Gradient estimates in generalized Orlicz spaces for quasilinear elliptic equations via extrapolation

1.   Introduction

Research has shown that fractional calculus is highly effective for capturing complex dynamics and accurately modeling real-life problems ^[9,40]. Fractional calculus extends integer-order derivatives and integrals, also known as differentiation and integration, to arbitrary orders ^[32], proving to be a powerful tool for understanding intricate systems. Numerous researchers have utilized the Lyapunov second method, or Lyapunov direct method, to analyze the qualitative and quantitative characteristics of dynamical systems. The Lyapunov direct method is particularly beneficial because it does not require knowledge of the differential equation's solution ^[35]. In ^[1,2,3,4], various fractional derivatives (FrDs) of Lyapunov functions (LF) were used in stability investigations, including the Caputo FrD, Dini FrD, and Caputo fractional Dini derivative. The most preferred is the Caputo FrD:

This derivative is easier to handle and more applicable, though it requires the function $\mathcal{L}(t, \nu(t))\in [\mathbb{R}\times\mathbb{R}^n, \mathbb{R}_+]$ to be continuously differentiable, which can be challenging. Other LF derivatives do not have this limitation, allowing sufficient conditions for these derivatives using a continuous LF that does not need to be continuously differentiable. The Dini FrD

where $\mathcal{L}:\mathbb{R}\times\mathbb{R}^n \to\mathbb{R}_+$, is continuous, $\mathfrak{f}\in C[\mathbb{R}\times\mathbb{R}^n, \mathbb{R}^n]$, $\kappa$ is a positive number and $^\alpha C_\mathfrak{r} = \frac{\alpha(\alpha-1)...(\alpha-\mathfrak{r}+1)}{\mathfrak{r}!}$, maintains the concept of FrDs, depending on both the present point ($t$) and the initial point ($t_0$) but not on the initial state $\mathcal{L}(t_0, \nu_0)$. This led to a more suitable definition

to be considered.

Several forms of stability for Caputo fractional differential equations (FrDE) with continuous domain have been investigated using this Caputo fractional Dini derivative (1.1) ^[1]. As noted in ^[7] and ^[25], a more holistic examination of stability can be achieved across time domains. In ^{[1,2,3,12,13,18]}, stability results were obtained for continuous time, ignoring discrete details, while in ^[8,23,26,28], discrete domains were considered. However, some systems undergo smooth and abrupt changes almost simultaneously, with multiple time scales or frequencies. Modeling such phenomena is better represented as dynamic systems that include continuous and discrete times, known as time scales or measure chains, denoted by $\mathbb{T}$ ^[14,17]. Dynamic equations on time scales, defined on discrete, continuous (connected), or combined domains, provide a broader analysis of difference and differential systems ^[16]. In order to extend stability properties from the classical to the fractional-order sense, we focus on the Lyapunov stability analysis of the Caputo fractional dynamic equations on time scale (FrDET) using a novel definition for the delta derivative of a LF, known as the Caputo Fr$\Delta$D on a time scale.

The study of fractional dynamic systems on time scales is recent and ongoing due to its advantages in modeling, mechanics, and population dynamics (see ^[39]). Recent literature on fractional dynamic systems on time scales focuses on the existence and uniqueness of solutions of FrDET, with Caputo-type derivatives being given more recognition (^{[10,11,20,29,30,33]}). However, in ^[24], the stability of fractional dynamic systems on time scales with applications to population dynamics were examined. Although the stability results were interesting, they applied Hyers-Ulam type stability, which is restrictive compared to Lyapunov stability, which has a broader application scope (^[19,27,37]). In ^[36,38], methods for solving discrete time scales in Caputo FrDET were developed.

Building on the existence and uniqueness results for Caputo-type FrDET established in ^[4], we extend the stability results in ^[21] to fractional order and the Lyapunov stability results for Caputo FrDE in ^[1] to a more generalized domain (time scale). This unification of continuous and discrete calculus gives rise to fractional difference equations (FrDfE) in discrete time, FrDE in continuous time, and fractional calculus on time scales in combined continuous and discrete time.

For this work, we consider the Caputo fractional dynamic system of order $\alpha$, with $0 < \alpha < 1$

where $\Upsilon\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}^n]$, $\Upsilon(t, 0)\equiv0$, and $^{C\mathbb{T}}D^\alpha \nu^\Delta$ is the Caputo Fr$\Delta$D of $\nu\in\mathbb{R}^n$ of order $\alpha$ with respect to $t\in\mathbb{T}$. Let $\nu(t) = \nu(t, t_0, \nu_0)\in C^\alpha_{rd}[\mathbb{T}, \mathbb{R}^n]$ be a solution of (1.2), assuming the solution exists and is unique (^[4,24]), this work aims to investigate the stability and asymptotic stability of the system (1.2).

To do this, we shall use the dynamic system of the form

where $\chi\in\mathbb{R_+}$, $\Xi: \mathbb{T}\times\mathbb{R_+} \to\mathbb{R_+}$ and $\Xi(t, 0)\equiv0$. System (1.3) is called the comparison system. For this work, we will assume that the system (1.3) with $\chi(t_0) = \chi_0$ has a solution $\chi(t) = \chi(t; t_0, \chi_0) \in C_{rd}^\alpha[ \mathbb{T}, \mathbb{R_+}]$ which is unique (^[4]).

In the next section (Section 2), we examine key terminologies, remarks, and a fundamental lemma laying the groundwork for the subsequent contributions. New definitions and crucial remarks are also introduced. In Section 3, we present Lemmas 3.1 and 3.2, which are essential components for proving the major results. In Section 4, practical examples demonstrate the significance and applicability of the newly introduced definitions and the established stability and asymptotic stability theorem. In Section 5, we provide the conclusion, summarizing the major findings and implications of the investigation.

2.   Preliminaries, definitions, and notations

The foundational principles of dynamic equations, encompassing derivatives and integrals, can be extended to non-integer orders by applying fractional calculus. This generalization to non-integer orders becomes particularly relevant when exploring dynamic equations on time scales, allowing for a versatile and comprehensive analysis of system behavior across continuous and discrete time domains. See ^[5,24,31,34]. In this section, we set the foundation, introduce notations, and definitions.

Definition 2.1. ^[7] For $t \in \mathbb{T}$, the forward jump operator $\sigma: \mathbb{T} \to \mathbb{T}$ is defined by

and the backward jump operator $\rho: \mathbb{T} \to \mathbb{T}$ is defined by

The following conditions hold:

(i) If $\sigma(t) > t$, then $t$ is termed right-scattered (rs).

(ii) If $\rho(t) < t$, then $t$ is termed left-scattered (ls).

(iii) If $t < \max\mathbb{T}$ and $\sigma(t) = t$, then $t$ is called right-dense (rd).

(iv) If $t > \min\mathbb{T}$ and $\rho(t) = t$, then $t$ is called left-dense (ld).

Definition 2.2. ^[7] The graininess function $\mu: \mathbb{T} \to [0, \infty)$ for $t \in \mathbb{T}$ is defined by

where $\sigma(t)$ is the forward jump operator.

Definition 2.3. ^[7] Let $\mathfrak{p}:\mathbb{T}\to\mathbb{R}$ and $t\in\mathbb{T}^k.$ The delta derivative $\mathfrak{p}^{\Delta}$ also known as the Hilger derivative is defined as:

provided the limit exist.

If $t$ is rd, the delta derivative of $\mathfrak{p}$, becomes

and if $t$ is rs, the Delta derivative becomes

where $\mathfrak{p}^\sigma$ denotes $\mathfrak{p}(\sigma(t))$.

Definition 2.4. ^[15] $\mathfrak{p}: \mathbb{T} \to \mathbb{R}$ is said to be rd-continuous if it remains continuous at each rd point within $\mathbb{T}$ and possesses finite left-hand limits at ld points in $\mathbb{T}$. The collection of all such rd-continuous functions is denoted as

Definition 2.5. ^[7] Let $\mathfrak{a}, \mathfrak{b} \in \mathbb{T}$ and $\mathfrak{p} \in C_{rd}$, then, the integration on the time scale $\mathbb{T}$ is defined as:

(i)

if $\mathbb{T} = \mathbb{R}$.

(ii) If the interval $[\mathfrak{a}, \mathfrak{b}]_\mathbb{T}$ contains only isolated points, then

(iii) If there exists a point $\sigma(t) > t$, then

Definition 2.6. ^[15] A function $\phi:[0, r]\to[0, \infty)$ is of class $\mathcal{K}$ if it is continuous, and strictly increasing on $[0, r]$ with $\phi(0) = 0$.

Definition 2.7. ^[15] $\mathcal{L}\in C[\mathbb{R}^n, \mathbb{R}]$ with $\mathcal{L}(0) = 0$ is called positive definite(negative definite) on the domain $D$ if $\exists$ a function $\phi\in\mathcal{K}$ $:$ $\phi(|\chi|)\leq \mathcal{L}(\chi)\; (\phi(|\chi|)\leq \mathcal{-L}(\chi))$ for $\chi\in D$.

Definition 2.8. ^[15] $\mathcal{L}\in C[\mathbb{R}^n, \mathbb{R}]$ with $\mathcal{L}(0) = 0$ is called positive semidefinite (negative semi-definite) on $D$ if $\mathcal{L}(\chi)\ge 0\; (\mathcal{L}(\chi) \leq 0)$ $\forall$ $\chi\in D$ and it can also vanish for some $\chi\ne0$.

Definition 2.9. ^[21] Assume $\mathcal{L}\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}_+]$, $\Upsilon\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}^n]$ and $\mu(t)$ is the graininess function, then the dini derivative of $\mathcal{L}(t, \nu)$ is defined as:

If $\mathcal{L}$ is differentiable, then $D_-\mathcal{L}^\Delta(t, \nu) = D^+\mathcal{L}^\Delta(t, \nu) = \mathcal{L}^\Delta(t, \nu)$.

Definition 2.10. ^[6] Consider $\alpha \in (0, 1)$, with $[\mathfrak{a}, \mathfrak{b}]$ being an interval on $\mathbb{T}$, and let $\Xi$ be a function that is integrable over $[\mathfrak{a}, \mathfrak{b}]$. The fractional integral of $\Xi$, w.r.t the order $\alpha$, is expressed as follows:

Definition 2.11. ^[4] Let $t\in\mathbb{T}, \; 0 < \alpha < 1$, and $\Xi:\mathbb{T}\to\mathbb{R}$. The Caputo FrD of order $\alpha$ of $\Xi$ is expressed as follows:

Lemma 2.1. ^[22] Let $\mathbb{T}$ represent a time scale with a minimal element $t_0 \ge 0$. Assume that for each $t \in \mathbb{T}$, there is a proposition $\mathbf{S}(t)$ such that the following conditions are satisfied:

(i) $\mathbf{S}(t_0)$ holds;

(ii) if $t$ is rs and $\mathbf{S}(t)$ holds, then $\mathbf{S}(\sigma(t))$ also holds;

(iii) for any rd $t$, there is a neighborhood $\mathcal{U}$ such that if $\mathbf{S}(t)$ is true, then $\mathbf{S}(t^*)$ is true for every $t^* \in \mathcal{U}$ with $t^* \ge t$;

(iv) for left-dense $t$, if $\mathbf{S}(t^*)$ holds for all $t^* \in [t_0, t)$, then $\mathbf{S}(t)$ also holds.

Therefore, $\mathbf{S}(t)$ is true for all $t \in \mathbb{T}$.

Remark 2.1. When $\mathbb{T} = \mathbb{N}$, Lemma 2.1 simplifies to the principle of mathematical induction. Specifically:

(1) $\mathbf{S}(t_0)$ being true corresponds to the statement holding for $n = 1$;

(2) $\mathbf{S}(t) \Rightarrow \mathbf{S}(\sigma(t))$ corresponds to: if the statement holds for $n = k$, then it also holds for $n = k + 1$.

Now, we give the following definitions and remarks.

Definition 2.12. Let $h\in C^\alpha_{rd}[\mathbb{T}, \mathbb{R}^n]$, the G-L Fr$\Delta$D is given by

and the G-L Fr$\Delta$DiD is given by

where $0 < \alpha < 1$, $^\alpha C_r = \frac{\alpha(\alpha-1)...(\alpha-r+1)}{r!}$, and $[\frac{(t- t_0)}{\mu}]$ represents the integer part of the fraction $\frac{(t-t_0)}{\mu}$.

Observe that if the domain is $\mathbb{R}$, then (2.4) becomes

Remark 2.2. It is necessary to note that the relationship between the Caputo Fr$\Delta$D and the G-L Fr$\Delta$D is given by

substituting (2.3) into (3.11) we have that the Caputo Fr$\Delta$D becomes

and the Caputo Fr$\Delta$DiD becomes

which is equivalent to

For notation simplicity, we represent the Caputo Fr$\Delta$D of order $\alpha$ as $^{C\mathbb{T}}D^\alpha$ and the Caputo Fr$\Delta$DiD of order $\alpha$ as $^{C\mathbb{T}}D_+^\alpha$.

Definition 2.13. The trivial solution of (1.2) is called:

$S_1$ Stable if for every $\epsilon > 0$ and $t_0\in \mathbb{T}$, $\exists$ $\delta = \delta (\epsilon, t_0) > 0$ $:$ for any $\nu_0\in \mathbb{R}^n$, $\|\nu_0\|\leq \delta$ $\implies$ $\|\nu(t; t_0, \nu_0)\| < \epsilon$ for $t\geq t_0$.

$S_2$ Asymptotically stable if it is stable and locally attractive, that is $\exists$ a $\delta_0 = \delta_0(t_0) > 0$ $:$ $\|\nu(t_0)\| < \delta_0$ $\implies$ $\lim\limits_{t\to\infty}\|\nu(t)\| = 0$ for $t_0, t\in\mathbb{T}$.

We now give the definition of the derivative of LF using the Fr$\Delta$DiD of $h(t)$, as provided in Eq (2.7).

Definition 2.14. The Caputo Fr$\Delta$DiD of the Lyapunov function $\mathcal{L}(t, \nu)\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R_+}]$ (which is locally Lipschitzian with respect to its second argument and $\mathcal{L}(t, 0)\equiv0$) along the trajectories of solutions of system (1.2) is defined as:

and can be expanded as

where $t\in \mathbb{T}$, and $\nu, \nu_0\in \mathbb{R}^n$, $\mu = \sigma(t)-t$ and $\nu(\sigma(t))-\mu^\alpha \Upsilon(t, \nu)\in\mathbb{R}^n$.

If $\mathbb{T}$ represents a discrete time scale and $\mathcal{L}(t, \nu(t))$ remains continuous at $t$, the Caputo Fr$\Delta$DiD of the LF for discrete times is expressed as:

and if $\mathbb{T}$ is continuous, that is $\mathbb{T} = \mathbb{R}$, and $\mathcal{L}(t, \nu(t))$ is continuous at $t$, we have that

Notice that (2.11) is the same in ^[1] where $\kappa > 0$.

Given that $\lim\limits_{N\to\infty}\sum_{r = 0}^{N}(-1)^r{^\alpha}C_r = 0$ where $\alpha\in(0, 1)$, and $\lim\limits_{\mu\to0^+}[\frac{(t-t_0)}{\mu}] = \infty$ therefore it is evident that

Also, based on Eq (2.7) and given that the Caputo and R-L definitions are equivalent when $h(t_0) = 0$ (see ^[1]), we can conclude that:

setting $h(\sigma(t)-r\mu = 1$ we obtain

3.   Results

Lemma 3.1. Assume $\mathfrak{h}$ and $\mathfrak{m} \in C_{rd}({\mathbb{T}}, \mathbb{R})$. Suppose $\exists$ $t_1 > t_0$, where $t_1 \in {\mathbb{T}}$, $:$ $\mathfrak{h}(t_1) = \mathfrak{m}(t_1)$ and $\mathfrak{h}(t) < \mathfrak{m}(t)$ for $t_0 \leq t < t_1$. Then, if the Caputo Fr$\Delta$DiD of $\mathfrak{h}$ and $\mathfrak{m}$ exist at $t_1$, the inequality $^{C\mathbb{T}}D^\alpha_+ \mathfrak{h}^\Delta(t_1) > ^{C\mathbb{T}}D^\alpha_+ \mathfrak{m}^\Delta(t_1)$ holds.

Proof. Applying (2.7), we have

at $t_1$, we have

Applying (2.14) to (3.1), we have

however, based on the Lemma's statement, we know that

then, we obtain

implying

Lemma 3.2. Assume that:

(1) $\nu^*(t; t_0, \nu_0)$, with $\nu^* \in C_{rd}^\alpha([t_0, T]_\mathbb{T}, \mathbb{R}^n)$, represents a solution to the system (1.2).

(2) $\mathcal{L}(t, \nu^*)\in C_{rd}[\mathbb{T}\times \mathbb{R}^n, \mathbb{R_+}]$ and for any $t\in [t_0, \infty)_\mathbb{T}, \; \nu^*\in \mathbb{R}^n$,

where $\phi\in \mathcal{K}$.

Then for $t\in [t_0, T]_\mathbb{T}$, the inequality

holds.

Proof. Let

and

Then, from (3.2) we have

Consider the system

A solution $\chi(t) = \chi(t, t_0, \chi_0)$ of (3.6) will also satisfy the Volterra delta integral equation

We claim that

In the event that this claim is untrue, there is a time $t_1\in(t_0, T]_\mathbb{T}$$:$

Lemma 3.1 is applied to (3.9) to obtain

Thus, based on (3.10) and for $t \in [t_0, T]_\mathbb{T}$, we obtain:

Clearly, (3.11) contradicts (3.5), so we conclude that the claim in (3.8) holds.

Combining (3.3), (3.4), (3.7) and (3.8) we get

$\nonumber \implies$

If $\mathbb{T}$ contains right scattered points, and $\mathcal{L}(t, \nu^*)$ is continuous, then (3.13) becomes

Theorem 3.1. Assume that:

(i) $\Xi\in C_{rd}[\mathbb{T}\times\mathbb{R_+}, \mathbb{R_+}]$ and $\Xi(t, \chi)\mu^\alpha$ is non-decreasing in $\chi$.

(ii) $\mathcal{L}\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}_+]$ be locally Lipschitz in the second variable such that

(iii) $z(t) = z(t; t_0, \chi_0)$ existing on $\mathbb{T}$ is the maximal solution of (1.3).

Then,

provided that

where $\nu(t) = \nu(t; t_0, \nu_0)$ is any solution of (1.2), $t\in\mathbb{T}$, $t\ge t_0$.

Proof. Utilizing the principle of induction as outlined in Lemma 2.1 for the assertion

(ⅰ) $\mathbf{S(t_0)}$ is true since $\mathcal{L}(t_0, \nu_0)\leq \chi_0$;

(ⅱ) Let $t$ be rs and $\mathbf{S(t)}$ be true. We need to show that $\mathbf{S}(\sigma(t))$ is true; that is

set $\mathfrak{p}(t) = \mathcal{L}(t, \nu(t))$, then, $\mathfrak{p}(\sigma(t)) = \mathcal{L}(\sigma(t), \nu(\sigma(t)))$, but from (2.7), we have

Also,

so that,

Given that $\Xi(t, \chi)\mu^\alpha$ is non-decreasing in $\mathfrak{u}$ and $\mathbf{S(t)}$ holds, it follows that $\mathfrak{p}(\sigma(t)) - z(\sigma(t)) \leq 0$, ensuring that (3.17) is satisfied.

(ⅲ) Let $t$ be rd and $\mathcal{N}$ denote the right neighborhood of $t \in \mathbb{T}$. We demonstrate that $\mathbf{S(t^*)}$ holds for $t^* \in \mathcal{N}$. This can be established by applying the comparison theorem for Caputo FrDEs, since at every rd-point $t^* \in \mathcal{N}$, $\sigma(t^*) = t^*$.

We shall make this proof in 3 parts. In Part 1, we show that the LF, $\mathcal{L}(t^*, \nu^*(t^*))$, is maximized by a solution of the comparison system; in Part 2, we show that the family of solutions of the comparison system is uniformly bounded and equi-continuous and therefore by the Arzela-Ascoli theorem, there would exist a sub-sequence that converges uniformly to a function $z(t)$; in Part 3, we show that this function $z(t)$ is indeed the maximal solution. This three parts put together concludes that $\mathcal{L}(t^*, \nu^*(t^*))\leq z(t^*)$ for $t^*\in\mathcal{N}$.

Let $\omega$ be a small enough arbitrary positive number such that $\omega\le B_\mathbb{T}$ (where $B_\mathbb{T}$ is a small enough number), for the IVP

where $t^*\in \mathcal{N}$, the function $\chi_\omega(t^*) = \chi(t^*)+\omega$ is a solution of (3.18) if and only if it satisfies:

Let $\mathfrak{p}(t^*)\in C_{rd}(\mathbb{T}, \mathbb{R_+})$ be such that $\mathfrak{p}(t^*) = \mathcal{L}(t^*, \nu^*(t^*))$.

We show that

the inequality (3.20) holds for $t^* = t_0$ since

Assuming that the inequality (3.20) is false, then, $\exists$ a point $t^*_1 > t_0$ $:$

From Lemma (3.1) it follows that

so,

and using (3.18), we arrive at

Therefore,

Now, for $t^*\in \mathcal{N}$.

but $\mathcal{L}(t^*, \nu)$ is Lipschitz in the second variable, so,

where $L > 0$ is the Lipschitz constant.

As $\mu \rightarrow 0$, $\|\nu^*(t^*-r\mu)-(\nu^*(t^*)-\mu^\alpha \Upsilon(t^*, \nu^*(t^*)))\| \rightarrow 0$, so that from (3.14) we have

Now, (3.22) with $t^* = t^*_1$ contradicts (3.21), hence (3.20) is true.

For $t^*\in \mathcal{N}$, we now show that whenever $\omega_1 < \omega_2$, then

Notice that (3.23) holds for $t^* = t_0$ since $\chi(t_0) + {\omega_1} < \chi(t_0) + {\omega_2}$ $\implies \omega_1 < \omega_2$. If the inequality (3.23) is false, there would exist a time $t^*_1$ such that $\chi_{\omega_1}(t^*_1) = \chi_{\omega_2}(t^*_1)$ and $\chi_{\omega_1}(t^*) < \chi_{\omega_2}(t^*)$ for $t_0 \le t^* < t^*_1$, $t^* \in \mathcal{N}$.

From Lemma (3.1), it follows that

However,

which is a contradiction, and so (3.23) is true. Now, from (3.23), and since $\omega \leq B_{\mathbb{T}}$, we determine that

and therefore we can say that the family of solutions $\{\chi_{\omega_i}(t^*)\}$ is uniformly bounded with bound $M > 0$ on $\mathbb{T}$. This means that $|\chi_{\omega_i}(t^*)| \leq M$ for $t^* \in \mathcal{N}$ and $\omega \in (0, B_{\mathbb{T}}]$.

We will now demonstrate that the family $\{\chi_{\omega_i}(t^*)\}$ is equicontinuous on $\mathbb{T}$. Suppose $\mathcal{S} = \sup \{|\Xi(t^*, \nu^*)| : (t^*, \nu^*) \in \mathcal{N} \times [-M, M]\}$. Next, consider $\{\omega_i\}_{i = 1}^\infty(t^*)$ as a decreasing sequence with $\lim_{i \to \infty} \omega_i = 0$, and a sequence of functions $\chi_{\omega_i}(t^*)$. Let $t^*_1, t^*_2 \in \mathcal{N}$ with $t^*_1 < t^*_2$, then the estimation that follows is valid:

A family of solutions $\{\chi_{\omega_i}(t^*)\}$ is said to be equicontinuous if given $\epsilon > 0$, we can find $\delta > 0$ such that $|\chi_{\omega_i}(t^*_2) - \chi_{\omega_i}(t^*_1)| < \epsilon$ whenever $|t^*_2 - t^*_1| < \delta$,

implying that $|\chi_{\omega_i}(t^*_2) - \chi_{\omega_i}(t^*_1)| \leq \frac{2\mathcal{S}}{\Gamma (\alpha+1)}\bigg[(t^*_2 - t_0)^\alpha\bigg] < \epsilon$ provided $|t^*_2 - t^*_1| < \delta$.

Now, we choose $\delta = \left (\frac{\epsilon \Gamma (\alpha+1)}{2\mathcal{S}}\right)^{\frac{1}{\alpha}}$, $\left (\frac{\epsilon \Gamma (\alpha+1)}{2\mathcal{S}}\right)^{\frac{1}{\alpha}} > \left(\frac{2\mathcal{S}(t^*_2 - t_0)^\alpha}{\Gamma(\alpha+1)} \times \frac{\Gamma (\alpha+1)}{2\mathcal{S}}\right)^{\frac{1}{\alpha}} = (t^*_2 - t_0)$ but $(t^*_2 - t_0) > |t^*_2 - t^*_1|$ so since $(t^*_2 - t_0) < \delta$, then $|t^*_2 - t^*_1| < \delta$. Proving that the family of solutions $\{\chi_\omega(t^*)\}$ is equicontinuous. According to the Arzelà-Ascoli theorem, the family $\{\chi_{\omega_i}(t^*)\}$ contains a subsequence $\{\chi_{\omega_{i_j}}(t^*)\}$ that converges uniformly to a function $z(t^*)$ on $\mathbb{T}$.

We then show that $z(t^*)$ is a solution of (1.3). Equation (3.19) becomes

Taking the limit as $i_j \rightarrow \infty$, then $\chi_{\omega_{i_j}}(t^*) \to z(t^*)$ on $\mathbb{T}$. Now (3.24) yields

Thus, $z(t^*)$ is a solution of (1.3) on $\mathbb{T}$. Since $\lim\limits_{j\to\infty}\chi_{\omega_{i_j}}(t^*) = z(t)$ exists, then for any $\chi_{\omega_i}$ that satisfies the dynamic equation (1.3), $\chi_\omega(t^*) \leq z(t^*)$. So from (3.20), we have that $\mathfrak{p}(t^*) < \chi_\omega(t^*) \leq z(t^*)$ on $\mathbb{T}$. Therefore by induction principle, the statement $\mathbf{S(t)}$ is true. Completing the proof.

Remark 3.1. Although comparison theorems for FrDE, FrDfE, and FrDET focus on understanding the behavior of solutions using a simpler comparison system, they differ in their time domains: FrDE has a continuous time domain, FrDfE has a discrete domain, and FrDET combines both. FrDE applies to continuous time systems, while FrDfE applies to discrete time systems. Theorem 3.1 examines the behavior of the LF concerning the maximal solution of the comparison system (1.3), considering an arbitrary time domain with a jump operator $\sigma(t)$ that can be discrete or continuous. This is illustrated in conditions (ii) and (iii) of Lemma 2.1 in the proof of Theorem 3.1. This suggests that the comparison theorems found in the literature ^[1,2] address only a specific case (case iii) of Theorem 3.1, particularly condition (iii), when $\mathbb{T} = \mathbb{R}$.

Theorem 3.2. (Stability) Assume that:

(1) $\mathcal{L}(t, \nu) \in C_{rd}[\mathbb{T} \times \mathbb{R}^n, \mathbb{R_+}]$, $\mathcal{L}(t, \nu(t))$ is locally Lipschitz with respect to $\nu$, $\mathcal{L}(t, 0) \equiv 0$, and

holds $\forall$ $(t, \nu) \in \mathbb{T} \times \mathbb{R}^n$ and $\phi \in \mathcal{K}$.

(2) $\Xi \in C_{rd}[\mathbb{T} \times \mathbb{R_+}, \mathbb{R_+}]$ is nondecreasing with respect to the second variable at all $t \in \mathbb{T}$, $\Xi(t, 0) \equiv 0$, and

(3) The zero solution of the comparison equation (1.3) is stable.

Then, the zero solution of the system (1.2) is stable.

Proof. By the assumption of stability of the zero solution of (1.3), let $\epsilon > 0$ be given, and for $\phi(\epsilon)$ and $t_0 \in \mathbb{T}$, there exists $\lambda = \lambda(t_0, \epsilon) > 0$ $:$

whenever $\chi_0 < \lambda$, where $z(t) = z(t, t_0, \chi_0)$ is the maximal solution of (1.3).

Given that $\mathcal{L}(t, 0) = 0$ and $\mathcal{L} \in C_{rd}$, which implies continuity of $\mathcal{L}$ at the origin, then given $\lambda > 0$, we can find a $\delta = \delta(t_0, \lambda) > 0$ $:$ for $\nu_0 \in \mathbb{R}^n$, if $\|\nu_0\| < \delta$, then $\mathcal{L}(t_0, \nu_0) < \lambda$.

Claim that $\|\nu_0\| < \delta$ implies $\|\nu(t)\| < \epsilon$, $\forall \, t \in \mathbb{T}$, where $\nu(t) = \nu(t, t_0, \nu_0)$ is any solution of (1.2). If this claim is incorrect, then there would exist a time $t_1 \in \mathbb{T}$, $t_1 > t_0$ $:$ the solution $\nu(t)$ of the dynamic system (1.2) at the instant time $t_1$ leaves the $\epsilon- \text{neighborhood}$ of the zero solution. That is $\|\nu(t)\| < \epsilon$ at $t_0 \leq t < t_1$ and

However, we know from Theorem 3.1 that

provided $\mathcal{L}(t_0, \nu_0) \leq \chi_0$, where $z(t)$ is the maximal solution of system (1.3).

Combining (3.26), (3.27), (3.29), and (3.28) for $t = t_1$, we obtain

The contradiction (3.30) shows that $t_1 \notin \mathbb{T}$ and therefore $\|\nu(t)\| < \epsilon$ $\forall \, t \in \mathbb{T}$ whenever $\|\nu_0\| < \delta$, and so the zero solution (1.2) is stable.

Theorem 3.3 (Asymptotic Stability). Assume the following:

(1) $\mathcal{L}(t, \nu)\in C_{rd}[\mathbb{R}_+\times \mathbb{R}^n, \mathbb{R}_+]$ is locally Lipschitz in $\nu$ for each $t\in\mathbb{T}$ and $\mathcal{L}(t, 0)\equiv 0$.

(2) For $t \geq t_0$,

(3) The inequality

Then, the zero solution $\nu = 0$ of the fractional dynamic system (1.2) is asymptotically stable.

Proof. By Theorem 3.2, the zero solution $\nu = 0$ of (1.2) is stable. It remains to show that

We shall make this proof in two phases.

Phase 1. Assuming (3.31) is not true, such that $\liminf\limits_{t\to \infty}\|\nu(t)\|\ne0$, then there would exist $T > 0$ such that for a given $\epsilon > 0$,

By condition 3, we deduce that $\mathcal{L}(\nu(t))$ is monotone decreasing and

since $\mathcal{L}(t, \nu(t))$ is positive definite and only $0$ at $\nu = 0$.

By utilizing Lemma 3.2 in relation to condition 3, we get

As $t\to \infty$, the RHS of (3.34) approaches $-\infty$. This is a contradiction, so $\liminf\limits_{t\to \infty}\|\nu(t)\| = 0$.

Phase 2. If $\limsup\limits_{t\to \infty}\|\nu(t)\|\ne0$, then given $\eta > 0$, there is a divergent sequence $\{t_k\}$, $:$ $\|\nu(t_k)\|\ge\eta$.Each $t_k\in\mathbb{T}$ could potentially be related to one of the following:

(ⅰ) $t_k$ is rs and ls (isolated points).

(ⅱ) $t_k$ is rs and ld.

(ⅲ) $t_k$ is rd and ls.

(ⅳ) $t_k$ is rd and ld (dense points).

Suppose $\exists$ a divergent subsequence $\{t_i\}$ of $\{t_k\}$, where each $t_i$ falls into one of the four cases mentioned above. Then, by Lemma 3.2 and Definition 2.5, we have that

for case (ⅰ),

for all $t_i, t_j\in \mathbb{T}$ and $r(t) = (t-s)^{\alpha-1}$.

This leads to a contradiction as $i \to \infty$, since $\mu(t_i)$ remains constant for each $i$. Therefore, $\limsup\limits_{t \to \infty} \|\nu(t)\| = 0$.

For case (ⅱ),

for $r(t) = (t-s)^{\alpha-1}$. This results in a contradiction as $i \to \infty$, since $\mu(t_i)$ is a constant for each $i$. So $\limsup\limits_{t\to \infty}\|\nu(t)\| = 0$.

For cases (ⅲ) and (ⅳ)

As $t_i\to \infty$, the right-hand side of (3.35) approaches $-\infty$ also contradicting the definition of $\mathcal{L}(t, \nu(t))$; $\implies\limsup\limits_{t\to \infty}\|\nu(t)\| = 0$.

Since $\liminf\limits_{t\to \infty}\|\nu(t)\| = \limsup\limits_{t\to \infty}\|\nu(t)\| = 0$, so, (3.31) holds. Then the zero solution $\nu = 0$ of (1.2) is asymptotically stable.

Remark 3.2. Theorems 3.2 and 3.3 represent a significant advancement in fractional calculus and stability analysis, building on research in FrDfE, FrDE, and integer-order dynamic equations. Although similar to the stability analysis in ^{[1,2,3,23,24,26,28,33,35,36,38]}, which addresses the stability of the zero solution of fractional dynamic systems, they differ by generalizing the time domain of system (1.2). This allows for stability analysis on various time scales, including discrete, continuous, and mixed. Moreover, the results extend to non-integer orders, enabling a more comprehensive analysis of system (1.2)'s behavior. Theorems 3.2 and 3.3 are better suited for analyzing the behavior of solutions in complex systems with multiple time scales and non-uniform grids, which can be challenging with FrDE and FrDfE.

4.   Discussion

Let us examine the dynamic system

for $t\geq t_0$, with initial conditions

where $\nu = (\nu_1, \nu_2)$ and $\mathcal{L} = (\mathcal{L}_1, \mathcal{L}_2)$.

Consider $\mathcal{L}(t, \nu_1, \nu_2) = |\nu_1|+|\nu_2|$, for $t\in\mathbb{T}$ and $(\nu_1, \nu_2)\in\mathbb{R}^2$. Then we compute the Dini derivative for $\mathcal{L}(t, \nu_1, \nu_2) = |\nu_1|+|\nu_2|$ as follows; from (2.2), we have that

Now, consider the consider the comparison equation

with solution

Even though conditions (ⅰ)–(ⅲ) of ^[21] are satisfied that is $\mathcal{L}\in C_{rd}[\mathbb{T}\times\mathbb{R}^n, \mathbb{R}_+]$, $D^+\mathcal{L}^\Delta(t, \nu_1, \nu_2)\leq \Xi(t, \mathcal{L}(t, \nu))$ and $\sqrt{\nu_1^2+\nu_2^2}\leq|\nu_1|+|\nu_2|\leq2(\nu_1^2+\nu_2^2)$, for $\mathfrak{b}(\|\nu\|) = r$ and $\mathfrak{a}(\|\nu\|) = 2r^2$, it is obvious to see that the solution (4.3) of the comparison system (4.2) is not stable, so we can not deduce the stability properties of the system (4.1) by applying the basic definition of the Dini-derivative to the LF $\mathcal{L}(t, \nu_1, \nu_2) = |\nu_1|+|\nu_2|$.

Now, we will apply our new definition on the same system but as a Caputo fractional dynamic system

for $t\geq t_0$, with initial conditions

where $\nu = (\nu_1, \nu_2)$ and $\Upsilon = (\Upsilon_1, \Upsilon_2)$.

Consider $\mathcal{L}(t, \nu_1, \nu_2) = |\nu_1|+|\nu_2|$, for $t\in\mathbb{T}$ and $(\nu_1, \nu_2)\in\mathbb{R}^2$. Then condition 1 of Theorem (3.2) is satisfied, for $\phi = \frac{1}{2}r$, where $\phi\in\mathcal{K}$, with $\nu = (\nu_1, \nu_2)\in \mathbb{R}^2$, so that the associated norm $\|\nu\| = \sqrt{\nu_1^2+\nu_2^2}$.

Since,

then, $\phi(\|\nu\|)\leq \mathcal{L}(t, \nu_1, \nu_2)$. From (2.9), we compute the Caputo Fr$\Delta$DiD for $\mathcal{L}(t, \nu_1, \nu_2) = |\nu_1|+|\nu_2|$ as follows:

Applying (2.12) and (2.14) we obtain

As $t\rightarrow \infty, \frac{(t-t_0)^{-\alpha}}{\Gamma(1-\alpha)} (\left|\nu_1(\sigma(t))\right|+\left|\nu_2(\sigma(t))\right|)\to 0$, then

Therefore,

Consider the comparison system

using the Laplace transform method

taking the inverse Laplace transform we have

where $E_{\alpha, 1}(z)$ is the Mittag-Leffler function, which can be approximated as:

Now, let $\left|\chi_0\right| < \delta$, then from (4.11), we have $\left|\chi(t)\right|$ = $\left|3\chi_0 E_{\alpha, 1}(-t^\alpha)\right|$ = $\left|3\chi_0\exp\left[-\frac{t^\alpha}{\Gamma(1+\alpha)}\right]\right|$ < $3\left|\exp\left[-\frac{t^\alpha}{\Gamma(1+\alpha)}\right]\right|\delta$ < $\epsilon$ whenever $\left|\chi_0\right|$ < $\delta = \frac{\epsilon}{3\left|\exp\left[-\frac{t^\alpha}{\Gamma(1+\alpha)}\right]\right|}$.

Therefore given $\epsilon > 0$, we can find a $\delta > 0$ such that $\left|\chi(t)\right| < \epsilon$ whenever $\left|\chi_0\right| < \delta$.

We conclude that the trivial solution of system (4.4) is stable as it satisfies all the conditions of Theorem 3.2 and the trivial solution of the comparison system (4.10) is stable.

Figure 1 below is the graphical representation of $E_{\alpha, 1}(-3t^\alpha)$. The behavior of the curve shows stability of the solution $\chi(t)$ over time.

Consider the Caputo fractional dynamic system

for $t \geq t_0$, with initial conditions

where $\upsilon = (\upsilon_1, \upsilon_2, \upsilon_3)$, $\Upsilon = (\Upsilon_1, \Upsilon_2, \Upsilon_3)$.

Consider $\mathcal{L}(t, \upsilon_1, \upsilon_2, \upsilon_3) = \upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2$, for $t \in \mathbb{T}$ and $(\upsilon_1, \upsilon_2, \upsilon_3) \in \mathbb{R}^3$. Then, condition 1 of Theorem (3.3) is satisfied, for $b(\|\upsilon\|) \leq \mathcal{L}(t, \upsilon) \leq a(\|\upsilon\|)$, with $b(r) = r$, $a(r) = 2r^2$, $a, b \in \mathcal{K}$, where the associated norm $\|\upsilon\| = \sqrt{\upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2}$.

Since

then, $\sqrt{\upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2} \leq \upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2 \leq 2(\upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2)$. From (2.9), we compute the Caputo Fr$\Delta$DiD for $\mathcal{L}(t, \upsilon_1, \upsilon_2, \upsilon_3) = \upsilon_1^2 + \upsilon_2^2 + \upsilon_3^2$ as follows:

Applying (2.12) and (2.14) we obtain

As $t\rightarrow \infty$, $\frac{(t-t_0)^{-\alpha}}{\Gamma(1-\alpha)} \left[(\upsilon_1(\sigma(t)))^2 + (\upsilon_2(\sigma(t)))^2 + (\upsilon_3(\sigma(t)))^2\right] \to 0$, then

applying $\upsilon(\sigma(t))\leq\mu^{C\mathbb{T}}D^\alpha \upsilon(t) + \upsilon(t)$

If $\mathbb{T} = \mathbb{R}$ we have that $\mu = 0$, so that (4.9) becomes;

Therefore,

Consider the comparison system

Applying the Laplace transform method, we obtain

Now, let $\chi_0 < \delta$, then from (4.11), we have $\chi(t) = 2\chi_0E_{\alpha, 1}(-t^\alpha) < 2E_{\alpha, 1}(-t^\alpha) < \epsilon$ whenever $\chi_0 < \delta = \frac{\epsilon}{2E_{\alpha, 1}(-t^\alpha)}$

Therefore given $\epsilon > 0$, we can find a $\delta(\epsilon) > 0$ (independent of $t$) $:$ $\chi(t) < \epsilon$ whenever $\chi_0 < \delta$ If $\mathbb{T} = \mathbb{N}_0$ we have that $\mu = 1$, so that (4.9) becomes;

considering the same comparison system as (4.10), we also arrive at the same conclusion as (4.11). Since all the conditions of Theorem 3.3 are satisfied, and zero solution of the comparison system (4.10) is stable, then we conclude that the zero solution of system (4.8) is stable and also asymptotically stable.

Figure 2 below is the graphical representation of $\chi(t) = E_{\alpha, 1}(-2t^\alpha)$. The behavior of the curves further buttresses the stability of $\chi(t)$ over time for $\alpha\in(0, 1)$.

5.   Conclusions

In conclusion, our study significantly advances the understanding of Lyapunov stability for Caputo FrDET. The novelty of our work is in the development of a new Dini derivative (the Caputo Fr$\Delta$DiD) for a Lyapunov function, which preserves the properties of FrD, requires only right dense continuity of the function and depends on the initial data $\mathcal{L}(t_0, \nu_0)$. Our novel definition generalizes existing definitions because it unifies the continuous ($\sigma(t) = t$) and discrete ($\sigma(t) > t$) time domain, as can be observed in Eqs (2.10) and (2.11). We have also shown the theoretical applicability of this definition in Theorems 3.1, 3.2, and 3.3 and the practical applicability as well as effectiveness in systems (4.4) and (4.8). Also, Figures 1 and 2 show a consistent behavior of the curves (a downward trend towards the trivial solution). This behavior reinforces the stability of the solutions obtained for systems (4.4) and (4.8), providing visual confirmation of the theoretical results. The new concept developed in this work successfully contributes to the advancement of fractional calculus in general and stability theory in particular from a continuous domain to a unified continuous and discrete domain, which is a breakthrough for modeling and other practical applications. By establishing comparison results and stability criteria, we have provided a solid theoretical foundation for analyzing the stability properties of these equations across time scales.

Author contributions

Michael Precious Ineh: Conceptualization, Methodology, Software, Investigation, Writing original draft; Edet Peter Akpan: Conceptualization, Methodology, Investigation, Supervision; Hossam A. Nabwey: Conceptualization, Software, Investigation, Validation, Funding acquisition. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors extend their appreciation to Prince Sattam Bin Abdulaziz University for funding this research through project number (PSAU/2024/01/ 921606).

Conflict of interest

The authors declare that they have no conflicts of interest.

Supplementary