Introduction

The concept behind fractional calculus is to change the natural numbers in a derivative’s order to rational ones. Despite being an elementary consideration, it has captivating correspondence that explains some physical phenomena, like viscoelasticity, phase locking loops, exchange signals, automatic control, etc. (See1,2). The study of differential equations with various boundary conditions under fractional derivatives (FDs) has seen significant advancements in recent years, (see3,4). This progress is largely due to the non-local nature of FDs, which gives rise to a wide range of phenomena across diverse fields of science and engineering. The slit-strip boundary conditions (see5) is a specific geometrically constrained domain that can arise in fracture mechanics, especially to describe the behavior of materials with cracks, heat conduction, and wave propagation in narrow regions. In the context of fractional differential equations (FDEs), they present challenging problems due to their complex geometry and the non-local effect of FDs. The qualitative analysis of hybrid FDEs is an emerging and expanding field, particularly in modeling continuous and discrete dynamics as well as memory effects. A significant breakthrough in this area came in 2011, when Zhao (see6) first combined fractional calculus with the theory of hybrid FDEs. Since then, these equations have been explored by several researchers, with the class of hybrid equations involving the fractional derivative of an unknown function whose nonlinearity depends on it. Recent developments and results on hybrid differential equations can be found in a series of papers7,8,9,10,11. For instance, Sitho et al.11 studied the hybrid fractional integro-differential equations (HFIDEs) given by

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\mathcal {D}}^{r}\Bigl [\frac{\varkappa (\varsigma )-\sum _{i=1}^{n} {\mathcal {I}}^{\omega _{i}}\kappa _{i}(\varsigma ,\varkappa (\varsigma ))}{ \zeta (\varsigma ,\varkappa (\varsigma ))}\Bigr ] & =m(\varsigma ,\varkappa (\varsigma )),\ \varsigma \in [0,T ], \\ \varkappa (0)=0, & \end{array} \right. \end{aligned}$$
(1)

where \(D^{r}\) represents the Riemann–Liouville FD of order r, \(0<r\le 1\), \({\mathcal {I}}^{\phi }\) is the Riemann–Liouville fractional integral (Riemann–Liouville FI) of order \(\phi >0,\ \phi \in \{\omega _{1},\omega _{2},\ldots ,\omega _{n}\}\), \(\zeta \in C([0,T] \times {\mathbb {R}}, {\mathbb {R}}-\{0\})\) and \(m,\kappa _{i}\in C([0,T] \times {\mathbb {R}},{\mathbb {R}})\), with \(\kappa _{i}(0,0))=0, \ i=0,1,\dots ,m^{*}\).

Ahmed et al.12 established the existence of solutions for (1) with a nonlocal boundary value problem. However, it is well known that the hybrid FDEs have several drawbacks in the literature see13,14,15,16,17,18,19,20 . In 2017 Almeida21 defined a generalized FD that extends the classical Caputo FD, also known as \(\vartheta -\)Caputo derivative. A function \(\vartheta (\varsigma )\) is included in this generalization to increase modeling systems with more flexibility and capture non-uniform scaling in time or space. The FDEs involving \(\vartheta -\)Caputo derivatives were extensively studied in the last decades by many researchers (see13,17,22,23,24,25,26,27).

Mathematical modeling through differential equations offers a robust framework for analyzing the spread and control of infectious diseases, as supported by recent studies such as28,29 and the references therein. Recent research has made tremendous progress in fractional modeling of cholera dynamics by utilizing novel methodologies and realistic complexities. Cui et al.30 highlighted the significance of memory effects in disease control techniques by putting out a fractional SVIR-B model that addresses saturated therapy and incomplete vaccination. The versatility and precision of FDs in epidemiological modeling was demonstrated by Baleanu et al.31, who presented a generalized fractional model to examine actual cholera epidemics. In a cholera epidemic model, Rashid et al.32 used a fractal-fractional operator technique, offering a thorough numerical analysis that successfully reflected the complex relationships between human populations and environmental reservoirs. These articles demonstrated how fractional calculus is increasingly being used to improve our comprehension and control of cholera outbreaks. Furthermore, researchers Atangana et al.33, Cetin et al.34 and Arik et al.35 developed mathematical models utilizing generalized Caputo fractional operators to analyze and predict the spread of COVID-19 in various regions. In contrast, the study by36 focuses on developing an epidemic model by employing the \(\vartheta\)-Caputo FD.

In light of ongoing research in this field, and motivated by the increasing interest in the development of generalized fractional operators, this study focuses on a novel class of hybrid fractional integral-differential equations (HFIDEs) that incorporate a mixture of generalized operators, specifically the Caputo and Riemann–Liouville operators defined with respect to another function. This generalized framework enables the unification of various existing models and allows for greater adaptability in the formulation of real-world problems. Therefore, motivated by both theoretical developments and practical relevance, we investigate the following hybrid fractional integral-differential equation (HFIDE),

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle ^{C}{\mathcal {D}}^{\vartheta ,r}\Bigl [\frac{\varkappa (\varsigma )-\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))}{\zeta (\varsigma ,\varkappa (\varsigma ))}\Bigr ]=m(\varsigma ,\varkappa (\varsigma )),\ \varsigma \in [0,1], & & \\ \displaystyle \varkappa (0)=0\ \ \varkappa ^{^{\prime }}(0)=0\ \varkappa ^{^{\prime \prime }}(0)=0 ...,\varkappa ^{m^{*}-2}(0)=0 & & \\ \displaystyle \varkappa (\mu )=a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds,\ 0<\rho<\mu<\theta <1, & & \end{array} \right. \end{aligned}$$
(2)

where \(^CD^{\vartheta ,r}\) represents the \(\vartheta\)-Caputo FD of order r (\(m^{*}-1<r\le m^{*}\)), \({\mathcal {I}}^{\phi ,\vartheta }\) is the \(\vartheta\)-Riemann-Liouville FI of order \(\phi >0,\ \phi \in \{\omega _1,\omega _2,\ldots , \omega _n\}\), \(\zeta \in C([0,T] \times {\mathbb {R}}, {\mathbb {R}}-\{0\})\), and \(m,\kappa _{i}\in C([0,T] \times {\mathbb {R}},{\mathbb {R}})\), with \(\kappa _{i}(0,0))=0, \ i=0,1,\dots ,m^{*}\). The definitions of the operators in problems (1) and (2) will be defined in the preliminaries section later.

This study makes several significant contributions to the theory and application of fractional differential equations. Firstly, it establishes the existence of solutions for a class of HFIDEs (2), formulated with slit-strip boundary conditions. The analysis employs a combination of the \(\vartheta\)-Caputo FD and the \(\vartheta\)-Riemann–Liouville FI, with the fixed-point theorem of Dhage’s serving as the primary tool for proving existence results. To demonstrate the validity and applicability of the theoretical findings, two illustrative examples are presented. A novel aspect of this work lies in the use of the generalized \(\vartheta\)-fractional operators, which enables the unification and extension of a wide range of existing models through appropriate choices of the function \(\vartheta\). This general framework significantly enhances the flexibility and scope of the studied equations. Furthermore, the developed HFIDE framework is applied to a cholera epidemic model characterized by memory-dependent transmission and environmental feedback mechanisms, focusing on the case \(0< r < 1\). Under appropriate initial conditions, numerical simulations are conducted to investigate the influence of the fractional-order parameters on the epidemic dynamics. These simulations highlight the role of memory effects in the spread of cholera and underscore the practical relevance of the proposed model.

This work is structured as follows: Section “Preliminaries” reviews key properties of \(\vartheta\)-fractional calculus and presents important definitions and results. Section “Existence results” focuses on establishing the existence of solutions. Section “Examples” presents illustrative examples to validate the theoretical findings. Section “Application: cholera outbreaks model” applies the developed framework to a practical case study on the dynamics of a cholera epidemic model. We conclude our results in Section “Conclusion and future work”.

Preliminaries

This section presents key lemmas and theorems utilized throughout this work. Specifically, we note that \(r>0\) represents the order of the fractional operators and \(\Gamma (\cdot )\) denotes the Euler gamma function.

Let \(\ \varkappa : [c, d]\longrightarrow {\mathbb {R}}\) be an integrable function, and \(\vartheta \, [c, d]\longrightarrow {\mathbb {R}}\) is an increasing function satisfy \(\vartheta ^{^{\prime }}(\tau )>0, \ \forall \tau \in [c,d]\).

Definition 1

2 Let \(r > 0\), \(\varkappa \in L^{1}([c,d], {\mathbb {R}})\), and \(\vartheta \in C^{n}([c,d], {\mathbb {R}})\) such that \(\vartheta '(\tau ) > 0\) for all \(\tau \in [c,d]\). Then, the \(\vartheta\)-Riemann–Liouville FI of order \(r\) of the function \(\varkappa\) is defined by

$$\begin{aligned} {\mathcal {I}}_{c^{+}}^{r,\vartheta } \varkappa (\tau ) = \frac{1}{\Gamma (r)} \int _{c}^{\tau } \vartheta '(\sigma ) \big ( \vartheta (\tau ) - \vartheta (\sigma ) \big )^{r-1} \varkappa (\sigma )\, d\sigma . \end{aligned}$$
(3)

Definition 2

2 Let \(r>0,\) and \(\varkappa \in L^{1}([c,d] ,{\mathbb {R}}),\) and \(\vartheta \in C^{n}([c,d] ,{\mathbb {R}})\) such that \(\vartheta ^{^{\prime }}(\tau )>0,\) for all \(\tau \in [c,d]\) . Then, \(\vartheta\)-Riemann–Liouville FD of order \(r\) of the function \(\varkappa\) is given by

$$\begin{aligned} {\mathcal {D}}_{c^{+}}^{r,\vartheta }\varkappa (\tau )=\Bigl (\frac{1}{\vartheta ^{^{\prime }}(\tau )}\frac{d}{d\tau }\Bigr )^{n}\ {\mathcal {I}} _{c^{+}}^{n-r,\vartheta }\varkappa (\tau ),\ n=[r]+1. \end{aligned}$$
(4)

Definition 3

21 Let \(\varkappa \in C^{n-1}([c, d], {\mathbb {R}})\), and n is defined as \(n = [r] + 1\). Then the \(\vartheta\)-Caputo FD is defined as:

$$\begin{aligned} ^C{\mathcal {D}}^{r, \vartheta }_{c^+} \varkappa (\tau ):= {\mathcal {D}}^{r, \vartheta }_{c^+} \left[ \varkappa (\tau )- \sum _{k=0}^{n-1} \frac{ x^{[k]}_{\vartheta }(c)}{k!} (\vartheta (\tau ) - \vartheta (c))^k \right] , \end{aligned}$$

where \(\varkappa ^{[k]}_{\vartheta }(\tau ) := \left( \frac{1}{ \vartheta ^{\prime }(\tau )} \frac{d}{d\tau } \right) ^k \varkappa (\tau )\). When \(\varkappa \in C^{n}([c, d], {\mathbb {R}})\), the \(\vartheta\)-Caputo FD can be written as

$$\begin{aligned} ^C{\mathcal {D}}^{r, \vartheta }_{c^+} \varkappa (\tau ) = {\mathcal {I}} ^{n-r,\vartheta }_{c^{+}} \ \varkappa ^{[n]}_{\vartheta }(\tau ). \end{aligned}$$
(5)

Remark 1

In particular, if \(\vartheta (\tau ) = \tau\), then equation (3) reduces to the classical Riemann-Liouville FI. Similarly, if \(\vartheta (\tau ) = \log \tau\), it corresponds to the Hadamard FI. Moreover, equations (4) and (5) reduce to the Riemann-Liouville and Caputo FDs or the Hadamard FDs, respectively, depending on the choice of the function \(\vartheta\).

Theorem 1

21,37 Let \(\varkappa :[c,d] \longrightarrow {\mathbb {R}}\) be a function.

  1. 1.

    If \(\varkappa\) is continuous, then

    $$\begin{aligned} ^{C}D^{r,\vartheta }_{c^+}{\mathcal {I}}^{r,\vartheta }_{c^+}\varkappa (\tau )= \varkappa (\tau ). \end{aligned}$$
  2. 2.

    If \(\varkappa \in C^{n-1}([c, d], {\mathbb {R}})\), then

    $$\begin{aligned} {\mathcal {I}}^{r,\vartheta }_{c^+} \ ^CD^{r,\vartheta }_{c^+} \varkappa (\tau )= \varkappa (\tau )-\sum _{k=0}^{n-1}\frac{\varkappa ^{[k]}_{\psi }(c)}{k!} (\vartheta (\tau )-\vartheta (c))^{k}. \end{aligned}$$

Lemma 1

21 Given \(\beta \in {\mathbb {R}}\) consider the functions \(\varkappa (\tau )= (\vartheta (\tau )-\vartheta (c))^{\beta -1}\), where \(\beta >n\) . Then, for \(r>0\),

$$\begin{aligned} ^{C}D^{r,\vartheta }_{c^+} \varkappa (\tau )=\frac{\Gamma (\beta )}{ \Gamma (\beta -r)}(\vartheta (\tau )-\vartheta (c))^{\beta -r-1}. \end{aligned}$$

Moreover, if \(\varkappa (\tau )= (\vartheta (\tau )-\vartheta (c))^{k}\), where \(k\in \{0,1,...m^{*}-1 \}\). Then, for \(r>0\),

$$\begin{aligned} ^{C}D^{r,\vartheta }_{c^+} \varkappa (\tau )=0. \end{aligned}$$

Let \({\mathcal {E}}=C(\Sigma ,{\mathbb {R}})\) be the space of continuous real-valued functions defined on \(\Sigma =[0,1]\), equipped with the norm \(||\cdot ||\) and a multiplication operation defined in \({\mathcal {E}}\) as follows:

$$\begin{aligned} ||\varkappa ||=\underset{\varsigma \in \Sigma }{\sup }|\varkappa (\varsigma )| \quad \text { and} \quad (\varkappa \nu )(\varsigma )=\varkappa (\tau )\nu (\varsigma ), \ \forall \varsigma \in \Sigma , \end{aligned}$$

where \({\mathcal {E}}\) forms a Banach algebra with respect to the supremum norm and the defined multiplication.Next, we present the Dhage fixed point theorem38, which will be applied in the Banach algebra \({\mathcal {E}}\).

Lemma 2

(Dhage fixed point theorem)38 Let \({\mathbb {B}}\) be a nonempty, closed, convex, and bounded subset of a Banach algebra \({\mathcal {E}}\). Furthermore, let \(\Theta _1, \ \Theta _2: {\mathcal {E}} \longrightarrow {\mathcal {E}}\) and \(\Lambda : {\mathbb {B}} \longrightarrow {\mathcal {E}}\) be three operators satisfying the following conditions:

  1. 1.

    \(\Theta _1\) and \(\Lambda\) are Lipschitz continuous with constants \(\sigma\) and \(\tau\) (Lipschitz constants), respectively;

  2. 2.

    \(\Theta _2\) is both continuous and compact;

  3. 3.

    For every \(\varkappa \in {\mathbb {B}}\), the relation \(\varkappa = \Theta _1 \varkappa \Theta _2 \nu + \Lambda \varkappa\) implies that \(\varkappa \in {\mathbb {B}}\);

  4. 4.

    The inequality \(\sigma M + \tau < 1\) holds, where \(M = ||\Theta _2( {\mathbb {B}})||\).

Then, the equation \(\Theta _1 \varkappa \Theta _2 \nu + \Lambda \varkappa = \varkappa\) has a solution in \({\mathbb {B}}\).

Existence results

This section is dedicated to establishing the existence theorem based on Dhage’s theory (Lemma 2). Before that, we focus on the following auxiliary lemma that will be the cornerstone of the solution to the problem (2).

Lemma 3

Let \(\varkappa \in AC([0,1],{\mathbb {R}})\). Then \(\varkappa\) is a solution of the HFIDE

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle ^{C}D^{\vartheta ,r}\Bigl [\frac{\varkappa (\varsigma )-\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))}{\zeta (\varsigma ,\varkappa (\varsigma ))}\Bigr ]=m(\varsigma ,\varkappa (\varsigma )),\ \varsigma \in \Sigma , & & \\ \displaystyle \varkappa (0)=0\ \ \varkappa ^{^{\prime }}(0)=0\ \varkappa ^{^{\prime \prime }}(0)=0 ...,\varkappa ^{m^{*}-2}(0)=0, & & \\ \displaystyle \varkappa ({\mu })=a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds,\ 0<\rho<\mu<\theta <1,\ & & \\ & & \end{array} \right. \end{aligned}$$
(6)

if and only if

$$\begin{aligned} \varkappa (\varsigma )= & \sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))+\zeta (\varsigma ,\varkappa (\varsigma ))\frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{\Delta } \nonumber \\ & \Biggl [a\int _{0}^{\rho }\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \nonumber \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \nonumber \\+ & \Bigl (\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{n}\int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})}\kappa _{i}(s,\varkappa (s))ds}{\zeta (\mu ,\varkappa (\mu ))} \nonumber \\- & \int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds\Bigr )\Biggr ], \end{aligned}$$
(7)

where

$$\begin{aligned} \Delta =\mu ^{m^{*}-1}-\frac{1}{m^{*}}(1-\mu ^{m^{*}})\ne 0. \end{aligned}$$

Proof

First, let \(\varkappa \in C^{n-1}([0,1])\) be a solution of (6). Then, by Theorem 1, we get

$$\begin{aligned} & \frac{\varkappa (t)-\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(s,\varkappa (s))}{\zeta (\varsigma ,\varkappa (\varsigma ))} \nonumber \\= & \int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)\frac{(\vartheta (\varsigma )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds+c_{0}(\vartheta (\varsigma )-\vartheta (0))+c_{1}(\vartheta (\varsigma )-\vartheta (0))^{2} \nonumber \\+ & c_{m^{*}-1}(\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}. \end{aligned}$$
(8)

Applying the boundary conditions:

$$\begin{aligned} \varkappa (0)=\varkappa ^{^{\prime }}(0)=...=\varkappa ^{m^{*}-2}(0)=0,\ and\ \varkappa ({\mu })=a\displaystyle \int _{0}^{\rho }\varkappa (s)ds+b \displaystyle \int _{\theta }^{1}\varkappa (s)ds.\ \end{aligned}$$

It follows that

$$\begin{aligned} c_{0}= & c_{1}=...=c_{m^{*}-2}=0, \ \text {and}\\ c_{m^{*}-1}= & \frac{\zeta (\varsigma ,\varkappa (\varsigma ))}{\Delta } \Biggl [a\int _{0}^{\eta }\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & \Bigl (\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{ (\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})} \kappa _{i}(s,\varkappa (s))ds}{\zeta (\mu ,\varkappa (\mu ))} \\- & \int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds\Bigr )\Biggr ]. \end{aligned}$$

Substituting the value of \(c_0,\ c_{1},\ldots , c_{m^*-1}\) into the equation (8), we obtain the required equation (7).

Conversely, we show that if \(\varkappa\) satisfies the integral equation (7), then it satisfies the problem (6).

Let \(y(\varsigma )=\frac{\varkappa (\varsigma )-\sum _{i=1}^{m^{*}} {\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))}{\zeta (\varsigma ,\varkappa (\varsigma ))}\). Substitute the integral equation into \(y(\varsigma )\),

$$\begin{aligned} y(\varsigma )=\frac{(\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{ \Delta }{\mathcal {T}}(\varsigma ), \end{aligned}$$

where

$$\begin{aligned} \mathcal {T(}\varsigma \mathcal {)}= & \Biggl [a\int _{0}^{\rho }{\mathcal {I}} ^{r,\vartheta }m(\varsigma ,\varkappa (\varsigma ))du+b\int _{\theta }^{1} {\mathcal {I}}^{r,\vartheta }m(\varsigma ,\varkappa (\varsigma ))du \\ & +\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{n}{\mathcal {I}}^{\omega _{i},\vartheta }\ \kappa _{i}(\mu ,\varkappa (\mu ))}{\zeta (\mu ,\varkappa (\mu ))} \\ & -{\mathcal {I}}^{r,\vartheta }m(\mu ,\varkappa (\mu ))\Biggr ]. \end{aligned}$$

Applying \(^{C}D^{\vartheta ,r}\) to both sides

$$\begin{aligned} ^{C}D^{\vartheta ,r}y(\varsigma )=\ ^{C}D^{\vartheta ,r}\left[ \frac{ (\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{\Delta }{\mathcal {T}} (\varsigma )\right] . \end{aligned}$$

Using Theorem 1 and Lemma 1, we have \(^{C}D^{\vartheta ,r}\) \({\mathcal {I}}^{r,\vartheta }m(\varsigma ,\varkappa (\varsigma ))=m(\varsigma ,\varkappa (\varsigma ))\), and \(^{C}D^{\vartheta ,r}(\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}=0\) (since \(m^{*}-1\ge \lceil r\rceil\)). Boundary terms vanish as they are constants with respect to \(\varsigma\). Thus

$$\begin{aligned} ^{C}D^{\vartheta ,r}y(\varsigma )= & m(\varsigma ,\varkappa (\varsigma )). \end{aligned}$$

Now, we verify boundary conditions. For \(k=0,1,\ldots ,m^{*}-2\), the term \((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}\) has \(\varkappa ^{(k)}(0)=0\) and the fractional integrals \({\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))\) and \({\mathcal {I}}^{r,\vartheta }m(\varsigma ,\varkappa (\varsigma ))\) inherit regularity from \(\kappa _{i},\zeta ,m\), ensuring \(\varkappa ^{(k)}(0)=0\). Substitute \(\varsigma =\mu\) into the integral equation:

$$\begin{aligned} \varkappa (\mu )=\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\mu ,\varkappa (\mu ))+\zeta (\mu ,\varkappa (\mu ))\frac{ (\vartheta (\mu )-\vartheta (0))^{m^{*}-1}}{\Delta }{\mathcal {T}}(\mu ). \end{aligned}$$

Simplify

$$\begin{aligned} {\mathcal {T}}(\mu )&=a\int _{0}^{\rho }{\mathcal {I}}^{r,\vartheta }m(\mu ,\varkappa (\mu ))du+b\int _{\theta }^{1}{\mathcal {I}}^{r,\vartheta }m(\mu ,\varkappa (\mu ))du \\&\quad +\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\mu ,\varkappa (\mu ))}{\zeta (\mu ,\varkappa (\mu ))}-{\mathcal {I}} ^{r,\vartheta }m(\mu ,\varkappa (\mu )). \end{aligned}$$

The polynomial term cancels via \(\Delta\), and boundary terms reduce to:

$$\begin{aligned} \varkappa (\mu )=a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds. \end{aligned}$$

Therefore, the integral equation (7) satisfies the HFIDE (6). The proof is completed. \(\square\)

Now, we need to introduce the following assumptions.

  1. (H1)

    Consider continuous, non-negative functions \(\zeta : \Sigma \times {\mathbb {R}} \longrightarrow {\mathbb {R}} \setminus \{0\}\) and \(\kappa _{i} : \Sigma \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\), for \(i = 1, 2, \ldots , m^{*}\). Let \(\Phi\) and \(\lambda _i\) (\(i = 1, 2, \ldots , m^{*}\)) be given functions with respective bounds \(\Vert \Phi \Vert\) and \(\Vert \lambda _i\Vert\), such that the following Lipschitz-type conditions hold:

    $$\begin{aligned} |\zeta (\varsigma ,\varkappa ) - \zeta (\varsigma ,\nu )| \le \Phi (\varsigma ) |\varkappa - \nu |, \end{aligned}$$

    and

    $$\begin{aligned} |\kappa _i (\varsigma ,\varkappa ) - \kappa _i (\varsigma ,\nu )| \le \lambda _i (\varsigma ) |\varkappa - \nu |,\quad i = 1, 2, \ldots , m^{*}, \end{aligned}$$

    for all \(\varsigma \in {\mathcal {I}}\) and \(\varkappa , \nu \in {\mathbb {R}}\).

  2. (H2)

    Considering a continuous function p(.) defined from \(\Sigma\) into \({\mathbb {R}},\) and a continuous nondecreasing function \(\Psi (.)\) defined from \({\mathbb {R}}^{+}\) into \({\mathbb {R}}^{+}\) such that

    $$\begin{aligned} |m(t,\varkappa (\varsigma )|\le p(\varsigma )\Psi (|\varkappa |),\ (\varsigma ,\varkappa )\in \Sigma \times {\mathbb {R}}. \end{aligned}$$
  3. (H3)

    There exists a constant \(M_{0}>0\) such that

    $$\begin{aligned} \frac{\varkappa (\mu )}{\zeta (\mu ,\varkappa (\mu ))}\le M_{0},\ \forall \varkappa \in C(\Sigma ,{\mathbb {R}}). \end{aligned}$$
  4. (H4)

    There exists a number \(R>0\) such that

    $$\begin{aligned} R\ge & (R||\Phi ||+F_{0})\frac{(\vartheta (\varsigma )-\vartheta (0)^{m^{*}-1}}{|\Delta |}\Bigl \{\Bigl (|a|\frac{(\rho )^{r+1}((\vartheta (1)-\vartheta (0))^{r+1}}{\Gamma (r+2)} \nonumber \\+ & |b|\frac{1-\theta ^{r+1}((\vartheta (1)-\vartheta (0))^{r-1}}{\Gamma (r+2) }+\frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)}\Bigr )||p||\Psi (R) \nonumber \\+ & M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}R||\lambda _{i}||+M_{1}}{\Gamma (\omega _{i}+1)\zeta (\mu ,\varkappa (\mu ))}\Bigr )\Bigr \}, \end{aligned}$$
    (9)

    where \(F_{0}=\underset{\varsigma \in \Sigma }{\sup }|\zeta (\varsigma ,0)|,\)\(M_{1}=\underset{\varsigma \in \Sigma }{\sup }|\omega _{i}(\varsigma ,0)|,\ i=1,2,\ldots ,m^{*}\), and

    $$\begin{aligned} ||\Phi ||M+\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}}{\Gamma (\omega _{i}+1)}||\lambda _{i}||<1. \end{aligned}$$
    (10)

Now, we prove the existence result of the problem (2) on Banach algebra based on Lemma 2.

Theorem 2

Assuming the assumptions (H1)–(H4) holds. Then there exists at least one solution for the problem (2).

Proof

Consider \({\mathbb {B}}=\{\varkappa \in {\mathcal {E}}:||\varkappa ||\le R\}\), where R verify the inequality (9). \({\mathbb {B}}\) is a closed, convex, and bounded subset of Banach space \({\mathcal {E}}\). The problem (2) can be transformed into the equation (7) using the Lemma 1. Now, we can write the following three operators \(\Theta ,\ \Xi ,\ and\ \Lambda\) defined from \({\mathcal {E}}\) into itself, such that

$$\begin{aligned} (\Theta \varkappa )(\varsigma )=\zeta (\varsigma ,\varkappa (\varsigma )),\forall \varsigma \in \Sigma , \\ (\Xi \varkappa )(\varsigma )= & \frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{\Delta }\Biggl [a\int _{0}^{\rho }\Bigl (\frac{1}{\Gamma (r) }\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & \Bigl (\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{ (\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})} \kappa _{i}(s,\varkappa (s))ds}{\zeta (\mu ,\varkappa (\mu ))}\\- & \int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds\Bigr )\Biggr ], \ \forall \varsigma \in \Sigma , \end{aligned}$$

and

$$\begin{aligned} (\Lambda \varkappa )(\varsigma )=\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))=\sum _{i=1}^{m^{*}}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(\varsigma )(\vartheta (\varsigma )-\vartheta (s))^{\omega _{i}-1}\kappa _{i}(s,\varkappa (s))ds,\ \forall \varsigma \in \Sigma . \end{aligned}$$

Next, we show that \(\Theta\), \(\Xi\), and \(\Lambda\) satisfy all the conditions of Lemma 2. The proof is given in the following steps.Step 1: \(\Theta\) and \(\Xi\) are Lipschitzian on \({\mathcal {E}}.\) Let \(\varkappa ,\nu \in {\mathcal {E}}\), so by (H1) we have

$$\begin{aligned} |(\Theta \varkappa )(\varsigma )-(\Theta \nu )(\varsigma )|= & |\zeta (\varsigma ,\varkappa (\varsigma ))-\zeta (\varsigma ,\nu (\varsigma )| \\\le & \Phi (\varsigma )||\varkappa -\nu ||\le ||\Phi ||||\varkappa -\nu ||, \end{aligned}$$

which implies that \(|(\Theta \varkappa )(\varsigma )-(\Theta \varkappa )(\varsigma )|\le ||\Phi ||||\varkappa -\nu ||\) for all \(\varkappa ,\nu \in {\mathcal {E}}\). So \(\Theta\) is a Lipschitzian on \({\mathcal {E}}\) . Analogously, for any \(\varkappa\), \(\nu \in {\mathcal {E}}\), we have

$$\begin{aligned} |(\Xi \varkappa )(\varsigma )-(\Xi \nu )(\varsigma )|= & \Bigl | \sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}\left( \varsigma ,\varkappa \left( \varsigma \right) \right) -\sum _{i=1}^{m^{*}} {\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\nu (\varsigma )) \Bigr | \\\le & \sum _{i=1}^{m^{*}}\frac{1}{\Gamma (\omega _{i})} \int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{\omega _{i}-1} \lambda _{i}(s) |\varkappa (s)-\nu (s)|ds \\\le & \sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}}{\Gamma (\omega _{i}+1)}||\lambda _{i}|| ||\varkappa -\nu ||. \end{aligned}$$

Consequently, we deduce that \(\Xi\) is Lipschitzian on \({\mathcal {E}}\) with a constant \(\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}}{\Gamma (\omega _{i})}||\lambda _{i}||\).

Step 2: \(\Xi\) is completely continuous on \({\mathbb {B}}.\)

Firstly, we will show that \(\Xi\) is continuous on \({\mathcal {E}}.\) Let \(\{\varkappa _{n}\}\) be a sequence in \({\mathbb {B}}\) converging to a point \(\varkappa \in {\mathbb {B}}.\) By applying the Lebesgue-dominated convergence theorem, we have for all \(\varsigma \in \Sigma\)

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\Xi \varkappa _{n}(\varsigma )= & \underset{n\rightarrow \infty }{\lim }\Biggl [\frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{\Delta }\Bigl [a\int _{0}^{\rho }\Bigl (\frac{1 }{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa _{n}(s))ds\Bigr )du \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa _{n}(s))ds\Bigr )du \\+ & \Bigl (\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{^{\prime }}(s)\frac{ (\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})} \kappa _{i}(\mu ,\varkappa _{n}(\mu ))d\mu }{\zeta (\mu ,\varkappa _{n}(\mu ))} \\- & \int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa _{n}(s))ds\Bigr )\Bigr ]\Biggr ] =\Xi \varkappa (\varsigma ). \end{aligned}$$

This implies that \(\Xi \varkappa _{n}\longrightarrow \Xi \varkappa\) pointwise on \(\Sigma\). Now, we aim to prove that the set \(\Xi ({\mathbb {B}})\) is uniformly bounded in \({\mathbb {B}}.\) For \(\varkappa \in {\mathbb {B}}\) and for all \(\varsigma \in \ \Sigma\), we have from (H2) that

$$\begin{aligned} |(\Xi \varkappa )(\varsigma )|= & \Bigl |\frac{((\vartheta (\varsigma )-\vartheta (0))^{n-1}}{\Delta }\Biggl [a\int _{0}^{\rho }\Bigl (\frac{1}{ \Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}m(s,\varkappa (s))ds\Bigr )du \\+ & \Bigl (\frac{a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{^{\prime }}(s)\frac{ (\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})} \kappa _{i}(\mu ,\varkappa (\mu ))d\mu }{\zeta (\mu ,\varkappa (\mu ))} \\- & \int _{0}^{\mu }\vartheta ^{^{\prime }}(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds\Bigr )\Biggr ]\Bigr | \\\le & \frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{|\Delta | }\Bigl \{\Bigl (|a|\frac{(\rho )^{r+1}((\vartheta (1)-\vartheta (0))^{r+1}}{ \Gamma (r+2)}+|b|\frac{1-\mu ^{r+1}((\vartheta (\varsigma )-\vartheta (0))^{r-1}}{\Gamma (r+2)}\Bigr ) \\\times & ||p||\Psi (R)+\frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)} ||p||\Psi (R) \\+ & \frac{\varkappa (\mu )-\sum _{i=1}^{n}\int _{0}^{\mu }\vartheta ^{^{\prime }}(\mu )\frac{(\vartheta (\mu )-\vartheta (s))^{\phi _{i}-1}}{\Gamma (\phi _{i})}\kappa _{i}(\mu ,\varkappa (\mu ))d\mu }{\zeta (\mu ,\varkappa (\mu ))} \Bigr \} \\\le & \frac{((\vartheta (\varsigma )-\vartheta (0))^{n-1}}{|\Delta |}\Bigl \{ \Bigl (|a|\frac{(\rho )^{\alpha +1}((\vartheta (1)-\vartheta (0))^{r+1}}{ \Gamma (r+2)}+|b|\frac{1-\mu ^{\alpha +1}((\vartheta (1)-\vartheta (0))^{r-1} }{\Gamma (r+2)} \\+ & \frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)}\Bigr )||p||\Psi (R)+M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}M_{1}}{\Gamma (\omega _{i}+1)\zeta (\mu ,\varkappa (\mu ))}=K. \end{aligned}$$

Therefore, \(||\Xi ||\le K\). Consequently, we deduce that \(\Xi\) is uniformly bounded on \({\mathbb {B}}.\) Finally, we will show that \(\Xi ({\mathbb {B}})\) is an equicontinuous set in \({\mathcal {E}}\). Let \(\varsigma _{1},\ \varsigma _{2}\in \Sigma\) with \(\varsigma _{1}<\varsigma _{2}\), by using the assumptions (H1) and (H2),  we obtain

$$\begin{aligned} & |(\Xi \varkappa )(\varsigma _{2})-(\Xi \varkappa )\varsigma _{1})| \\ & \left. \le \right. \Bigl |\Bigl (\frac{\varkappa (\mu )-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{^{\prime }}(\mu )\frac{(\vartheta (\mu )-\vartheta (s))^{\phi _{i}-1}}{\Gamma (\phi _{i})}\kappa _{i}(\mu ,\varkappa (\mu ))d\mu }{\zeta (\mu ,\varkappa (\mu ))} \\ & \left. -\right. \int _{0}^{\mu }\vartheta ^{^{\prime }}(\mu )\frac{ (\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}m(s,\varkappa (s))ds\Bigr ) \frac{((\vartheta (\varsigma _{2})-\vartheta (0))^{m^{*}-1}-((\vartheta (\varsigma _{1})-\vartheta (0))^{m^{*}-1}}{\Delta } \\ & \left. \Bigl \{\right. a\int _{0}^{\rho }\Bigl (\frac{1}{\Gamma (r)} \int _{0}^{\varsigma _{2}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{2})-\vartheta (s))^{r-1}-\int _{0}^{\varsigma _{1}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{1})-\vartheta (s))^{r-1})m(s,\varkappa (s))ds \Bigr )du \\ & \left. +\right. b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)} (\int _{0}^{\varsigma _{2}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{2})-\vartheta (s))^{r-1}-\int _{0}^{\varsigma _{1}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{1})-\vartheta (s))^{r-1})m(s,\varkappa (s))ds \Bigr )du\Bigr \}\Bigr | \\ & \left. \le \right. M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\phi _{i}}M_{1}}{\Gamma (\omega _{i}+1)\zeta (\mu ,\varkappa (\mu ))}\frac{((\vartheta (\varsigma _{2})-\vartheta (0))^{n-1}-((\vartheta (\varsigma _{1})-\vartheta (0))^{n-1}}{\Delta } \\ & \left. \Bigl \{\right. \frac{|a|(\rho )^{r+1}}{\Gamma (r)}\Bigl ( \int _{0}^{\varsigma _{1}}\text {sign}(r)\vartheta ^{^{\prime }}(s)[(\vartheta (\varsigma _{2})-\vartheta (s))^{r-1}-(\vartheta (\varsigma _{1})-\vartheta (s))^{r-1}]ds \\ & \left. +\right. \int _{\varsigma _{1}}^{\varsigma _{2}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{2})-\vartheta (s))^{r-1}ds\Bigr ) \\ & \left. +\right. \frac{|b|(1-\mu ^{r+1})}{\Gamma (r)}\Bigl ( \int _{0}^{\varsigma _{1}}\text {sign}(r)\vartheta ^{^{\prime }}(s)[(\vartheta (\varsigma _{2}-\vartheta (s))^{r-1}-(\vartheta (\varsigma _{1})-\vartheta (s))^{r-1}]ds \\ & \left. +\right. \int _{\varsigma _{1}}^{\varsigma _{2}}\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma _{2})-\vartheta (s))^{r-1}ds\Bigr )\Bigr \} ||p||\Psi (R) \\ & \left. \le \right. M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\phi _{i}}M_{1}}{\Gamma (\phi _{i}+1)\zeta (\mu ,\varkappa (\mu ))}\frac{((\vartheta (\varsigma _{2})-\vartheta (0))^{m^{*}-1}-((\vartheta (\varsigma _{1})-\vartheta (0))^{m^{*}-1} }{\Delta } \\ & \left. \Bigl \{\right. \frac{|a|(\rho )^{r+2}}{\Gamma (r+1)}+\frac{ |b|(1-\mu ^{r+2})}{\Gamma (r+1)}\Bigl (\text {sign}(r+1)[(\vartheta (\varsigma _{2}-\vartheta (0))^{r}-(\vartheta (\varsigma _{2})-\vartheta (\varsigma _{1}))^{r}-(\vartheta (\varsigma _{1})-\vartheta (0))^{r}]\Bigr \} \\\times & ||p||\Psi (R). \end{aligned}$$

Since \(\vartheta (\cdot )\) is a continuous function and \(\varsigma _{1}\rightarrow \varsigma _{2}\), it follows that the right-hand side of the above inequality approaches zero. As a result, by applying the Arzela-Ascoli theorem, we deduce that \(\Xi\) is completely continuous on \({\mathbb {B}}\).

Step 3: The condition (3) of Lemma 2 holds. Let \(\varkappa \in {\mathcal {E}}\) and \(\nu \in {\mathbb {B}}\) be arbitrary elements such that \(\varkappa =\Theta \varkappa \Xi \nu +\Lambda \varkappa .\) Then

$$\begin{aligned} |\varkappa (\varsigma )|\le & |(\Theta \varkappa )(\varsigma )||(\Xi \nu )(\varsigma )|+|(\Lambda \varkappa )(\varsigma )| \\\le & |\zeta (\varsigma ,\varkappa (\varsigma ))|\Bigl \{\frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{\Delta }\Biggl [a\int _{0}^{\rho } \Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}|m(s,\varkappa (s))|ds\Bigr ) du \\+ & b\int _{\theta }^{1}\Bigl (\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}|m(s,\varkappa (s))|ds\Bigr )du \\+ & \Bigl (\frac{\varkappa (\mu )-\sum _{i=1}^{m^{*}}\int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{\omega _{i}-1}}{\Gamma (\omega _{i})}|\kappa _{i}(\mu ,\varkappa (\mu ))|d\mu }{ |\zeta (\mu ,\varkappa (\mu ))|} \\- & \int _{0}^{\mu }\vartheta ^{\prime }(s)\frac{(\vartheta (\mu )-\vartheta (s))^{r-1}}{\Gamma (r)}|m(s,\varkappa (s))|ds\Bigr )\Biggr ] +\sum _{i=1}^{m^{*}}\frac{1}{\Gamma (\omega _{i})}\int _{0}^{\varsigma }\vartheta ^{^{\prime }}(\varsigma )(\vartheta (\varsigma )-\vartheta (s))^{\omega _{i}-1}|\kappa _{i}(s,\varkappa (s))|ds \\\le & (|\zeta (\varsigma ,\varkappa (\varsigma ))-\zeta (\varsigma ,0)|+|\zeta (\varsigma ,0)|)\frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{|\Delta |}\Bigl \{\Bigl (|a|\frac{(\rho )^{r+1}((\vartheta (1)-\vartheta (0))^{r+1}}{\Gamma (r+2)} \\+ & |b|\frac{1-\mu ^{r+1}((\vartheta (1)-\vartheta (0))^{r-1}}{\Gamma (r+2)}+ \frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)}\Bigr )||p||\Psi (R)+M_{0} -\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\phi _{i}}R||\lambda _{i}||+M_{1}}{\Gamma (\phi _{i}+1)\zeta (\mu ,\varkappa (\mu ))}\Bigr )\Bigr \} \\\le & (R||\Phi ||+F_{0})\frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{|\Delta |}\Bigl \{\Bigl (|a|\frac{(\rho )^{r+1}((\vartheta (1)-\vartheta (0))^{r+1}}{\Gamma (r+2)} \\+ & |b|\frac{1-\mu ^{r+1}((\vartheta (1)-\vartheta (0))^{r-1}}{\Gamma (r+2)}+ \frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)}\Bigr )||p||\Psi (R) \\+ & M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}R||\lambda _{i}||+M_{1}}{\Gamma (\omega _{i}+1)\zeta (\mu ,\varkappa (\mu ))}\Bigr )\Bigr \} \le R. \end{aligned}$$

Hence \(\varkappa \in {\mathbb {B}}.\)

Step 4: Finally, to fulfill condition (4) of Lemma 2, we show that \(\sigma M+\tau <1\). Since "Note: Step 4 should be the beginning of a new line". 

$$\begin{aligned} M= & ||\Xi ({\mathbb {B}})||=\underset{\varkappa \in {\mathbb {B}}}{\sup }\Bigl \{ \underset{\varsigma \in \Sigma }{\sup }|(\Xi \varkappa )(\varsigma )|\Bigr \} \\\le & \frac{((\vartheta (\varsigma )-\vartheta (0))^{m^{*}-1}}{|\Delta | }\Bigl \{\Bigl (|a|\frac{(\rho )^{r+1}((\vartheta (1)-\vartheta (0))^{r+1}}{ \Gamma (r+2)}+|b|\frac{1-\mu ^{r+1}((\vartheta (1)-\vartheta (0))^{r-1}}{ \Gamma (r+2)} \\+ & \frac{(\vartheta (1)-\vartheta (0))^{r}}{\Gamma (r+1)}\Bigr ) ||p||\vartheta (R)+M_{0}-\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}M_{1}}{\Gamma (\omega _{i}+1)\zeta (\mu ,\varkappa (\mu ))}, \end{aligned}$$

and by (H3), we obtain

$$\begin{aligned} ||\Phi ||M+\sum _{i=1}^{m^{*}}\frac{(\vartheta (1)-\vartheta (0))^{\omega _{i}}}{\Gamma (\omega _{i}+1)}||\lambda _{i}||<1, \end{aligned}$$

with \(\sigma =||\Phi ||\) and \(\tau =\sum _{i=1}^{m^*}\frac{(\vartheta (1)- \vartheta (0))^{\omega _{i}}}{\Gamma (\omega _{i}+1)}||\lambda _{i}||\). Thus all the conditions of Lemma 2 are satisfied. Hence the operator equation \(\varkappa = \Theta \varkappa \ \Xi \varkappa + \Lambda \varkappa\) has a solution \(\varkappa ^{*}\) in \({\mathbb {B}}\). In consequence, problem 1 has a solution on \({\mathbb {B}}\). This completes the proof. \(\square\)

Examples

This section presents two examples to validate the above theoretical outcomes.

Example 1

Consider the HFIDE defined as:

$$\begin{aligned} \left\{ \begin{array}{ll} ^{C}{\mathcal {D}}^{\vartheta ,r}\Bigl [\frac{\varkappa (\varsigma )-\sum _{i=1}^{m^{*}}{\mathcal {I}}^{\omega _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))}{\zeta (\varsigma ,\varkappa (\varsigma ))}\Bigr ]=m(\varsigma ,\varkappa (\varsigma )),\ \varsigma \in [0,1 ], & \\ \varkappa (0)=0\ \ \varkappa ^{^{\prime }}(0)=0\ \varkappa ^{^{\prime \prime }}(0)=0 ...,\varkappa ^{m^{*}-2}(0)=0, & \\ \varkappa ({\mu })=a\int _{0}^{\rho }\varkappa (s)ds+b\int _{\theta }^{1}\varkappa (s)ds.\ & \end{array} \right. \end{aligned}$$
(11)

Here, \(r=\frac{1}{2},\) \(m^{*}=2,\) \(\vartheta (\varsigma )=\varsigma ,\) \(m(\varsigma ,\varkappa (\varsigma ))=\displaystyle |\varkappa |e^{-\varsigma },\) where \(p(\varsigma )=e^{-\varsigma }\) and \(\Psi (|\varsigma |)=|\varsigma |.\)Then, to prove the assumption (H1), we put \(\zeta (\varsigma ,\varkappa (\varsigma ))=\displaystyle \frac{\varsigma }{100}\Bigl (\frac{|\varkappa (\varsigma )|}{1+|\varkappa (\varsigma )|}\Bigr )\) such that

$$\begin{aligned} |\zeta (\varsigma ,\varkappa )-\zeta (\varsigma ,\nu )|\le \frac{\varsigma }{100}|\varkappa -\nu |, \end{aligned}$$

and

$$\begin{aligned} \sum _{i=1}^{2}{\mathcal {I}}^{\phi _{i},\vartheta }\kappa _{i}(\varsigma ,\varkappa (\varsigma ))={\mathcal {I}}^{\frac{1}{2},\vartheta }\Bigl (\frac{ \varsigma }{5}(\varkappa (\varsigma )+e^{{\varsigma }^{2}}\Bigr )+{\mathcal {I}} ^{\frac{3}{2},\vartheta }\Bigl (\frac{\varsigma \sin \varsigma }{3+\varsigma } \Bigl (\frac{\varkappa (\varsigma )}{1+\varkappa (\varsigma )}+\sin \varsigma \Bigr )\Bigr ), \end{aligned}$$

where

$$\begin{aligned} |\kappa _{1}(\varsigma ,\varkappa )-\kappa _{1}(\varsigma ,\nu )|\le \frac{ \varsigma }{5}|\varkappa -\nu |,\ |\kappa _{2}(\varsigma ,\varkappa )-\kappa _{2}(\varsigma ,\nu )|\le \frac{1}{3+\varsigma }|\varkappa -\nu |. \end{aligned}$$

It follows from (10) that the constant r satisfies (9). Since all the assumptions of Theorem 2 are fulfilled. The equation (11) has a solution on \({\mathbb {B}} .\)

Example 2

Consider the HFIDE defined as:

$$\begin{aligned} \left\{ \begin{array}{ll} ^{C}{\mathcal {D}}^{\vartheta ,0.8}\left[ \frac{\varkappa (\varsigma )-\mathcal { I}^{0.5,\vartheta }(\sin (\varsigma )\varkappa (\varsigma ))}{1+\varsigma ^{2}+\varkappa ^{2}(\varsigma )}\right] =e^{-\varsigma }\cos (\varkappa (\varsigma )),\quad \varsigma \in [0,1], & \\ \varkappa (0)=0,\quad \varkappa ^{\prime }(0)=0,\quad \varkappa ^{\prime \prime }(0)=0, & \\ \varkappa (\mu )=\frac{1}{2}\int _{0}^{\mu }\varkappa (s)\,ds+\frac{1}{3} \int _{\mu }^{1}\varkappa (s)\,ds,\quad 0<\mu <1. & \end{array} \right. \end{aligned}$$
(12)

Define \(\zeta (\varsigma ,\varkappa )=1+\varsigma ^{2}+\varkappa ^{2}\), which is continuous, non-negative, and satisfies the Lipschitz condition:

$$\begin{aligned} |\zeta (\varsigma ,\varkappa _{1})-\zeta (\varsigma ,\varkappa _{2})|\le 2|\varkappa _{1}-\varkappa _{2}|, \end{aligned}$$

with \(\Phi (\varsigma )=2\). Define \(\kappa _{1}(\varsigma ,\varkappa )=\sin (\varsigma )\varkappa (\varsigma )\), which satisfies:

$$\begin{aligned} |\kappa _{1}(\varsigma ,\varkappa _{1})-\kappa _{1}(\varsigma ,\varkappa _{2})|\le |\sin (\varsigma )||\varkappa _{1}-\varkappa _{2}|, \end{aligned}$$

with \(\lambda _{1}(\varsigma )=|\sin (\varsigma )|\). Thus, condition (H1) holds.

For condition (H2), we define \(m(\varsigma ,\varkappa )=e^{-\varsigma }\cos (\varkappa (\varsigma ))\), which satisfies:

$$\begin{aligned} |m(\varsigma ,\varkappa )|\le e^{-\varsigma }, \end{aligned}$$

with \(p(\varsigma )=e^{-\varsigma }\) and \(\Psi (u)=1\).

To verify the condition (H3), let \(\zeta (\varsigma ,\varkappa )=1+\varsigma ^{2}+\varkappa ^{2}\), we have:

$$\begin{aligned} \frac{\varkappa (\mu )}{\zeta (\mu ,\varkappa (\mu ))}\le M_{0}\quad \text { for a sufficiently large }M_{0}>0. \end{aligned}$$

Finally, for condition (H4), let \(\vartheta (\varsigma )=\varsigma ,\) and \(R=1\), then, we compute the terms to verify the relations (9) and (10), where \(F_{0}=\sup _{\varsigma \in [0,1]}|\zeta (\varsigma ,0)|=2\). A detailed computation ensures this inequality holds for suitably chosen constants a, b, and \(\mu\). Therefore, all assumptions (H1-H4) are satisfied for the given example. By Theorem 2, there exists at least one solution \(\varkappa (\varsigma )\) to the problem (12).

Application: cholera outbreaks model

In this section, we model the dynamics of cholera outbreaks by incorporating fractional derivatives into the system’s equations. We focus on three components: the susceptible population \(S(\varsigma )\), infected individuals \(I(\varsigma )\), and the concentration of vibrio cholera bacteria in the water source \(B(\varsigma )\). To achieve this, we employ a fractional-order system (2) with considering \(m^{*}=1\) (\(0<r<1\)), \(\kappa _{i}(\varsigma ,\varkappa (\varsigma ))=0\), for all i, and \(\zeta (\varsigma ,\varkappa (\varsigma ))=1\), as described by the following system of equations:

$$\begin{aligned} \left\{ \begin{array}{ll} ^{C}{\mathcal {D}}^{r,\vartheta }S(\varsigma )=-\beta S(\varsigma )\dfrac{ I(\varsigma )}{N}-\mu _{S}S(\varsigma ), & \\ ^{C}{\mathcal {D}}^{r,\vartheta }I(\varsigma )=\beta S(\varsigma )\dfrac{ I(\varsigma )}{N}+\lambda S(\varsigma )B(\varsigma )-(\mu _{I}+\gamma )I(\varsigma ), & \\ ^{C}{\mathcal {D}}^{r,\vartheta }B(\varsigma )=\xi I(\varsigma )-\delta B(\varsigma ), & \end{array} \right. \end{aligned}$$
(13)

with initial conditions \(S(0)\ge 0,\quad I(0)\ge 0,\quad B(0)\ge 0\), where

  • \(\beta\): Transmission rate due to direct contact.

  • \(\lambda\): Transmission rate via the environment (waterborne).

  • \(\xi\): Shedding rate of cholera bacteria into water.

  • \(\delta\): Decay rate of cholera bacteria.

  • \(\mu _S, \mu _I\): Mortality rates of susceptible and infected individuals.

  • \(\gamma\): Recovery rate of infected individuals.

  • \(\vartheta (\varsigma )=\varsigma ^{2}\).

The model (13) can be written as

$$\begin{aligned} \left\{ \begin{array}{ll} ^{C}{\mathcal {D}}^{r,\vartheta }S(\varsigma )=F_{S}(\varsigma ,S,I,B), & \\ ^{C}{\mathcal {D}}^{r,\vartheta }I(\varsigma )=F_{I}(\varsigma ,S,I,B), & \\ ^{C}{\mathcal {D}}^{r,\vartheta }B(\varsigma )=F_{B}(\varsigma ,S,I,B), & \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{c} F_{S}(\varsigma ,S,I,B)=-\beta S(\varsigma )\dfrac{I(\varsigma )}{N}-\mu _{S}S(\varsigma ),\qquad \qquad \qquad \ \ \ \\ F_{I}(\varsigma ,S,I,B)=\beta S(\varsigma )\dfrac{I(\varsigma )}{N}+\lambda S(\varsigma )B(\varsigma )-(\mu _{I}+\gamma )I(\varsigma ), \\ F_{B}(\varsigma ,S,I,B)=\xi I(\varsigma )-\delta B(\varsigma ).\qquad \qquad \qquad \qquad \qquad \ \end{array} \right. \end{aligned}$$

Numerical scheme

In this subsection, we introduce the Adams-Bashforth-Moulton method for approximating the solution to the proposed Caputo cholera model. For further details on this method and its convergence, refer to the works of Diethelm & Ford39,40,41. The Adams-Bashforth-Moulton method is one of the most widely employed numerical techniques for solving fractional-order initial value problems. We now examine the fractional differential equation governing the model:

$$\begin{aligned} ^{C}D^{r,\vartheta }X_{j}(\varsigma )=F_{j}(\varsigma ,X_{j}(\varsigma )),\quad X_{j}(0)=X_{j}^{0}, \quad j\in \{S,I,B\}, \end{aligned}$$
(14)

where \(X_{j}^{0}\) is the arbitrary real number, \(0<r<1\) and the generalized fractional differential operator \(^{C}D^{r,\vartheta }\) is identical to the well-known Volterra integral equation in the \(\vartheta\)-Caputo sense.

$$\begin{aligned} X_{j}(\varsigma )=X_{j}^{0}+\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }\vartheta ^{\prime }(s)(\vartheta (\varsigma )-\vartheta (s))^{r-1}F_{j}(s,X_{j}(s))ds,\quad j\in \{S,I,B\}. \end{aligned}$$

Putting \(\vartheta (\varsigma )=\varsigma ^{2}\), we obtain

$$\begin{aligned} X_{j}(\varsigma )=X_{j}^{0}+\frac{1}{\Gamma (r)}\int _{0}^{\varsigma }2s(\varsigma ^{2}-s^{2}))^{r-1}F_{j}(s,X_{j}(s))ds,\quad j\in \{S,I,B\}. \end{aligned}$$
(15)

Using Adams–Bashforth–Moulton method, we explain the numerical solution of a fractional order colera model. The algorithm is described in the following manner. Let \(\varsigma _{j}=jh,\ h=\frac{T}{N}\), \(j=0,1,2,\dots ,N\in \mathbb {Z^{+}}\) for \(0\le \varsigma \le T\). We discretize equation (15) at \(\varsigma _{n+1}\) as follows

$$\begin{aligned} \left\{ \begin{array}{c} S(\varsigma _{n+1})=S(0)+\frac{h^{r}}{\Gamma (r+2)}\Big [F_{S}(\varsigma _{n+1},S^{p}(\varsigma _{n+1}),I^{p}(\varsigma _{n+1}),B^{p}(\varsigma _{n+1})) +\sum _{j=0}^{n}a_{j,n+1}F_{S}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j}))\Big ] \\ I(\varsigma _{n+1})=I(0)+\frac{h^{r}}{\Gamma (r+2)}\Big [F_{I}(\varsigma _{n+1},S^{p}(\varsigma _{n+1}),I^{p}(\varsigma _{n+1}),B^{p}(\varsigma _{n+1})) +\sum _{j=0}^{n}a_{j,n+1}F_{I}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j}))\Big ] \\ B(\varsigma _{n+1})=B(0)+\frac{h^{r}}{\Gamma (r+2)}\Big [F_{B}(\varsigma _{n+1},S^{p}(\varsigma _{n+1}),I^{p}(\varsigma _{n+1}),B^{p}(\varsigma _{n+1})) +\sum _{j=0}^{n}a_{j,n+1}F_{B}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j}))\Big ], \end{array} \right. \end{aligned}$$
(16)

where

$$\begin{aligned} a_{j,n+1}= {\left\{ \begin{array}{ll} n^{r+1}-(n-r)(n+1)^{r}, & j=0, \\ (n-j+2)^{r+1}+(n-j)^{r+1}-2(n-j+1)^{r+1}, & 0<j\le n, \\ 1, & j=n+1, \end{array}\right. } \end{aligned}$$

and the predicted values are given by

$$\begin{aligned} S^{p}(\varsigma _{n+1})=S(0)+\frac{1}{\Gamma (r)} \sum _{j=0}^{n}b_{j,n+1}F_{S}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j})) \\ I^{p}(\varsigma _{n+1})=I(0)+\frac{1}{\Gamma (r)} \sum _{j=0}^{n}b_{j,n+1}F_{I}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j})) \\ B^{p}(\varsigma _{n+1})=B(0)+\frac{1}{\Gamma (r)} \sum _{j=0}^{n}b_{j,n+1}F_{B}(\varsigma _{j},S(\varsigma _{j}),I(\varsigma _{j}),B(\varsigma _{j})) \end{aligned}$$

where \(b_{j,n+1}=\frac{h^{r}}{r}(n+1-j)^{r}-(n-j)^{r}\).

Numerical simulation and discussion

In this part, we have discussed the obtained numerical results for the system (16) with units of time per day through the above suggested numerical technique and mathematical software MATLAB. To simulate the results graphically for the compartments S, I, and B with different values of fractional orders \(r=0.6,0.9\) with \(\vartheta (\varsigma ) =\varsigma ^{2}\), we have used values of the model parameter in the numerical simulations as \(\beta =0.2, \ \lambda =0.02,\ \xi =0.05,\ \delta =0.01\), \(\mu _{S}=0.05,\ \mu _{I}=0.02,\ \gamma =0.15\), \([N,T,h]=[15,20,0.01]\), and initial conditions \([S_{0},I_{0},B_{0}]=[14,8,2.5]\) where N is the total population.

Figs. 1 and 2 represent models of cholera epidemics with fractional order dynamics (\(r=0.6\) and \(r=0.9\)) over a time of 20 days, showing the evolution of the components: susceptible population \(S(\varsigma )\), infected individuals \(I(\varsigma )\) and concentration of cholera bacteria \(B(\varsigma )\) in water. In both cases, the susceptible population \(S(\varsigma )\) declines as more individuals get infected, while the infected population \(I(\varsigma )\) increases initially and then declines as the disease progresses. The bacteria concentration \(B(\varsigma )\) increases over time due to the shedding from infected individuals, but its growth rate slows as the dynamics of the disease change. The different fractional orders (\(r=0.6\) and \(r=0.9\)) influence the steepness and timing of the transitions in the model, with a higher fractional order \(r=0.9\)) showing more abrupt changes in the infected population and bacteria concentration, while the \(r=0.6\) case shows a smoother progression, reflecting different rates of disease spread and environmental impact.

Fig. 1
figure 1

Cholera dynamics of the susceptible population \(S(\varsigma )\), infected \(I(\varsigma )\), and bacteria concentration \(B(\varsigma )\) over time with \(r=0.6\) and \(\vartheta (\varsigma )=\varsigma ^2\).

Full size image
Fig. 2
figure 2

Cholera dynamics of the susceptible population \(S(\varsigma )\), infected \(I(\varsigma )\), and bacteria concentration \(B(\varsigma )\) over time with \(r=0.9\) and \(\vartheta (\varsigma )=\varsigma ^2\).

Full size image

Remark 2

The selection of \(\vartheta (\varsigma ) = \varsigma ^2\) in the \(\vartheta\)-Caputo fractional derivative aims to reflect an accelerated memory effect over time, where the impact of past states increases nonlinearly. This is particularly relevant for cholera dynamics, where delayed feedback through environmental reservoirs (e.g., contaminated water) significantly influences disease progression. The quadratic kernel emphasizes the cumulative effect of long-term bacterial shedding and waterborne transmission.

This choice generalizes the classical Caputo derivative (\(\vartheta (\varsigma ) = \varsigma\)) and connects to other known fractional operators such as the Caputo-Hadamard (\(\vartheta (\varsigma ) = log(\varsigma\))) and Katugampola ((\(\vartheta (\varsigma ) = \varsigma ^{\rho }\)) ) types. Numerical simulations show that this formulation affects both the transient and steady-state behaviors, highlighting the importance of kernel selection in fractional epidemic modeling.

Conclusion and future work

In this work, we analyzed a class of fractional hybrid integro-differential equations (HFIDEs) involving the \(\vartheta\)-Caputo derivative and slit-strip boundary conditions. We established the existence of solutions using Dhage’s fixed point theorem under appropriate Lipschitz-type conditions, providing a robust analytical framework for hybrid fractional systems. This approach offers significant advantages over previous studies, such as27, which examined coupled hybrid systems using more restrictive assumptions and relied on the Shaefer fixed point theorem. The validity of our findings was demonstrated through illustrative examples that highlight the versatility and applicability of the developed framework.

This study explored the dynamics of a cholera epidemic model using fractional-order derivatives (\(r=0.6 \ \text {and} \ r=0.9\)) to capture the memory-dependent behavior of disease transmission. By incorporating fractional calculus into the system of equations that govern susceptible individuals (\(S\)), infected individuals (\(I\)), and bacterial concentration (\(B\)), we have observed distinct dynamical patterns compared to classical models of integer order. Simulations demonstrated that lower fractional orders (e.g., \(r = 0.6\)) introduced slower progression and reduced peak magnitudes in the infected population and bacterial concentration, reflecting stronger memory effects and sub-diffusive processes. Conversely, higher fractional orders (e.g., \(r = 0.9\)) approached the behavior of classical ODE models but retained subtle differences in transient dynamics. These results highlight the importance of fractional calculus in modeling biological systems with inherent memory and nonlocal interactions, providing a more flexible framework to describe complex epidemiological phenomena.

A key contribution of this study is the use of the generalized \(\vartheta\)-Caputo derivative with \(\vartheta (\varsigma ) = \varsigma ^2\), capturing nonlinear memory effects relevant to cholera dynamics. Numerical results confirm that varying the fractional order significantly affects disease behavior. The framework is flexible, encompassing classical operators such as Caputo, Caputo-Hadamard, and Katugampola, making it suitable for broader epidemiological modeling.

In the future, this study opens avenues for further exploration of hybrid fractional systems. Future research could focus on extending the theoretical and computational aspects of this work by investigating the stability and periodicity of solutions under more intricate boundary conditions and developing efficient numerical methods for high-dimensional hybrid fractional systems. Additionally, exploring fractional models that incorporate time-delay kernels or variable-order derivatives for more nuanced memory effects.