The Fractional Lyapunov Direct Method for ψ-Caputo Fractional-Order Systems

1. Introduction

The Lyapunov stability theory (see, e.g., [1,2,3]) is quite old but relatively unknown to many different mathematicians and potential researchers dealing with mathematics, sciences, and engineering. This theory builds new mathematical insights to predict general stability dynamics of equilibria of continuous-time ordinary differential systems. In the absence of complete new mathematical theories, the legacy of Lyapunov’s work [4] is still blooming in the modern stability theory of ordinary differential systems, nonlinear dynamical systems, and control theory. One of the greatest ideas of Lyapunov concerns the Lyapunov direct method that connects the notion of derivative to a non-negative Lyapunov function and builds a uniformly negative definite or semi-definite characterization along many such differential systems’ non-trivial trajectories. This suffices to say that the trajectories of many such systems either tend to equilibria or stay near to them as time $t\to \infty$. The novel pursuit of findings of Lyapunov functions is interesting, especially for complicated nonlinear systems, and can be promising to build an entirely new scalar equation or inequality for an effective dynamical stability analysis beyond equilibrium points. An application of such a Lyapunov method gives a quantification tool to measure exponential growth-decay properties of solutions to many such systems in nonlinear dynamics and control theory.

The Lyapunov direct method was believed to be hard in general to go beyond such ordinary differential systems, especially for fractional-order systems that use knowledge of global fractional calculus operators. The ideas of fractional calculus are ancient and date back to 1695 with a thoughtful question from mathematician Guillaume de l’Hôpital to Gottfried Wilhelm Leibniz: can the order of integer derivatives and integrals be extended? Their perceptions of non-trivial concepts as rational, irrational, or real orders have shown new light in diverse areas of modern sciences and engineering [5,6]. See, for instance, Valério et al. [7] discussed some pioneers of applications of fractional calculus that motivated diverse mathematicians, physicists, and engineers. Call these concepts fractional derivatives or integrals, which continually aim to build potential fractional-order systems that allow effective initial conditions subject to initial time, and the motion of state evolution in Euclidean space ${\mathbb{R}}^n$ could be all past history dependent. There exist many different potential fractional derivative operators in recent decades that build phenomenal characterization to the rational, irrational, real orders, etc., attached to them, which seem complicated to memorize and can be learned in review works [9,10]. The most basic ones that use the knowledge of continuity and differentiability are well-known Riemann-Liouville and Caputo fractional derivative operators. Both these kinds of operators develop deep insights into symbolic formulas when one wishes to represent them in adequate formulations.

There are some problems in the stability theory of fractional-order systems; Lyapunov’s brilliant ideas are instinctively needed to characterize the behavior of solutions as time $t\to \infty$. One of the goals is to see what the fractional Lyapunov theory is in the basic qualitative stability theory of importance in our universal fractional calculus. The complete fractional Lyapunov direct method still seems far from reach, but it does not mean impossible to formulate. For some typical problems, it actually concerns the computation of fractional derivatives of Lyapunov functions, which seem obviously non-trivial. This is indeed true in the sense that the chain rule and product rules, like classical ordinary derivatives, are not valid for fractional derivatives, making it reasonably difficult to guess intuitive Lyapunov functions to work with (see, e.g., [11,12]). Call the fractional Lyapunov direct method that concerns many different versions of Lyapunov theorems that have been studied by notable authors in some basic understanding of different governing fractional-order systems in qualitative stability theory. See, for instance, Caputo-type fractional-order systems [13,14,15,16,17,18,19,20,21], tempered fractional dynamical systems [28,29], Hadamard-type fractional-order systems [30], Caputo-Fabrizio fractional-order systems [24], power fractional differential systems [25], generalized proportional fractional-order systems [31,32], and time-varying order derivative systems [26,27], where the authors have proved some new results on Lyapunov stability theory by considering a single fractional derivative of a Lyapunov function with respect to an order attached to many of these systems. Their results ultimately give rise to limiting behaviors of solutions to many such governing systems. We do not know how to extend beyond their extraordinary results and method to advance stability theory in respective directions of studies of fractional-order systems, which seem crucial and quite challenging. In [22] and a subsequent work [23], the author addresses multiple fractional derivatives of different orders to the components of a scalar Lyapunov function to predict stability dynamics of equilibria of incommensurate Caputo-type fractional-order systems and proposes a fractional Lyapunov direct method.

The chain rule, like easier ordinary derivatives, was believed to be true at first glance, but Tarasov [11] demonstrated that this rule is not valid for global fractional derivatives. Osler [33] introduced the fractional derivative of a complex-valued function with respect to another complex-valued function and represented it in complex integral form. On the other hand, Kilbas et al. [8] demonstrated that the fractional derivatives of a function with respect to another function can be entirely formulated in a unified manner and suggested the Riemann-Liouville fractional integral and fractional derivative. The $\psi$-Riemann-Liouville fractional derivative seems prominent, and there could be a vast, deep mathematical constructive theory. Quite often, the mentioned fractional derivative is not tied up with many different applications that seek representative $\psi$-Riemann-Liouville fractional-order systems. This is obvious in the sense that a vast majority of research concerns physical systems that amend fractional derivative operators with physically coherent initial conditions in a similar form as in ordinary derivatives’ counterparts. For instance, the position, velocity, acceleration, etc., of the state of any physical systems or control systems at some reasonable specified initial time could be promising to capture the nature of defining representative fractional-order systems. It is quite surprising that Almeida [34] independently introduced the $\psi$-Caputo fractional extension of the standard Caputo fractional derivative, which develops a promising operator making a difference to the $\psi$-Riemann-Liouville fractional derivative. Their work managed to build a $\psi$-Mittag-Leffler function that explicitly posed an analytic solution to a $\psi$-Caputo linear fractional differential equation. Recently, Sousa and Oliveira [35] developed a typical generalization to the Hilfer fractional derivative and proposed the notion of the $\psi$-Hilfer fractional derivative, which gives a unified generalization to both $\psi$-Caputo and $\psi$-Riemann-Liouville fractional derivatives.

The theories and applications of $\psi$-Caputo fractional derivatives seem crucial to some frontier areas of infinitely many problems arising in new mathematics and allied areas of sciences and engineering. In scientific demonstrations, the $\psi$-Caputo fractional derivatives are essential to some problems on control theory, synchronizing dynamics, and chaotic dynamics to uncover unexplored dynamics proposed by different researchers in recent works [46,47,48,49,50]. In short, Lenka [46] proposed $\psi$-Mittag-Leffler asymptotic stability and developed sufficient conditions to control problems designed for a class of $\psi$-Caputo fractional-order systems. Lenka and Upadhyay [47] demonstrated synchronization of memory chaos arising in nonlinear $\psi$-Caputo fractional-order systems. Omri and Mabrouk [48] considered $\psi$-Caputo fractional-order homogeneous systems and studied the stabilization problem of autonomous single-input, affine control systems by means of homogeneous feedback. Omri [49] studied the problem of stabilization of $\psi$-Caputo fractional homogeneous polynomial systems and constructed stabilizing feedback laws by using Lyapunov functions. N’Gbo et al. [50] proposed theoretical fractional Lyapunov exponents to measure the complexity of attractors arising in nonlinear $\psi$-Caputo fractional-order systems. On the other hand, the stability dynamics of understanding $\psi$-Caputo fractional-order systems remain crucial to qualitative stability theory, and some progress has been made by different authors in mathematical works [36,37,38,39,40,51]. For instance, Almeida et al. [38] studied commensurate $\psi$-Caputo fractional-order systems and introduced a typical integral unboundedness condition with the nonlinear function whenever $t\to \infty$ that gives long-term asymptotic behavior of non-trivial solutions. Later, Almeida et al. [40] developed a sufficient condition to commensurate the linear class of $\psi$-Caputo systems, which guarantees not only asymptotic stability of the zero solution but also its components’ decay like ${\left(\psi \left(t\right)\right)}^{-\alpha }$ with $\alpha \in (0,1]$. Li and Li [36] found trajectories of $\psi$-Caputo fractional-order systems can obey $\psi$-algebraic decay, and they developed some new local stability results for autonomous nonlinear systems. Lenka and Bora [37] studied Lyapunov functions and formulated some Lyapunov stability theorems under very restricted assumptions for $\psi$-Caputo fractional-order systems. They have introduced so-called autonomous and non-autonomous Lyapunov functions, which seem essential to stability dynamics of some complicated $\psi$-Caputo fractional-order systems. Recently, Lenka [39] established comparison theories for incommensurate $\psi$-Caputo fractional-order systems and showed that a Metzler matrix can be effective in some problems to completely predict the global asymptotic stability of such systems’ equilibrium points. Lien et al. [51] studied a class of $\psi$-Caputo fractional-order time-delay systems by including bounded delay arguments and developed some sufficient conditions for Mittag-Leffler stability by the applications of reasonable comparison techniques.

The Lyapunov direct method has a territory that has shown some new hope for the basic stability analysis of Caputo-type fractional-order systems in the literature (see, for example, Lenka’s work [53]). This method has potential; it demands adequate knowledge of fractional derivatives of Lyapunov functions. As we can see, going beyond the Lyapunov stability theory of standard Caputo fractional-order systems still remains unknown and partially extension known, as suggested in mathematical work by Lenka and Bora [37]. Is it possible to expand the Lyapunov direct method for stability analysis of $\psi$-Caputo fractional-order systems? We are interested in this question and wish to address a reasonable Lyapunov stability theory in a deeper notion.

The prime goal of this paper is to establish Lyapunov direct theorems for $\psi$-Caputo fractional-order systems that are linked with the properties of some generalized $\psi$-Gronwall inequalities and Class-${\mathcal{K}}_{\infty }$ functions. In this context, we introduce two new lemmas that develops non-negative solutions with a final limit of 0 as $t\to \infty$ of a class of $\psi$-Caputo fractional differential equations. Then, we give new formulations of some generalized $\psi$-Gronwall inequalities that involve non-positive, non-negative, or both types of singular kernels that provide reasonable bounds in terms of the $\psi$-Mittag-Leffler function. The mentioned lemma and some generalized $\psi$-Gronwall inequalities provide new characterizations to our improved Lyapunov stability theorems. Moreover, we also establish higher versions of Lyapunov theorems by using the knowledge of negative definite or semi-definite characterizations of fractional derivatives of Lyapunov functions along such systems. Finally, we consider some exemplary models to demonstrate applicable Lyapunov theorems for predicting stability dynamics to their equilibrium points.

In short, this paper highlights new developments in the sections below. In Section 2, the standard $\psi$-Caputo fractional-order system is considered, and some definitions that serve this system are recalled. Two new lemmas that build non-negativity and an estimate whenever $t\to \infty$ are proposed. In Section 3, we introduce new formulations to some generalized $\psi$-Gronwall inequalities that contain both non-negative and non-positive singular kernels. In Section 4, we put forward new Lyapunov theorems for $\psi$-Caputo fractional-order systems that concern stability dynamics of their equilibrium points. In Section 5, we consider some modified dynamic models in the sense of $\psi$-Caputo fractional derivatives and discuss the novelty of our new results. In Section 6, we close this paper with a further remark.

Notations: Let $\mathbb{N}$ be the set of natural numbers, ${\mathbb{R}}_{+}$ be the set of positive real numbers, ${\mathbb{R}}_{\ge 0}$ be the set of non-negative real numbers, $\mathbb{R}$ be the set of real numbers, and $U^T$ be the transpose of $U\in {\mathbb{R}}^{m\times n}$. ${\mathbb{R}}^n$ is the Euclidean space, $\parallel ·\parallel$ is the standard Euclidean norm, ${\mathcal{C}}^0$ is the space of continuous functions, and ${\mathcal{C}}^n$ is the space of $\mathit{n}$-times continuously differentiable functions.

2. Preliminary Background

This section has two main goals. The first one is to give a compact notion to so-called $\psi$-Caputo fractional-order systems. The second one is to find limiting behavior of a class of $\psi$-Caputo fractional differential equations.

In a prior work [34], Almeida expanded some basics of $\psi$-Caputo fractional derivatives, and later Li and Li [36] exploited the asymptotics of linear and linearization of nonlinear $\psi$-Caputo fractional-order systems. In short, we recall the left $\psi$-fractional operators below that serve our considered system (3).

Definition 1.

[34,36] Let $n\in \mathbb{N}$, $\gamma \in {\mathbb{R}}_{+}$, and $I:-\infty \le t^{★}<t^{•}\le \infty$. Let $\eta \in {\mathcal{C}}^0\left(I\right)$. Let $\psi \in {\mathcal{C}}^1\left(I\right)$ be increasing such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in \left(t^{★},t^{•}\right)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. The γ-order left ψ-Riemann-Liouville fractional integral of η is defined by

$$\begin{array}{c}\hfill {}^{RL}{\mathfrak{I}}_{t^{★},t}^{-\gamma ,\psi }\eta \left(t\right):={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\vartheta \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\vartheta \right)\eta \left(\vartheta \right)d\vartheta ,t>t^{★}.\end{array}$$

(1)

Definition 2.

[34,36] Let $n\in \mathbb{N}$, $\gamma \in {\mathbb{R}}_{+}$, and $I:-\infty \le t^{★}<t^{•}\le \infty$. Let $\eta \in {\mathcal{C}}^0\left(I\right)$, which is ${\mathcal{C}}^n\left(\left(t^{★},t^{•}\right)\right)$. Let $\psi \in {\mathcal{C}}^n\left(I\right)$ be increasing such that ${\psi }^{\prime }\left(t\right)\ne 0$, for all $t\in \left(t^{★},t^{•}\right)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. The γ-order left ψ-Caputo fractional derivative of η is defined by

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }\eta \left(t\right):=\{\begin{array}{cc}{}^{RL}{\mathfrak{I}}_{t^{★},t}^{-(n-\gamma ),\psi }{\eta }_{\psi }^{\left[n\right]}\left(t\right),t>t^{★},\hfill & \mathit{when}\gamma \in (n-1,n),\hfill \\ {\eta }_{\psi }^{\left[n\right]}\left(t\right),\hfill & \mathit{when}\gamma =n,\hfill \end{array}\end{array}$$

(2)

where ${\eta }_{\psi }^{\left[n\right]}\left(t\right):={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n\eta \left(t\right)$.

Throughout this paper, we consider the $\psi$-Caputo fractional-order system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }x\left(t\right)& =q(t,x(t\left)\right)\hfill \\ \hfill x\left(t^{★}\right)& =x_{t^{★}}\hfill \end{array}\end{array}$$

(3)

where initial time $t^{★}\in \mathbb{R}$, order $\gamma \in (0,1]$, state variable $x={\left(x_1,x_2,\cdots ,x_n\right)}^T\in {\mathbb{R}}^n$, left $\psi$-Caputo fractional derivative operator ${}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }x\left(t\right)={\left({}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }{x}_1\left(t\right),{}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }{x}_2\left(t\right),\cdots ,{}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }{x}_n\left(t\right)\right)}^T\in {\mathbb{R}}^n$, and the function $q:[t^{★},\infty )\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous in its arguments.

In this system (3), the only fractional derivative order is $\gamma$; we thus call it a commensurate-order system. Whenever $q(t,x(t\left)\right)$ is independent of time, meaning $q(t,x(t\left)\right)=q\left(x\right(t\left)\right)$, it becomes an autonomous system; otherwise, it is a non-autonomous system.

In many applications of interest, the constant vectors are needed for stability analysis of many such systems. We put them in a definition below.

Definition 3.

We say a constant vector $x_{eq}\in {\mathbb{R}}^n$ is an equilibrium point of system (3) iff it satisfies the equation in (3).

Remark 1.

Any non-zero equilibrium $x_{eq}\ne 0$ of (3) can always be translated to a zero equilibrium system of this type.

The Mittag-Leffler function is better known as the queen function of fractional calculus, which is stated below in the Definition 4. We need this function, which can be very useful to understand typical bounds to generalized $\psi$-Gronwall inequalities and some key concepts of Lyapunov stability theory addressed in Section 3 and Section 4, respectively.

Definition 4.

[41] The Mittag-Leffler function $E_{{\beta }_1,{\beta }_2}:\mathbb{R}\to \mathbb{R}$ with two parameters is defined by

$$\begin{array}{c}\hfill E_{{\beta }_1,{\beta }_2}\left(t\right):={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{t^k}{\Gamma ({\beta }_{1}k+{\beta }_2)}},t\in \mathbb{R},{\beta }_1>0,{\beta }_2>0.\end{array}$$

(4)

A class function Class-${\mathcal{K}}_{\infty }$ is known to reach ∞ radially is stated below. We need such a class function to prove our main Lyapunov stability theorems in Section 4.

Definition 5.

[2] We say a function $\chi :[0,\infty )\to [0,\infty )$ belongs to Class-${\mathcal{K}}_{\infty }$if

i) 

it is continuous and strictly increasing,

ii) 

it satisfies $\chi \left(0\right)=0$ and $\chi \left(r\right)\to \infty$ as $r\to \infty$.

This class function can be used to estimate the limit of an unknown solution associated with some $\psi$-Caputo fractional differential equations. For our purpose, we consider the two lemmas below.

Lemma 1.

Let $\eta :[t^{★},\infty )\to \mathbb{R}$ be an unknown continuous function that satisfies the initial value problem

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }\omega \left(\eta \left(t\right)\right)=-a\left(t\right)\xi \left(\eta \left(t\right)\right),\omega \left(\eta \left(t^{★}\right)\right)={\omega }_{t^{★}}\ge 0,\end{array}$$

(5)

where $\gamma \in (0,1]$, $a:[t^{★},\infty )\to [0,\infty )$ is continuous and satisfies $a\left(t\right)\ge a^{\#}>0$ for all $t\ge t^{★}$ with some constant $a^{\#}$, $\omega :\mathbb{R}\to \mathbb{R}$ is continuous function, and $\xi :[0,\infty )\to [0,\infty )$ belongs toClass-${\mathcal{K}}_{\infty }$. Then, we have $\omega \left(\eta \left(t\right)\right)$ remains non-negative and obeys $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)=0$.

Proof. 

First, we prove that $\omega \left(\eta \left(t\right)\right)$ remains non-negative. Here, we take the Riemann-Liouville fractional integral on both sides of (5) and obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \omega \left(\eta \left(t\right)\right)& =\omega \left(\eta \left(t^{★}\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)a\left(\tau \right)\xi \left(\eta \left(\tau \right)\right)d\tau \hfill \\ \hfill & \le \omega \left(\eta \left(t^{★}\right)\right),\forall t\ge t^{★},\hfill \end{array}\end{array}$$

(6)

where non-negativity of the integrand functions were utilized. Set $e\left(t\right)=\omega \left(\eta \left(t^{★}\right)\right)-\omega \left(\eta \left(t\right)\right)$. Then, one has $e\left(t\right)\ge 0$ for all $t\ge t^{★}$. If $\omega \left(\eta \left(t\right)\right)\le 0$, ∀$t\ge t^{★}$, then $e\left(t\right)\ge \omega \left(\eta \left(t^{★}\right)\right)$, $\forall t\ge t^{★}$. Due to continuity of $\omega$, it follows that $\omega \left(\eta \left(t^{★}\right)\right)\le 0$. This contradicts our assumption that $\omega \left(\eta \left(t^{★}\right)\right)\ge 0$. Thus, one must have $\omega \left(\eta \left(t\right)\right)\ge 0$ for all $t\ge t^{★}$.

We remain to prove that $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)=0$. For that, on the contrary, we assume that $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)\ne 0$. Then ∃ a finite constant $L_{+}>0$ such that $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)=L_{+}$.

Since $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)=L_{+}$, it follows that for every $ϵ>0$ there exists a $T_c>t^{★}$ such that

$$\begin{array}{c}\hfill L_{+}-ϵ<\omega \left(\eta \left(t\right)\right)<L_{+}+ϵ,\forall t>T_c.\end{array}$$

(7)

Then, by using $a\left(t\right)\ge a^{\#}>0$, in the equation of (6), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \omega \left(\eta \left(t\right)\right)& \le \omega \left(\eta \left(t^{★}\right)\right)-{\displaystyle \frac{a^{\#}}{\Gamma \left(\gamma \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\xi \left(\eta \left(\tau \right)\right)d\tau \hfill \\ \hfill & =\omega \left(\eta \left(t^{★}\right)\right)-{\displaystyle \frac{a^{\#}}{\Gamma \left(\gamma \right)}}{\int }_{t^{★}}^{T_c}{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\xi \left(\eta \left(\tau \right)\right)d\tau \hfill \\ \hfill & -{\displaystyle \frac{a^{\#}}{\Gamma \left(\gamma \right)}}{\int }_{T_c}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\xi \left(\eta \left(\tau \right)\right)d\tau \hfill \\ \hfill & \le \omega \left(\eta \left(t^{★}\right)\right)+{\displaystyle \frac{a^{\#}}{\Gamma (\gamma +1)}}\left(\underset{t\in [t^{★},T_c)}{sup}\xi \left(\eta \left(t\right)\right)\right){\left(\psi \left(T_c\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\hfill \\ \hfill & -{\displaystyle \frac{a^{\#}}{\Gamma \left(\gamma \right)}}{\int }_{T_c}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\xi \left(\eta \left(\tau \right)\right)d\tau .\hfill \end{array}\end{array}$$

(8)

Substituting (7) in (8), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \omega \left(\eta \left(t\right)\right)+{\displaystyle \frac{a^{\#}}{\Gamma (\gamma +1)}}\xi \left(L_{+}-ϵ\right){\int }_{T_c}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & \le \omega \left(\eta \left(t^{★}\right)\right)+{\displaystyle \frac{a^{\#}}{\Gamma (\gamma +1)}}\left(\underset{t\in [t^{★},T_c)}{sup}\xi \left(\eta \left(t\right)\right)\right){\left(\psi \left(T_c\right)-\psi \left(t^{★}\right)\right)}^{\gamma }.\hfill \end{array}\end{array}$$

(9)

Consequently, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \omega \left(\eta \left(t\right)\right)+{\displaystyle \frac{a^{\#}}{\Gamma (\gamma +1)}}\xi \left(L_{+}-ϵ\right){\left(\psi \left(t\right)-\psi \left(T_c\right)\right)}^{\gamma }\hfill \\ \hfill & \le \omega \left(\eta \left(t^{★}\right)\right)+{\displaystyle \frac{a^{\#}}{\Gamma (\gamma +1)}}\left(\underset{t\in [t^{★},T_c)}{sup}\xi \left(\eta \left(t\right)\right)\right){\left(\psi \left(T_c\right)-\psi \left(t^{★}\right)\right)}^{\gamma }.\hfill \end{array}\end{array}$$

(10)

Letting $t\to \infty$ in (10), one arrives at a contradiction by noticing the left side of the inequality in (10) is tending to ∞ while the right side remains finite. Thus, one must have $\underset{t\to \infty }{lim}\omega \left(\eta \left(t\right)\right)=0$. This completes the proof. □

Lemma 2.

Let $\eta :[t^{★},\infty )\to \mathbb{R}$ be an unknown continuous function that satisfies the initial value problem

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\gamma ,\psi }\eta \left(t\right)=-a\left(t\right)\xi \left(\eta \left(t\right)\right),\eta \left(t^{★}\right)={\eta }_{t^{★}}\ge 0,\end{array}$$

(11)

where $\gamma \in (0,1]$, $a:[t^{★},\infty )\to [0,\infty )$ is continuous and satisfies $a\left(t\right)\ge a^{\#}>0$ for all $t\ge t^{★}$ with some constant $a^{\#}$, and $\xi :[0,\infty )\to [0,\infty )$ belongs toClass-${\mathcal{K}}_{\infty }$. Then, we have $\eta \left(t\right)$ remains non-negative and obeys $\underset{t\to \infty }{lim}\eta \left(t\right)=0$.

Proof. 

Following the proof strategy in Lemma 1, we obtain the result. □

Remark 2.

The proofs of the new Lemma 1 and Lemma 2 are analogous to the author’s previous result [Lemma 2, [23]. The lemmas generalize the standard class of Caputo-type fractional differential equations, as they seem non-trivial.

3. The Generalized $\psi$-Gronwall Inequalities

This section has one goal that establishes some new generalized $\psi$-Gronwall inequalities associated with non-positive and non-negative singular kernels and sharp reasonable $\psi$-Mittag-Leffler bounds.

The generalized $\psi$-Gronwall inequalities having only non-negative singular kernels have appeared in Sousa and Oliveira’s work [45]. In contrast, our results below assist rigorous formulations for many such inequalities in the cases of non-positive singular kernels and of both types of kernels. In particular, they reduced the standard generalized Gronwall inequalities that involve non-positive singular kernels by the analogy of reductio ad absurdum and have appeared recently in Lenka’s works [42,43,44].

The following result below develops a $\psi$-Mittag-Leffler bound to a generalized $\psi$-Gronwall inequality that is associated with a typical $\psi$-Riemann-Liouville fractional integral.

Theorem 1.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $v\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$, and suppose that $g_1,g_2\in {\mathbb{R}}_{\ge 0}$ are continuous on $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+g_1\left(t\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau -g_2\left(t\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(12)

for all $t\in [t^{★},T)$, where constant $\gamma \in (0,1]$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(t\right)-g_2\left(t\right)\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(13)

for all $t\in [t^{★},T)$.

Proof. 

The inequality (13) obviously holds when

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& =E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(t\right)-g_2\left(t\right)\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right].\hfill \end{array}\end{array}$$

(14)

We thus proceed on the contrary by assuming that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& >E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(t\right)-g_2\left(t\right)\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(15)

for all $t\in [t^{★},T)$. Next, we define

$$\begin{array}{c}\hfill z\left(t\right)=\eta \left(t\right)-E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(t\right)-g_2\left(t\right)\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)F\left(t\right),\forall t\ge t^{★},\end{array}$$

(16)

where $F\left(t\right)=\left[v\left(t\right)+h\left(t\right)\right]$ and $h\left(t\right)=\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)$. Then, inequality (12) gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+\left(g_1\left(t\right)-g_2\left(t\right)\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)z\left(\tau \right)d\tau \hfill \\ \hfill & +\left(g_1\left(t\right)-g_2\left(t\right)\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)F\left(\tau \right)E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(\tau \right)-g_2\left(\tau \right)\right){\left(\psi \left(\tau \right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\forall t\ge t^{★}.\hfill \end{array}\end{array}$$

(17)

Since v is continuous, bounded and non-decreasing, and h is non-negative, (17) simplifies to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+|\left(g_1\left(t\right)-g_2\left(t\right)\right)|{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left|z\left(\tau \right)\right|d\tau \hfill \\ \hfill & +|\left(g_1\left(t\right)-g_2\left(t\right)\right)|v_c{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(\tau \right)-g_2\left(\tau \right)\right){\left(\psi \left(\tau \right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)d\tau \hfill \\ \hfill & +|\left(g_1\left(t\right)-g_2\left(t\right)\right)|{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left|h\left(\tau \right)\right|E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left(g_1\left(\tau \right)-g_2\left(\tau \right)\right){\left(\psi \left(\tau \right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)d\tau \hfill \\ \hfill & +\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right),\forall t\ge t^{★},\hfill \end{array}\end{array}$$

(18)

where $v\left(t\right)\le v_c$, for some finite constant $v_c>0$. Due to continuity of the functions $\eta$, v, w, $g_1$, and $g_2$, letting $t\to t^{★}$ in (18), one has $\eta \left(t^{★}\right)\le v\left(t^{★}\right)$. It contradicts our assumption that $\eta \left(t^{★}\right)>v\left(t^{★}\right)$. Thus, the inequality (13) must be true. This completes the proof. □

Two corollaries of Theorem 1 can be stated, which seem basic but are useful in many areas of studies of $\psi$-Caputo fractional-order systems.

Corollary 1.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $v\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+{\lambda }_1{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau -{\lambda }_2{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(19)

for all $t\in [t^{★},T)$, where constants $\gamma \in (0,1]$, ${\lambda }_1\ge 0$ and ${\lambda }_2\ge 0$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left({\lambda }_1-{\lambda }_2\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(20)

for all $t\in [t^{★},T)$.

Proof. 

Take the constant functions $g_1\left(t\right)={\lambda }_1$ and $g_2\left(t\right)={\lambda }_2$ in Theorem 1. □

Corollary 2.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v_c+{\lambda }_1{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau -{\lambda }_2{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(21)

for all $t\in [t^{★},T)$, where constants $v_c>0$, $\gamma \in (0,1]$, ${\lambda }_1\ge 0$ and ${\lambda }_2\ge 0$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(\Gamma \left(\gamma \right)\left({\lambda }_1-{\lambda }_2\right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v_c+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(22)

for all $t\in [t^{★},T)$.

Proof. 

Take the constant function $v\left(t\right)=v_c$ in Corollary 1. □

A new analog of the generalized $\psi$-Gronwall inequality is given below in Theorem 2. It holds kernel functions, main ingredients, and time-dependent functions inside the integrand, unlike the previous Theorem 1.

Theorem 2.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $\tilde{w}\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $\tilde{v}\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$. Suppose that $g_{\ge 0}\in \mathbb{R}$ is continuous on $[t^{★},T)$ and satisfies $g\left(t\right)\ge \tilde{C}>0$ for all $t\ge t^{★}$ with some constant $\tilde{C}$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that $\widehat{h}\in \mathbb{R}$ is continuous and bounded by constant $\tilde{M}>0$ on $[t^{★},T)$, and all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le \tilde{v}\left(t\right)-{\lambda }^{★}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)\eta \left(\tau \right)d\tau +{\lambda }^{\#}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\widehat{h}\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\tilde{w}\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(23)

for all $t\in [t^{★},T)$, where constants ${\lambda }^{★}\ge 0$, ${\lambda }^{\#}>0$, and $\gamma \in (0,1]$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(-{\lambda }^{★}\tilde{C}\Gamma \left(\gamma \right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }+{\lambda }^{\#}\tilde{M}\Gamma \left(\gamma \right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[\tilde{v}\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\tilde{w}\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(24)

for all $t\in [t^{★},T)$.

Proof. 

The inequality (23) gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le \tilde{v}\left(t\right)-{\lambda }^{★}\tilde{C}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau +{\lambda }^{\#}\tilde{M}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\tilde{w}\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(25)

for all $t\ge b$, where $g\left(t\right)\ge \tilde{C}>0$ and $|\widehat{h}\left(t\right)|\le \tilde{M}$ were utilized. Then, by applying Corollary 1 to (25), one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(-{\lambda }^{★}\tilde{C}\Gamma \left(\gamma \right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }+{\lambda }^{\#}\tilde{M}\Gamma \left(\gamma \right){\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[\tilde{v}\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\tilde{w}\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(26)

for all $t\in [t^{★},T)$. This finished the proof. □

Our next results improve in a manner to express $\psi$-Mittag-Leffler bounds rigorously different from previous results in Theorem 1 and Theorem 2. The obtained bounds include ingredients with fractional integrals, and integral functions with the Mittag-Leffler function.

Theorem 3.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $v\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$, and suppose that $g_1,g_2\in {\mathbb{R}}_{\ge 0}$ are non-constant, continuous, and increasing on $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+{\lambda }_1{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)\eta \left(\tau \right)d\tau -{\lambda }_2{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(27)

for all $t\in [t^{★},T)$, where constants ${\lambda }_1\ge 0$, ${\lambda }_2\ge 0$, and $\gamma \in (0,1]$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left({\lambda }_1\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)d\tau \right)}^{\gamma }-{\lambda }_2\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)d\tau \right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(28)

for all $t\in [t^{★},T)$.

Proof. 

The inequality (28) is obvious when

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& =E_{\gamma ,1}\left({\lambda }_1\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)d\tau \right)}^{\gamma }-{\lambda }_2\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)d\tau \right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(29)

for all $t\in [t^{★},T)$. Assume on contrary that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& >E_{\gamma ,1}\left({\lambda }_1\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)d\tau \right)}^{\gamma }-{\lambda }_2\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)d\tau \right)}^{\gamma }\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(30)

for all $t\in [t^{★},T)$. Define

$$\begin{array}{c}\hfill F^{+}\left(t\right)=\eta \left(t\right)-E\left(t\right)\left[v\left(t\right)+\widehat{h}\left(t\right)\right],\end{array}$$

(31)

for all $t\in [t^{★},T)$, where $E\left(t\right)=E_{\gamma ,1}\left({\lambda }_1\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)d\tau \right)}^{\gamma }-{\lambda }_2\Gamma \left(\gamma \right){\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)d\tau \right)}^{\gamma }\right)$ and $\widehat{h}\left(t\right)=\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)$ were used. Now, the inequality (27) gives that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left({\lambda }_1{g}_1\left(\tau \right)-{\lambda }_2{g}_2\left(\tau \right)\right)F^{+}\left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left({\lambda }_1{g}_1\left(\tau \right)-{\lambda }_2{g}_2\left(\tau \right)\right)\left[v\left(\tau \right)+\widehat{h}\left(\tau \right)\right]E\left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\forall t\ge t^{★}.\hfill \end{array}\end{array}$$

(32)

Since v is continuous, bounded and non-decreasing, and $g_1,g_2$ are increasing, inequality (32) develops a new bound below

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+\left({\lambda }_1|g_1\left(t\right)|+{\lambda }_2\left|g_2\left(t\right)\right|\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left|F^{+}\left(\tau \right)\right|d\tau \hfill \\ \hfill & +\left({\lambda }_1|g_1\left(t\right)|+{\lambda }_2\left|g_2\left(t\right)\right|\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left[v_c+\left|\widehat{h}\left(\tau \right)\right|\right]\left|E\left(\tau \right)\right|d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\forall t\ge t^{★},\hfill \end{array}\end{array}$$

(33)

where $v\left(t\right)\le v_c<\infty$ with constant $v_c>0$. Subsequently, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+\left({\lambda }_1|g_1\left(t\right)|+{\lambda }_2\left|g_2\left(t\right)\right|\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left|F^{+}\left(\tau \right)\right|d\tau \hfill \\ \hfill & +\left({\lambda }_1|g_1\left(t\right)|+{\lambda }_2\left|g_2\left(t\right)\right|\right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left[v_c+\left|\widehat{h}\left(\tau \right)\right|\right]\left|E\left(\tau \right)\right|d\tau \hfill \\ \hfill & +\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right),\hfill \end{array}\end{array}$$

(34)

for all $t\ge t^{★}$. Since $\eta$, v, w, $g_1$, and $g_2$ were continuous, by letting $t\to t^{★}$ in (34), one obtains $\eta \left(t^{★}\right)\le v\left(t^{★}\right)$. It contradicts to priori assumption $\eta \left(t^{★}\right)>v\left(t^{★}\right)$. Therefore, the inequality (28) should be true. This finished the proof. □

The following results are associated with power-$\gamma$ integrals and fractional integral ingredients with Mittag-Leffler functions in the respective inequality bounds.

Theorem 4.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $v\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$, and suppose that $g_1,g_2\in {\mathbb{R}}_{\ge 0}$ are non-constant, continuous, and decreasing on $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+{\lambda }_1{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)\eta \left(\tau \right)d\tau -{\lambda }_2{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(35)

for all $t\in [t^{★},T)$, where constants ${\lambda }_1\ge 0$, ${\lambda }_2\ge 0$, and $\gamma \in (0,1]$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(\Gamma \left(\gamma \right){\left(1+\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\gamma }\left[{\lambda }_1\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)d\tau \right)-{\lambda }_2\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)d\tau \right)\right]\right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(36)

for all $t\in [t^{★},T)$.

Proof. 

See the proof of Theorem 3. □

Theorem 5.

Let $\eta \in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$, where $t^{★}\in \mathbb{R}$ and $t^{★}<T\le \infty$. Let $w\in {\mathbb{R}}_{\ge 0}$ be continuous on the interval $[t^{★},T)$. Let $v\in {\mathbb{R}}_{\ge 0}$ be continuous, bounded, and non-decreasing on $[t^{★},T)$, and suppose that $g_1,g_2\in {\mathbb{R}}_{\ge 0}$ are continuous on $[t^{★},T)$. Let ψ be continuously differentiable, increasing on $[t^{★},T)$ such that ${\psi }^{\prime }\left(t\right)\ne 0$, ∀$t\in (t^{★},T)$, and $\psi \left(t\right)\to \infty$ as $t\to \infty$. Assume that all the mentioned functions obey the integral inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le v\left(t\right)+{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_1\left(\tau \right)\eta \left(\tau \right)d\tau -{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)g_2\left(\tau \right)\eta \left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(37)

for all $t\in [t^{★},T)$, where constant $\gamma \in (0,1]$. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \eta \left(t\right)& \le E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)\left[g_1\left(\tau \right)-g_2\left(\tau \right)\right]d\tau \right)\hfill \\ \hfill & \times \left[v\left(t\right)+\left(E_{\gamma ,1}\left(\gamma \Gamma \left(\gamma \right){\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\gamma -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(38)

for all $t\in [t^{★},T)$.

Proof. 

See the proof of Theorem 3. □

Remark 3.

The results of Theorems 1, 2, 3, 4, and 5 may seem trivial, but proving them by means of any different approach remains an exercise problem.

4. The Lyapunov Stability Theorems

This section has three small goals that deal with fractional Lyapunov direct theorems (also called fractional Lyapunov stability theorems) for $\psi$-Caputo fractional-order systems. We establish new theorems by introducing a Lyapunov function and a $\psi$-Caputo fractional derivative of a Lyapunov function that fulfills some distinctive inequalities.

The goals are tied up with the characterization of linear inequality, nonlinear inequality, and negative definite or semi-definite characterization, which were addressed in Subsection 4.1, Section 4.2, and Section 4.3, respectively. They develop promising mathematical tools in the sense of the so-called fractional Lyapunov direct method.

4.1. Linear Inequality Characterization

This first part builds the basic foundation for small Lyapunov theorems and introduces some conceptual definitions that are interesting stability theories needed for $\psi$-Caputo fractional-order systems.

Definition 6.

The zero equilibrium point of system (3) is said to be

i) 

locally asymptotically stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $\parallel x\left(t\right)\parallel \to 0$ as $t\to \infty$,

ii) 

globally asymptotically stable if for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, the non-trivial solution $\parallel x\left(t\right)\parallel \to 0$ as $t\to \infty$.

Theorem 6.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -g\left(t\right)\mathcal{V}(t,x\left(t\right))+w\left(t\right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(39)

where

i) 

fractional-order $\delta \in (0,1]$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$,

iii) 

the function $w\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$.

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

By using (39) of $A_2$, we define

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))& =-g\left(t\right)\mathcal{V}(t,x(t\left)\right)+w\left(t\right)-N\left(t\right)\hfill \\ \hfill \mathcal{V}(t^{★},x\left(t^{★}\right))& ={\mathcal{V}}_{t^{★}}\hfill \end{array}\end{array}$$

(40)

where N is non-negative continuous function on $[t^{★},\infty )$. We take the $\psi$-Riemann-Liouville integral on (40) [Theorem 4, [34]] and obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& =\mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau -{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)N\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(41)

Since N is non-negative, expression (41) gives

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le \mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(42)

Applying Theorem 5 to inequality (42), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le E_{\delta ,1}\left(-\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\hfill \\ \hfill & \times \left[\mathcal{V}(t^{★},x\left(t^{★}\right))+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right].\hfill \end{array}\end{array}$$

(43)

Combining $A_1$ and (43), one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\hfill \\ \hfill & \times \left[k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right)+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(44)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, and ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$, it follows from (44) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(45)

Thus, the equilibrium $x=0$ of system (3) should be locally asymptotically stable on $\mathbb{D}$. Now take the domain $\mathbb{D}={\mathbb{R}}^n$ and choose the Class-${\mathcal{K}}_{\infty }$ function ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. Thus, the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, the aforementioned proof strategy must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally asymptotically stable on ${\mathbb{R}}^n$. This completes the proof. □

Definition 7.

The zero equilibrium point of system (3) is said to be locally fractional integral ψ-Mittag-Leffler asymptotically stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $x\left(t\right)$ obeys the following:

i) 

the limiting value $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0$,

ii) 

the inequality

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le C^{\#}{E}_{\delta ,1}\left(-\mu {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\end{array}\end{array}$$

(46)

for all $t\ge t^{★}$, where order $\delta \in (0,1]$, constants $\mu >0$ and $C^{\#}\ge 1$, the function $g\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, and the functions ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs toClass-${\mathcal{K}}_{\infty }$ .

Whenever $\mathbb{D}={\mathbb{R}}^n$ and $i)$ and $ii)$ hold for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, it is called globally fractional integral ψ-Mittag-Leffler asymptotically stable.

Corollary 3.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -g\left(t\right)\mathcal{V}(t,x\left(t\right)),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(47)

where

i) 

fractional-order $\delta \in (0,1]$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$.

Then, the equilibrium $x=0$ to system (3) is locally fractional integral ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally fractional integral ψ-Mittag-Leffler asymptotically stable.

Proof. 

Following the proof of Theorem 6, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(48)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, it follows from (48) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(49)

Thus, the equilibrium $x=0$ of system (3) should be locally/globally fractional integral $\psi$-Mittag-Leffler asymptotically stable. This completes the proof. □

Theorem 7.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs toClass-${\mathcal{K}}_{\infty }$and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\lambda }^{+}g\left(t\right)\mathcal{V}(t,x\left(t\right))+w\left(t\right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(50)

where

i) 

fractional-order $\delta \in (0,1]$, and some constant ${\lambda }^{+}>0$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and increasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$,

iii) 

the function $w\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$.

Then, the equilibrium $x=0$ to system (3) is locally fractional integral ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally fractional integral ψ-Mittag-Leffler asymptotically stable.

Proof. 

We first modify inequality (50) of $A_2$ and define

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))& =-{\lambda }^{+}g\left(t\right)\mathcal{V}(t,x\left(t\right))+w\left(t\right)-N\left(t\right)\hfill \\ \hfill \mathcal{V}(t^{★},x\left(t^{★}\right))& ={\mathcal{V}}_{t^{★}}\hfill \end{array}\end{array}$$

(51)

where N is non-negative continuous function on $[t^{★},\infty )$. By taking the $\psi$-Riemann-Liouville integral on (51) [Theorem 4, [34]], we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& =\mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right){\lambda }^{+}g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau -{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)N\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(52)

Since N is non-negative, equation (52) reduces to inequality

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le \mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right){\lambda }^{+}g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(53)

Then, applying Theorem 3 to inequality (53), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le E_{\delta ,1}\left(-{\lambda }^{+}{\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)}^{\delta }\right)\hfill \\ \hfill & \times \left[\mathcal{V}(t^{★},x\left(t^{★}\right))+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right].\hfill \end{array}\end{array}$$

(54)

Consequently, it follows from $A_1$ and (54) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-{\lambda }^{+}{\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)}^{\delta }\right)\hfill \\ \hfill & \times \left[k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right)+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(55)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$ and ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$, it follows from (55) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(56)

It implies that equilibrium $x=0$ of system (3) should be locally asymptotically stable on $\mathbb{D}$. Now we set the domain $\mathbb{D}={\mathbb{R}}^n$ and take the Class-${\mathcal{K}}_{\infty }$ function ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. Observe that the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, the our proof procedure must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally asymptotically stable on ${\mathbb{R}}^n$. This completes the proof. □

Definition 8.

The zero equilibrium point of system (3) is said to be locally integral ψ-Mittag-Leffler asymptotically stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $x\left(t\right)$ obeys the following:

i) 

the limiting value $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0$,

ii) 

the inequality

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le C^{\#}{E}_{\delta ,1}\left(-\mu {\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)}^{\delta }\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\end{array}\end{array}$$

(57)

for all $t\ge t^{★}$, where order $\delta \in (0,1]$, constants $\mu >0$ and $C^{\#}\ge 1$, the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and increasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, and the functions ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs toClass-${\mathcal{K}}_{\infty }$ .

Whenever $\mathbb{D}={\mathbb{R}}^n$ and $i)$ and $ii)$ hold for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, it is called globally integral ψ-Mittag-Leffler asymptotically stable.

Corollary 4.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\lambda }^{+}g\left(t\right)\mathcal{V}(t,x\left(t\right)),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(58)

where

i) 

fractional-order $\delta \in (0,1]$, and some constant ${\lambda }^{+}>0$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and increasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$.

Then, the equilibrium $x=0$ to system (3) is locally integral ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally integral ψ-Mittag-Leffler asymptotically stable.

Proof. 

Following the proof of Theorem 7, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-{\lambda }^{+}{\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)}^{\delta }\right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(59)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, it follows from (59) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(60)

Thus, the equilibrium $x=0$ of system (3) should be locally/globally integral $\psi$-Mittag-Leffler asymptotically stable. This completes the proof. □

Theorem 8.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\lambda }^{+}g\left(t\right)\mathcal{V}(t,x\left(t\right))+w\left(t\right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(61)

where

i) 

fractional-order $\delta \in (0,1]$, and some constant ${\lambda }^{+}>0$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and decreasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$,

iii) 

the function $w\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$.

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

It follows from inequality (61) of $A_2$ that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le \mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right){\lambda }^{+}g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(62)

Then, applying Theorem 4 to inequality (62), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le E_{\delta ,1}\left(-{\lambda }^{+}{\left(1+\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\right)\hfill \\ \hfill & \times \left[\mathcal{V}(t^{★},x\left(t^{★}\right))+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right].\hfill \end{array}\end{array}$$

(63)

Consequently, by $A_1$ and (63), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-{\lambda }^{+}{\left(1+\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\right)\hfill \\ \hfill & \times \left[k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right)+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(64)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, and ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$, it follows from (64) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(65)

Hence, the equilibrium $x=0$ of system (3) should be locally asymptotically stable on $\mathbb{D}$. Now we put the domain $\mathbb{D}={\mathbb{R}}^n$ and set the Class-${\mathcal{K}}_{\infty }$ function ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. Thus, the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, the aforementioned proof strategy must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally asymptotically stable on ${\mathbb{R}}^n$. This completes the proof. □

Definition 9.

The zero equilibrium point of system (3) is said to be locally power-δ integral ψ-Mittag-Leffler asymptotically stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $x\left(t\right)$ obeys the following:

i) 

the limiting value $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0$,

ii) 

the inequality

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le C^{\#}{E}_{\delta ,1}\left(-\mu {\left(1+\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\end{array}\end{array}$$

(66)

for all $t\ge t^{★}$, where order $\delta \in (0,1]$, constants $\mu >0$ and $C^{\#}\ge 1$, the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and decreasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, and the functions ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ .

Whenever $\mathbb{D}={\mathbb{R}}^n$ and $i)$ and $ii)$ hold for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, it is called globally power-δ integral ψ-Mittag-Leffler asymptotically stable.

Corollary 5.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\lambda }^{+}g\left(t\right)\mathcal{V}(t,x\left(t\right)),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(67)

where

i) 

fractional-order $\delta \in (0,1]$, and some constant ${\lambda }^{+}>0$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is non-constant, continuous, and decreasing on $[t^{★},\infty )$, and satisfies ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$.

Then, the equilibrium $x=0$ to system (3) is a locally power-δ integral ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally power-δ integral ψ-Mittag-Leffler asymptotically stable.

Proof. 

Following the previous Theorem 8, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-{\lambda }^{+}{\left(1+\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\left({\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \right)\right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(68)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)d\tau \to \infty$ as $t\to \infty$, it follows from (68) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(69)

Thus, the equilibrium $x=0$ of system (3) should be locally/globally power-$\delta$ integral $\psi$-Mittag-Leffler asymptotically stable. This completes the proof. □

Theorem 9.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -g\left(t\right)\mathcal{V}(t,x\left(t\right))+\tilde{h}\left(t\right)\mathcal{V}(t,x\left(t\right))+w\left(t\right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(70)

where

i) 

fractional-order $\delta \in (0,1]$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$, and satisfies $g\left(t\right)\ge \lambda >0$, with some constant λ,

iii) 

the function $\tilde{h}\in \mathbb{R}$ is continuous on $[t^{★},\infty )$, and satisfies $|\tilde{h}\left(t\right)|\le M$, with some constant $M>0$,

iv) 

the function $w\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$ and satisfies ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$,

v) 

the condition $\lambda >M$ holds.

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

Immediately following from inequality (70) of $A_2$ is that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le \mathcal{V}(t^{★},x\left(t^{★}\right))-{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)g\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)\tilde{h}\left(\tau \right)\mathcal{V}(\tau ,x\left(\tau \right))d\tau +{\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau .\hfill \end{array}\end{array}$$

(71)

Then, applying Theorem 2 to inequality (71), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{V}(t,x(t\left)\right)& \le C^{\#}{E}_{\delta ,1}\left(-\lambda {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }+M{\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right)\hfill \\ \hfill & \times \left[\mathcal{V}(t^{★},x\left(t^{★}\right))+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right].\hfill \end{array}\end{array}$$

(72)

Consequently, it follows from $A_1$ and (72) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\lambda {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }+M{\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right)\hfill \\ \hfill & \times \left[k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right)+\left(E_{\delta ,1}\left(\delta {\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \right)-1\right)\right],\hfill \end{array}\end{array}$$

(73)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using ${\int }_{t^{★}}^t{\left(\psi \left(t\right)-\psi \left(\tau \right)\right)}^{\delta -1}{\psi }^{\prime }\left(\tau \right)w\left(\tau \right)d\tau \to 0$ as $t\to \infty$, and $\lambda >M$, it follows from (73) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(74)

Thus, the equilibrium $x=0$ of system (3) should be locally asymptotically stable on $\mathbb{D}$. Now take the domain $\mathbb{D}={\mathbb{R}}^n$ and choose the Class-${\mathcal{K}}_{\infty }$ function ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. Note that the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, the aforementioned proof strategy must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally asymptotically stable on ${\mathbb{R}}^n$. This completes the proof. □

Definition 10.

The zero equilibrium point of system (3) is said to be locally ψ-Mittag-Leffler asymptotically stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $x\left(t\right)$ obeys the following:

i) 

the limiting value $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0$,

ii) 

the inequality

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le C^{\#}{E}_{\delta ,1}\left(-\mu {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\end{array}\end{array}$$

(75)

for all $t\ge t^{★}$, where order $\delta \in (0,1]$, constants $\mu >0$ and $C^{\#}\ge 1$, and the functions ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ .

Whenever $\mathbb{D}={\mathbb{R}}^n$ and $i)$ and $ii)$ hold for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, it is called globally ψ-Mittag-Leffler asymptotically stable.

Corollary 6.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -g\left(t\right)\mathcal{V}(t,x\left(t\right))+\tilde{h}\left(t\right)\mathcal{V}(t,x\left(t\right)),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(76)

where

i) 

fractional-order $\delta \in (0,1]$,

ii) 

the function $g\in {\mathbb{R}}_{\ge 0}$ is continuous on $[t^{★},\infty )$, and satisfies $g\left(t\right)\ge \lambda >0$, with some constant λ,

iii) 

the function $\tilde{h}\in \mathbb{R}$ is continuous on $[t^{★},\infty )$, and satisfies $|\tilde{h}\left(t\right)|\le M$, with some constant $M>0$,

iv) 

the condition $\lambda >M$ holds.

Then, the equilibrium $x=0$ to system (3) is locally ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally ψ-Mittag-Leffler asymptotically stable.

Proof. 

Following the proof strategy in Theorem 9, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\lambda {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }+M{\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(77)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$, and using $\lambda >M$, it follows from (77) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(78)

Thus, the equilibrium $x=0$ of system (3) should be locally/globally $\psi$-Mittag-Leffler asymptotically stable. This completes the proof. □

Corollary 7.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -\lambda \mathcal{V}(t,x\left(t\right)),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(79)

where $\delta \in (0,1]$ and constant $\lambda >0$.

Then, the equilibrium $x=0$ to system (3) is locally ψ-Mittag-Leffler asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally ψ-Mittag-Leffler asymptotically stable.

Proof. 

Following the proof strategy in Theorem 9, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\lambda {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(80)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Letting $t\to \infty$ in (80), one obtains

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(81)

Thus, the equilibrium $x=0$ of system (3) should be locally/globally $\psi$-Mittag-Leffler asymptotically stable. This completes the proof. □

Definition 11.

The zero equilibrium point of system (3) is said to be locally ψ-Mittag-Leffler stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, the non-trivial solution $x\left(t\right)$ obeys the the inequality

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le C^{†}{E}_{\delta ,1}\left(-\mu {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\end{array}\end{array}$$

(82)

for all $t\ge t^{★}$, where order $\delta \in (0,1]$, constants $\mu \ge 0$ and $C^{†}\ge 1$, and the functions ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$. Whenever $\mathbb{D}={\mathbb{R}}^n$ and (82) holds for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, it is called globally ψ-Mittag-Leffler stable.

Corollary 8.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -\lambda \mathcal{V}(t,x\left(t\right)),\forall t\ge t^{★},\forall x\in \mathbb{D},\end{array}$$

(83)

where $\delta \in (0,1]$ and constant $\lambda \ge 0$.

Then, the equilibrium $x=0$ to system (3) is locally ψ-Mittag-Leffler stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally ψ-Mittag-Leffler stable.

Proof. 

In short, by following the proof strategy in Theorem 9, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)& \le C^{\#}{E}_{\delta ,1}\left(-\lambda {\left(\psi \left(t\right)-\psi \left(t^{★}\right)\right)}^{\delta }\right)k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\hfill \end{array}\end{array}$$

(84)

for all $t\ge t^{★}$, where constant $C^{\#}\ge 1$. Thus, the equilibrium $x=0$ of system (3) should be locally/globally $\psi$-Mittag-Leffler stable. This completes the proof. □

4.2. Nonlinear Inequality Characterization

In this second part, we establish theorems that consider the $\psi$-Caputo fractional derivative of a Lyapunov function bounded above by minus multiplied by a nonlinear Class-${\mathcal{K}}_{\infty }$ function. It develops a new characterization for the Lyapunov function.

Theorem 10.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous and bounded function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_3\left(\parallel x\left(t\right)\parallel \right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(85)

where $\delta \in (0,1]$ and ${\alpha }_3\left(·\right)$ belongs toClass-${\mathcal{K}}_{\infty }$ .

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

Since $\mathcal{V}(t,x)$ satisfy $A_1$, we have

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le M^{★}{\alpha }_2\left(\parallel x\parallel \right),\end{array}$$

(86)

where $|k_2\left(t\right)|\le M^{★}$, for some constant $M^{★}>0$. Combining (86) and (85) of $A_2$ simplifies to the inequality

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_3\left({\alpha }_2^{-1}\left({\displaystyle \frac{1}{M^{★}}}\mathcal{V}(t,x\left(t\right))\right)\right).\end{array}$$

(87)

Define a Class-${\mathcal{K}}_{\infty }$ function by ${\alpha }_4\left(\mathcal{V}(t,x(t\left)\right)\right)={\alpha }_3\left({\alpha }_2^{-1}\left({\displaystyle \frac{1}{M^{★}}}\mathcal{V}(t,x\left(t\right))\right)\right)$. Then, we rewrite the inequality (87) as

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_4\left(\mathcal{V}(t,x(t\left)\right)\right).\end{array}$$

(88)

Now, we consider a comparison $\psi$-fractional differential equation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{W}(t,y\left(t\right))& =-{\alpha }_4\left(\mathcal{W}(t,y(t\left)\right)\right)\hfill \\ \hfill \mathcal{W}(t^{★},y\left(t^{★}\right))& =\mathcal{V}(t^{★},x\left(t^{★}\right))\ge 0.\hfill \end{array}\end{array}$$

(89)

Then, by applying comparison principle [Lemma $2.3$, [39]] to (88) and (89), we obtain

$$\begin{array}{c}\hfill \mathcal{V}(t,x\left(t\right))\le \mathcal{W}(t,y\left(t\right)),\forall t\ge t^{★}.\end{array}$$

(90)

By the application of Lemma 2 to (89), we see that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mathcal{W}(t,y\left(t\right))=0.\end{array}$$

(91)

Consequently, by using $A_1$, (90) and (91), we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0.\end{array}$$

(92)

Thus, the equilibrium $x=0$ of system (3) should be locally asymptotically stable on $\mathbb{D}$. Now take the domain $\mathbb{D}={\mathbb{R}}^n$ and choose the Class-${\mathcal{K}}_{\infty }$ functions ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$ and ${\rho }_2\left(r\right)=M^{★}{\alpha }_2\left(r\right)$, where $r=\parallel x\parallel$. Note that the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, the aforementioned proof strategy must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally asymptotically stable on ${\mathbb{R}}^n$. This completes the proof. □

Corollary 9.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous and bounded function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -\lambda \mathcal{V}(t,x\left(t\right))-{\alpha }_3\left(\parallel x\left(t\right)\parallel \right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(93)

where $\delta \in (0,1]$, constant $\lambda >0$, and ${\alpha }_3\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ .

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

The proof is a consequence of Theorem 10. □

Corollary 10.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

$k_1{\parallel x\parallel }^p\le \mathcal{V}(t,x)\le k_2\left(t\right){\parallel x\parallel }^{pq}$, where some positive constants $k_1,p,q$, and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous and bounded function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_3\left(\parallel x\left(t\right)\parallel \right),\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(94)

where $\delta \in (0,1]$ and ${\alpha }_3\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ .

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

The result follows from Theorem 10. □

4.3. Negative Definite or Semi-Definite Characterization

This is the third part; we extend the previous subsection results to a general case. A sharp inequality that develops a negative definite characterization of the $\psi$-Caputo fractional derivative of a Lyapunov function is proposed in Theorem 11 below. A negative semi-definite characterization of the $\psi$-Caputo fractional derivative of the Lyapunov function is proposed in Theorem 12.

Theorem 11.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous and bounded function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))<0,\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(95)

where $\delta \in (0,1]$.

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

We assume on the contrary that the equilibrium point $x=0$ of (3) is not asymptotically stable on $\mathbb{D}$. In what follows, ∃ a finite constant $L\ne 0$ such that $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =L$. Then, from (95), we construct

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))& =-{\alpha }_3(\parallel x\left(t\right)\parallel )-N\left(t,\mathcal{V}(t,x(t\left)\right)\right),\hfill \end{array}\end{array}$$

(96)

for all $t>t^{★}$, $\forall x\in \mathbb{D}-\left\{0\right\}$, where the function $N:[t^{★},\infty )\times [0,\infty )\to [0,\infty )$ is continuous, and the function ${\alpha }_3\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$. Consequently, we can majorize the expression in (96) by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))& \le -{\alpha }_3(\parallel x\left(t\right)\parallel ),\hfill \end{array}\end{array}$$

(97)

for all $t>t^{★}$, $\forall x\in \mathbb{D}-\left\{0\right\}$. Since $\mathcal{V}(t,x)$ satisfy $A_1$, one has

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le M^{★}{\alpha }_2\left(\parallel x\parallel \right),\end{array}$$

(98)

where $|k_2\left(t\right)|\le M^{★}$, for some constant $M^{★}>0$. Then, we combine (97) and (98) to obtain

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_3\left({\alpha }_2^{-1}\left({\displaystyle \frac{1}{M^{★}}}\mathcal{V}(t,x\left(t\right))\right)\right).\end{array}$$

(99)

Define ${\alpha }_4\left(\mathcal{V}(t,x(t\left)\right)\right)={\alpha }_3\left({\alpha }_2^{-1}\left({\displaystyle \frac{1}{M^{★}}}\mathcal{V}(t,x\left(t\right))\right)\right)$, which is a Class-${\mathcal{K}}_{\infty }$ function. As a result, the inequality (99) is now simplified to

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le -{\alpha }_4\left(\mathcal{V}(t,x(t\left)\right)\right).\end{array}$$

(100)

In order to find an upper bound to inequality (100), we consider a comparison differential equation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{W}(t,y\left(t\right))& =-{\alpha }_4\left(\mathcal{W}(t,y(t\left)\right)\right)\hfill \\ \hfill \mathcal{W}(t^{★},y\left(t^{★}\right))& =\mathcal{V}(t^{★},x\left(t^{★}\right))\ge 0.\hfill \end{array}\end{array}$$

(101)

By applying comparison principle [Lemma $2.3$, [39]] to (100) and (101), we get

$$\begin{array}{c}\hfill 0\le \mathcal{V}(t,x\left(t\right))\le \mathcal{W}(t,y\left(t\right)),\forall t\ge t^{★}.\end{array}$$

(102)

By the application of Lemma 2 to (101), we see that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mathcal{W}(t,y\left(t\right))=0.\end{array}$$

(103)

From (102) and (103), one obtains that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mathcal{V}(t,x\left(t\right))=0.\end{array}$$

(104)

Thus, it follows from $A_1$ and (104) that

$$\begin{array}{c}\hfill 0\le {\alpha }_1\left(L\right)\le 0.\end{array}$$

(105)

Since ${\alpha }_1\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$, the inequality (105) leads to a contradiction for $L\ne 0$. Therefore, we should have $\underset{t\to \infty }{lim}\parallel x\left(t\right)\parallel =0$. Hence, the equilibrium point $x=0$ of (3) should be asymptotically stable on $\mathbb{D}$. For global asymptotic stability, we let the domain $\mathbb{D}={\mathbb{R}}^n$ and Class-${\mathcal{K}}_{\infty }$ function $\alpha (\parallel x\parallel )={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. It enables a pathway that the level curve of $V(t,x)$ must tend to ∞ as $\parallel x\parallel \to \infty$. Consequently, the aforementioned procedure should be true on $\mathbb{D}={\mathbb{R}}^n$. Thus, $x=0$ of system (3) should be globally asymptotically stable. Here we close the proof. □

Corollary 11.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

$k_1{\parallel x\parallel }^p\le \mathcal{V}(t,x)\le k_2\left(t\right){\parallel x\parallel }^{pq}$, where some positive constants $k_1,p,q$, and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous and bounded function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))<0,\forall t>t^{★},\forall x\in \mathbb{D}-\left\{0\right\},\end{array}$$

(106)

where $\delta \in (0,1]$.

Then, the equilibrium $x=0$ for system (3) is locally asymptotically stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally asymptotically stable.

Proof. 

The result follows from Theorem 11. □

Definition 12.

The zero equilibrium point of system (3) is said to be

i) 

locally stable if for any $x\left(t^{★}\right)\in \mathbb{D}\subset {\mathbb{R}}^n$, there exist a $\delta >0$ such that $\parallel x\left(t^{★}\right)\parallel \le \delta \Rightarrow \parallel x\left(t\right)\parallel \le ϵ$, where $ϵ>0$,

ii) 

globally stable if for all $x\left(t^{★}\right)\in {\mathbb{R}}^n$, there exist a $\delta >0$ with $\delta \to \infty$ such that $\parallel x\left(t^{★}\right)\parallel \le \delta \Rightarrow \parallel x\left(t\right)\parallel \le ϵ$, where $ϵ>0$.

Theorem 12.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

${\alpha }_1\left(\parallel x\parallel \right)\le \mathcal{V}(t,x)\le k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right)$, where ${\alpha }_1\left(·\right),{\alpha }_2\left(·\right)$ belongs to Class-${\mathcal{K}}_{\infty }$ and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le 0,\forall t\ge t^{★},\forall x\in \mathbb{D},\end{array}$$

(107)

where $\delta \in (0,1]$.

Then, the equilibrium $x=0$ for system (3) is locally stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally stable.

Proof. 

We begin with the inequality (107) and consider a comparison $\psi$-fractional differential equation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{W}(t,y\left(t\right))& =0\hfill \\ \hfill \mathcal{W}(t^{★},y\left(t^{★}\right))& =\mathcal{V}(t^{★},x\left(t^{★}\right))\ge 0.\hfill \end{array}\end{array}$$

(108)

Then, by applying comparison principle [Lemma $2.3$, [39]] to (107) and (108), we have

$$\begin{array}{c}\hfill \mathcal{V}(t,x\left(t\right))\le \mathcal{W}(t,y\left(t\right)),\forall t\ge t^{★}.\end{array}$$

(109)

Since the analytic solution of (108) is $\mathcal{W}(t,y\left(t\right))=\mathcal{V}(t^{★},x\left(t^{★}\right))$, it follows from (109) that

$$\begin{array}{c}\hfill \mathcal{V}(t,x\left(t\right))\le \mathcal{V}(t^{★},x\left(t^{★}\right)),\forall t\ge t^{★}.\end{array}$$

(110)

By using $A_1$ in (110), we can easily obtain

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\left(t\right)\parallel \right)\le k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right),\forall t\ge t^{★}.\end{array}$$

(111)

Thus, the inequality (111) is simplified to

$$\begin{array}{c}\hfill \parallel x\left(t\right)\parallel \le {\alpha }_1^{-1}\left(k_2\left(t^{★}\right){\alpha }_2\left(\parallel x\left(t^{★}\right)\parallel \right)\right)\le ϵ,\forall t\ge t^{★},\end{array}$$

(112)

whenever $\parallel x\left(t^{★}\right)\parallel \le \delta$, where $\delta ={\alpha }_2^{-1}\left({\displaystyle \frac{1}{k_2\left(t^{★}\right)}}{\alpha }_1\left(ϵ\right)\right)$ and $ϵ>0$. Therefore, the equilibrium $x=0$ of (3) should be stable on $\mathbb{D}$. Now take the domain $\mathbb{D}={\mathbb{R}}^n$ and choose the Class-${\mathcal{K}}_{\infty }$ function ${\rho }_1\left(r\right)={\alpha }_1\left(r\right)$, where $r=\parallel x\parallel$. Note that the function $\mathcal{V}(t,x)\to \infty$ as $r\to \infty$. As a result, by letting $\delta \to \infty$, the inequality (112) must hold on ${\mathbb{R}}^n$. Therefore, the equilibrium $x=0$ of system (3) should be globally stable on ${\mathbb{R}}^n$. This completes the proof. □

Corollary 12.

Let $x=0$ be an equilibrium of (3). Let $\mathbb{D}\subset {\mathbb{R}}^n$ be a domain containing $x=0$. Let $\mathcal{V}:[t^{★},\infty )\times \mathbb{D}\to {\mathbb{R}}_{\ge 0}$ be a ${\mathcal{C}}^1$ function and satisfy

$A_1.$

$k_1{\parallel x\parallel }^p\le \mathcal{V}(t,x)\le k_2\left(t\right){\parallel x\parallel }^{pq}$, where some positive constants $k_1,p,q$, and $k_2:[t^{★},\infty )\to {\mathbb{R}}_{+}$ is a continuous function,

$A_2.$

along system (3) non-trivial solution $x\left(t\right)$:

$$\begin{array}{c}\hfill {}^C{\mathfrak{D}}_{t^{★},t}^{\delta ,\psi }\mathcal{V}(t,x\left(t\right))\le 0,\forall t\ge t^{★},\forall x\in \mathbb{D},\end{array}$$

(113)

where $\delta \in (0,1]$.

Then, the equilibrium $x=0$ for system (3) is locally stable. If $\mathbb{D}={\mathbb{R}}^n$ and $A_1,A_2$ hold globally on ${\mathbb{R}}^n$, then $x=0$ to (3) is globally stable.

Proof. 

The proof follows from Theorem 12. □

Remark 4.

Theorems 11 and 12 may be viewed as new extensions to Lyapunov stability theorems of Caputo-type fractional-order systems: [Theorems 3 and 4, [14], [Theorems 1 and 2, [13]], [Theorem 11, [17]], and [Theorem $6.2$, [18]]. These theorems strengthen Lyapunov function assumptions using Class-${\mathcal{K}}_{\infty }$ and improve the author’s previous results, [Theorem 1 and Theorem 2, [37]]. They can be applicable to stability analysis of Caputo fractional-order systems whenever one sets $\psi \left(t\right)=t-t^{★}$.

Remark 5.

The proofs of theorems in Subsection 4.1, Section 4.2, and Section 4.3 are rather constructive and analytic in nature and give some new insights to understanding fractional Lyapunov stability theory for systems (3). The readers may discover reasonable new proofs to these theorems, which remain an open exercise problem.

5. Worked-Out Dynamic Models

The applications of Lyapunov stability theorems can be very basic for future studies. This section’s goal is to predict stability dynamics for some $\psi$-Caputo generalized dynamic models.

We begin with a fractional-order Van der Pol oscillator that describes a physical system that was considered in a work by Tavazoie et al. [52]. We give a $\psi$-Caputo extension of such a dynamic model in Example 1 to demonstrate one of our theoretical results.

Example 1.

Below we consider the ψ-Caputo fractional-order Van der Pol oscillator system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{1,t}^{\gamma ,\psi }{x}_1\left(t\right)& =x_2\left(t\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{1,t}^{\gamma ,\psi }{x}_2\left(t\right)& =-x_1\left(t\right)+\mu \left(1-x_1^2\left(t\right)\right)x_2\left(t\right)\hfill \end{array}\end{array}$$

(114)

subject to $x_1\left(1\right)=x_1^1$ and $x_2\left(1\right)=x_1^2$, where $\gamma \in (0,1]$, $\psi \left(t\right)=1+(t-1)-e^{-(t-1)}$ and the parameter μ is negative.

Since it is easy to notice that the system (114) has an equilibrium point ${\left(0,0\right)}^T\in {\mathbb{R}}^2$, we consider the quadratic Lyapunov function $\mathcal{V}(t,x)={\parallel x\parallel }^2$, where $x={\left(x_1,x_2\right)}^T\in {\mathbb{R}}^2$. We remark that the choice of Lyapunov function may not be unique, and one might discover a completely different Lyapunov function. However, note that our choice of Lyapunov function obeys

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)={\displaystyle \frac{1}{2}}{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le {\parallel x\parallel }^2=k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right),\forall t\ge 1,\forall x\in {\mathbb{R}}^2,\end{array}$$

(115)

where a constant function $k_2\left(t\right)=1$. By using Lemma 6 of [37], we obtain along system (114) non-trivial solution $x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right)\right)}^T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{1,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \le 2x_1\left(t\right){}^C{\mathfrak{D}}_{1,t}^{\gamma ,\psi }{x}_1\left(t\right)+2x_2\left(t\right){}^C{\mathfrak{D}}_{1,t}^{\gamma ,\psi }{x}_2\left(t\right)\hfill \\ \hfill & =2\mu \left(1-x_1^2\left(t\right)\right)x_2^2\left(t\right)\hfill \\ \hfill & \le 0,\forall t\ge 1,\forall x\in \mathbb{D}\subset {\mathbb{R}}^2,\hfill \end{array}\end{array}$$

(116)

where the domain $\mathbb{D}=\{x\in {\mathbb{R}}^2:x_1^2+x_2^2\le 1\}$ was taken. When we set $\delta =\gamma \in (0,1]$, we see that $A_1$ and $A_2$ of Theorem 12 are satisfied. Thus, by Theorem 12, the equilibrium point ${\left(0,0\right)}^T$ should be locally stable on $\mathbb{D}\subset {\mathbb{R}}^2$.

Next, we thought of a well-known fractional-order Lorenz system that describes a typical extension of the classical Lorenz system, which was previously known in a work by Lenka and Bora [13]. The Lorenz system is a qualified simplified model of atmospheric convection first developed by E. N. Lorenz [54] and discovered moving particles in a trajectory behave in an unpredictable manner, giving rise to so-called "chaos." We propose a $\psi$-Caputo extension of such a dynamic model that begins at some positive initial time and look for such system stability dynamics in Example 2. Our exhibition shows that the zero equilibrium of such an extended system is globally asymptotically stable under a reasonable condition.

Example 2.

We consider the following ψ-Caputo fractional-order Lorenz system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_1\left(t\right)& =\sigma \left(x_2\left(t\right)-x_1\left(t\right)\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_2\left(t\right)& =\rho x_1\left(t\right)-x_2\left(t\right)-x_1\left(t\right)x_3\left(t\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_3\left(t\right)& =x_1\left(t\right)x_2\left(t\right)-\beta x_3\left(t\right)\hfill \end{array}\end{array}$$

(117)

subject to $x_1\left(10\right)=x_{10}^1$, $x_2\left(10\right)=x_{10}^2$, and $x_3\left(10\right)=x_{10}^3$, where $\gamma \in (0,1]$, $\psi \left(t\right)=1+{(t-10)}^2$ and the parameters $\sigma ,\rho ,\beta$ are positive.

First, we see that this system (117) has an equilibrium at ${\left(0,0,0\right)}^T\in {\mathbb{R}}^3$. By inspection, we take the Lyapunov function $\mathcal{V}(t,x)={\displaystyle \frac{1}{\sigma }}{x}_1^2+x_2^2+x_3^2$, where $x={\left(x_1,x_2,x_3\right)}^T\in {\mathbb{R}}^3$. Notice that this function obeys a typical inequality below

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)=min\{1,{\displaystyle \frac{1}{\sigma }}\}{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le max\{1,{\displaystyle \frac{1}{\sigma }}\}{\parallel x\parallel }^2=k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right),\forall t\ge 10,\forall x\in {\mathbb{R}}^3,\end{array}$$

(118)

where a constant function $k_2\left(t\right)=1$. By Lemma 6 of [37], it follows that along system (117) non-trivial solution $x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right)\right)}^T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \le {\displaystyle \frac{2}{\sigma }}{x}_1\left(t\right){}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_1\left(t\right)+2x_2\left(t\right){}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_2\left(t\right)+2x_3\left(t\right){}^C{\mathfrak{D}}_{10,t}^{\gamma ,\psi }{x}_3\left(t\right)\hfill \\ \hfill & =2x_1\left(t\right)x_2\left(t\right)-2x_1^2\left(t\right)+2\rho x_1\left(t\right)x_2\left(t\right)-2x_2^2\left(t\right)-2x_1\left(t\right)x_2\left(t\right)x_3\left(t\right)\hfill \\ \hfill & +2x_1\left(t\right)x_2\left(t\right)x_3\left(t\right)-2\beta x_3^2\left(t\right)\hfill \\ \hfill & \le \left(\rho -1\right)x_1^2\left(t\right)+\left(\rho -1\right)x_2^2\left(t\right)-2\beta x_3^2\left(t\right)\hfill \\ \hfill & <0,\forall t>10,\forall x\in {\mathbb{R}}^3-\left\{{\left(0,0,0\right)}^T\right\},\hfill \end{array}\end{array}$$

(119)

where the condition $0<\rho <1$ was used. Now, we are in a position to set $\delta =\gamma \in (0,1]$. Consequently, we see $A_1$ and $A_2$ of Theorem 11 are satisfied. Thus, by Theorem 11, the equilibrium point ${\left(0,0,0\right)}^T\in {\mathbb{R}}^3$ of system (117) should be globally asymptotically stable on ${\mathbb{R}}^3$ provided the obtained condition $0<\rho <1$ holds.

Remark 6.

In system (117), one can consider any non-positive initial time to experiment with how roughly trajectories of such a system behave as $t\to \infty$. The rough computational trajectories are currently out of scope in this research. The Lyapunov theorems might provide some guidance on stability dynamics to the observed experiments.

In the following Example 3, we construct a $\psi$-Caputo fractional-order nonlinear system, and it is shown that the zero equilibrium point is locally asymptotically stable.

Example 3.

Consider the nonlinear ψ-Caputo fractional-order system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{-5,t}^{\gamma ,\psi }{x}_1\left(t\right)& =x_1\left(t\right)\left(x_1^2\left(t\right)+x_2^2\left(t\right)-4\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{-5,t}^{\gamma ,\psi }{x}_2\left(t\right)& =x_2\left(t\right)\left(x_1^2\left(t\right)+x_2^2\left(t\right)-4\right)\hfill \end{array}\end{array}$$

(120)

subject to $x_1(-5)=x_{-5}^1$ and $x_2(-5)=x_{-5}^2$, where $\gamma \in (0,1)$ and $\psi \left(t\right)=5+(t+5)$.

In our construction one can observe that the system (120) has an equilibrium point ${\left(0,0\right)}^T\in {\mathbb{R}}^2$. Here we use the Lyapunov function $\mathcal{V}(t,x)={\parallel x\parallel }^2$, where $x={\left(x_1,x_2\right)}^T\in {\mathbb{R}}^2$. Note that one can bound this Lyapunov function as stated below

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)={\displaystyle \frac{1}{4}}{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le 2{\parallel x\parallel }^2=k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right),\forall t\ge -5,\forall x\in {\mathbb{R}}^2,\end{array}$$

(121)

where a constant function $k_2\left(t\right)=2$. By using Lemma 6 of [37], one obtains along system (120) non-trivial solution $x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right)\right)}^T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{-5,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \le 2x_1\left(t\right){}^C{\mathfrak{D}}_{-5,t}^{\gamma ,\psi }{x}_1\left(t\right)+2x_2\left(t\right){}^C{\mathfrak{D}}_{-5,t}^{\gamma ,\psi }{x}_2\left(t\right)\hfill \\ \hfill & =2\left(x_1^2\left(t\right)+x_2^2\left(t\right)-4\right)\left(x_1^2\left(t\right)+x_2^2\left(t\right)\right)\hfill \\ \hfill & <0,\forall t>-5,\forall x\in \mathbb{D}-\left\{{\left(0,0\right)}^T\right\},\hfill \end{array}\end{array}$$

(122)

where the domain $\mathbb{D}=\{x\in {\mathbb{R}}^2:x_1^2+x_2^2<4\}\subset {\mathbb{R}}^2$. Set $\delta =\gamma \in (0,1)$. Then, we see that $A_1$ and $A_2$ of Theorem 11 are satisfied. Thus, by Theorem 11, the equilibrium point ${\left(0,0\right)}^T$ should be locally asymptotically stable on $\mathbb{D}\subset {\mathbb{R}}^2$.

Following Lenka’s work [46], we construct a $\psi$-Caputo fractional-order system (123) and examine this system by using one of our results below.

Example 4.

Consider the nonlinear ψ-Caputo fractional-order system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_1\left(t\right)& =-7x_1\left(t\right)+sin\left(x_1\left(t\right)+x_2\left(t\right)\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_2\left(t\right)& =-11x_2\left(t\right)+sin\left(x_3\left(t\right)+x_2\left(t\right)\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_3\left(t\right)& =-17x_3\left(t\right)+sin\left(x_1\left(t\right)+x_3\left(t\right)\right)\hfill \end{array}\end{array}$$

(123)

subject to $x_1\left(0\right)=x_0^1$, $x_2\left(0\right)=x_0^2$, and $x_3\left(0\right)=x_0^3$, where $\gamma \in (0,1]$ and $\psi \left(t\right)=7+t$.

First, note that ${\left(0,0,0\right)}^T\in {\mathbb{R}}^3$ is an equilibrium for this system (123). We take the Lyapunov function $\mathcal{V}(t,x)=x_1^2+x_2^2+x_3^2$, where $x={\left(x_1,x_2,x_3\right)}^T\in {\mathbb{R}}^3$. This Lyapunov function obviously satisfies the inequality

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)={\displaystyle \frac{1}{2}}{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le {\parallel x\parallel }^2=k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right),\forall t\ge 0,\forall x\in {\mathbb{R}}^3,\end{array}$$

(124)

where a constant function $k_2\left(t\right)=1$. By Lemma 6 of [37], along system (123) non-trivial solution $x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right)\right)}^T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \le 2x_1\left(t\right){}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_1\left(t\right)+2x_2\left(t\right){}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_2\left(t\right)+2x_3\left(t\right){}^C{\mathfrak{D}}_{0,t}^{\gamma ,\psi }{x}_3\left(t\right)\hfill \\ \hfill & =-14x_1^2\left(t\right)-22x_2^2\left(t\right)-34x_3^2\left(t\right)+2x_1\left(t\right)sin\left(x_1\left(t\right)+x_2\left(t\right)\right)\hfill \\ \hfill & +2x_2\left(t\right)sin\left(x_3\left(t\right)+x_2\left(t\right)\right)+2x_3\left(t\right)sin\left(x_1\left(t\right)+x_3\left(t\right)\right)\hfill \\ \hfill & \le -13x_1^2\left(t\right)-21x_2^2\left(t\right)-33x_3^2\left(t\right)+{sin}^2\left(x_1\left(t\right)+x_2\left(t\right)\right)\hfill \\ \hfill & +{sin}^2\left(x_3\left(t\right)+x_2\left(t\right)\right)+{sin}^2\left(x_1\left(t\right)+x_3\left(t\right)\right)\hfill \\ \hfill & \le -9x_1^2\left(t\right)-17x_2^2\left(t\right)-29x_3^2\left(t\right)\hfill \\ \hfill & <0,\forall t>0,\forall x\in {\mathbb{R}}^3-\left\{{\left(0,0,0\right)}^T\right\}.\hfill \end{array}\end{array}$$

(125)

Set $\delta =\gamma \in (0,1]$. Then, we see that $A_1$ and $A_2$ of Theorem 11 are satisfied. Thus, by Theorem 11, the equilibrium point ${\left(0,0,0\right)}^T\in {\mathbb{R}}^3$ of system (123) should be globally asymptotically stable on ${\mathbb{R}}^3$.

The following Example 5 provides a case study scenario where the Lyapunov stability theorems seem to fail to predict a reasonable understanding of the stability of zero equilibrium of system (126).

Example 5.

Consider the linear ψ-Caputo fractional-order system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }{x}_1\left(t\right)& =x_1\left(t\right)-x_2\left(t\right)\hfill \\ \hfill {}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }{x}_2\left(t\right)& =x_1\left(t\right)+x_2\left(t\right)\hfill \end{array}\end{array}$$

(126)

subject to $x_1\left(50\right)=x_{50}^1$ and $x_2\left(50\right)=x_{50}^2$, where $\gamma \in (0,{\displaystyle \frac{1}{2}})$ and $\psi \left(t\right)=(t-50)$.

Note that if we assume a candidate function, likewise $\mathcal{V}(t,x)={\parallel x\parallel }^2$, where $x={\left(x_1,x_2\right)}^T\in {\mathbb{R}}^2$, obviously, one obtains

$$\begin{array}{c}\hfill {\alpha }_1\left(\parallel x\parallel \right)={\displaystyle \frac{1}{2}}{\parallel x\parallel }^2\le \mathcal{V}(t,x)\le 2{\parallel x\parallel }^2=k_2\left(t\right){\alpha }_2\left(\parallel x\parallel \right),\forall t\ge 50,\forall x\in {\mathbb{R}}^2,\end{array}$$

(127)

where a constant function $k_2\left(t\right)=2$. By using Lemma 6 of [37], along system (126) non-trivial solution $x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right)\right)}^T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \le 2x_1\left(t\right){}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }{x}_1\left(t\right)+2x_2\left(t\right){}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }{x}_2\left(t\right)\hfill \\ \hfill & =2\left(x_1^2\left(t\right)+x_2^2\left(t\right)\right)\hfill \\ \hfill & =2\mathcal{V}(t,x\left(t\right)),\forall t>50,\forall x\in {\mathbb{R}}^2-\left\{{\left(0,0\right)}^T\right\}.\hfill \end{array}\end{array}$$

(128)

Obviously, one can see that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \nless 0,\forall t>50,\forall x\in {\mathbb{R}}^2-\left\{{\left(0,0\right)}^T\right\}.\hfill \end{array}\end{array}$$

(129)

Similarly, one obtains that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^C{\mathfrak{D}}_{50,t}^{\gamma ,\psi }\mathcal{V}(t,x\left(t\right))& \nleqq 0,\forall t\ge 50,\forall x\in {\mathbb{R}}^2.\hfill \end{array}\end{array}$$

(130)

Thus, our Lyapunov stability theorems (see Theorem 11 and Theorem 12) seem ineffective for the stability analysis for such a simple system. It indicates that there should be more to the unknown than what we so far understood of fractional Lyapunov stability theory. The novel pursuit of these might bring creative directions to further advance fractional Lyapunov stability theory. Here we close the worked-out dynamic models’ demonstration.

Remark 7.

The selected models in Examples 1, 2, 3, and 4 are quite representative, and we do not see yet their rough computational trajectories associated with respective systems. The computational trajectories are currently out of scope in our demonstration. The theoretical results suggested in our work might provide reasonable knowledge to predict when their trajectories should converge or stay near to typical equilibria of such systems in some domain of interest, especially whenever $t\to \infty$.

6. Further Remarks

The stability problem of equilibrium points of $\psi$-Caputo fractional-order systems whenever the order lies in the interval $(0,1]$ still seems challenging, although some basic foundations for the rigorous constructive analytic Lyapunov stability theorems are proposed. The formulations of new generalized $\psi$-Gronwall inequalities and the notions of definitions of fractional integral $\psi$-Mittag-Leffler asymptotic stability, integral $\psi$-Mittag-Leffler asymptotic stability, power-$\delta$ integral $\psi$-Mittag-Leffler asymptotic stability, and $\psi$-Mittag-Leffler asymptotic stability develop some prospects for Lyapunov stability theory. The inclusion of Class-${\mathcal{K}}_{\infty }$ functions and negative definite or semi-definite characterizations of the $\psi$-Caputo fractional derivative of the Lyapunov function further enhances new generalized Lyapunov stability theorems to the considered problem. The implications of our special Lyapunov stability theorems demonstrate effectiveness to typical dynamic models such as the $\psi$-Caputo fractional-order Van der Pol oscillator system [see Example 1] and the $\psi$-Caputo fractional-order Lorenz system [see Example 2].

The state of the art of this work provides some reasonable new mathematical insights to proofs and conceptual definitions, which adds new knowledge to Lyapunov stability theory for $\psi$-Caputo fractional-order systems. Our investigation shows that the Lyapunov method has a distinction and also a sharp limitation. In short, the method tackles difficult problems [see Examples 1- 4] but does not go beyond to predict asymptotic stability and stability of typical systems [see, e.g., Example 5].

We pose the following open problems.

• Is it possible to restate the higher versions of Lyapunov stability theorems that can efficiently tackle the system (3)?
• Does there exist any elegant alternative method to the Lyapunov direct method to predict the local and global stability of equilibrium of the system (3)?