Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method

1.   Introduction

Fractional calculus is a well-known theory regarding fractional differential equations (FDEs) which has received much consideration and attention during the past decades and also has became the most important branch in applied analysis because of its extensive applications in a vast range of applied sciences ^[1,2,3].

Meanwhile, the variety of fractional operators defined by mathematicians has led researchers to focus on the differences and outputs of mathematical models designed by these operators and to use a wide range of fractional derivation operators in their studies. Some of the prominent works in this field are different types of fractional mathematical models in which the effects of the order of fractional derivatives on the dynamic behavior of the solutions of the assumed systems are carefully simulated. Some examples include the following: In ^[4,5], the use of a Caputo derivative; in ^[6,7], the use of a Caputo-conformable derivative; in ^[8,9,10], the use of a generalized derivative; in ^[11,12], the use of a quantum Caputo derivative; in ^[13,14], the use of a nonsingular Caputo-Fabrizio derivative; in ^{[15,16,17,18]}, the use of a nonsingular Mittag-Leffler kernel-type derivative.

One of the fractional derivatives that is defined by the combination of the properties of the Caputo and Hadamard operators is the Caputo-Hadamard fractional derivative. There are limited fractional models and problems designed by this operator. Examples can be seen in ^{[19,20,21,22,23,24]}.

Hence, as we see, the existence and uniqueness problems for FDEs have many forms according to the shape of the differential model and of course the form of the initial or boundary conditions. In the newly published works, the role of fractional calculus in the topics of control theory can be widely observed. In the meantime, the fractional order controller is one of the key concepts in the field of control problems. One of the most important specifications of the control problems is stability analysis which is considered to be a fundamental condition for every control problem. In 1996, Matignon ^[25] was one of the first mathematicians to conduct research on the stability of linear differential systems using a Caputo operator. Since then, many researchers have implemented further investigations into the stability of such linear fractional systems ^[26,27]. In regard to the nonlinear fractional systems, the stability criterion is much more difficult. The direct method attributed to Lyapunov gives a way to study a special type of stability tremed the Mittag-Leffler stability for a given fractional nonlinear system without solving it explicitly ^[28,29]. Such a direct method due to Lyapunov is a sufficient condition to confirm the stability of the nonlinear systems; in other words, the given systems may still be stable even if we cannot choose a Lyapunov's mapping to fulfill the stability property for the mentioned system.

In this paper, the main properties such as the existence , uniqueness and different types of stability are studied for the fractional system involving the nonlinear Caputo-Hadamard FIVP as given by

Where $\;1<\ell<2$,$\; 0<\beta<\ell-1$, $\phi_{0}, \phi_1 \in \mathbb{R}^{n}$ , $A\in\mathbb{R}^{n\times n}$, $\Theta=t\frac{d}{dt}$ and $\psi:[c,\infty)\times \mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a given function. $_{CH}D^{\ell}_{c}$ and $_{CH}D^{\beta}_{c}$ are the Caputo-Hadamard derivatives of orders $\ell$ and $\beta$, respectively. The basic motivation and novelty of this work is that we attempt to use some specifications of the modified Laplace transform to the Caputo-Hadamard FIVP to derive the corresponding Hadamard integral equation in terms of one-parametric and two-parametric Mittag-Leffler functions. Also, there is no work about the generalized Mittag-Leffler stability for a fractional system designed by using a Caputo-Hadamard operator so far. Thus, with the help of Lyapunov functions and the aid of $\mathcal{K}$-class functions, we will prove this type of stability.

The manuscript is structured as follows. Section 2 is devoted to recalling definitions, theorems, lemmas and remarks that will be applied throughout the next sections. In Section 3, we shall give several sufficient conditions confirming the existence of the solution and its uniqueness for the nonlinear Caputo-Hadamard FIVP given by (1.1) using the Banach contraction principle. In Section 4, by using a Lyapunov-like function and a $\mathcal{K}$-class function, the generalized Mittag-Leffler stability for the Caputo-Hadamard system (1.1) is established. We validate our findings in Section 5 and end the paper in Section 6.

2.   Preliminaries

At first, the fundamental notions related to the scope of the present paper are recollected in this section. Let the space

be so that $\Theta=t\frac{d}{dt}$ stands for the Hadamard derivative, and $AC([c,b],\mathbb{R})$ consists of all functions on $[c,b]$ with the absolute continuity property.

Definition 2.1. ^[1,30] The Hadamard integral of a given function $\psi (t): [c,b]\to \mathbb{R}$ of the order $\ell>0$ is defined by

Definition 2.2. ^[1] The Hadamard derivative of a function $\psi (t):[c,b]\to \mathbb{R}$ belonging to $AC^{n}_{\Theta}$ of the order $\ell$ is defined by

where $\Theta=t\frac{d}{dt}$, and $n-1<\ell<n\in \mathbb{Z}^{+}.$

Lemma 2.3. ^[31] Let $\ell>0,\; n=[\ell]+1$. If $\psi(t)\in AC^{n}_{\Theta},$ then the Hadamard fractional derivative $_{H}D^{\ell}_{c^{+}}$ exists almost everywhere on $[c,b]$ and can be represented in the following form:

In particular, when $0<\ell<1$, then, for $\psi(t)\in AC[c,b],$

Definition 2.4 ^[32] The Caputo-Hadamard derivative of the function $\psi (t)$ of the order $\ell (n-1<\ell<n)$ is defined by

Lemma 2.5. ^[32] If $\psi(t)\in AC^{n}_{\Theta}$ is a function such that $_{CH}D^{\ell}\psi(t)$ and $_{H}D^{\ell}\psi(t)$ exist, then

and when $0<\ell<1,$ then

In view of the aforementioned definitions related to the Hadamard operators (integral and derivative operators), we can not obtain the corresponding Laplace transforms due to the initial value starting at the time $t=c >0$. For this reason, it is necessary that we provide a new type of definition for the case with the starting value at the time $t=c>0.$

Definition 2.6. ^[33,34] For a mapping $\psi (t)$ given on $[c,\infty)(c>0)$, the modified Laplace transform of $\psi$ is defined by

Also, the inverse modified Laplace transform of $\tilde \psi (s)$ is defined by

The following properties are fulfilled for these modified transforms.

Proposition 2.7. ^[34] If $\mathbb{L}_{c}\{\psi(t)\}= \tilde{\psi}(s)$, then

where $\Theta=t\frac{d}{dt}$.

Lemma 2.8. ^[34] Let $n-1<\ell<n$. Then

Definition 2.9. ^[34] Let $\psi$ and $h$ be defined on $[c,\infty)$. Then the integral $\int_{c}^{t} \psi(c\frac{t}{w}) h(w)\, \frac{dw}{w}$ is termed the convolution of $\psi$ and $h$, that is,

Proposition 2.10. ^[34] If $\mathbb{L}_{c}\{\psi(t)\} =\tilde{\psi}(s)$ and $\mathbb{L}_{c}\{h(t)\} =\tilde{h}(s)$, then

conversely,

Definition 2.11. ^[35] The one-parametric Mittag–Leffler function is defined as

Clearly, $\mathbb{E}_{\ell}(z)=e^{z}$ for $\ell=1$ The two-parametric Mittag-Leffler function is of the following form

The derivative of the Mittag-Leffler function is given by

and

Subsequently, we present the modified Laplace transform of a Mittag-Leffler function. By utilizing the formula ^[36]

and by using the change of the variable $t=\ln\frac{w}{c}$, we get

Definition 2.12. ^[37] For a normed space $||\mathbb{B}||=(\mathbb{B},||.||)$, the operator $N:\mathbb{B}\to \mathbb{B}$ satisfies the Lipschitz condition, if there is a positive real constant $K$ such that for all $\phi$ and $y$ in $\mathbb{B}$,

Remark 2.13. ^[37] Given Definition 2.12, if $0<K<1$, the operator $N$ is called a contraction mapping on the normed space $||\mathbb{B}||=(\mathbb{B},||.||)$.

Theorem 2.14 (Banach fixed point theorem).^[38] Let $\mathbb{B}$ be a Banach space and $N$ be a contraction mapping with the Lipschitz constant $K$. Then $N$ has an unique fixed point.

3.   Existence and uniqueness of solution

For a given $T>c>0$, let $\mathbb{E}=C([c,T],\mathbb{R}^{n})$ be a Banach space consisting of continuous $n$-vector mappings given on $[c,T]$ furnished with the norm

Notice that the norm of an n-vector $\phi(t)=(\phi_{1}(t),\phi_{2}(t),\dots,\phi_{n}(t)) \in \mathbb{R}^{n}$ is presented as

Based on the problem given by (1.1), we introduce the Banach space $\mathbb{B}=\{\phi; \phi\in \mathbb{E}, {}_{CH}D^{\beta}_{c}\phi\in \mathbb{E}\}$ via the norm

Now, we first derive the equivalent solution to our system.

Lemma 3.1. For $1<\ell<2$, $0<\beta<\ell -1$ and invertible matrix $[Is^{\ell}-A]$, the solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is given as

Proof. Let $\Psi(s)$ and $\Phi(s)$ be the modified Laplace transforms of $\psi(t)$ and $\phi(t)$, respectively. Then, by using the modified Laplace transform and its properties for the nonlinear Caputo-Hadamard FIVP given by (1.1), we have

so

By applying the inverse modified Laplace transform to the above relation, we obtain

and this concludes the proof.

We will use the Banach's contraction principle to prove the existence of a solution of the nonlinear Caputo-Hadamard FIVP given by (1.1).

Theorem 3.2. Let $\psi:[c,\infty)\times \mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuous function that fulfills the following Lipschitz inequality

Then the nonlinear Caputo-Hadamard FIVP given by (1.1) has a solution uniquely on $[c,T]$ if

where $||\psi(t,0,0)||\le M_{0}$ and $\big|\big|\mathbb{E}_{\ell,i}\left(A \left(\ln \frac{t}{c}\right)^{\ell}\right)\big|\big| \le M_{i}, i \in \{ 1,2,\ell\}.$

Proof. Consider the operator $N:\mathbb{B}\to \mathbb{B}$ formulated by

We follow the proof in some steps:

Step 1: $N$ is well–defined: Given $\phi\in \mathbb{B}$ and $t\in [c,T]$, we have

Consequently, we obtain

Applying the first derivative of $N \phi(t)$ and using (2.2) and (2.3), we have

Hence,

where $M'_{\ell}=\frac{M_{\ell}}{c}$.

Now, one can estimate that

Consequently, we obtain

From (3.2) and (3.3), we find that

This implies that $N$ is well defined.

Step 2: $N$ is a contraction on $\mathbb{B}$: For $\phi,y\in \mathbb{B}$ and $t\in [c,T]$, we get

On the other hand, $||N' \phi(t)-N'y(t)|| \le \frac{1}{t}K M_{\ell}\left(\ln \frac{t}{c}\right)^{\ell-1}||\phi-y||_{\mathbb{B}}.$

So,

Then,

The contractive property for $N$, thanks to (3.1), is established. As a consequence, Theorem 2.14 confirms the existence of a unique solution for the nonlinear Caputo-Hadamard FIVP given by (1.1) on $[c,T]$. This completes the proof.

4.   Generalized Mittag-Leffler stability

In this section, we follow our study in relation to the stability of the nonlinear Caputo-Hadamard FIVP given by (1.1) by using terms of a Lyapunov-like function and $\mathcal{K}$-class function. For more information, see ^[39,40,41].

From now on, we suppose that the Lyapunov function $\mathbb{V}:[c,\infty)\times\mathbb{R}^{n}\to \mathbb{R}^{+}$ is continuously differentiable with respect to the time variable $t$, Lipschtiz with respect to the unknown function $\phi$, and also $\mathbb{V}(t,0)=0.$

Definition 4.1. ^[42] The solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is said to be as follows:

● Stable if for all $\phi_{0}$, there exists $\varepsilon>0$ such that $||\phi(t)||\le \varepsilon$ for $t\ge0.$

● Asymptotically stable if $||\phi(t)||\to 0$ as $t\to \infty.$

Definition 4.2. ^[42] The solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is Mittag-Leffler stable if

where $\ell\in(1,2),\, \lambda\ge 0,\, \gamma>0,\,m(0)=0,\,m(\phi)\ge 0$ and $m(\phi)$ is locally Lipschitz on $\phi\in \mathbb{B}\in \mathbb{R}^{n}$ with a constant $m_{0}$.

Definition 4.3. ^[42] The solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is generalized Mittag-Leffler stable if

such that $\ell\in(1,2),\, -\ell<\rho<1-\ell,\, \gamma\ge 0,\, \lambda>0,\,m(0)=0,\,m(\phi)\ge 0$ and $m(\phi)$ is locally Lipschitz on $\phi\in \mathbb{B}\in \mathbb{R}^{n}$ with a constant $m_{0}$.

Remark 4.4. ^[42] Mittag-Leffler stability and generalized Mittag-Leffler stability imply asymptotic stability.

4.1.   Lyapunov method

Theorem 4.5. Let $\phi=0$ be an equilibrium point of the nonlinear Caputo-Hadamard FIVP given by (1.1), and assume that $\mathbb{V}$ satisfies

such that $\phi\in \mathbb{R}^{n}$, $c,b,q>0$. Then, the zero solution is Mittag-Leffler stable if $\mathbb{V}( c,\phi(c)) \ge 0$ and $\Theta \mathbb{V}( c,\phi(c)) = 0$, where $\Theta = \frac{d}{dt}$.

Proof. Using the inequality given by (4.2), a nonnegative function $M(t)$ exists and satisfies

Let $\mathbb{L}_{c} \{\mathbb{V}(t,\phi(t))\}=\mathbb{V}(s).$ Then, the application of the Laplace transform given by (4.3) gives

By applying the inverse modified Laplace transform to (4.4), we obtain

Since both $\left(\ln\frac{t}{c}\right)^{\ell-1}$ and $\mathbb{E}_{\ell,\ell}\left(-q\left(\ln\frac{t}{c}\right)^{\ell}\right)$ are nonnegative functions and $\mathbb{V}_{1}=\Theta \mathbb{V}(c,\phi(c))=0$, we deduce that

In accordance with (4.1), we obtain

for $m= \dfrac{\mathbb{V}_{0}}{c}\ge 0$. In this case, the zero solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is Mittag-Leffler stable.

4.2.   Stability via $\mathcal{K}$-class functions

Definition 4.6. ^[43] If $\varphi \in C([0,\infty), [0,\infty))$ is strictly increasing, and $\varphi(c)=0,\; c>0$, then $\varphi$ is termed a $\mathcal{K}$-class function, as illustrated by $\varphi\in \mathcal{K}$.

Theorem 4.7. Let $\phi=0$ be an equilibrium point of the nonlinear Caputo-Hadamard FIVP given by (1.1). Suppose that there exists a $\mathcal{K}$-class function $\varphi$ that satisfies

for $M \geq 0$. Then, the zero solution is stable.

Proof. By applying (4.6), there exists some $M \geq 0$ so that

By using the Laplace transform and its inverse, we obtain

where $\mathbb{V}_{0}=\mathbb{V} (c,\phi(c) )$, and $\mathbb{V}_{1}=\Theta \mathbb{V} (c,\phi(c) ) .$

Substituting (4.8) into (4.5) yields

Therefore

Then, by Eq (4.7), we get $||\phi(t)||\le M,\;t>c,$ which confirms that the zero solution of the nonlinear Caputo-Hadamard FIVP given by (1.1) is stable.

5.   Example

Here, we validate our results by providing the following example.

Example 5.1. According to (1.1), consider the nonlinear Caputo-Hadamard FIVP

Here, we have $A=0$ and $\psi\left(t,\phi(t), {}_{CH}D^{1/2}_{1,t}\phi(t)\right)=\dfrac{1}{10}\left(-|\phi(t)|- {}_{CH}D^{1/2}_{1,t}|\phi(t)|\right),$ where $\psi:[1,e]\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}.$ In order to show that (5.1) has a unique solution, we simply check that

which is satisfying the Lipschitz condition with $K= \frac{1}{10}$. Since $|\mathbb{E}_{\ell,\ell}\left(A(\ln \frac{t}{c})^{\ell}\right)|\le M_{\ell} ,$ for $A=0$, we have $\mathbb{E}_{\frac{3}{2},\frac{3}{2}}(0) =\frac{2}{\sqrt{\pi}}$ and

From Theorem 3.2, the nonlinear Caputo-Hadamard FIVP given by (5.1) has a unique solution.

On the other hand, consider the Lyapunov function $\mathbb{V}(t,\phi(t))=|\phi(t)|.$ In this case,

Hence, the hypotheses of Theorem 4.5 hold with $c=0,b=1$ and $q=\dfrac{1}{10}$. Accordingly, the zero solution of the given nonlinear Caputo-Hadamard FIVP given by (5.1) is Mittag-Leffler stable.

6.   Conclusions

In this paper, we provided several hypotheses that demonstrate the existence of a solution and its uniqueness for the nonlinear Caputo-Hadamard FIVP given by (1.1) by using the Banach contraction principle. To do this, the modified Laplace transform played a main role in finding the corresponding integral equation by using Mittag-Leffler functions with one and two parameters to derive the Hadamard integrals. Subsequently, we used a Lyapunov-like function and $\mathcal{K}$-class function to prove the generalized Mittag-Leffler stability for the Caputo-Hadamard system given by (1.1). Further, we examined the theoretical results by designing an illustrative example. In subsequent works, the notion of generalized Mittag-Leffler stability can be discussed for different nonlinear systems furnished with non-singular derivation operators. Also, one can focus on the generalized Mittag-Leffler stability problem of $q$-FDEs in a variety of different forms.

Acknowledgments

The third and fifth authors would like to thank Azarbaijan Shahid Madani University. Also, the authors would like to thank the dear reviewers for their valuable and constructive comments to improve the quality of the paper.

Conflict of interest

The authors declare no conflicts of interest.