Generalized exponential function and initial value problem for conformable dynamic equations

1.   Introduction

Fractional calculus generalized the classical calculus to an arbitrary (non-integer) order. The history of this theory goes back to mathematicians like Leibniz (1646–1716), Liouville (1809–1882), Riemann (1826–1866), Letnikov (1837–1888), Grünwald (1838–1920) and others ^[16,19]. For the last three centuries, fractional calculus is getting famous, like all fields of science. It is one of the most intensively developed fields of mathematical assessment. Because of its various applications in designing, financial aspects, account, geography, likelihood and measurements, compound designing, physical science, splines, thermodynamics and neural organizations ^[5,9,22].

There are some recent developments in fractional calculus and its applications. In ^[8], the authors studied the fractional second and third-order nonlinear Schrödinger equations. They studied symmetric and antisymmetric solutions and analyzed the influence of the Lêvy index on different solutions. Lu et al. ^[14] solved the fractional discrete coupled nonlinear Schrödinger equations on account of the modified Riemann-Liouville fractional derivative and Mittag Leffler function. In ^[13], Li et al. studied the existence, bifurcation and stability of two-dimensional optical solutions in the framework of fractional nonlinear Schrödinger equation.

From the literature review reader can see several definitions of fractional operators like Riemann-Liouville, Caputo, Grünwald-Letnikov, Weyl, Hadamard, Marchaud and Riesz ^[12,15]. These types of derivatives do not satisfy the fundamental formulas of differentiation like the product rule, the quotient rule, the chain rule, etc. Khalil ^[12] introduced a new well-behaved simple fractional derivative known as the conformable fractional derivative (CFD) based on the derivative's basic limit concept: Suppose $\Phi:\left[0, \infty\right) \longrightarrow\mathbb{R}$ be a function, then for all $\upsilon > 0$ and $\alpha\in\left(0, 1\right]$,

where $\Phi^{\left(\alpha\right) }\left(\upsilon\right)$ is known as the CFD of $\Phi$ of order $\alpha$. The definition of CFD introduced by Khalil retaines all the classical characteristics of the derivative, and satisfy the chain rule. This new definition attracted many researchers, some results have been obtained for the fundamental properties of the CFD in ^[1].

In 1988, Hilger in his Ph. D. thesis introduced the time scale theory that has recently received a great deal of attraction to integrate and extend the discrete and continuous analysis ^[10]. The investigation of dynamic equations on arbitrary time scales show such distinction and helps avoid proving outcomes twice: Once for the differential equation and once for the difference equation. The basic idea is to prove a result for a dynamic equation where the domain of the function is a so-called time scale $\mathbb{T}$, which is an arbitrary nonempty closed subset of reals ^[7].

Bastos in his Ph. D. thesis developed fractional calculus on time ^[18]. The theory of fractional (non-integer order) calculus on time scales is a topic of great interest for researchers nowadays. In ^[6], Benkhettou et al. developed a conformable fractional calculus theory on an arbitrary time scale, which extends the conformable fractional calculus and the fundamental techniques for fractional differentiation and integration on time scales.

Ahmed ^[2] discussed the class of non-local stochastic differential equations involving conformable fractional time derivative operator and the existence of mild solution for the non-local conformable stochastic differential equation. In ^[3], Ahmed discussed the class of conformable fractional stochastic differential equations driven by the Rosenblatt process. In ^[4], Ahmed studied a non-instantaneous impulsive conformable fractional stochastic delay integro-differential system driven by the Rosenblatt process.

The exponential function on the time scale introduced by Hilger in ^[11] and has been devoted to solving the first-order linear dynamic equations, and second-order linear dynamic equations with constant and variable coefficients ^[11]. Euler-Cauchy dynamic equations on time scales have been solved using an exponential function. Exponential function for conformable fractional calculus has been defined in ^[17]. The conformable exponential function with respect to operator $\Delta^{\alpha}$ has been defined ^[20]. This definition is implicit. In this paper, we define the generalized conformable fractional exponential function by following the conformable calculus outline and also developing its fundamental characteristics. However, our definition is explicit. The generalized exponential function is consistent with the exponential function on time scales for $\alpha = 1$ and consistent with CF exponential function if $\mathbb{T} = \mathbb{R}$. The generalized exponential function and these theorems are the generalizations of the exponential function on time scales discussed by Bohner et al. ^[7] and the conformable fractional exponential function in ^[17].

In ^[24], the global existence, extension, boundedness and stabilities of solutions have been discussed for the following conformable fractional differential equation:

corresponding to the local and non-local initial condition $x\left(a\right) = x_{a}$ and $x\left(a\right) +g\left(x\right) = x_{a}$ respectively. We generalized the theory established in ^[24] to the conformable fractional dynamic equation:

We consider the global existence, extension, boundedness and stability of the solutions corresponding to the conformable fractional dynamic equation:

according to the local initial condition

and non-local initial condition

where $\psi^{\left(\alpha\right) }\left(s\right)$ refers to the CFD of order $\alpha\in\left(0, 1\right]$ for functions defined on arbitrary time scales $\mathbb{T}$, $k:\left[0, \infty\right) _{\mathbb{T}}\times \mathbb{R\rightarrow R}$. Also, $k$ is right dense continuous and on a suitable function space $g$ is functional.

We structure the remaining of the paper: Section 2 recalls some essential definitions and results. Section 3 aims to define the generalized exponential function and develop its fundamental characteristics. In Section 4, we generalize the Grönwall type inequalities in the conformable setting. In Section 5, we set some rules for the global existence, extension and boundedness of solutions to the local initial value problem and then discuss the stability of solutions. Section 6 examines the existence of solutions to the nonlocal initial value problem. Finally, we conclude our findings in the last section.

2.   Basic notions

Here, we recall some basic definitions and results which are essential to the sequel ^[21]. Throughout this manuscript, let us denote the time scale by $\mathbb{T}$ and the set of all rd-continuous functions by $C_{rd}.$

Lemma 2.1. If $h\in C_{rd}$ and $u\in\mathbb{T}^{k}$, then

Definition 2.1. Assume that $k > 0$, the Hilger complex number is defined by

Definition 2.2. For $k > 0$, the strip is defined by

Definition 2.3. For $k > 0$, the cylindrical transformation $\xi _{k}:\mathbb{C}_{k}\rightarrow \mathbb{Z}_{k}$ is defined by

where $\log$ is the principal logarithm function. Notice that

Let us denote by $\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$ the set of all functions whose conformable fractional differentiable of order $\alpha$ is continuous, where $\alpha\in\left(0, 1\right]$. The following lemmas give some meaningful connections concerning the conformable fractional derivative on time scales ^[6].

Lemma 2.2. Suppose $l\in\mathcal{C}\left(\mathbb{T}, \mathbb{R}\right)$ and assume $v\in\mathbb{T}^{k}$. If $l$ is continuous at $v$ where $v$ isright-scattered, implies that $l\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$ at $v$ such that

where $\left(l\right) ^{\left(\alpha\right) }\left(\cdot\right)$represents the conformable fractional derivative of order $\alpha\in\left(0, 1\right].$

Lemma 2.3. Suppose $h, l\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$. Then

(1) If $h$, $l\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$, then the product $hl\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$ with

(2) If $h\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$, then $\frac{1}{h}\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$ such that

applicable at each points $s\in\mathbb{T}^{k}$ for which $h\left(s\right)h\left(\sigma\left(s\right) \right) \neq0.$

(3) If $h$, $l\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$, then $\frac{h}{l}\in\mathcal{C}^{\alpha}\left(\mathbb{T}, \mathbb{R}\right)$ such that

applicable at each points $s\in\mathbb{T}^{k}$ for which $l\left(s\right)l\left(\sigma\left(s\right) \right) \neq0$.

Lemma 2.4. If $\left(g\left(s\right) \right) ^{\left(\alpha\right) }$ is continuous on $\left[c, d\right] _{\mathbb{T}}$, then

where $I_{\alpha}$ represents the conformable fractional integral of order $\alpha\in\left(0, 1\right].$

Lemma 2.5. Let $\alpha\in\left(0, 1\right]$. Then for allright-dense continuous function $g:\mathbb{T\longrightarrow R}$, a function$G_{\alpha}:\mathbb{T\rightarrow R}$ exists in such a way that

for each $s\in\mathbb{T}^{k}$. Function $G_{\alpha}$ is called an $\alpha$-antiderivative of $g$.

3.   Generalized exponential function

In this section, we use Definition 2.3 to define a generalized exponential function. Let us generalize the regressive concept in a conformable setting.

Definition 3.1. A function $l:\mathbb{T\rightarrow R}$ is said to be "$\alpha$-regressive" provided

holds. The set of all $\alpha$-regressive and right-dense continuous functions $l:\mathbb{T\rightarrow R}$ [7,Definition 1.58] is referred to as $\mathcal{R}^{\alpha}$.

Definition 3.2. In $\mathcal{R}^{\alpha}$ "$\alpha$-circle plus" addition $\oplus_{\alpha}$ is defined as below:

Definition 3.3. For $h\in\mathcal{R}^{\alpha}$, define

Definition 3.4. Define "$\alpha$-circle minus" subtraction $\ominus_{\alpha}$ on $\mathcal{R}^{\alpha}$ as below:

For $k, l\in\mathcal{R}^{\alpha}$, we have

Lemma 3.1. Show that $\left(\mathcal{R}^{\alpha}, \oplus_{\alpha}\right)$ is an Abelian group.

Proof. The proof is straight-forward, so it is left as an exercise.

The group $\left(\mathcal{R}^{\alpha}, \oplus_{\alpha}\right)$ is also called the $\alpha$-regressive group.

Remark 3.1. For $\alpha = 1$, it becomes regressive group ^[7].

Definition 3.5. The set $\mathcal{R}^{\alpha^{+}}$ of all positively $\alpha$-regressive elements of $\mathcal{R}^{\alpha}$ is defined by

Theorem 3.1. Suppose $h, k\in\mathcal{R}^{\alpha}$. Then

(1) $\left(h\ominus_{\alpha}h\right) \left(\tau\right) = 0,$

(2) $\ominus_{\alpha}\left(\ominus_{\alpha}h\right) \left(\tau\right) = \varphi\left(\tau\right),$

(3) $\left(k\ominus_{\alpha}h\right) \left(\tau\right) \in\mathcal{R}^{\alpha},$

(4) $\ominus_{\alpha}\left(h\ominus_{\alpha}k\right) \left(\tau\right) = \left(k\ominus_{\alpha}h\right) \left(\tau\right),$

(5) $\ominus_{\alpha}\left(k\oplus_{\alpha}h\right) \left(\tau\right) = \left[\left(\ominus_{\alpha}k\right) \oplus_{\alpha}\left(\ominus_{\alpha}h\right) \right] \left(\tau\right),$

(6) $\left[h\oplus_{\alpha}\dfrac{k}{1+\mu h\tau^{\alpha-1}}\right]\left(\tau\right) = h\left(\tau\right) +k\left(\tau\right).$

Proof. (1) By using Definitions 3.2–3.4 respectively, it follows that

(2) Definition 3.3 yields that

(3) By using Definitions 3.1 and 3.4 respectively, we have

We note that $\dfrac{k\left(\tau\right) -h\left(\tau\right) } {1+\mu\left(\tau\right) h\left(\tau\right) \tau^{\alpha-1}}$ is right-dense continuous. Therefore, $\left(k\ominus_{\alpha}h\right) \left(\tau\right) \in\mathcal{R}^{\alpha}.$

(4) Using Definitions 3.3 and 3.4, it implies that

(5) By using Definitions 3.2 and 3.3 respectively, we get

On the other side, using Definitions 3.2 and 3.3 respectively, we obtain

Hence,

(6) From Definition 3.2, we have

Next, we define the generalized exponential function.

Definition 3.6. Let $h\in\mathcal{R}^{\alpha}$, then the generalized exponential function is defined by

To be more precise, using the definition for the cylindrical transformation Definition 2.3 we obtain

Example 3.1. When $\mathbb{T = R}$, then

Example 3.2. When $\mathbb{T = Z}$, then

Example 3.3. When $\mathbb{T = }q^{\mathbb{N}}$, $q > 1$, then

Remark 3.2. The exponential function on arbitrary time scale ^[7] is obtained by choosing $\alpha = 1$.

Remark 3.3. Definition 2.3 in ^[17] is obtained by choosing $\mathbb{T = R}$.

3.1.   Characteristics of generalized exponential function

Theorem 3.2. (Semigroup property) If $f\in\mathcal{R}^{\alpha}$, then

Proof. By using Definition 3.6, we have

Theorem 3.3. Suppose $k\in\mathcal{R}^{\alpha}$. Then

and

Proof. Let $\sigma\left(v\right) > v$. Then

From Lemma 2.1, it implies that

From Definition 2.3, it follows that

Hence we have

Corollary 3.1. Suppose $h\in\mathcal{R}^{\alpha}$. Then $E_{h}\left(u, 0\right)$ is a solution of the following Cauchy problem:

Proof. Let $x\left(\cdot\right) = E_{h}\left(\cdot, 0\right)$ be a solution of Eq (3.2). First note that

It remains to show that $E_{h}\left(u, 0\right)$ satisfies Eq (3.2). By Theorem 3.3, it follows that

Therefore,

Hence $E_{h}\left(u, 0\right)$ is a solution to the Cauchy problem (3.2).

Corollary 3.2. Suppose $h\in\mathcal{R}^{\alpha}$. Then $E_{h}\left(v, 0\right)$ is the unique solution of the IVP (3.2).

Proof. Suppose $x\left(\cdot\right)$ be any solution of the IVP (3.2). Then by using part (3) of Lemma 2.3, it follows that

According to our assumption and Theorem 3.3, it implies that

Consequently, $x\left(v\right) = bE_{h}\left(v, 0\right)$, where $b$ is a constant. Thus,

Hence, $x\left(v\right) = E_{h}\left(v, 0\right)$ is the unique solution.

Theorem 3.4. Suppose $h\in\mathcal{R}^{\alpha}$. Then

Proof. Since for right-dense points $\left(\sigma\left(u\right) = u\right)$, the case is trivial. And for right-scattered points $\left(\sigma\left(u\right) > u\right)$, by Lemma 2.2, we have

Theorem 3.3 implies that

which proves the desired result.

Theorem 3.5. If $l\in\mathcal{R}^{\alpha}$, then

Proof. Let us consider the following IVP:

First note that

Differentiating

By part (2) of Lemma 2.3, Theorems 3.3, 3.4 and Definition 3.3 we obtain

Hence,

which proves the required result

Theorem 3.6. If $h, p\in\mathcal{R}^{\alpha}$, then

Proof. Let us consider the IVP:

We show that its solution is $E_{h}\left(\omega, 0\right) E_{p}\left(\omega, 0\right)$. We have

Now by using part (1) of Lemma 2.3 it implies that

By Theorems 3.3, 3.4 and Definition 3.2, we have

Therefore,

which proves the desired result.

Theorem 3.7. Assume $h, p\in\mathcal{R}^{\alpha}$. Then

Proof. By using Theorems 3.5 and 3.6, we have

By using Definitions 3.2–3.4, we have

which proves the required result.

Theorem 3.8. Suppose $h\in\mathcal{R}^{\alpha}$. Then

Proof. By taking $\alpha$-conformable fractional derivative on time scales, we have

According to Theorems 3.3 and 3.5, it follows that

By using Definition 3.3 and Theorems 3.4, 3.5, we can obtain

which proves the required result.

Theorem 3.9. Let $h, l\in\mathcal{R}^{\alpha}$. Then

Proof. By taking $\alpha$-conformable fractional derivative on time scales, we have

Theorem 3.7 and using part (3) of Lemma 2.3 implies that

By applying Theorem 3.3, we obtain

which proves the required result.

Theorem 3.10. Suppose $h\in\mathcal{R}^{\alpha}$ and $c, d, 0\in\mathbb{T}$.Then

and

Proof. Firstly, according to Theorems 3.4 and 3.5, we obtain

According to Definition 3.3, Theorems 3.3 and 3.5, it follows that

which proves our first part.

Now, taking both sides the $\alpha$-conformable fractional integral, we have

which gives the desired result.

Theorem 3.11. Let $k\in\mathcal{R}^{\alpha}$. Then

Proof. By Lemma 2.2, it implies that

From Theorem 3.3, we obtain

Similarly, we find

Theorem 3.12. Let $k\in\mathcal{R}^{\alpha}$. Then

and

Proof. From Theorems 3.5 and 3.11, we obtain

Definition 3.3, Theorems 3.3 and 3.5 implies that

which proves our first part.

Therefore, conformable $\alpha$-fractional integration on time scales implies

which proves the required result.

4.   Grönwall's inequality

Theorem 4.1. Let $y, f\in C_{rd}$ and $p\in\mathcal{R}^{\alpha^{+}}$. Then

implies

Proof. By using part (1) of Lemma 2.3, Theorems 3.3 and 3.4, we have

Definition 3.3 implies that

Now taking the $\alpha$-conformable fractional integral on time scales, and by using Lemma 2.4, it implies that

By given assumption and using Theorem 3.5, we obtain

And hence the assertion follows by applying Theorem 3.2:

which proves the required result.

Theorem 4.2. Let $y, f\in C_{rd}$ and $p\in\mathcal{R}^{\alpha^{+}}$, $p\geq0$. Then

implies that

Proof. Define

Then $z\left(0\right) = 0$ and

By using Lemma 2.5 and Eq (4.1), we obtain

Theorem 4.1 yields

And hence the claim follows because of Eq (4.1). Therefore,

which completes the proof.

Theorem 4.3. (Grönwall's inequality) Let $y, p\in C_{rd}$, $p\in\mathcal{R}^{\alpha^{+}}$and $\lambda\geq0$ such that

then

Proof. Let $f\left(t\right) = \lambda$. Then by Theorem 4.2, it follows that

From Theorem 3.12, we obtain

Therefore,

which completes the proof.

Remark 4.1. For $\alpha = 1$ and $\mathbb{T = R}$, we obtained the Grönwall's inequalities in classical calculus.

5.   Local initial value problems

We will develop some conditions for the global existence, extension and boundedness of solutions related to the local initial value problem (LIVP) in the following section.

The following assumptions will be needed throughout the following section.

Suppose $\Omega = \left[0, \infty\right) _{\mathbb{T}}\times\mathbb{R}$.

(H1) The mapping $k:\Omega\rightarrow \mathbb{R}$ is right-dense continuous.

(H2) A positive constant $K > 0$ exists in such manner that, for all $\left(\tau, x\right)$, $\left(\tau, x\right)$ in $\Omega$,

(H3) A nonnegative mapping $l\geq0$ exists in such manner that, for all $\left(\tau, x\right)$ in $\Omega$,

for which

is bounded on $\left[0, \infty\right) _{\mathbb{T}}$.

(H4) A nonnegative mapping $h\geq0$ and a positive constant $K > 0$ exists in such manner that, for all $\left(\tau, x\right)$, $\left(\tau, y\right)$ in $\Omega$,

for which

is bounded on $\left[0, \infty\right) _{\mathbb{T}}$.

Through Lemmas 2.4 and 2.5, the LIVP (1.1) and (1.2) is simply transformed into an Integral Equation (IE).

Lemma 5.1. If (H1) holds, then a function $\psi$ in $C\left(\left[0, b\right]_{\mathbb{T}}\right)$ is a solution of local initial value problem (1.1) and (1.2) if and only if $\psi$ is acontinuous solution of the following integral equation:

We can now demonstrate the existence and uniqueness of the solution to the LIVP (1.1) and (1.2) as a consequence of Definition 3.6.

Theorem 5.1. The local initial value problem (1.1) and (1.2)has unique solution defined on $\left[0, a\right] _{\mathbb{T}}$ wheneverthe assumptions (H1) and (H2) hold.

Proof. The claim will be verified via Banach's contraction principle on $C\left(\left[0, a\right] _{\mathbb{T}}\right)$. Let $k > 0$ be a constant and $k\in R^{\alpha^{+}}$ and let $\left\Vert \cdot\right\Vert$ denote the Euclidean norm on $\mathbb{R}^{n}$. Define the interval $\left[0, a\right] _{\mathbb{T}}$. Let us denote by $C\left(\left[0, a\right] _{\mathbb{T} }\right)$ the space of continuous functions along with a suitable norm. Define the term "TZ-norm"

where $E_{k}\left(\tau, 0\right)$ in Defintion 3.6. The well-known sup-norm

It is simple to prove that $\left\Vert \cdot\right\Vert _{k}$ is equivalent to $\left\Vert \cdot\right\Vert _{0}$. Hence $\left(C\left(\left[0, a\right] _{\mathbb{T}}\right), \left\Vert \cdot\right\Vert _{k}\right)$ is Banach space.

Define an operator

as

Lemma 5.1 assures that the fixed points of the operator $T$ are the solutions of local IVP (1.1) and (1.2).

For any $\psi, \varphi\in\left(C\left(\left[0, a\right] _{\mathbb{T} }\right), \left\Vert \cdot\right\Vert _{k}\right)$, then

By (H2), it follows that

Thus, it implies that

Now we have to find

By Theorem 3.3, it implies that

where $k > 0$ be a positive constant. Then by taking $\alpha$-conformable fractional integral, it follows that

By using Lemma 2.4, we obtain

Now, using Eq (5.3) in Eq (5.2), it follows that

Therefore,

As $0 < \dfrac{K}{k} < 1$, we observe that $T$ is a contractive map and the Banach contraction principle assures that there exists only one solution $\psi$ in $C\left(\left[0, a\right] _{\mathbb{T}}\right)$ such that $T\left(\psi\right) = \psi$, and therefore the LIVP (1.1) and (1.2) has unique $\psi$ in $C\left(\left[0, a\right] _{\mathbb{T}}\right)$. This completes the proof.

Next, we investigate the expansion to the right of the solutions of LIVP (1.1) and (1.2).

Lemma 5.2. Suppose $\psi\left(\tau\right)$ is a solution to thelocal IVP (1.1) and (1.2) defined on $\left[0, \tau^{+}\right) _{\mathbb{T}}$ such that $\tau^{+}\neq\infty$. If $\lim\limits_{\tau\rightarrow \tau^{+}}\psi\left(\tau\right)$ exists, then $\psi\left(\tau\right)$ can be expanded to $\left[0, \tau^{+}\right]_{\mathbb{T}}$ provided the hypothesis (H1) holds.

Proof. Here, $\tau^{+}$ is a right-dense point. Let $\lim\limits_{\tau\longrightarrow \tau^{+}}\psi\left(\tau\right) = \psi^{+}$. Now suppose ${\bf{J}} = \left[0, \tau^{+}\right) _{\mathbb{T}}$ and define a function $\tilde{\psi}\left(\tau\right)$ by

By [6,part (ⅰ) of Theorem 4] and since $\lim\limits_{\tau \longrightarrow\tau^{+}}\psi\left(\tau\right) = \psi^{+}$, therefore the function $\tilde{\psi}\left(\tau\right)$ is surely continuous on $\left[0, \tau^{+}\right] _{\mathbb{T}}.$ We next demonstrate that the function $\tilde{\psi}\left(\tau\right)$ is also a solution of the LIVP (1.1) and (1.2) defined on $\left[0, \tau^{+}\right] _{\mathbb{T}}$, and obviously, it is sufficient to prove

Note that the equation

And the continuities of $\tilde{\psi}$ and $k$ gives that

Moreover, using mean value theorem [23,Theorem 15], we see that for all $\tau$ in $\left[0, \tau^{+}\right) _{\mathbb{T}}$, $\exists$ a point $\zeta$ in $\left[\tau, \tau^{+}\right] _{\mathbb{T}}^{k}$ such that

Now taking the $\lim\limits_{\tau\rightarrow \tau^{+}}$ on both sides and using Eq (5.4), we obtain

By Lemma 2.2, we conclude that

Hence, we have shown that the function $\tilde{\psi}\left(\tau\right)$ is also a solution of the LIVP (1.1) and (1.2) defined on $\left[0, \tau^{+}\right] _{\mathbb{T}}$, and it is an extension of the solution $\psi\left(\tau\right)$ to $\left[0, \tau^{+}\right] _{\mathbb{T}}$. Therefore, the required result follows.

Definition 5.1. Suppose ${\bf{I}}$ is the maximal existence interval of the solution $\psi\left(\tau\right)$ of the LIVP (1.1) and (1.2), then $\psi\left(\tau\right)$ is said to be come arbitrarily close to the boundary of $\Omega = \left[0, \infty\right) _{\mathbb{T}}\times\mathbb{R}$ to the right if it is not possible for every closed and bounded domain $\Omega_{0}$ in $\Omega$, the point $\left(\tau, \psi\left(\tau\right) \right)$ always remains in $\Omega_{0}$ for all $\tau$ in ${\bf{I}}$.

Theorem 5.2. If (H1) and (H2) hold, then the solution of the local initial value problem (1.1) and (1.2) comes arbitrarily close to the boundary of $\Omega = \left[0, \infty\right)_{\mathbb{T}}\times\mathbb{R}$ to the right.

Proof. The local IVP (1.1) and (1.2) has a unique solution by Theorem 5.1, and refers to the solution by $\psi\left(\tau\right)$. Suppose $\boldsymbol{I}$ refers to the maximal existence interval of $\psi\left(\tau\right)$. Again, we conclude that using Theorem 5.1, ${\bf{I}} = \left[0, \infty\right) _{\mathbb{T}}$ or $\left[0, \tau^{+}\right) _{\mathbb{T}}$ with $\tau^{+}\neq\infty$. The required result is clear when ${\bf{I}} = \left[0, \infty\right) _{\mathbb{T}}$. Now, assume the case ${\bf{I}} = \left[0{\bf{, }}\tau ^{+}\right) _{\mathbb{T}}$ with $\tau^{+}\neq\infty$. Conversely, suppose that the desired result is not true. That is, the solution $\psi\left(\tau\right)$ of the LIVP (1.1) and (1.2) does not go arbitrarily near to the boundary of $\Omega = \left[0, \infty\right) _{\mathbb{T}}\times\mathbb{R}$ to the right implies that $\Omega_{0} \subset\Omega$ exists such that $\Omega_{0}$ is closed and bounded with $\left(\tau, \psi\left(\tau\right) \right) \in\Omega_{0}$, $\forall$ $\tau\in{\bf{I}}$. Because $k$ on $\Omega_{0}\subset\Omega$ is continuous, a positive number $C$ must exist such that

Furthermore, mean value theorem ^[23] ensures that, for any $\tau_{1}, \tau_{2}$ in ${\bf{I}}$ with $\tau_{1} < \tau_{2}$, a point $\xi$ exists in $\left[\tau_{1}, \tau_{2}\right] _{\mathbb{T}}^{k}$ such that

From Eqs (1.1) and (5.5), it follows that

Therefore,

which shows that $\psi\left(\tau\right)$ is uniformly continuous on ${\bf{I}}$, and hence $\lim\limits_{\tau\longrightarrow\tau^{+}}\psi\left(\tau\right)$ exists. And thus by using Lemma 5.2, the solution $\psi\left(\tau\right)$ can be extended to the closed interval $\left[0, \tau^{+}\right] _{\mathbb{T}}$, it violates the statement that $\left[0, \tau^{+}\right) _{\mathbb{T}}$ is the maximal existence interval of $\psi\left(\tau\right)$. Therefore, it follows the required result.

By the use of Theorems 5.1, 5.2 and Grönwall's inequality (4.3), we now present a result to ensure that the solution of Eq (1.1) with the local initial condition (1.2) is defined and bounded on $\left[0, \infty\right) _{\mathbb{T}}$.

Theorem 5.3. The solution of the local IVP (1.1) and (1.2) is defined and bounded on $\left[0, \infty\right)_{\mathbb{T}}$ whenever (H1)–(H3) hold.

Proof. The LIVP has only one solution, according to Theorem 5.1. Define the solution as $\psi\left(\tau\right)$, with $\left[0, \tau^{+}\right) _{\mathbb{T}}$ as its maximal existence interval. Now we have to show that $\tau^{+} = \infty$ and $\psi\left(\tau\right)$ is bounded on $\left[0, \tau^{+}\right) _{\mathbb{T}}$. Based on assumption (H3), Eq (5.1) follows that

And so, Eq (5.6) follows using Grönwall's inequality (4.3):

It should be noted that the hypothesis of boundedness of ${ \int_{0}^{\tau}} \xi_{\mu\left(\nu\right) }\left(k\left(\nu\right) \right) \nu ^{\alpha-1}\Delta\nu$ would imply that a positive number $C$ exists so that ${ \int_{0}^{\tau}} \xi_{\mu\left(\nu\right) }\left(k\left(\nu\right) \right) \nu ^{\alpha-1}\Delta\nu\leq C$. Therefore,

Thus $\psi\left(\tau\right)$ is bounded on $\left[0, \tau^{+}\right) _{\mathbb{T}}$. When $\tau^{+}\neq\infty$, then by Theorem 5.2 clearly follows that

This is in contradiction with the boundedness of $\psi\left(\tau\right)$ on $\left[0, \tau^{+}\right) _{\mathbb{T}}$. Therefore, $\tau^{+} = \infty$, and consequently follows the required result.

Through the Grönwall's inequality (4.3), we are further investigating the stability of the solutions to the LIVP (1.1) and (1.2).

Definition 5.2. Assume $\psi\left(\tau\right)$ be a solution to Eq (1.1) defined on $\left[0, \infty\right) _{\mathbb{T}}$ with $\psi\left(0\right) = \psi_{0}$. The solution $\psi\left(\tau\right)$ is called stable if for all positive number $\epsilon$, $\exists$ a positive number $\delta$ such that each solution $\varphi\left(\tau\right)$ with $\left\vert \varphi\left(0\right) -\psi\left(0\right) \right\vert < \delta$ holds for any $\tau\geq0$ and satisfies the inequality

Theorem 5.4. Every solution of the LIVP (1.1) and (1.2) is always stable whenever the hypothesis (H1), (H3) and (H4) holds.

Proof. By Theorem 5.3, the solution of LIVP (1.1) and (1.2) always exists and is defined on $\left[0, \infty\right)_{\mathbb{T}}$. Let $\psi\left(\tau\right)$ be a solution with$\psi\left(0\right) = \psi_{0}$ and $\varphi\left(\tau\right)$ asolution with $\varphi\left(0\right) = \varphi_{0}$. Then

and

By (H4), it follows that

And Eq (5.7) follows, using Grönwall's inequality(4.3) and the hypothesis of boundedness of ${ \int_{0}^{\tau}}\xi_{\mu\left(\nu\right) }\left(h\left(\nu\right) \right)\Delta^{\alpha}\nu$ would imply that a positive number $C$ exists so that ${ \int_{0}^{\tau}}\xi_{\mu\left(\upsilon\right) }\left(h\left(\upsilon\right) \right)\Delta^{\alpha}\upsilon\leq C$. Therefore,

By Definition 5.2, it follows that

Hence $\psi\left(\tau\right)$ is stable on $\left[0, \infty\right)_{\mathbb{T}}$.

6.   Non-local initial value problems

The existence of solutions to the non-local initial value problem (NLIVP) is addressed in this section. Then, in order to prove the main result, we recall a fixed point theorem that will be used in this section.

Lemma 6.1. (Fixed point theorem) $U$ is an open set in a Banach space$B$'s closed, convex set $C$. Suppose $0\in U$. It is also assumed that$A\left(\bar{U}\right)$ is bounded and that $A:\bar{U}\rightarrow C$ isgiven by

in which

and

That is, a nonnegative nondecreasing function $\phi:\left[0, \infty\right)_{\mathbb{T}}\mathbb{\rightarrow }\left[0, \infty\right) _{\mathbb{T}}$ exists satisfying $\phi\left(z\right) < z$ for $z > 0$, such that

for any $x, y\in\bar{U}$. Then either (C1) $A$ has a fixed point $u\in\bar{U}$; or (C2) there exist a point $u\in\partial U$ and $\lambda\in\left(0, 1\right)$ such that $u = \lambda A\left(u\right)$, where $\bar{U}$ and $\partial U$ refers to the closure and boundary of $U$, respectively.

Further we need the following hypothesis.

(H5) $k$ is a right-dense continuous function defined on $\left[0, a\right] _{\mathbb{T}}\times\mathbb{R}.$

(H6) There exists a positive constant $\gamma$ in $\left(0, 1\right)$ and a nonnegative and nondecreasing function $\phi$ in $C\left(\left[0, \infty\right) _{\mathbb{T}}\right)$ with $\phi\left(\varsigma\right) < \gamma\varsigma$, $\varsigma > 0$ and $\left\vert g\left(\tau\right) -g\left(\nu\right) \right\vert _{k}\leq\phi\left(\left\Vert \tau -\nu\right\Vert _{k}\right)$ for all $\tau, \nu$ in $C\left(\left[0, a\right] _{\mathbb{T}}\right)$.

(H7) There is a nonnegative function $\varphi\in C\left(\left[0, a\right] _{\mathbb{T}}\right)$ such that $\varphi > 0$ on a subinterval of $\left[0, a\right] _{\mathbb{T}}$, as well as a nonnegative and nondecreasing function $\Psi\in C\left(\left[0, \infty\right) _{\mathbb{T}}\right)$,

for each $\left(t, u\right)$ in $\left[0, a\right] _{\mathbb{T}} \times\mathbb{R}$ and

The following lemma is easy to verify by Lemmas 2.4 and 2.5.

Lemma 6.2. A function $\psi$ in $C\left(\left[0, a\right]_{\mathbb{T}}\right)$ satisfies the NLIVP (1.1) and (1.3)provided that assumption (H5) holds iff $\psi$ is a continuous as well assolution of IE:

We first define some sets of functions in $C\left(\left[0, a\right] _{\mathbb{T}}\right)$ and operators in order to use FPT to address the existence of solutions to the NLIVP.

Given a positive number $r$ and $k > 0$ be a positive constant, define the subset ${\large u}_{r}$ of $C\left(\left[0, a\right] _{\mathbb{T}}\right)$ by

Similarly, we define

Also, define three operators from the space $C\left(\left[0, a\right] _{\mathbb{T}}\right)$ to itself, respectively, by

Using the standard arguments, the complete continuity of the operator $A_{1}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T} }\right)$ can be verified, and it is also easy to check that the operator $A_{2}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T} }\right)$ is a nonlinear contraction under the condition (H6). Here we omit their proofs.

Lemma 6.3. The operator $A_{1}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T}}\right)$ is completely continuous provided(H5) holds.

Lemma 6.4. The operator $A_{2}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T}}\right)$ is a nonlinear contractionprovided (H6) holds.

In this section, we are now presenting the main result.

Theorem 6.1. There exists at least one solution defined on $\left[0, a\right] _{\mathbb{T}}$ of the NLIVP (1.1) and (1.3) whenever (H5)–(H7) holds$.$

Proof. A positive number $r$ exists in view of the hypothesis of supremum in (H7), such that

And then we define the set ${\Large u}_{r}$ by

We first show that the operators $A$, $A_{1}$, $A_{2}$ satisfy the corresponding conditions of FPT and owing to Lemmas 6.3 and 6.4, we just need to demonstrate the ßoundedness of $A\left({\Large \bar{u}}_{r}\right)$. Infact, for each $\psi$ in ${\Large \bar{u} }_{r}$ it implies under the assumptions (H5) and (H6) that

and

According to Eq (6.3), it implies that

By using Eq (6.10) in Eq (6.9), it follows that

Hence, according the definition of the operator $A$ and using Eqs (6.8) and (6.11), we have

which justifies the uniform boundedness of the set $A\left({\Large \bar{u} }_{r}\right)$.

Lastly, it remains to be shown that the case (C2) does not occur in the FPT (6.1). We claim, by contradiction. Suppose (C2) holds implies that $\lambda\in\left(0, 1\right)$ and $\psi\in\partial{\Large u}_{r}$ exists with $\psi = \lambda A\left(\psi\right)$, that is,

Under the hypothesis (H6) and (H7), it further follows that

By using Eq (6.3), it implies that

Using Eqs (6.10) and (6.13) in Eq (6.12), it becomes

Hence,

But

This implies that

which is in contradiction with inequality (6.7). We have therefore shown that the operators $A, \; A_{1}$ and $A_{2}$ meet all the conditions in FPT (6.1), and hence we deduce that the operator $A$ has at least one fixed point $\psi$ in ${\Large \bar{u}}_{r}$, which satisfies the NLIVP.

Remark 6.1 For $\alpha = 1$ and $\mathbb{T = R}$, the classical results corresponding to ordinary differential equations will be yielded.

Example 6.2 Assume $\mathbb{T = R}$, $r_{0} = 0$ and $p\left(s\right) = -1$, then $E_{p}\left(s, s_{0}\right) = e^{-\frac{s^{\alpha}}{\alpha}}$. Also let $\Omega{ = }\left[0, \infty\right) \times\mathbb{R}$, $k\left(s, x\right) = e^{-\frac{s^{\alpha}}{\alpha}}\left(x+\sin x\right)$, $l\left(s\right) = h\left(s\right) = 2e^{-\frac{s^{\alpha}}{\alpha}}$ and $K = 2.$

(I) Local initial value problems.

For all $\left(s, x\right)$, $\left(s, y\right) \in\Omega$,

It is easy to see that $\sin x$ is Lipschitz: $\left\vert \sin x-\sin y\right\vert \leq\left\vert x-y\right\vert$ with Lipschitz constant $L = 1$. It implies that

So

and

Since $\left\vert \sin x\right\vert \leq\left\vert x\right\vert$, therefore,

Thus,

Since $\mathbb{T = R}$, therefore $\mu\left(s\right) = 0$ and $\xi_{\mu\left(s\right) }\left[l\left(s\right) \right] = l\left(s\right)$, for which

Since we know that

Hence, we can write

which implies that

Thus, for the above-mentioned functions and variables, hypotheses (H1)–(H3) in Theorem 5.3 are met, indicating that the solution to the LIVP (1.1) and (1.2) is defined and bounded on $\left[0, \infty\right) _{\mathbb{T}}$.

(II) Stabilities.

We observe that, analogous to the case (I), for each $\left(s, x\right)$, $\left(s, y\right) \in\Omega$,

for which $I_{\alpha}h\left(s\right) \leq2$. Hence, all the requirements are satisfied for Theorem 5.4. We can derive that each solution to the LIVP (1.1) and (1.2) is always stable using Theorem 5.4.

(III) Non-local initial value problems.

Select $\left[0, a\right]$ such that $2 > e^{\frac{a^{\alpha}}{\alpha}}$. Let us define the functions

such that $0 < \gamma < \varphi\left(a\right) -1$. For $u\in C\left(\left[0, a\right] _{\mathbb{T}}\right)$, define the functional

and then it's simple to determine whether $g$ is a contraction:

Moreover, observe that

for any $\left(s, x\right) \in\left[0, a\right] \times\mathbb{R}$ as well as the fact that a direct calculation yields

As

Thus,

We can deduce that the corresponding non-local IVP (1.1) and(1.3) has at least one solution defined on $\left[0, a\right]$ because assumptions (H5)–(H7) in Theorem 6.1 are fulfilled for the aforementioned functions, functionals and parameters.

Remark 6.2. For $\alpha = 1$ and $\mathbb{T = R}$, the classical results corresponding to ordinary differential equations will be yielded.

7.   Conclusions

Based on the theory of conformable fractional calculus on time scales, we defined generalized exponential function. Also, we proved some of its fundamental properties. Furthermore, we introduced Grönwall's type inequalities in the considered frame. Through Grönwall's inequality, we investigate the stability of the solution to the LIVP. In addition, some conditions for the global existence, extension and boundedness of LIVP's solutions as well as their stabilities are established by using conformable fractional calculus on time scales and FPT. Moreover, we obtained the existence result of the nonlocal initial value problem.

Acknowledgments

The author M. A. Alqudah was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project (No. PNURSP2022R14). The author T. Abdeljawad would like to thank Prince Sultan University for the support through the TAS research lab.

Conflict of interest

The authors declare no conflicts of interest regarding this article.