\to t}{lim}\frac{h^{\varrho }\left(t\right)-h^{\varrho }\left(s\right)}{p\left(t\right)-p\left(s\right)}.$$

(3) 

If h is structural differentiable at t then

$$h^{\varrho }\left(\upsilon \left(t\right)\right)=h^{\varrho }\left(t\right)+(p\left(\upsilon \left(t\right)\right)-p\left(t\right))h^{{\Delta }_p^{\varrho }}\left(t\right).$$

Lemma 1.

Let $a,b\in \left[0,+\infty \right)$ and $\lambda \in \left(0,1\right];$ we have the following inequality:

$$\left|x^{\lambda }-y^{\lambda }\right|\le {\left|x-y\right|}^{\lambda }.$$

(1)

Lemma 2.

Let E be a Banach space and $A:E\to E$ an application admitting to the fixed point, such that there exists $r>0,$ $\alpha >1$ and $\kappa >0$, with

$$∥Ax∥<r,\forall x\in E,$$

(2)

$$∥Ax-Ay∥\le \kappa {∥x-y∥}^{\alpha },\forall x,y\in E.$$

(3)

If ${\kappa }^{1-\alpha }>2r$ then the application A has a unique fixed point.

Proof. 

Suppose that application A has two fixed points $x_1$ and $x_2$, such that $x_1\ne x_2$; then, $A{x}_1=x_1$ and $A{x}_2=x_2$. By inequality (2), we deduce that $∥x_1∥<r$ and $∥x_2∥<r$. From inequality (3), we have ${\kappa }^{1-\alpha }\le 2r,$ which contradicts.

This completes the proof. □

2. Timescales and Structural Nabla Derivatives

We begin by introducing the notion of structural Nabla derivatives on timescales (for a function given on an arbitrary timescale $\mathbb{T}$).

Definition 5

(The structural Nabla derivative). Assume $h,p:\mathbb{T}\to \mathbb{R},$ $t\in {\mathbb{T}}_{\kappa },\varrho >0$. We clarify $h^{{\nabla }_p^{\varrho }}\left(t\right)\in \mathbb{R}$, provided it exists, with the attribute that provided $\forall \epsilon >0$; there is a neighborhood $\mathcal{U}$ of t, such that

$$\left|[h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)]-h^{{\nabla }_p^{\varrho }}\left(t\right)[p\left(\tau \left(t\right)\right)-p\left(s\right)]\right|\le \epsilon \left|p\left(\tau \right(t\left)\right)-p\left(s\right)\right|,\forall s\in \mathcal{U}.$$

(4)

The $h^{{\nabla }_p^{\varrho }}\left(t\right)$ is called the structural Nabla derivative of h at t (associated with ϱ and $p(.)$). On the other hand, h is said to be the structural Nabla differentiable on ${\mathbb{T}}_{\kappa }$, provided $h^{{\nabla }_p^{\varrho }}\left(t\right)$ exists $\forall t\in {\mathbb{T}}_{\kappa }$.

Theorem 2.

Assume $h,p:\mathbb{T}\to \mathbb{R}$, $t\in {\mathbb{T}}_{\kappa }$ and $\varrho >0$. Then, h is ${\nabla }_p^{\varrho }$-differentiable if and only if the limit

$$\underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)},$$

exists and is finite. Thus,

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=\underset{s\to t,s\in \mathbb{T}}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}.$$

(5)

Proof. 

Assuming that h is ${\nabla }_p^{\varrho }$-differentiable at t, we have that, for any $\epsilon >0$, there exists a neighborhood ${\mathcal{U}}_1$ of t, such as

$$\left|\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}-h^{{\nabla }_p^{\varrho }}\left(t\right)\right|<\frac{\epsilon }{2},\forall s\in {\mathcal{U}}_1.$$

Setting

$$L_t=\underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)},$$

by (5), then there is a neighborhood ${\mathcal{U}}_2$ of t, such as

$$\left|\frac{h^{\varrho }\left(\upsilon \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \left(t\right)\right)-p\left(s\right)}-L_t\right|<\frac{\epsilon }{2},\forall s\in {\mathcal{U}}_2,$$

by the last inequality, we obtain $\left|h^{{\nabla }_p^{\varrho }}\left(t\right)-L_t\right|<\epsilon$, for $t\in {\mathcal{U}}_1\cap {\mathcal{U}}_2$, and when $\epsilon$ tends to 0 we have $h^{{\nabla }_p^{\varrho }}\left(t\right)=L_t.$ The proof is now completed. □

Corollary 1.

Let $h,p:\mathbb{T}\to \mathbb{R}$, $t\in {\mathbb{T}}_{\kappa },\varrho >0$ be assumed. If $h^{\varrho }$, p are ∇-differentiable at t, then h is ${\nabla }_p^{\varrho }$-differentiable at t and

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=\frac{{\left(h^{\varrho }\right)}^{\nabla }\left(t\right)}{p^{\nabla }\left(t\right)}.$$

(6)

Proof. 

We have, by Theorem 2,

$$\begin{array}{ccc}\hfill \underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}& =& \underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{\tau \left(t\right)-s}\frac{\tau \left(t\right)-s}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{\tau \left(t\right)-s}\underset{s\to t}{lim}\frac{p\left(\tau \right(t\left)\right)-p\left(s\right)}{\tau \left(t\right)-s}\hfill \\ & =& \frac{{\left(h^{\varrho }\right)}^{\nabla }\left(t\right)}{p^{\nabla }\left(t\right)}.\hfill \end{array}$$

The proof is now completed. □

Theorem 3.

Let $h,p:\mathbb{T}\to \mathbb{R}$, $t\in {\mathbb{T}}_{\kappa },\varrho >0$. Then, the characteristics enumerated above are valid:

1. 

If $h,$ p are continuous at t and $t\in {\mathcal{L}}_s$ then h is ${\nabla }_p^{\varrho }$-differentiable at t and

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=\frac{{\left(h^{\varrho }\right)}^{\nabla }\left(t\right)}{p^{\nabla }\left(t\right)}=\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(t\right)}{p\left(\tau \right(t\left)\right)-p\left(t\right)}.$$

(7)

2. 

If h is a structural Nabla differentiable at t then

$$h^{\varrho }\left(\tau \left(t\right)\right)=h^{\varrho }\left(t\right)+(p\left(\tau \left(t\right)\right)-p\left(t\right))h^{{\nabla }_p^{\varrho }}\left(t\right),$$

and if p is ∇-differentiable at t then

$$h^{\varrho }\left(\tau \left(t\right)\right)=h^{\varrho }\left(t\right)-(\nu \left(t\right)p^{\nabla }\left(t\right)h^{{\nabla }_p^{\varrho }}\left(t\right).$$

Proof. 

$\left(1\right)$ Assuming $h,$ p are continuous at $t\in {\mathcal{L}}_s$, then $h,$ p are ∇-differentiable at t. By Theorem 2, we find that h is ${\nabla }_p^{\varrho }$-differentiable at t and is given by (7)

$\left(2\right)$ If $\tau \left(t\right)=t$, then $p\left(\tau \right(t\left)\right)-p\left(t\right)=0$ and

$$h^{\varrho }\left(\tau \left(t\right)\right)=h^{\varrho }\left(t\right)=h^{\varrho }\left(t\right)+(p\left(\tau \left(t\right)\right)-p\left(t\right))h^{{\nabla }_p^{\varrho }}\left(t\right).$$

If $t\in {\mathcal{L}}_s$, by 2, we have

$$\begin{array}{ccc}\hfill h^{\varrho }\left(\tau \left(t\right)\right)& =& h^{\varrho }\left(t\right)+(p\left(\tau \left(t\right)\right)-p\left(t\right))\frac{h^{\varrho }\tau \left(t\right))-h^{\varrho }\left(t\right)}{p\left(\tau \right(t\left)\right)-p\left(t\right)}\hfill \\ & =& h^{\varrho }\left(t\right)+(p\left(\tau \left(t\right)\right)-p\left(t\right))h^{{\nabla }_p^{\varrho }}\left(t\right).\hfill \end{array}$$

This completes the proof. □

Theorem 4.

Let $h,g:\mathbb{T}\to \mathbb{R}$ be continuous and a structural Nabla differentiable at $t\in {\mathbb{T}}^{\kappa }$. Then, we have

1. 

For all $\lambda \in \mathbb{R}$, the function $\lambda h:\mathbb{T}\to \mathbb{R}$ is a structural differentiable at t with

$${\left(\lambda h\right)}^{{\nabla }_p^{\varrho }}\left(t\right)={\lambda }^{\varrho }{h}^{{\nabla }_p^{\varrho }}\left(t\right).$$

2. 

The product function $hg:\mathbb{T}\to \mathbb{R}$ is a structural differentiable at t where

$$\begin{array}{cc}\hfill {\left(hg\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =h^{{\nabla }_p^{\varrho }}\left(t\right)g^{\varrho }\left(t\right)+h^{\varrho }\left(\upsilon \left(t\right)\right)g^{{\nabla }_p^{\varrho }}\left(t\right)\hfill \\ \hfill & =h^{{\nabla }_p^{\varrho }}\left(t\right)g^{\varrho }\left(\upsilon \left(t\right)\right)+h^{\varrho }\left(t\right)g^{{\nabla }_p^{\varrho }}\left(t\right).\hfill \end{array}$$

3. 

If $h\left(t\right)h\left(\tau \right(t\left)\right)\ne 0$ then $\frac{1}{h}$ is a structural differentiable at t where

$${\left(\frac{1}{h}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)=\frac{-h^{{\nabla }_p^{\varrho }}\left(t\right)}{h^{\varrho }\left(\upsilon \left(t\right)\right)h^{\varrho }\left(t\right)}.$$

4. 

If $g\left(t\right)g\left(\tau \right(t\left)\right)\ne 0$ then $\frac{h}{g}$ is a structural differentiable at t with

$${\left(\frac{h}{g}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)=\frac{h^{{\nabla }_p^{\varrho }}\left(t\right)g^{\varrho }\left(t\right)-h^{\varrho }\left(t\right)g^{{\nabla }_p^{\varrho }}\left(t\right)}{g^{\varrho }\left(\upsilon \left(t\right)\right)g^{\varrho }\left(t\right)}.$$

Proof. 

$\left(1\right)$ From Theorem 2, we have

$$\begin{array}{ccc}\hfill {\left(\lambda h\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{{\left(\lambda h\right)}^{\varrho }\left(\tau \left(t\right)\right)-{\left(\lambda h\right)}^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& {\lambda }^{\varrho }\underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& {\lambda }^{\varrho }{h}^{{\nabla }_p^{\varrho }}\left(t\right).\hfill \end{array}$$

$\left(2\right)$ Applying equality (5) of Theorem 2, we have

$$\begin{array}{ccc}\hfill {\left(hg\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{{\left(hg\right)}^{\varrho }\left(\tau \left(t\right)\right)-{\left(hg\right)}^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to t}{lim}{g}^{\varrho }\left(\tau \left(t\right)\right)\frac{\left[h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}+h^{\varrho }\left(s\right)\frac{\left[g^{\varrho }\left(\tau \left(t\right)\right)-g^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}.\hfill \end{array}$$

If h, g are ${\nabla }_p^{\varrho }$-differentiable at t then by equality (5) of Theorem 2, we obtain

$$\begin{array}{ccc}\hfill {\left(hg\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& g^{\varrho }\left(\tau \left(t\right)\right)\underset{s\to t}{lim}\frac{\left[h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}+\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)\frac{\left[g^{\varrho }\left(\tau \left(t\right)\right)-g^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& g^{\varrho }\left(\tau \left(t\right)\right)h^{{\nabla }_p^{\varrho }}\left(t\right)+\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)\frac{\left[g^{\varrho }\left(\tau \left(t\right)\right)-g^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}.\hfill \end{array}$$

As h is continuous, then

$$\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)=h^{\varrho }\left(t\right),$$

and we have

$$\begin{array}{ccc}\hfill {\left(hg\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& g^{\varrho }\left(\tau \left(t\right)\right)h^{{\nabla }_p^{\varrho }}\left(t\right)+\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)\underset{s\to t}{lim}\frac{\left[g^{\varrho }\left(\tau \left(t\right)\right)-g^{\varrho }\left(s\right)\right]}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& g^{\varrho }\left(\tau \left(t\right)\right)h^{{\nabla }_p^{\varrho }}\left(t\right)+h^{\varrho }\left(t\right)g^{{\nabla }_p^{\varrho }}\left(t\right).\hfill \end{array}$$

$\left(3\right)$ By equality (5) of Theorem 2, we obtain

$$\begin{array}{ccc}\hfill {\left(\frac{1}{h}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{{\left(\frac{1}{h}\right)}^{\varrho }\left(\tau \left(t\right)\right)-{\left(\frac{1}{h}\right)}^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to t}{lim}\frac{-1}{h^{\varrho }\left(\tau \left(t\right)\right)h^{\varrho }\left(s\right)}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)},\hfill \end{array}$$

by the hypothesis, $h^{\varrho }\left(\tau \left(t\right)\right)h^{\varrho }\left(t\right)\ne 0,$ then

$$\underset{s\to t}{lim}\frac{-1}{h^{\varrho }\left(\tau \left(t\right)\right)h^{\varrho }\left(s\right)},$$

exists and equals $\frac{-1}{h^{\varrho }\left(\tau \left(t\right)\right)h^{\varrho }\left(t\right)}$. In view of h being ${\nabla }_p^{\varrho }$-differentiable at t, then

$$\begin{array}{ccc}\hfill {\left(\frac{1}{h}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{-1}{h^{\varrho }\left(\tau \left(t\right)\right)h^{\varrho }\left(s\right)}\underset{s\to t}{lim}\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \frac{-h^{{\nabla }_p^{\varrho }}\left(t\right)}{h^{\varrho }\left(\upsilon \left(t\right)\right)h^{\varrho }\left(t\right)}.\hfill \end{array}$$

$\left(4\right)$ We calculate a quotient formula using 2 and 3:

$$\begin{array}{ccc}\hfill {\left(\frac{h}{g}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& {\left(h·\frac{1}{g}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)\hfill \\ & =& h^{\varrho }\left(t\right){\left(\frac{1}{g}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)+h^{{\nabla }_p^{\varrho }}\left(t\right)\frac{1}{g^{\varrho }\left(\tau \left(t\right)\right)}\hfill \\ & =& -h^{\varrho }\left(t\right)\frac{g^{{\nabla }_p^{\varrho }}\left(t\right)}{g^{\varrho }\left(\tau \left(t\right)\right)g^{\varrho }\left(t\right)}+h^{{\nabla }_p^{\varrho }}\left(t\right)\frac{1}{g^{\varrho }\left(\tau \left(t\right)\right)}\hfill \\ & =& \frac{h^{{\nabla }_p^{\varrho }}\left(t\right)g^{\varrho }\left(t\right)-h^{\varrho }\left(t\right)g^{{\nabla }_p^{\varrho }}\left(t\right)}{g^{\varrho }\left(\tau \left(t\right)\right)g^{\varrho }\left(t\right)}.\hfill \end{array}$$

The proof is now completed. □

Example 2.

Let

$$\mathbb{T}:=\left\{1-\frac{1}{n}:n\in \mathbb{N}\right\}\cup \left\{1\right\},$$

and $f,p:\mathbb{T}\to \mathbb{R}$ are functions given by

$$h\left(t\right)=\left\{\begin{array}{c}{t}^2,\mathrm{if}t\ne 1\\ 4,\mathrm{if}t=1\end{array}\right.,p\left(t\right)=\left\{\begin{array}{c}\sqrt{t},\mathrm{if}t\ne 1\\ \sqrt{2},\mathrm{if}t=1\end{array}\right.,\forall t\in \mathbb{T}.$$

So, h is ${\nabla }_p^{1/2}$-differentiable at 1 and $h^{{\nabla }_p^{1/2}}\left(1\right)$ is given by

$$\begin{array}{ccc}\hfill h^{{\nabla }_p^{1/2}}\left(1\right)& =& \underset{s\to 1}{lim}\frac{\sqrt{h\left(\tau \right(1\left)\right)}-\sqrt{h\left(s\right)}}{p\left(\tau \right(1\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to 1}{lim}\frac{2-s}{\sqrt{2}-\sqrt{s}}\hfill \\ & =& 2\sqrt{2}.\hfill \end{array}$$

Remark 1.

By Example 2, we deduce that if h is ${\nabla }_p^{\varrho }$-differentiable at $t\in {\mathbb{T}}_{\kappa };$ this does not imply that h is continuous at $t.$

Theorem 5.

Let $h,p:\mathbb{T}\to \mathbb{R}$, and ϱ is a quotient of odd positive integers. If h is ${\nabla }_p^{\varrho }$-differentiable on ${\mathbb{T}}_{\kappa }$ and p is ld-continuous, then h is ld-continuous on ${\mathbb{T}}_{\kappa }$.

Proof. 

Assuming that h is ${\nabla }_p^{\varrho }$-differentiable at t, by Definition 2, for $\epsilon >0$, there exists a neighborhood $\mathcal{U}$ of t, such as (4) holds. We have the next two cases:

• If $t\in {\mathcal{L}}_d$, we have p is ld-continuous, then p is continuous at t. By Formula (4), we deduce that

  $$\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)=h^{\varrho }\left(t\right),$$

  which implies that h is continuous at $t.$
• If $t\in {\mathcal{R}}_d$, then $\underset{s\to t}{lim}p\left(s\right)$ exist and is equal to l. Since $s\to t$ and $\epsilon \to 0$ in (4), we obtain

  $$\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)=h^{\varrho }\left(\tau \left(t\right)\right)-h^{{\nabla }_p^{\varrho }}\left(t\right)\left[p\left(\tau \left(t\right)\right)-l\right].$$
  Then, $\underset{s\to t}{lim}{h}^{\varrho }\left(s\right)$ exists. In the end, we find that h is ld-continuous on ${\mathbb{T}}_{\kappa }$.

This completes the proof. □

Proposition 1.

Let $p:\mathbb{T}\to \mathbb{R}$ with $\mathbb{T}$ a timescale and $\varrho >0$; we have

(1) 

p is ${\nabla }_p^{\varrho }$-differentiable on ${\mathbb{T}}_{\kappa }$ and $p^{{\nabla }_p^{\varrho }}$ is defined by

$$p^{{\nabla }_p^{\varrho }}\left(t\right)=\sum _{l=1}^{l=\varrho -1}{p}^l\left(\tau \left(t\right)\right)p^{\varrho -1-l}\left(t\right).$$

(2) 

If $p\left(t\right)\ne 0,\forall t\in {\mathbb{T}}_{\kappa }$, then $\frac{1}{p}$ is ${\nabla }_p^{\varrho }$-differentiable on ${\mathbb{T}}_{\kappa }$ and ${\left(\frac{1}{p}\right)}^{{\nabla }_p^{\varrho }}$ is defined by

$${\left(\frac{1}{p}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)=-\sum _{l=1}^{l=\varrho -1}{p}^{l-\varrho }\left(\tau \left(t\right)\right)p^{-\left(1+l\right)}\left(t\right).$$

Proof. 

$\left(1\right)$ By equality (5) of Theorem 2, we obtain

$$\begin{array}{ccc}\hfill p^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{p^{\varrho }\left(\tau \left(t\right)\right)-p^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}=\underset{s\to t}{lim}\sum _{l=1}^{l=\varrho -1}{p}^l\left(\tau \left(t\right)\right)p^{\varrho -1-l}\left(s\right)\hfill \\ & =& \sum _{l=1}^{l=\varrho -1}{p}^l\left(\tau \left(t\right)\right)p^{\varrho -1-l}\left(t\right).\hfill \end{array}$$

$\left(2\right)$ Since $p\left(t\right)\ne 0$ for all $t\in {\mathbb{T}}_{\kappa }$, then $p\left(t\right)p\left(\tau \left(t\right)\right)\ne 0$ for all $t\in {\mathbb{T}}_{\kappa }$, and by item 3 of Theorem 4 we conclude that $\frac{1}{p}$ is ${\nabla }_p^{\varrho }$-differentiable, and we have

$$\begin{array}{ccc}\hfill {\left(\frac{1}{p}\right)}^{{\nabla }_p^{\varrho }}\left(t\right)& =& -\frac{p^{{\nabla }_p^{\varrho }}\left(t\right)}{p^{\varrho }\left(t\right)p^l\left(\tau \left(t\right)\right)}=-\frac{1}{p^{\varrho }\left(t\right)p^{\varrho }\left(\tau \left(t\right)\right)}\sum _{l=1}^{l=\varrho -1}{p}^l\left(\tau \left(t\right)\right)p^{\varrho -1-l}\left(t\right)\hfill \\ & =& -\sum _{l=1}^{l=\varrho -1}{p}^{l-\varrho }\left(\tau \left(t\right)\right)p^{-\left(1+l\right)}\left(t\right).\hfill \end{array}$$

This completes the proof. □

Example 3.

Assume $p:\mathbb{T}\to \mathbb{R}$ and $\varrho >0$. Let the functions $h\left(t\right)=\lambda \in \mathbb{R}$, $g\left(t\right)=t$ and $f\left(t\right)=\frac{1}{t}.$ By Theorem 2 we obtain

$$\begin{array}{ccc}\hfill h^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{{\lambda }^{\varrho }-{\lambda }^{\varrho }}{p\left(\upsilon \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& 0,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill g^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{g^{\varrho }\left(\tau \left(t\right)\right)-g^{\varrho }\left(s\right)}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to t}{lim}\frac{{\tau }^{\varrho }\left(t\right)-s^{\varrho }}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & \ne & 1.\hfill \end{array}$$

Indeed, by item 3 of Theorem 4 we have

$$\begin{array}{ccc}\hfill f^{{\Delta }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{f^{\varrho }\left(\tau \left(t\right)\right)-f^{\varrho }\left(t\right)}{p\left(\tau \right(t\left)\right)-p\left(t\right)}\hfill \\ & =& -\underset{s\to t}{lim}\frac{{\left(\tau \left(t\right)\right)}^{\varrho }-s^{\varrho }}{{\left(s\tau \left(t\right)\right)}^{\varrho }\left(p\left(\upsilon \right(t\left)\right)-p\left(t\right)\right)}\hfill \\ & =& -\frac{{\left(t\right)}^{{\Delta }_p^{\varrho }}}{t^{\varrho }{\tau }^{\varrho }\left(t\right)}.\hfill \end{array}$$

Example 4.

Assume $p:\mathbb{T}\to \mathbb{R}$ and $\varrho \in \mathbb{N}$, such as p is ∇-differentiable on ${\mathbb{T}}_{\kappa }$ and $p^{\nabla }\left(t\right)\ne 0,$ let $g,h:\mathbb{T}\to \mathbb{R}$ functions be given by $g\left(t\right)=t$ and $h\left(t\right)=\frac{1}{t}$; then, g and h are ${\nabla }_p^{\varrho }$-differentiable, by Example 3, and we have

$$\begin{array}{ccc}\hfill g^{{\nabla }_p^{\varrho }}\left(t\right)& =& \underset{s\to t}{lim}\frac{{\tau }^{\varrho }\left(t\right)-s^{\varrho }}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\hfill \\ & =& \underset{s\to t}{lim}\frac{\tau \left(t\right)-t}{p\left(\tau \right(t\left)\right)-p\left(s\right)}\sum _{l=1}^{l=\varrho -1}{\left(\tau \left(t\right)\right)}^l{s}^{\varrho -1-l}\hfill \\ & =& \frac{t^{\varrho -1}}{p^{\nabla }\left(t\right)}\sum _{l=1}^{l=\varrho -1}{\left(\tau \left(t\right)t^{-1}\right)}^l,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill h^{{\nabla }_p^{\varrho }}\left(t\right)& =& -\frac{{\left(t\right)}^{{\Delta }_p^{\varrho }}}{t^{\varrho }{\tau }^{\varrho }\left(t\right)}\hfill \\ & =& -\frac{1}{t{\tau }^{\varrho }\left(t\right)p^{\nabla }\left(t\right)}\sum _{l=1}^{l=\varrho -1}{\left(\tau \left(t\right)t^{-1}\right)}^l.\hfill \end{array}$$

We will give the relationship between the usual structural Nabla derivative and the structural derivative given in Definition 4.

Theorem 6.

Let $h:\mathbb{T}\to \mathbb{R}$ be ${\Delta }_p^{\varrho }$-differentiable on ${\mathbb{T}}^{\kappa }$, and if $h^{{\Delta }_p^{\varrho }}$ is continuous on ${\mathbb{T}}^{\kappa }$ then h is ${\nabla }_p^{\varrho }$- differentiable on ${\mathbb{T}}_{\kappa }$ where

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=h^{{\Delta }_p^{\varrho }}\left(\tau \left(t\right)\right),\forall t\in {\mathbb{T}}_{\kappa }.$$

Proof. 

Let $t\in {\mathbb{T}}_{\kappa }$ be fixed. We will firstly consider the case where t is left-scattered. Since h is a structural differentiable, it is continuous. Then, h is a Nabla structural differentiable at t where

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=\frac{h^{\varrho }\left(\tau \left(t\right)\right)-h^{\varrho }\left(t\right)}{p\left(\tau \right(t\left)\right)-p\left(t\right)}.$$

Since $\tau \left(t\right)\in {\mathcal{R}}_s$, we have

$$h^{{\Delta }_p^{\varrho }}\left(\tau \left(t\right)\right)=\frac{h^{\varrho }\left(\upsilon \left(\tau \left(t\right)\right)\right)-h^{\varrho }\left(\tau \left(t\right)\right)}{p\left(\upsilon \right(\tau \left(t\right)\left)\right)-p\left(\tau \right(t\left)\right)}=\frac{h^{\varrho }\left(t\right)-h^{\varrho }\left(\tau \left(t\right)\right)}{p\left(t\right)-p\left(\tau \right(t\left)\right)}.$$

Then,

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=h^{{\Delta }_p^{\varrho }}\left(\tau \left(t\right)\right),$$

which gives the desired result.

Let $t\in {\mathcal{L}}_d\cap {\mathcal{R}}_d$, simultaneously. Here, since the existence of $h^{{\Delta }_p^{\varrho }}\left(t\right)$, we have

$$\underset{s\to t}{lim}\frac{h^{\varrho }\left(t\right)-h^{\varrho }\left(s\right)}{p\left(t\right)-p\left(s\right)},$$

(8)

existing and finite, which is equal to $h^{{\Delta }_p^{\varrho }}\left(t\right)$. Since $t\in {\mathcal{L}}_d$, by the existence of (8), we have $h^{{\nabla }_p^{\varrho }}\left(t\right)$ existing as a limit. Then,

$$h^{{\nabla }_p^{\varrho }}\left(t\right)=h^{{\Delta }_p^{\varrho }}\left(t\right).$$

This completes the proof. □

3. Structural Nabla Integration on Timescales

Now, we introduce the structural Nabla integration on timescales.

Definition 6

(The structural Nabla integration). Let ${\left[\alpha ,\beta \right]}_{\mathbb{T}}\subseteq \mathbb{T}$ and $h,p:\mathbb{T}\to \mathbb{R}$, such that h is a regulated function and p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$. Let ϱ be a quotient of odd positive integers and $\varrho >0$; we define the ${\nabla }_p^{\varrho }$-integrable function on ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$ by

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }f\left(t\right)& =& {\left({\int }_{\alpha }^{t}h\left(s\right){\nabla }_{p}s\right)}^{\frac{1}{\varrho }}\hfill \\ & =& {\left({\int }_{\alpha }^{t}h\left(s\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }},\hfill \end{array}$$

Here, $\int h\left(t\right)\nabla t$ is an usual indefinite integral of timescales (see [2]).

Example 5.

Let $h:\mathbb{T}\to \mathbb{R}$ be a function defined by $h\left(t\right)=c\in \mathbb{R}$. Let ϱ be a quotient of odd positive integers, and p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$. From Definition 6, we have

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }\left(c\right)& =& {\left(c{\int }_{\alpha }^t{p}^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}\hfill \\ & =& c^{\frac{1}{\varrho }}{\left[p\left(t\right)-p\left(\alpha \right)\right]}^{\frac{1}{\varrho }}.\hfill \end{array}$$

Indeed, if $p^{\nabla }\left(t\right)\ne 0,\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}},$ we obtain

$${\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }{\left(p^{\nabla }\right)}^{-1}\left(t\right)={\left(t-\alpha \right)}^{\frac{1}{\varrho }},\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

Theorem 7.

Let ${\left[\alpha ,\beta \right]}_{\mathbb{T}}\subseteq \mathbb{T}$ and $h,p:\mathbb{T}\to \mathbb{R}$, such that h is a regulated function and p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$. Let ϱ be a quotient of odd positive integers. Then,

$${\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }\left(h^{{\nabla }_p^{\varrho }}\right)\left(t\right)={\left[h^{\varrho }\left(t\right)-h^{\varrho }\left(\alpha \right)\right]}^{\frac{1}{\varrho }},\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

If $h\left(\alpha \right)=0,$ we have

$${\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }\left(h^{{\nabla }_p^{\varrho }}\right)\left(t\right)=h\left(t\right),\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

(9)

Proof. 

From Definitions 5 and 6, we obtain

$${\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla \varrho }\left(h^{{\nabla }_p^{\varrho }}\right)\left(t\right)={\left({\int }_{\alpha }^t{h}^{{\nabla }_p^{\varrho }}\left(s\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }},\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

By Corollary 1, we find

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla \varrho }\left(h^{{\nabla }_p^{\varrho }}\right)\left(t\right)& =& {\left({\int }_{\alpha }^t{\left(h^{\varrho }\right)}^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}\hfill \\ & =& {\left[h^{\varrho }\left(t\right)-h^{\varrho }\left(\alpha \right)\right]}^{\frac{1}{\varrho }}.\hfill \end{array}$$

If

$$h\left(\alpha \right)=0,$$

we obtain (9).

The proof is now completed. □

Theorem 8.

Let ${\left[\alpha ,\beta \right]}_{\mathbb{T}}\subseteq \mathbb{T}$ and $h,p:\mathbb{T}\to \mathbb{R}$, such that h is a regulated function and p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$. Let ϱ be a quotient of odd positive integers and $\varrho >0$; then,

$${\left({\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }h\right)}^{{\nabla }_p^{\varrho }}\left(t\right)=h\left(t\right),\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

Proof. 

We pose $g={\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }f$; by Definition 5, we have

$$\begin{array}{ccc}\hfill g^{{\nabla }_p^{\varrho }}\left(t\right)& =& \frac{1}{p^{\nabla }\left(t\right)}{\left({\int }_{\alpha }^{t}h\left(s\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\nabla }\hfill \\ & =& \frac{1}{p^{\nabla }\left(t\right)}h\left(t\right)p^{\nabla }\left(t\right)\hfill \\ & =& h\left(t\right),\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}},\hfill \end{array}$$

That is to say,

$${\left({\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }f\right)}^{{\nabla }_p^{\varrho }}\left(t\right)=h\left(t\right),\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

The proof is now completed. □

4. An Application

Below, we use $C\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)$ as a Banach space of all continuous functions from ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$ into $\mathbb{R}$, where ${\left[\alpha ,\beta \right]}_{\mathbb{T}}=\left[\alpha ,\beta \right]\cap \mathbb{T}$ with the norm

$${∥h∥}_{\infty }=sup\left\{\left|h\left(t\right)\right|:t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}\right\}.$$

Let $p:\mathbb{T}\to \mathbb{R}$, where p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$ and $\varrho$ is a quotient of odd positive integers, such as $\varrho \ne 0$. We consider the following initial value problem:

$$\left\{\begin{array}{c}{x}^{{\nabla }_p^{\varrho }}\left(t\right)=h\left(t,x\left(t\right)\right),t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}},\hfill \\ x\left(a\right)=0,\hfill \end{array}\right.$$

(10)

where $x^{{\nabla }_p^{\varrho }}$ is the ${\nabla }_p^{\varrho }$-differentiable on timescales, and $h:{\left[\alpha ,\beta \right]}_{\mathbb{T}}\times \mathbb{R}\to \mathbb{R}$ is a left-dense continuous function. Our next aim is to show the minimal requirements to obtain the existence of a unique solution to (10).

Lemma 3.

Let $p:\mathbb{T}\to \mathbb{R}$, such that p is ∇-differentiable in ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$, $\varrho \in \mathbb{R}$, $\varrho >0$. The function $x\in \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)$ is the said solution of (10) if and only if it is the solution of the next integral equation:

$$x\left(t\right)={\left({\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }},\forall t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}.$$

(11)

Proof. 

Let x be a solution to problem (10); then,

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }\left(x^{{\nabla }_p^{\varrho }}\right)\left(t\right)& =& {\left({\int }_{\alpha }^t{x}^{{\nabla }_p^{\varrho }}\left(t\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}\hfill \\ & =& {\left({\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}.\hfill \end{array}$$

(12)

By Theorem 8, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{p,\alpha }^{\varrho ,\nabla }\left(x^{{\nabla }_p^{\varrho }}\right)\left(t\right)& =& {\left[x^{\varrho }\left(t\right)-x^{\varrho }\left(\alpha \right)\right]}^{\frac{1}{\varrho }}\hfill \\ & =& x\left(t\right).\hfill \end{array}$$

(13)

Substituting (13) in (7), we find (11).

Conversely, let x be a solution tp problem (11); then, $x\left(\alpha \right)=0$. Moreover, by Corollary 1, we find

$$\begin{array}{ccc}\hfill x^{{\nabla }_p^{\varrho }}\left(t\right)& =& \frac{1}{p^{\nabla }\left(t\right)}{\left({\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\nabla }\hfill \\ & =& h\left(t,x\left(t\right)\right).\hfill \end{array}$$

This completes the proof. □

Theorem 9.

Let $h\in \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}}\times \mathbb{R},\mathbb{R}\right)$; there exist functions $G:{\left[\alpha ,\beta \right]}_{\mathbb{T}}\to \left[0,\infty \right)$, such that

$$\left|h\left(t,x\right)\right|\le G\left(t\right),\forall \left(t,x\right)\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}\times \mathbb{R}.$$

(14)

Then, (10) admits a solution on ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$.

Proof. 

We will convert problem (10) as a fixed-point problem. Let

$$A:\mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)\to \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right),$$

given by

$$Ax\left(t\right)={\left({\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}.$$

(15)

We show the existence of a fixed point for A defined by (15) by applying Schauder’s fixed-point theorem [11]. There are four steps to this end.

Step $\mathit{1}:$ We have A is continuous. Let ${\left(x_n\right)}_n$ be a sequence, such that $x_n\to x$ in $\mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)$; then, $\exists \delta >0,$ so that ${∥x_n∥}_{\infty }\le \delta ,\forall n\in \mathbb{N}$. Since h is uniformly continuous in $\left[-\delta ,\delta \right]\times {\left[\alpha ,\beta \right]}_{\mathbb{T}}$, then, given $\epsilon >0$, and there is a $\theta >0$, if $\left|x_1-x_2\right|\le \theta ,$ we have

$$\left|h\left(t,x_1\right)-h\left(t,x_2\right)\right|\le \epsilon {\left[{\int }_{\alpha }^b\left|p^{\nabla }\left(s\right)M\left(s\right)\right|\nabla s\right]}^{-1}.$$

Moreover, for each $t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}$, we have

$$\left|{\left(Ax\right)}^{\varrho }\left(t\right)-{\left(A{x}_n\right)}^{\varrho }\left(t\right)\right|\le {\int }_{\alpha }^t\left|h\left(t,x\left(t\right)\right)-h\left(t,x_n\left(t\right)\right)\right|\left|p^{\nabla }\left(s\right)G\left(s\right)\right|\nabla s.$$

Thus, there is a $n_0\in \mathbb{N}$, such that $∥x_n-x∥\le \theta$, for all $n\ge n_0$. Hence,

$$∥{\left(Ax\right)}^{\varrho }-{\left(A{x}_n\right)}^{\varrho }∥\le \epsilon ,\forall n\ge n_0.$$

(16)

Let

$$B,L:\mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)\to \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right),$$

be operators defined by

$$B_{\varrho }x={\left(Ax\right)}^{\varrho },$$

and

$$L_{\varrho }x=x^{\frac{1}{\varrho }}.$$

By (16), we deduce that B is continuous. Moreover, by Lemma 1, we conclude that L is continuous. Since

$$A=L_{\varrho }\circ B_{\varrho },$$

then A is continuous.

Step $2:$ It is enough to see that $\exists \delta >0,$ so that $∥Ax∥\le \delta .$ For each $t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}$, we have

$$\begin{array}{ccc}\hfill \left|Ax\left(t\right)\right|& \le & {\left[{\int }_{\alpha }^t\left|h\left(t,x\left(t\right)\right)\right|\left|p^{\nabla }\left(s\right)\right|\nabla s\right]}^{\frac{1}{\varrho }}\hfill \\ & \le & {\left[{\int }_{\alpha }^{\beta }\left|p^{\nabla }\left(s\right)G\left(s\right)\right|\nabla s\right]}^{\frac{1}{\varrho }}\hfill \\ & =& \delta .\hfill \end{array}$$

Step $\mathit{3}:$ Let $t_1,t_2\in {\left[\alpha ,\beta \right]}_{\mathbb{T}},t_1\le t_2$; then,

$$\begin{array}{ccc}\hfill \left|{\left(Ax\right)}^{\varrho }\left(t_1\right)-{\left(Ax\right)}^{\varrho }\left(t_2\right)\right|& \le & \left|{\int }_{t_1}^{t_2}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right|\hfill \\ & \le & G\left|{\int }_{t_1}^{t_2}{p}^{\nabla }\left(s\right)\nabla s\right|.\hfill \end{array}$$

As $t_1\to t_2$, the RHS of the above inequality tends to zero. By Step 1 to Step 3, and owing to the Arzela–Ascoli theorem, we find that

$$B_{\varrho }:\mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)\to \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right),$$

is completely continuous, since $A=L_{\varrho }\circ B_{\varrho }$, which implies A is completely continuous. Using Schauder’s fixed point theorem, we find that A has a fixed point, which is a solution to (10).

The proof is now completed. □

Theorem 10.

Let $h\in \mathcal{C}\left({\left[\alpha ,\beta \right]}_{\mathbb{T}}\times \mathbb{R},\mathbb{R}\right)$; there exist two functions,

$$M,r:{\left[\alpha ,\beta \right]}_{\mathbb{T}}\to \left[0,\infty \right),$$

and a constant $\lambda >0$, such that (14) holds, and

$$\left|h\left(t,x\right)-h\left(t,y\right)\right|\le r\left(t\right){\left|x-y\right|}^{\lambda },t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}},x,y\in \mathbb{R},$$

(17)

with

$$\lambda \ge \varrho \mathrm{and}{\left[{\mathcal{I}}_{\left|p\right|,\alpha }^{\varrho ,\nabla }r\left(\beta \right)\right]}^{\varrho -\lambda }>2^{\varrho }{\left[{\mathcal{I}}_{\left|p\right|,\alpha }^{\varrho ,\nabla }M\left(\beta \right)\right]}^{\varrho }.$$

(18)

Then, problem (10) admits a unique solution to ${\left[\alpha ,\beta \right]}_{\mathbb{T}}$.

Proof. 

From Theorem 9, we find that A has a fixed point in $C\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)$. To show that A admits a unique fixed point, it is enough to satisfy the conditions of Lemma 3 on the application A. We mean to show that there is a positive constant r, so that $∥Ax∥<r$. For $t\in {\left[a,b\right]}_{\mathbb{T}}$, one has

$$\begin{array}{ccc}\hfill \left|Ax\left(t\right)\right|& \le & {\left|{\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right|}^{\frac{1}{\varrho }}\hfill \\ & \le & {\mathcal{I}}_{\left|p\right|,\alpha }^{\varrho ,\nabla }M\left(\beta \right)\hfill \\ & =& r.\hfill \end{array}$$

By Lemma 2, for every $x,y\in C\left({\left[\alpha ,\beta \right]}_{\mathbb{T}},\mathbb{R}\right)$ and $t\in {\left[\alpha ,\beta \right]}_{\mathbb{T}}$, we have

$$\begin{array}{ccc}\hfill \left|Ax\left(t\right)-Ay\left(t\right)\right|& \le & \left|{\left({\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}-{\left({\int }_{\alpha }^{t}h\left(t,y\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right)}^{\frac{1}{\varrho }}\right|\hfill \\ & \le & {\left|{\int }_{\alpha }^{t}h\left(t,x\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s-{\int }_{\alpha }^{t}h\left(t,y\left(t\right)\right)p^{\nabla }\left(s\right)\nabla s\right|}^{\frac{1}{\varrho }}\hfill \\ & \le & {\left[{\int }_{\alpha }^b\left|h\left(t,x\left(t\right)\right)-h\left(t,y\left(t\right)\right)\right|\left|p^{\nabla }\left(s\right)\right|\nabla s\right]}^{\frac{1}{\varrho }}.\hfill \end{array}$$

Substituting (17) to obtain

$$∥Ax-Ay∥\le {\mathcal{I}}_{\left|p\right|,\alpha }^{\varrho ,\nabla }r\left(\beta \right){∥x-y∥}^{\frac{\lambda }{\varrho }}.$$

By inequality (18), we deduce that inequality (3) is satisfied. Finally, we deduce that A has a unique fixed point. This fixed point is exactly the unique solution to (10).

The proof is now completed. □

5. An Example

• Let the next initial value problem,

  $$\left\{\begin{array}{c}{x}^{{\nabla }_t^{\frac{1}{3}}}\left(t\right)={\left(t^2+t\right)}^{-1}sin\left(x\left(t\right)\right),\mathrm{for}\mathrm{all}t\in {\left[0,m\right]}_{\mathbb{N}},\hfill \\ x\left(0\right)=0.\hfill \end{array}\right.$$

  (19)
  Here, $\mathbb{T}=\mathbb{N}$, $\alpha =0,$ $\beta =m$, $\varrho =\frac{1}{3}$, $p\left(t\right)=t$, and

  $$h\left(t,x\right)={\left(t^2+t\right)}^{-1}sin\left(x\right).$$
  Let

  $$\eta \left(t\right)={\left(t^2+t\right)}^{-1},t\in {\left[0,m\right]}_{\mathbb{N}}.$$
  Then, $p^{\nabla }\left(t\right)=1,$ for $\left(t,x\right)\in {\left[0,m\right]}_{\mathbb{N}}\times \mathbb{R}$, and we have

  $$\left|h\left(t,x\right)\right|\le \eta \left(t\right)\mathrm{and}\left|h\left(t,x\right)-h\left(t,y\right)\right|\le \eta \left(t\right)\left|x-y\right|.$$
  Then, one can deduce that

  $$M\left(t\right)=r\left(t\right)=\eta \left(t\right),$$

  and $\lambda =1$. We have

  $$\begin{array}{ccc}\hfill {\mathcal{I}}_{\left|p\right|,\alpha }^{\varrho ,\nabla }r\left(\beta \right)& =& {\mathcal{I}}_{\left|t\right|,0}^{\frac{1}{3},\nabla }r\left(m\right)\hfill \\ & =& {\left(\frac{m}{m+1}\right)}^3.\hfill \end{array}$$
  If $m\le 12,$ then (18) is verified and then all the requirements of Theorem 10 are satisfied, which implies that (19) has a unique solution to ${\left[0,m\right]}_{\mathbb{N}}$.
• Assume $p,f:\mathbb{T}\to \mathbb{R}$, t is left-dense and $\lambda \in \mathbb{N}$. By Theorem 2, we obtain

  $$f^{{\nabla }_p^{\lambda }}\left(t\right)=\underset{s\to t}{lim}\frac{f^{\lambda }\left(t\right)-f^{\lambda }\left(s\right)}{p\left(t\right)-p\left(s\right)}=\underset{s\to t}{lim}\frac{f\left(t\right)-f\left(s\right)}{p\left(t\right)-p\left(s\right)}\sum _{\ell =0}^{\ell =n-1}{f}^{\ell }\left(t\right)f^{\lambda -1-\ell }\left(s\right).$$
  If $p,f$ are ∇-differentiable at point t, we find that

  $$f^{{\nabla }_p^{\lambda }}\left(t\right)=\lambda \frac{f^{\nabla }\left(t\right)}{p^{\nabla }\left(t\right)}{f}^{\lambda -1}\left(t\right).$$

6. Conclusions

We studied in the present article more general properties of structural Nabla derivatives on a timescale. The concept that is being discussed here by “The definition of derivatives and integrals” is a generalization of the structural derivative, which is crucial to understanding experimental outcomes in biomedicine that pertain to the structure of neuro-imaging signal propagation in different parts of the human brain (see [22]). This new structural derivative on timescales has the benefit of being more than just a straightforward mathematical generalization, since it enables us to address significant ideas like self-similarity and indistinguishability that might arise in complex systems.