The Stability and Global Attractivity of Fractional Differential Equations with the Ψ-Hilfer Derivative in the Context of an Economic Recession

Abstract

Fractional differential equations (FDEs) are employed to describe the physical universe. This article investigates the attractivity of solutions for FDEs and Ulam–Hyers–Rassias stability, involving the $\Psi$-Hilfer fractional derivative. Important results are presented using Krasnoselskii’s fixed point theorem, which provides a framework for analyzing the stability and attractivity of solutions. Novel results on the attractiveness of solutions to nonlinear FDEs in Banach spaces are derived, and the existence of solutions, stability properties, and behavior of system equilibria are examined. The application of $\Psi$-Hilfer fractional derivatives in modeling financial crises is explored, and a financial crisis model using $\Psi$-Hilfer fractional derivatives is proposed, providing more general and global results. Furthermore, we also perform a numerical analysis to validate our theoretical findings.

1. Introduction

The subject of fractional calculus had applications in diverse and widespread fields of engineering and science, such as electro-magnetic, visco-elasticity, fluid mechanics, electro-chemistry, biological population models, optics, and signals processing. Fractional calculus extended conventional differentiation and integration to non-integer orders, offering a powerful framework for analyzing complex dynamics. The non-local nature of fractional operators enabled them to capture the historical dynamics of systems, making fractional calculus is an indispensable tool for studying dynamical systems. In recent years, fractional-order differential equations (FODEs) had garnered significant attention from researchers, driven by their potential to model real-world phenomena more accurately, as seen in [1,2,3]. A FODE represented a generalized extension of traditional integer-order differential equations. The FODE had proven to be a valuable tool in various fields, particularly in modeling complex physical phenomena in pure and applied sciences, offering a more accurate and detailed representation of real-world systems.

FODEs could be obtained in the time space with a power law memory kernel of the non-local relationship. They provided a powerful tool for describing the memory of different substances and the nature of inheritance. A fractional derivative in applied mathematics and mathematical analysis was a derivative of any arbitrary order, real or complex. It initially appeared in a 1965 letter from Gottfried Wilhelm Leibniz to Guillaume de L’Hopital. There were two important types of fractional derivatives: the Riemann–Liouville derivative (RL-derivative) and the Caputo fractional derivative. The RL-derivative of a constant was not zero. Additionally, the fractional derivative of any function, such as the exponential and Mittag–Leffler functions, had a singularity at the origin if the function was constant at the origin.

The study of FDEs is becoming more popular across a wide range of scientific fields. The selection of a phase space has a significant impact on how these equations are studied. In fact, there are a variety of phase spaces described in the literature, and each one has unique characteristics. One of the most fascinating areas of the qualitative theory of FDEs is the study of existence, uniqueness, stability and controllability. The Russian mathematician and engineer Lyapunov established attractiveness as a key concept in the science of dynamical systems in his thesis. However, since its introduction in 1892, this field has been greatly developed (see [4,5,6] for a recent survey of findings on the existence and attractiveness of mild solutions to many kinds of differential problems in infinite dimensional spaces). Some authors like S. Abbas et al. [7,8] discussed implicit impulsive conformable fractional differential equations with infinite delay in b-metric spaces and periodic solutions of nonlinear fractional pantograph integro-differential equations with the $\psi$-Caputo derivative.

Recently, researchers have explored the $\Psi$-Hilfer operator, yielding fascinating results on the existence, uniqueness, stability, and attractivity of solutions to FDEs involving integro-differential operators. These findings were achieved by employing the fixed point theory and leveraging the renowned Gronwall inequality, a powerful tool for establishing bounds on solutions to differential equations (see [9,10]). FDEs are an essential component of models for computer graphics, agricultural research, environmental science, and vision. FDEs are also a fundamental tool for defining the nature of the physical world. Examples include light rays, which take the path with the shortest distance, and are easily represented by the Euler–Lagrange equations. Abdelhedi [11] examined FDEs with a $\Psi$-Hilfer fractional derivative, and Sugumaran et al. [12] established the results on $\Psi$-Hilfer FDEs with a complex order. Classical (SI) models have garnered substantial interest in mathematical biology, particularly in the context of applying fractional calculus to biological systems.

This integration has enabled researchers to better capture the complexities of biological phenomena and model the dynamics of infectious diseases more accurately. Developed in the early 20th century, this pioneering model serves as a discrete-time deterministic framework in biology, empowering researchers to elucidate the dynamics of infectious disease transmission and treatment. By leveraging this model, scientists can gain valuable insights into the spread of diseases and inform the development of effective control strategies (see [13]). Adjimi et al. [14] discussed existence results for the $\psi$-Caputo hybrid fractional integro-differential equations, and Hammoumi et al. [15] examined the mild solutions for impulsive fractional differential inclusions with a Hilfer derivative in Banach spaces. Hafeez et al. [16,17] studied the existence of global and local mild solutions for fractional Navier–Stokes equations and the existence and uniqueness results for mixed derivative involving fractional operators. Li et al. [18] investigated analytical approaches on the attractivity of solutions for multi-term fractional functional evolution equations.

Building upon the existing literature, this study investigates the attractivity of solutions for FDEs, including Ulam–Hyers–Rassias stability, within the framework of $\Psi$-Hilfer fractional derivatives. Furthermore, we provide a rigorous proof of the existence of solutions for the financial crisis model introduced by Korobeinikov, shedding new light on the mathematical modeling of complex economic phenomena (see [19]). Korobeinikov leveraged mathematical modeling to analyze the dynamics of financial crises. He proposed a simplified qualitative mathematical model to simulate the progression of a crisis. By utilizing this model, researchers can identify key factors that contribute to global economic instability and inform strategies to enhance economic security worldwide.

The present study is motivated by the increasing complexity of economic systems and the need for more accurate models to capture their dynamics. Fractional calculus, particularly the $\Psi$-Hilfer fractional derivative, has shown promise in modeling complex systems. However, its application in modeling financial crises is still limited. This study aims to fill this gap by exploring the utility of the $\Psi$-Hilfer fractional derivative in capturing the complexities of economic systems and providing insights into the dynamics of financial instability. In this article, we employ the fixed point theory to investigate the global attractivity and stability of solutions, utilizing the $\Psi$-Hilfer fractional derivative. This mathematical framework enables us to analyze the dynamic behavior of complex systems and derive conditions for stability and attractivity.

$$\left\{\begin{array}{c}{}^{\mathfrak{H}}D_{t_0}^{\beta ,\gamma ;\Psi }w\left(t\right)=f(t,w\left(t\right)),t_0\le t,\hfill \\ I_0^{1-p;\Psi }w\left(t_0\right)=w_0,\hfill \end{array}\right.$$

(1)

where ${}^{\mathfrak{H}}D_{t_0}^{\beta ,\gamma ;\Psi }(·)$ represent the $\Psi$-Hilfer fractional derivative of order $0<\beta <1$ and $0\le \gamma \le 1$, and $I_{t_0}^{1-p;\Psi }(·)$ is the $\Psi$-Hilfer RL-fractional integral of order $0<1-p<1$, where $p=\beta +\gamma (1-\beta )$.

This paper’s structure is as follows: Section 2 lays the groundwork with fundamental definitions, crucial lemmas, and pivotal theorems that facilitate a deeper understanding of the subsequent analysis. In Section 3, we establish the results about global attractivity to nonlinear FDEs in Banach spaces. In Section 4, we present Ulam–Hyers–Rassias stability, involving the $\Psi$-Hilfer fractional derivative. In Section 5, we explore a practical application of the $\Psi$-Hilfer fractional derivative in modeling financial crises. This section demonstrates the utility of fractional calculus in capturing the complexities of economic systems and providing insights into the dynamics of financial instability. Section 6 presents the numerical analysis, which validates the theoretical findings through computational simulations.

2. Preliminaries

We indicate ${\mathfrak{C}}^m(\mathbf{J},\omega )$ to be the space of the m-times continuously differential function in $\mathbf{J}:=[t_0,T]$ and ${\mathbf{J}}^{\prime }:=(t_0,T)$. The weighted space is the ${\mathfrak{C}}_{1-p;\Psi }$ part of a function $\omega$ on ${\mathbf{J}}^{\prime }$, which is defined in [20].

$${\mathfrak{C}}_{1-p;\Psi }(\mathbf{J},\omega )=\{w:{\mathbf{J}}^{\prime }\to \omega ;{(\Psi \left(t\right)-\Psi \left(0\right))}^{1-p}w\left(t\right)\in \mathfrak{C}(\mathbf{J},\omega )\},0\le p<1,$$

with

$$\begin{array}{ccc}\hfill {\parallel w\parallel }_{{\mathfrak{C}}_{1-p;\Psi }}& =& {\parallel (\Psi \left(t\right)-\Psi \left(0\right))}^{1-p}{w\left(t\right)\parallel }_{\mathfrak{C}}\hfill \\ & =& \underset{t\in \mathbf{J}}{max}\left|{(\Psi \left(t\right)-\Psi \left(0\right))}^{1-p}w\left(t\right)\right|.\hfill \end{array}$$

It is clear that ${\mathfrak{C}}_{1-p;\Psi }(\mathbf{J},\omega )$ is a Banach space. The RL-fractional integral with the function $\Psi$ on $[t_0,T]$ is defined by (see [20])

$$I_{t_0}^{\beta ;\Psi }w\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\Psi }^{\prime }\left(s\right){(\Psi \left(t\right)-\Psi \left(s\right))}^{\beta -1}w\left(s\right)ds.$$

The RL-fractional derivative with respect to the function $\Psi \left(t\right)$ is defined by

$$D_{t_0}^{\beta ;\Psi }w\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{\int }_{t_0}^t{\Psi }^{\prime }\left(s\right){(\Psi \left(t\right)-\Psi \left(s\right))}^{-\beta }w\left(s\right)ds,$$

and, in particular, this is the RL-fractional derivative for $\Psi \left(t\right)=t$, which is

$$D_{t_0}^{\beta }w\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{dt}}{\int }_{t_0}^t{(t-s)}^{-\beta }w\left(s\right)ds.$$

(2)

Let $m-1<\beta <m$ with $m\in \mathbf{N},I=[t_0,T]$ be the interval in such a way that $-\infty \le t_0<T\le \infty$ and $w,\Psi \in {\mathfrak{C}}^m([t_0,T],\mathbb{R})$ are the two functions in such a way that $\Psi$ is non-decreasing and ${\Psi }^{\prime }\left(t\right)\ne 0,\forall t\in I$. The $\Psi$-Hilfer FD ${}^{\mathfrak{H}}D_{t_0}^{\beta ,\gamma ;\Psi }(·)$ of order $0\le \beta \le 1$ is defined by

$${}^{\mathfrak{H}}D_{t_0}^{\beta ,\gamma ;\Psi }w\left(t\right)=I_{t_0}^{\gamma (m-\beta );\Psi }{\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^m{I}_{t_0}^{(1-\gamma )(m-\beta );\Psi }w\left(t\right).$$

(3)

Remark 1.

By taking $\Psi \left(t\right)=t$ and $\gamma \to 0$ on both sides of Equation (3), we obtain the RL-fractional derivative given by (2).

Theorem 1

([20]). If $w\in {\mathfrak{C}}_{p,\Psi }^m[t_0,T],m-1<\beta <m,0\le \gamma \le 1$, then

$$I_{t_0}^{\beta ;\Psi }{}^{\mathfrak{H}}D_{t_0}^{\beta ,\gamma ;\Psi }w\left(t\right)=w\left(t\right)-\sum _{l=1}^m{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(0\right))}^{p-l}}{\Gamma (p-l+1)}}{w}_{\Psi }^{[m-l]}{I}^{(1-\gamma )(m-\beta );\Psi }w\left(t\right).$$

Definition 1

([21]). Since the zero solution of $w\left(t\right)$ of (1) is globally attractive, if for each solution of (1) $\to 0$, we have $t\to \infty$.

Theorem 2

([22]). Let M be a closed, bounded, non-empty, and convex subset of a Banach space ω, and let $Q:\omega \to \omega$ and $W:M\to \omega$ be the two operators as follows:

(1) 

Q is a contraction with $P<1$ as a constant;

(2) 

$WM$ is contained in a compact subset of ω and W is continuous;

(3) 

$[w_3=Q{w}_1+W{w}_2{,}_1,w_2\in M]\Rightarrow w_3\in M$;

Therefore, the operator equation $w_1=Q{w}_1+W{w}_1$ has a solution in M.

Theorem 3

([23] (Schauder fixed point theorem)). If V is a closed, convex, non-empty and bounded subset of Banach space ω and if $T:V\to V$ is totally continuous at that point, T is said to be a fixed point.

Definition 2

([24]). Assume that F is the vector space of $\mathbf{K}$. A function ${\parallel ·\parallel }_{\delta }:F\to [t_0,\infty )$ is said to be the δ norm if it satisfies the following:

 (1)

${\parallel w\parallel }_{\delta }=0$, if $w=0$;

 (2)

${\parallel \Lambda w\parallel }_{\delta }={|\Lambda |}^{\delta }{\parallel w\parallel }_{\delta }\forall \Lambda \in \mathbf{K}$ and $w\in F$;

 (3)

${\parallel w+z\parallel }_{\delta }\le {\parallel w\parallel }_{\delta }+{\parallel z\parallel }_{\delta }$.

Definition 3.

Equation (1) is generalized δ-Ulam–Hyers–Rassias-fixed with $(\Phi ,\zeta )$ if there exists ${\mathfrak{C}}_{f,\delta ,w_0,\Phi }>0$ in such a way that for every solution $z\in \tilde{\mathcal{U}}$ of the relation (4), there exists a unique solution $w\in \tilde{\mathcal{U}}$ of Equation (1) with

$${|z\left(t\right)-w\left(t\right)|}^{\delta }\le {\mathfrak{C}}_{f,\delta ,w_0,\Phi }({\zeta }^{\delta }+{\Phi }^{\delta }\left(t\right)),t\in \mathbf{J}.$$

3. Global Attractivity Analysis with $\Psi$-Hilfer Fractional Derivative

This article’s primary finding is covered in this section, where we talk about the attractivity of the solution for (1) involving the $\Psi$-Hilfer fractional derivative. Let us suppose that $f(t,w)$ fulfills the condition given below before we start with the main points:

$\left(H1\right)$ $f(t,w(t\left)\right)$ is Lebesgue-measurable with regard to t on interval $[t_0,\infty )$, and there exists ${\beta }_1\in (0,\beta )$ in such a way that ${\int }_{t_0}^k{\left|f(t,w\left(t\right))\right|}^{{\displaystyle \frac{1}{{\beta }_1}}}dt<\infty \forall t_0<k<\infty$ and $f(t,w(t\left)\right)$ must be continuous with w on interval $[t_0,\infty )$. By Supposition $\left(H1\right)$, the fractional integral equation of (1) is given below:

$$w\left(t\right)={\mathcal{M}}^{p,\Psi }(t,t_0)w_0+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,w\left(s\right))ds,t_0<t,$$

where

$${\mathfrak{N}}^{\Psi ,\beta }(s,t):={\Psi }^{\prime }{(\Psi \left(t\right)-\Psi \left(s\right))}^{\beta -1},$$

as well as

$${\mathcal{M}}^{p,\Psi }(t,t_0):={\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{p-1}}{\Gamma \left(p\right)}}.$$

The following operators are defined:

$$qw\left(t\right)=q_{1}w\left(t\right)+q_{2}w\left(t\right).$$

where

$$q_{1}w\left(t\right)={\mathcal{M}}^{p,\Psi }(t,t_0)w_0,$$

and

$$q_{2}w\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,w\left(s\right))ds.$$

It is obvious that $w\left(t\right)$ is a solution of (1).

Lemma 1.

Suppose that the function $f(t,w(t\left)\right)$ fulfills Supposition $\left(H1\right)$ and ${\left({\Psi }^{\prime }\left(t\right)\right)}^{{\displaystyle \frac{1}{1-{\beta }_1}}}\le {\Psi }^{\prime }\left(t\right)$ with $t\in (t_0,\infty )$, as well as ${\beta }_1\in (0,\beta )$. Suppose that the condition given below:

$\left(H2\right)$ $\left|f(t,w\left(t\right))\right|\le \nu {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-{\beta }_1}$ for $t\in (t_0,\infty )$ as well as $w\left(t\right)\in {\mathfrak{C}}_{1-p}((t_0,\infty )$, $\mathbb{R}),$$\nu \ge 0$, as well as $\beta <{\gamma }_1<1$.

At that point, the operator $q_2$ is continuous and $q_2{M}_{1,\Psi }$ is a compact subset of $\mathbb{R}$ for $t_0+T\le t$, as follows:

$$M_{1,\Psi }=\{w\left(t\right)\in {\mathfrak{C}}_{1-p,\Psi }((t_0,\infty ),\mathbb{R}),\left|w\left(t\right)\right|\le {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p_1},t_0+T_1\le t\},$$

$p_1={\displaystyle \frac{1}{2}}({\gamma }_1-\beta )$, and $\Psi \left(T_1\right)$ fulfills

$$|w_0|{\displaystyle \frac{\Psi {\left(T_1\right)}^{\frac{1}{2}(p-1)}}{\Gamma \left(p\right)}}+{\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (1+\beta -{\gamma }_1)}}\Psi {\left(T_1\right)}^{-{\displaystyle \frac{1}{2}}(\gamma -\beta )}\le 1.$$

(4)

Proof. 

$q:M_{1,\Psi }\to M_{1,\Psi }$ for $t_0+T_1\le t$. According to the above discussion about $M_{1,\Psi }$, it is not difficult to see that $M_{1,\Psi }$ is a convex, bounded and closed subset of $\mathbb{R}$. Apply $\left(H2\right)$. Thus, for $t_0\le t$, we obtain

$$\begin{array}{ccc}\hfill |q_2{w}_2\left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s){\mathfrak{N}}_{\Psi }^{\beta }(t,s)\left|f(s,w_2\left(s\right))\right|ds\hfill \\ \hfill & \le & {\displaystyle \frac{\nu }{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s){(\Psi \left(s\right)-\Psi \left(t_0\right))}^{-{\gamma }_1}ds\hfill \\ \hfill & \le & {\displaystyle \frac{\nu }{\Gamma \left(\beta \right)}}{\int }_0^{\Psi \left(t\right)-\Psi \left(t_0\right)}{(\Psi \left(s\right)-\Psi \left(t_0\right))}^{\beta -1}{\left(1-{\displaystyle \frac{v}{\Psi \left(t\right)-\Psi \left(t_0\right)}}\right)}^{\beta -1}{v}^{-{\gamma }_1}dv\hfill \\ \hfill & \le & {\displaystyle \frac{\nu {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{\beta -1}}{\Gamma \left(\beta \right)}}{\int }_0^1{(1-\beta )}^{\beta -1}{\beta }^{-{\gamma }_1}(\Psi \left(t\right)-\Psi \left(t_0\right))dv\hfill \\ \hfill & =& {\displaystyle \frac{\nu \Gamma (1-{\gamma }_1){(\Psi \left(t\right)-\Psi \left(t_0\right))}^{\beta }}{\Gamma (\beta -{\gamma }_1+1)}}\hfill \\ & \le & {\displaystyle \frac{\nu \Gamma (1-{\gamma }_1){(\Psi \left(t\right)-\Psi \left(t_0\right))}^{\beta -{\gamma }_1}}{\Gamma (\beta -{\gamma }_1+1)}};\hfill \end{array}$$

(5)

for $t_0+T_1\le t$, the inequality (4) and $\beta <{\gamma }_1$ yield

$${\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\le {\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}\Psi {\left(T_1\right)}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\le 1.$$

At that level, for $t_0+T\le t$, we obtain

$$\begin{array}{ccc}\hfill |q_2{w}_2\left(t\right)|& \le & {\displaystyle \frac{\nu \Gamma (1-{\gamma }_1){(\Psi \left(t\right)-\Psi \left(t_0\right))}^{\beta -{\gamma }_1}}{\Gamma (\beta -{\gamma }_1+1)}}\hfill \\ & =& \left[{\displaystyle \frac{\nu \Gamma (1-{\gamma }_1){(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-\frac{1}{2}({\gamma }_1-\beta )}}{\Gamma (\beta -{\gamma }_1+1)}}\right]{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\hfill \\ & \le & {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}={(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p_1},\hfill \end{array}$$

which shows that $q_2{M}_{1,\Psi }\subset M_{1,\Psi }$ for $t_0+T\le t$. W is continuous for any $w_{2,n}\left(t\right),w_2\left(t\right)\in M_{1,\Psi },n=1,2,\dots$, with

$$\underset{n\to \infty }{lim}|w_{2,n}\left(t\right)-w_2\left(t\right)|=0;$$

thus, we have ${lim}_{n\to \infty }{w}_{2,n}\left(t\right)=w_2\left(t\right)$ and

$$\underset{n\to \infty }{lim}f(t,w_{2,n}\left(t\right))=f(t,w_2\left(t\right)),t_0+T_1\le t.$$

Suppose that $ϵ>0$ gives us $t_0+T_1\le T$ in such a way that

$${\displaystyle \frac{\nu \Gamma (1-{\gamma }_1){(\Psi \left(T\right)-\Psi \left(t_0\right))}^{-\left({\gamma }_1-\beta \right)}}{\Gamma (\beta -{\gamma }_1+1)}}<{\displaystyle \frac{ϵ}{2}}.$$

As opposed to that, let $\mu ={\displaystyle \frac{\beta -1}{1-{\gamma }_1}}$. Then, $1+\mu >0$, since ${\beta }_1\in (0,\beta )$. Hence, for $T\ge t\ge t_0+T_1$, we obtain

$$\begin{array}{ccc}\hfill |q_2{w}_{2,n}\left(t\right)-q_2{w}_2\left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)|f(t,w_{2,n}\left(s\right))-f(t,w_2\left(s\right))|ds\hfill \\ & \le & {\displaystyle \frac{1}{p\left(\beta \right)}}{\left({\int }_{t_0}^t{\Psi }^{\prime }\left(s\right){(\Psi \left(t\right)-\Psi \left(s\right))}^{{\displaystyle \frac{\beta -1}{1-{\beta }_1}}}ds\right)}^{1-{\beta }_1}\hfill \\ & \times & {\left({\int }_{t_0}^t{|f(t,w_{2,n}\left(s\right))-f(t,w_2\left(s\right))|}^{{\displaystyle \frac{1}{{\beta }_1}}}ds\right)}^{{\beta }_1}\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\left({\displaystyle \frac{{(\Psi \left(T\right)-\Psi \left(t_0\right))}^{1+\mu }}{1+\mu }}\right)}^{1-{\beta }_1}{\parallel f(·,w_{2,n}(·))-f(·,w_2(·))\parallel }_{{\mathfrak{C}}_p}\hfill \\ & \times & {\left({\int }_{t_0}^t{(\Psi \left(t\right)-\Psi \left(s\right))}^{\prime }ds\right)}^{{\beta }_1}\hfill \\ \hfill |q_2{w}_{2,n}\left(t\right)-q_2{w}_2\left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\left({\displaystyle \frac{{(\Psi \left(T\right)-\Psi \left(t_0\right))}^{1+\mu }}{1+\mu }}\right)}^{1-{\beta }_1}{\displaystyle \frac{{(\Psi \left(T\right)-\Psi \left(t_0\right))}^{1+p}}{{\Psi }^{\prime }\left(T\right)(1+p)}}\hfill \\ & \times & \parallel f(·,w_{2,n}(·))-f(·,w_2(·)){\parallel }_{{\mathfrak{C}}_p}\to 0asn\to \infty .\hfill \end{array}$$

For $T<t$, we changes the variables in Equation (5). Thus, we obtain

$$\begin{array}{ccc}\hfill |q_2{w}_{2,n}\left(t\right)-q_2{w}_2\left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)|f(t,w_{2,n}\left(s\right))-f(t,w_2\left(s\right))|ds\hfill \\ & \le & {\displaystyle \frac{2\nu }{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s){(\Psi \left(s\right)-\Psi \left(t_0\right))}^{-{\gamma }_1}ds\hfill \\ & \le & {\displaystyle \frac{2\nu \Gamma (1-{\gamma }_1)}{\Gamma (1+\beta -{\gamma }_1)}}{(\Psi \left(T\right)-\Psi \left(t_0\right))}^{-({\gamma }_1-\beta )}<ϵ.\hfill \end{array}$$

At that point, $T+t_0\le t,q_2{w}_{2,n}\left(t\right)-q_2{w}_2\left(t\right)\to 0asn\to \infty$, which shows that $q_2{M}_{1,\Psi }$ is equicontinuous. Hence, we prove that $q_2{M}_{1,\Psi }$ is continuous. Assume that $ϵ>0$ is given. Since

$$\underset{t\to \infty }{lim}{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p_1}=0,$$

there exists $t_0+T_1<T^{\prime }$ in such a way that ${(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p_1}<{\displaystyle \frac{ϵ}{2}}$ for $T^{\prime }<t$. Assume that $t_0+T_1\le t_1,t_2$ and $t_1<t_2$. If $t_1,t_2\in [t_0+T_1,T^{\prime }],{\int }_0^{T^{\prime }}{\left|f(s,w\left(s\right))\right|}^{{\displaystyle \frac{1}{{\beta }_1}}}ds$ exists by Supposition $\left(H1\right)$, we have

$$\begin{array}{ccc}\hfill |q_{2}w\left(t_2\right)-q_{2}w\left(t_1\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_2}{\mathfrak{N}}^{\Psi ,\beta }(t_2,s)|f(s,w\left(s\right))|ds+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_1}{\mathfrak{N}}^{\Psi ,\beta }(t_1,s)\left|f(s,w\left(s\right))\right|ds\hfill \\ & +& {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_1}{\mathfrak{N}}^{\Psi ,\beta }(t_2,s)|f(s,w\left(s\right))|ds-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_1}{\mathfrak{N}}^{\Psi ,\beta }(t_2,s)\left|f(s,w\left(s\right))\right|ds\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\left({\displaystyle \frac{1}{1+\mu }}\right)}^{1-{\beta }_1}[{(\Psi \left(t_1\right)-\Psi \left(t_0\right))}^{1+\mu }-{(\Psi \left(t_2\right)-\Psi \left(t_0\right))}^{1+\mu }\hfill \\ & +& {(\Psi \left(t_2\right)-\Psi \left(t_1\right))}^{1+\mu }{]}^{1-{\beta }_1}\times {\left[{\int }_{t_0}^{T^{\prime }}{\left|f(s,w\left(s\right))\right|}^{{\displaystyle \frac{1}{{\beta }_1}}}ds\right]}^{{\beta }_1}+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\left({\displaystyle \frac{1}{1+\mu }}\right)}^{1-{\beta }_1}\hfill \\ & \times & {\left[{(\Psi \left(t_2\right)-\Psi \left(t_1\right))}^{1+\mu }\right]}^{1-{\beta }_1}{\left[{\int }_{t_0}^{T^{\prime }}{\left|f(s,w\left(s\right))\right|}^{{\displaystyle \frac{1}{{\beta }_1}}}ds\right]}^{{\beta }_1}\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\left({\displaystyle \frac{1}{1+\mu }}\right)}^{1-{\beta }_1}{\left[{\int }_{t_0}^{T^{\prime }}{\left|f(s,w\left(s\right))\right|}^{{\displaystyle \frac{1}{{\beta }_1}}}ds\right]}^{{\beta }_1}{(\Psi \left(t_2\right)-\Psi \left(t_1\right))}^{\beta -{\beta }_1}\to 0\hfill \\ & & as{t}_2\to t_1.\hfill \end{array}$$

If $T^{\prime }<t_1,t_2$, we change the variables in Equation (5). Thus, we obtain

$$\begin{array}{ccc}\hfill |q_{2}w\left(t_2\right)-q_{2}w\left(t_1\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_2}{\mathfrak{N}}^{\Psi ,\beta }(t_2,s)|f(s,w\left(s\right))|ds+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_1}{\mathfrak{N}}^{\Psi ,\beta }(t_1,s)\left|f(s,w\left(s\right))\right|ds\hfill \\ & \le & {\displaystyle \frac{U}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_2}{\mathfrak{N}}^{\Psi ,\beta }(t_2,s){(\Psi \left(s\right)-\Psi \left(t_0\right))}^{-{\gamma }_1}ds\hfill \\ & +& {\displaystyle \frac{U}{\Gamma \left(\beta \right)}}{\int }_{t_0}^{t_1}{\mathfrak{N}}^{\Psi ,\beta }(t_1,s){(\Psi \left(s\right)-\Psi \left(t_0\right))}^{-{\gamma }_1}ds\hfill \\ & \le & {(\Psi \left(t_1\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}+{(\Psi \left(t_2\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\hfill \\ & =& {(\Psi \left(t_1\right)-\Psi \left(t_0\right))}^{-p_1}+{(\Psi \left(t_2\right)-\Psi \left(t_0\right))}^{-p_1}<ϵ,as{t}_2\to t_1.\hfill \end{array}$$

If $t_2>T^{\prime }>t_1\ge t_0+T_1$, it must be noted that if we have $t_2\to t_1$ at that point, then we also have $t_2\to T^{\prime }$ and $T^{\prime }\to t_1$. □

Lemma 2.

Suppose that $f(t,w(t\left)\right)$ fulfills Condition $\left(H1\right)$ and $t^{{\displaystyle \frac{1}{1-{\beta }_1}}}\le t$ with ${\beta }_1\in (0,\beta )$ and $t\in (t_0,\infty )$. Suppose that the following condition holds:

$\left(H3\right)$  $\left|f(t,w\left(t\right))\right|\le \nu {(t-t_0)}^{-{\gamma }_1},w\left(t\right)\in {\mathfrak{C}}_{1-\beta ;t}((t_0,\infty ),\mathbb{R}),t\in (t_0,\infty ),\nu \ge 0$ and $1>{\gamma }_1>\beta$.

Therefore, the operator $q_2$ is continuous and $q_2{M}_{1,\Psi }$ is located in a compact subset of $\mathbb{R}$ for $t_0+T\le t$, where

$$M_{1,\Psi }=\{w\left(t\right)\in {\mathfrak{C}}_{1-\beta ;t}((t_0,\infty ),\mathbb{R}),\left|w\left(t\right)\right|\le {(t-t_0)}^{-p_1},t_0+T_1\le t\}$$

$p_1={\displaystyle \frac{1}{2}}({\gamma }_1-\beta )$, and $T_1$ fulfils

$$|w_0|{\displaystyle \frac{{\left(T_1\right)}^{\frac{1}{2}(\beta -1)}}{\Gamma \left(\beta \right)}}+{\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (1+\beta -{\gamma }_1)}}{T}_1^{{\displaystyle \frac{\beta }{2}}}\le 1.$$

Proof. 

The proof proceeds in the same manner from the specific decision of $\Psi \left(t\right)=t$, and we take $\gamma \to 0$ in Lemma (1). Note that we can obtain some specific cases by selecting the $\Psi (·)$-function in the condition of Lemma (1). □

Lemma 3.

If Assumptions $\left(H1\right)$ and $\left(H2\right)$ are true, the solution of (1) at that point is in $M_{1,\Psi }$ for $t_0+T_1\le t$.

Proof. 

It must be noted that System (1) has a solution if $w\left(t\right)$ is a fixed point of q. It needs to be shown that in order to have a fixed $w_2\in M_{1,\Psi }$, we need to ensure that $w\in {\mathfrak{C}}_{1-\beta ;\Psi }((t_0,\infty ),\mathbb{R}),w=q_{1}w+q_2{w}_2\Rightarrow w\in M_{1,\Psi }$ holds. Using Condition $\left(H2\right)$ and following the same steps results in (5) if $w=q_{1}w+q_2{w}_2$, we have

$$\begin{array}{ccc}\hfill \left|w\left(t\right)\right|=|q_{1}w+q_2{w}_2|& \le & {\mathcal{M}}^{p,\Psi }(t,t_0)\left|w_0\right|+{\displaystyle \frac{U}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t_1,s){(\Psi \left(s\right)-\Psi \left(t_0\right))}^{-{\gamma }_1}ds\hfill \\ & \le & {\mathcal{M}}^{p,\Psi }(t,t_0)\left|w_0\right|+{\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-({\gamma }_1-\beta )}\hfill \end{array}$$

Now, for $t_0+T_1\le t$ from (4) and $1>{\gamma }_1>\beta >0$, we have

$$\begin{array}{c}\hfill {\mathcal{M}}_{{\displaystyle \frac{1}{2}}}^{p,\Psi }(t,t_0)\left|w_0\right|+{\displaystyle \frac{\nu \Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\\ \hfill \le {\displaystyle \frac{\Psi {\left(T_1\right)}^{\frac{1}{2}(p-1)}}{\Gamma \left(p\right)}}+{\displaystyle \frac{U\Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}\Psi {\left(t\right)}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\le 1\end{array}$$

where ${\mathcal{M}}_{{\displaystyle \frac{1}{2}}}^{p,\Psi }(t,t_0):={\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(t_0\right))}^{\frac{1}{2}(p-1)}}{\Gamma \left(p\right)}}$.

For $t_0+T_1\le t$, we have

$$\begin{array}{ccc}\hfill \left|w\right(t\left)\right|& \le & \left[{\mathcal{M}}_{{\displaystyle \frac{1}{2}}}^{p,\Psi }(t,t_0)\left|w_0\right|+{\displaystyle \frac{U\Gamma (1-{\gamma }_1)}{\Gamma (\beta -{\gamma }_1+1)}}\Psi {\left(t\right)}^{-{\displaystyle \frac{1}{2}}({\gamma }_1-\beta )}\right]\hfill \\ & \times & {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p}\le {(\Psi \left(t\right)-\Psi \left(t_0\right))}^{-p};\hfill \end{array}$$

then, $w\left(t\right)\in M_{1,\Psi }$ for $t_0+T_1\le t$. According to $t_0$, Theorem (1) and Lemma (1), there exists $w_2\in M_{1,\Psi }$ in such a way that $w_2=q_1{w}_2+q_2{w}_2$. Thus, we prove that q has a fixed point in $M_{1,\Psi }$, which is the solution of (1). □

4. δ-Ulam–Hyers–Rassias Stability

Assume that the space of a continuous function is piecewise-weighted, as follows:

$$Z=\{h:\mathbf{J}\to \mathbb{R}/h\in P{\mathfrak{C}}_{1-\beta ;\Psi }(\mathbf{J},\mathbb{R}),0\le \beta <1\},$$

with the generalized metric space on Z being given below:

$$d(h,k)=inf\left\{{\mathfrak{C}}_1+{\mathfrak{C}}_2\in [t_0,\infty )/{|h\left(t\right)-k\left(t\right)|}^{\delta }\le ({\mathfrak{C}}_1+{\mathfrak{C}}_2)({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }),t\in \mathbf{J}\right\}$$

(6)

where ${\mathfrak{C}}_1\in \left\{\mathfrak{C}\in [t_0,\infty )/{|h\left(t\right)-k\left(t\right)|}^{\delta }\le \mathfrak{C}{\varrho }^{\delta }\left(t\right)\right\}$ and ${\mathfrak{C}}_2\in \left\{\mathfrak{C}\in [t_0,\infty )/{|h\left(t\right)-k\left(t\right)|}^{\delta }\le \mathfrak{C}{\zeta }^{\delta }\right\}$.

Suppose that $1\ge \delta >0,\zeta \ge 0,\varrho \in P{\mathfrak{C}}_{1-\beta ;\Psi }(\mathbf{J}\times \mathbb{R})$ is increasing and that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{|}^{\mathfrak{H}}{D}_{t_0}^{\beta ,\gamma ;\Psi }-f(t,z\left(t\right))|\le \varrho \left(t\right),t_0<t,\hfill \\ |z\left(t\right)-w_0\left(z\left(t\right)\right)|\le \zeta .\hfill \end{array}\right.\end{array}$$

(7)

Remark 2.

Check whether $z\in \tilde{\mathcal{U}}$ is the solution to Equation (7) and z is the solution of the following inequality given below:

$$|z\left(t\right)-w_0\left(z\left(t_0\right)\right)|\le \zeta ,$$

(8)

and

$$|z\left(t\right)-{\mathcal{M}}^{p,\Psi }(t,t_0)z\left(t_0\right)-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,z\left(s\right))\le {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)\varrho \left(s\right)ds,$$

and

$$|z\left(t\right)-w_0-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,z\left(s\right))\le \zeta +{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)\varrho \left(s\right)ds,$$

(9)

By using Banach’s fixed point theorem, the impulsive fractional differential Equation (1) demonstrates the stability of the generalized Ulam–Hyers–Rassias type. Before we discuss our results on stability, we develop some suppositions:

$\left(H4\right)$ $f\in {\mathfrak{C}}_{1-\beta ;\Psi }(\mathbf{J}\times \mathbb{R},\mathbb{R})$;

$\left(H5\right)$ There exists a non-negative consistent term $P_f$ in such a way that

$$|f(t,v_1)-f(t,v_2)|\le P_f|v_1-v_2|fort\in \mathbf{J}and{v}_1,v_2\in \mathbb{R}.$$

$\left(H6\right)$ $w_0\in {\mathfrak{C}}_{1-\beta ;\Psi }(\mathbf{J}\times \mathbb{R})$, and there exists a non-negative constant $P_{w_0}$ in such a way that

$$|w_0\left(v_1\right)-w_1\left(v_2\right)|\le P_{w_0}|v_1-v_2|,t_0<t,v_1,v_2\in \mathbb{R}.$$

$\left(H7\right)$ Let $\varrho \in {\mathfrak{C}}_{1-\beta ;\Psi }(\mathbf{J},{\mathbb{R}}_{+})$ be the increasing function. Then, there exists ${\mathfrak{C}}_{\varrho }>0$ in such a way that

$${\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)\varrho \left(s\right)ds\le {\mathfrak{C}}_{\varrho }\varrho \left(t\right),t\in \mathbf{J}.$$

Theorem 4.

Suppose that Suppositions (H4)–(H7) are satisfied. If there exists $z\in \tilde{\mathcal{U}}$ that satisfies Equation (7), there exists a solution $z_0:\mathbf{J}\to \mathbb{R}$ in such a way that

$$\begin{array}{ccc}\hfill z_0\left(t\right)& =& {\mathcal{M}}^{p,\Psi }w\left(t_0\right)+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,z_0\left(s\right))ds,t_0<t,\hfill \\ & =& w_0+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,z_0\left(s\right))ds,\hfill \end{array}$$

and

$$|z\left(t\right)-z_0{\left(t\right)|}^{\delta }\le {\displaystyle \frac{(1+{\mathfrak{C}}_{\varrho }^{\delta })({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta })}{1-\Phi }},$$

(10)

and

$$\Phi =max\{P_{w_0}^{\delta }+P_f^{\delta }{\mathfrak{C}}_{\varrho }^{\delta }\}$$

(11)

Proof. 

Consider $\omega :z\to z$ the operator given below:

$$\begin{array}{c}\hfill \omega w\left(t\right)=\left\{\begin{array}{c}{\mathcal{M}}^{p,\Psi }w\left(t_0\right)+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,w_0\left(s\right))ds,t_0<t,\hfill \\ w_0+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,z_0\left(s\right))ds.\hfill \end{array}\right.\end{array}$$

(12)

For every $w\in Z$, $\omega$ is a well-defined operator according to Condition $\left(H4\right)$. We show that $\omega$ is strictly contractive on Z in three situations in order to apply the Banach fixed point theorem. Remember that

$${|h\left(t\right)-k\left(t\right)|}^{\delta }\le \left\{\begin{array}{c}{\mathfrak{C}}_1{\varrho }^{\delta }\left(t\right),t\in \mathbf{J},\hfill \\ {\mathfrak{C}}_2{\zeta }^{\delta }.\hfill \end{array}\right.$$

(13)

In Equation (12), $\left(H5\right),\left(H6\right)$ and Equation (13), we use the definition of $\omega$. Thus, the following case exists:

Case 1.

Using Assumptions $\left(H5\right),\left(H7\right)$ and Equation (13), we obtain

$$\begin{array}{ccc}\hfill {|\omega h\left(t\right)-\omega k\left(t\right)|}^{\delta }& \le & |{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,h\left(s\right))ds-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,k\left(s\right))ds{|}^{\delta }\hfill \\ & \le & P_f^{\delta }{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)|h\left(s\right)-k\left(s\right)|ds\right)}^{\delta }\hfill \\ & \le & P_f^{\delta }{\mathfrak{C}}_1{\mathfrak{C}}_{\varrho }^{\delta }{\varrho }^{\delta }\left(t\right).\hfill \end{array}$$

Case 2.

Using Assumption $\left(H6\right)$ and Equation (12), we obtain

$$\begin{array}{ccc}\hfill {|\omega h\left(t\right)-\omega k\left(t\right)|}^{\delta }& \le & |w_0\left(h\left(t\right)\right)-w_0{\left(k\left(t\right)\right)|}^{\delta }\hfill \\ & \le & {\left(P_{w_0}|h\left(t\right)-k\left(t\right)|\right)}^{\delta }\hfill \\ & \le & P_{w_0}^{\delta }{\mathfrak{C}}_2{\zeta }^{\delta }.\hfill \end{array}$$

Case 3.

Using Assumptions $\left(H4\right),\left(H5\right),\left(H6\right)$ and Equation (12), we obtain

$$\begin{array}{ccc}\hfill {|\omega h\left(t\right)-\omega k\left(t\right)|}^{\delta }& \le & |w_0\left(h\left(t\right)\right)-w_0\left(k\left(t\right)\right){|}^{\delta }+\left|{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)\right(f(s,h\left(s\right))-f(s,k\left(s\right))ds{|}^{\delta }\hfill \\ & \le & P_{w_0}^{\delta }{\mathfrak{C}}_2{\zeta }^{\delta }+P_f^{\delta }{\mathfrak{C}}_1{\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)\varrho \left(s\right)\right)}^{\delta }\hfill \\ & \le & (P_{w_0}^{\delta }+P_f^{\delta }{\mathfrak{C}}_{\varrho }^{\delta })({\mathfrak{C}}_1+{\mathfrak{C}}_2)({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }).\hfill \end{array}$$

At last, we obtain

$$\begin{array}{ccc}\hfill {|\omega h\left(t\right)-\omega k\left(t\right)|}^{\delta }& \le & max(P_{w_0}^{\delta }+P_f^{\delta }{\mathfrak{C}}_{\varrho }^{\delta })({\mathfrak{C}}_1+{\mathfrak{C}}_2)({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta })\hfill \\ & =& \Phi ({\mathfrak{C}}_1+{\mathfrak{C}}_2)({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }).\hfill \end{array}$$

Hence, we obtain

$$d({\omega }_h,{\omega }_k)\le \Phi d(h,k),$$

for any $h,k\in Z$. Taking $h_0\in Z$ and the piecewise continuous property of $h_0$ and ${\omega }_{h_0}$, there exists a constant $0<S_1<\infty$ at that point such that

$$\begin{array}{ccc}\hfill |\omega h_0\left(t\right)-h_0{\left(t\right)|}^{\delta }& \le & |{\mathcal{M}}^{p,\Psi }(t,t_0)w\left(t_0\right)+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,h_0\left(s\right))ds-h_0\left(t\right){|}^{\delta }\hfill \\ & \le & S_1{\varrho }^{\delta }\left(t\right)\le S_1({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }),t_0<t.\hfill \end{array}$$

In contrast, $S_2$ and $S_3$ with $0<S_2<\infty$ and $0<S_3<\infty$ hold in such a way that

$$\begin{array}{ccc}\hfill |\omega h_0\left(t\right)-h_0{\left(t\right)|}^{\delta }& =& |w_0\left(h_0\left(t\right)\right)-h_0{\left(t\right)|}^{\delta }\hfill \\ & \le & S_2{\zeta }^{\delta }\le S_2({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }),t_0<t,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill |\omega h_0\left(t\right)-h_0{\left(t\right)|}^{\delta }& =& |w_0\left(h_0\left(t\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_{t_0}^t{\mathfrak{N}}^{\Psi ,\beta }(t,s)f(s,h_0\left(s\right))ds-h_0\left(t\right){|}^{\delta }\hfill \\ & \le & S_3({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }),t_0<t.\hfill \end{array}$$

Since $w_0,f$ and $h_0$ must be bounded on $\mathbf{J}$ and $\varrho (·)+{\zeta }^{\delta }>0$, (6) implies that

$$d(\omega h_0,h_0)<\infty .$$

It must be noted that there exists a continuous function $z_0:\mathbf{Z}\to \mathbb{R}$ in such a way that ${\omega }^m{z}_0\to z_0$ in $(Z,d)$ as $m\to \infty$ and $\omega z_0=z_0$, which enables $z_0$ to fulfill (12) for each $t\in \mathbf{J}$. Finally, for the proof of this theorem, we consider that $0<{\mathfrak{C}}_h<\infty$ holds in such a way that

$$|h_0{\left(t\right)-h\left(t\right)|}^{\delta }\le {\mathfrak{C}}_h({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta }),t\in \mathbf{J}.$$

Suppose that $h,h_0$ are bounded in $\mathbf{J}$ and ${min}_{t\in \mathbf{J}}({\varrho }^{\delta }\left(t\right)+{\zeta }^{\delta })>0$. Then, we have $d(h_0,h)<\infty$ for every $h\in Z$, i.e., $Z=\{h\in Z/d(h_0,h)<\infty \}$. By Suppositions (H4)–(H7), Equations (8) and (9) show that

$$d(z,\omega z)\le 1+{\mathfrak{C}}_{\varrho }^{\delta }.$$

(14)

Hence, according to Equation (14), we obtain

$$d(z,z_0)\le {\displaystyle \frac{d(\omega z,z)}{1-\Phi }}\le {\displaystyle \frac{1+{\mathfrak{C}}_{\varrho }^{\delta }}{1-\Phi }},$$

Thus, it is proven that (10) is true for $t\in \mathbf{J}$. □

5. Application on Financial Crisis Using $\Psi$-Hilfer Fractional Derivative

In this part, we describe the model of the financial crisis using $\Psi$-Hilfer fractional derivatives, which are presented in Korobeinikov’s study [19]. In order to determine the usual derivative of the two sub-populations Y and Z, we use the financial crisis model that is as follows:

$$\begin{array}{ccc}\hfill \dot{Y}& =& -\gamma Y{Z}^{\beta },\hfill \\ \hfill \dot{Z}& =& \gamma Y{Z}^{\beta }-{\displaystyle \frac{1}{\rho }}Z\hfill \end{array}$$

(15)

This model divides the population into two groups: one is a healthy sub-population $Y\left(t\right)$ and the other is an activated sub-population $Z\left(t\right)$. We consider the latter population to be homogeneous. Furthermore, we assume that an average active unit must interact with activated units in such a way that $\beta >1$ in order for the population to be activated. $1<\beta <10$ for the modern population and $1<\beta <2$ for the majority population. We suppose that $\gamma$ is a positive activation rate coefficient and that activation happens at a rate of $\gamma Y{Z}^{\beta }$. The activation rate coefficient, representing the proportion of information dissemination relative to working capital turnover, serves as a key indicator of economic efficiency. Upon activation, units remain in an active state and transmit their activation rate to neighboring healthy units. Following a duration of $\rho$ time units, activated units typically enter a dormant phase, withdrawing from the population and ceasing participation.

We can use a $\Psi$-Hilfer fractional derivative to express the model (15) in this way. In the sections that follow, we utilize the model given below to examine the existence, uniqueness, and stability of solutions.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathfrak{H}}D^{\beta ,\gamma ;\Psi }Y\left(t\right)=-\gamma Y{Z}^{\beta }.\hfill \\ {}^{\mathfrak{H}}D^{\beta ,\gamma ;\Psi }Z\left(t\right)=\gamma Y{Z}^{\beta }-{\displaystyle \frac{1}{\rho }}Z.\hfill \end{array}\right.\end{array}$$

(16)

To establish the local existence of solutions for problem (16), we first present the following corollary:

Corollary 1.

If we take $E={\mathcal{B}}_r$, which is a ball with radius r, then (16) has a unique solution, and $P\le {\displaystyle \frac{\Gamma \left(\gamma \right)}{2\Psi {\left(c\right)}^{1-\beta }}}-L{\parallel X\parallel }_{P\left(U\right)}$, where $P=max\{{\displaystyle \frac{1}{\rho }}+2\gamma r^{n+1},\gamma r^{\beta }\}$.

Proof. 

Suppose that $f:E\to \mathbb{R}$ is continuous on E and must be a closed subset ${\mathbb{R}}_m^{+}$. Suppose that $P=max\{{\displaystyle \frac{1}{\rho }}+2\gamma r^{n+1},\gamma r^{\beta }\}$. Consider $Y_1=(y_1,z_1)$ and $Y_2=(y_2,z_2)$.

$$\begin{array}{ccc}\hfill {\mathcal{I}}_1+{\mathcal{I}}_2& =& |-\gamma y_1{z}_1^{\beta }+\gamma y_2{z}_2^{\beta }|+|\gamma y_1{z}_1^{\beta }-{\displaystyle \frac{1}{\rho }}{z}_1-\gamma y_2{z}_2^{\beta }+{\displaystyle \frac{1}{\rho }}{z}_2|.\hfill \\ \hfill {\mathcal{I}}_1& =& |-\gamma y_1{z}_1^{\beta }+\gamma y_2{z}_2^{\beta }|=|-\gamma (y_1{z}_1^{\beta }-y_2{z}_2^{\beta })|\hfill \\ & \le & |-\gamma ||y_1{z}_1^{\beta }-y_2{z}_2^{\beta }|\hfill \\ & \le & \gamma |y_1{z}_1^{\beta }-y_2{z}_2^{\beta }-y_1{z}_2^{\beta }+y_1{z}_2^{\beta }|\hfill \\ & \le & \gamma |y_1(z_1^{\beta }-z_2^{\beta })|+\gamma |z_2^{\beta }(y_1-y_2)|\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \gamma |y_1\left|\right|z_1^{\beta }-z_2^{\beta }|+\gamma |z_2^{\beta }\left|\right|y_1-y_2|\hfill \\ & \le & \gamma |y_1\left|\right|z_1-z_2\left|\right|z_1^{\beta -1}+z_1^{\beta -2}{z}_2+\dots +z_1{z}_2^{\beta -2}+z_2^{\beta -1}|+\gamma |z_2{|}^{\beta }|y_1-y_2|\hfill \\ & \le & \gamma |y_1\left|\right|z_1-z_2\left|\right(|z_1{|}^{\beta -1}+|z_1{|}^{\beta -2}|z_2|+\dots +|z_1\left|\right|z_2{|}^{\beta -2}+|z_2{|}^{\beta -1})\hfill \\ & +& \gamma |z_2{|}^{\beta }|y_1-y_2|.\hfill \\ \hfill {\mathcal{I}}_2& =& |\gamma y_1{z}_1^{\beta }-{\displaystyle \frac{1}{\rho }}{z}_1-\gamma y_2{z}_2^{\beta }+{\displaystyle \frac{1}{\rho }}{z}_2|\hfill \\ & =& |\gamma (y_1{z}_1^{\beta }-y_2{z}_2^{\beta })|+|{\displaystyle \frac{1}{\rho }}(z_1-z_2)|\hfill \\ & \le & \left|\gamma \right||y_1{z}_1^{\beta }-y_2{z}_2^{\beta }\left|\right|{\displaystyle \frac{1}{\rho }}\left|\right|z_1-z_2|\hfill \\ & \le & \gamma |y_1{z}_1^{\beta }-y_2{z}_2^{\beta }-y_1{z}_2^{\beta }+y_1{z}_2^{\beta }|+{\displaystyle \frac{1}{\rho }}|z_1-z_2|\hfill \\ & \le & \gamma |y_1(z_1^{\beta }-z_2^{\beta })|+\gamma |z_2^{\beta }(y_1-y_2)|+{\displaystyle \frac{1}{\rho }}|z_1-z_2|\hfill \\ & \le & \gamma |y_1\left|\right|z_1^{\beta }-z_2^{\beta }|+\gamma |z_2^{\beta }\left|\right|y_1-y_2|+{\displaystyle \frac{1}{\rho }}|z_1-z_2|\hfill \\ & \le & \gamma |y_1\left|\right|z_1-z_2\left|\right|z_1^{\beta -1}+z_1^{\beta -2}{z}_2+\dots +z_1{z}_2^{\beta -2}+z_2^{\beta -1}|+\gamma |z_2{|}^{\beta }|y_1-y_2|+{\displaystyle \frac{1}{\rho }}|z_1-z_2|\hfill \\ & \le & \left({\displaystyle \frac{1}{\rho }}+\gamma |y_1\left|\right(|z_1{|}^{\beta -1}+|z_1{|}^{\beta -2}|z_2|+\dots +|z_1\left|\right|z_2{|}^{\beta -2}+|z_2{|}^{\beta -1})\right)|z_1-z_2|+\gamma |z_2{|}^{\beta }|y_1-y_2|.\hfill \end{array}$$

Assume that $E={\mathcal{B}}_r:={\{(y,z):|(y,z)|}_2\le r\}$ and that there exists $0<n\le \beta$. Then, we have the following:

$$\begin{array}{ccc}\hfill {\mathcal{I}}_1+{\mathcal{I}}_2& \le & \left({\displaystyle \frac{1}{\rho }}+2\gamma |y_1\left|\right(|z_1{|}^{\beta -1}+|z_1{|}^{\beta -2}|z_2|+\dots +|z_1\left|\right|z_2{|}^{\beta -2}+|z_2{|}^{\beta -1})\right)|z_1-z_2|+\gamma |z_2{|}^{\beta }|y_1-y_2|\hfill \\ & \le & \left({\displaystyle \frac{1}{\rho }}+2\gamma r^{n+1}\right)|z_1-z_2|+\gamma r^{\beta }|y_1-y_2|\hfill \\ & \le & P\left(\right|y_1-y_2|+|z_1-z_2\left|\right),\hfill \end{array}$$

where $P=max\{{\displaystyle \frac{1}{\rho }}+2\gamma r^{n+1},\gamma r^{\beta }\}$.

Hence, the proof is complete. □

6. Numerical Analysis

Here, we suggest a numerical analysis for the system described by the FDE (15), based on the Caputo definition.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathfrak{C}}D_t^{e_1}Y\left(t\right)=-\gamma Y{Z}^{\beta }\hfill \\ {}^{\mathfrak{C}}D_t^{e_2}Z\left(t\right)=\beta Y{Z}^{\beta }-{\displaystyle \frac{1}{\rho }}Z,\hfill \end{array}\right.\end{array}$$

(17)

where $e_1$ and $e_2$ are fractional orders. The Adams–Bashforth–Moulton predictor-corrector method, described in [25], was utilized to conduct numerical simulations on integer-order and fractional-order models. The simulation parameters were set as $\beta =1.95, \gamma =8.25$, and $\rho =500, \nu =4, {\nu }_{tol}={10}^{-6}, \mathfrak{h}=2^{-6},t\in [0,1000]$, with initial conditions $(Y_0,Z_0)=(0.25,0.75)$ and $(Y_0^{\prime },Z_0^{\prime })=(0.25,0.75)$ representing the first derivatives of Y and Z, respectively. The chosen parameters were based on Korobeinikov’s [19] financial crisis model, where $\beta$ is the mean value for a large population, which is a real number rather than an integer. Typically, $1<\beta <10$ for modern populations, with most populations having $1<\beta <10$. When $\beta \le 1$, the economy lacks a safety margin, making it susceptible to financial crises. Additionally, $\gamma$ and $\rho$ represent the market efficiency and the average time an activated unit affects other units, respectively. A higher $\Gamma =\gamma \rho$ product is indicative of a more stable economy. The selection of suitable orders $e_1$ and $e_2$ was facilitated by calculating the eigenvalues of Matrix A, associated with System (17), given by $A=\left[\begin{array}{cc}0& 0\\ 0& -{\displaystyle \frac{1}{\rho }}\end{array}\right]$ By applying the stability criterion of Ibrahim et al. [26], we find that $\beta <2$ ensures the asymptotic stability of System (17).

7. Conclusions

In this paper, we examine the stability and global attractivity of solutions for FDEs with the $\Psi$-Hilfer fractional derivative in the frame of an economic recession. We obtain some major findings on the global attractivity of solutions. We investigate the Ulam–Hyers–Rassias stability, involving the $\Psi$-Hilfer fractional derivative. By using Krasnoselskii’s fixed point theorem, we provide some solutions for FDEs in a Banach space. We discuss an important application on financial crises using the $\Psi$-Hilfer fractional derivative. The population we study is considered to be homogeneous. Furthermore, we assume that an average active unit must interact with activated units. To investigate the existence, uniqueness, and stability of solutions, a mathematical model is developed. Future studies can extend this work by considering non-homogeneous populations and incorporating additional factors that affect an economic recession. Moreover, the application of the $\Psi$-Hilfer fractional derivative can be explored in other fields, such as biology and physics. The results obtained in this paper can serve as a foundation for further research in the area of FDEs and their applications. Additionally, the theoretical findings presented in this paper can be validated through numerical simulations and empirical data analysis, providing a more comprehensive understanding of the subject matter.