Analysis of an ABC-Fractional Asset Flow Model for Financial Markets

Abstract

This paper proposes a novel fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative to analyze asset price dynamics in financial markets. Compared to classical models, the proposed model incorporates a nonlocal and non-singular fractional operator, allowing for a more accurate representation of investor behavior and market adjustment processes. The model captures both short-term trend-driven responses and long-term valuation-based decisions. We establish key theoretical properties of the system, including the existence and uniqueness of solutions, positivity, boundedness, and both local and global stability using Lyapunov functions. Numerical simulations under varying fractional orders demonstrate how the ABC derivative governs the convergence speed and equilibrium behavior of the system. Compared to classical integer-order models, the ABC-based approach provides smoother dynamics, greater flexibility in modeling behavioral heterogeneity, and better alignment with observed long-term financial phenomena.

1. Introduction

The financial and economic system is one of the most critical pillars of global stability and development. It governs the flow of capital, the allocation of resources, and the dynamics of consumption and production. Over the past century, the global economy has become increasingly interconnected, where disruptions in one market can rapidly cascade across borders. Events such as the 2008 global financial crisis [1,2], the COVID-19 economic shock [3,4], and recent banking instabilities [5] have highlighted how fragile and complex these systems can be. Understanding and predicting the behavior of financial systems is thus essential—not only for economists and policymakers but also for investors, regulators, and society at large.

To meet this challenge, mathematical modeling has become an indispensable tool for exploring the mechanisms underlying financial systems. These models help capture nonlinear behaviors such as asset bubbles, liquidity dry-ups, and sudden crashes. Among various approaches, the asset flow differential equations (AFDEs) have gained prominence for their ability to model asset prices through demand–supply feedback and heterogeneous trader behaviors. Initially introduced by Caginalp and co-authors, AFDEs consider both trend-following and value-based investor strategies, incorporating real-world behavioral effects that drive asset price fluctuations [6,7].

Unlike classical equilibrium models, AFDEs generate dynamic, sometimes chaotic trajectories that reflect the endogenous instability of markets. These models have been successfully used to study speculative bubbles, momentum effects, and liquidity crises [8,9,10]. Enhancements to the basic AFDE framework have incorporated time delays, behavioral inertia, and heterogeneity, broadening their relevance to modern financial applications [11,12].

However, many existing asset flow differential equation (AFDE) models are formulated using integer-order derivatives, which may not adequately reflect the memory effects typically observed in financial markets. In practice, financial behavior often exhibits long-term dependence, where current dynamics are influenced by an extended history of past states. To address this, various studies have explored the use of fractional-order differential equations (FDEs), which naturally incorporate memory and hereditary properties [13,14,15,16]. For instance, Olayiwola et al. [17] applied a fractional-order Rössler system to examine the effects of interest rates and investment demand, while Chauhan et al. [18] investigated a financial system with memory-driven dynamics under external influences.

There are various type of fractional-order derivatives such as Riemann–Liouville [19,20], Liouville–Caputo [21,22,23], Hadamard [24], Caputo–Fabrizio [25,26], and tempered [27,28] fractional derivatives. Fractional derivatives such as the Caputo and Caputo–Fabrizio (CF) operators have been widely used in modeling real-world systems with memory and hereditary effects. The Caputo derivative, though classical, involves a singular power-law kernel, which can lead to difficulties in capturing smooth transition behavior and may introduce numerical instabilities near the origin. The CF derivative addressed this by introducing an exponential kernel, which is non-singular and provides improved numerical behavior; however, it lacks the flexibility to accurately capture both short- and long-term memory effects simultaneously. However, the Atangana–Baleanu–Caputo (ABC) derivative [29] employs a non-singular, nonlocal kernel based on the Mittag–Leffler function, which offers a smooth interpolation between short-memory and long-memory behavior. This feature makes the ABC derivative particularly advantageous for modeling systems where memory effects persist but decay gradually, such as in economic and financial markets.

Among the recent advancements in fractional calculus, the Atangana–Baleanu–Caputo (ABC) derivative has emerged as a powerful modeling tool. It employs a non-singular and nonlocal Mittag–Leffler kernel, offering a more realistic and smooth representation of memory decay [29,30]. This makes ABC derivatives especially suitable for economic and financial systems, where market responses are shaped not by isolated events but by cumulative past experiences. The ABC derivative is widely used in many mathematical models to explain real phenomena. Rehman et al. [31] proposed a fractional financial model based on the ABC derivative to express the intricate phenomena in finance. Wang and Khan [32] studied the dynamics of the competition between rural and commercial banks by using the fractal-fractional ABC derivative. Liping et al. [33] improved the financial chaotic model using the Atangana–Baleanu stochastic fractional derivative to generalize the model. They also provided the method that can find the numerical solution of the proposed model. Farman et al. [34] constructed the fractional-order model of disorderly finance system by adding the critical minimum interest rate in ABC derivative sense. Li et al. [35] studied the dynamics of the emerging three-compartment financial bubble problem in the ABC derivative operator to study the financial bubbles. Thabet et al. [36] proposed a mathematical model based on the generalized ABC derivative to study the behaviors of bubbles and collapses in the financial bubbles.

Motivated by the need to better capture memory effects in financial systems, this study introduces a fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative. Unlike classical or Caputo-type formulations, the ABC derivative features a nonlocal and non-singular kernel, allowing for a more realistic representation of persistent behavioral influences and gradual market adjustments. The proposed model extends existing frameworks by incorporating this operator into the asset flow structure and by addressing key theoretical properties of the system. We establish the existence and uniqueness of solutions using fixed point theory [37,38], and further demonstrate the positivity and boundedness of all state variables to ensure their economic interpretability. In addition, we identify the model’s equilibrium points, analyze their local stability, and prove global asymptotic stability through a Lyapunov-based approach. Numerical simulations are provided to support the analytical results and to illustrate how variations in the fractional order affect convergence behavior and system dynamics. These contributions offer a more flexible and mathematically rigorous approach to modeling asset market behavior under memory-dependent dynamics.

2. Fractional Calculus

In this section, we introduce the Atangana–Baleanu–Caputo fractional derivative that is used to improve the mathematical model of asset flow differential equations [6]. The basic definitions and theories involving ABC-fractional calculus are given in the following.

ABC-Fractional Derivative

Definition 1.

Let $g\in C^1(a,b),a<b,$ be a function, and let $\xi \in [0,1]$. The Atangana–Baleanu fractional derivative in Caputo-type of order ξ is given by [29,39]

$${}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(t\right)={\displaystyle \frac{\Psi \left(\xi \right)}{1-\xi }}{\int }_a^t{\displaystyle \frac{dg}{ds}}{E}_{\xi }\left[-{\displaystyle \frac{\xi }{1-\xi }}{(t-s)}^{\xi }\right]ds,$$

(1)

where $\Psi \left(\xi \right)$ is the normalization function given by $\Psi \left(\xi \right)=1-\xi +\xi /\Gamma \left(\xi \right)$, characterized by $\Psi \left(0\right)=\Psi \left(1\right)=1,$ and the Mittag–Leffler function $E_{\xi }\left(z\right)$ with $\mathbb{C}$, the set of the complex number, is given by

$$E_{\xi }\left(z\right)=\sum _{\beta =0}^{\infty }{\displaystyle \frac{z^{\beta }}{\Gamma (1+\xi \beta )}},\xi ,z\in \mathbb{C},Re\left(z\right)>0.$$

Definition 2.

The ABC-fractional integral of the function $g\in C^1(a,b)$ is given by [29,39]

$${}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}I_t^{\xi }g\left(t\right)={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}g\left(t\right)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_a^{t}g\left(s\right){(t-s)}^{\xi -1}ds.$$

(2)

Lemma 1

([40]). The ABC-fractional derivative and ABC-fractional integral of the function $g\in C^1(a,b)$ satisfies the Newton–Leibniz equality

$${}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}I_t^{\xi }\left({}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(t\right)\right)=g\left(t\right)-g\left(a\right).$$

Lemma 2

([29,41]). For two functions $f,g\in {\Lambda }^1(a,b),b>a,$ the ABC-fractional derivative satisfies the following inequality

$$\Vert {}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }f\left(t\right)-{}_{\mathit{a\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(t\right)\Vert \le \Lambda \Vert f\left(t\right)-g\left(t\right)\Vert .$$

Lemma 3

(Banach’s fixed point theorem [42]). Let D be a non-empty closed subset of a Banach space E. Then any contraction mapping Q from D into itself has a unique fixed point.

3. Mathematical Model

An improved mathematical model is based on Gaginalp et al. [43], which proposed the dynamical system of AFDEs to describe those four state variables in the market price:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dP}{dt}}& =P\left(t\right)F\left({\displaystyle \frac{D}{S}}\right),\hfill \\ \hfill {\displaystyle \frac{dI}{dt}}& =k\left(t\right)(1-I\left(t\right))-(1-k\left(t\right))I\left(t\right)+I\left(t\right)(1-I\left(t\right))F\left({\displaystyle \frac{D}{S}}\right),\hfill \\ \hfill {\displaystyle \frac{dZ}{dt}}& =a_1\left(a_{2}F\left({\displaystyle \frac{D}{S}}\right)-Z\left(t\right)\right),\hfill \\ \hfill {\displaystyle \frac{dV}{dt}}& =a_3\left(a_4{\displaystyle \frac{P_f-P\left(t\right)}{P_f}}-V\left(t\right)\right).\hfill \end{array}$$

(3)

In the above, $P\left(0\right)=P_0,I\left(0\right)=I_0,Z\left(0\right)=Z_0,$ and $V\left(0\right)=V_0$ are positive constants.

The market price of an asset at time t, denoted by $P\left(t\right)$, is assumed to be strictly positive and bounded above, meaning it cannot tend toward infinity. The quantity $I\left(t\right)$, representing the fraction of the total asset invested in the stock market, is restricted to the closed interval $[0,1]$. The trend-based component of the investor preference at time is defined by $Z\left(t\right)$ and the value-based component of the investor preference at time is defined by $V\left(t\right)$ [10,44].

The demand and supply in the model are defined as follows:

$$D\left(t\right)=k\left(t\right)(1-I(t\left)\right),S\left(t\right)=(1-k(t\left)\right)I\left(t\right),$$

where $k\left(t\right)$ is the transition rate between demanders and suppliers, defined by:

$$k\left(t\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh(Z\left(t\right)+V\left(t\right)).$$

The price adjustment function F is the function of demand and supply [9,43]. For this work, we define function $F\left(x\right)=log\left({\displaystyle \frac{1}{2}}\left(1+exp\{x-1\}\right)\right)$ as a continuous, strictly increasing function that satisfies $F\left(1\right)=0$, ensuring that when demand equals supply, the market price remains unchanged.

Moreover, all parameters $P_f,a_1,a_2,a_3,a_4$ in the model are assumed to be positive real numbers.

In this article, the fractional-order asset flow differential equations (FAFDEs) based on the Atangana–Baleanu–Caputo-fractional derivative are introduced as follows:

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P\left(t\right)& =G_1(t,P),\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I\left(t\right)& =G_2(t,I),\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z\left(t\right)& =G_3(t,Z),\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\left(t\right)& =G_4(t,V).\hfill \end{array}$$

(4)

The kernels are given by the following:

$$\begin{array}{cc}\hfill G_1(t,P)& =P\left(t\right)F\left(H\right(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right)\left)\right),\hfill \\ \hfill G_2(t,I)& =k(t,P(t),I(t),Z(t),V(t\left)\right)-I\left(t\right)+I\left(t\right)(1-I(t\left)\right)F\left(H\right(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right)\left)\right),\hfill \\ \hfill G_3(t,Z)& =b_{1}F\left(H(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right))\right)-b_{2}Z\left(t\right),\hfill \\ \hfill G_4(t,V)& =b_3-b_{4}P\left(t\right)-b_{5}V\left(t\right).\hfill \end{array}$$

(5)

In the above, ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }$ is the Atangana–Baleanu–Caputo fractional derivative of order $0<\xi <1,b_i$ are positive constants for $i=1,2,3,4,5$, and $P\left(0\right)=P_0,I\left(0\right)=I_0,Z\left(0\right)=Z_0,$ and $V\left(0\right)=V_0$ are positive constants.

By comparing to classical AFDE models, the proposed ABC-fractional model introduces several important improvements. The proposed model incorporates a non-singular, nonlocal memory kernel, allowing it to more accurately reflect the cumulative influence of past market states. Classical models assume instantaneous adjustment based solely on current variables, while real financial systems often exhibit delayed and persistent reactions—features that are naturally captured by the fractional-order framework. The ABC-fractional formulation offers greater flexibility in modeling both short-term and long-term investor behavior. The inclusion of a trend-based component $Z\left(t\right)$ and a value-based component $V\left(t\right)$, modulated by the fractional order $\xi$, enables the model to differentiate between transient speculative responses and slowly adjusting valuation strategies.

4. Model Analysis

4.1. Existence of the Solution

To show the existence of the solution, we proceed as follows. Applying the ABC-fractional integral to model (5), we have

$$\begin{array}{cc}\hfill P\left(t\right)& =P\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t,P)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_1(s,P){(t-s)}^{\xi -1}ds,\hfill \\ \hfill I\left(t\right)& =I\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_2(t,I)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_2(s,I){(t-s)}^{\xi -1}ds,\hfill \\ \hfill Z\left(t\right)& =Z\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_3(t,Z)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_3(s,Z){(t-s)}^{\xi -1}ds,\hfill \\ \hfill V\left(t\right)& =V\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_4(t,V)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_4(s,V){(t-s)}^{\xi -1}ds,\hfill \end{array}$$

(6)

or

$$\begin{array}{cc}\hfill P\left(t\right)-P\left(0\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t,P)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_1(s,P){(t-s)}^{\xi -1}ds,\hfill \\ \hfill I\left(t\right)-I\left(0\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_2(t,I)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_2(s,I){(t-s)}^{\xi -1}ds,\hfill \\ \hfill Z\left(t\right)-Z\left(0\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_3(t,Z)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_3(s,Z){(t-s)}^{\xi -1}ds,\hfill \\ \hfill V\left(t\right)-V\left(0\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_4(t,V)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_4(s,V){(t-s)}^{\xi -1}ds.\hfill \end{array}$$

(7)

Let set $B={\left(C\left(\mathcal{J}\right)\right)}^4$ be the Banach space of real-valued continuous functions defined on an interval $\mathcal{J}=[0,T]$ with the corresponding norm defined by $\parallel (P,I,Z,V)\parallel =\parallel P\parallel +\parallel I\parallel +\parallel Z\parallel +\parallel V\parallel ,$ where the following hold:

$$\begin{array}{cc}\hfill \parallel P\parallel & =\underset{t\in \mathcal{J}}{sup}\left|P\left(t\right)\right|={\Omega }_1,\hfill \\ \hfill \parallel I\parallel & =\underset{t\in \mathcal{J}}{sup}\left|I\left(t\right)\right|={\Omega }_2,\hfill \\ \hfill \parallel Z\parallel & =\underset{t\in \mathcal{J}}{sup}\left|Z\left(t\right)\right|={\Omega }_3,\hfill \\ \hfill \parallel V\parallel & =\underset{t\in \mathcal{J}}{sup}\left|V\left(t\right)\right|={\Omega }_4.\hfill \end{array}$$

(A1)

The function $F\left(x\right)=log\left({\displaystyle \frac{1}{2}}(1+exp\{x-1\})\right)$ is increasing and continuously differentiable, satisfies $F\left(1\right)=0$, and there exists a constant $L_1>0$ such that for all $p,q\in [0,\infty )$,

$$|F\left(p\right)-F\left(q\right)|\le L_1|p-q|.$$

(A2)

Let the market adjustment function be defined by

$$k\left(t\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh(Z\left(t\right)+V\left(t\right)),$$

where $Z\left(t\right)$ and $V\left(t\right)$ are continuous functions representing investor preferences. Then $k:(0,\infty )\to (0,1)$ is continuously differentiable with respect to time t.

Define the function

$$H(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right))={\displaystyle \frac{k\left(t\right)(1-I(t\left)\right)}{(1-k(t\left)\right)I\left(t\right)}},$$

for $I\left(t\right)\in (0,1)$. Then $H:(0,\infty )\times (0,1)\to [0,\infty )$ is continuous, and there exists a constant $L_2>0$ such that for all $t>0$, and all $(P_1,I_1,Z_1,V_1),(P_2,I_2,Z_2,V_2)\in {\mathbb{R}}^4$,

$$|H(t,P_1,I_1,Z_1,V_1)-H(t,P_2,I_2,Z_2,V_2)|\le L_2\sum _{i=1}^4|{\theta }_i^{\left(1\right)}-{\theta }_i^{\left(2\right)}|,$$

where $({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)=(P,I,Z,V)$.

Theorem 1

(Lipschitz condition and contraction). For each of the kernels $G_1,G_2,G_3,G_4$ in (5), there exists $L_i>0,i=3,4,5,6$ such that the following hold:

$$\begin{array}{cc}\hfill \parallel G_1(t,P)-G_1(t,P_1)\parallel & \le L_3\parallel P\left(t\right)-P_1\left(t\right)\parallel ,\hfill \\ \hfill \parallel G_2(t,I)-G_2(t,I_1)\parallel & \le L_4\parallel I\left(t\right)-I_1\left(t\right)\parallel ,\hfill \\ \hfill \parallel G_3(t,Z)-G_3(t,Z_1)\parallel & \le L_5\parallel Z\left(t\right)-Z_1\left(t\right)\parallel ,\hfill \\ \hfill \parallel G_4(t,V)-G_4(t,V_1)\parallel & \le L_6\parallel V\left(t\right)-V_1\left(t\right)\parallel .\hfill \end{array}$$

And the above are contractions for $0\le L_i<1,i=3,4,5,6.$

The following also hold:

$$\begin{array}{cc}\hfill L_3& =log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))(1-{\Omega }_2)}{(1-(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))){\Omega }_2}}-1\}\right)\right),\hfill \\ \hfill L_4& =(1+2{\Omega }_2)C_F+(1+{\Omega }_2){\Omega }_2{L}_1{L}_2,\hfill \\ \hfill L_5& =b_1{L}_1{L}_2+b_2,\hfill \\ \hfill L_6& =b_5.\hfill \end{array}$$

Proof. 

$$\begin{array}{cc}\hfill \parallel G_1(t,P)-G_1(t,P_1)\parallel & =∥P\left[log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh(Z+V))(1-I)}{(1-(\frac{1}{2}+\frac{1}{2}tanh(Z+V)))I}}-1\}\right)\right)\right]\hfill \\ \hfill & -P_1\left[log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh(Z+V))(1-I)}{(1-(\frac{1}{2}+\frac{1}{2}tanh(Z+V)))I}}-1\}\right)\right)\right]∥\hfill \\ \hfill & =∥log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh(Z+V))(1-I)}{(1-(\frac{1}{2}+\frac{1}{2}tanh(Z+V)))I}}-1\}\right)\right)(P-P_1)∥\hfill \\ \hfill & \le \left(log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))(1-{\Omega }_2)}{(1-(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))){\Omega }_2}}-1\}\right)\right)\right)\parallel P-P_1\parallel \hfill \\ \hfill & =L_3\parallel P-P_1\parallel ,\hfill \end{array}$$

where $L_3=log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))(1-{\Omega }_2)}{(1-(\frac{1}{2}+\frac{1}{2}tanh({\Omega }_3+{\Omega }_4))){\Omega }_2}}-1\}\right)\right).$

$$\begin{array}{cc}\hfill \parallel G_2(t,I)-G_2(t,I_1)\parallel & =∥{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh(Z+V)-I+I(1-I)F\left(H\left(I\right)\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh(Z+V)-I_1+I_1(1-I_1)F\left(H\left(I_1\right)\right)\right)∥\hfill \\ \hfill & =\parallel -I+I(1-I)F\left(H\left(I\right)\right)-(-I_1+I_1(1-I_1)F\left(H\left(I_1\right)\right))\parallel \hfill \\ \hfill & =\parallel -I+IF\left(H\left(I\right)\right)-I^{2}F\left(H\left(I\right)\right)+I_1-I_{1}F\left(H\left(I_1\right)\right)+I_1^{2}F\left(H\left(I_1\right)\right)\parallel \hfill \\ \hfill & \le \parallel I-I_1\parallel +\parallel I-I_1\parallel \parallel F\left(H\left(I_1\right)\right)\parallel +\parallel I\parallel \parallel F\left(H\left(I\right)\right)-F\left(H\left(I_1\right)\right)\parallel \hfill \\ \hfill & +\parallel I^2\parallel \parallel F\left(H\left(I\right)\right)-F\left(H\left(I_1\right)\right)\parallel +\parallel I^2-I_1^2\parallel \parallel F\left(H\left(I_1\right)\right)\parallel \hfill \\ \hfill & \le \parallel I-I_1\parallel +\parallel I-I_1\parallel C_F+{\Omega }_2[L_1{L}_2\parallel I-I_1\parallel ]\hfill \\ \hfill & +{\Omega }_2^2[L_1{L}_2\parallel I-I_1\parallel ]+\parallel I-I_1\parallel \parallel I+I_1\parallel C_F\hfill \\ \hfill & \le [(1+2{\Omega }_2)C_F+(1+{\Omega }_2){\Omega }_2{L}_1{L}_2]\parallel I-I_1\parallel \hfill \\ \hfill & =L_4\parallel I-I_1\parallel ,\hfill \end{array}$$

where $L_4=(1+2{\Omega }_2)C_F+(1+{\Omega }_2){\Omega }_2{L}_1{L}_2.$

$$\begin{array}{cc}\hfill \parallel G_3(t,Z)-G_3(t,Z_1)\parallel & =\parallel b_{1}F\left(H\left(Z\right)\right)-b_{2}Z-(b_{1}F\left(H\left(Z_1\right)\right)-b_2{Z}_1)\parallel \hfill \\ \hfill & =\parallel b_1[F\left(H\left(Z\right)\right)-F\left(H\left(Z_1\right)\right)]+b_2(Z-Z_1)\parallel \hfill \\ \hfill & \le \parallel b_1[F\left(H\left(Z\right)\right)-F\left(H\left(Z_1\right)\right)]\parallel +\parallel b_2(Z-Z_1)\parallel \hfill \\ \hfill & \le b_1{L}_1{L}_2\parallel Z-Z_1\parallel +b_2\parallel Z-Z_1\parallel \hfill \\ \hfill & =(b_1{L}_1{L}_2+b_2)\parallel Z-Z_1\parallel \hfill \\ \hfill & =L_5\parallel Z-Z_1\parallel ,\hfill \end{array}$$

where $L_5=b_1{L}_1{L}_2+b_2.$

$$\begin{array}{cc}\hfill \parallel G_4(t,V)-G_4(t,V_1)\parallel & =\parallel b_3-b_{4}P-b_{5}V-(b_3-b_{4}P-b_5{V}_1)\parallel \hfill \\ \hfill & =\parallel b_5(V_1-V)\parallel \hfill \\ \hfill & \le b_5\parallel V-V_1\parallel \hfill \\ \hfill & =L_6\parallel V-V_1\parallel ,\hfill \end{array}$$

where $L_6=b_5.$

If $0\le L_i<1,i=1,2,...,6$, then $G_j(t,X_j),j=1,2,3,4$ is a contraction.

Now for $t=t_n,n=1,2,...$ define the following recursive form of (7)

$$\begin{array}{cc}\hfill P_n\left(t\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t,P_{n-1})+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_1(s,P_{n-1}){(t-s)}^{\xi -1}ds,\hfill \\ \hfill I_n\left(t\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_2(t,I_{n-1})+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_2(s,I_{n-1}){(t-s)}^{\xi -1}ds,\hfill \\ \hfill Z_n\left(t\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_3(t,Z_{n-1})+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_3(s,Z_{n-1}){(t-s)}^{\xi -1}ds,\hfill \\ \hfill V_n\left(t\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_4(t,V_{n-1})+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_4(s,V_{n-1}){(t-s)}^{\xi -1}ds,\hfill \end{array}$$

(8)

with the initial conditions $P_0\left(t\right)=P\left(0\right),I_0\left(t\right)=I\left(0\right),Z_0\left(t\right)=Z\left(0\right),$ and $V_0\left(t\right)=V\left(0\right).$

The differences between successive terms in (8) are expressed as follows:

$$\begin{array}{cc}\hfill {\Phi }_{1n}\left(t\right)& =P_n\left(t\right)-P_{n-1}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}(G_1(t,P_{n-1})-G_1(t,P_{n-2}))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t(G_1(t,P_{n-1})-G_1(t,P_{n-2})){(t-s)}^{\xi -1}ds,\hfill \\ \hfill {\Phi }_{2n}\left(t\right)& =I_n\left(t\right)-I_{n-1}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}(G_2(t,I_{n-1})-G_2(t,I_{n-2}))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t(G_2(t,I_{n-1})-G_2(t,I_{n-2})){(t-s)}^{\xi -1}ds,\hfill \\ \hfill {\Phi }_{3n}\left(t\right)& =Z_n\left(t\right)-Z_{n-1}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}(G_3(t,Z_{n-1})-G_3(t,Z_{n-2}))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t(G_3(t,Z_{n-1})-G_3(t,Z_{n-2})){(t-s)}^{\xi -1}ds,\hfill \\ \hfill {\Phi }_{4n}\left(t\right)& =V_n\left(t\right)-V_{n-1}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}(G_4(t,V_{n-1})-G_4(t,V_{n-2}))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t(G_4(t,V_{n-1})-G_4(t,V_{n-2})){(t-s)}^{\xi -1}ds,\hfill \end{array}$$

(9)

Taking the norm on both side of each equation in (9), we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{1n}\left(t\right)\parallel & =\parallel P_n\left(t\right)-P_{n-1}\left(t\right)\parallel \hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel (G_1(t,P_{n-1})-G_1(t,P_{n-2}))\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel (G_1(t,P_{n-1})-G_1(t,P_{n-2}))\parallel {(t-s)}^{\xi -1}ds,\hfill \\ \hfill \parallel {\Phi }_{2n}\left(t\right)\parallel & =\parallel I_n\left(t\right)-I_{n-1}\left(t\right)\parallel \hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel (G_2(t,I_{n-1})-G_2(t,I_{n-2}))\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel (G_2(t,I_{n-1})-G_2(t,I_{n-2}))\parallel {(t-s)}^{\xi -1}ds,\hfill \\ \hfill \parallel {\Phi }_{3n}\left(t\right)\parallel & =\parallel Z_n\left(t\right)-Z_{n-1}\left(t\right)\parallel \hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel (G_3(t,Z_{n-1})-G_3(t,Z_{n-2}))\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel (G_3(t,Z_{n-1})-G_3(t,Z_{n-2}))\parallel {(t-s)}^{\xi -1}ds,\hfill \\ \hfill \parallel {\Phi }_{4n}\left(t\right)\parallel & =\parallel V_n\left(t\right)-V_{n-1}\left(t\right)\parallel \hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel (G_4(t,V_{n-1})-G_4(t,V_{n-2}))\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel (G_4(t,V_{n-1})-G_4(t,V_{n-2}))\parallel {(t-s)}^{\xi -1}ds,\hfill \end{array}$$

(10)

Furthermore, the first equality in (10) can be reduced to the following expressions:

$$\begin{array}{cc}\hfill \parallel {\Phi }_{1n}\left(t\right)\parallel & =\parallel P_n\left(t\right)-P_{n-1}\left(t\right)\parallel \hfill \\ \hfill & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel (G_1(t,P_{n-1})-G_1(t,P_{n-2}))\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel (G_1(t,P_{n-1})-G_1(t,P_{n-2}))\parallel {(t-s)}^{\xi -1}ds,\hfill \\ \hfill & \le {\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{L}_3\parallel P_{n-1}-P_{n-2}\parallel +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{L}_3{\int }_0^t\parallel P_{n-1}-P_{n-2}\parallel {(t-s)}^{\xi -1}ds\hfill \\ \hfill & \le L_3\parallel {\Phi }_{1(n-1)}\left(t\right)\parallel \left|{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right|.\hfill \end{array}$$

As a result, we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{1n}\left(t\right)\parallel & \le L_3\left|{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right|\parallel {\Phi }_{1(n-1)}\left(t\right)\parallel .\hfill \end{array}$$

(11)

Analogously,

$$\begin{array}{cc}\hfill \parallel {\Phi }_{2n}\left(t\right)\parallel & \le L_4\left|{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right|\parallel {\Phi }_{2(n-1)}\left(t\right)\parallel ,\hfill \\ \hfill \parallel {\Phi }_{3n}\left(t\right)\parallel & \le L_5\left|{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right|\parallel {\Phi }_{3(n-1)}\left(t\right)\parallel ,\hfill \\ \hfill \parallel {\Phi }_{4n}\left(t\right)\parallel & \le L_6\left|{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right|\parallel {\Phi }_{4(n-1)}\left(t\right)\parallel .\hfill \end{array}$$

(12)

□

Theorem 2.

The ABC-fractional model given in (5) has a solution if we can find $T_0^{\xi }$ satisfying the inequality

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_i<1,i=3,4,5,6.\end{array}$$

Proof. 

From (11) and (12) we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{1n}\left(t\right)\parallel & \le \parallel P\left(0\right)\parallel {\left[\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_3\right]}^n\hfill \\ \hfill \parallel {\Phi }_{2n}\left(t\right)\parallel & \le \parallel I\left(0\right)\parallel {\left[\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_4\right]}^n\hfill \\ \hfill \parallel {\Phi }_{3n}\left(t\right)\parallel & \le \parallel Z\left(0\right)\parallel {\left[\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_5\right]}^n\hfill \\ \hfill \parallel {\Phi }_{4n}\left(t\right)\parallel & \le \parallel V\left(0\right)\parallel {\left[\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_6\right]}^n\hfill \end{array}$$

(13)

The existence of the solution is confirmed by Theorem 1 and we have to show that the functions $P\left(t\right),I\left(t\right),Z\left(t\right),$ and $V\left(t\right)$ are solutions of model (5).

Assume the following:

$$\begin{array}{cc}\hfill P\left(t\right)-P\left(0\right)& =P_n\left(t\right)-{\varphi }_{1n}\left(t\right),\hfill \\ \hfill I\left(t\right)-I\left(0\right)& =I_n\left(t\right)-{\varphi }_{2n}\left(t\right),\hfill \\ \hfill Z\left(t\right)-Z\left(0\right)& =Z_n\left(t\right)-{\varphi }_{3n}\left(t\right),\hfill \\ \hfill V\left(t\right)-V\left(0\right)& =V_n\left(t\right)-{\varphi }_{4n}\left(t\right).\hfill \end{array}$$

(14)

From (14) we obtain

$$\begin{array}{cc}\hfill \parallel {\varphi }_{1n}\left(t\right)\parallel & ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\parallel G_1(t,P_n)-G_1(t,P_{n-1})\parallel \hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\parallel G_1(t,P_n)-G_1(t,P_{n-1})\parallel {(t-s)}^{\xi -1}ds\hfill \\ \hfill & \le {\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{L}_3\parallel P_n-P_{n-1}\parallel +{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{L}_3\parallel P_n-P_{n-1}\parallel .\hfill \end{array}$$

Repeating the process recursively leads to

$$\begin{array}{c}\hfill \parallel {\varphi }_{1n}\left(t\right)\parallel \le {\left[{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right]}^{n+1}{L}_3^n{\parallel P_n-P_{n-1}\parallel }^n,\end{array}$$

which at $t=T_0^{\xi }$ yields

$$\begin{array}{cc}\hfill \parallel {\varphi }_{1n}\left(t\right)\parallel & \le {\left[{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{T_0^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right]}^{n+1}{L}_3^n{\parallel P_n-P_{n-1}\parallel }^n,\hfill \\ \hfill \parallel {\varphi }_{1n}\left(t\right)\parallel & \to 0.\hfill \end{array}$$

(15)

Applying the limit to both sides of (15) as $n\to \infty ,$ we see that $\parallel {\varphi }_{1n}\left(t\right)\parallel \to 0$ for

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_3<1.\end{array}$$

Similarly, we can show that $\parallel {\varphi }_{2n}\left(t\right)\parallel \to 0,\parallel {\varphi }_{3n}\left(t\right)\parallel \to 0,\parallel {\varphi }_{4n}\left(t\right)\parallel \to 0,$

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_i<1,i=4,5,6.\end{array}$$

Theorem 1 and 2 guarantee the existence of the solution of model (5) by the Banach fixed point theorem. □

Theorem 3

(Uniqueness of solution). The ABC-fractional model (5) has a unique solution, provided that

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_i<1,i=3,4,5,6.\end{array}$$

Proof. 

Assume that $P_1\left(t\right),I_1\left(t\right),Z_1\left(t\right),$ and $V_1\left(t\right)$ are also solutions to (5). Then

$$\begin{array}{cc}\hfill P\left(t\right)-P_1\left(t\right)& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}\left(G_1(t,P)\right)-G_1(t,P_1))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t\left(G_1(t,P)\right)-G_1(t,P_1)){(t-s)}^{\xi -1}ds.\hfill \end{array}$$

Taking the norm of both sides, we obtain

$$\begin{array}{c}\hfill \parallel P\left(t\right)-P_1\left(t\right)\parallel \le {\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{L}_3\parallel P-P_1\parallel +{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{L}_3\parallel P-P_1\parallel .\end{array}$$

That is

$$\begin{array}{c}\hfill \parallel P\left(t\right)-P_1\left(t\right)\parallel \left(1-\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_3\right)\le 0.\end{array}$$

Sine $\left(1-\left({\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}+{\displaystyle \frac{t^{\xi }}{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\right)L_3\right)>0,$ we obtain $\parallel P\left(t\right)-P_1\left(t\right)\parallel =0$. Thus we have $P\left(t\right)=P_1\left(t\right)$.

Similarly, we can show that $I\left(t\right)=I_1\left(t\right),Z\left(t\right)=Z_1\left(t\right),$ and $V\left(t\right)=V_1\left(t\right),$ which completes the proof of Theorem 3. □

4.2. Positivity and Boundedness of the Solution

Lemma 4

(Gronwall–Bellman inequality). Suppose $\gamma >0,a\left(t\right)$ is a nonnegative function that is locally integrable on $0\le t<T\le \infty ,$ and $g\left(t\right)$ is a nonnegative, nondecreasing continuous bounded function defined on $0\le t<T.$ If $u\left(t\right)$ is nonnegative and locally integrable on $0\le t<T$ with

$$\begin{array}{c}\hfill u\left(t\right)\le a\left(t\right)+g\left(t\right){\int }_0^t{(t-s)}^{\gamma -1}u\left(s\right)ds\end{array}$$

on this interval, then

$$\begin{array}{c}\hfill u\left(t\right)\le a\left(t\right)+g\left(t\right){\int }_0^t\left[\sum _{n=1}^{\infty }{\displaystyle \frac{{(g\left(t\right)\Gamma \left(\gamma \right))}^n}{\Gamma \left(n\gamma \right)}}{(t-s)}^{n\gamma -1}a\left(s\right)\right]ds.\end{array}$$

Furthermore, if a(t) is nondecreasing on $0\le t<T,$ then

$$\begin{array}{c}\hfill u\left(t\right)\le a\left(t\right)E_{\gamma }(g\left(t\right)\Gamma \left(\gamma \right)t^{\gamma }),\end{array}$$

where $E_{\gamma }\left(z\right)$ is the Mittag–Leffler function defined by

$$\begin{array}{c}\hfill E_{\gamma }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (k\gamma +1)}},forz>0.\end{array}$$

Lemma 5

(Generalized mean value theorem, [45]). Let $g\left(x\right)\in C[a,b],$ and let ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(x\right)\in C[a,b]$ when $0<\xi \le 1$. Then we have $g\left(x\right)=g\left(a\right)+{\displaystyle \frac{1}{\Gamma \left(\xi \right)}}{}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(\eta \right){(x-a)}^{\xi }$, when $0\le \eta \le x$,$\forall x\in (a,b]$.

Let

$$\begin{array}{c}\hfill X\left(t\right)=(X_1\left(t\right),X_2\left(t\right),X_3\left(t\right),X_4\left(t\right))=(P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right)).\end{array}$$

Note that by Lemma 5, if $g\left(x\right)\in [0,b],{}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(x\right)\in (0,b]$, and ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(x\right)\ge 0,\forall x\in (0,b]$ when $0<\xi \le 1$, then the function $g\left(x\right)$ is nondecreasing, and if ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }g\left(x\right)\le 0,\forall x\in (0,b]$, then the function $g\left(x\right)$ is nonincreasing $\forall x\in [0,b]$. To show that $X\left(t\right)$ is positively invariant, using Lemma 5, we have

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }{P\left(t\right)|}_{P=0}& =0\ge 0,\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }{I\left(t\right)|}_{I=0}& =k(t,P(t),I(t),Z(t),V(t\left)\right)\ge 0,\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }{Z\left(t\right)|}_{Z=0}& =b_{1}F\left(H(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right))\right)\ge 0,\mathrm{if}D\ge S\hfill \\ \hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }{V\left(t\right)|}_{V=0}& =b_3-b_{4}P\left(t\right)\ge 0, \mathrm{if}{b}_3\ge b_{4}P\left(t\right).\hfill \end{array}$$

(16)

Theorem 4.

The ABC-fractional system is

$$\begin{array}{c}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }{X}_i\left(t\right)=G_i(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right)),i=1,2,3,4\end{array}$$

and is subject to the initial conditions $X\left(0\right)\ge 0$.

Assume that $G_i\left(t\right),P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right))\ge 0$ whenever $X\left(t\right)=0;$ there exists a constant $C_i>0$ such that

$$\begin{array}{c}\hfill |G_i(t,P\left(t\right),I\left(t\right),Z\left(t\right),V\left(t\right))|\le C_i(1+|X_i\left(t\right)\left|\right)\mathit{for}\mathit{all}t\ge 0.\end{array}$$

Then, the solutions $X_i\left(t\right)$ satisfy

$$\begin{array}{c}\hfill 0\le X_i\le {\displaystyle \frac{|X_i\left(0\right)|+D_0}{1-D_1}}{E}_{\xi }\left({\displaystyle \frac{D_2}{1-D_1}}{t}^{\xi }\right)\mathit{for}\mathit{all}t\ge 0,\end{array}$$

where

$$\begin{array}{cc}\hfill D_0& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)\Gamma (\xi +1)}}{C}_i{t}^{\xi },\hfill \\ \hfill D_1& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i,\hfill \\ \hfill D_2& ={\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{C}_i.\hfill \end{array}$$

Proof. 

Start from the integral form of the ABC-fractional differential equation:

$$\begin{array}{c}\hfill X_i\left(t\right)=X_i\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_i(t,X\left(t\right))+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{(t-s)}^{\xi -1}{G}_i(s,X\left(s\right))ds.\end{array}$$

Sine Lemma 5, that is $X_i\left(t\right)\ge 0$ for all $t\ge 0$, and using the assumption that $|G_i(t,X\left(t\right))|\le C_i(1+|X_i\left(t\right)\left|\right)$, we obtain

$$\begin{array}{cc}\hfill |X_i\left(t\right)|& \le |X_i\left(0\right)|+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i(1+|X_i\left(t\right)\left|\right)+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{C}_i{\int }_0^t{(t-s)}^{\xi -1}(1+|X_i\left(s\right)\left|\right)ds\hfill \\ \hfill & = |X_i\left(0\right)|+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i\left|X_i\left(t\right)\right|+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)\Gamma (\xi +1)}}{C}_i{t}^{\xi }\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{C}_i{\int }_0^t{(t-s)}^{\xi -1}\left|X_i\left(s\right)\right|ds.\hfill \end{array}$$

Grouping terms, we obtain

$$\begin{array}{c}\hfill (1-D_1)|X_i\left(t\right)|\le |X_i\left(0\right)|+D_0+D_2{\int }_0^t{(t-s)}^{\xi -1}\left|X_i\left(s\right)\right|ds,\end{array}$$

where

$$\begin{array}{cc}\hfill D_0& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)\Gamma (\xi +1)}}{C}_i{t}^{\xi },\hfill \\ \hfill D_1& ={\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{C}_i,\hfill \\ \hfill D_2& ={\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{C}_i.\hfill \end{array}$$

Assuming $D_1<1,$ we divide through

$$\begin{array}{c}\hfill |X_i\left(t\right)|\le {\displaystyle \frac{|X_i\left(0\right)|+D_0}{1-D_1}}+{\displaystyle \frac{D_2}{1-D_1}}{\int }_0^t{(t-s)}^{\xi -1}\left|X_i\left(s\right)\right|ds\end{array}$$

by Lemma 4 we obtain

$$\begin{array}{c}\hfill |X_i\left(t\right)|\le {\displaystyle \frac{|X_i\left(0\right)|+D_0}{1-D_1}}{E}_{\xi }\left({\displaystyle \frac{D_2}{1-D_1}}{t}^{\xi }\right)\mathrm{for}\mathrm{all}t\ge 0.\end{array}$$

Thus, $X_i\left(t\right)$ remains positive and bounded as claimed. □

4.3. Equilibrium Points

We analyze the equilibrium points of the proposed fractional-order asset model, where all fractional derivatives vanish. These points represent steady states of the system and offer insight into its long-term behavior. To find the equilibrium point $C^{\ast }=(P^{\ast },I^{\ast },Z^{\ast },V^{\ast }),$ we set all equations of system (5) to zero:

$$\begin{array}{c}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P=0,{}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I=0,{}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z=0,{}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V=0.\end{array}$$

Then

$$\begin{array}{c}\hfill C_1^{\ast }=(P_1^{\ast },I_1^{\ast },Z_1^{\ast },V_1^{\ast }),\end{array}$$

where the following hold:

$$\begin{array}{cc}\hfill P_1^{\ast }& =P^{\ast }\hfill \\ \hfill I_1^{\ast }& =k\left(P^{\ast },I^{\ast },0,{\displaystyle \frac{b_3-b_4{P}^{\ast }}{b_5}}\right),\hfill \\ \hfill Z_1^{\ast }& =0,\hfill \\ \hfill V_1^{\ast }& ={\displaystyle \frac{b_3-b_4{P}^{\ast }}{b_5}}.\hfill \end{array}$$

$$\begin{array}{c}\hfill C_2^{\ast }=(P_2^{\ast },I_2^{\ast },Z_2^{\ast },V_2^{\ast }),\end{array}$$

where the following hold:

$$\begin{array}{cc}\hfill P_2^{\ast }& =0\hfill \\ \hfill I_2^{\ast }& ={\displaystyle \frac{(F^{\ast }-1)+\sqrt{4F^{\ast }k\left(0,I^{\ast },\frac{b_1}{b_2}{F}^{\ast },\frac{b_3}{b_4}\right)+{(F^{\ast }-1)}^2}}{2F^{\ast }}},\hfill \\ \hfill Z_2^{\ast }& ={\displaystyle \frac{b_1}{b_2}}{F}^{\ast },\hfill \\ \hfill V_2^{\ast }& ={\displaystyle \frac{b_3}{b_5}}.\hfill \end{array}$$

$$\begin{array}{c}\hfill C_3^{\ast }=(P_3^{\ast },I_3^{\ast },Z_3^{\ast },V_3^{\ast }),\end{array}$$

where the following hold:

$$\begin{array}{cc}\hfill P_3^{\ast }& =0\hfill \\ \hfill I_3^{\ast }& ={\displaystyle \frac{(F^{\ast }-1)-\sqrt{4F^{\ast }k\left(0,I^{\ast },\frac{b_1}{b_2}{F}^{\ast },\frac{b_3}{b_4}\right)+{(F^{\ast }-1)}^2}}{2F^{\ast }}},\hfill \\ \hfill Z_3^{\ast }& ={\displaystyle \frac{b_1}{b_2}}{F}^{\ast },\hfill \\ \hfill V_3^{\ast }& ={\displaystyle \frac{b_3}{b_5}}.\hfill \end{array}$$

4.4. Local Stability Analysis

In this subsection, we analyze the local stability of the equilibrium points of the fractional-order system by evaluating the Jacobian matrix at each equilibrium. The eigenvalues of the Jacobian matrix indicate whether small perturbations decay or grow over time. In the course of this analysis, a square root expression appears, and it is required that the quantity

$$\begin{array}{c}\hfill 4F^{\ast }k\left(0,I^{\ast },{\displaystyle \frac{b_1}{b_2}}{F}^{\ast },{\displaystyle \frac{b_3}{b_4}}\right)+{(F^{\ast }-1)}^2\ge 0,\end{array}$$

holds to ensure that the square root remains real and the solution remains within the domain of real numbers.

The Jacobian matrix of system (5) is computed by evaluating the partial derivatives of the right-hand side with respect to the state variables. It is given by

$$\begin{array}{c}\hfill J=\left[\begin{array}{cccc}F& P{F}_I& P{F}_Z& P{F}_V\\ 0& d_1& k_Z+(1-I)I{F}_Z& k_V+(1-I)I{F}_V\\ 0& b_1{F}_I& b_1{F}_Z-b_2& b_1{F}_V\\ -b_4& 0& 0& -b_5\end{array}\right]\end{array}$$

where the following hold:

$$\begin{array}{cc}{d}_1\hfill & =-1+I{F}_I+F-I^2{F}_I-2IF,\hfill \\ F\hfill & =log\left({\displaystyle \frac{1}{2}}\left(1+exp\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh(Z+V))(1-I)}{(1-(\frac{1}{2}+\frac{1}{2}tanh(Z+V)))I}}-1\}\right)\right),\hfill \\ k_Z\hfill & =k_V={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh(Z+V),\hfill \\ F_I\hfill & ={\displaystyle \frac{(1+tanh{(Z+V)}^2)exp{\displaystyle \left\{\frac{1-I-tanh(Z+V)}{I-2+tanh(Z+V)I}\right\}}}{{(I-2+tanh(Z+V))}^2\left(1+exp\left\{{\displaystyle \frac{1-I-tanh(Z+V)}{I-2+tanh(Z+V)I}}\right\}\right)}},\hfill \\ F_Z\hfill & =F_V={\displaystyle \frac{({(I-1)}^2+1)exp\left\{{\displaystyle \frac{1-I-tanh(Z+V)}{I-2+tanh(Z+V)I}}\right\}}{{((I-2)cosh(Z+V)+Isinh(Z+V))}^2\left(1+A\right)}}.\hfill \end{array}$$

Additionally,

$$\begin{array}{c}\hfill A=exp\left\{-{\displaystyle \frac{Icosh(Z+V)+sinh(Z+V)-cosh(Z+V)}{Isinh(Z+V)+Isinh(Z+V)-2cosh(Z+V)}}\right\}.\end{array}$$

We now consider the Jacobian matrix at the equilibrium point $C_1^{\ast },$ denoted by $J\left(C_1^{\ast }\right)$.

$$\begin{array}{c}\hfill J\left(C_1^{\ast }\right)=\left[\begin{array}{cccc}{F}_1^{\ast }& P^{\ast }{F}_{I_1}^{\ast }& P^{\ast }{F}_{Z_1}^{\ast }& P_1^{\ast }{F}_{V_1}^{\ast }\\ 0& d_1^{\ast }& k_{Z_1}^{\ast }+(1-I_1^{\ast })I_1^{\ast }{F}_{Z_1}^{\ast }& k_{V_1}^{\ast }+(1-I_1^{\ast })I_1^{\ast }{F}_{V_1}^{\ast }\\ 0& b_1{F}_{I_1^{\ast }}^{\ast }& b_1{F}_{Z_1}^{\ast }-b_2& b_1{F}_{V_1}^{\ast }\\ -b_4& 0& 0& -b_5\end{array}\right]\end{array}$$

(17)

where the following hold:

$$\begin{array}{cc}\hfill d_1^{\ast }& =-1+I_1^{\ast }{F}_{I_1}^{\ast }+F_1^{\ast }-{\left(I_1^{\ast }\right)}^2{F}_{I_1^{\ast }}^{\ast }-2I_1^{\ast }{F}_1^{\ast },\hfill \\ \hfill F_1^{\ast }& =log\left({\displaystyle \frac{1}{2}}\left(1+exp\left\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh\left(V_1^{\ast }\right))(1-I_1^{\ast })}{(1-(\frac{1}{2}+\frac{1}{2}tanh\left(V_1^{\ast }\right)))I_1^{\ast }}}-1\right\}\right)\right),\hfill \\ \hfill k_{Z_1}^{\ast }& =k_{V_1}^{\ast }={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left(V_1^{\ast }\right),\hfill \\ \hfill F_{I_1}^{\ast }& ={\displaystyle \frac{(1+tanh{\left(V_1^{\ast }\right)}^2)exp\left\{{\displaystyle \frac{1-I_1^{\ast }-tanh\left(V_1^{\ast }\right)}{I_1^{\ast }-2+tanh\left(V_1^{\ast }\right)I_1^{\ast }}}\right\}}{{(I_1^{\ast }-2+tanh\left(V_1^{\ast }\right))}^2\left(1+exp\left\{{\displaystyle \frac{1-I_1^{\ast }-tanh\left(V_1^{\ast }\right)}{I_1^{\ast }-2+tanh\left(V_1^{\ast }\right)I_1^{\ast }}}\right\}\right)}},\hfill \\ \hfill F_{Z_1}^{\ast }& =F_{V_1}^{\ast }={\displaystyle \frac{({(I_1^{\ast }-1)}^2+1)exp\left\{{\displaystyle \frac{1-I_1^{\ast }-tanh\left(V_1^{\ast }\right)}{I_1^{\ast }-2+tanh\left(V_1^{\ast }\right)I_1^{\ast }}}\right\}}{{((I_1^{\ast }-2)cosh\left(V_1^{\ast }\right)+I_1^{\ast }sinh\left(V_1^{\ast }\right))}^2\left(1+A_1^{\ast }\right)}},\hfill \end{array}$$

Additionally,

$$\begin{array}{c}\hfill A_1^{\ast }=exp\left\{-{\displaystyle \frac{I_1^{\ast }cosh\left(V_1^{\ast }\right)+sinh\left(V_1^{\ast }\right)-cosh\left(V_1^{\ast }\right)}{I_1^{\ast }sinh\left(V_1^{\ast }\right)+I_1^{\ast }sinh\left(V_1^{\ast }\right)-2cosh\left(V_1^{\ast }\right)}}\right\}.\end{array}$$

The characteristic equation of $J\left(C_1^{\ast }\right)$ is

$$\begin{array}{c}\hfill {\lambda }^4+{\rho }_1{\lambda }^3+{\rho }_2{\lambda }^2+{\rho }_3\lambda +{\rho }_4=0,\end{array}$$

(18)

where the following hold:

$$\begin{array}{cc}\hfill {\rho }_1& =b_2+b_5-d_1^{\ast }-F_1^{\ast }-b_1{F}_{Z_1}^{\ast },\hfill \\ \hfill {\rho }_2& =P^{\ast }{b}_4{F}_{V_1}^{\ast }+(F_1^{\ast }-b_2)d_1^{\ast }+(b_2-F_1^{\ast }-d_1^{\ast })b_5-b_2{F}_1^{\ast }\hfill \\ \hfill & +b_1((d_1^{\ast }+F_1^{\ast }-b_5+({(I_1^{\ast })}^2-I_1^{\ast })F_{I_1}^{\ast })F_{Z_1}^{\ast }-k_{Z_1}^{\ast }{F}_{I_1}^{\ast }),\hfill \\ \hfill {\rho }_3& =(-(F_1^{\ast }-b_5)(({(I_1^{\ast })}^2-I_1^{\ast })F_{Z_1}^{\ast }-k_{Z_1}^{\ast })b_1-b_4{P}^{\ast }(F_{V_1}^{\ast }{(I_1^{\ast })}^2-F_{V_1}^{\ast }{I}_1^{\ast }-k_{V_1}^{\ast }))F_{I_1}^{\ast }\hfill \\ \hfill & +F_{Z_1}^{\ast }((F_1^{\ast }+d_1^{\ast })b_5-d_1^{\ast }{F}_1^{\ast })b_1+((-b_2+d_1^{\ast })F_1^{\ast }-b_2{d}_1^{\ast })b_5\hfill \\ \hfill & +F_1^{\ast }{b}_2{d}_1^{\ast }+b_4{P}^{\ast }{F}_{V_1}^{\ast }(B_2-d_1^{\ast }),\hfill \\ \hfill {\rho }_4& =(-b_4{P}^{\ast }(F_{V_1}^{\ast }{(I_1^{\ast })}^2-F_{V_1}^{\ast }{I}_1^{\ast }-k_{V_1}^{\ast })b_2-b_1{b}_5{F}_1^{\ast }(F_{Z_1}^{\ast }{(I_1^{\ast })}^2-F_{Z_1}^{\ast }{I}_1^{\ast }-k_{Z_1}^{\ast }))F_{I_1}^{\ast }\hfill \\ \hfill & -d_1^{\ast }((P^{\ast }{F}_{V_1}^{\ast }{b}_4-b_5{F}_1^{\ast })b_2+b_1{b}_2{F}_1^{\ast }{F}_{Z_1}^{\ast }.\hfill \end{array}$$

The equilibrium points are locally asymptotically stable if all the eigenvalues $\lambda$ of the Jacobian matrix evaluated at the equilibrium points satisfy: Therefore, by the Routh–Hurwitz-type stability conditions for fractional-order systems [46], a necessary and sufficient condition for local asymptotic stability is

$$\begin{array}{c}\hfill |arg\left({\lambda }_i\right)|>\xi {\displaystyle \frac{\pi }{2}},i=1,2,3,4,\end{array}$$

(19)

where ${\lambda }_i$ are the eigenvalues of the Jacobian matrix $J\left(C_1^{\ast }\right)$ associated with the equilibrium point $C_1^{\ast }$. Hence, the equilibrium is locally asymptotically stable if all eigenvalues of $J\left(C_1^{\ast }\right)$ satisfy condition (19). This confirms that the fractional-order asset model exhibits stability properties that are at least comparable to those of its integer-order counterpart.

We now consider the Jacobian matrix at the equilibrium point $C_2^{\ast },$ denoted by $J\left(C_2^{\ast }\right)$.

$$\begin{array}{c}\hfill J\left(C_2^{\ast }\right)=\left[\begin{array}{cccc}{F}_2^{\ast }& 0& 0& 0\\ 0& d_2^{\ast }& k_{Z_2}^{\ast }+(1-I_2^{\ast })I_2^{\ast }{F}_{Z_2}^{\ast }& k_{V_2}^{\ast }+(1-I_2^{\ast })I_2^{\ast }{F}_{V_2}^{\ast }\\ 0& b_1{F}_{I_2^{\ast }}^{\ast }& b_1{F}_{Z_2}^{\ast }-b_2& b_1{F}_{V_2}^{\ast }\\ -b_4& 0& 0& -b_5\end{array}\right]\end{array}$$

(20)

where the following hold:

$$\begin{array}{cc}\hfill d_2^{\ast }& =-1+I_2^{\ast }{F}_{I_2}^{\ast }+F_2^{\ast }-{\left(I_2^{\ast }\right)}^2{F}_{I_2^{\ast }}^{\ast }-2I_2^{\ast }{F}_2^{\ast },\hfill \\ \hfill F_2^{\ast }& =log\left({\displaystyle \frac{1}{2}}\left(1+exp\left\{{\displaystyle \frac{(\frac{1}{2}+\frac{1}{2}tanh(Z_2^{\ast }+V_2^{\ast }))(1-I_2^{\ast })}{(1-(\frac{1}{2}+\frac{1}{2}tanh(Z_2^{\ast }+V_2^{\ast })))I_2^{\ast }}}-1\right\}\right)\right),\hfill \\ \hfill k_{Z_2}^{\ast }& =k_{V_2}^{\ast }={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh(Z_2^{\ast }+V_2^{\ast }),\hfill \\ \hfill F_{I_2}^{\ast }& ={\displaystyle \frac{(1+tanh{(Z_2^{\ast }+V_2^{\ast })}^2)exp\left\{{\displaystyle \frac{1-I_2^{\ast }-tanh(Z_2^{\ast }+V_2^{\ast })}{I_2^{\ast }-2+tanh(Z_2^{\ast }+V_2^{\ast })I_2^{\ast }}}\right\}}{{(I_2^{\ast }-2+tanh(Z_2^{\ast }+V_2^{\ast }))}^2\left(1+exp\left\{{\displaystyle \frac{1-I_2^{\ast }-tanh(Z_2^{\ast }+V_2^{\ast })}{I_2^{\ast }-2+tanh(Z_2^{\ast }+V_2^{\ast })I_2^{\ast }}}\right\}\right)}},\hfill \\ \hfill F_{Z_2}^{\ast }& =F_{V_2}^{\ast }={\displaystyle \frac{({(I_2^{\ast }-1)}^2+1)exp\left\{{\displaystyle \frac{1-I_2^{\ast }-tanh(Z_2^{\ast }+V_2^{\ast })}{I^{\ast }{2}_2-2+tanh(Z_2^{\ast }+V_2^{\ast })I_2^{\ast }}}\right\}}{{((I_2^{\ast }-2)cosh(Z_2^{\ast }+V_2^{\ast })+I_2^{\ast }sinh(Z_2^{\ast }+V_2^{\ast }))}^2\left(1+A_2^{\ast }\right)}}.\hfill \end{array}$$

Additionally,

$$\begin{array}{c}\hfill A_2^{\ast }=exp\left\{-{\displaystyle \frac{I_2^{\ast }cosh(Z_2^{\ast }+V_2^{\ast })+sinh(Z_2^{\ast }+V_2^{\ast })-cosh(Z_2^{\ast }+V_2^{\ast })}{I_2^{\ast }sinh(Z_2^{\ast }+V_2^{\ast })+I_2^{\ast }sinh(Z_2^{\ast }+V_2^{\ast })-2cosh(Z_2^{\ast }+V_2^{\ast })}}\right\}.\end{array}$$

The characteristic equation of $J\left(C_2^{\ast }\right)$ is

$$\begin{array}{c}\hfill {\lambda }^4+{\kappa }_1{\lambda }^3+{\kappa }_2{\lambda }^2+{\kappa }_3\lambda +{\kappa }_4=0\end{array}$$

(21)

where the following hold:

$$\begin{array}{cc}\hfill {\kappa }_1& =b_2+b_5-d_2^{\ast }-F_2^{\ast }-b_1{F}_{Z_2}^{\ast },\hfill \\ \hfill {\kappa }_2& =(F_2^{\ast }-b_2)d_2^{\ast }+(b_2-F_2^{\ast }-d_2^{\ast })b_5-b_2{F}_2^{\ast }\hfill \\ \hfill & +b_1((d_2^{\ast }+F_2^{\ast }-b_5+({\left(I_2^{\ast }\right)}^2-I_2^{\ast })F_{I_2}^{\ast })F_{Z_2}^{\ast }-k_{Z_2}^{\ast }{F}_{I_2}^{\ast }),\hfill \\ \hfill {\kappa }_3& =((F_2^{\ast }{F}_{Z_2}^{\ast }+(({\left(I_2^{\ast }\right)}^2-I_2^{\ast })F_{I_2}^{\ast }+d_2^{\ast })F_{Z_2}^{\ast }-k_{Z_2}^{\ast }{F}_{I_2}^{\ast })b_5\hfill \\ \hfill & -F_2^{\ast }(({\left(I_2^{\ast }\right)}^2-I_2^{\ast })F_{I_2}^{\ast }+d_2^{\ast })F_{Z_2}^{\ast }-k_{Z_2}^{\ast }{F}_{I_2}^{\ast })b_1+((d_2^{\ast }-b_2)F^{\ast }-b_2{d}_2^{\ast })b_5+b_2{d}_2^{\ast }{F}_2^{\ast },\hfill \\ \hfill {\kappa }_4& =-(((({\left(I_2^{\ast }\right)}^2-I_2^{\ast })F_{Z_2}^{\ast }-k_{Z_2}^{\ast })F_{I_2}^{\ast }+d_2^{\ast }{F}_{Z_2}^{\ast })b_1-b_2{d}_2^{\ast })b_5{F}_2^{\ast }.\hfill \end{array}$$

The local asymptotic stability of the equilibrium point $C_2^{\ast }$ follows directly from the eigenvalue condition stated in Equation (19), and the equilibrium point $C_3^{\ast }$ can be shown to be locally asymptotically stable in the same manner as $C_2^{\ast }$.

4.5. Global Stability Analysis

After identifying the equilibrium points of the system, we investigate the global stability of the fractional-order asset model. A Lyapunov function is constructed to reflect the economic structure of the system and shown to decrease along solution trajectories. Under suitable conditions, it is proven that the system converges to the equilibrium regardless of initial states, indicating stable long-term market behavior.

To analyze the global stability of the system, we consider the equilibrium point $C_1^{\ast }=(P_1^{\ast },I_1^{\ast },Z_1^{\ast },V_1^{\ast })$ and construct a suitable Lyapunov function around it.

Define Lyapunov function M as

$$\begin{array}{c}\hfill M={\displaystyle \frac{1}{2}}{(P-P_1^{\ast })}^2+{\displaystyle \frac{1}{2}}{(I-I_1^{\ast })}^2+{\displaystyle \frac{1}{2}}{(Z-Z_1^{\ast })}^2+{\displaystyle \frac{1}{2}}{(V-V_1^{\ast })}^2.\end{array}$$

It is observed that the function M is always positive and equals zero only at the equilibrium point $C_1^{\ast }$.

Consider

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }M& =(P-P_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+(I-I_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+(Z-Z_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z+(V-V_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =(P-P_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+(I-I_1^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+\left(Z\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z+\left(V-{\displaystyle \frac{b_3-b_4{P}_1^{\ast }}{b_5}}\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =(P-P_1^{\ast })PF+(I-I_1^{\ast })(k-I+I(1-I)F)\hfill \\ \hfill & +\left(Z\right)(b_{1}F-b_{2}Z)+\left(V-{\displaystyle \frac{b_3-b_4{P}_1^{\ast }}{b_5}}\right)(b_3-b_{4}P-b_{5}V)\hfill \\ \hfill & =P^{2}F-P{P}_1^{\ast }F+I^2{I}_1^{\ast }F-I^{3}F-I{I}_1^{\ast }F+I^{2}F-I_1^{\ast }k+Ik+I{I}_1^{\ast }-I^2+b_{1}ZF\hfill \\ \hfill & -b_2{Z}^2-b_{4}PV-b_5{V}^2+2b_{3}V-{\displaystyle \frac{b_4^{2}P{P}_1^{\ast }}{b_5}}-b_{4}V{P}_1^{\ast }+{\displaystyle \frac{b_3{b}_4{P}_1^{\ast }}{b_5}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}}-{\displaystyle \frac{b_3^2}{b_5}}.\hfill \end{array}$$

To simplify the analysis, the equation can be rearranged as ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }M={\eta }_1-{\eta }_2,$ where the following hold:

$$\begin{array}{cc}\hfill {\eta }_1& =P^{2}F+I^2{I}_1^{\ast }F+I^{2}F+Ik+I{I}_1^{\ast }+b_{1}ZF+2b_{3}V+{\displaystyle \frac{b_3{b}_4{P}_1^{\ast }}{b_5}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}},\hfill \\ \hfill {\eta }_2& =P{P}_1^{\ast }F+I^{3}F+I{I}_1^{\ast }F+I_1^{\ast }k+I^2+b_2{Z}^2+b_{4}PV+b_5{V}^2+{\displaystyle \frac{b_4^{2}P{P}_1^{\ast }}{b_5}}+b_{4}V{P}_1^{\ast }+{\displaystyle \frac{b_3^2}{b_5}}.\hfill \end{array}$$

Thus, ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }M<0$ if ${\eta }_1<{\eta }_2$ and ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }M=0$ when $P=P_1^{\ast },I=I_1^{\ast },Z=Z_1^{\ast },$ and $V=V_1^{\ast }$. Therefore, it can be concluded that the equilibrium point $C_1^{\ast }$ is globally asymptotically stable if ${\eta }_1<{\eta }_2$.

Following the stability analysis at $C_1^{\ast }$, we investigate the global asymptotic stability of the second equilibrium point $C_2^{\ast }=(P_2^{\ast },I_2^{\ast },Z_2^{\ast },V_2^{\ast })$ using the Lyapunov method.

Although $I_2^{\ast }={\displaystyle \frac{(F^{\ast }-1)+\sqrt{4F^{\ast }k\left(0,I^{\ast },\frac{b_1}{b_2}{F}^{\ast },\frac{b_3}{b_4}\right)+{(F^{\ast }-1)}^2}}{2F^{\ast }}}$ by definition, for the purpose of this analysis, we set $I_2^{\ast }={\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}+\psi$. Define Lyapunov function R as

$$\begin{array}{c}\hfill R={\displaystyle \frac{1}{2}}{(P-P_2^{\ast })}^2+{\displaystyle \frac{1}{2}}{(I-I_2^{\ast })}^2+{\displaystyle \frac{1}{2}}{(Z-Z_2^{\ast })}^2+{\displaystyle \frac{1}{2}}{(V-V_2^{\ast })}^2.\end{array}$$

Consider

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }M& =(P-P_2^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+(I-I_2^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+(Z-Z_2^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z+(V-V_2^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =\left(P\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+\left(I-{\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}-\psi \right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+\left(Z-{\displaystyle \frac{b_1}{b_2}}{F}^{\ast }\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z\hfill \\ \hfill & +\left(V-{\displaystyle \frac{b_3}{b_5}}\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =\left(P\right)PF+\left(I-{\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}-\psi \right)(k-I+I(1-I)F)\hfill \\ \hfill & +\left(z-{\displaystyle \frac{b_1}{b_2}}{F}^{\ast }\right)(b_{1}F-b_{2}Z)+\left(V-{\displaystyle \frac{b_3}{b_5}}\right)(b_3-b_{4}P-b_{5}V)\hfill \\ \hfill & =-{\displaystyle \frac{k}{2}}-I^2+{\displaystyle \frac{3I^{2}F}{2}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}}+{\displaystyle \frac{I}{2}}-{\displaystyle \frac{IF}{2}}+{\displaystyle \frac{k}{2F^{\ast }}}-{\displaystyle \frac{I}{2F^{\ast }}}-\psi k+\psi I\hfill \\ \hfill & -{\displaystyle \frac{I^{2}F}{2F^{\ast }}}+{\displaystyle \frac{IF}{2F^{\ast }}}+\psi I^{2}F-\psi IF+b_{1}ZF-b_{4}PV+2b_{3}V+P^{2}F-I^{3}F\hfill \\ \hfill & +Ik-b_2{Z}^2-b_5{V}^2-{\displaystyle \frac{b_3^2}{b_5}}+b_{1}Z{F}^{\ast }-{\displaystyle \frac{b_1^{2}F{F}^{\ast }}{b_2}}.\hfill \end{array}$$

To simplify the analysis, the equation can be rearranged as ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }R={\eta }_3-{\eta }_4,$ where the following hold:

$$\begin{array}{cc}\hfill {\eta }_3& ={\displaystyle \frac{3I^{2}F}{2}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}}+{\displaystyle \frac{I}{2}}+{\displaystyle \frac{k}{2F^{\ast }}}+\psi I+{\displaystyle \frac{IF}{2F^{\ast }}}+\psi I^{2}F+b_{1}ZF+2b_{3}V+P^{2}F+Ik+b_{1}Z{F}^{\ast },\hfill \\ \hfill {\eta }_4& ={\displaystyle \frac{k}{2}}+I^2+{\displaystyle \frac{IF}{2}}+{\displaystyle \frac{I}{2F^{\ast }}}+\psi k+{\displaystyle \frac{I^{2}F}{2F^{\ast }}}+\psi IF+b_{4}PV+I^{3}F+b_2{Z}^2+b_5{V}^2+{\displaystyle \frac{b_3^2}{b_5}}+{\displaystyle \frac{b_1^{2}F{F}^{\ast }}{b_2}}.\hfill \end{array}$$

Thus, ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }R<0$ if ${\eta }_3<{\eta }_4$ and ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }R=0$ when $P=P_2^{\ast },I=I_2^{\ast },Z=Z_2^{\ast },$ and $V=V_2^{\ast }$. Therefore, it can be concluded that the equilibrium point $C_2^{\ast }$ is globally asymptotically stable if ${\eta }_3<{\eta }_4$.

Following the stability analysis at $C_2^{\ast }$, we investigate the global asymptotic stability of the second equilibrium point $C_3^{\ast }=(P_3^{\ast },I_3^{\ast },Z_3^{\ast },V_3^{\ast })$ using the Lyapunov method.

Although $I_3^{\ast }={\displaystyle \frac{(F^{\ast }-1)-\sqrt{4F^{\ast }k\left(0,I^{\ast },\frac{b_1}{b_2}{F}^{\ast },\frac{b_3}{b_4}\right)+{(F^{\ast }-1)}^2}}{2F^{\ast }}}$ by definition, for the purpose of this analysis, we set $I_3^{\ast }={\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}-\psi$. Define Lyapunov function R as

$$\begin{array}{c}\hfill Q={\displaystyle \frac{1}{2}}{(P-P_3^{\ast })}^2+{\displaystyle \frac{1}{2}}{(I-I_3^{\ast })}^2+{\displaystyle \frac{1}{2}}{(Z-Z_3^{\ast })}^2+{\displaystyle \frac{1}{2}}{(V-V_3^{\ast })}^2.\end{array}$$

Consider

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Q& =(P-P_3^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+(I-I_3^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+(Z-Z_3^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z+(V-V_3^{\ast }){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =\left(P\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P+\left(I-{\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}+\psi \right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }I+\left(Z-{\displaystyle \frac{b_1}{b_2}}{F}^{\ast }\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Z\hfill \\ \hfill & +\left(V-{\displaystyle \frac{b_3}{b_5}}\right){}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }V\hfill \\ \hfill & =\left(P\right)PF+\left(I-{\displaystyle \frac{(F^{\ast }-1)}{2F^{\ast }}}+\psi \right)(k-I+I(1-I)F)\hfill \\ \hfill & +\left(z-{\displaystyle \frac{b_1}{b_2}}{F}^{\ast }\right)(b_{1}F-b_{2}Z)+\left(V-{\displaystyle \frac{b_3}{b_5}}\right)(b_3-b_{4}P-b_{5}V)\hfill \\ \hfill & =-{\displaystyle \frac{k}{2}}-I^2+{\displaystyle \frac{3I^{2}F}{2}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}}+{\displaystyle \frac{I}{2}}-{\displaystyle \frac{IF}{2}}+{\displaystyle \frac{k}{2F^{\ast }}}-{\displaystyle \frac{I}{2F^{\ast }}}+\psi k-\psi I\hfill \\ \hfill & -{\displaystyle \frac{I^{2}F}{2F^{\ast }}}+{\displaystyle \frac{IF}{2F^{\ast }}}-\psi I^{2}F+\psi IF+b_{1}ZF-b_{4}PV+2b_{3}V+P^{2}F-I^{3}F\hfill \\ \hfill & +Ik-b_2{Z}^2-b_5{V}^2-{\displaystyle \frac{b_3^2}{b_5}}+b_{1}Z{F}^{\ast }-{\displaystyle \frac{b_1^{2}F{F}^{\ast }}{b_2}}.\hfill \end{array}$$

To simplify the analysis, the equation can be rearranged as ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Q={\eta }_5-{\eta }_6,$ where the following hold:

$$\begin{array}{cc}\hfill {\eta }_5& ={\displaystyle \frac{3I^{2}F}{2}}+{\displaystyle \frac{b_3{b}_{4}P}{b_5}}+{\displaystyle \frac{I}{2}}+{\displaystyle \frac{k}{2F^{\ast }}}+\psi k+{\displaystyle \frac{IF}{2F^{\ast }}}+\psi IF+b_{1}ZF+2b_{3}V+P^{2}F+Ik+b_{1}Z{F}^{\ast },\hfill \\ \hfill {\eta }_6& ={\displaystyle \frac{k}{2}}+I^2+{\displaystyle \frac{IF}{2}}+{\displaystyle \frac{I}{2F^{\ast }}}+\psi I+{\displaystyle \frac{I^{2}F}{2F^{\ast }}}+\psi I^{2}F+b_{4}PV+I^{3}F+b_2{Z}^2+b_5{V}^2+{\displaystyle \frac{b_3^2}{b_5}}+{\displaystyle \frac{b_1^{2}F{F}^{\ast }}{b_2}}.\hfill \end{array}$$

Thus, ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Q<0$ if ${\eta }_5<{\eta }_6$ and ${}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }Q=0$ when $P=P_3^{\ast },I=I_3^{\ast },Z=Z_3^{\ast },$ and $V=V_3^{\ast }$. Therefore, it can be concluded that the equilibrium point $C_3^{\ast }$ is globally asymptotically stable if ${\eta }_5<{\eta }_6$.

5. Numerical Examples

In this section, a numerical scheme is developed to solve the mathematical model (5).

Consider

$$\begin{array}{cc}\hfill {}_{\mathrm{0\hspace{1em} }}{}^{\mathit{ABC}}D_t^{\xi }P& =G_1(t,P\left(t\right)),\hfill \\ \hfill P\left(0\right)& =P_0.\hfill \end{array}$$

We can write

$$\begin{array}{c}\hfill P\left(t\right)=P\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t,P\left(t\right))+{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}{\int }_0^t{G}_1(s,P\left(s\right)){(t-s)}^{\xi -1}ds.\end{array}$$

(22)

Applying Lagrange’s interpolation polynomial on the interval $[t_k,t_{k+1}]$ to the equality $G_1(y,P\left(y\right))=P\left(y\right)F(y,P\left(y\right),I\left(y\right),Z\left(y\right),V\left(y\right))$ leads to

$$\begin{array}{cc}\hfill P_K& \approx {\displaystyle \frac{1}{h}}\left((y-t_{k-1})G_1(t_k,P\left(t_k\right),I\left(t_k\right),Z\left(t_k\right),V\left(t_k\right))\right.\hfill \\ \hfill & \left.-(y-t_k)G_1(t_{k-1},P\left(t_{k-1}\right),I\left(t_{k-1}\right),Z\left(t_{k-1}\right),V\left(t_{k-1}\right))\right),\hfill \end{array}$$

(23)

where $h=t_k-t_{k-1}.$

Now substituting (23) into (22), we have

$$\begin{array}{cc}\hfill P\left(t_{n+1}\right)& =P\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t_k,P\left(t_k\right),I\left(t_k\right),Z\left(t_k\right),V\left(t_k\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left({\displaystyle \frac{t_k,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{h}}{\int }_{t_j}^{t_{j+1}}(y-t_{j-1}){(t_{n+1}-y)}^{\xi -1}dy\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{G_1(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{h}}{\int }_{t_j}^{t_{j+1}}(y-t_j){(t_{n+1}-y)}^{\xi -1}dy\right)\hfill \\ \hfill & =P\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t_n,P\left(t_n\right),I\left(t_n\right),Z\left(t_n\right),V\left(t_n\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left({\displaystyle \frac{G_1(t_j,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{h}}{\Delta }_{j-1}\right.\hfill \\ & \left.-{\displaystyle \frac{G_1(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{h}}{\Delta }_j\right),\hfill \end{array}$$

(24)

where

$$\begin{array}{cc}\hfill {\Delta }_{j-1}& ={\int }_{t_j}^{t_{j+1}}(y-t_{j-1}){(t_{n+1}-y)}^{\xi -1}dy\hfill \\ \hfill & =-{\displaystyle \frac{1}{\xi }}\left((t_{j+1}-t_{j-1}){(t_{n+1}-t_{j+1})}^{\xi }-(t_j-t_{j-1}){(t_{n+1}-t_j)}^{\xi }\right)\hfill \\ & -{\displaystyle \frac{1}{\xi (\xi +1)}}\left({(t_{n+1}-t_{j+1})}^{\xi +1}{(t_{n+1}-t_{j+1})}^{\xi }-{(t_{n+1}-t_j)}^{\xi +1}\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\Delta }_j& ={\int }_{t_j}^{t_{j+1}}(y-t_j){(t_{n+1}-y)}^{\xi -1}dy\hfill \\ \hfill & =-{\displaystyle \frac{1}{\xi }}\left((t_{j+1}-t_{j-1}){(t_{n+1}-t_{j+1})}^{\xi }\right)\hfill \\ & -{\displaystyle \frac{1}{\xi (\xi +1)}}\left({(t_{n+1}-t_{j+1})}^{\xi +1}-{(t_{n+1}-t_j)}^{\xi }\right).\hfill \end{array}$$

(26)

Furthermore, substituting $t_j=jh$ into (25) and (26) leads to

$$\begin{array}{cc}\hfill {\Delta }_{j-1}& ={\displaystyle \frac{h^{\xi +1}}{\xi (\xi +1)}}\left({(n+1-j)}^{\xi }(n-j+2+\xi )-{(n-j)}^{\xi }(n-j+2+2\xi )\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\Delta }_j& ={\displaystyle \frac{h^{\xi +1}}{\xi (\xi +1)}}\left({(n+1-j)}^{\xi +1}-{(n-j)}^{\xi }(n-j+1+\xi )\right).\hfill \end{array}$$

(28)

We can express (24) in terms of (27) and (28) as follows:

$$\begin{array}{cc}\hfill P\left(t_{n+1}\right)& =P\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_1(t_n,P\left(t_n\right),I\left(t_n\right),Z\left(t_n\right),V\left(t_n\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left(\begin{array}{c}{\displaystyle \frac{G_1(t_j,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi }(n-j+2+\xi )-{(n-j)}^{\xi }(n-j+2+2\xi )\right)\\ -{\displaystyle \frac{G_1(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi +1}-{(n-j)}^{\xi }(n-j+1+\xi )\right)\end{array}\right).\hfill \end{array}$$

In the same way, we have the following equations for the remaining state variables

$$\begin{array}{cc}\hfill I\left(t_{n+1}\right)& =I\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_2(t_n,P\left(t_n\right),I\left(t_n\right),Z\left(t_n\right),V\left(t_n\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left(\begin{array}{c}{\displaystyle \frac{G_2(t_j,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi }(n-j+2+\xi )-{(n-j)}^{\xi }(n-j+2+2\xi )\right)\\ -{\displaystyle \frac{G_2(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi +1}-{(n-j)}^{\xi }(n-j+1+\xi )\right)\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z\left(t_{n+1}\right)& =Z\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_3(t_n,P\left(t_n\right),I\left(t_n\right),Z\left(t_n\right),V\left(t_n\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left(\begin{array}{c}{\displaystyle \frac{G_3(t_j,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi }(n-j+2+\xi )-{(n-j)}^{\xi }(n-j+2+2\xi )\right)\\ -{\displaystyle \frac{G_3(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi +1}-{(n-j)}^{\xi }(n-j+1+\xi )\right)\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill V\left(t_{n+1}\right)& =V\left(0\right)+{\displaystyle \frac{1-\xi }{\Psi \left(\xi \right)}}{G}_4(t_n,P\left(t_n\right),I\left(t_n\right),Z\left(t_n\right),V\left(t_n\right))\hfill \\ \hfill & +{\displaystyle \frac{\xi }{\Psi \left(\xi \right)\Gamma \left(\xi \right)}}\sum _{j=1}^n\left(\begin{array}{c}{\displaystyle \frac{G_4(t_j,P\left(t_j\right),I\left(t_j\right),Z\left(t_j\right),V\left(t_j\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi }(n-j+2+\xi )-{(n-j)}^{\xi }(n-j+2+2\xi )\right)\\ -{\displaystyle \frac{G_4(t_{j-1},P\left(t_{j-1}\right),I\left(t_{j-1}\right),Z\left(t_{j-1}\right),V\left(t_{j-1}\right))}{\Gamma (\xi +2)}}\\ \times h^{\xi }\left({(n+1-j)}^{\xi +1}-{(n-j)}^{\xi }(n-j+1+\xi )\right)\end{array}\right).\hfill \end{array}$$

To illustrate the dynamic behavior of the ABC-fractional-order asset model, we numerically simulate the system for various values of the fractional order parameter $\xi \in \{0.5,0.6,0.7,0.8\}$. This parameter governs the strength of memory effects within the system: lower values correspond to weaker memory and thus faster dynamic adjustment, while higher values introduce stronger memory, resulting in slower convergence to equilibrium. The numerical method employed is a modified predictor–corrector scheme tailored for the Atangana–Baleanu–Caputo derivative, ensuring stability and accuracy in capturing the memory-dependent dynamics.

The initial conditions and parameter values for the asset model are set as follows:

$$\begin{array}{cccccc}& P\left(0\right)=0.49,\hfill & & I\left(0\right)=0.59,\hfill & & Z\left(0\right)=0.001,\hfill \\ & V\left(0\right)=0.18,\hfill & & b_1=0.002,\hfill & & b_2=0.03,\hfill \\ & b_3=0.014,\hfill & & b_4=0.01,\hfill & & b_5=0.05.\hfill \end{array}$$

All simulations begin from identical initial conditions and evolve over a common time interval. Across all values of $\xi$, the trajectories of the variables are smooth and monotonic, showing no signs of oscillation or divergence. This numerical behavior confirms the model’s stability and supports the existence of the unique equilibrium point $C_1^{\ast }=(P^{\ast },I^{\ast },Z^{\ast },V^{\ast })=(0.49,0.5901,0,0.182)$, in agreement with the theoretical analysis of global stability.

Figure 1, Figure 2, Figure 3 and Figure 4 present the time evolution of four key state variables of the model. These include the market price $P\left(t\right)$, which represents the observed price of the asset in the market; the fraction of total asset $I\left(t\right)$, which indicates the portion of assets actively held or invested and reflects overall investor sentiment or market participation; the trend-based component $Z\left(t\right)$, capturing short-term responses to recent price movements or momentum effects; and the value-based component $V\left(t\right)$, which models long-term valuation behavior influenced by trading costs, market frictions, and fundamental assessments.

All simulations begin from identical initial conditions and evolve over a common time interval. Across all values of $\xi$, the trajectories of the variables are smooth and monotonic, showing no signs of oscillation or divergence. This numerical behavior confirms the model’s stability and supports the existence of the unique equilibrium point $C_1^{\ast }=(P^{\ast },I^{\ast },Z^{\ast },V^{\ast })=(0.49,0.5901,0,0.182)$, which is in agreement with the theoretical analysis of global stability.

Although the equilibrium point $C_1^{\ast }=(P^{\ast },I^{\ast },Z^{\ast },V^{\ast })=(0.49,0.5901,0,0.182)$ is derived mathematically, it also carries clear economic interpretation. The component $P^{\ast }=0.49$ represents the long-run market price toward which the system stabilizes, assuming that behavioral and market forces are balanced. The value $I^{\ast }=0.5901$ reflects the steady-state level of asset holding or investor participation, suggesting that approximately 59% of the asset pool remains active in the market at equilibrium—indicating a moderately active financial environment that avoids both excessive speculation and market inactivity. The equilibrium value $Z^{\ast }=0$ implies that, in the long term, trend-following behavior becomes neutralized, meaning there is no persistent directional pressure from recent price movements. This reflects a market that has absorbed short-term momentum effects. In contrast, the positive value $V^{\ast }=0.182$ indicates that valuation-based trading remains present at equilibrium, modeling the continued influence of long-term investors who base decisions on deviations from intrinsic value. The equilibrium reflects a financial system in which speculative forces subside, valuation principles endure, and asset prices converge to a stable level under the influence of both behavioral dynamics and market mechanisms.

As the fractional order $\xi$ increases, the convergence rate of each variable to its equilibrium becomes progressively slower. This is a direct manifestation of the memory effect: stronger memory (larger $\xi$) causes past states to influence the current dynamics more persistently, thereby delaying the return to equilibrium. The trend-based component $Z\left(t\right)$ exhibits rapid decay toward zero for all values of $\xi$, suggesting that short-term speculative behavior has a transient influence on the system. In contrast, the value-based component $V\left(t\right)$ converges more gradually to its steady state, implying that long-term valuation is more susceptible to the influence of memory. The market price $P\left(t\right)$ and the asset fraction $I\left(t\right)$ both follow a similar pattern of monotonic convergence, with observable delays in reaching their respective equilibria as $\xi$ increases.

These numerical observations are consistent with the theoretical predictions of the model. The absence of oscillatory or chaotic behavior reinforces the global asymptotic stability established via Lyapunov analysis, and demonstrates that the ABC-fractional framework provides a robust structure for modeling memory-dependent financial systems.

From an economic perspective, the simulations offer several meaningful insights. In markets characterized by stronger memory—i.e., with higher values of $\xi$—the system tends to respond more gradually to shocks or behavioral changes. As a result, deviations from equilibrium persist longer, reflecting inertia in price adjustment and investor reaction. Investor sentiment, represented by $I\left(t\right)$, adapts more slowly in high-memory regimes, suggesting delayed collective behavioral responses. Moreover, the gradual convergence of the value-based component $V\left(t\right)$ indicates that investors with a valuation-based strategy are influenced more heavily by historical trends, adjusting cautiously over extended horizons. Notably, the smooth and stable nature of all system trajectories, regardless of the value of $\xi$, implies that while memory prolongs convergence, it may also exert a stabilizing influence by damping abrupt fluctuations and preventing volatility escalation.

6. Conclusions

In this work, we proposed and analyzed an asset flow model governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative to capture memory effects inherent in financial markets. The model extends traditional asset pricing frameworks by incorporating nonlocal and non-singular dynamics, offering a more realistic depiction of investor behavior and market adjustment. We established the existence, uniqueness, positivity, and boundedness of solutions, and demonstrated both local and global stability using rigorous mathematical techniques. Numerical simulations confirmed these theoretical findings and revealed that stronger memory (larger $\xi$) leads to slower convergence toward equilibrium, reflecting the delayed adjustment typical of real-world financial systems. The results highlight the stabilizing role of fractional memory and validate the ABC approach as a robust tool for modeling long-term financial dynamics.

While the proposed model offers several theoretical and computational advantages, it also has limitations. The parameter values were selected based on theoretical considerations rather than calibration to empirical data, and the model does not currently incorporate stochastic influences or multi-asset interactions that are often present in real markets. Additionally, the assumed functional forms for demand, supply, and adjustment dynamics, while mathematically tractable, may not fully capture the heterogeneity and complexity of investor behavior across different market regimes. Future work may address these limitations through data-driven parameter estimation, extension to stochastic or networked financial systems, and further empirical validation of the model structure.