Existence and Stability Analysis of Nonlinear Systems with Hadamard Fractional Derivatives

Abstract

This paper investigates the existence, uniqueness, and finite-time stability of solutions to a class of nonlinear systems governed by the Hadamard fractional derivative. The analysis is carried out using two fundamental tools from fixed point theory: the Krasnoselskii fixed point theorem and the Banach contraction principle. These methods provide rigorous conditions under which solutions exist and are unique. Furthermore, criteria ensuring the finite-time stability of the system are derived. To demonstrate the practicality of the theoretical results, a detailed example is presented. This paper also discusses certain assumptions and presents corollaries that naturally emerge from the main theorems.

1. Introduction

Fractional calculus has found increasing applications across many scientific and engineering domains due to its ability to model memory and hereditary properties in complex systems. For instance, in fluid dynamics, fractional models provide improved accuracy in capturing non-local behaviors [1]. In mathematical biology, fractional differential equations have been employed to model population dynamics and disease transmission with memory effects [2,3]. Furthermore, in the field of image processing, fractional-order operators have shown remarkable results in tasks such as image denoising [4,5] and image super-resolution [6,7]. These applications highlight the growing importance of fractional calculus as a modeling tool in both theoretical and applied research.

The study of the existence and stability of solutions to differential equations is one of the most crucial topics in understanding natural phenomena such as those in physics, chemistry, engineering, and other sciences. In particular, fractional differential equations have garnered significant attention from researchers due to their ability to model complex systems more accurately than classical differential equations (see [8,9,10,11,12,13,14,15,16]).

Among the various fractional derivative operators, the Hadamard derivative stands out due to its logarithmic kernel, which makes it particularly suitable for modeling phenomena with multiplicative or scale-invariant behavior. Unlike the Caputo or Riemann–Liouville derivatives, which are more commonly used in fractional calculus, the Hadamard derivative captures dynamics that evolve on a logarithmic timescale. This feature is advantageous in many applied contexts, such as systems with memory effects that depend on the ratio of times rather than their difference. Motivated by these properties, we focus this study on nonlinear systems involving the Hadamard fractional derivative. Several recent works have explored equations incorporating this operator and highlighted its distinctive analytical features (see [17,18,19,20,21,22,23,24,25,26]).

The concept of finite-time stability was introduced in the 1950s, distinguishing itself from classical stability in two key aspects. First, it applies to systems that operate within a predefined, finite time interval. Second, it requires the system variables to remain within predefined limits. The finite-time settling properties of systems are discussed in references [27,28,29,30,31].

In [32], the authors obtained existence and finite-time stability results for the following fractional q-difference equations

$$\left\{\begin{array}{c}{}^c\mathfrak{D}_q^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=g_2\left(ϵ,\varpi \left(ϵ\right)\right),0\le ϵ\le 1,\hfill \\ \varpi \left(0\right)={\varpi }_0,{\varpi }_0\in \mathbb{R},0<\varsigma \le 1.\hfill \end{array}\right.$$

(1)

Agarwal et al. [33] extended the previous problem (1) to the subsequent form

$$\left\{\begin{array}{c}{}^c\mathfrak{D}_q^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=a_1{g}_2\left(ϵ,\varpi \left(ϵ\right)\right)+a_2{\mathfrak{I}}_q^{\gamma }{g}_3\left(ϵ,\varpi \left(ϵ\right)\right),0\le ϵ\le 1,\hfill \\ \varpi \left(0\right)={\displaystyle \frac{a_3}{{\Gamma }_q\left(\tau \right)}}{\int }_0^{\eta }{\left(\eta -qs\right)}_q^{\left(\tau -1\right)}\varpi \left(\nu \right){\mathrm{d}}_q\nu ,\tau >0,0<\eta <1,\hfill \end{array}\right.$$

(2)

where $0<q\le 1$, ${}^c\mathfrak{D}_q^{\varsigma }$ denote the q-fractional derivative of Caputo type of order $\varsigma \in \left(0,1\right]$ and ${\mathfrak{I}}_q^{\gamma }$ denote q-integral of Riemann–Liouville such that $\gamma \in \left(0,1\right)$, $a_1$, $a_2$, and $a_3$ are real constants, and $g_1,g_2,g_3:\left[0,1\right]\times \mathbb{R}\to \mathbb{R}$ are given continuous functions. The authors investigate the existence, uniqueness and finite time stability.

So, according to the above works, we propose in this paper another direction of the following form

$$\left\{\begin{array}{c}{}^H\mathfrak{D}^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=A_1{g}_2\left(ϵ,\varpi \left(ϵ\right)\right)+A_2\left({}^H\mathfrak{I}^{\gamma }\right)g_3\left(ϵ,\varpi \left(ϵ\right)\right),1\le ϵ\le e,\hfill \\ \varpi \left(1\right)={\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }},\tau >0,1<\eta <e,\hfill \end{array}\right.$$

(3)

where $\varsigma \in \left(0,1\right]$, ${}^H\mathfrak{D}^{\varsigma }$ denote the Hadamard fractional derivative of order $\varsigma$; ${\mathfrak{I}}^{\gamma }$ denote the Hadamard integral of order $\gamma \in \left(0,1\right)$; $A_1$, $A_2$, and $A_3$ are constant square matrices of order n; and $g_1,g_2,g_3$ are given continuous functions from $\left[1,e\right]\times {\mathbb{R}}^n$ to ${\mathbb{R}}^n$ such that $g_1\left(1,0\right)=0$ for simplicity.

The novelty of this research lies in its exploration of a nonlinear system involving Hadamard fractional derivatives, an area that remains underdeveloped despite the growing interest in fractional calculus. While earlier studies have analyzed linear and simple nonlinear systems, the current work addresses more complex structures, extending the theoretical results to accommodate broader classes of nonlinearities. By addressing the existence, uniqueness, and finite-time stability of solutions, this study not only extends previous results but also provides a solid theoretical foundation for future explorations of nonlinear systems or fractional partial differential equations governed by Hadamard fractional derivatives.

The structure of the paper is organized as follows: In the upcoming section, we will outline the key components of the Hadamard fractional derivative calculus. Section 3 focuses on establishing specific criteria to determine the existence and uniqueness of solutions for the system described in Equation (3). Section 4 addresses finite-time stability. In Section 5, we provide a numerical example to support our theoretical analysis. Finally, the concluding section presents a comprehensive discussion of the implications of our findings.

2. Essential Materials

This section introduces the necessary materials for our research. It commences with basic definitions and findings related to the Hadamard fractional derivative, which can be found in [8,9,12].

Lemma 1

([9]). The Hadamard fractional integral of order $\varsigma >0$ for a continuous function $\varpi :\left[1,+\infty \right)\to \mathbb{R}$ is defined as

$${}^H\mathfrak{I}^{\varsigma }\varpi \left(ϵ\right)={\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }},$$

where $\Gamma \left(·\right)$ represents the gamma function which be read as

$$\Gamma \left(\varsigma \right)={\int }_0^{\infty }{ϵ}^{\varsigma -1}{e}^{-ϵ}dϵ.$$

Lemma 2

([9]). The Hadamard fractional derivative of order $\varsigma >0$ for a function $\varpi :\left[1,+\infty \right)\to \mathbb{R}$ is defined as

$${}^H\mathfrak{D}^{\varsigma }={\displaystyle \frac{1}{\Gamma \left(n-\varsigma \right)}}{\left(ϵ{\displaystyle \frac{d}{dϵ}}\right)}^n{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{n-\varsigma -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }},n-1<\varsigma <n.$$

Lemma 3

([9]). Let $n-1<\varsigma <n$, $n\in \mathbb{N}$. The equality ${}^H\mathfrak{I}^{\varsigma }\left({}^H\mathfrak{D}^{\varsigma }\right)\varpi \left(ϵ\right)=0$ is valid if and only if

$$\varpi \left(ϵ\right)=\sum _{k=1}^n{c}_k{\left(logϵ\right)}^{\varsigma -k},\mathit{for}\mathit{each}ϵ\in \left[1,e\right],$$

(4)

where $c_k\in \mathbb{R}$, $k=1,\dots ,n$ are arbitrary constants.

Lemma 4.

Let $\varpi \in C\left(\left[1,e\right],\mathbb{R}\right)$. Then, the unique solution of the problem (3) is given by

$$\begin{array}{cc}\hfill \varpi \left(ϵ\right)& =g_1\left(ϵ,\varpi \left(ϵ\right)\right)+{\displaystyle \frac{A_1}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{g}_2\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & +{\displaystyle \frac{A_2}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{g}_3\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & +{\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right).\hfill \end{array}$$

(5)

Proof. 

We apply the operator integral ${}^H\mathfrak{I}^{\varsigma }$ on both sides of (3) and by utilizing Lemma 3 with $n=1$, one has

$$\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)=c_0{\left(logϵ\right)}^{\varsigma -1}+A_1\left({}^H{\mathfrak{I}}^{\varsigma }\right)g_2\left(ϵ,\varpi \left(ϵ\right)\right)+A_2\left({}^H\mathfrak{I}^{\varsigma }\right)\left({}^H\mathfrak{I}^{\gamma }\right)g_3\left(ϵ,\varpi \left(ϵ\right)\right),$$

where $c_0=\varpi \left(1\right)-g_1\left(1,\varpi \left(1\right)\right)$ is a constant deduced from the expansion (4) and that $\varsigma \in \left(0,1\right]$, next we use integral properties we obtain

$$\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)=c_0{\left(logϵ\right)}^{\varsigma -1}+A_1\left({}^H{\mathfrak{I}}^{\varsigma }\right)g_2\left(ϵ,\varpi \left(ϵ\right)\right)+A_2\left({}^H\mathfrak{I}^{\varsigma +\gamma }\right)g_3\left(ϵ,\varpi \left(ϵ\right)\right),$$

and by the given nonlocal condition we have

$$c_0={\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right).$$

Hence, we obtain the expression (5).

When performing the reverse operation, we apply the Hadamard fractional derivative of order $\varsigma \in \left(0,1\right]$ on the both sides of (5) and we obtain

$$\begin{array}{cc}\hfill {}^H\mathfrak{D}^{\varsigma }\varpi \left(ϵ\right)& {=}^H{\mathfrak{D}}^{\varsigma }{g}_1\left(ϵ,\varpi \left(ϵ\right)\right)+A_1^H{\mathfrak{D}}^{\varsigma }\left({}^H\mathfrak{I}^{\varsigma }\right)g_2\left(ϵ,\varpi \left(ϵ\right)\right)+A_2^H{\mathfrak{D}}^{\varsigma }\left({}^H\mathfrak{I}^{\varsigma +\gamma }\right)g_3\left(ϵ,\varpi \left(ϵ\right)\right)\hfill \\ \hfill & {+}^H{\mathfrak{D}}^{\varsigma }\left({\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right)\right),\hfill \end{array}$$

then, we obtain

$${}^H\mathfrak{D}^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=A_1{g}_2\left(ϵ,\varpi \left(ϵ\right)\right)+A_2\left({}^H\mathfrak{D}^{\gamma }{g}_3\right)\left(ϵ,\varpi \left(ϵ\right)\right),$$

and clearly

$$\varpi \left(1\right)={\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right).$$

□

Definition 1.

Let $[1,e]$, and consider the Hadamard fractional differential system given by Equation (3). A function

$$x:[1,e]\to {\mathbb{R}}^n$$

is said to be a solution of the system (3) if

• $x\in C([1,e],{\mathbb{R}}^n)$, i.e., x is continuous on $[1,e]$,
• $x\left(t\right)$ satisfies the system (3) with initial condition and its equivalent integral form (5), for all $t\in [1,e]$.

3. Existence and Uniqueness Results

Let $\varpi =\left({\varpi }_1,{\varpi }_2,\dots ,{\varpi }_n\right)\in {\mathbb{R}}^n$, we denote the Euclidean vector norm $∥\varpi ∥={\sum }_{i=1}^n\left|{\varpi }_i\right|$ and for $A={\left(a_{i,j}\right)}_{\stackrel{1⩽i⩽n}{1⩽j⩽n}}$, the matrix norm $∥A∥=\underset{\stackrel{1⩽i⩽n}{1⩽j⩽n}}{max}\left|a_{ij}\right|$. Let the Banach space $\mathfrak{C}$ of all vector–value continuous functions from $\left[1,e\right]$ into ${\mathbb{R}}^n$ equipped by the norm

$${∥\varpi ∥}_{\mathfrak{C}}=\underset{ϵ\in \left[1,e\right]}{max}∥\varpi \left(ϵ\right)∥.$$

To apply fixed point theory, we first rewrite the given system (3) in an equivalent integral form by using of Lemma 4 as the form $\varpi =\Psi \varpi$, with the operator $\Psi :\mathfrak{C}\to \mathfrak{C}$ being defined by

$$\begin{array}{cc}\hfill \left(\Psi \varpi \right)\left(ϵ\right)& =g_1\left(ϵ,\varpi \left(ϵ\right)\right)+{\displaystyle \frac{A_1}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{g}_2\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & +{\displaystyle \frac{A_2}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{g}_3\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & +{\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right).\hfill \end{array}$$

(6)

The problem’s solutions (3) are entirely determined by the existence of fixed points for $\Psi$.

The first outcome of our investigation, utilizing the Banach contraction mapping principle, focuses on the existence of a unique solution to the system under consideration.

Theorem 1.

Assume that for all $ϵ\in \left[1,e\right]$ and $\varpi ,\varkappa \in {\mathbb{R}}^n$, there exist constants $M_j,j=1,2,3$ such that ${∥g_j\left(ϵ,\varpi \right)-g_j\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_j{∥\varpi -\varkappa ∥}_{\mathfrak{C}}$. If

$$\Theta :=M_1+{\displaystyle \frac{∥A_1∥M_2}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥M_3}{\Gamma \left(\varsigma +\gamma +1\right)}}+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}<1,$$

(7)

then problem (3) has a unique continuous solution on the interval $\left[1,e\right]$.

Proof. 

To validate the hypotheses of the Banach contraction mapping principle, a two-step verification procedure will be implemented. Therefore, set

$$L_j=\underset{ϵ\in \left[1,e\right]}{max}∥g_j\left(ϵ,0\right)∥<\infty ,j=1,2,3,$$

and consider the closed ball $\mathcal{B}=\left\{\varpi \in \mathfrak{C}:{∥\varpi ∥}_{\mathfrak{C}}\le r\right\}$ where

$$L_1+{\displaystyle \frac{∥A_1∥L_2}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥L_3}{\Gamma \left(\varsigma +\gamma +1\right)}}\le \left(1-\Theta \right)r.$$

First, we show $\Psi \mathcal{B}\subset \mathcal{B}$. Since the function $g_j$ satisfies the Lipschitz conditions,

$${∥g_j\left(ϵ,\varpi \right)∥}_{\mathfrak{C}}\le {∥g_j\left(ϵ,\varpi \right)-g_j\left(ϵ,0\right)∥}_{\mathfrak{C}}+{∥g_j\left(ϵ,0\right)∥}_{\mathfrak{C}}\le M_{j}r+L_j,j=1,2,3.$$

Then, for $ϵ\in \left[1,e\right]$, $\varpi \in \mathcal{B}$, we have

$$\begin{array}{cc}\hfill ∥\left(\Psi \varpi \right)\left(ϵ\right)∥& \le ∥g_1\left(ϵ,\varpi \left(ϵ\right)\right)∥+{\displaystyle \frac{∥A_1∥}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}∥g_2\left(\nu ,\varpi \left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}∥g_3\left(\nu ,\varpi \left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_3∥}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}∥\varpi \left(\nu \right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}+∥g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right)∥\hfill \\ \hfill & \le M_{1}r+L_1+{\displaystyle \frac{∥A_1∥\left(M_{2}r+L_2\right)}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥\left(M_{3}r+L_3\right)}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & {\displaystyle \frac{∥A_3∥r}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}+∥g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right)∥\hfill \\ \hfill & \le \left(M_1+{\displaystyle \frac{∥A_1∥M_2}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥M_3}{\Gamma \left(\varsigma +\gamma +1\right)}}+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}\right)r\hfill \\ \hfill & + L_1+{\displaystyle \frac{∥A_1∥L_2}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥L_3}{\Gamma \left(\varsigma +\gamma +1\right)}}\hfill \\ \hfill & \le \Theta r+\left(1-\Theta \right)r,\hfill \end{array}$$

hence ${∥\Psi \varpi ∥}_{\mathfrak{C}}\le r$. Then $\Psi \mathcal{B}\subset \mathcal{B}$.

Secondly, for all $ϵ\in \left[1,e\right]$ and any $\varpi ,\varkappa \in {\mathbb{R}}^n$, we obtain

$$\begin{array}{cc}\hfill & ∥\left(\Psi \varpi \right)\left(ϵ\right)-\left(\Psi \varkappa \right)\left(ϵ\right)∥\hfill \\ \hfill & \le ∥g_1\left(ϵ,\varpi \left(ϵ\right)\right)-g_1\left(ϵ,\varkappa \left(ϵ\right)\right)∥\hfill \\ \hfill & + {\displaystyle \frac{∥A_1∥}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}∥g_2\left(\nu ,\varpi \left(\nu \right)\right)-g_2\left(\nu ,\varkappa \left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}∥g_3\left(\nu ,\varpi \left(\nu \right)\right)-g_3\left(\nu ,\varkappa \left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & \le M_1{∥\varpi -\varkappa ∥}_{\mathfrak{C}}+{\displaystyle \frac{∥A_1∥M_2{∥\varpi -\varkappa ∥}_{\mathfrak{C}}}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥M_3{∥\varpi -\varkappa ∥}_{\mathfrak{C}}}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }},\hfill \end{array}$$

hence,

$${∥\Psi \varpi -\Psi \varkappa ∥}_{\mathfrak{C}}\le \Theta {∥\varpi -\varkappa ∥}_{\mathfrak{C}}.$$

As $\Theta$ is less than 1, it implies that $\Psi$ is a contraction. Therefore, according to the Banach contraction mapping principle, we can deduce that the operator $\Psi$ possesses a unique fixed point. This fixed point serves as the only continuous solution to the system (3). □

Remark 1.

The constant Θ represents the Lipschitz-type bound of the integral operator defined in our setting. It is computed from the norm estimates based on the assumed Lipschitz condition on the nonlinear functions, the properties of the Hadamard fractional integral, and the bounds of the integral domain. Under the given hypotheses, we show that Θ is less than 1, which guarantees that the operator is a strict contraction and thus allows the application of Banach’s fixed point theorem

The result of existence that follows is deduced from Krasnoselskii’s fixed point theorem.

Lemma 5

(Krasnoselskii [34]). Let Π be a nonempty, convex, closed, and bounded subset of a Banach space S. Assume that ${\Psi }_1$ and ${\Psi }_2$ map Π into S such that $\left(i\right)$ ${\Phi }_1$ is a contraction mapping on Π; $\left(ii\right)$ ${\Psi }_2$ is completely continuous on Π; $\left(iii\right)$ $\nu ,w\in \mathrm{\Pi }$, implies ${\Psi }_1\nu +{\Psi }_{2}w\in \mathrm{\Pi }$. Then there exists $\nu \in \mathrm{\Pi }$ with $\nu ={\Psi }_1\nu +{\Psi }_2\nu$.

Consider the set $\mathrm{\Pi }=\left\{\varpi \in \mathfrak{C}:{∥\varpi ∥}_{\mathfrak{C}}\le R\right\}$ for an arbitrary constant R. According to Lemma 4 we define operators ${\Psi }_1$ and ${\Psi }_2$ on $\mathrm{\Pi }$ to ${\mathbb{R}}^n$ as

$$\left({\Psi }_1\varpi \right)\left(ϵ\right)=g_1\left(ϵ,\varpi \left(ϵ\right)\right)+{\displaystyle \frac{A_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-g_1\left(1,A_3\left({}^H\mathfrak{I}^{\tau }\right)\varpi \left(\eta \right)\right),$$

(8)

and

$$\begin{array}{cc}\hfill \left({\Psi }_2\varpi \right)\left(ϵ\right)& ={\displaystyle \frac{A_1}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{g}_2\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{A_2}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{g}_3\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}.\hfill \end{array}$$

(9)

Theorem 2.

Assume that for all $\varpi ,\varkappa \in {\mathbb{R}}^n$ and $ϵ\in \left[1,e\right]$, there exist constants $M_1$ satisfies ${∥g_1\left(ϵ,\varpi \right)-g_1\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_1{∥\varpi -\varkappa ∥}_{\mathfrak{C}}$ such that

$$M_1+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}<1.$$

(10)

In addition, if the functions $g_2\left(ϵ,\varpi \right)$ and $g_3\left(ϵ,\varpi \right)$ are bounded for all $\left(ϵ,\varpi \right)\in \left[1,e\right]\times \mathrm{\Pi }$, then problem (3) has at least one solution on the interval $\left[1,e\right]$.

Proof. 

Due to the boundedness of the functions $g_2\left(ϵ,\varpi \right)$ and $g_3\left(ϵ,\varpi \right)$. Then, for any $\varpi \in \mathrm{\Pi }$ and $ϵ\in \left[1,e\right]$ we denote by

$$\widetilde{M_2}=\underset{\left(ϵ,\varpi \right)\in \left[1,e\right]\times \mathrm{\Pi }}{max}∥g_2\left(ϵ,\varpi \right)∥,\widetilde{M_3}=\underset{\left(ϵ,\varpi \right)\in \left[1,e\right]\times \mathrm{\Pi }}{max}∥g_3\left(ϵ,\varpi \right)∥,$$

and take R such that

$$R\ge \left(L_1+{\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥\widetilde{M_2}}{\Gamma \left(\varsigma +\gamma +1\right)}}\right){\left(1-M_1-{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}\right)}^{-1}.$$

Step 1. For all $ϵ\in \left[1,e\right]$ and any $\varpi ,\varkappa \in \mathrm{\Pi }$ we obtain

$$\begin{array}{cc}\hfill {∥{\Psi }_1\varpi +{\Psi }_2\varkappa ∥}_{\mathfrak{C}}& \le M_{1}R+L_1+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}R\hfill \\ \hfill & + {\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥\widetilde{M_3}}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & \le M_{1}R+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}R\hfill \\ \hfill & + L_1+{\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{∥A_2∥\widetilde{M_2}}{\Gamma \left(\varsigma +\gamma +1\right)}}\hfill \\ \hfill & \le R\hfill \end{array}$$

Thus, ${\Psi }_1\varpi +{\Psi }_2\varkappa \in \mathrm{\Pi }$. The continuity of $g_2$ and $g_3$ implies that the operator ${\Psi }_2$ is continuous. Furthermore, since

$${∥{\Psi }_2\varkappa ∥}_{\mathfrak{C}}\le {\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma \right)}}+{\displaystyle \frac{∥A_2∥\widetilde{M_2}}{\Gamma \left(\varsigma +\gamma \right)}},$$

then, ${\Psi }_2$ is uniformly bounded on $\mathrm{\Pi }$.

Step 2. For any all ${ϵ}_1,{ϵ}_2\in \left[1,e\right]$ and any $\varkappa \in \mathrm{\Pi }$ we obtain

$$\begin{array}{cc}\hfill & ∥\left({\Psi }_2\varkappa \right)\left({ϵ}_2\right)-\left({\Psi }_2\varkappa \right)\left({ϵ}_1\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}\left|{\int }_1^{{ϵ}_2}{\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma -1}{g}_2\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-{\int }_1^{{ϵ}_1}{\left(log{\displaystyle \frac{{ϵ}_1}{\nu }}\right)}^{\varsigma -1}{g}_2\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\right|\hfill \\ \hfill & + {\displaystyle \frac{1}{\Gamma \left(\varsigma +\gamma \right)}}\left|{\int }_1^{{ϵ}_2}{\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma +\gamma -1}{g}_3\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}-{\int }_1^{{ϵ}_1}{\left(log{\displaystyle \frac{{ϵ}_1}{\nu }}\right)}^{\varsigma +\gamma -1}{g}_3\left(\nu ,\varpi \left(\nu \right)\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}\right|\hfill \\ \hfill & \le {\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma \right)}}\left({\int }_1^{{ϵ}_1}\left({\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma -1}-{\left(log{\displaystyle \frac{{ϵ}_1}{\nu }}\right)}^{\varsigma -1}\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}+{\int }_{{ϵ}_1}^{{ϵ}_2}{\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\right)\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥\widetilde{M_3}}{\Gamma \left(\varsigma +\gamma \right)}}\left({\int }_1^{{ϵ}_1}\left({\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma +\gamma -1}-{\left(log{\displaystyle \frac{{ϵ}_1}{\nu }}\right)}^{\varsigma +\gamma -1}\right){\displaystyle \frac{\mathrm{d}\nu }{\nu }}+{\int }_{{ϵ}_1}^{{ϵ}_2}{\left(log{\displaystyle \frac{{ϵ}_2}{\nu }}\right)}^{\varsigma +\gamma -1}{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\right)\hfill \\ \hfill & \le {\displaystyle \frac{∥A_1∥\widetilde{M_2}}{\Gamma \left(\varsigma \right)}}\left({\left(log{ϵ}_2\right)}^{\varsigma -1}-{\left(log{ϵ}_1\right)}^{\varsigma -1}\right)\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥\widetilde{M_3}}{\Gamma \left(\varsigma +\gamma \right)}}\left({\left(log{ϵ}_2\right)}^{\varsigma +\gamma -1}-{\left(log{ϵ}_1\right)}^{\varsigma +\gamma -1}\right),\hfill \end{array}$$

which infer that the last expression is independent of $\varkappa$ and since ${ϵ}_1\to {ϵ}_2$ tends to zero then it also tends to zero. Hence, ${\Psi }_2$ is relatively compact on $\mathrm{\Pi }$. Hence, due to the Arzelà–Ascoli Theorem, ${\Psi }_2$ is compact on $\mathrm{\Pi }$.

Step 3. For all $ϵ\in \left[1,e\right]$ and any $\varpi ,\varkappa \in \mathrm{\Pi }$ we obtain

$${∥{\Psi }_1\varpi -{\Psi }_1\varkappa ∥}_{\mathfrak{C}}\le \left(M_1+{\displaystyle \frac{∥A_3∥\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}\right){∥\varpi -\varkappa ∥}_{\mathfrak{C}}.$$

Based on the given condition (10), it can be inferred that ${\Psi }_1$ is a contraction operator. By satisfying the hypothesis stated in Lemma 5, we can ensure the existence of at least one continuous solution for the problem (3) within the interval $\left[1,e\right]$. □

Remark 2.

The use of Banach’s contraction principle and Krasnoselskii’s fixed point theorem is motivated by the structure of the problem and the nature of the associated integral operator. These tools provide sharper conditions for existence and uniqueness, whereas other methods such as Schauder’s or Leray–Schauder’s fixed point theorems, although applicable in more general contexts, are not necessary here and may require additional assumptions not aligned with our setting.

4. Finite-Time Stability

Herein, we will explicate the stability criteria relevant to the solutions of problem (3).

Definition 2.

The system (3) is finite-time stable with respect to the nonlocal values if for ψ and ϖ two solutions of the problem (3) there exists $\gamma >0$, such that

$$∥\psi \left(ϵ\right)-\varpi \left(ϵ\right)∥\le \gamma ∥\psi \left(\eta \right)-\varpi \left(\eta \right)∥,\forall ϵ\in \left[1,e\right],$$

where $1<\eta <e$ related with the nonlocal condition.

Theorem 3.

Assume that for all $\varpi ,\varkappa \in {\mathbb{R}}^n$ and $ϵ\in \left[1,e\right]$, there exist positive constants $M_j,j=1,2,3$ such that ${∥g_j\left(ϵ,\varpi \right)-g_j\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_j{∥\varpi -\varkappa ∥}_{\mathfrak{C}}$ and the condition (7) holds. Then, the finite-time stability of the solution of the problem (3) is achieved with respect to the nonlocal values.

Proof. 

Consider ${\varpi }_1$ and ${\varpi }_2$ two solutions of the problem (3) and satisfy the fixed point equation $\Psi \varpi =\varpi$, where $\Psi$ is defined by (6). By applying the conditions outlined in Theorem 3, we achieve

$$\begin{array}{cc}\hfill ∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥& \le ∥g_1\left(ϵ,{\varpi }_1\left(ϵ\right)\right)-g_1\left(ϵ,{\varpi }_2\left(ϵ\right)\right)∥\hfill \\ \hfill & + {\displaystyle \frac{∥A_1∥}{\Gamma \left(\varsigma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma -1}∥g_2\left(\nu ,{\varpi }_1\left(\nu \right)\right)-g_2\left(\nu ,{\varpi }_2\left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥}{\Gamma \left(\varsigma +\gamma \right)}}{\int }_1^{ϵ}{\left(log{\displaystyle \frac{ϵ}{\nu }}\right)}^{\varsigma +\gamma -1}∥g_3\left(\nu ,{\varpi }_2\left(\nu \right)\right)-g_3\left(\nu ,{\varpi }_2\left(\nu \right)\right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ \hfill & + ∥{\varpi }_1\left(1\right)-{\varpi }_2\left(1\right)∥+∥g_1\left(1,{\varpi }_1\left(1\right)\right)-g_1\left(1,{\varpi }_2\left(1\right)\right)∥\hfill \\ \hfill & \le M_1∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥+{\displaystyle \frac{∥A_1∥M_2}{\Gamma \left(\varsigma +1\right)}}∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥\hfill \\ \hfill & + {\displaystyle \frac{∥A_2∥M_3}{\Gamma \left(\varsigma +\gamma +1\right)}}∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥+\left(1+M_1\right)∥{\varpi }_1\left(1\right)-{\varpi }_2\left(1\right)∥\hfill \\ \hfill & =\left(M_1+K_1+K_2\right)∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥+\left(1+M_1\right)∥{\varpi }_1\left(1\right)-{\varpi }_2\left(1\right)∥,\hfill \end{array}$$

(11)

where

$$K_1={\displaystyle \frac{∥A_1∥M_2}{\Gamma \left(\varsigma +1\right)}}\mathrm{and}{K}_2={\displaystyle \frac{∥A_2∥M_3}{\Gamma \left(\varsigma +\gamma +1\right)}}.$$

Moreover, taking into account the nonlocal condition of our problem, we acquire

$$\begin{array}{ccc}\hfill ∥{\varpi }_1\left(1\right)-{\varpi }_2\left(1\right)∥& \le & {\displaystyle \frac{∥A_3∥}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}∥{\varpi }_1\left(\nu \right)-{\varpi }_2\left(\nu \right)∥{\displaystyle \frac{\mathrm{d}\nu }{\nu }}\hfill \\ & \le & {\displaystyle \frac{∥A_3∥}{\Gamma \left(\tau +1\right)}}∥{\varpi }_1\left(\eta \right)-{\varpi }_2\left(\eta \right)∥,\hfill \end{array}$$

then

$$∥{\varpi }_1\left(1\right)-{\varpi }_2\left(1\right)∥\le K_3∥{\varpi }_1\left(\eta \right)-{\varpi }_2\left(\eta \right)∥,$$

(12)

where $K_3={\displaystyle \frac{∥A_3∥}{\Gamma \left(\tau +1\right)}}$. Substitute (12) in (11) we obtain

$$∥{\varpi }_1\left(ϵ\right)-{\varpi }_2\left(ϵ\right)∥\le {\displaystyle \frac{K_3\left(1+M_1\right)}{1-\left(M_1+K_1+K_2\right)}}∥{\varpi }_1\left(\eta \right)-{\varpi }_2\left(\eta \right)∥,$$

which prove the finite-time stability of solution for the system (3) with respect to the nonlocal values. □

5. Illustrative Example

In this section, let consider the system (3) with the following quantities $\varpi ={\left({\varpi }_1,{\varpi }_2\right)}^T$, $\varsigma =\gamma ={\displaystyle \frac{1}{4}}$, $\tau ={\displaystyle \frac{1}{2}}$, $\eta ={\displaystyle \frac{4}{3}}$, and

$$A_1=\left(\begin{array}{cc}{\displaystyle \frac{1}{16}}& 0\\ {\displaystyle \frac{1}{16}}& {\displaystyle \frac{1}{8}}\end{array}\right),A_2=\left(\begin{array}{cc}0& {\displaystyle \frac{1}{12}}\\ {\displaystyle \frac{1}{6}}& {\displaystyle \frac{1}{12}}\end{array}\right),A_3=\left(\begin{array}{cc}0& 0\\ {\displaystyle \frac{1}{10}}& 0\end{array}\right),$$

and for the functions, let

$$g_1\left(ϵ,\varpi \right)={\displaystyle \frac{ϵ}{16}}{\left(sin{\varpi }_1,sin{\varpi }_2\right)}^T,$$

$$g_2\left(ϵ,\varpi \right)={\displaystyle \frac{{ϵ}^2}{8}}{\left(cos{\varpi }_1,cos{\varpi }_2\right)}^T,$$

and

$$g_3\left(ϵ,\varpi \right)={\left({\displaystyle \frac{\frac{1}{7}\left|{\varpi }_1\right|}{\left|{\varpi }_1\right|+1}},tanϵ\right)}^T.$$

Therefore, a plain calculation demonstrates that $∥A_1∥={\displaystyle \frac{1}{8}}$, $∥A_2∥={\displaystyle \frac{1}{6}}$, $∥A_3∥={\displaystyle \frac{1}{10}}$, $M_1={\displaystyle \frac{1}{16}}$, $M_2={\displaystyle \frac{1}{8}}$, $M_3={\displaystyle \frac{1}{7}}$, and

$$\Gamma \left(\varsigma +1\right)=\Gamma \left({\displaystyle \frac{5}{4}}\right)=0.9064,$$

$$\Gamma \left(\varsigma +\gamma +1\right)=\Gamma \left(\tau +1\right)=\Gamma \left({\displaystyle \frac{3}{2}}\right)=0.8862.$$

Furthermore, upon disregarding all the aforementioned calculations in (7), we obtain

$$\begin{array}{ccc}\hfill \Theta & =& {\displaystyle \frac{1}{16}}+{\displaystyle \frac{\frac{1}{8}\times \frac{1}{8}}{0.9064}}+{\displaystyle \frac{\frac{1}{6}\times \frac{1}{7}}{0.8862}}+{\displaystyle \frac{\frac{1}{10}\left(1+\frac{1}{16}\right)}{0.8862}}\hfill \\ & \simeq & 0.2265<1.\hfill \end{array}$$

By virtue of Theorem 1, the considered system is proven to possess a unique solution within the range $\left[1,e\right]$. Additionally, the fulfillment of the conditions outlined in Theorem 3 allows us to infer that the solution of this problem exhibits finite-time stability.

6. Restrictions and Corollaries

Our investigation focused on a more universal problem rather than the research conducted by [32,33] but our problem is equipped by Hadamard fractional derivative. Therefore, if we equipped the problems in [32,33] by Hadamard fractional derivatives and integrals, the problems (1) and (2) will be as the following forms

$$\left\{\begin{array}{c}{}^H\mathfrak{D}^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=g_2\left(ϵ,\varpi \left(ϵ\right)\right),0\le ϵ\le 1,\hfill \\ \varpi \left(0\right)={\varpi }_0,{\varpi }_0\in \mathbb{R},0<\varsigma \le 1.\hfill \end{array}\right.$$

(13)

and

$$\left\{\begin{array}{c}{}^H\mathfrak{D}^{\varsigma }\left(\varpi \left(ϵ\right)-g_1\left(ϵ,\varpi \left(ϵ\right)\right)\right)=a_1{g}_2\left(ϵ,\varpi \left(ϵ\right)\right)+a_2\left({}^H\mathfrak{I}^{\gamma }\right)g_3\left(ϵ,\varpi \left(ϵ\right)\right),0\le ϵ\le 1,\hfill \\ \varpi \left(0\right)={\displaystyle \frac{a_3}{\Gamma \left(\tau \right)}}{\int }_1^{\eta }{\left(log{\displaystyle \frac{\eta }{\nu }}\right)}^{\tau -1}\varpi \left(\nu \right){\displaystyle \frac{\mathrm{d}\nu }{\nu }},\tau >0,1<\eta <e,\hfill \end{array}\right.$$

(14)

such that $0<\varsigma ,\gamma \le 1$, $\gamma \in \left(0,1\right)$, $a_1$, $a_2$, and $a_3$ are real constant, and $g_1,g_2,g_3:\left[1,e\right]\times \mathbb{R}\to \mathbb{R}$ are given continuous functions and. Next, there are certain limitations that need to be considered.

Corollary 1.

Assume that for all $ϵ\in \left[1,e\right]$ and $\varpi ,\varkappa \in \mathbb{R}$, there exist Lipschitz constants $M_j$ such that

$${∥g_j\left(ϵ,\varpi \right)-g_j\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_j{∥\varpi -\varkappa ∥}_{\mathfrak{C}},j=1,2,3.$$

If

$$\Theta :=M_1+{\displaystyle \frac{\left|a\right|M_2}{\Gamma \left(\varsigma +1\right)}}+{\displaystyle \frac{\left|b\right|M_3}{\Gamma \left(\varsigma +\gamma +1\right)}}+{\displaystyle \frac{\left|c\right|\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}<1,$$

(15)

then problem (14) has a unique solution on the interval $\left[1,e\right]$.

Corollary 2.

Assume that for all $ϵ\in \left[1,e\right]$ and $\varpi ,\varkappa \in \mathbb{R}$, there exist constants $M_1$ satisfies ${∥g_1\left(ϵ,\varpi \right)-g_1\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_1{∥\varpi -\varkappa ∥}_{\mathfrak{C}}$ such that

$$M_1+{\displaystyle \frac{\left|c\right|\left(1+M_1\right)}{\Gamma \left(\tau +1\right)}}<1.$$

(16)

In addition, if the functions $g_2\left(ϵ,\varpi \right)$ and $g_3\left(ϵ,\varpi \right)$ are bounded for all $\left(ϵ,\varpi \right)\in \left[1,e\right]\times \mathrm{\Pi }$, then there is at least one solution for the problem (14) on the interval $\left[1,e\right]$.

Corollary 3.

Assume that for all $ϵ\in \left[1,e\right]$ and $\varpi ,\varkappa \in \mathbb{R}$, there exist a positive constants $M_j$ such that

$${∥g_j\left(ϵ,\varpi \right)-g_j\left(ϵ,\varkappa \right)∥}_{\mathfrak{C}}\le M_j{∥\varpi -\varkappa ∥}_{\mathfrak{C}},j=1,2,3$$

and the condition (15) holds. Then, the attainment of stability within a finite time for the solution to the problem (14) is accomplished by taking into account the nonlocal values.

Remark 3.

It should be noted that Corollaries 1–3 are findings that relate to the system form (14), which are more generally than (13).

Remark 4.

It is worth mentioning that the applicability of the main results is subject to certain structural assumptions, such as the Lipschitz continuity of the nonlinear terms and the definition of the problem over a fixed compact interval. These conditions, while standard in the literature, may limit the extension of the results to more general settings such as systems with state-dependent delays or unbounded domains. Future work may aim to relax these assumptions and investigate broader classes of fractional systems.

7. Conclusions

In this paper, we have investigated the existence, uniqueness, and finite-time stability of solutions for a class of nonlinear systems involving Hadamard fractional derivatives. By employing Krasnoselskii’s fixed point theorem and Banach’s contraction principle, sufficient conditions for the existence and uniqueness of solutions were derived. Additionally, a finite-time stability analysis was conducted, which is particularly important for systems operating over finite time intervals and within constrained domains.

The results presented here generalize earlier findings in the literature and provide a broader framework for analyzing fractional-order systems governed by Hadamard-type operators. A numerical example was provided to illustrate the validity and applicability of the theoretical findings. Moreover, the framework introduced in this study allows for further extensions to more complex systems.

The generalization developed in this work has potential applications in various scientific and engineering domains such as control systems, viscoelastic material modeling, biological systems with memory effects, and signal processing, where fractional-order models often provide more accurate descriptions than classical integer-order ones.

Future work may focus on systems with time-varying coefficients, external perturbations, or more generalized boundary conditions, as well as on developing numerical methods tailored specifically for Hadamard-type fractional differential equations.