Hyers–Ulam and Hyers–Ulam–Rassias Stability for a Class of Fractional Evolution Differential Equations with Neutral Time Delay

Abstract

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann–Liouville and Caputo fractional derivatives with orders $0<\alpha <1$ and $1<\beta <2$. We identify the infinitesimal generator of the cosine family and analyze the stability of the mild solution using both Hyers–Ulam–Rassias and Hyers–Ulam stability methodologies, ensuring robust and reliable results for fractional dynamic systems with delay. In order to guarantee that the features of invariance under transformations, such as rotations or reflections, result in the presence of fixed points that remain unchanging and represent the consistency and balance of the underlying system, fixed-point theorems employ the symmetry idea. Lastly, the results obtained are applied to a fractional order nonlinear wave equation with finite delay with respect to time.

1. Introduction

Fractional evolution differential equations (FEDEs) constitute a specialized category of differential equations that expand upon the traditional framework of evolution equations by integrating fractional derivatives. In contrast to integer-order derivatives, which denote instantaneous rates of change, fractional derivatives encapsulate memory effects and long-range dependencies inherent in dynamical systems. In recent years, FEDEs have garnered considerable scholarly attention owing to their capacity to model intricate phenomena characterized by anomalous diffusion, non-local interactions, and fractal-like behavior (see [1,2,3,4]).

The versatility of FEDEs spans a wide array of disciplines, including physics, biology, finance, and engineering, especially in cases where conventional differential equations fall short in describing complex dynamics. In physics, FEDEs are instrumental in analyzing viscoelastic materials and anomalous diffusion and sub-diffusion processes. In biological sciences, they provide valuable insights into population dynamics, patterns of disease transmission, and neural system behaviors. Meanwhile, in finance, these equations effectively model market dynamics shaped by long-term dependencies and memory-driven pheno- mena [5,6,7,8,9,10].

The study of neutral evolution within fractional differential equations in FDEs is of paramount importance for several reasons. Firstly, it provides valuable insights into the robustness and stability of systems governed by FDEs. By examining how small variations in fractional order influence system behavior, researchers can develop more resilient and adaptable systems. Secondly, such investigations contribute to the theoretical advancement of fractional calculus, highlighting the sensitivity of solutions to parameter changes and expanding the analytical tools available for exploring these complex systems [11,12,13,14,15].

A key aspect of this exploration is the stability analysis of FDEs, which is critical for understanding the response of solutions to disturbances. Stability considerations are essential for designing and controlling reliable systems. Among various approaches, Hyers–Ulam–Rassias stability has gained prominence due to its broader and more flexible framework. This concept extends Ulam stability, originally formulated for functional equations, by incorporating generalized perturbation constraints as proposed by Th. M. Rassias. This generalized approach significantly enriches the mathematical framework for analyzing the stability of FDEs and their applications [16,17].

Hyers–Ulam–Rassias stability plays a pivotal role in the context of FDEs by providing a quantitative measure of solution resilience to perturbations. This stability framework evaluates whether small changes in initial conditions or parameters result in proportionally minor deviations in solutions, confined within a predefined range that adapts to the characteristics of the perturbation. This capability is critical for understanding and ensuring the robustness of systems modeled using FDEs. For an in-depth discussion, see [18,19,20,21,22].

In the study of fractional evolution differential equations, the idea of symmetry is crucial, especially when determining the existence and uniqueness of their solutions. Because symmetry reduces complexity and makes it easier to identify invariant features, it frequently simplifies the structure of these equations. To demonstrate the existence and uniqueness of solutions to such equations, the fixed-point theorem—a key technique in functional analysis—is frequently used. Banach’s fixed-point theorem, for example, is used in contraction maps to show that, under certain circumstances, such as Lipschitz continuity of the fractional derivative operator, there is a single fixed point (solution). Further encouraging the applicability of fixed-point methods is the symmetry inherent in some fractional differential operators or boundary conditions, which guarantees that solutions match the geometric or physical invariance of the system. By utilizing the interaction between symmetry and fixed-point theorems, researchers have successfully solved a wide class of fractional evolution equations that arise in a variety of fields, including biology, engineering, and physics.

Mixed fractional derivatives combine fractional derivatives of multiple orders, such as the Caputo and Riemann–Liouville derivatives, into a single differential equation. These derivatives extend the classical integer-order derivatives, thereby facilitating the modeling of systems characterized by complex, non-local, and memory-dependent behaviors. A mixed fractional differential equation may include terms using both $\left({\mathcal{D}}^{\alpha }\right)$ and $\left({\mathcal{D}}^{\beta }\right)$ derivatives, where $\alpha ,\beta$ are non-integer orders, indicating various physical processes or scales of impact.

Mixed fractional derivatives have been found in research to be very beneficial in capturing the dynamics of systems with many types of anomalous diffusion or variable rates of temporal evolution occurring at the same time. Advanced mathematical methods are necessary to analyze the existence, uniqueness, and stability of solutions, which are essential for ensuring the robustness and reliability of models across diverse scenarios [23,24]. M. Fečkan and J. Wang’s work [25] on mixed-order fractional differential equations demonstrates the applicability of these models in various fields, highlighting their ability to more accurately represent real-world phenomena compared to traditional models. This method is especially important in domains such as physics and engineering, where accurate models of dynamic systems are required.

Overall, the investigation of mixed fractional derivatives offers a rich and flexible framework for dealing with complex differential equations, opening the door to more accurate and complete models in applied mathematics and diverse scientific disciplines [26,27].

Recently, Telli et al. investigated [28] the existence and stability of solutions for a broad class of Riemann–Liouville (RL) fractional differential equations (FDEs) distinguished by variable order and finite delay for the flowing problem:

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{0+}^{\eta \left(\mathit{r}\right)}\xi \left(r\right)& =\upsilon (\mathit{r},{\xi }_{\mathit{r}}),\mathit{r}\in [0,\mathcal{M}],\hfill \\ \hfill \xi \left(r\right)& =\varpi \left(\mathit{r}\right),\mathit{r}\in [-\nu ,0],\nu >0,\hfill \end{array}$$

where $\mathcal{M}\in (0,\infty )$, $\eta \left(\mathit{r}\right)\in (0,1]$, and ${\mathfrak{D}}_{0+}^{\eta \left(\mathit{r}\right)}$ represent fractional differential operators with variable order $\eta (·)$. By establishing a partition founded on generalized intervals, the authors introduced a piecewise-constant function, denoted as $\eta \left(\mathit{r}\right)$. This innovative approach allowed them to transform the original variable-order Riemann–Liouville (RL) fractional differential equations (FDEs), which incorporate both variable order and finite delay, into equivalent standard Riemann–Liouville (RL) fractional differential equations (FDEs) with a constant fractional order for each interval delineated by the partition.

Moreover, Qien Li and Yong Zhou [29] examined the existence of mild solutions for a particular class of Hilfer fractional stochastic evolution equations with order $1<\gamma <2$ and type $0\le \delta \le 1$ that have the following form:

$$\begin{array}{cc}\hfill {\displaystyle {}^{\mathcal{H}}\mathfrak{D}_{0^{+}}^{\gamma ,\delta }\zeta \left(\nu \right)=\mathcal{A}\zeta \left(\nu \right)+\mathcal{F}(\nu ,\zeta )+\mathcal{H}(\nu ,\zeta ,\zeta )\frac{d\eta }{d\nu },}& \nu \in \mathcal{J}=[0,\mathsf{b}],\hfill \\ \hfill \left({\mathcal{I}}_{0^{+}}^{2-\alpha }\zeta \right)\left(0\right)={\zeta }_0\in L_0^2(\Omega ,\mathbb{H}),\\ \hfill {\left({\mathcal{I}}_{0^{+}}^{2-\alpha }\zeta \right)}^{\prime }\left(0\right)={\zeta }_1\in L_0^2(\Omega ,\mathbb{H}).\end{array}$$

Thus, ${}^{\mathcal{H}}\mathfrak{D}_{0^{+}}^{\gamma ,\delta }(·)$ is the Hilfer fractional derivative with order $1<\gamma \le 2$ and type $0<\delta <1$. Moreover, ${\mathcal{I}}_{0^{+}}^{2-\alpha }(·)$ is the Riemann–Liouville integral operator with order $(2-\alpha )$ where $\alpha =\gamma -\delta (2-\gamma )$. The findings demonstrate the existence of mild solutions in both compact and non-compact contexts.

The authors of [30] investigated the existence and uniqueness of solutions to initial value problems associated with delay fractional differential equations by employing a combination of Riemann–Liouville and Caputo fractional derivatives.

$$\begin{array}{ccc}\hfill {\displaystyle {}^{\mathcal{R}\mathcal{L}}\mathfrak{D}^{\mathfrak{a}}{[}^{\mathcal{C}}{\mathcal{D}}^{\mathfrak{b}}\zeta \left(\nu \right)-\mathcal{H}(\nu ,{\zeta }_{\nu })]}& =\mathcal{F}(\nu ,{\zeta }_{\nu }),\hfill & \hfill \nu \in [0,\mathsf{d}],\\ \hfill \zeta \left(\nu \right)& =\mathcal{P}\left(\nu \right),\hfill & \hfill \nu \in [-\mathsf{c},0],\\ \hfill \underset{\nu \to 0}{lim}{\nu }^{1-\mathfrak{a}}{[}_{\mathcal{C}}{\mathcal{D}}^{\mathfrak{b}}\zeta \left(\nu \right)]& =0,\zeta \left(0\right)={\zeta }_0.\hfill \end{array}$$

Here, ${}^{\mathcal{R}\mathcal{L}}\mathfrak{D}^{\mathfrak{a}}(·)$, and ${}^{\mathcal{C}}\mathfrak{D}^{\mathfrak{b}}(·)$ denote the Riemann–Liouville and Caputo fractional derivatives, respectively, with orders $0<\mathfrak{b}<1,1<\mathfrak{a}<2$. Moreover, by applying Krasnoselskii’s fixed-point theorem within a weighted Banach space, the authors establish sufficient conditions for the existence of solutions to initial value problems concerning delay fractional differential equations that incorporate both Riemann–Liouville and Caputo fractional derivatives over an infinite interval.

Building on the insights and methodologies from these foundational studies, this article delves into the existence and stability of mild solutions for the following mixed neutral evolution fractional differential problem.

$$\left\{\begin{array}{cc}{\displaystyle {}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)-\gamma (h,\sigma ,{\sigma }_h)]=\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h),}\hfill & h\in J=[0,b],\hfill \\ \sigma \left(h\right)=\delta \left(h\right),\hfill & h\in [-a,0],\hfill \\ \underset{h\to 0}{lim}{h}^{1-\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)]=0,{\sigma }^{\prime }\left(0\right)={\sigma }_0,\hfill \end{array}\right.$$

(1)

(i) 

${}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }(·)$, and ${}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }(·)$ are the Riemann–Liouville and Caputo fractional derivatives, respectively, where the orders are $0<\alpha <1,1<\beta <2$ and $1<\alpha +\beta <2$;

(ii) 

$\mathcal{A}:D\left(\mathcal{A}\right)\subset \mathbb{H}\to \mathbb{H}$ is an infinitesimal generator of a strongly continuous cosine family, where $\mathbb{H}$ signifies a Banach space;

(iii) 

${\left\{\mathcal{R}\left(h\right)\right\}}_{t\ge 0}(·)$ is a cosine family operator such that $\parallel \mathcal{R}\left(h\right)\parallel \le \mathcal{M},\mathcal{M}\le 1$;

(iv) 

$\kappa (·),\gamma (·):J\times \mathbb{H}\times \mathbb{H}\to \mathbb{H}$ is a specified function that meets certain conditions;

(v) 

$\delta \left(h\right):[-a,0]\to \mathbb{H}$ is a continuous function.

This article focuses on the stability of mild solutions for fractional evolution equations incorporating neutral dynamics and finite delay. The limited research leveraging mixed fractional differentiation in this context highlights a significant gap in the literature, making it imperative to address this area to deepen the scientific understanding and broaden the practical applications of such equations. For the following reasons, we decided on two different derivatives:

• Physical Analysis of Procedures: Different physical events are linked to different kinds of fractional derivatives:
  • Caputo derivatives: They are helpful in the physical sciences and engineering because they may be used in systems whose beginning circumstances are expressed in terms of integer derivatives.
  • Riemann–Liouville derivatives capture non-local effects and are helpful for modeling viscoelastic materials and anomalous diffusion. By selecting both, initial-value issues and long-range memory effects can be represented together.
• Improved Modeling Accuracy: One fractional derivative type is insufficient to adequately simulate the dynamics of some systems. For example, viscoelastic material relaxation behavior could be modeled by a Caputo derivative. The memory-driven diffusion in porous media could be modeled using a Riemann–Liouville derivative. By combining them, the model is guaranteed to support several dynamic system components.

Despite the importance of mixed fractional differentiation and its various applications, we have not found any research that studies the conditions necessary for the existence of solutions and their uniqueness on a fractional evolution equation with a finite time delay. Therefore, we presented this research as a comprehensive study in addition to studying the stability of solutions using Hyers–Ulam–Rassias and Hyers–Ulam stability. Finally, we applied the results we reached to a nonlinear wave equation with a fractional order with respect to time and a time delay.

The article is organized as follows:

• Section 2 introduces fundamental concepts and preliminary results essential for the subsequent analysis.
• Section 3 and Section 4 delve into the structure of mild solutions utilizing the cosine family framework, and present the results on the existence of solutions, respectively.
• Section 5 examines the Hyers–Ulam–Rassias stability and Hyers–Ulam stability for system (1), providing critical insights into the system’s resilience to perturbations. Finally, this article concludes with key findings supplemented by computational applications to illustrate the theoretical results.

2. Preliminaries

This section provides a detailed overview of the concepts and terminology pertinent to the study, with a focus on fractional calculus and the operators associated with cosine and sine families. Additionally, a series of lemmas are presented to support and strengthen the findings of this research. These foundational elements lay the groundwork for the analytical and theoretical developments discussed in subsequent sections.

Definition 1 

([31]). For at least the nth continuously differentiable function $\mathcal{F}:[0,\infty )\to \mathbb{R}$, the Caputo derivative of fractional order α is defined as

$$\begin{array}{c}\hfill {}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\alpha }\mathcal{F}\left(h\right)={\displaystyle \frac{1}{\Gamma (\mathfrak{n}-\alpha )}}{\int }_0^h{(h-\ell )}^{\mathfrak{n}-\alpha -1}{\mathcal{F}}^{\left(\mathfrak{n}\right)}(\ell )d\ell ,\mathfrak{n}-1<\alpha <\mathfrak{n},\mathfrak{n}\in \mathbb{N}.\end{array}$$

Definition 2 

([31]). Let $\mathcal{F}:[0,\infty )\to \mathbb{R}$ be the nth continuously differentiable function. Then, the derivative of fractional order α due to the Riemann–Liouville is presented as

$$\begin{array}{c}\hfill {}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }\mathcal{F}(\ell )={\displaystyle \frac{1}{\Gamma (\mathfrak{n}-\alpha )}}{\displaystyle \frac{d^{\mathfrak{n}}}{d{h}^{\mathfrak{n}}}}\left({\int }_0^{\ell }{(\ell -h)}^{\mathfrak{n}-\alpha -1}\mathcal{F}\left(h\right)dh\right),\mathfrak{n}-1<\alpha <\mathfrak{n};\mathfrak{n}\in \mathbb{N}.\end{array}$$

Definition 3 

([31]). The left fractional integrals of the function $\mathcal{F}$ are

$${\mathcal{I}}_a^{\alpha }\mathcal{F}\left(h\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^h{\left(h-\ell \right)}^{\mathfrak{d}-1}\mathcal{F}(\ell )d\ell ,h>a,\alpha >0.$$

Lemma 1 

([30]). Let $\mathfrak{n}\in \mathbb{N},\mathfrak{n}-1<\alpha <\mathfrak{n}$, and $\mathcal{F}\left(h\right)\in C^{\mathfrak{n}}[0,1]$. Then,

$$\begin{array}{c}\hfill {\mathcal{I}}^{\alpha }\left({}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\alpha }\mathcal{F}\left(h\right)\right)=\zeta \left(h\right)+a_0+a_{1}h+\dots +a_{\mathfrak{n}-1}{h}^{\mathfrak{n}-1}.\end{array}$$

Lemma 2 

([30]). The unique solution of the linear fractional differential equation,

$${}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }\mathcal{F}\left(h\right)=0,$$

is given by

$$\begin{array}{c}\hfill \mathcal{F}\left(h\right)=a_1{h}^{\alpha -1}+a_2{h}^{\alpha -2}+a_3{h}^{\alpha -3}+\cdots +a_{\mathfrak{n}}{h}^{\alpha -\mathfrak{n}},a_i\in \mathbb{R},i=1,2,\cdots ,\mathfrak{n},\end{array}$$

such that $\mathfrak{n}=\left[\alpha \right]+1$, and $\left[\alpha \right]$ denotes the integer part of the real number α.

Definition 4 

([32]). Let $0<\mathfrak{R}\left(\alpha \right)<1$, $\mathcal{F}\left(\zeta \right)\in A{C}^n[0,b]$ for any $b>0$, then $\left|\mathcal{F}\left(\zeta \right)\right|\le B{e}^{q_0\zeta },(\zeta >b>0),B,q_0>0$, and the finite limits are ${lim}_{\zeta \to 0+}{[}^{\mathcal{R}L}{\mathcal{D}}_{0+}^k{\mathcal{I}}_{0^{+}}^{\mathfrak{n}-\alpha }\mathcal{F}\left(\zeta \right)]$ and ${lim}_{\zeta \to \infty }{[}^{\mathcal{R}L}{\mathcal{D}}_{0+}^k{\mathcal{I}}_{0^{+}}^{\mathfrak{n}-\alpha }\mathcal{F}\left(\zeta \right)]$, where $k=0,1,\cdots ,n-1.$ The Laplace transform is as follows:

$$\begin{array}{c}\hfill L\left\{{}^{\mathcal{R}L}\mathcal{D}_{0^{+}}^{\alpha }\mathcal{F}\left(\zeta \right)\right\}\left(\lambda \right)={\lambda }^{\alpha }\mathcal{L}\left\{\mathcal{F}\left(\zeta \right)\right\}\left(\lambda \right)-\left({\mathcal{I}}_{0^{+}}^{\mathfrak{n}-\alpha }\mathcal{F}\left(\zeta \right)\right)(0+).\end{array}$$

(2)

Definition 5 

([33]). If ${\left\{\mathcal{Q}\left(h\right)\right\}}_{h\in \mathbb{R}}:\mathbb{H}\to \mathbb{H}$ is a 1-parameter family of bounded linear operators where $\mathbb{H}$ denotes the Banach space, the below properties can be realized:

(a) 

$\mathcal{Q}\left(0\right)=I$;

(b) 

$\mathcal{Q}(\ell +h)+\mathcal{Q}(\ell -h)=2\mathcal{Q}(\ell )\mathcal{Q}\left(h\right)$ for all $\ell ,h\in \mathbb{R}$;

(c) 

$\mathcal{Q}\left(h\right)\zeta$ is a continuous on $\mathbb{R}$ for each $\zeta \in \mathbb{H}$.

Then, ${\left\{\mathcal{Q}\left(h\right)\right\}}_{h\in \mathbb{R}}:\mathbb{H}\to \mathbb{H}$ is called a strongly continuous cosine family. Additionally, a 1-parameter family ${\left\{\mathcal{T}\left(h\right)\right\}}_{h\in \mathbb{R}}$ that satisfies

$$\mathcal{Q}\left(h\right)\zeta ={\int }_0^h\mathcal{R}(\ell )\zeta d\ell ,\zeta \in \mathbb{H},h\in \mathbb{R},$$

is referred to as a sine family related to the strongly continuous cosine family.

Lemma 3 

([33]). Consider ${\left\{\mathcal{R}\left(h\right)\right\}}_{\mathbf{h}\in \mathbb{R}}$, a strongly continuous cosine family in $\mathbb{H}$ with the property that ${\parallel \mathcal{Q}\left(h\right)\parallel }_{\mathcal{B}\left(\mathbb{H}\right)}\le M{e}^{\xi \left|h\right|},h\in \mathbb{R}$, where $\left({\mathcal{B}\left(\mathbb{H}\right);\parallel .\parallel }_{\mathcal{B}\left(\mathbb{H}\right)}\right)$ denotes the Banach space of all bounded linear operators from $\mathbb{H}$ to itself. Let $\mathcal{A}$ be an infinitesimal generator of ${\left\{\mathcal{R}\left(h\right)\right\}}_{h\in \mathbb{R}}$. Then, for $Re\lambda >\xi ,{\lambda }^2\in \rho \left(\mathcal{A}\right)$, we have

$$\lambda \mathrm{R}({\lambda }^2;\mathcal{A})\zeta ={\int }_0^{\infty }{e}^{-\lambda \mathbf{h}}\mathcal{R}\left(h\right)\zeta dh,\mathrm{R}({\lambda }^2;\mathcal{A})\zeta ={\int }_0^{\infty }{e}^{-\lambda h}\mathcal{Q}\left(h\right)\zeta dh,\zeta \in \mathbb{H},$$

where $\mathrm{R}(\lambda ;\mathcal{A})={(\lambda I-\mathcal{A})}^{-1}$ is the resolvent of the operator $\mathcal{A}$ and $\lambda \in \varrho \left(\mathcal{A}\right)$ belongs to the resolvent set of $\mathcal{A}$.

The infinitesimal generator of ${\left\{\mathcal{R}\left(h\right)\right\}}_{h\in \mathbb{R}}$ is defined by

$$\mathcal{A}\zeta =\underset{h\to 0}{lim}{\displaystyle \frac{d^2}{d{h}^2}}\mathcal{R}\left(h\right)\zeta ,\mathrm{for}\mathrm{all}\zeta \in \mathcal{D}\left(\mathcal{A}\right),$$

where the domain $\mathcal{D}\left(\mathcal{A}\right)$ is provided by

$$\mathcal{D}\left(\mathcal{A}\right)=\{\zeta \in \mathbb{H}:\mathcal{Q}\left(h\right)\zeta \in {\mathcal{C}}^2(\mathbb{R},\mathbb{H})\}.$$

It is well established that the infinitesimal generator $\mathcal{A}$ is a closed, densely defined operator in $\mathbb{H}.$ Subsequently, to present our findings, the following is required.

Definition 6 

([33]). Given $\nu >0$, and

$${\psi }_{\varrho }\left(\nu \right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\nu )}^k}{\Gamma (k+1)\Gamma (1-\varrho (k+1\left)\right)}},\varrho \in (0,1),\nu \in \mathbb{C},$$

then ${\psi }_{\varrho }\left(\nu \right)$ is called Mainardi’s Wright-type function and satisfies

$${\psi }_{\varrho }\left(\nu \right)\ge 0,{\int }_0^{\infty }{\theta }^{\xi }{\psi }_{\varrho }\left(\theta \right)d\theta ={\displaystyle \frac{\Gamma (1+\xi )}{\Gamma (1+\varrho \xi )}},\xi >-1.$$

3. Structure of Mild Solution

At this stage, a mild solution is formulated by leveraging the cosine family operator in conjunction with the Laplace transform. This approach facilitates the representation of solutions to fractional evolution equations by combining the dynamic properties of the cosine family with the analytical power of the Laplace transform, enabling a comprehensive framework for addressing the problem at hand.

Lemma 4. 

The problem (1) is addressed in the corresponding integral form, as shown below:

$$\begin{array}{c}\hfill \sigma \left(h\right)=\left\{\begin{array}{cc}{\mathcal{I}}^{\alpha +\beta }[\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)]+{\mathcal{I}}^{\beta }\gamma (h,\sigma ,{\sigma }_h)+\delta \left(0\right)+{\sigma }_{0}h,\hfill & h\in [0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.\end{array}$$

(3)

Proof. 

According to Definitions 2, we have

$$\begin{array}{c}\hfill {}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }\sigma \left(h\right)={\mathcal{I}}^{\alpha }[\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)]+\gamma (h,\sigma ,{\sigma }_h)+c_0{h}^{\alpha -1},h\in (0,b].\end{array}$$

(4)

Since the condition ${lim}_{h\to 0}{h}^{1-\alpha }{[}_{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)]=0$, $c_0=0$, we can write (4) as

$$\begin{array}{c}\hfill {}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }\sigma \left(h\right)={\mathcal{I}}^{\alpha }[\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)]+\gamma (h,\sigma ,{\sigma }_h).\end{array}$$

Now, both sides are affected by ${\mathcal{I}}^{\beta }$, so we have

$$\begin{array}{c}\hfill \sigma \left(h\right)={\mathcal{I}}^{\alpha +\beta }[\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)]+{\mathcal{I}}^{\beta }\gamma (h,\sigma ,{\sigma }_h)+c_1+c_{2}h.\end{array}$$

From the initial conditions, we obtain

$$\begin{array}{c}\hfill \sigma \left(h\right)={\mathcal{I}}^{\alpha +\beta }[\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)]+{\mathcal{I}}^{\beta }\gamma (h,\sigma ,{\sigma }_h)+\delta \left(0\right)+{\sigma }_{0}h.\end{array}$$

The proof is completed. □

Lemma 5. 

Given that $\sigma \left(h\right)$ satisfies (3), then

$$\sigma \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill & h\in (0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0],\hfill \end{array}\right.$$

such that $\iota =\alpha +\beta \in (1,2),\iota =2\rho$. Also,

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\rho }\left(h\right)& ={\int }_0^{\infty }{\psi }_{\rho }\left(\theta \right)\mathcal{R}\left(h^{\rho }\theta \right)d\theta ,\hfill \\ \hfill {\overline{\mathcal{G}}}_{\rho }(h,\ell )& ={\int }_0^{\infty }\rho \theta {\psi }_{\rho }\left(\theta \right)\mathcal{T}\left({(h-\ell )}^{\rho }\theta \right)d\theta ,\hfill \end{array}$$

according to Definition 6, ${\psi }_{\rho }\left(\theta \right)$ is defined as a probability density function.

Proof. 

• For $h\in [-a,0]$, we acquire

  $$\sigma \left(h\right)=\delta \left(h\right).$$
• For $\mathbf{h}\in J$, before taking the Laplace transformation of both sides of System (3), assuming $Re\xi >0$, then we attain

  $$\begin{array}{cc}\hfill \overline{\sigma }\left(\xi \right):=& \mathcal{L}\left\{\sigma \left(h\right)\right\}\left(\xi \right)={\int }_0^{\infty }{e}^{-\xi \ell }\sigma (\ell )d\ell ,\hfill \\ \hfill \overline{\kappa }\left(\xi \right):=& \mathcal{L}\left\{\kappa (h,\sigma ,{\sigma }_h)\right\}\left(\xi \right)={\int }_0^{\infty }{e}^{-\xi \ell }\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill \\ \hfill \overline{\gamma }\left(\xi \right):=& \mathcal{L}\left\{\gamma (h,\sigma ,{\sigma }_h)\right\}\left(\xi \right)={\int }_0^{\infty }{e}^{-\xi \ell }\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell .\hfill \end{array}$$

Using the Laplace transform on either side of (3), we obtain

$$\begin{array}{cc}\hfill \overline{\sigma }\left(\xi \right)& ={\xi }^{-\iota }[\mathcal{A}\overline{\sigma }\left(\xi \right)+\overline{\kappa }\left(\xi \right)]+{\xi }^{-\beta }\overline{\gamma }\left(\xi \right)+{\xi }^{-1}\delta \left(0\right)+{\xi }^{-1}\delta \left(0\right)+{\xi }^{-2}{\sigma }_0,\hfill \\ \hfill ({\xi }^{\iota }I-\mathcal{A})\overline{\sigma }\left(\xi \right)& =\overline{\kappa }\left(\xi \right)+{\xi }^{\alpha }\overline{\gamma }\left(\xi \right)+{\xi }^{\iota -1}\delta \left(0\right)+{\xi }^{\iota -2}{\sigma }_0,\hfill \\ \hfill \overline{\sigma }\left(\xi \right)& ={({\xi }^{\iota }I-\mathcal{A})}^{-1}\left[\overline{\kappa }\left(\xi \right)+{\xi }^{\alpha }\overline{\gamma }\left(\xi \right)+{\xi }^{\iota -1}\delta \left(0\right)+{\xi }^{\iota -2}{\sigma }_0\right].\hfill \end{array}$$

From Lemma 3 and since $\rho ={\displaystyle \frac{\iota }{2}}$, we can write

$$\begin{array}{cc}\hfill \overline{\sigma }\left(\xi \right)& ={\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{T}\left(h\right)\overline{\kappa }\left(\xi \right)dh+{\xi }^{\alpha }{\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{T}\left(h\right)\overline{\gamma }\left(\xi \right)dh\hfill \\ \hfill & +{\xi }^{\rho -1}{\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{R}\left(h\right)\delta \left(0\right)dh+{\xi }^{\rho -2}{\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{R}\left(h\right){\sigma }_{0}dh.\hfill \end{array}$$

Let $\theta \in (0,\infty )$; the Mainardi’s Wright-type function defined in Definition 6 refers to ${\psi }_{\rho }\left(\theta \right)$, and

$${\psi }_{\rho }\left(\theta \right)={\displaystyle \frac{1}{\rho }}{\theta }^{{\displaystyle \frac{-1}{\rho }}-1}{\omega }_{\rho }({\theta }^{{\displaystyle \frac{-1}{\rho }}}),$$

where ${\omega }_{\rho }\left(\theta \right)$ is the 1-sided stable probability and is denoted as in [34]

$${\omega }_{\rho }\left(\theta \right)={\displaystyle \frac{1}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k-1}{\theta }^{-\rho k-1}{\displaystyle \frac{\Gamma (\rho k+1)}{k!}}sin\left(\rho k\pi \right),\theta \in (0,\infty ).$$

In the interim, the following is possible:

$${\int }_0^{\infty }{e}^{-\xi \theta }{\omega }_{\rho }\left(\theta \right)d\theta =e^{-{\xi }^{\rho }}, for\rho \in \left({\displaystyle \frac{1}{2}},1\right).$$

Hence, our situation

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{T}\left(h\right)\overline{\kappa }\left(\xi \right)dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-{\left(\xi h\right)}^{\rho }}\mathcal{T}\left(h^{\rho }\right)\rho h^{\rho -1}\overline{\kappa }\left(\xi \right)dh\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }\rho h^{\rho -1}{e}^{-{\left(\xi h\right)}^{\rho }}\mathcal{T}\left(h^{\rho }\right)e^{-\xi h}\left[\kappa (\ell ,\sigma ,{\sigma }_{\ell })\right]d\ell dh\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\rho h^{\rho -1}{e}^{-\xi h\theta }{\omega }_{\rho }\left(\theta \right)\mathcal{T}\left(h^{\rho }\right)e^{-\xi h}\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell dh\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }\rho {\displaystyle \frac{h^{\rho -1}}{{\theta }^{\rho }}}{e}^{-\xi (h+\ell )}{\omega }_{\rho }\left(\theta \right)\mathcal{T}\left({\displaystyle \frac{h^{\rho }}{{\theta }^{\rho }}}\right){\omega }_{\rho }\left(\theta \right)\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\theta d\ell dh\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^h{\int }_0^{\infty }\rho {\displaystyle \frac{{(h-\ell )}^{\rho -1}}{{\theta }^{\rho }}}{e}^{-\xi h}{\omega }_{\rho }\left(\theta \right)\mathcal{T}\left({\displaystyle \frac{{(h-\ell )}^{\rho }}{{\theta }^{\rho }}}\right)\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\theta d\ell dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}\left\{{\int }_0^h{\int }_0^{\infty }\rho \theta {(h-\ell )}^{\rho -1}{\psi }_{\rho }\left(\theta \right)\mathcal{T}\left({(h-\ell )}^{\rho }\theta \right)\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\theta d\ell \right\}dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}\left\{{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h-\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}dh\hfill \\ \hfill =& \mathcal{L}\left\{{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}\left(\xi \right).\hfill \end{array}$$

Similarly, we consider that ${lim}_{h\to 0}{\mathcal{I}}^{1-\beta }\left\{{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h-\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}=0$. The term can be expressed as the final term in accordance with Definition (2), leading us to the conclusion that

$$\begin{array}{cc}\hfill & {\xi }^{\beta }{\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{T}\left(h\right)\overline{\gamma }\left(\xi \right)dh={\xi }^{\beta }\mathcal{L}\left\{{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h-\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}\left(\xi \right)\hfill \\ \hfill & ={\xi }^{\beta }\mathcal{L}\left\{{\int }_0^t{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h-\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}\left(\xi \right)-\underset{h\to 0}{lim}{\mathcal{I}}^{1-\beta }\left\{{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h-\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}\hfill \\ \hfill & =\mathcal{L}\left\{{}^{\mathcal{R}L}\mathcal{D}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right\}\left(\xi \right).\hfill \end{array}$$

In the same vein, we may deduce that

$$\begin{array}{cc}\hfill & {\xi }^{\rho -1}{\int }_0^{\infty }{e}^{-{\xi }^{\rho }t}\mathcal{R}\left(h\right)\delta \left(0\right)dh\hfill \\ \hfill =& \rho {\int }_0^{\infty }{\left(\xi t\right)}^{\rho -1}{e}^{-{\left(\xi h\right)}^{\rho }}\mathcal{R}\left(h^{\rho }\right)\delta \left(0\right)dh\hfill \\ \hfill =& {\displaystyle \frac{-1}{\xi }}{\int }_0^{\infty }{\displaystyle \frac{d}{dh}}\left(e^{-{(\xi s\ell )}^{\rho }}\right)\mathcal{R}\left(h^{\rho }\right)\delta \left(0\right)dh\hfill \\ \hfill =& {\int }_0^{\infty }{\int }_0^{\infty }\theta {\omega }_{\rho }\left(\theta \right)e^{-\xi h\theta }\mathcal{R}\left(h^{\rho }\right)\delta \left(0\right)d\theta dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}\left\{{\int }_0^{\infty }{\rho }^{-1}{\theta }^{{\displaystyle \frac{-1}{\rho }}-1}{\omega }_{\rho }({\theta }^{{\displaystyle \frac{-1}{\rho }}})\mathcal{R}\left(h^{\rho }\theta \right)\delta \left(0\right)d\theta \right\}dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}\left\{{\int }_0^{\infty }{\psi }_{\rho }\left(\theta \right)\mathcal{R}\left(h^{\rho }\theta \right)\delta \left(0\right)d\theta \right\}dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)dh\hfill \\ \hfill =& \mathcal{L}\left\{{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)\right\}\left(\xi \right).\hfill \end{array}$$

Furthermore, as $\mathcal{L}\left(\xi \right)={\xi }^{-1}$, the following is yielded:

$$\begin{array}{cc}\hfill & {\xi }^{-1}{\xi }^{\rho -1}{\int }_0^{\infty }{e}^{-{\xi }^{\rho }h}\mathcal{R}\left(h\right){\sigma }_{0}dh\hfill \\ \hfill =& {\int }_0^{\infty }{e}^{-\xi h}\left\{{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell \right\}dh\hfill \\ \hfill =& \mathcal{L}\left\{{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell \right\}\left(\xi \right).\hfill \end{array}$$

Ultimately, the desired result is achieved through the application of the inverse Laplace transform, which allows for the reconstruction of the solution in the time domain. This step completes the analytical process and concludes the proof, confirming the validity of the proposed solution framework. □

Definition 7. 

If $\sigma \left(h\right)\in \left(\mathcal{C}\right[-a,b];\mathbb{H})$ and satisfies the following equation:

$$\sigma \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill & h\in [0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0],\hfill \end{array}\right.$$

(5)

then it is the mild solution of (1)

Consequently, the proofs of all subsequent Lemmas follow a similar pattern as $\rho$ approaches 1.

Lemma 6 

([35]). The following estimates for ${\mathcal{G}}_{\rho }\left(h\right)$ and ${\overline{\mathcal{G}}}_{\rho }\left(h\right)(h,\ell )$ are verified for any fixed $h\ge 0$ and $0<\ell <h$

$$\left|{\mathcal{G}}_{\rho }\left(h\right)\zeta \right|\le \mathcal{M}\left|\zeta \right|\mathit{and}\left|{\overline{\mathcal{G}}}_{\rho }(h,\ell )\zeta \right|\le {\mathcal{M}}_1\left|\zeta \right|,{\mathcal{M}}_1={\displaystyle \frac{\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)}}.$$

Lemma 7 

([35]). For any $0<\ell <h$ and $h>0$, the operators ${\mathcal{G}}_{\rho }\left(h\right)$ and ${\overline{\mathcal{G}}}_{\rho }(h,\ell )$ are strongly continuous.

Lemma 8 

([35]). Suppose that $\mathcal{R}\left(h\right)$ and $\mathcal{T}(h,\ell )$ are compact for every $0<\ell <v$. In that case, for any $0<\ell <h$, the operators ${\mathcal{G}}_{\rho }\left(h\right)$ and ${\overline{\mathcal{G}}}_{\rho }(h,\ell )$ are compact.

Lemma 9. 

The inequality below holds for any $y\in \mathbb{H}$.

$$\parallel {\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )y\parallel =\parallel {\displaystyle \frac{d}{dt}}{\overline{\mathcal{G}}}_{\rho }(h,\ell )y\parallel \le {\displaystyle \frac{2\rho b^{2\rho -1}\mathcal{M}}{\Gamma \left(3\rho \right)}}\parallel y\parallel .$$

In addition, it is uniformly continuous, that is, for any $h_2>h_1\ge 0,$

$$\parallel {\overline{\mathcal{G}}}_{{\rho }^2}(h_2,\ell )-{\overline{\mathcal{G}}}_{{\rho }^2}(h_1,\ell )\parallel \to 0\mathit{as}{h}_2\to h_1.$$

Proof. 

Since ${\displaystyle \frac{d}{dh}}\mathcal{T}\left(h^{\rho }\theta \right)y=\rho \theta h^{\rho -1}\mathcal{R}\left(h^{\rho }\right)y$ for $h>0$ and $y\in \mathbb{H}$, it can be calculated that

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dh}}{\overline{\mathcal{G}}}_{\rho }(h,\ell )y& ={\displaystyle \frac{d}{dh}}{\int }_0^{\infty }\rho \theta {\psi }_{\rho }\left(\theta \right)\mathcal{T}\left({(h-\ell )}^{\rho }\theta \right)yd\theta \hfill \\ \hfill & ={\int }_0^{\infty }\rho \theta {\psi }_{\rho }\left(\theta \right){\displaystyle \frac{d}{dh}}\mathcal{T}\left({(h-\ell )}^{\rho }\theta \right)yd\theta \hfill \\ \hfill & ={(h-\ell )}^{\rho -1}{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right)\mathcal{R}\left({(h-\ell )}^{\rho }\theta \right)yd\theta \hfill \\ \hfill & ={\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )y.\hfill \end{array}$$

Now, from Definition (6), we obtain

$$\begin{array}{cc}\hfill \parallel {\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )y\parallel =& {\parallel (h-\ell )}^{\rho -1}{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right)\mathcal{R}\left({(h-\ell )}^{\rho }\theta \right)yd\theta \parallel \hfill \\ \hfill & ={(h-\ell )}^{\rho -1}{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right){\int }_0^{{(h-\ell )}^{\rho }\theta }\parallel \mathcal{T}\left(h\right)y\parallel dhd\theta \hfill \\ \hfill & \le {\rho }^2{(h-\ell )}^{2\rho -1}\mathcal{M}\parallel y\parallel {\int }_0^{\infty }{\theta }^3{\psi }_{\rho }\left(\theta \right)d\theta \hfill \\ \hfill & ={\displaystyle \frac{2{\rho }^2\mathcal{M}}{\rho \Gamma \left(3\rho \right)}}{(h-\ell )}^{2\rho -1}\parallel y\parallel .\hfill \end{array}$$

Since $0<\ell <h\le b$, then

$${(h-\ell )}^{2\rho -1}\le h^{2\rho -1}\le b^{2\rho -1},$$

which implies that

$$\parallel {\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )y\parallel \le {\displaystyle \frac{2\rho b^{2\rho -1}\mathcal{M}}{\Gamma \left(3\rho \right)}}\parallel y\parallel .$$

Now, we shall show that the operator ${\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )$ has uniform continuity for $h_2>h_1\ge 0$:

$$\begin{array}{cc}\hfill & ∥{\overline{\mathcal{G}}}_{{\rho }^2}(h_2,\ell )-{\overline{\mathcal{G}}}_{{\rho }^2}(h_1,\ell )∥=∥{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right){(h_2-\ell )}^{\rho -1}\mathcal{R}\left({(h_2-\ell )}^{\rho }\theta \right)d\theta -\hfill \\ \hfill & {\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right){(h_1-\ell )}^{\rho -1}\mathcal{R}\left({(h_1-\ell )}^{\rho }\theta \right)d\theta ∥\hfill \\ \hfill & \le {\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right){(h_2-\ell )}^{\rho }\parallel \mathcal{R}\left({(h_2-\ell )}^{\rho }\theta \right)-\mathcal{R}\left({(h_1-\ell )}^{\rho }\theta \right)\parallel d\theta \hfill \\ \hfill & +\mathcal{M}{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right)\parallel h_2^{\rho -1}-h_1^{\rho -1}\parallel d\theta \hfill \\ \hfill & ={\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right){(h_2-\ell )}^{\rho }\parallel \mathcal{R}\left({(h_2-\ell )}^{\rho }\theta \right)-\mathcal{R}\left({(h_1-\ell )}^{\rho }\theta \right)\parallel d\theta +{\displaystyle \frac{\mathcal{M}\rho }{\Gamma \left(2\rho \right)}}∥h_2^{\rho -1}-h_1^{\rho -1}∥.\hfill \end{array}$$

Since the operator $\mathcal{R}\left({(h,\ell )}^{\rho }\theta \right)$ is strongly continuous based on Lemma $\left(3.3\right)$ in [35] as $\rho \to 1$, we might conclude that $\parallel \mathcal{R}\left({(h_2-\ell )}^{\rho }\theta \right)-\mathcal{R}\left({(h_1-\ell )}^{\rho }\theta \right)\parallel \to 0$ as $h_2\to h_1$ Thus, it is proved that ${\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )$ is uniformly continuous. The proof is complete. □

4. Existence and Uniqueness Results

This section summarizes our findings concerning the existence and uniqueness of the mild solution to Problem (1). Additionally, we introduce a methodology that addresses the finite delay associated with a mild solution.

For the treatment of the finite delay, we define the operator $\Omega :\left(\mathcal{C}\right[-a,b];\mathbb{H})\to \left(\mathcal{C}\right[-a,b];\mathbb{H})$ in the following manner:

$$(\Omega \sigma )\left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill & h\in [0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Let $u(·):[-a,b]\to \mathbb{H}$ be the function denoted by

$$u\left(h\right)=\left\{\begin{array}{cc}0,\hfill & h\in (0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Subsequently, $u\left(0\right)=\delta \left(0\right)$, and we provide the function defined by $v(·)$ for each $w\in \left(\mathcal{C}\right[-a,b];\mathbb{H})$ with $w\left(0\right)=0$ via

$$v\left(h\right)=\left\{\begin{array}{cc}w\left(h\right),\hfill & h\in [0,b],\hfill \\ 0,\hfill & h\in [-a,0].\hfill \end{array}\right.$$

If $\sigma (·)$ satisfies that $\sigma \left(h\right)=(\Omega \sigma )\left(h\right)$ for all $h\in [-a,b]$, we can decompose $\sigma \left(h\right)=u\left(h\right)+v\left(h\right),$ for all $h\in [-a,b]$. It is denoted as ${\sigma }_h=u_h+v_h,$ for all $h\in [-a,b]$, and the function $w(·)$ satisfies the following equation:

$$\begin{array}{cc}\hfill w\left(h\right)& ={\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell \hfill \\ \hfill & +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,u+v,u_{\ell }+v_{\ell })d\ell .\hfill \end{array}$$

Now, we define the space $\mathfrak{D}$ such that

$$\mathfrak{D}=\{w\in (\mathcal{C}[-0,b];\mathbb{H}),w(0)=0\},$$

with $\parallel ·\parallel$.

$${\parallel w\parallel }_{\mathfrak{D}}=\underset{h\in [0,b]}{max}\parallel w\left(h\right)\parallel .$$

Here, $(\mathfrak{D},\parallel ·{\parallel }_{\mathfrak{D}})$ is a Banach space. Let us define an operator $\Gamma :\mathfrak{D}\to \mathfrak{D}$ as

$$\begin{array}{cc}\hfill (\Gamma w)\left(h\right)& ={\mathcal{G}}_{\rho }\left(t\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell \hfill \\ \hfill & +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,u+v,u_{\ell }+v_{\ell })d\ell .\hfill \end{array}$$

It appears that the fixed point of $\Gamma$ is identical to the fixed point of the operator $\Omega$.

Let us now consider that

$$\Lambda =\{w\in \mathfrak{D}:\parallel w{\parallel }_{\mathfrak{D}}\le \mathsf{ϝ}\}.$$

For clarity, let $w\in \Lambda$ and $h\in [0,b]$; then,

$$\begin{array}{cc}\hfill \parallel u_h{\parallel }_{[-a,0]}& =\underset{-a\le \eta \le 0}{max}\left|u(h+\eta )\right|\hfill \\ \hfill & \le \underset{-a\le \eta +h\le b}{max}\left|u\left(\xi \right)\right|\hfill \\ \hfill & \le \underset{-a\le \xi \le b}{max}\left|u\left(\xi \right)\right|\hfill \\ \hfill & =\underset{-a\le \xi \le 0}{max}\left|\delta \left(\xi \right)\right|={\parallel \delta \parallel }_{[-a,0]}.\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}\hfill \parallel v_h{\parallel }_{[-a,0]}& =\underset{-a\le \eta \le 0}{max}\left|v(h+\eta )\right|\hfill \\ \hfill & \le \underset{-a\le \eta +h\le b}{max}\left|v\left(\xi \right)\right|\hfill \\ \hfill & =\underset{-a\le \xi \le b}{max}\left|v\left(\xi \right)\right|\hfill \\ \hfill & =\underset{0\le \xi \le b}{max}\left|w\left(\xi \right)\right|={\parallel w\parallel }_{\mathfrak{D}}.\hfill \end{array}$$

Prior to presenting our main findings, we introduce the following hypotheses.

$\left(\mathbb{A}{\mathbb{S}}_1\right)$

$\gamma :J\times \mathbb{H}\times \mathbb{H}\to \mathbb{H}$ be a continuous function, and there exists a constant $\mathcal{Z}>0$ so that for any $(h,\sigma ,{\sigma }_h),(h,{\sigma }^{*},{\sigma }_h^{*})\in J\times \mathbb{H}\times \mathbb{H}$, we have

$$\parallel \gamma (h,\sigma ,{\sigma }_h)-\gamma (h,{\sigma }^{*},{\sigma }_h^{*})\parallel \le \mathcal{Z}\left(\parallel \sigma -{\sigma }^{*}\parallel +\parallel {\sigma }_h-{\sigma }_h^{*}{\parallel }_{[-a,0]}\right).$$

$\left(\mathbb{A}{\mathbb{S}}_2\right)$

$\kappa :J\times \mathbb{H}\times \mathbb{H}\to \mathbb{H}$ is a continuous function, and there is a positive constant $\mathcal{O}$ such that for any $(h,\sigma ,{\sigma }_h),(h,{\sigma }^{*},{\sigma }_h^{*})\in J\times \mathbb{H}\times \mathbb{H}$, we have

$$\parallel \kappa (h,\sigma ,{\sigma }_h)-\kappa (h,{\sigma }^{*},{\sigma }_h^{*})\parallel \le \mathcal{O}\left(\parallel \sigma -{\sigma }^{*}\parallel +\parallel {\sigma }_h-{\sigma }_h^{*}{\parallel }_{[-a,0]}\right).$$

Lemma 10. 

For each $h>0$ and $y\in \mathbb{H}$, the following formula is true if (3) holds.

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dh}}\left({\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right)=\hfill \\ \hfill & {\int }_0^h\left[{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )+(\rho -1){(h-\ell )}^{\rho -2}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\right]\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell .\hfill \end{array}$$

Furthermore, according to assumption $\left(\mathbb{A}{\mathbb{S}}_1\right)$ and letting $\parallel \gamma (h,0,0)\parallel ={\mathcal{Z}}^{*}$,

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{d}{dh}}\left({\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right)∥\le \hfill \\ \hfill & \left[{\displaystyle \frac{2b^{3\rho -1}}{\Gamma \left(3\rho \right)}}\mathcal{M}+b^{\rho -1}{\mathcal{M}}_1\right]\left(\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right).\hfill \end{array}$$

Proof. 

According to the Leibniz rule, we can write

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dh}}\left({\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right)\hfill \\ \hfill =& {\int }_0^h\left[{(h-\ell )}^{2(\rho -1)}{\int }_0^{\infty }{\rho }^2{\theta }^2{\psi }_{\rho }\left(\theta \right)\mathcal{R}\left({(h-\ell )}^{\rho }\theta \right)+(\rho -1){(h-\ell )}^{\rho -2}{\overline{\mathcal{G}}}_{\rho }h,\ell )\right]\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ \hfill =& {\int }_0^h\left[{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )+(\rho -1){(h-\ell )}^{\rho -2}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\right]\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell .\hfill \end{array}$$

From Lemmas (6) and (9), we can obtain

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{d}{dh}}\left({\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \right)∥\hfill \\ \hfill & =∥{\int }_0^h\left[{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )+(\rho -1){(h-\ell )}^{\rho -2}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\right]\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell ∥\hfill \\ \hfill & \le {\int }_0^h\left[{(h-\ell )}^{\rho -1}\parallel {\overline{\mathcal{G}}}_{{\rho }^2}{(h,\ell )\parallel +(\rho -1)(h-\ell )}^{\rho -2}\parallel {\overline{\mathcal{G}}}_{\rho }(h,\ell )\parallel \right]\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (h,0,0)+\gamma (h,0,0)\parallel d\ell \hfill \\ \hfill & \le {\int }_0^h\left[{(h-\ell )}^{\rho -1}\parallel {\overline{\mathcal{G}}}_{{\rho }^2}{(h,\ell )\parallel +(\rho -1)(h-\ell )}^{\rho -2}\parallel {\overline{\mathcal{G}}}_{\rho }(h,\ell )\parallel \right]\left(\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (h,0,0)\parallel +\parallel \gamma (h,0,0)\parallel \right)d\ell \hfill \\ \hfill \le & {\int }_0^{\mathbf{h}}\left[{(h-\ell )}^{\rho -1}\parallel {\overline{\mathcal{G}}}_{{\rho }^2}{(h,\ell )\parallel +(\rho -1)(h-\ell )}^{\rho -2}\parallel {\overline{\mathcal{G}}}_{\rho }(h,\ell )\parallel \right]\left(\mathcal{Z}(\parallel \sigma \parallel +\parallel {\sigma }_{\ell }{\parallel }_{[-a,0]})+{\mathcal{Z}}^{*}\right)d\ell \hfill \\ \hfill \le & {\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)}}{\int }_0^h{(h-\ell )}^{\rho -1}\left(\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right)d\ell \hfill \\ \hfill & +(\rho -1){\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -2}\left(\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right)d\ell \hfill \\ \hfill & \le \left[{\displaystyle \frac{2b^{3\rho -1}}{\Gamma \left(3\rho \right)}}\mathcal{M}+b^{\rho -1}{\mathcal{M}}_1\right]\left(\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right).\hfill \end{array}$$

Ultimately, the proof is fulfilled. □

Theorem 1. 

Given the assumptions $\left(\mathbb{A}{\mathbb{S}}_1\right)$, $\left(\mathbb{A}{\mathbb{S}}_2\right)$, and

$$\begin{array}{c}\hfill \mathcal{E}={\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\Gamma \left(\rho \right)}}\mathcal{O}+({\mathcal{M}}_{\mathcal{B}}+{\overline{\mathcal{M}}}_{\mathcal{B}})\mathcal{Z}<{\displaystyle \frac{1}{2}},\end{array}$$

(6)

there exists a unique mild solution for the fractional neutral problem (1) on $[-a,b],$ where

$$\begin{array}{c}\hfill {\mathcal{M}}_{\mathcal{B}}={\displaystyle \frac{2\mathcal{M}{b}^{3\rho +\beta -1}}{\Gamma (3\rho +\beta +1)\mathcal{B}(3\rho ,\beta +1)}},{\overline{\mathcal{M}}}_{\mathcal{B}}={\displaystyle \frac{\rho \mathcal{M}{b}^{2\rho +\beta -1}}{\Gamma (2\rho +\beta +1)\mathcal{B}(2\rho ,\beta +1)}},\end{array}$$

such that $\mathcal{B}(n,m)$ is a beta function.

Proof. 

For each $\mathsf{ϝ}\ge 0$, we demonstrate that $\Gamma$ transfers bounded sets of $\mathfrak{D}$ to bounded sets in $\mathfrak{D}$. We proceed by setting

$$\Lambda =\{w\in \mathfrak{D}:\parallel w{\parallel }_{\mathfrak{D}}\le \mathsf{ϝ}\},$$

whereby, with the following reduction,

$$\mathsf{ϝ}\ge {\displaystyle \frac{\mathcal{M}\parallel \left(\delta \left(0\right)\parallel +b\parallel {\sigma }_0\parallel \right)+\mathcal{E}{\parallel \delta \parallel }_{[-a,0]}+{\mathcal{E}}^{*}}{1-\mathcal{E}}},$$

where

$${\mathcal{E}}^{*}={\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\Gamma \left(\rho \right)}}{\mathcal{O}}^{*}+({\mathcal{M}}_{\mathcal{B}}+{\overline{\mathcal{M}}}_{\mathcal{B}}){\mathcal{Z}}^{*}.$$

Consequently, for any $w\in \Lambda$, we obtain

$$\begin{array}{cc}\hfill & ∥(\Gamma w)\left(h\right)∥\le \hfill \\ \hfill & ∥{\mathcal{G}}_{\rho }\left(\mathbf{h}\right)\delta \left(0\right)∥+∥{\int }_0^{\mathbf{h}}{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell ∥+∥{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,u+v,u_{\ell }+v_{\ell })d\ell ∥\hfill \\ \hfill & +∥{}^{\mathcal{R}L}\mathcal{D}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell ∥.\hfill \end{array}$$

Given the well-established relationships between the Riemann–Liouville and the Caputo fractional derivatives, for $0<\beta <1$, we obtain the following:

$$\begin{array}{c}\hfill {}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell {=}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell .\end{array}$$

After that, we can write

$$\begin{array}{cc}\hfill & ∥(\Gamma w)\left(h\right)∥\le \hfill \\ \hfill & \mathcal{M}\parallel \left(\delta \left(0\right)\parallel +b\parallel {\sigma }_0\parallel \right)+∥{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,u+v,u_{\ell }+v_{\ell })ds∥\hfill \\ \hfill & +∥{\mathcal{I}}^{\odot }\left({\displaystyle \frac{d}{dh}}{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\ell \right)∥\hfill \\ \hfill & =\mathcal{M}\parallel \left(\delta \left(0\right)\parallel +b\parallel {\sigma }_0\parallel \right)+∥{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,u+v,u_{\ell }+v_{\ell })d\ell ∥\hfill \\ \hfill & +∥{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}{\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\tau d\ell ∥\hfill \\ \hfill & +∥{\displaystyle \frac{(\rho -1)}{\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,u+v,u_{\ell }+v_{\ell })d\tau d\ell ∥\hfill \\ \hfill & \le \mathcal{M}\parallel \left(\delta \left(0\right)\parallel +b\parallel {\sigma }_0\parallel \right)+{\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\left[\mathcal{O}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{O}}^{*}\right]d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\left[\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right]d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\left[\mathcal{Z}\left({\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}\right)+{\mathcal{Z}}^{*}\right]d\tau d\ell \hfill \\ \hfill & \le \mathcal{M}\parallel \left(\delta \left(0\right)\parallel +b\parallel {\sigma }_0\parallel \right)+{\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\Gamma \left(\rho \right)}}\left[{\mathcal{O}\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}+{\mathcal{O}}^{*}\right]+({\mathcal{M}}_{\mathcal{B}}+{\overline{\mathcal{M}}}_{\mathcal{B}})\left[{\mathcal{Z}\parallel w\parallel }_{\mathfrak{D}}+{\parallel \delta \parallel }_{[-a,0]}+{\mathcal{Z}}^{*}\right]\hfill \\ \hfill & \le \mathsf{ϝ}.\hfill \end{array}$$

Here, we demonstrate that the operator $\Gamma$ is a contraction map. We enquire about $w,w^{*}\in \Lambda$. For any $\mathbf{h}\in [0,b]$, we have

$$\begin{array}{cc}\hfill & ∥(\Gamma w)\left(h\right)-(\Gamma w^{*})\left(h\right)∥\le {\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\parallel \kappa (\ell ,u+v,u_{\ell }+v_{\ell })-\kappa (\ell ,u+v^{*},u_{\ell }+v_{\ell }^{*})\parallel d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,u+v,u_{\ell }+v_{\ell })-\gamma (\ell ,u+v^{*},u_{\ell }+v_{\ell }^{*})\parallel d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,u+v,u_{\ell }+v_{\ell })-\gamma (\ell ,u+v^{*},u_{\ell }+v_{\ell }^{*})\parallel d\tau d\ell \hfill \\ \hfill & \le {\mathcal{M}}_1\mathcal{O}{\int }_0^h{(h-\ell )}^{\rho -1}\left(\parallel v-v^{*}{\parallel }_{\mathfrak{D}}+{\parallel v_{\ell }-v_{\ell }^{*}\parallel }_{[-a,0]}\right)d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}\mathcal{Z}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\left(\parallel v-v^{*}{\parallel }_{\mathfrak{D}}+{\parallel v_{\ell }-v_{\ell }^{*}\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }\mathcal{Z}}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\left(\parallel v-v^{*}{\parallel }_{\mathfrak{D}}+{\parallel v_{\ell }-v_{\ell }^{*}\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le 2{\mathcal{M}}_1(\mathcal{O}\parallel v-v^{*}{\parallel }_{\mathfrak{D}}){\int }_0^h{(h-\ell )}^{\rho -1}d\ell \hfill \\ \hfill & +{\displaystyle \frac{4\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}(\mathcal{Z}\parallel v-v^{*}{\parallel }_{\mathfrak{D}}){\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}(\mathcal{Z}\parallel v-v^{*}{\parallel }_{\mathfrak{D}}){\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}d\tau d\ell \hfill \\ \hfill & \le 2\left({\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\Gamma \left(\rho \right)}}\mathcal{O}+({\mathcal{M}}_{\mathcal{B}}+{\overline{\mathcal{M}}}_{\mathcal{B}})\mathcal{Z}\right)\parallel v-v^{*}{\parallel }_{\mathfrak{D}}=2\mathcal{E}{\parallel v-v^{*}\parallel }_{\mathfrak{D}}.\hfill \end{array}$$

which presumes that $\mathcal{E}<{\displaystyle \frac{1}{2}}$ is relevant. Thus, assuming $\left(\mathbb{A}{\mathbb{S}}_1\right),\left(\mathbb{A}{\mathbb{S}}_2\right)$, we can use Banach’s contraction mapping principle to deduce that $\Gamma$ has a unique fixed point and that (5) is the mild solution to the problem (1) on $[-a,b]$. The proof is complete. □

5. Hyers–Ulam–Rassias and Hyers–Ulam Stability Results

The present section details the findings related to Hyers–Ulam–Rassias stability and Hyers–Ulam stability for the system under consideration. These stability results are crucial for understanding how the system responds to perturbations, providing insights into its robustness and resilience in the face of small disturbances (1).

Definition 8. 

Given that $\varphi \left(h\right):J\to [0,\infty )$ is a continuous function and $\sigma \left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$ satisfies the following inequality:

$$\begin{array}{c}\hfill \left|{}_{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)-\gamma (h,\sigma ,{\sigma }_h)]-\mathcal{A}\sigma \left(h\right)-\kappa (h,\sigma ,{\sigma }_h)\right|\le \varphi \left(h\right),h\in J,\end{array}$$

(7)

Additionally, there is a solution $\varkappa \left(h\right)$ of (1) and a constant $\mathcal{K}>0$ independent of $\sigma \left(h\right),\varkappa \left(h\right)$ in such a manner that

$$\begin{array}{c}\hfill \left|\sigma \right(h)-\varkappa (h\left)\right|\le \mathcal{K}\varphi \left(h\right),h\in J,\end{array}$$

(8)

then, we say that (1) has Hyers–Ulam–Rassias stability.

Remark 1. 

A function $\sigma \left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$ is the solution of (7) if and only if there is $\mathcal{S}\left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$, which depends on a solution σ in such a manner that

(i) 

$\left|\mathcal{S}\right(h\left)\right|\le \varphi \left(h\right),\mathit{for}\mathit{all}h\in [-a,b].$

(ii) 

${}_{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}_{\mathcal{C}}{\mathcal{D}}^{\beta }\sigma \left(h\right)-\gamma (h,\sigma ,{\sigma }_h)]=\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)+\mathcal{S}\left(h\right),h\in J.$

Theorem 2. 

If hypotheses $\left(\mathbb{A}{\mathbb{S}}_1\right)$, $\left(\mathbb{A}{\mathbb{S}}_2\right)$ and relation (6) are fulfilled, assume that there is a constant $\mathcal{Q}\in [0,1]$ such that given a non-decreasing continuous function $\varphi \left(h\right):J\to (0,\infty )$ satisfies

$${\int }_0^h{(h-\ell )}^{\rho -1}\varphi \left(h\right)\le \mathcal{Q}\varphi \left(h\right).$$

(9)

Then, the problem (1) has Hyers–Ulam–Rassias stability.

Proof. 

Assume that the unique solution $\sigma \left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$ to the problem (1) corresponds to any solution $\varkappa \left(h\right)$ and is the solution of inequality (7). Then, in light of Lemma (4), we have

$$\sigma \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\left(h\right)\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill & h\in (0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Further, if $\varkappa$ is the solution of inequality (7), using Remark (1), we obtain

$$\left\{\begin{array}{cc}{\displaystyle {}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\varkappa \left(h\right)-\gamma (h,\varkappa ,{\varkappa }_h)]=\mathcal{A}\varkappa +\kappa (h,\varkappa ,{\varkappa }_h)+\mathcal{S}\left(h\right),}\hfill & h\in J=[0,b]\hfill \\ \varkappa \left(h\right)=\delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Moreover,

$$\varkappa \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d{\ell }^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\varkappa ,{\varkappa }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\varkappa ,{\varkappa }_{\ell })d\ell +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\mathcal{S}(\ell )d\ell ,\hfill & h\in [0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

For each $h\in [0,b]$, we consider

$$\begin{array}{cc}\hfill & ∥\sigma \left(h\right)-\varkappa \left(h\right)∥\le {\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\parallel \kappa (\ell ,\sigma ,{\sigma }_{\ell })-\kappa (\ell ,\varkappa ,{\varkappa }_{\ell })\parallel d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{a}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (\ell ,\varkappa ,{\varkappa }_{\ell })\parallel d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{a}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (\ell ,\varkappa ,{\varkappa }_{\ell })\parallel d\tau d\ell \hfill \\ \hfill & +{\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\left|\mathcal{S}(\ell )\right|d\ell \hfill \\ \hfill & \le {\mathcal{M}}_1\mathcal{O}{\int }_0^h{(h-\ell )}^{\rho -1}\left({\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\varkappa }_{\ell }\parallel }_{[-a,0]}\right)d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}\mathcal{Z}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\left({\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\varkappa }_{\ell }\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }\mathcal{Z}}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^{\mathbf{h}}{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\left({\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\varkappa }_{\ell }\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \\ \hfill & +{\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\varphi \left(h\right)d\ell \hfill \\ \hfill & \le 2{\mathcal{M}}_1{(\mathcal{O}\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}){\int }_0^h{(h-\ell )}^{\rho -1}d\ell \hfill \\ \hfill & +{\displaystyle \frac{4\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{(\mathcal{Z}\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}){\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{(\mathcal{Z}\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}){\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}d\tau d\ell +{\mathcal{M}}_1\mathcal{Q}\varphi \left(h\right)\hfill \\ \hfill & \le {2\mathcal{E}\parallel \sigma -\varkappa \parallel }_{\mathfrak{D}}+{\mathcal{M}}_1\mathcal{Q}\varphi \left(h\right).\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill ∥\sigma \left(h\right)-\varkappa \left(h\right)∥\le \mathcal{K}\varphi \left(h\right),\mathcal{K}={\displaystyle \frac{{\mathcal{M}}_1\mathcal{Q}}{1-2\mathcal{E}}},\end{array}$$

for all h in J. In other words, Equation (1) is Hyers–Ulam–Rassias-stable. The proof is complete. □

Definition 9. 

If $\sigma \left(\mathbf{h}\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$, satisfying the following inequality

$$\begin{array}{c}\hfill \left|{}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)-\gamma (h,\sigma ,{\sigma }_h)]-\mathcal{A}\sigma \left(h\right)-\kappa (h,\sigma ,{\sigma }_h)\right|\le \varrho ,h\in J,\end{array}$$

(10)

where $\mathbf{\varrho }>0$, and there is a solution $\vartheta \left(h\right)$ of (1) and constant $\mathfrak{r}>0$ such that

$$\begin{array}{c}\hfill \left|\sigma \right(h)-\vartheta (h\left)\right|\le \mathfrak{r}\mathbf{\varrho },h\in J,\end{array}$$

(11)

then we say that (1) has Hyers–Ulam stability.

Remark 2. 

A function $\sigma \left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$ is a solution of inequality (10) if and only if there is $\mathcal{S}\left(h\right)\in \left(\mathcal{C}\right[-a,b],\mathbb{H})$, which depends on a solution σ such that

(i) 

$\left|\mathcal{S}\right(h\left)\right|\le \mathbf{\varrho },\mathrm{for}\mathrm{all}h\in [-a,b].$

(ii) 

${}_{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}_{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\sigma \left(h\right)-\gamma (h,\sigma ,{\sigma }_h)]=\mathcal{A}\sigma \left(h\right)+\kappa (h,\sigma ,{\sigma }_h)+\mathcal{S}\left(h\right),h\in J.$

Theorem 3. 

If the hypotheses $\left(\mathbb{A}{\mathbb{S}}_1\right)$, $\left(\mathbb{A}{\mathbb{S}}_2\right)$ and relation (6) are satisfied, then the problem (1) exhibits Hyers–Ulam stability.

Proof. 

Assuming the existence of a unique solution to problem (1), let $\sigma \left(h\right)$ be a solution in the space $\left(\mathcal{C}\right[-a,b],\mathbb{H})$. The solution to the inequality (7) is denoted as $\vartheta \left(h\right)$. Subsequently, in light of Lemma (4), we have

$$\sigma \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\sigma ,{\sigma }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\sigma ,{\sigma }_{\ell })d\ell ,\hfill & h\in (0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Furthermore, by employing Remark (2) and considering $\vartheta$ as the solution to inequality (7), we obtain

$$\left\{\begin{array}{cc}{\displaystyle {}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\beta }\vartheta \left(h\right)-\gamma (h,\vartheta ,{\vartheta }_h)]=\mathcal{A}\vartheta +\kappa (h,\vartheta ,{\vartheta }_h)+\mathcal{S}\left(h\right),}\hfill & h\in J=[0,b],\hfill \\ \vartheta \left(h\right)=\delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

Likewise,

$$\vartheta \left(h\right)=\left\{\begin{array}{c}{\mathcal{G}}_{\rho }\delta \left(0\right)+{\int }_0^h{\mathcal{G}}_{\rho }(\ell ){\sigma }_{0}d\ell {+}^{\mathcal{R}L}{\mathcal{D}}_{0^{+}}^{\beta }{\int }_0^h{(h\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\gamma (\ell ,\vartheta ,{\vartheta }_{\ell })d\ell \hfill \\ +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\kappa (\ell ,\vartheta ,{\vartheta }_{\ell })d\ell +{\int }_0^h{(h-\ell )}^{\rho -1}{\overline{\mathcal{G}}}_{\rho }(h,\ell )\mathcal{S}(\ell )d\ell ,\hfill & h\in [0,b],\hfill \\ \delta \left(h\right),\hfill & h\in [-a,0].\hfill \end{array}\right.$$

We take precautions for each $h\in [0,b]$:

$$\begin{array}{cc}\hfill & ∥\sigma \left(h\right)-\vartheta \left(h\right)∥\le {\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\parallel \kappa (\ell ,\sigma ,{\sigma }_{\ell })-\kappa (\ell ,\vartheta ,{\vartheta }_{\ell })\parallel d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (\ell ,\vartheta ,{\vartheta }_{\ell })\parallel d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\parallel \gamma (\ell ,\sigma ,{\sigma }_{\ell })-\gamma (\ell ,\vartheta ,{\vartheta }_{\ell })\parallel d\tau d\ell \hfill \\ \hfill & +{\mathcal{M}}_1{\int }_0^h{(h-\ell )}^{\rho -1}\parallel \mathcal{S}(\ell )\parallel d\ell \hfill \\ \hfill & \le {\mathcal{M}}_1\mathcal{O}{\int }_0^h{(h-\ell )}^{\rho -1}\left({\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\varkappa }_{\ell }\parallel }_{[-a,0]}\right)d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho \mathcal{M}{b}^{2\rho -1}\mathcal{Z}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}\left({\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\vartheta }_{\ell }\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{\rho (\rho -1)\mathcal{M}{b}^{\rho }\mathcal{Z}}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}\left({\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}+{\parallel {\sigma }_{\ell }-{\vartheta }_{\ell }\parallel }_{[-a,0]}\right)d\tau d\ell \hfill \\ \hfill & +{\mathcal{M}}_1\mathbf{\varrho }{\int }_0^h{(h-\ell )}^{\rho -1}d\ell \hfill \\ \hfill & \le 2{\mathcal{M}}_1{(\mathcal{O}\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}){\int }_0^h{(h-\ell )}^{\rho -1}d\ell \hfill \\ \hfill & +{\displaystyle \frac{4\rho \mathcal{M}{b}^{2\rho -1}}{\Gamma \left(3\rho \right)\Gamma \left(\beta \right)}}{(\mathcal{Z}\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}){\int }_0^h{\int }_0^h{(h-\ell )}^{\rho -1}{(h-\tau )}^{\beta -1}d\tau d\ell \hfill \\ \hfill & +{\displaystyle \frac{2\rho (\rho -1)\mathcal{M}{b}^{\rho }}{\Gamma \left(2\rho \right)\Gamma \left(\beta \right)}}{(\mathcal{Z}\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}){\int }_0^{\mathbf{h}}{\int }_0^h{(h-\ell )}^{\rho -2}{(h-\tau )}^{\beta -1}d\tau d\ell +{\displaystyle \frac{b^{\rho }}{\rho }}{\mathcal{M}}_1\mathbf{\varrho }\hfill \\ \hfill & \le {2\mathcal{E}\parallel \sigma -\vartheta \parallel }_{\mathfrak{D}}+{\displaystyle \frac{b^{\rho }}{\rho }}{\mathcal{M}}_1\mathbf{\varrho }.\hfill \end{array}$$

This suggests that

$$\begin{array}{c}\hfill ∥\sigma \left(h\right)-\vartheta \left(h\right)∥\le \mathfrak{r}\mathbf{\varrho },\mathfrak{r}={\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\rho (1-2\mathcal{E})}}.\end{array}$$

In this regard, Equation (1) is Hyers–Ulam-stable for every h in J. The proof is complete. □

6. An Application

This section presents an application that demonstrates how our primary findings can be effectively utilized. Consider the fractional inhomogeneous wave equation:

$$\left\{\begin{array}{cc}{\displaystyle {}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\frac{2}{5}}{[}^{\mathcal{C}}{\mathcal{D}}_{0^{+}}^{\frac{2}{3}}\sigma (h,\zeta )-\gamma (h,\zeta ,\sigma ,{\sigma }_h)]=\frac{{\partial }^2}{\partial {\zeta }^2}\sigma (h,\zeta )+\kappa (h,\zeta ,\sigma ,{\sigma }_h),}\hfill & h\in J=[0,1],\zeta \in [0,\pi ],\hfill \\ \sigma (h,\zeta )=\sqrt{1-h},\hfill & h\in [{\displaystyle \frac{-1}{2}},0],\hfill \\ \underset{h\to 0}{lim}{h}^{-{\displaystyle \frac{1}{2}}}{[}_{\mathcal{C}}{\mathcal{D}}^{{\displaystyle \frac{2}{3}}}\sigma (h,\zeta )]=0,\hfill \\ \sigma (h,0)=\sigma (h,\pi )=0,\hfill & h\in [0,1].\hfill \end{array}\right.$$

(12)

Let $\mathbb{H}=\mathcal{C}\left(\right[0,1]\times \mathbb{R},\mathbb{R})$ be the space of all continuous functions equipped with the norm ${\parallel \sigma \parallel }_{\mathbb{H}}=\underset{h\in [0,1]}{sup}\underset{\zeta \in [0,\pi ]}{sup}\left|\sigma (h,\zeta )\right|$, and the operator $\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subseteq \mathbb{H}\to \mathbb{H}$ is defined as $\mathcal{A}={\displaystyle \frac{{\partial }^2\sigma (h,\zeta )}{\partial {\zeta }^2}}$ with a domain

$$\mathcal{D}\left(\mathcal{A}\right)=\left\{\sigma \in \mathbb{H}:{\displaystyle \frac{\partial \sigma }{\partial \zeta }},{\displaystyle \frac{{\partial }^2\sigma }{\partial {\zeta }^2}}\in \mathbb{H}\right\}.$$

It follows that the operator $\mathbb{A}$ serves as densely defined in $\mathbb{H}$ and is the infinitesimal generator of a resolvent cosine family $\mathcal{R}\left(h\right),h>0$ on $\mathbb{H}$. Consider that $\alpha ={\displaystyle \frac{2}{5}},\beta ={\displaystyle \frac{2}{3}}$, which implies that $\iota ={\displaystyle \frac{16}{15}}=1.0667$ and $\rho =0.5333$

$$\begin{array}{cc}\hfill ∥{\mathcal{G}}_{\rho }(h,\ell )∥& \le 1⟹\mathcal{M},\hfill \\ \hfill ∥{\overline{\mathcal{G}}}_{\rho }(h,\ell )∥& \le {\displaystyle \frac{1}{\Gamma \left(1.0666\right)}}=1.0355⟹{\mathcal{M}}_1,\hfill \\ \hfill ∥{\overline{\mathcal{G}}}_{{\rho }^2}(h,\ell )∥& \le 1.1937.\hfill \end{array}$$

We also assume the values of

$$\begin{array}{cc}\hfill {\mathcal{M}}_{\mathcal{B}}& =2.4804,\hfill \\ \hfill {\overline{\mathcal{M}}}_{\mathcal{B}}& =0.6117.\hfill \end{array}$$

Let the continuous functions $\kappa (h,\sigma ,{\sigma }_h),\gamma (h,\sigma ,{\sigma }_h):[0,1]\times \mathbb{H}\times \mathbb{H}\to \mathbb{H}$ be defined as

$$\begin{array}{cc}\hfill \kappa (h,\sigma ,{\sigma }_h)& ={\displaystyle \frac{tanh\left(\frac{{\sigma }_h}{9}\right)}{\sqrt{9+h^2}}}+{\displaystyle \frac{cos\left(\frac{2\sigma }{3}\right)}{3e^{3h+1}}},\hfill \\ \hfill \gamma (h,\sigma ,{\sigma }_h)& ={\displaystyle \frac{arctan\frac{5{\sigma }_h}{2}}{85|{\sigma }_h|}}+{\displaystyle \frac{\pi sin\left(2\sigma \right)}{85\sqrt{2-h}}}.\hfill \end{array}$$

It is evident that the functions $\kappa (h,\sigma ,{\sigma }_h),\gamma (h,\sigma ,{\sigma }_h)$ are continuous and satisfy the hypotheses $\left(\mathbb{A}{\mathbb{S}}_1\right)$ and $\left(\mathbb{A}{\mathbb{S}}_2\right)$ as demonstrated in the following:

$$\begin{array}{cc}\hfill ∥\kappa (h,\sigma ,{\sigma }_h)-\kappa (h,\vartheta ,{\vartheta }_h)∥& \le {\displaystyle \frac{∥tanh\left(\frac{{\sigma }_h}{9}\right)-tanh\left(\frac{{\vartheta }_h}{9}\right)∥}{∥\sqrt{9+h^2}∥}}+{\displaystyle \frac{∥cos\left(\frac{2\sigma }{3}\right)-cos\left(\frac{2\vartheta }{3}\right)∥}{\parallel 3e^{3h+1}\parallel }}\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}\left(∥tanh\left({\displaystyle \frac{{\sigma }_h}{9}}\right)-tanh\left({\displaystyle \frac{{\vartheta }_h}{9}}\right)∥\right)+{\displaystyle \frac{1}{3}}\left(∥cos\left({\displaystyle \frac{2\sigma }{3}}\right)-cos\left({\displaystyle \frac{2\vartheta }{3}}\right)∥\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}\left(\parallel {\sigma }_h-{\vartheta }_h\parallel +\parallel \sigma -\vartheta \parallel \right).\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}\hfill ∥\gamma (h,\sigma ,{\sigma }_h)-\gamma (h,\vartheta ,{\vartheta }_h)∥& \le ∥{\displaystyle \frac{arctan\frac{5{\sigma }_h}{2}-arctan\frac{5{\vartheta }_h}{2}}{85|{\vartheta }_h|}}∥+∥{\displaystyle \frac{\pi sin\left(2\sigma \right)-\pi sin\left(2\vartheta \right)}{85\sqrt{2-h}}}∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{85}}∥arctan{\displaystyle \frac{5{\sigma }_h}{2}}-arctan{\displaystyle \frac{5{\vartheta }_h}{2}}∥+{\displaystyle \frac{\pi }{85}}∥sin\left(2\sigma \right)-sin\left(2\vartheta \right)∥\hfill \\ \hfill & \le {\displaystyle \frac{\pi }{85}}\left(∥{\sigma }_h-{\vartheta }_h∥+∥\sigma -\vartheta ∥\right).\hfill \end{array}$$

Thus, the conditions $\left(\mathbb{A}{\mathbb{S}}_1\right)$ and $\left(\mathbb{A}{\mathbb{S}}_2\right)$ of Theorem 1 are satisfied with

$$\mathcal{O}={\displaystyle \frac{1}{3}},\mathcal{Z}={\displaystyle \frac{\pi }{85}}.$$

Theorem 4 

(Application to Theorem 1). With the goal of demonstrating Theorem 1, we take

$$\mathcal{E}=\left({\displaystyle \frac{1.0779}{\Gamma \left(0.5333\right)}}\right)\left({\displaystyle \frac{1}{3}}\right)+\left(2.4804+0.6117\right)\left({\displaystyle \frac{\pi }{85}}\right)=0.3216.$$

This leads to the conclusion that

$$2\mathcal{E}=0.6433<1.$$

Theorem 5 

(Application to Theorem 2). The stability of (12) is subsequently examined using the Hyers–Ulam and Hyers–Ulam–Rassias techniques. Given that $\varphi \left(\mathbf{h}\right)=e^{\mathbf{h}}$, we obtain

$$\begin{array}{cc}\hfill & ∥{\int }_0^h{(h-\ell )}^{\rho -1}{e}^{\ell }d\ell ∥\le ∥{\int }_0^h{h}^{\rho -1}{e}^{\ell }d\ell ∥\hfill \\ \hfill & \le {\int }_0^h∥h^{\rho -1}{e}^{\ell }∥d\ell \hfill \\ \hfill & \le {\int }_0^h{e}^{\ell }d\ell =e^h-1<{\displaystyle \frac{4}{5}}{e}^h,h\in [0,1].\hfill \end{array}$$

Such that, $∥h^{\rho -1}∥\le 1$. Thus, $\varphi \left(h\right)$ satisfies Equation (9) with $\mathcal{Q}={\displaystyle \frac{4}{5}}$. As a result, Theorem 2 assures that Equation (12) possesses Hyers–Ulam–Rassias stability where

$$\mathcal{K}={\displaystyle \frac{\left(1.0355\right)\left(\frac{4}{5}\right)}{1-0.6433}}=2.3222>0.$$

This leads to the conclusion that problem (12) is Hyers–Ulam–Rassias-stable.

Theorem 6 

(Application to Theorem 3). To verify Theorem 3, we require

$$∥\sigma \left(h\right)-\vartheta \left(h\right)∥\le \mathfrak{r}\mathbf{\varrho },h\in [0,1],$$

where ϱ is any positive real constant. Ultimately, we have

$$\begin{array}{c}\hfill \mathfrak{r}={\displaystyle \frac{b^{\rho }{\mathcal{M}}_1}{\rho (1-2\mathcal{E})}}={\displaystyle \frac{1.0355}{0.5333(1-0.6433)}}=5.4430>0.\end{array}$$

Subsequently, there exists $\mathfrak{r}=5.4430$ such that

$$∥\sigma \left(h\right)-\vartheta \left(h\right)∥\le \mathfrak{r}\mathbf{\varrho }.$$

As a result, problem (12) is Hyers–Ulam-stable.

7. Conclusions

In this article, we conducted an explicit examination of neutral functional evolution equations incorporating the mixed fractional derivative ${}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }\left({}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }\right)$, where ${}^{\mathcal{R}\mathcal{L}}\mathcal{D}_{0^{+}}^{\alpha }(·)$ and ${}^{\mathcal{C}}\mathcal{D}_{0^{+}}^{\beta }(·)$ denote the Riemann–Liouville and Caputo fractional derivatives, respectively, and where $1<\alpha +\beta <2$. We established both Hyers–Ulam–Rassias and Hyers–Ulam stability for the system. By applying the Banach contraction principle, we derived the conditions required for the existence and uniqueness of mild solutions. These results were then demonstrated through simulation data obtained from the inhomogeneous fractional wave equation. Future research will explore the controllability and Hyers–Ulam stability of Hilfer–Katugampola fractional differential inclusions, utilizing nonlinear Leray–Schauder methods and measurements of non-compactness.