A Novel Lyapunov Asymptotic Eventual Stability Approach for Nonlinear Impulsive Caputo Fractional Differential Equations

Abstract

This paper investigates the asymptotic eventual stability (AE-S) for nonlinear impulsive Caputo fractional differential equations (ICFDEs) with fixed impulse moments, employing auxiliary Lyapunov functions (ALF) which are specifically constructed as analogues of vector Lyapunov functions (VLF). A novel derivative tailored for VLF is introduced, offering a more robust framework than existing approaches based on scalar Lyapunov functions (SLF). Adequate conditions for AE-S involving ICFDEs are provided. We also used the predictor corrector method to implement a numerical solution for a given impulsive Caputo fractional differential equation. These findings extend and improve upon existing results, providing significant advancements in the stability analysis of systems with memory effects and impulsive dynamics. The study holds practical relevance for modeling and analyzing real-world systems, including control processes, biological systems, and economic dynamics where fractional-order behavior and impulses play a crucial role.

1. Introduction

Impulsive differential equations (IDEs) serve as effective mathematical models for representing real-world processes and phenomena in fields such as physics, biology, population dynamics, neural networks, industrial robotics, and economics. As observed by Stamova [1,2] and others [3,4,5], it is generally accepted that certain impulsive effects are inevitable in population interactions (see [6]).

Again, many evolutionary processes such as those in optimal control of economic systems, frequency-modulated signal processing systems, and the motion of certain flying objects, are distinguished by abrupt state changes at specific moments in time [7,8].

Over the years, considerable efforts have been made in studying fractional differential equations (FDEs), which have been considered as powerful tool in the modeling of real world phenomena as it extends the classical integer order derivatives to fractional order [9]. As observed in some previously published results [10,11,12,13,14], a key focus in the qualitative theory of fractional derivatives is the stability of solutions, as it allows for the comparison of the behavior of solutions originating from different initial conditions.

However, despite advancements and promising applications, the development of the theory of impulsive fractional differential equations (IFDEs) is still not fully realized. In some reports [15,16,17], the existence and uniqueness results for a class of IFDEs were established. Also, some authors [18,19,20] have developed existence theorems for IDEs with a measurable right side for handling delay problems.

Global stability for ICFDEs with fixed impulse moments using the Lyapunov’s second method was examined in a previous report [1]. Suffice to state here that, the use of the Lyapunov’s second method has a restriction in its application to impulsive differential systems. Now, the Lyapunov method has been widely explored for analyzing the stability of solutions to differential equations [21], but as rightly observed by Stamova [1], the application of classical (continuous) Lyapunov functions significantly limits the potential of the method, but since the solutions of the systems under consideration are piecewise continuous functions, it becomes necessary to use Lyapunov function analogous with first-kind discontinuities. These functions greatly enhance the applicability of Lyapunov’s direct method to impulsive systems.

However, in numerous practical scenarios, the stability of certain sets must be studied even though these sets are not invariant under a given system of differential equations, ruling out Lyapunov stability. As stated in Ref. [21], such sets can be encountered in the study of self-regulating management systems (see [22]). Hence, to address the challenges that may arise, Lakshmikantham et al. [23] proposed a new concept called eventual stability (E-S), asserting that while the set in question is not invariant in the conventional sense, it exhibits invariance in an asymptotic sense (see also [24]). As observed in Ref. [25], there are numerous perturbation and adaptive control problems where the point of interest is not an equilibrium (invariant) point but rather an eventually stable set that is asymptotically invariant. This perspective allows Lyapunov stability to be viewed as a special case of E-S. Consequently, the E-S of solutions for IDEs has been widely studied [1]. For instance, Ref. [21] established results on the E-S and eventual boundedness of IDEs involving the “supremum”, while Ref. [26] provided sufficient conditions for maintaining E-S in perturbed IDEs.

In this paper, motivated by the researches in Refs. [10,11,21], and by appropriately adopting the vector form of the Lyapunov functions which is extended to a class of piecewise continuous Lyapunov functions, along with comparison results, we provide sufficient conditions for ensuring AE-S of the set of trivial solution with an illustrative example.

2. Preliminary Notes and Definitions

Let $R^n$ be the n-dimensional Euclidean space with norm $‖.‖,$ and let $\mathrm{\Omega }$ be a domain in $R^n$ containing the origin; $R_{+}=[0,\infty ),$$\hspace{0.33em}R=(-\infty ,\infty )$, $t_0\in R_{+},\hspace{0.17em}\hspace{0.17em}t>0$.

Let $J\subset R_{+}.$ Define the following class of functions $P{C}^{\beta }[J,\mathrm{\Omega }]=\alpha :J\to \mathrm{\Omega },\hspace{0.17em}\hspace{0.17em}$$\hspace{0.17em}\alpha (t)$ which is a $\beta$—times piecewise continuous mapping (with a fractional order $\beta$) from the interval $J$ into the range $\mathrm{\Omega }$ containing discontinuous points $\hspace{0.33em}{t}_k\in J$ at which $\alpha (t_k^{+})$ exists.

The basic concept of calculus such as the derivative and integrals can be generalized to non-integer order using fractional calculus. This allows for more in-depth understanding of behavior of functions, particularly when they have complex or irregular behavior. There are multiple ways to define fractional derivatives and the integrals and the choice of definitions depends on the specific applications [9,26,27].

Various definitions exist for fractional derivatives and integrals.

General case. Consider a given number, $n-1<\beta <n,\hspace{0.17em}$$\beta >0$ where $n$ represents a natural number, and $\mathrm{\Gamma }(.)$ signifies the Gamma function.

Definition 1. 

The Riemann Liouville fractional derivative of order $\beta$ of $\hspace{0.17em}\gamma (t)$ is given by Ref. [1]:

$${}_{t_0}{}^{RL}D{}_t{}^{\beta }\gamma (t)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\beta )}}{\displaystyle \frac{d^n}{d{t}^n}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{n-\beta -1}\gamma (s)}ds,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\ge t_0.$$

Definition 2. 

The Caputo fractional derivative of order $\beta$ of $\hspace{0.17em}\gamma (t)$ is defined as follows (see [1]):

$${}_{t_0}{}^{C}D{}_t{}^{\beta }\gamma (t)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\beta )}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{n-\beta -1}{\gamma }^{(n)}(s)}ds,\hspace{0.17em}\hspace{0.17em}t\ge t_0$$

The Caputo derivative shares many properties with standard derivatives, making it more intuitive and easier to apply. Additionally, the initial conditions of the Caputo fractional derivative are more straightforward to interpret in a physical context.

Definition 3. 

The Grunwald-Letnikov fractional derivative of order $\beta$ of $\hspace{0.17em}\gamma (t)$ is defined by Agarwal et al. as follows [10]:

$${}^{GL}D{}_0{}^{\beta }\gamma (t)=\underset{h\to 0^{+}}{\lim }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=0}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C{}_r\gamma (t-rh),\hspace{0.17em}\hspace{0.17em}t\ge t_0}$$

and

Definition 4. 

The Grunwald-Letnikov fractional Dini derivative of order $\beta$ of $\hspace{0.17em}\gamma (t)$ is expressed by Ref. [10]:

$${}^{GL}D{}_0{}^{\beta }\gamma (t)=\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=0}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C{}_r\gamma (t-rh),\hspace{0.17em}\hspace{0.17em}t\ge t_0}$$

where ${}^{\beta }C_r$ denotes the binomial coefficients and $\left[{\displaystyle \frac{t-t_0}{h}}\right]$ is the integer part of ${\displaystyle \frac{t-t_0}{h}}$.

Particular case (when n = 1). In the specific case, the order of $\beta$ is typically less than 1 in most applications so that $\beta \in (0,1).$ To simplify the notation, we will use ${}^{C}D^{\beta }$ instead of ${}_{t_0}{}^{C}D^{\beta }$ and the Caputo fractional derivative of order $\beta$ of the function $\gamma (t)$ is given by

$${}^{C}D{}^{\beta }\gamma (t)={\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{-\beta }{\gamma }^{\prime }\hspace{0.17em}ds,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\ge t_0}$$

(1)

3. Impulsive Dynamics in Fractional Order Derivatives

Let us explore the initial value problem (IVP) for the system of fractional order with a Caputo derivative for $\beta \in (0,1).$

$$\begin{array}{l}{}^{C}D{}^{\beta }\gamma (t)=f(t,\gamma ),\hspace{0.17em}\hspace{0.17em}t\ge t_0,\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma (t_0)={\gamma }_0,\end{array}$$

(2)

$where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma ,{\gamma }_0\in R^N,\hspace{0.17em}\hspace{0.17em}f\in C[R_{+}\times R^N,R^N],\hspace{0.17em}f(t,0)\equiv 0\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}(t_0,x_0)\in R_{+}\times R^N.$

Some sufficient conditions for the existence of the global solutions to (2) are considered in Refs. [1,7,17,22,28,29].

The IVP for FDE (2) is equivalent to the following Volterra integral equation (see Ref. [11]),

$$\gamma (t)={\gamma }_0+{\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{\beta -1}f(s,\gamma (s))ds,\hspace{0.17em}\hspace{0.17em}t\ge t_0}.$$

(3)

Consider the IVP for the system of IFDEs with a Caputo derivative for $0<\beta <1,$

$$\begin{array}{l}{}^{C}D{}^{\beta }\gamma (t)=f(t,\gamma ),\hspace{0.17em}\hspace{0.17em}t\ge t_0,\hspace{0.17em}\hspace{0.17em}t\ne t_k,\hspace{0.17em}\hspace{0.17em}k=1,2,\dots \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta \gamma =I_k(\gamma (t_k)),\hspace{0.17em}\hspace{0.17em}t=t_k,\hspace{0.17em}\hspace{0.17em}k\in N,\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma (t_0^{+})={\gamma }_0,\end{array}$$

(4)

$where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma ,{\gamma }_0\in R^N,\hspace{0.17em}\hspace{0.17em}f\in C[R_{+}\times R^N,R^N],\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{t}_0\in R_{+},\hspace{0.17em}\hspace{0.17em}{I}_k:R^N\to R^N,\hspace{0.17em}\hspace{0.17em}k=1,2,\dots$ under the following assumptions:

(i) 

$\hspace{0.17em}0<t_1<t_2<\dots <t_k<\dots ,\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{t}_k\to \infty \hspace{0.17em}\hspace{0.17em}as\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\to \infty ;$

(ii) 

$f:R_{+}\times R^N\to R^N$ is piecewise continuous in $(t_{k-1},t_k]$ and for each $x\in R^N,$$\hspace{0.17em}k=1,2,\dots ,$ and $\underset{(t,y)\to (t_k^{+},\gamma )}{\lim }\hspace{0.17em}f(t,y)=f(t_k^{+},\gamma )$ exists;

(iii) 

$I_k\times R^N\to R^N.$

In this paper, we assume that $\hspace{0.17em}f(t,0)\equiv 0,\hspace{0.17em}\hspace{0.17em}{I}_k(0)\equiv 0$ for all $k$ ensuring that the system (4) has a trivial solution, and the points $t_k,\hspace{0.17em}\hspace{0.17em}k=1,2,\dots$ are fixed such that $t_1<t_2<\dots \hspace{0.17em}\hspace{0.17em}$ and $\underset{k\to \infty }{\lim }{t}_k=\infty .$ The system (4) with the initial condition $\gamma (t_0)={\gamma }_0$ is assumed to have a solution $\gamma (t;t_0,{\gamma }_0)\in P{C}^{\beta }([t_0,\infty ),R^N).$

Remark 1. 

The second equation in (4) is referred to as the impulsive condition, and the function $I_k(\gamma (t_k))$ represents the magnitude of the jump in the solution at the point $t_k.$

Definition 5. 

Let $\mathrm{\Omega }:R_{+}\times R^N\to R^N.$ Then $\mathrm{\Omega }$ is said to belong to class $\chi$ if,

(i) 

$\mathrm{\Omega }$ is continuous in $(t_{k-1},t_k]$ and for each $\gamma \in R^N$ and $\underset{(t,y)\to (t_k^{+},\gamma )}{\lim }\mathrm{\Omega }(t,y)=\mathrm{\Omega }(t_k^{+},\gamma )$ exists;

(ii) 

$\mathrm{\Omega }$ is locally Lipschitz in its second variable $\gamma$ and $\mathrm{\Omega }\hspace{0.17em}(t,0)\equiv 0$.

Now, for any function $\mathrm{\Omega }(t,\gamma )\in PC([t_0,\infty )\times \xi ,R_{+}^N).$ Let us define the Caputo fractional Dini derivative (CFDD) as:

$$\begin{array}{l}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,\gamma )=\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}\{\mathrm{\Omega }(t,\gamma )-\mathrm{\Omega }(t_0,{\gamma }_0)-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^{r+1}{}^{\beta }C{}_r[\mathrm{\Omega }(t-\hspace{0.17em}rh,\hspace{0.17em}\gamma -h^{\beta }f(t,\gamma )-\mathrm{\Omega }(t_0,{\gamma }_0)]\}}\hspace{0.17em},\hspace{0.17em}t\ge t_0,\end{array}$$

(5)

where $t\in [t_0,\infty ),\hspace{0.17em}\hspace{0.17em}(\gamma ,{\gamma }_0)\in \xi ,\hspace{0.17em}\hspace{0.17em}\xi \in R^N$ and there is $h>0$ such that $t-rh\in [t_0,T]$ and the binomial coefficient ${}^{\beta }C{}_r=\frac{\mathrm{\Gamma }(\beta +1)}{r!(\beta -r)!}.$

Definition 6. 

A function $g\in PC[R^n,R^n]$ is said to be quasimonotone nondecreasing in $\gamma ,$ if $\gamma \le y$ and ${\gamma }_i=y_i$ for $1\le i\le n$ implies $g_i(\gamma )=g_i(y).$

Definition 7. 

The zero solution $\gamma (t;t_0,{\gamma }_0)\in P{C}^{\beta }([t_0,\infty ),R^N)$ (which means the zero solution belongs to a space of $\beta -$times piecewise continuous mapping from the interval $[t_0,\infty )$ into $R^N$) of (4) is said to be:

(E-S1) eventually stable if for every $\epsilon >0\hspace{0.17em}\hspace{0.17em}$and$\hspace{0.17em}{t}_0\in R_{+}$ there exist $\delta =\delta (\epsilon ,t_0)>0\hspace{0.17em}$ continuous in $t_0$ such that for any$\hspace{0.17em}{\gamma }_0\in R^N,\hspace{0.17em}\hspace{0.17em}‖{\gamma }_0‖\le \delta$ implies $\hspace{0.17em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon$ for $t\ge t_0;$

(E-S2) uniformly eventually stable if for every $\epsilon >0\hspace{0.17em}\hspace{0.17em}$and$\hspace{0.17em}{t}_0\in R_{+}$ there exist $\delta =\delta (\epsilon )>0\hspace{0.17em},$ continuous in $t_0$ such that for any$\hspace{0.17em}{\gamma }_0\in R^N,$$\hspace{0.17em}‖{\gamma }_0‖\le \delta$ implies$\hspace{0.17em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon$ for $t\ge t_0;$

(E-S3) asymptotically eventually stable if it is eventually stable and if for each $\epsilon >0\hspace{0.33em}$ and $t_0\in R_{+}$ there exist positive numbers ${\delta }_0={\delta }_0(t_0)>0$ and $T=T(t_0,\epsilon )$ such that for $t\ge t_0+T$ and $\hspace{0.17em}‖{\gamma }_0‖\le \delta$ implies$\hspace{0.17em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon ;$

(E-S4) uniformly asymptotically eventually stable if it is uniformly eventually stable and ${\delta }_0={\delta }_0(\epsilon )$ and $T=T(\epsilon )$ such that for $t\ge t_0+T,$ the inequality$\hspace{0.17em}‖{\gamma }_0‖\le \delta$ implies$\hspace{0.17em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon .$ Example of such functions are Mittag-Leffler functions, etc.

Definition 8. 

A function $a(r)$ is said to belong to the class $K$ if $a\in PC([0,\psi ),R_{+}),\hspace{0.17em}$$\hspace{0.17em}a(0)=0,$ and $a(r)$ is strictly monotone increasing in $r.$

In this paper, we define the following sets:

$${\overline{S}}_{\psi }=\left\{\gamma \in R^N:\hspace{0.17em}‖\gamma ‖\le \psi \right\}$$

$$S_{\psi }=\left\{\gamma \in R^N:\hspace{0.17em}‖\gamma ‖<\psi \right\}$$

Suffice to say that the inequalities between vectors are understood to be component-wise inequalities.

We shall adopt the comparison system for the ICFDE of the type

$$\begin{array}{l}{}_{t_0}{}^{C}D{}^{\beta }u=g(t,u),\hspace{0.17em}\hspace{0.17em}t\ge t_0,\hspace{0.17em}\hspace{0.17em}t\ne t_k,\hspace{0.17em}\hspace{0.17em}k=1,2,\dots \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta u={\psi }_k(u(t_k)),\hspace{0.17em}\hspace{0.17em}t=t_k,\hspace{0.17em}\hspace{0.17em}k\in N,\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u(t_0^{+})=u_0,\end{array}$$

(6)

existing for $t\ge t_0$, $u\in R^n,\hspace{0.17em}\hspace{0.17em}{R}_{+}=[t_0,\infty ),\hspace{0.17em}\hspace{0.17em}g:R_{+}\times R_{+}^n\to R^n,\hspace{0.17em}\hspace{0.17em}g(t,0)\equiv 0,$ where $g$ is the continuous mapping of $R_{+}\times R_{+}^n\hspace{0.17em}$ into $R_{+}\times R_{+}^n\to R^n$. The function $g\in PC[R_{+}\times R_{+}^n,R^n]$ is such that for any initial data $(t_0,u_0)\in R_{+}\times R^n,$ the system (6) with initial condition $u(t_0)=u_0$ is assumed to have a solution $u(t,t_0,u_0)\in P{C}^{\beta }([t_0,\infty ),R^n).$

Lemma 1. 

Assume $m\in PC([t_0,T]\times {\overline{S}}_{\psi },R^N),$ and suppose there exists $t^{\ast }\in [t_0,T]$

Such that for ${\alpha }_1<{\alpha }_2,$$m(t^{\ast },{\alpha }_1)=m(t^{\ast },{\alpha }_2)$ and $m(t,{\alpha }_1)<m(t,{\alpha }_2)$ for $t_0\le t<t^{\ast }.$ Then if the CFDD of $m$ exists at $t^{\ast },$ then the inequality

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_1)-{}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_2)>0,$ holds.

Proof of Lemma 1. 

Let $\mathrm{\Omega }(t,\gamma )=m(t,{\alpha }_1)-m(t,{\alpha }_2).$ Applying (5), we have

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D_{+}^{\beta }(m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2))=\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}\{[m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2)]-[m(t_0,{\alpha }_1)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-m(t_0,{\alpha }_2)]\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^{r+1}{}^{\beta }C_r[m(t^{\ast }-rh,{\alpha }_1)-m(t^{\ast }-rh,{\alpha }_2)]}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\},\end{array}$$

when ${\alpha }_1={\alpha }_2$ we have

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D_{+}^{\beta }(m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2))=\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}\{-[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^{r+1}{}^{\beta }C_r[m(t^{\ast }-rh,{\alpha }_1)-m(t^{\ast }-rh,{\alpha }_2)]}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\}\end{array}$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D_{+}^{\beta }(m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2))=-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C_r[m(t^{\ast }-rh,{\alpha }_1)-m(t^{\ast }-rh,{\alpha }_2)]}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C_r[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]}\end{array}$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D_{+}^{\beta }(m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2))=-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C_r[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]}\end{array}$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D_{+}^{\beta }(m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2))=-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h^{\beta }}}{\displaystyle \sum _{r=1}^{\left[\frac{t-t_0}{h}\right]}{(-1)}^r{}^{\beta }C_r[m(t_0,{\alpha }_1)-m(t_0,{\alpha }_2)].}\end{array}$$

Applying (3.8) in Ref. [5], where $\underset{h\to 0^{+}}{\lim \sup }{\displaystyle \frac{1}{h}}{\displaystyle \sum _{r=0}^{[\frac{t-t_0}{h}]}{(-1)}^r=D_0^{\beta }(1)=\frac{{(t-t_0)}^{-\beta }}{\mathrm{\Gamma }(1-\beta )}},$ we have

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_1)-{}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_2)=-{\displaystyle \frac{{(t-t_0)}^{-\beta }}{\mathrm{\Gamma }(1-\beta )}}\left[m(t^{\ast },{\alpha }_1)-m(t^{\ast },{\alpha }_2)\right].$$

By the statement of Lemma 1, we have that

$$m(t,{\alpha }_1)-m(t,{\alpha }_2)<0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{t}_0\le t\le t^{\ast }.$$

And so, it follows that

$${}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_1)-{}^{C}D{}_{+}{}^{\beta }m(t^{\ast },{\alpha }_2)>0$$

Note that some existence results for (6) are given in Refs. [18,22].□

Remark 2. 

Lemma 1 generalizes Lemma 1 in Ref. [10], in which the vectors $m(t,{\alpha }_1)$ and $m(t,{\alpha }_2)$ are compared on a component-by-component basis.

In the following, we establish the comparison results for the system (4).

4. Vector Fractional Differential Equations: Inequalities and Comparison Results

In this section, let us suppose that $0<\beta <1$.

Theorem 1. 

(Comparison Results). Suppose that:

(i) 

$g\in PC[R_{+}\times R_{+}^n,\hspace{0.33em}{R}^n]$and is continuous in $(t_{k-1},t_k],\hspace{0.17em}\hspace{0.17em}k=1,2,\dots$  and  $g(t,u)$  is

Quasimonotone non-decreasing in   $u$  for each  $u\in R^n$and

$\underset{(t,y)\to (t_k^{+},u)}{\lim }\hspace{0.17em}g(t,u)=g(t_k^{+},u)$  exists;

(ii) 

$\mathrm{\Omega }\in PC[R_{+}\times R^N,R_{+}{}^N]$  and   $\mathrm{\Omega }\in \chi$  such that

${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,\gamma )\le g(t,\mathrm{\Omega }(t,\gamma )),\hspace{0.33em}\hspace{0.33em}(t,\gamma )\in R_{+}\times R^N$  and

$\mathrm{\Omega }(t_k,\gamma +I_k(\gamma (t_k)))\le {\omega }_k(\mathrm{\Omega }(t,\gamma (t))),\hspace{0.33em}t=t_k,\hspace{0.17em}\gamma \in S_{\psi }$  and the function

${\omega }_k:R_{+}^N\to R_{+}^N$ is non decreasing for $k=1,2,\dots$

(iii) 

$\vartheta (t)=\vartheta (t,t_0,u_0)\in P{C}^{\beta }([t_0,T],R^n)$ be the maximal solution of the IVP for the IFDEsystem (6)

Then,

$$\mathrm{\Omega }(t,\gamma (t))\le \vartheta (t),\hspace{0.33em}\hspace{0.33em}t\ge t_0,$$

(7)

where $\gamma (t)=\gamma (t,t_0,{\gamma }_0)\in P{C}^{\beta }([t_0,T],R^N)$ represents any solution of (4) that lies on$[t_0,\infty ),$ if only that

$$\mathrm{\Omega }(t_0^{+},{\gamma }_0)\le u_0.$$

(8)

Proof of Theorem 1. 

Assume $\eta \in S_{\psi }$ is an arbitrarily small vector, and examine the IVP for the following system of FDEs,

$$\begin{array}{l}{}^{C}D{}^{\beta }u=g(t,u)+\eta ,\hspace{0.33em}\hspace{0.33em}for\hspace{0.33em}t\in [t_0,\infty )\hspace{0.33em}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}u(t_0^{+})=u_0+\eta ,\end{array}$$

(9)

for $t\in [t_0,\infty ).$

The function $u_{\eta }(t,\alpha )$ is a solution of (9), where $\alpha >0$ for the FDE (6) if and only if it satisfies the Volterra fractional integral equation,

$$u_{\eta }(t,\alpha )=u_0+\eta +\frac{1}{\mathrm{\Gamma }(\beta )}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{\beta -1}(g(s,u_{\eta }(s,\alpha ))+\alpha )ds,\hspace{0.33em}\hspace{0.33em}t\in [t_0,\infty )}.$$

(10)

Suppose that the function $m(t,\alpha )\in PC([t_0,T]\times S_{\psi },\hspace{0.33em}{R}_{+}^N)$ is defined as $m(t,\alpha )=\mathrm{\Omega }(t,{\gamma }^{\ast }(t)).$

We now prove that

$$m(t,\alpha )\hspace{0.33em}<u_{\eta }(t,\alpha )\hspace{0.33em}\hspace{0.33em}for\hspace{0.33em}\hspace{0.33em}t\in [t_0,\infty ).$$

(11)

It can be seen that the inequality (11) is valid whenever $t=t_0,\hspace{0.33em}\hspace{0.33em}i.e.\hspace{0.33em}$,

$$m(t_0,\alpha )=\mathrm{\Omega }(t_0,{\gamma }_0)\le u_0<u_{\eta }(t_0,\alpha )\hspace{0.17em}\hspace{0.17em}.$$

Assume that the inequality (11) is not true, then there exists a point $t_1>t_0$ such that $m(t_1,\alpha )=u_{\eta }(t_1,\alpha )\hspace{0.33em}\hspace{0.33em}and\hspace{0.33em}\hspace{0.33em}m(t,\alpha )<u_{\eta }(t,\alpha )$ for $t\in [t_0,t_1).$

As a result of Lemma 1, it follows that

$${}^{C}D_{+}^{\beta }(m(t_1,\alpha )-u_{\eta }(t_1,\alpha ))>0\hspace{0.33em}\hspace{0.33em}i.e.$$

$${}^{C}D_{+}^{\beta }(m(t_1,\alpha ))\hspace{0.33em}>{}^{C}D{}_{+}{}^{\beta }u_{\eta }(t_1,\alpha )\hspace{0.33em}$$

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t_1,\gamma (t_1))\hspace{0.33em}>{}^{C}D{}_{+}{}^{\beta }u_{\eta }(t_1,\alpha )\hspace{0.33em}.$$

And using (9) we arrive at

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t_1,\gamma (t_1))\hspace{0.33em}>g(t_1,u(t_1,\alpha ))\hspace{0.33em}+\eta >g(t_1,u(t_1,\alpha ).$$

Therefore,

$${}^{C}D{}_{+}{}^{\beta }m(t_1,\alpha )>g(t_1,u(t_1,\alpha )).$$

(12)

From Theorem 1, the function ${\gamma }^{\ast }(t)=\gamma (t,t_0,{\gamma }_0)$ satisfies the IVP (9) and the equality

$$\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}[{\gamma }^{\ast }(t)-{\gamma }_0-S({\gamma }^{\ast }(t),h)]=f(t,{\gamma }^{\ast }(t)),\hspace{0.33em}\hspace{0.33em}holds,$$

(13)

where ${\gamma }^{\ast }(t)=\gamma (t,t_0,{\gamma }_0)$ is any other solution of (6), and

$$S({\gamma }^{\ast }(t),h)={\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[{\gamma }^{\ast }(t-rh)-{\gamma }_0\right].$$

(14)

Multiply (13) through by $\hspace{0.17em}{h}^{\beta }$

$$\underset{h\to 0^{+}}{\lim \sup }[{\gamma }^{\ast }(t)-{\gamma }_0-S({\gamma }^{\ast }(t),h)]=h^{\beta }f(t,{\gamma }^{\ast }(t))$$

$${\gamma }^{\ast }(t)-{\gamma }_0-[S({\gamma }^{\ast }(t),h)+\rho (h^{\beta })]=h^{\beta }f(t,{\gamma }^{\ast }(t))$$

$${\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))=[S({\gamma }^{\ast }(t),h)+{\gamma }_0+\rho (h^{\beta })].$$

(15)

Thus,

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m(t,\alpha )-m(t_0,\alpha )-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[m(t-rh,\alpha )-m(t_0,\alpha )\right]=\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh,{\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]\}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}t\in [t_0,\infty )\end{array}$$

$$\begin{array}{l}=\mathrm{\Omega }(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh)-{\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]\}+\\ \hspace{0.33em}\hspace{0.33em}{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\{[\mathrm{\Omega }(t-rh),S({\gamma }^{\ast }(t),h)+{\gamma }_0+\rho (h^{\beta })-\mathrm{\Omega }(t_0,{\gamma }_0)]-\hspace{0.33em}[\mathrm{\Omega }(t-rh,{\gamma }^{*}(t-rh))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\mathrm{\Omega }(t_0,{\gamma }_0)]\}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}$$

(16)

Since $\mathrm{\Omega }(t,\gamma )$ is locally Lipschitzian with respect to the second variable, we have,

$$\le L\left|{(-1)}^{r+1}\right|‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}({}^{\beta }C_r)}[S({\gamma }^{\ast }(t),h)+{\gamma }_0+\rho (h^{\beta })-{\gamma }^{*}(t-rh)]‖.$$

where $L>0$ is a Lipschitz constant

$$\le L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}({}^{\beta }C_r)}[S({\gamma }^{\ast }(t),h)+\rho (h^{\beta }))-({\gamma }^{*}(t-rh)-{\gamma }_0)]‖.$$

(17)

Using (14); (17) becomes,

$$\le L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}({}^{\beta }C_r)({\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}}[({\gamma }^{\ast }(t-rh)-{\gamma }_0)+\rho (h^{\beta }))-({\gamma }^{*}(t-rh)-{\gamma }_0)]‖$$

$$\le L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}({}^{\beta }C_r){(-1)}^{r+1}[{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{}^{\beta }C_r}}[({\gamma }^{\ast }(t-rh)-{\gamma }_0)]+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{}^{\beta }C_r}\rho (h^{\beta })-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{}^{\beta }C_r}({\gamma }^{*}(t-rh)-{\gamma }_0)]‖$$

$$\le L{(-1)}^{r+1}‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}({}^{\beta }C_r)}[({\gamma }^{\ast }(t-rh)-{\gamma }_0)]+[{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{}^{\beta }C_r}-1]+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{}^{\beta }C_r}\rho (h^{\beta })‖.$$

(18)

Substituting (18) into (16) we have

$$\begin{array}{l}=\mathrm{\Omega }(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh)-{\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]+\\ \hspace{0.33em}\hspace{0.33em}L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}({\gamma }^{\ast }(t-rh)-{\gamma }_0)[{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}{}^{\beta }C_r}-1]+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}{}^{\beta }C_r}\rho (h^{\beta })‖.\end{array}$$

$$\begin{array}{l}=\mathrm{\Omega }(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh)-{\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]+\\ \hspace{0.33em}\hspace{0.33em}L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}({\gamma }^{\ast }(t-rh)-{\gamma }_0)[-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r{}^{\beta }C_r}-1]+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}{}^{\beta }C_r}\rho (h^{\beta })‖.\end{array}$$

Dividing through by $h^{\beta }>0$ and taking the $\lim \sup \hspace{0.17em}\hspace{0.17em}as\hspace{0.17em}\hspace{0.17em}h\to 0^{+}$ we have,

$$\begin{array}{l}{}^{C}D{}_{+}{}^{\beta }m(t,\alpha )=\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{\mathrm{\Omega }(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh)-{\gamma }^{\ast }(t)-h^{\beta }f(t,{\gamma }^{\ast }(t))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]\right\}+\\ \hspace{0.33em}\hspace{0.33em}\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}({\gamma }^{\ast }(t-rh)-{\gamma }_0)[-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r{}^{\beta }C_r}-1]+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}{}^{\beta }C_r}\rho (h^{\beta })‖.\end{array}$$

Recall that,

$$\underset{h\to 0^{+}}{\lim }{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}=-1\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\rho (h^{\beta })=0.$$

From (3.6) and (3.7) in [10], we have that

$${}^{C}D{}_{+}{}^{\beta }m(t,\alpha )={}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,{\gamma }^{\ast }(t))+L‖{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}({}^{\beta }C_r)}({\gamma }^{\ast }(t-rh)-{\gamma }_0)[-(-1)-1]+0‖.$$

Using condition (ii) of the Theorem 1, we obtain the estimate

$${}^{C}D{}_{+}{}^{\beta }m(t,\alpha )\le g(t,\mathrm{\Omega }(t,{\gamma }^{\ast }(t))=g(t,m(t,\alpha )).$$

(19)

Also,

$$m(t_0^{+},\alpha )\le u_0\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m(t_k^{+},\alpha )=\mathrm{\Omega }(t_k^{+},\gamma (t_k)+I_k(\gamma (t_k))\le {\rho }_k(m(t_k)).$$

(20)

Now Equation (20) with $t=t_1$ contradicts (12), hence (11) is true.□

For $\hspace{0.33em}t\in [t_0,T],$ we now establish that

$$u_{{\eta }_1}(t,\alpha )<u_{{\eta }_2}(t,\alpha )\hspace{0.33em}\hspace{0.33em}\mathrm{whenever}{\eta }_1<{\eta }_2.$$

(21)

Observe that the inequality (21) holds for $t=t_0.$

Assume that (21) is not true. Then there exists a point $t_1$ such that $u_{{\eta }_1}(t_1,\alpha )=u_{{\eta }_2}(t_2,\alpha )$ and $u_{{\eta }_1}(t,\alpha )<u_{{\eta }_2}(t,\alpha )\hspace{0.33em}\hspace{0.33em}for\hspace{0.17em}\hspace{0.17em}t\in [t_0,t_1).$

By Lemma 1, we have that

$${}^{C}D_{+}^{\beta }(u_{{\eta }_1}(t_1,\alpha )-u_{{\eta }_2}(t_2,\alpha ))>0\hspace{0.33em}.$$

However,

$${}^{C}D_{+}^{\beta }(u_{{\eta }_1}(t_1,\alpha )-u_{{\eta }_2}(t_2,\alpha ))=\hspace{0.33em}{}^{C}D_{+}^{\beta }(u_{{\eta }_1}(t_1,\alpha )-u_{{\eta }_2}(t_2,\alpha ))$$

$$\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=g(t_1,u(t_1,\alpha ))+{\eta }_1-[g(t_1,u(t_1,\alpha ))+{\eta }_2]\hspace{0.33em}$$

$$\hspace{0.33em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\eta }_1-{\eta }_2<0.$$

which is a contradiction, and so (21) is true. Thus, (11) and (21) guarantee that the family of solutions $\{u_{\eta }(t,\alpha )\},\hspace{0.17em}\hspace{0.17em}t\in [t_0,T]$ of (9) is uniformly bounded, i.e. there exists $\lambda >0$ with $\left|u_{\eta }(t,\alpha )\right|\le \lambda$, with bound $\lambda$ on $[t_0,T].$ We now show that the family $\{u_{\eta }(t,\alpha )\}$ is equicontinuous on $[t_0,T].$ Let $k=\sup \left\{|g(t,\gamma )|:(t,\gamma )\in \left[t_0,T\right]\times \left[-\lambda ,\lambda \right]\right\},$ where $\lambda$ is the bound on the family $\{u_{\eta }(t,\alpha )\}.$ Fix a decreasing sequence ${\left\{{\eta }_i\right\}}_{i=0}^{\infty }(t)$, to the end that $\underset{i\to \infty }{\lim }{\eta }_i=0,$ and let $\{u_{\eta }(t,\alpha )\}$ be a sequence of functions. Also, let $t_1,t_2\in [t_0,T]$ with $t_1<t_2,$ then we have the following estimates,

$$\begin{array}{l}‖u_{\eta }(t_2,\alpha )-u_{\eta }(t_1,\alpha )‖\le \hspace{0.17em}||u_0+{\eta }_i+{\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}{\displaystyle \underset{t_0}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}}(g(s,u(s,{\eta }_i))+{\eta }_i)ds\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\left(u_0+{\eta }_i+{\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}{\displaystyle \underset{t_0}{\overset{t_2}{\int }}{\left(t_1-s\right)}^{\beta -1}}(g(s,u(s,{\eta }_i))+{\eta }_i)ds\right)||\end{array}$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}‖\left[{\displaystyle \underset{t_0}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}}-{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_1-s\right)}^{\beta -1}}\right](g(s,u(s,{\eta }_i))u(t_2,{\eta }_i)ds‖\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\mathrm{\Gamma }(\beta )}}‖\left({\displaystyle \underset{t_0}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}}-\hspace{0.17em}{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_1-s\right)}^{\beta -1}}\right)ds‖\end{array}$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\mathrm{\Gamma }(\beta )}}‖\left({\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_2-s\right)}^{\beta -1}}+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}}-{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_1-s\right)}^{\beta -1}}\right)ds‖\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\mathrm{\Gamma }(\beta )}}‖{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_2-s\right)}^{\beta -1}}-{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_1-s\right)}^{\beta -1}}+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}ds}‖\end{array}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\mathrm{\Gamma }(\beta )}}\left[‖{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_2-s\right)}^{\beta -1}}-{\displaystyle \underset{t_0}{\overset{t_1}{\int }}{\left(t_1-s\right)}^{\beta -1}}ds‖+‖{\displaystyle \underset{t_1}{\overset{t_2}{\int }}{\left(t_2-s\right)}^{\beta -1}}ds‖\right]$$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\beta \mathrm{\Gamma }(\beta )}}\left[‖{\left(t_2-t_0\right)}^{\beta }-{\left(t_1-t_0\right)}^{\beta }-{\left(t_2-t_1\right)}^{\beta }‖\hspace{0.17em}+‖{\left(t_2-t_1\right)}^{\beta }‖\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\displaystyle \frac{k}{\beta \mathrm{\Gamma }(\beta )}}\left[{\left(t_2-t_1\right)}^{\beta }+{\left(t_2-t_1\right)}^{\beta }\right]={\displaystyle \frac{2k}{\mathrm{\Gamma }(\beta +1)}}{\left(t_2-t_1\right)}^{\beta }<\epsilon .\end{array}$$

provided $‖t_2-t_1‖<\delta (\epsilon )={\left({\displaystyle \frac{\mathrm{\Gamma }(\beta +1)\epsilon }{2k}}\right)}^{{\displaystyle \frac{1}{\beta }}},$ verifying that the family of solutions $\{u_{\eta }(t,\alpha )\}$ is equicontinuous. According to Arzela-Ascoli theorem, $\{u_{{\eta }_i}(t,\alpha )\}$ ensures the existence of a subsequence $\{u_{{\eta }_{ij}}(t,\alpha )\}$ that converges uniformly to the function $\vartheta (t)$ on $[t_0,T].$ Then we show that $\vartheta (t)$ is a solution of (10). Thus, (10) becomes

$$u_{{\eta }_{i_j}}(t,\alpha )=u_{0i_j}+{\eta }_{i_j}+\frac{1}{\mathrm{\Gamma }(\beta )}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{\beta -1}(g_{i_j}(s,u_{\eta i_j}(s,\alpha ))+{\eta }_{i_j})ds,\hspace{0.33em}\hspace{0.33em}t\in [t_0,\infty )}.$$

(22)

Taking the $\lim \hspace{0.17em}\hspace{0.17em}as\hspace{0.17em}\hspace{0.17em}{i}_j\to \infty$ in (22) yields,

$$\vartheta (t)=u_0+{\eta }_{i_j}+\frac{1}{\mathrm{\Gamma }(\beta )}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{\beta -1}(g(s,\vartheta (t))ds,\hspace{0.33em}\hspace{0.33em}t\in [t_0,\infty )}.$$

(23)

Hence, $\vartheta (t)$ is a solution of (6) on $[t_0,T].$ We assert that $\vartheta (t)$ is the maximal solution of (6). Therefore, from (11), it follows that

$m(t,\alpha )<u_{\eta }(t,\alpha )\le \vartheta (t)\hspace{0.17em}\hspace{0.17em}on\hspace{0.17em}\hspace{0.17em}[t_0,T].$□

Suppose that in Theorem 1, $g(t,u)\equiv 0,$ then we have the following results

Corollary 1. 

Assume that Condition (i) of Theorem 1 holds and,

$(i)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }\in PC[R_{+}\times R^N,R_{+}{}^N]$  and   $\mathrm{\Omega }\in \chi$  such that

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,\gamma )\le 0,$$

(24)

holds, and

$\mathrm{\Omega }(t_k,\gamma +I_k(\gamma (t_k)))\le {\omega }_k(\mathrm{\Omega }(t,\gamma (t))),\hspace{0.33em}t=t_k,\hspace{0.17em}\hspace{0.17em}\gamma \in S_{\psi }$ and the function,

${\omega }_k:R_{+}^N\to R_{+}^N$ is nondecreasing for $k=1,2,\dots$. Then for$t\in [t_0,\infty ),$ the inequality,

$\mathrm{\Omega }(t,\gamma (t))\le \mathrm{\Omega }(t_0^{+},{\gamma }_0)$ holds.

5. Main Results

This section provides sufficient conditions for E-S as well as AE-S of the system (4). Again, assume that $0<\beta <1$.

Theorem 2. 

Assume that:

(i) 

$g\in PC[R_{+}\times R_{+}^n,\hspace{0.33em}{R}^n]$ is piecewise continuous in $\hspace{0.17em}(t_{k-1},t_k]$ and for each $u\in R^n,\hspace{0.17em}\hspace{0.17em}k=1,2,\dots ,$ and $\underset{(t,y)\to (t_k^{+},u)}{\lim }\hspace{0.17em}g(t,y)=g(t_k^{+},u)$ exists, and $\hspace{0.33em}g(t,u)$ is

quasimonotone nondecreasing in $u.$

(ii) 

$\mathrm{\Omega }\in PC[R_{+}\times R^N,R_{+}{}^N]$ and $\mathrm{\Omega }\in \chi$ such that

${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,\gamma )\le g(t,\mathrm{\Omega }(t,\gamma )),\hspace{0.33em}t\ne t_k,\hspace{0.17em}\hspace{0.17em}holds\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}all\hspace{0.17em}\hspace{0.17em}(t,\gamma )\in R_{+}\times S_{\Psi }.$

There exists ${\psi }_0>0$ such that ${\gamma }_0\in S_{\psi }$ implies that $\gamma (t)+I_k(\gamma (t_k))\in S_{\psi }$ and

$\mathrm{\Omega }(t_k,\gamma +I_k(\gamma (t_k)))\le {\omega }_k(\mathrm{\Omega }(t,\gamma (t))),\hspace{0.33em}t=t_k,\hspace{0.17em}\hspace{0.17em}\gamma \in S_{\psi }$ and the function

${\omega }_k:R_{+}^N\to R_{+}^N$ is nondecreasing for $k=1,2,\dots$

(iii) 

$b\left(‖\gamma ‖\right)\le {\mathrm{\Omega }}_0(t,\gamma ),$ where $b\in K$ and ${\mathrm{\Omega }}_0(t,\gamma )={\displaystyle \sum _{i=1}^N{\mathrm{\Omega }}_i(t,\gamma )}.$

Then the E-S of the set of trivial solution $u=0$ of (6) implies the E-S of the set of trivialsolution $\gamma =0$ of (4).

Proof of Theorem 2. 

Let $0<\epsilon <\rho$ and $t_0\in R_{+}$ be given.

Assume that the solution (6) is eventually stable. Then given $b(\epsilon )>0$ and $t_0\in R_{+}$, there exists a positive function $\delta =\delta (t_o,\epsilon )>0$ which is continuous in $t_0$ for each $\epsilon$ such that

$${\displaystyle \sum _{i=1}^N{u}_{i0}\le \delta }\mathrm{implies}{\displaystyle \sum _{i=1}^N{u}_i(t,t_0,u_0)<b(\epsilon ),\hspace{0.33em}t\ge t_0},$$

(25)

where $u(t,t_0,u_0)$ is any solution of (6).

Choose $u_0=\mathrm{\Omega }(t_0^{+},{\gamma }_0).$

Since $\mathrm{\Omega }(t,x)$ is continuous, then by the property of continuity, given $\epsilon >0$ there exists a positive function ${\delta }_1={\delta }_1(t_0,\delta )>0$ that is continuous in $t_0$ for each $\delta$ such that the inequalities

$$‖\mathrm{\Omega }(t,\gamma )-\mathrm{\Omega }(t_0,{\gamma }_0)‖<\delta \mathrm{implies}‖\gamma -{\gamma }_0‖<{\delta }_1,$$

and as $‖\mathrm{\Omega }(t,\gamma )‖\to 0,\hspace{0.17em}\hspace{0.17em}‖\gamma ‖\to 0$ then the inequalities

$$‖{\gamma }_0‖<{\delta }_1\hspace{0.33em}\hspace{0.33em}and\hspace{0.33em}\hspace{0.33em}‖\mathrm{\Omega }(t_0,{\gamma }_0)‖<\delta ,$$

(26)

are satisfied simultaneously.

We claim that, if $‖{\gamma }_0‖<{\delta }_1$ then $\hspace{0.33em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon$.

Suppose that this claim is false, then there would exists a point $t_1\in [t_0,t)$ and the solution $\gamma (t,t_0,{\gamma }_0)$ with $‖{\gamma }_0‖<{\delta }_1$ such that

$$‖\gamma (t_1)‖=\epsilon \hspace{0.33em}\hspace{0.33em}and\hspace{0.33em}\hspace{0.33em}‖\gamma (t)‖<\epsilon \mathrm{for}t\in [t_0,t_1).$$

(27)

So that using (27); condition (iii) of Theorem 2 reduces to the form

$b\left(‖\gamma (t_1)‖\right)\le {\displaystyle \sum _{i=1}^N{\mathrm{\Omega }}_i(t_1,\gamma (t_1))}$, implying

$$b(\epsilon )\le {\displaystyle \sum _{i=1}^N{\mathrm{\Omega }}_i(t_1,\gamma (t_1))}.$$

(28)

for $t\in [t_0,t_1)$ and from Theorem 1,

$$\mathrm{\Omega }(t,\gamma (t)\le \vartheta (t).$$

(29)

where $\vartheta (t)={\displaystyle \sum _{i=1}^n{\vartheta }_i(t,t_0,u_0)}$ is the maximal solution of (6).

Then, using (27)–(29) and condition $(iii)$ of Theorem 2 we arrive at the estimate

$$b(\epsilon )\le {\mathrm{\Omega }}_0(t_1,\gamma (t_1))\le {\displaystyle \sum _{i=1}^N{\vartheta }_i(t,t_0,u_0)}<b(\epsilon ).$$

which leads to a contradiction.

Thus, the E-S of the set of trivial solution $\gamma =0$ of (4) is implied by the E-S of the set of trivial solution $u=0$ of (6).□

Theorem 3. 

Assume that:

(i) 

$g\in PC[R_{+}\times R_{+}^n,\hspace{0.33em}{R}^n]$  is piecewise continuous in $\hspace{0.17em}(t_{k-1},t_k]$ and for each $u\in R^n,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,2,\dots$ and $\underset{(t,y)\to (t_k^{+},u)}{\lim }\hspace{0.17em}g(t,y)=g(t_k^{+},u)$ exists, and $\hspace{0.33em}g(t,u)$ is quasimonotone nondecreasing in $u,$

(ii) 

$\mathrm{\Omega }\in PC[R_{+}\times R^N,R_{+}{}^N]$  and  $\hspace{0.33em}\mathrm{\Omega }\in \chi$, such that

${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,\gamma )\le g(t,\mathrm{\Omega }(t,\gamma )),$ $t\ne t_k,\hspace{0.17em}$  holds for all  $(t,\gamma )\in R_{+}\times S_{\psi }.$

There exists  ${\psi }_0>0$  such that  ${\gamma }_0\in S_{\psi }$  implies that  $\gamma (t)+I_k(\gamma (t_k))\in S_{\psi }$ and

$\mathrm{\Omega }(t_k,\gamma +I_k(\gamma (t_k)))\le {\omega }_k(\mathrm{\Omega }(t,\gamma (t))),\hspace{0.33em}t=t_k,\hspace{0.17em}\gamma \in S_{\psi }$and the function

${\omega }_k:R_{+}^N\to R_{+}^N$  is nondecreasing for  $k=1,2,\dots$

(iii) 

$b\left(‖\gamma ‖\right)\le {\mathrm{\Omega }}_0(t,\gamma ),$  for all  $(t,\gamma )\in R_{+}\times S_{\Psi },$  where  $b\in K$  and  ${\mathrm{\Omega }}_0(t,\gamma )={\displaystyle \sum _{i=1}^N{\mathrm{\Omega }}_i(t,\gamma )}.$

Therefore, the AE-S of the set of trivial solution $\gamma =0$ of (4) is implied by the AE-S of the set of trivial solution $u=0$ of (6).

Proof of Theorem 3. 

Let $0<\epsilon <\rho$ and $t_0\in R_{+}$ be given.

Suppose that the set of trivial solution of (6) is asymptotically eventually stable. This means that it is eventually stable, and given $b(\epsilon )>0$ and $t_0\in R_{+}$, a positive function $\delta =\delta (t_o,\epsilon )>0$ exists which is continuous in $t_0$ for each $\epsilon$ such that

$${\displaystyle \sum _{i=1}^N{u}_{i0}\le \delta }\mathrm{implies}{\displaystyle \sum _{i=1}^N{u}_i(t,t_0,u_0)<b(\epsilon ),\hspace{0.33em}t\ge t_0}.$$

(30)

Since $\mathrm{\Omega }(t,x)$ is continuous, then by the property of continuity, given $\epsilon >0$ a positive function ${\delta }_1={\delta }_1(t_o,\delta )>0$ exists which is continuous on $t_0$ for each $\delta ,$ such that the inequalities

$$‖\mathrm{\Omega }(t,\gamma )-\mathrm{\Omega }(t_0,{\gamma }_0)‖<\delta \mathrm{implies}‖\gamma -{\gamma }_0‖<{\delta }_1,$$

and as $‖\mathrm{\Omega }(t,\gamma )‖\to 0,\hspace{0.17em}\hspace{0.17em}‖\gamma ‖\to 0$ then the inequalities

$$‖{\gamma }_0‖<{\delta }_1\mathrm{and}‖\mathrm{\Omega }(t_0,{\gamma }_0)‖<\delta ,$$

(31)

are satisfied simultaneously.

We claim that, if $‖{\gamma }_0‖<{\delta }_1,$ then $\hspace{0.33em}‖\gamma (t,t_0,{\gamma }_0)‖<\epsilon$.

Now, suppose this claim is false, then there exists a sequence $\left\{t_n\right\}$ such that $t_n\to \infty$ as $n\to \infty$ and $t_n\ge t_0+T$ such that

$‖\gamma (t_k,t_0,{\gamma }_0)‖\ge \epsilon$, where $\gamma (t_k,t_0,{\gamma }_0)$ is some solution of (4) starting in $‖{\gamma }_0‖<{\delta }_0$.

Now, from condition (iii) of the Theorem 3, we have that

$$b(‖\gamma (t_k,t_0,{\gamma }_0)‖)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}{\mathrm{\Omega }}_0(t_k,\gamma (t_k,t_0,{\gamma }_0)),\hspace{0.33em}\hspace{0.17em}{t}_k\ge t_0+T$$

$$b(\epsilon )\le \hspace{0.17em}\hspace{0.17em}{\mathrm{\Omega }}_0(t_k,\gamma (t_k,t_0,{\gamma }_0)),\hspace{0.33em}{t}_k\ge t_0+T.$$

(32)

This implies that $\gamma (t)\in S_{\psi }\hspace{0.33em}for\hspace{0.33em}t\in [t_0,t_1)$ and for $t\in [t_0,t_1)$ from Theorem 1,

$${\mathrm{\Omega }}_0(t_k,\gamma (t_k))\le {\displaystyle \sum _{i=1}^N{\vartheta }_i(t_k,t_0,u_0)}.$$

(33)

Combining (29), (30), (32) and (33) gives the estimate

$$b(\epsilon )\le {\mathrm{\Omega }}_0(t_k,\gamma (t_k,t_0,{\gamma }_0))\le {\displaystyle \sum _{i=1}^N{\vartheta }_i(t_k,t_0,u_0)}<b(\epsilon ),$$

which leads to a contradiction.

Therefore, the AE-S of the set of trivial solution $\gamma =0$ of (4) is implied by the AE-S of the trivial solution $u=0$ of (6).□

6. Example 1

Consider the system of FDEs

$$\begin{array}{l}{}^{C}D{}^{\beta }\gamma _1(t)=-6{\gamma }_1-\frac{{\gamma }_2^2\cos {\gamma }_1}{2{\gamma }_1}+{\gamma }_1\sin {\gamma }_2+\frac{{\gamma }_2^2\tan {\gamma }_1}{{\gamma }_1},\hspace{0.17em}\hspace{0.17em}t\ne t_k\\ {}^{C}D{}^{\beta }\gamma _2(t)=\frac{3{\gamma }_1^2}{{\gamma }_2}-{\gamma }_2\sin {\gamma }_1-{\gamma }_2\sec {\gamma }_1-{\gamma }_1^2\cos {\gamma }_2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\ne t_k\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta {\gamma }_1={\xi }_k(\gamma (t_k)),\hspace{0.17em}\hspace{0.17em}t=t_k\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta {\gamma }_2={\tau }_k(\gamma (t_k)),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t=t_k\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}k=1,2,\dots \end{array}$$

(34)

with initial conditions

$${\gamma }_1(t_0^{+})={\gamma }_{10}\mathrm{and}{\gamma }_2(t_0^{+})={\gamma }_{20}\hspace{0.17em}.$$

where ${\gamma }_1,{\gamma }_2\in R^N$ are arbitrary functions.

(34) is equivalent to (4) and $f=(f_1{f}_2),$ where

$$f_1(t,{\gamma }_1)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{}^{C}D{}^{\beta }\gamma _1(t)=-6{\gamma }_1-\frac{{\gamma }_2^2\cos {\gamma }_1}{2{\gamma }_1}+{\gamma }_1\sin {\gamma }_2+\frac{{\gamma }_2^2\tan {\gamma }_1}{{\gamma }_1}\hspace{0.17em}$$

and

$$f_2(t,{\gamma }_2)={}^{C}D{}^{\beta }\gamma _2(t)=\frac{3{\gamma }_1^2}{{\gamma }_2}-{\gamma }_2\sin {\gamma }_1-{\gamma }_2\sec {\gamma }_1-{\gamma }_1^2\cos {\gamma }_2.$$

Consider a vector Lyapunov function of the form $\mathrm{\Omega }={({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2)}^T,$ where ${\mathrm{\Omega }}_1(t,{\gamma }_1,{\gamma }_2)={{\gamma }_1}^2,\hspace{0.17em}\hspace{0.17em}{\mathrm{\Omega }}_2(t,{\gamma }_1,{\gamma }_2)={{\gamma }_2}^2.$ So that $\mathrm{\Omega }={({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2)}^T$ with $\gamma =({\gamma }_1,{\gamma }_2)\in R^2,$ so that $‖\gamma ‖=\sqrt{{\gamma }^2+y^2}$

$${\displaystyle \sum _{i=1}^2{\mathrm{\Omega }}_i(t,{\gamma }_1,{\gamma }_2)}={\gamma }_1^2+{\gamma }_2^2.$$

The assumption, $b\left(‖\gamma ‖\right)\le {\displaystyle \sum _{i=1}^n{\mathrm{\Omega }}_i(\gamma ,y)\le a\left(t,‖\gamma ‖\right)}$ reduces to

$\sqrt{{\gamma }^2+y^2}\le {\gamma }^2+y^2\le 2{\left(\sqrt{{\gamma }^2+y^2}\right)}^2.$ With the proviso that $b(\wp )=\wp ,$ and $a(\wp )=2{\wp }^2.$

Furthermore, we deduce that using (5) and ${\mathrm{\Omega }}_1(t,{\gamma }_1,{\gamma }_2)={{\gamma }_1}^2,$

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }(t,{\gamma }_1,{\gamma }_2)=\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\{\mathrm{\Omega }(t,{\gamma }_1,{\gamma }_2)-\mathrm{\Omega }(t_0,{\gamma }_0)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^{r+1}}{}^{\beta }C{}_r\left[\mathrm{\Omega }(t-rh,\gamma -h^{\beta }{f}_i(t,{\gamma }_1,{\gamma }_2))-\mathrm{\Omega }(t_0,{\gamma }_0)\right]\},t\ge t_0\end{array}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)=\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{{{\gamma }_1}^2-{\gamma }_{10}^2)+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)\left[\mathrm{\Omega }(t-rh,\gamma -h^{\beta }{f}_1(t,{\gamma }_1))-{\gamma }_{10}^2\right]\right\},t\ge t_0$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)=\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{{{\gamma }_1}^2-{\gamma }_{10}^2)+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)[{({\gamma }_1-h^{\beta }{f}_1(t,{\gamma }_1))}^2-{\gamma }_{10}^2]\right\},t\ge t_0$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le \underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{{{\gamma }_1}^2-{\gamma }_{10}^2)+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)[{\gamma }_1^2-2{\gamma }_1{h}^{\beta }{f}_1(t,{\gamma }_1)+h^{2\beta }{f}_1^2(t,{\gamma }_1))-{\gamma }_{10}^2]\right\},t\ge t_0$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le \underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{{{\gamma }_1}^2+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r){\gamma }_1^2-{\gamma }_{10}^2-+{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r){\gamma }_{10}^2-2{\gamma }_1{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)h^{\beta }{f}_1(t,{\gamma }_1)\right\},t\ge t_0$$

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le \underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}\left\{{\displaystyle \sum _{r=0}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r){\gamma }_1^2-{\displaystyle \sum _{r=0}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r){\gamma }_{10}^2-2{\gamma }_1{\displaystyle \sum _{r=1}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)h^{\beta }{f}_1(t,{\gamma }_1)\right\},t\ge t_0\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(35)

Recall from (3.7) and (3.8) in Ref. [10] that

$$\underset{h\to 0^{+}}{\lim \sup }\frac{1}{h^{\beta }}{\displaystyle \sum _{r=0}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)={\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underset{h\to 0^{+}}{\lim }{\displaystyle \sum _{r=0}^{[\frac{t-t_0}{h}]}{(-1)}^r}({}^{\beta }C_r)=-1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(36)

and substituting (36) into (35), we have,

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le {\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{{\gamma }_{10}^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}+2{\gamma }_1{f}_1(t,{\gamma }_1)$$

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le {\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\hspace{0.17em}\hspace{0.17em}+2{\gamma }_1{f}_1(t,{\gamma }_1)\hspace{0.17em}\hspace{0.17em}$$

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le {\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\hspace{0.17em}\hspace{0.17em}+2{\gamma }_1(-6{\gamma }_1-\frac{{\gamma }_2^2\cos {\gamma }_1}{2{\gamma }_1}+{\gamma }_1\sin {\gamma }_2+\frac{{\gamma }_2^2\tan {\gamma }_1}{{\gamma }_1})$$

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le {\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\hspace{0.17em}\hspace{0.17em}-12{\gamma }_1^2-{\gamma }_2^2\cos {\gamma }_1+2{\gamma }_1^2\sin {\gamma }_2+{\gamma }_2^2\tan {\gamma }_1.\hspace{0.17em}$$

$As\hspace{0.17em}\hspace{0.17em}t\to \infty ,\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{{\gamma }_1^2}{t^{\beta }\mathrm{\Gamma }(1-\beta )}}\to 0,\hspace{0.17em}\hspace{0.17em}$ so we now have that

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le \hspace{0.17em}\hspace{0.17em}-12{\gamma }_1^2-{\gamma }_2^2\cos {\gamma }_1+2{\gamma }_1^2\sin {\gamma }_2+{\gamma }_2^2\tan {\gamma }_1$$

$$=\hspace{0.17em}\hspace{0.17em}2{\gamma }_1^2(-6+\sin {\gamma }_2)+{\gamma }_2^2(-\cos {\gamma }_1+2\tan {\gamma }_1)$$

$$\hspace{0.17em}\le 2{\gamma }_1^2(-6+|\sin {\gamma }_2|)+{\gamma }_2^2(\frac{2\left|\sin {\gamma }_1\right|}{\left|\cos {\gamma }_1\right|}-|\cos {\gamma }_1|)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

$$\hspace{0.17em}\le 2{\gamma }_1^2(-6+1)+{\gamma }_2^2(2-1)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

$$={\gamma }_1^2(-10)+{\gamma }_2^2(1)\hspace{0.17em}.\hspace{0.17em}$$

Therefore,

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_1(t,{\gamma }_1)\le -10{\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2.$$

(37)

Also, for ${\gamma }_0\in S_{\psi },\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}t=t_k,$ whence $k=1,2,\dots$ we arrive at,

$${\mathrm{\Omega }}_1(t,\gamma (t)+{\xi }_k)=\left|{\xi }_k+\gamma (t)\right|\le {\mathrm{\Omega }}_1(t,\gamma (t)).$$

Similarly, we compute for the Dini derivative for ${\mathrm{\Omega }}_2(t,{\gamma }_2)={\gamma }_2^2$ and follow through the same argument by substituting for

$f_2(t,{\gamma }_2)={}^{C}D^{\beta }{\gamma }_2(t)=\frac{3{\gamma }_1^2}{{\gamma }_2}-{\gamma }_2\sin {\gamma }_1-{\gamma }_2\sec {\gamma }_1-{\gamma }_1^2\cos {\gamma }_2$ in (3.4) to have that,

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }_2(t,{\gamma }_2)\le 4{\mathrm{\Omega }}_1-4{\mathrm{\Omega }}_2.$$

(38)

Also for ${\gamma }_0\in S_{\psi },\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}t=t_k,$ whence $k=1,2,\dots$ we arrive at,

$${\mathrm{\Omega }}_2(t,\gamma (t)+{\tau }_k)=\left|{\tau }_k+\gamma (t)\right|\le {\mathrm{\Omega }}_2(t,\gamma (t)).$$

By combining (37) and (38), we have

$${}^{C}D{}_{+}{}^{\beta }\mathrm{\Omega }\le \left(\begin{array}{l}-10\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\\ 4\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-4\end{array}\right)\left(\begin{array}{l}{\mathrm{\Omega }}_1\\ {\mathrm{\Omega }}_2\end{array}\right)=g(t,\mathrm{\Omega }).$$

(39)

Next, let’s take the following comparison system into account,

$${}^{C}D{}^{\beta }u=g(t,\mathrm{\Omega })=Au,\mathrm{where}A=\left(\begin{array}{l}-10\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\\ 4\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-4\end{array}\right).$$

The vector inequality (39) and all the conditions of Theorem 3 are met because the eigenvalues of A have negative real parts. Therefore, the system (34) is asymptotically eventually stable. Hence, the trivial solution ${\gamma }_0=0$ of the set $\gamma (t)=0$ of ICFDE (34) is asymptotically eventually stable.

Example 2

Consider the ICFDE for a typical Predictor-Corrector model of the form:

$${}^{C}D{}^{\beta }y(t)=-y(t)\hspace{0.17em}+t,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\in [0,2],\hspace{0.17em}\hspace{0.17em}0<\beta <1,$$

(40)

with the initial condition:

$$y(0)=0,$$

(41)

and an impulse at $t=1$ defined as:

$$y(1^{+})=y(1^{-})+2.$$

(42)

Here, ${}^{C}D{}^{\beta }y(t)$ is the Caputo fractional derivative of order $\beta .$

Systems (40)–(42) can be rewritten as:

$$y(t)=y(0)+{\displaystyle \frac{1}{\mathrm{\Gamma }(\beta )}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{(t-s)}^{\beta -1}(-y(s)+s))ds,}$$

(43)

where $\mathrm{\Gamma }(\beta )$ is the Gamma function.

Solving numerically using the predictor-corrector method:

(i)

We will discretize the time interval [0, 2] into N points: $t_1,t_2,\dots ,t_N$ with step size h.

(ii)

At $t=1,$ we would apply the impulse condition $y(1^{+})=y(1^{-})+2$.

(iii)

Between the jump points, we compute the solution using the Caputo fractional derivative and integral approximations.

The solution at each step is computed as:

$$y_n=y_0+{\displaystyle \frac{h^{\beta }}{\mathrm{\Gamma }\left(\beta \right)}}{\sum }_{k=0}^{n-1}(t_n-t_k{)}^{\beta -1}(-y_k+t_k),$$

(44)

as illustrated in

Table 1. Table of results of Example 2.

Time	Solution
0	0
0.1	0.0056
0.2	0.0221
1	1.3452
1.1	3.1573
2	4.7839

and Figure 1.

7. Results

The numerical solution is computed and displayed both as a table and a graph. The Solution $y(t)$ is continuous except at the impulse point $t=1,$ where a jump of size 2 occurs.

8. Conclusions

This paper addressed the AE-S for nonlinear ICFDE with fixed impulse moments using the ALF - generalized by a class of piecewise continuous Lyapunov functions, which are analogues of VLF. We also used the predictor-corrector method to implement a numerical solution for a given ICFDE. The novelty in the use of the auxiliary (vector) Lyapunov functions lies in the fact that the “restrictions” encountered by the SLF is safely handled especially for large scale dynamical systems, since it involves splitting the Lyapunov functions into components so that each of the components can easily describe the behavior of the solution state. By using the comparison results, sufficient conditions for AE-S of ICFDEs are presented. Results obtained are extension and improvements on existing results.