Fractional-Order Delay Differential Equations: Existence, Uniqueness, and Ulam–Hyers Stability

Abstract

This article presents several key findings for fractional-order delay differential equations. First, we establish the existence and uniqueness of solutions using two distinct approaches, the Chebyshev norm and the Bielecki norm, thereby providing a comprehensive understanding of the solution space. Notably, the uniqueness of the solution is rigorously demonstrated using the Lipschitz condition, ensuring a single solution under specific constraints. Additionally, we examine a specific form of constant delay and apply Burton’s method to further confirm the uniqueness of the solution. Furthermore, we conduct an in-depth investigation into the Hyers–Ulam stability of the problem, providing valuable insights into the behavior of solutions under perturbations. Notably, our results eliminate the need for contraction constant conditions that are commonly imposed in the existing literature. Finally, numerical simulations are performed to illustrate and validate the theoretical results obtained in this study. Fractional-order delay differential equations play a crucial role in real-life applications in systems where memory and delayed effects are essential. In biology and epidemiology, they model disease spread with incubation delays and immune memory. In control systems and robotics, they help design stable controllers by accounting for time-lagged responses and past behavior.

1. Introduction

Fractional delay differential equations (FDDEs) have a rich history, intertwined with the development of fractional calculus and delay differential equations (DDEs). The concept of fractional calculus, which involves derivatives and integrals of non-integer orders, dates back to the 17th century when mathematicians like Leibniz and L’Hopital explored its potential. However, it was not until the 19th and 20th centuries that fractional calculus began to take shape as a distinct field, with significant contributions from mathematicians such as Riemann–Liouville and Caputo; see [1,2,3]. The introduction of delay into differential equations, on the other hand, has its roots in the study of systems where the rate of change in the systems’ states depends not only on the current state but also on past states. This concept plays a pivotal role in modeling real-world phenomena where time delays are inherent, such as in population dynamics, epidemiology, and engineering systems; see [4,5,6].

The synthesis of these two concepts, fractional calculus and DDEs, into FDDEs represents a relatively recent advancement. FDDEs are particularly effective in modeling systems that exhibit memory and hereditary properties, where the future state of the system involves non-integer-order dependence on past states. This makes FDDEs a powerful tool in the analysis and modeling of complex systems in various fields, including physics, biology, and finance; see [7,8,9]. Research into FDDEs has been gaining momentum, with studies focusing on their mathematical properties, numerical solutions, and applications; see [10,11,12]. The complexity of solving FDDEs, due to their non-local and memory-dependent nature, has led to the development of new numerical methods and analytical techniques. These advancements are crucial for understanding and predicting the behavior of systems governed by FDDEs, thereby opening new avenues for research and applications in science and engineering.

The notion of stability in functional equations was pioneered by Ulam in 1940, when he presented his groundbreaking ideas at a conference. Subsequently, Hyers made significant contributions to Ulam’s work in 1941, which led to the development of the Hyers–Ulam stability concept. This concept has since become a cornerstone in the study of functional equations, enabling researchers to examine the stability properties of various mathematical models. This concept assesses the sensitivity of solutions to small perturbations, determining whether they remain close to the original solution. Dragicevic [13] discussed generalized dichotomies and Hyers–Ulam stability, while Selvam et al. [14] examined the Ulam-type stability of a linear differential equation with an integral transform. This framework has far-reaching implications in mathematics, allowing scientists to analyze and predict the behavior of complex systems with greater precision. The stability concept has been extensively applied in various fields, including differential equations, functional equations, and control theory, and continues to be a vital tool in advancing our understanding of mathematical models and their applications; see [15,16].

Ulam–Hyers stability analysis plays a pivotal role in the study of FDDEs, providing a robust framework for examining the stability properties of these complex systems; see [17,18]. Recent studies on Hyers–Ulam stability have been conducted by several prominent researchers. For instance, An et al. [19] investigated the relative controllability and Hyers–Ulam stability of Riemann–Liouville fractional delay differential systems. Similarly, Asadzade et al. [4] examined the finite-time stability of fractional stochastic neutral DDEs. Furthermore, Ali et al. [20] explored the existence and Ulam-type stability of coupled fractal fractional differential equations. Additionally, Kiskinov et al. [21] analyzed the Hyers–Ulam stability and shed light on the Hyers–Ulam–Rassias stability for linear fractional systems with Riemann–Liouville derivatives and distributed delays. These studies demonstrate the ongoing interest in Hyers–Ulam stability and its applications in various fields, including mathematics, physics, and engineering.

In addition to continuous-time fractional differential systems, recent research has also focused on discrete fractional difference equations involving nabla (∇) and delta ($\Delta$) fractional operators. These discrete fractional models have been extensively studied regarding existence, uniqueness, and Ulam–Hyers stability properties. For instance, Dimitrov and Jonnalagadda [22] established existence, uniqueness, and stability results for nabla fractional difference equations. Jonnalagadda [23] analyzed coupled systems of fractional difference equations with anti-periodic boundary conditions, while Abbas et al. [24] investigated the existence and Ulam-type stability of implicit fractional q-difference equations. These studies demonstrate that discrete fractional systems exhibit stability characteristics analogous to their continuous counterparts, albeit requiring distinct analytical approaches. In the continuous setting, several researchers have recently examined concrete problems involving fractional-order delay systems. For example, Balaji et al. [25] investigated Galerkin approximations for fractional-order delay differential equations with constant or time-periodic coefficients, and Zhang and Li [26] studied the existence and Ulam-type stability of fractional multi-delay differential systems. However, many of these contributions are limited to specific fractional orders or rely solely on continuous formulations.

Our study significantly advances the existing literature on fractional-order DDEs by investigating the existence and uniqueness of solutions within the order $(1,2)$ and conducting a comprehensive Ulam–Hyers stability analysis. Unlike previous research, which often relied on a single approach, we employ a dual-method strategy, utilizing both Chebyshev and Bielecki norms to establish existence and uniqueness. Furthermore, our work distinguishes itself through the application of Lipschitz conditions and Burton’s method to prove uniqueness, thereby providing a more robust framework for understanding solution behavior. Notably, our Ulam–Hyers stability analysis offers new insights into the stability properties of these equations, setting our research apart from earlier studies. By exploring this specific order range and employing a multifaceted approach, our study contributes a unique perspective to the field, enhancing the understanding of fractional order delay differential equations and their stability characteristics.

The fractional-order delay differential equations examined in this study have direct relevance in real-world systems characterized by both memory and time-delay effects. A notable application lies in epidemiological modeling, where the delay term represents the incubation period of infectious diseases, and the fractional derivative captures memory effects related to immune response and prior exposure. Such models are crucial for accurately predicting and managing the spread of diseases such as COVID-19 and tuberculosis. Similar structures also arise in engineering contexts, particularly in control systems where delayed feedback and historical states influence current system behavior. The analytical results presented in this paper, including the uniqueness and Hyers–Ulam stability of solutions, contribute to ensuring the reliability and robustness of such applied models.

Motivated by the above work, we examine the existence and uniqueness of solutions as well as Hyers–Ulam stability for a specific Caputo fractional-order delay differential equation. The analysis focuses on determining the conditions under which the solutions exist, are unique, and exhibit stability properties

$$\left\{\begin{array}{cc}{}^{\mathrm{C}}D^{\rho }\phi \left(t\right)=f(t,\phi \left(t\right),\phi \left(g\left(t\right)\right)),\hfill & t\in [0,I],1<\rho <2,\hfill \\ \phi \left(t\right)=\omega \left(t\right)={\omega }_0,{\phi }^{\prime }\left(t\right)={\omega }^{\prime }\left(t\right)={\omega }_1,\hfill & t\in [-\gamma ,0],\hfill \end{array}\right.$$

(1)

where $\phi \in \mathrm{C}([-\gamma ,0],\mathbb{R}),f\in \mathrm{C}([0,I]\times {\mathbb{R}}^2,\mathbb{R}),$ verifying $t\ge g\left(t\right)$ as well as ${}^{\mathrm{C}}D^{\rho }$ as the Caputo fractional derivative (CFD) of order $1<\rho <2$. The function $\omega \left(t\right)$ is a continuous function defined on the interval $[-\gamma ,0]$, which specifies the initial conditions for the solution $\phi \left(t\right)$. Specifically, ${\omega }_0$ and ${\omega }_1$ represent the values of $\omega \left(t\right)$ and ${\omega }^{\prime }\left(t\right)$ at $t=0$, respectively.

This paper is structured as follows: Section 2 presents foundational concepts, including the Caputo fractional-order derivative and Hyers–Ulam stability, along with key properties and inequalities relevant to the analysis. Section 3 explores the existence and uniqueness of solutions for the given problem. Section 4 delves into Hyers–Ulam stability, utilizing Picard operator theory to derive stability results for the equation. Section 5 provides numerical simulations to illustrate and validate the theoretical findings.

2. Preliminaries

Definition 1 

([27,28]). The Riemann–Liouville integral operator of order ${\mathcal{I}}^{\rho }\phi \left(t\right)$ is well-defined for $t\in [0,I]$ provided that $\phi \left(t\right)$ is Lebesgue integrable on the interval $[0,I]$. Under this assumption, the kernel ${(t-s)}^{\rho -1}$ is locally integrable for $\rho >0$, and the convolution integral

$${\mathcal{I}}^{\rho }\phi \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\phi \left(s\right)ds,t\in [0,I],$$

is finite for all $t\in [0,I]$. Furthermore, ${\mathcal{I}}^{\rho }\phi$ is continuous on $[0,I]$ and if $\phi \in \mathcal{C}\left(\right[0,I\left]\right)$ then ${\mathcal{I}}^{\rho }\phi \in \mathcal{C}\left([0,I]\right)$ as well. These properties ensure that the operator is well-posed for a wide class of functions, particularly those commonly encountered in fractional differential equations.

Definition 2 

([27,28]). The CFD of order ρ for the function φ is expressed as

$$D^{\rho }\phi \left(t\right)={\displaystyle \frac{1}{\Gamma (\mathfrak{n}-\rho )}}{\int }_0^t{(t-s)}^{\mathfrak{n}-\rho -1}{\phi }^{\mathfrak{n}}\left(s\right)ds,t\in [0,I],$$

where $\mathfrak{n}=\left[\rho \right]+1$ as well as $\left[\rho \right]$ represent the integer part of ρ.

Definition 3 

([29,30]). Consider a metric space $(\mathcal{Z},b)$. A mapping $N:\mathcal{Z}\to \mathcal{Z}$ is called a Picard operator (PO) if it satisfies two conditions:

(i): 

N has a unique fixed point $z^{*}\in \mathcal{Z}$, i.e., $X_N=\left\{z^{*}\right\}$ where $X_N=\{z\in \mathcal{Z}:N\left(z\right)=z\}$.

(ii): 

the sequence of iterates ${\left(N^{\mathfrak{n}}\left(z_0\right)\right)}_{\mathfrak{n}\in \mathbb{N}}$ converges to $z^{*}$ for every initial point $z_o\in \mathcal{Z}$.

Lemma 1 

([29,30]). Consider an ordered metric space $(\mathcal{Z},b,\le )$, where $N:\mathcal{Z}\to \mathcal{Z}$ is an increasing PO with a unique fixed point $z^{*}$ (i.e., $X_N=\left\{z^{*}\right\}$). If $z\le N\left(z\right)$ for some $z\in \mathcal{Z}$, then $z\le z^{*}$. Conversely, if $z\ge N\left(z\right)$, then $z\ge z^{*}$.

Lemma 2 

([31]). Suppose $\xi :[0,I]\to [0,\infty )$ is a real valued function as well as e is a non-negative, locally integrable function on $[0,I]$. Assume there exists constant $h>0$ and $1<\rho <2$ in such a way that

$$\xi \left(t\right)=\vartheta \left(t\right)+h{\int }_0^t{(t-s)}^{-\rho }\xi \left(s\right)ds,$$

then there exists a constant $\sigma =\sigma \left(\rho \right)$ in such a way that

$$\xi \left(t\right)=\vartheta \left(t\right)+\sigma h{\int }_0^t{(t-s)}^{-\rho }\vartheta \left(s\right)ds.$$

Definition 4 

([32]). The Hyers–Ulam stability of Equation (1) holds if there is a positive real constant u such that for any $\epsilon >0$ and every solution $\xi \in \mathcal{C}\left(\right[-\gamma ,I],\mathbb{R})$ satisfying the given inequality

$$|D^{\rho }\xi \left(t\right)-f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))|\le \epsilon ,t\in [0,I].$$

(2)

There is a solution $\phi \in \mathcal{C}\left(\right[-\gamma ,I],\mathbb{R})$ to Equation (1), such that

$$\left|\xi \right(t)-\phi (t\left)\right|\le u\epsilon ,t\in [-\gamma ,I].$$

Remark 1 

([32]). A function $\xi \in \mathcal{C}\left(\right[0,I],\mathbb{R})$ satisfies inequality (2) if there exists a function $\varphi \in \mathcal{C}(d,\mathbb{R})$ in such a way that

(i):  

$\left|\varphi \right(t\left)\right|\le \epsilon ,\forall t\in [0,I],$

(ii): 

${}^{\mathcal{C}}D^{\rho }\xi \left(t\right)=f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))+\varphi \left(t\right),\forall t\in [0,I].$

Remark 2 

([27]). A function $\xi \in \mathcal{C}(d,\mathbb{R})$ satisfies inequality (2), then, by assumption, $\xi \left(t\right)$ is uniformly bounded on $[0,I]$.

Consider the Caputo-type fractional differential equation

$$D^{\rho }\xi \left(t\right)=f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))+\varrho \left(t\right),with\left|\varrho \left(t\right)\right|\le \epsilon ,$$

apply the Riemann–Liouville integral operator ${\mathcal{I}}^{\rho }$ and using the identity for the Caputo derivative $(\rho \in (1,2\left)\right)$

$$\xi \left(t\right)-\xi \left(0\right)+{\xi }^{\prime }\left(0\right)t={\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))ds+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\varrho \left(s\right)ds$$

Taking absolute values and using $\left|\varrho \right(s\left)\right|\le \epsilon$, we obtain the exact bound

$$\begin{array}{ccc}\hfill |\xi \left(t\right)-\xi \left(0\right)+{\xi }^{\prime }\left(0\right)t-{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))ds|& =& {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left|\varrho \left(s\right)\right|ds\hfill \\ & \le & {\displaystyle \frac{\epsilon }{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}ds\hfill \\ & \le & {\displaystyle \frac{\epsilon t^{\rho }}{\Gamma (\rho +1)}}.\hfill \end{array}$$

Alternatively, for a uniform bound valid for all $t\in [0,I]$, the inequality becomes

$$|\xi \left(t\right)-\xi \left(0\right)+{\xi }^{\prime }\left(0\right)t-{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))ds|\le {\displaystyle \frac{\epsilon I^{\rho }}{\Gamma (\rho +1)}}.$$

The following inequality will be useful in our analysis: for $\rho ,\vartheta >0$,

$${\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}\le {\displaystyle \frac{e^{\vartheta t}}{{\vartheta }^{\rho }}}\Gamma \left(\rho \right).$$

Suppose that if we put $y=t-s$ in the above integral, we obtain

$${\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}=e^{\vartheta t}{\int }_0^t{y}^{\rho -1}{e}^{-\vartheta y}dy,$$

if we put $z=\vartheta y$ in above equality then we obtain

$$\begin{array}{ccc}\hfill {\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}& =& {\displaystyle \frac{e^{\vartheta t}}{{\vartheta }^{\rho }}}{\int }_0^{\vartheta t}{z}^{\rho -1}{e}^{-z}dz\hfill \\ & \le & {\displaystyle \frac{e^{\vartheta t}}{{\vartheta }^{\rho }}}{\int }_0^{\infty }{z}^{\rho -1}{e}^{-z}dz\hfill \\ & =& {\displaystyle \frac{e^{\vartheta t}}{{\vartheta }^{\rho }}}\Gamma \left(\rho \right).\hfill \end{array}$$

3. Main Results

In this section, we overcome the constraints imposed by the contractivity constants and establish the existence and uniqueness of solutions to Equation (1).

Assume that

$Q_1$:

$f\in \mathrm{C}(d\times {\mathbb{R}}^2,\mathbb{R}),g\in \mathcal{C}(d,[-\gamma ,I]$ verifying $t\ge g\left(t\right)$ on $[0,I]$.

$Q_2$:

There exists a positive constant P such that the function f satisfies the Lipschitz condition:

$$|f(t,{\phi }_1,{\xi }_1)-f(t,{\phi }_2,{\xi }_2)|\le P(|{\phi }_1-{\phi }_2|+|{\xi }_1-{\xi }_2\left|\right),\forall {\phi }_k,{\xi }_k\in \mathbb{R},andt\in [0,I].$$

$Q_3$:

${\displaystyle \frac{2P{I}^{\rho }}{\Gamma \left({\rho }_{+}1\right)}}<1$.

Theorem 1. 

Under the assumption that conditions $Q_1,Q_2$, and $Q_3$ are satisfied, Equation (1) admits a unique solution.

Proof. 

We begin by transforming Equation (1) into an equivalent fixed-point problem. Subsequently, we introduce the operator

$$M:\mathcal{C}\left(\right[-\gamma ,I],\mathbb{R})\to \mathcal{C}\left(\right[-\gamma ,I],\mathbb{R}),$$

defined as

$$\begin{array}{c}\hfill M\phi \left(t\right)=\left\{\begin{array}{cc}\phi \left(t\right),\hfill & t\in [-\gamma ,0]\hfill \\ \phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(s,\phi \left(s\right),\phi \left(g\left(s\right)\right))ds,\hfill & t\in [0,I].\hfill \end{array}\right.\end{array}$$

Our goal is to determine the unique fixed point of the operator F. To achieve this, we consider the Banach space $\mathfrak{Z}:=\mathcal{C}\left(\right[-\gamma ,I],\mathbb{R})$ equipped with the Chebyshev norm defined by

$${\parallel \phi \parallel }_{\mathfrak{C}}=\underset{t\in [-\gamma ,I]}{max}\left|\phi \left(t\right)\right|.$$

For every $\phi \left(t\right),\xi \left(t\right)\in \mathfrak{Z}$. For $t\in [-\gamma ,0]$, we get

$$\left|M\phi \right(t)-M\xi (t\left)\right|=0,t\in [0,I].$$

Therefore

$$\begin{array}{ccc}\hfill \left|M\phi \right(t)-M\xi (t\left)\right|& \le & {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left|f\right(s,\phi \left(s\right),\phi \left(g\left(s\right)\right)-f(s,\xi \left(s\right),\xi \left(g\left(s\right)\right))|ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left(\underset{-\gamma \le s\le I}{max}|\phi \left(s\right)-\xi \left(s\right)|+\underset{-\gamma \le s\le I}{max}|\phi \left(g\left(s\right)\right)-\xi \left(g\left(s\right)\right)|\right)ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\parallel \phi -\xi \parallel }_{\mathfrak{C}}{\int }_0^t{(t-s)}^{\rho -1}ds\hfill \\ & \le & {\displaystyle \frac{2P{I}^{\rho }}{\Gamma \left({\rho }_{+}1\right)}}{\parallel \phi -\xi \parallel }_{\mathfrak{C}}.\hfill \end{array}$$

Therefore, we obtain

$${\parallel M\phi -M\xi \parallel }_{\mathfrak{C}}\le {\displaystyle \frac{2P{I}^{\rho }}{\Gamma \left({\rho }_{+}1\right)}}{\parallel \phi -\xi \parallel }_{\mathfrak{C}},\phi \left(t\right),\xi \left(t\right)\in \mathfrak{Z}.$$

Thus, M is a contraction with respect to the Chebyshev norm on $\mathfrak{Z}$. By the Banach Contraction Principle, there exists a unique fixed point of M. This completes the proof. □

Theorem 2. 

Suppose that the assumptions $Q_1$ and $Q_2$ hold, Equation (1) possesses a unique solution.

Proof. 

Following a similar approach to Theorem 1, Let $\mathfrak{Z}:=\mathcal{C}\left(\right[-\gamma ,I],\mathbb{R})$ be the Banach space endowed with the Bielecki norm given by

$${\parallel \phi \parallel }_{\mathfrak{B}}=\underset{t\in [-\gamma ,I]}{max}\left|\phi \left(t\right)\right|e^{-\vartheta t}.$$

(3)

Given the similarity to the previous proof of Theorem 1, we highlight the key differences below:

$$\begin{array}{ccc}\hfill \left|M\phi \right(t)-M\xi (t\left)\right|& \le & {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left|f\right(s,\phi \left(s\right),\phi \left(g\left(s\right)\right)-f(s,\xi \left(s\right),\xi \left(g\left(s\right)\right))|ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left(\underset{-\gamma \le s\le I}{max}|\phi \left(s\right)-\xi \left(s\right)|e^{-\vartheta s}+\underset{-\gamma \le s\le I}{max}|\phi \left(g\left(s\right)\right)-\xi \left(g\left(s\right)\right)|e^{-\vartheta s}\right)ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\parallel \phi -\xi \parallel }_{\mathfrak{B}}{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}ds\hfill \\ & \le & {\displaystyle \frac{2P}{{\vartheta }^{\rho }}}{\parallel \phi -\xi \parallel }_{\mathfrak{B}}{e}^{\vartheta t}.\hfill \end{array}$$

Therefore, we obtain

$${\parallel M\phi -M\xi \parallel }_{\mathfrak{B}}\le \Lambda {\parallel \phi -\xi \parallel }_{\mathfrak{B}},where\Lambda ={\displaystyle \frac{2P}{{\vartheta }^{\rho }}}.$$

By selecting $\vartheta >0$ sufficiently large such that $\Lambda <1$, we apply the Banach Contraction Principle to conclude that M has a unique fixed point, which completes the proof. □

Remark 3. 

Consider the special case of Equation (1) with $g\left(t\right)=t-p$, where $p>0$ is a constant delay.

$$\left\{\begin{array}{cc}{}^{\mathrm{C}}D^{\rho }\phi \left(t\right)=f(t,\phi \left(t\right),\phi (t-p)),\hfill & t\in [0,I],1<\rho <2,\hfill \\ \phi \left(t\right)=\omega \left(t\right)={\omega }_0,{\phi }^{\prime }\left(t\right)={\omega }^{\prime }\left(t\right)={\omega }_1,\hfill & t\in [-\gamma ,0].\hfill \end{array}\right.$$

(4)

Next, we demonstrate the existence and uniqueness of solutions for the fractional-order differential equation, utilizing progressive contractions and a modified Lipschitz condition that differs from Theorem 1.

Theorem 3. 

Consider a continuous function $f:d\times {\mathbb{R}}^2\to \mathbb{R}$. Suppose there exists a positive constant P in such a way that

$$|f(t,{\phi }_1,\xi )-f(t,{\phi }_2,\xi )|\le P|{\phi }_1-{\phi }_2|,$$

for all ${\phi }_1,{\phi }_2,\xi \in \mathbb{R}$ as well as $t\in [0,I]$. In this case, Equation (4) admits a unique solution.

Proof. 

It is clear that Equation (4) is equivalent to the integral system given below:

$$\begin{array}{c}\hfill \phi \left(t\right)=\left\{\begin{array}{cc}\phi \left(t\right),\hfill & -p\le t\le 0,\hfill \\ \phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(s,\phi \left(s\right),\phi (s-p))ds,\hfill & 0\le t\le I,r=0,1.\hfill \end{array}\right.\end{array}$$

For progressive contractions, we divide $[0,I]$ into $\mathfrak{n}$ equal parts, each with length A, satisfying $0<A<p$ and $\mathfrak{n}A=I$. The partition is defined as follows:

$$0=A_0<A_1<···<A_{\mathfrak{n}}=I,A_k-A_{k-1}=A.$$

Furthermore, observe that if $t\le A_{k+1}$, then $t-p\le A_k$, which follows from the argument given below:

$$t\le A_{k+1}\Rightarrow t-p\le A_{k+1}-p\le A_{k+1}-A=A_k.$$

$\mathbf{Case}\mathbf{1}$: Consider the complete normed space $(E,\parallel ·{\parallel }_1)$ consisting of continuous functions $\phi :[-\gamma ,A_1]\to \mathbb{R}$, with the norm defined as follows:

$${\parallel \phi \parallel }_1=\underset{t\in [-\gamma ,A_1]}{max}\left|\phi \left(t\right)\right|e^{-\vartheta t},$$

we also take $\phi \left(t\right)=\omega \left(t\right),{\phi }^{\prime }\left(t\right)={\omega }^{\prime }\left(t\right)$ for $-p\le t\le 0$. Now, consider the mapping $M_1:E\to E$, given by the following:

$$\begin{array}{c}\hfill M\phi \left(t\right)=\left\{\begin{array}{cc}\phi \left(t\right),\hfill & -p\le t\le 0,\hfill \\ \phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(s,\phi \left(s\right)),\phi (s-p)ds,\hfill & 0\le t\le A_1.\hfill \end{array}\right.\end{array}$$

For $\phi \left(t\right),\xi \left(t\right)\in E,M_1\phi \left(t\right)=M_1\xi \left(t\right)$ if $t\in [-p,0]$, thus we take $t\in [0,A_1].$ Therefore

$$|M_1\phi \left(t\right)-M_1\xi \left(t\right)|\le {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}|f(s,\phi \left(s\right),\phi (s-p))-f(s,\xi \left(s\right),\xi (s-p))|ds.$$

Since $0\le s\le A_1$, we have $(s-p)\in [-p,0]$. Therefore, based on the definition of E, we obtain

$$\begin{array}{ccc}\hfill |M_1\phi \left(t\right)-M_1\xi \left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left[\right|f(s,\phi \left(s\right),\omega (s-p))-f(s,\xi \left(s\right),\omega (s-p))\hfill \\ & +& f(s,\phi \left(s\right),{\omega }^{\prime }(s-p))-f(s,\xi \left(s\right),{\omega }^{\prime }(s-p))\left|\right]ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}\left(\underset{-\gamma \le s\le A_1}{max}|\phi \left(s\right)-\xi \left(s\right)|e^{-\vartheta s}\right)ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\parallel \phi -\xi \parallel }_1{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}\le {\displaystyle \frac{P}{{\vartheta }^{\rho }}}{\parallel \phi -\xi \parallel }_{\mathfrak{B}}{e}^{\vartheta t}.\hfill \end{array}$$

Thus, we get

$$\parallel M_1\phi \left(t\right)-M_1{\xi \left(t\right)\parallel }_1\le \Lambda {\parallel \phi -\xi \parallel }_1,$$

where $\Lambda ={\displaystyle \frac{P}{{\vartheta }^{\rho }}}$. For $\vartheta >0$ satisfying $\Lambda <1$, $M_1$ becomes a contraction mapping, implying the existence of a unique fixed point ${\omega }_1,{\omega }_1^{\prime }\in M_1$, satisfying Equation (4) on $[-p,A_1]$.

• $\mathbf{Case}\mathbf{2}$: Next, we extend the interval to $[-p,A_2]$. Consider the complete normed space $(E_1,\parallel ·{\parallel }_2)$ of continuous functions $\phi :[-p,A_2]\to \mathbb{R}$, with the norm defined as follows:

$${\parallel \phi \parallel }_2=\underset{t\in [-\gamma ,A_1]}{max}\left|\phi \left(t\right)\right|e^{-\vartheta t},$$

we also take $\phi \left(t\right)={\omega }_1\left(t\right),{\phi }^{\prime }\left(t\right)={\omega }_1^{\prime }\left(t\right)$ for $-p\le t\le A_1$. Now, consider the mapping $M_2:E_1\to E_1$, given by

$$\begin{array}{c}\hfill M\phi \left(t\right)=\left\{\begin{array}{cc}{\phi }_1\left(t\right),\hfill & -p\le t\le 0,\hfill \\ \phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(s,\phi \left(s\right),\phi (s-p))ds,\hfill & A_1\le t\le A_2.\hfill \end{array}\right.\end{array}$$

For $\phi \left(t\right),\xi \left(t\right)\in E_1,M_2\phi \left(t\right)=M_2\xi \left(t\right)$ if $t\in [-p,A_1]$, thus we take $t\in [A_1,A_2].$ Therefore

$$|M_2\phi \left(t\right)-M_2\xi \left(t\right)|\le {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}|f(s,\phi \left(s\right),\phi (s-p))-f(s,\xi \left(s\right),\xi (s-p))|ds.$$

Since $0\le s\le A_2$, we have $(s-p)\in [-p,A_1]$. Therefore, based on the definition of $E_1$, we obtain

$$\begin{array}{ccc}\hfill |M_2\phi \left(t\right)-M_2\xi \left(t\right)|& \le & {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left[\right|f(s,\phi \left(s\right),{\omega }_1(s-p))-f(s,\xi \left(s\right),{\omega }_1(s-p))\hfill \\ & +& f(s,\phi \left(s\right),{\omega }_1^{\prime }(s-p))-f(s,\xi \left(s\right),{\omega }_1^{\prime }(s-p))\left|\right]ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}\left(\underset{-\gamma \le s\le A_1}{max}|\phi \left(s\right)-\xi \left(s\right)|e^{-\vartheta s}\right)ds\hfill \\ & \le & {\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\parallel \phi -\xi \parallel }_2{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}\le {\displaystyle \frac{P}{{\vartheta }^{\rho }}}{\parallel \phi -\xi \parallel }_{\mathfrak{B}}{e}^{\vartheta t}.\hfill \end{array}$$

Thus, we get

$$\parallel M_1\phi \left(t\right)-M_1{\xi \left(t\right)\parallel }_2\le \Lambda {\parallel \phi -\xi \parallel }_2,$$

where $\Lambda ={\displaystyle \frac{P}{{\vartheta }^{\rho }}}$. For $\vartheta >0$ satisfying $\Lambda <1$, $M_2$ becomes a contraction mapping, implying the existence of a unique fixed point ${\omega }_2,{\omega }_2^{\prime }\in M_2$, satisfying Equation (4) on $[-p,A_2]$.

• $\mathbf{Case}\mathbf{3}$: By iterating this process up to the nth step, we can construct a continuous mapping ${\omega }_n$ and ${\omega }_n^{\prime }$, similar to the previous steps, which yields a unique solution to Equation (4) on the interval $[-p,A_n]=[-p,I]$. □

4. Ulam–Hyers Stability Analysis

A stability analysis of Equation (1) is carried out within the Ulam–Hyers framework, as also addressed in [18,32].

Theorem 4. 

The conditions $Q_1$ and $Q_2$ hold, then the Equation (1) is Ulam–Hyers Stable.

Proof. 

Assuming $\xi$ satisfies Equation (2), Theorem 2 ensures the existence of a unique solution $\phi$ to the problem stated below:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^{\mathrm{C}}D^{\rho }\phi \left(t\right)=f(t,\phi \left(t\right),\phi \left(g\left(t\right)\right)),\hfill & t\in [0,I],1<\rho <2,\hfill \\ \phi \left(t\right)=\xi \left(t\right)={\xi }_0,{\phi }^{\prime }\left(t\right)={\xi }^{\prime }\left(t\right)={\xi }_1,\hfill & t\in [-\gamma ,0],\hfill \end{array}\right.\end{array}$$

consequently, we obtain

$$\begin{array}{c}\hfill \phi \left(t\right)=\left\{\begin{array}{cc}\xi \left(t\right),\hfill & -p\le t\le [-\gamma ,0],\hfill \\ \xi \left(0\right)+{\xi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(s,\phi \left(s\right),\phi \left(g\left(s\right)\right))ds,\hfill & t\in [0,I].\hfill \end{array}\right.\end{array}$$

From Remark 2, we also have

$$|\xi \left(t\right)-\xi \left(0\right)-{\xi }^{\prime }\left(0\right)t-{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))ds|\le {\displaystyle \frac{\epsilon I^{\rho }}{\Gamma (\rho +1)}},\forall t\in [0,I],$$

since $\left|\xi \right(t)-\phi (t\left)\right|=0$ on $[-\gamma ,0]$, it follows from $Q_2$ that for all $t\in [0,I]$,

$$\begin{array}{ccc}\hfill \left|\xi \right(t)-\phi (t\left)\right|& =& |\xi \left(t\right)-\xi \left(0\right)-{\xi }^{\prime }\left(0\right)t-{\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}f(t,\xi \left(t\right),\xi \left(g\left(t\right)\right))ds|\hfill \\ & +& {\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}|f(t,\xi \left(t\right),\xi \left(g\left(s\right)\right))-f(t,\phi \left(t\right),\phi \left(g\left(s\right)\right))|ds\hfill \\ & \le & {\displaystyle \frac{\epsilon I^{\rho }}{\Gamma (\rho +1)}}+{\displaystyle \frac{P}{\Gamma \left(\rho \right)}}{\int }_0^t{(t-s)}^{\rho -1}\left(\left|\xi \right(s)-\phi (s\left)\right|+\left|\xi \right(g\left(s\right))-\phi (g\left(s\right)\left)\right|\right)ds\hfill \\ & \le & {\displaystyle \frac{\epsilon I^{\rho }}{\Gamma (\rho +1)}}+{\displaystyle \frac{2P}{\Gamma \left(\rho \right)}}{\parallel \xi -\phi \parallel }_{\mathfrak{B}}{\int }_0^t{(t-s)}^{\rho -1}{e}^{\vartheta s}ds\hfill \\ & \le & {\displaystyle \frac{\epsilon I^{\rho }}{\Gamma (\rho +1)}}+{\displaystyle \frac{2P}{{\vartheta }^{\rho }}}{\parallel \xi -\phi \parallel }_{\mathfrak{B}}{e}^{\vartheta t}.\hfill \end{array}$$

Thus, we get

$${(1-\Lambda )\parallel \xi -\phi \parallel }_{\mathfrak{B}}\le {\displaystyle \frac{\epsilon I^{\rho }{e}^{\vartheta \gamma }}{\Gamma (\rho +1).}}$$

For a sufficiently large $\vartheta >0$ such that $\Lambda <1$, it follows that

$$|\xi \left(t\right)-\phi \left(t\right)|e^{-\vartheta t}\le {\parallel \xi -\phi \parallel }_{\mathfrak{B}}\le {\displaystyle \frac{\epsilon I^{\rho }{e}^{\vartheta \gamma }}{(1-\Lambda )\Gamma (\rho +1)}},\forall t\in [-\gamma ,I].$$

Therefore, we conclude that Equation (1) is stable in the sense of Ulam–Hyers. □

5. Numerical Simulations

5.1. Example

Consider the FDDE

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^{\mathrm{C}}D^{1.5}\phi \left(t\right)=-\phi \left(t\right)+0.5\phi (t-0.2),\hfill & t\in [0,0.3],\hfill \\ \phi \left(t\right)=1,{\phi }^{\prime }\left(t\right)=0,\hfill & t\in [-0.2,0].\hfill \end{array}\right.\end{array}$$

Define the operator $M:\mathcal{C}\left(\right[-0.2,0.3],\mathbb{R})\to \mathcal{C}\left(\right[-0.2,0.3],\mathbb{R})$ by

$$\begin{array}{c}\hfill M\phi \left(t\right)=\left\{\begin{array}{cc}1,\hfill & t\in [-0.2,0],\hfill \\ 1+{\displaystyle \frac{1}{\Gamma \left(1.5\right)}}{\int }_0^t{(t-s)}^0.5(-\phi \left(t\right)+0.5\phi (t-0.2))ds,\hfill & t\in [0,0.3].\hfill \end{array}\right.\end{array}$$

We proceed to demonstrate that M acts as a contraction mapping on the Banach space $\mathfrak{Z}=\mathcal{C}\left(\right[-0.2,0.3],\mathbb{R})$ equipped with Chebyshev norm

$$\parallel \phi \parallel =\underset{t\in [-0.2,0.3]}{max}\left|\phi \left(t\right)\right|.$$

For arbitrary function $\phi ,\psi \in \mathfrak{Z}$, and for any $t\in [0,0.3]$, it follows that

$$\begin{array}{ccc}\hfill \left|M\phi \right(t)-M\psi (t\left)\right|& =& \left|{\displaystyle \frac{1}{\Gamma \left(1.5\right)}}{\int }_0^t{(t-s)}^{0.5}[-\phi \left(s\right)+0.5\phi (s-0.2)+\psi \left(s\right)-0.5\psi (s-0.2)]ds\right|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(1.5\right)}}{\int }_0^t{(t-s)}^{0.5}\left(\right|\phi \left(s\right)-\psi \left(s\right)|+0.5|\phi (s-0.2)-\psi (s-0.2)\left|\right)ds\hfill \\ & \le & {\displaystyle \frac{1.5}{\Gamma \left(1.5\right)}}\parallel \phi -\psi \parallel {\int }_0^t{(t-s)}^{0.5}ds,\hfill \end{array}$$

where we use the uniform norm property and linearity of the integral operator. Since the integral evaluates to

$${\int }_0^t{(t-s)}^{0.5}ds={\displaystyle \frac{t^{1.5}}{1.5}},$$

we obtain the inequality

$$|M\phi \left(t\right)-M\psi \left(t\right)|\le {\displaystyle \frac{t^{1.5}}{\Gamma \left(1.5\right)}}\parallel \phi -\psi \parallel .$$

Taking the supremum over $t\in [0,0.3]$, we deduce that

$$\parallel M\phi \left(t\right)-M\psi \left(t\right)\parallel \le {\displaystyle \frac{{\left(0.3\right)}^{1.5}}{\Gamma \left(1.5\right)}}\parallel \phi -\psi \parallel \approx 0.1855\parallel \phi -\psi \parallel .$$

Since the contraction constant $0.1855$ is strictly less than one, the operator M is a contraction on $\mathfrak{Z}$. Since the operator M satisfies all the conditions of Theorem 1, the Banach Fixed Point Theorem guarantees the existence of a unique fixed point ${\phi }^{*}\in \mathfrak{Z}$,

$${\phi }^{*}=M{\phi }^{*},$$

which corresponds to the unique solution of the given FDDE.

5.2. Example

Let us consider the following nonlinear Caputo fraction order DDE of order $\rho =1.8$, defined over the interval $[0,1]$

$${}^{\mathrm{C}}D^{1.8}\phi \left(t\right)=f(t,\phi \left(t\right),\phi \left(g\left(t\right)\right))=a\phi \left(t\right)+b\phi \left(g\left(t\right)\right),$$

subject to the initial conditions

$$\phi \left(t\right)={\omega }_0=1,{\phi }^{\prime }\left(t\right)={\omega }_1=0,t\in [-0.3,0],$$

where the delay term is given by $g\left(t\right)=t-\tau$ with $\tau =0.3$, and the constants $a=0.5, b=0.3$. Clearly, the delay satisfies $g\left(t\right)\le t$ for all $t\in [0,1]$, preserving causality. Define the nonlinear mapping $f(t,x,y)=ax+by$, which is continuous and satisfies the global Lipschitz condition in both x and y. That is, for any $x_1,x_2,y_1,y_2\in \mathbb{R}$,

$$|f(t,x_1,y_1)-f(t,x_2,y_2)|\le |a\left|\right|x_1-x_2|+|b\left|\right|y_1-y_2|\le P(|x_1-x_2|+|y_1-y_2\left|\right),$$

where $P=max\left(\right|a|,|b\left|\right)=0.5,$ satisfying the Lipschitz condition required for uniqueness. Now, let $\mathfrak{Z}=\mathcal{C}\left(\right[-0.3,1],\mathbb{R})$ be the Banach space of continuous function equipped with the Bielecki norm

$${\parallel \phi \parallel }_{\mathfrak{B}}=\underset{t\in [-0.3,1]}{max}\left|\phi \left(t\right)\right|e^{-\vartheta t},$$

with $\vartheta >0$. To ensure that the corresponding integral operator derived from the equation is a contraction, we compute the contraction constant:

$$\Lambda ={\displaystyle \frac{2P}{{\vartheta }^{\rho }}}.$$

Taking $\vartheta =4,$ we compute

$$\Lambda ={\displaystyle \frac{2\times 0.5}{4^{1.8}}}={\displaystyle \frac{1}{e^{1.8ln\left(4\right)}}}={\displaystyle \frac{1}{e^{2.494}}}\approx {\displaystyle \frac{1}{12.12}}\approx 0.0825,$$

which clearly satisfies $\Lambda <1$. Therefore, the operator defined by the equivalent integral equation

$$\begin{array}{c}\hfill M\phi \left(t\right)=\left\{\begin{array}{cc}\phi \left(t\right),\hfill & t\in [-0.3,0],\hfill \\ \phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(1.8\right)}}{\int }_0^t{(t-s)}^{0.8}f(s,\phi \left(s\right),\phi \left(g\left(s\right)\right))ds,\hfill & t\in [0,1],\hfill \end{array}\right.\end{array}$$

is a strict contraction in the Bielecki norm on $\mathfrak{Z}$. Since the operator M satisfies all the conditions of Theorem 2.

5.3. Example

Consider the Caputo-type FDDE of order $\rho =1.7$, defined on the interval $[0,1.2]$, with a discrete delay of $p=0.3$. The equation takes the form:

$${}^{\mathrm{C}}D^{1.7}\phi \left(t\right)=f(t,\phi \left(t\right),\phi (t-p)),$$

with the initial conditions:

$$\phi \left(t\right)=ln(t+2),{\phi }^{\prime }\left(t\right)={\displaystyle \frac{1}{t+2}},fort\in [-p,0].$$

Define the function

$$f(t,x,y)={\displaystyle \frac{1}{5}}(x+2y)+sin\left(t\right).$$

This function is continuous, and the partial Lipschitz condition in the first arguments is satisfied with Lipschitz constant $P={\displaystyle \frac{3}{5}}$, since

$$|f(t,x_1,y_1)-f(t,x_2,y_2)|\le {\displaystyle \frac{1}{5}}|x_1-x_2|+{\displaystyle \frac{2}{5}}|y_1-y_2|\le {\displaystyle \frac{3}{5}}\left(\right|x_1-x_2|+|y_1-y_2\left|\right).$$

To apply the fixed point theorem under the Bielecki norm, we choose $\vartheta =3$. Then the contraction constant becomes

$$\Lambda ={\displaystyle \frac{3P}{{\vartheta }^{1.7}}}={\displaystyle \frac{3\left(0.6\right)}{3^{1.7}}}={\displaystyle \frac{1.8}{6.47}}\approx 0.278<1.$$

Since $\Lambda <1$, the contraction condition is fulfilled. We divide the interval $[0,1.2]$ into subintervals of length A = 0.2 < p, i.e,

$$A_0=0,A_1=0.2,A_2=0.4,A_3=0.6,A_4=0.8,A_5=1.0,A_6=1.2.$$

For $t\in [0,A_1]$, the solution is given by the integral form

$$\phi \left(t\right)=\phi \left(0\right)+{\phi }^{\prime }\left(0\right)t+{\displaystyle \frac{1}{\Gamma \left(1.7\right)}}{\int }_0^t{(t-s)}^{0.7}f(s,\phi \left(s\right),\phi (s-0.3))ds,$$

since $\Gamma \left(1.7\right)\approx 0.9086,$ and $\phi (s-p)$ lies within the known initial segment for $s\le A_1,$ the mapping constructed is well-defined. By applying the progressive contraction principal, the unique solution on $[0,A_1]$ extends recursively to $[0,1.2]$, yielding a unique global solution over $[-0.3,1.2]$. Therefore, all the conditions of Theorem 3 are satisfied.

5.4. Example (Demonstrating Ulam–Hyers Stability)

Let us consider the following nonlinear FDDE

$${}^{\mathrm{C}}D^{1.65}\phi \left(t\right)=f(t,\phi \left(t\right),\phi (t-p)),t\in [0,1],$$

subject to initial condition

$$\phi \left(t\right)=1.2+cos\left(t\right),{\phi }^{\prime }\left(t\right)=-sin\left(t\right),t\in [-p,0],$$

where $p=0.3$, and the function f is defined as

$$f(t,x,y)=0.5x+0.3y+e^{-t}.$$

For any $x_1,x_2\in \mathbb{R}$ and fixed $y\in \mathbb{R}$, we compute

$$|f(t,x_1,y)-f(t,x_2,y)|=|0.5x_1-0.5x_2|=0.5|x_1-x_2|.$$

Thus, the Lipschitz constant $P=0.5$, which satisfy the required condition of the Theorem 4. Now assume that there exists a function $\xi \left(t\right)$ to satisfy the perturbed inequality

$$|\xi \left(t\right)-\xi \left(0\right)-{\xi }^{\prime }\left(0\right)t-{\displaystyle \frac{1}{\Gamma \left(1.65\right)}}{\int }_0^t{(t-s)}^{0.65}f(s,\xi \left(s\right),\xi (s-p))ds|\le {\displaystyle \frac{\epsilon t^{1.65}}{\Gamma (1.65+1)}},$$

let us take $\epsilon =0.01$, and work in the weighted norm space

$${\parallel \psi \parallel }_{\mathfrak{B}}=\underset{t\in [-p,1]}{sup}\left|\psi \left(t\right)\right|e^{-4t},with\vartheta =4.$$

Since

$$\Lambda ={\displaystyle \frac{2P}{{\vartheta }^{\rho }}}={\displaystyle \frac{2\times 0.5}{4^{1.65}}}.$$

Using $4^{1.65}\approx e^{1.65·ln\left(4\right)}=e^{1.65\left(1.386\right)}\approx e^{2.2869}\approx 9.84,$ we get

$$\Lambda \approx {\displaystyle \frac{1}{9.84}}\approx 0.1016<1.$$

This verifies the contraction requirement. Hence according to Theorem 4, there exists a unique solution $\phi \left(t\right)$ such that

$${\parallel \xi -\phi \parallel }_{\mathfrak{B}}\le {\displaystyle \frac{\epsilon I^{\rho }{e}^{\vartheta p}}{(1-\Lambda )\Gamma (\rho +1)}},$$

where $I=1,\rho =1.65,p=0.3,$ and $\Gamma \left(2.65\right)\approx 1.49,$ we get

$${\parallel \xi -\phi \parallel }_{\mathfrak{B}}\le {\displaystyle \frac{0.01·1^{1.65}·e^{1.2}}{(1-0.1016)·1.49}}\approx {\displaystyle \frac{0.01·3.32}{0.8984·1.49}}\approx {\displaystyle \frac{0.0332}{1.3386}}\approx 0.0248.$$

This shows that the approximate solution $\xi \left(t\right)$ remains close to the exact solution $\phi \left(t\right)$, and the distance is quantitatively bounded, which proves Ulam–Hyers stability.

6. Conclusions

This research aimed to pioneer novel existence and uniqueness criteria for Caputo fractional solutions by leveraging the Chebyshev and Bielecki norms more efficiently, thereby ensuring uniqueness. A special case involving constant delay was examined using Burton’s method, which facilitated the existence and uniqueness of solutions under mitigated Lipschitz conditions for the third variable. Furthermore, the study explored Ulam–Hyers stability and demonstrated it without relying on traditional techniques, such as Picard operators or Gronwall-type inequalities. In addition, numerical simulations have been carried out to validate and illustrate the theoretical results obtained in this work. The findings have significant implications for future research in fractional calculus, particularly in modeling real-world systems with memory effects. Potential applications include improved predictive models in fields such as control theory, signal processing, and viscoelastic materials. These results offer a more generalized framework for analyzing fractional systems, enabling future studies to handle complex models with delays and weaker regularity assumptions. The approach also introduces alternative analytical tools, encouraging the development of new techniques beyond traditional fixed-point and inequality-based methods.