Introduction

Fractional calculus (FC) is an important discipline of mathematics which is mainly investigate non-integer order integrals and derivatives. For the first time in history FC was discussed by two great mathematician Leibnitz and L. Hospital by asking half order derivative of a function1. Because of its complexity researchers did not study the subject at that time. With the passage of time when the technology get advancement and new function have been introduced, the subject got proper attention from the mathematician it deserved. Currently, FC has attracted attention of many researchers in the fields of science and engineering. For example author have used FC for the solutions of problem in signal processing2, complex systems related to control theory3, diffusion problem in physics4, problem related economic terms like interest rates, commodity prices and dynamics of market fluctuations5 and accurate dimensional analysis of image processing6. Many applications of FC can also be found in biology, chemistry and other diverse fields of sciences. For instance, authors6 studied a super-twisting sliding mode control of robotic manipulator using the concepts of FC. Shah, et.al7 studied an epidemic model of dengue fever disease using fractional non-singular derivative. Ahmad, et.al8 investigated an adaptive control of a robotic manipulator with a delay in the input by applying the fractional order derivative. Shaikh, et.al9 conducted a detailed mathematical analysis of HIV/AIDS disease by using non-singular type derivative. In the same way, authors10 applied FC concepts to study mathematical of COVID-19 using real data of Nigeria. Proceeding with the mentioned process, researchers11 studied mathematical model for the evolution of COVID-19 outbreak in India.

One of the interesting property of fractional derivative is that it does not have a unique definition. Various definitions, like the Riemann-Liouville (R-L) and the Caputo derivatives12, the conformable derivative13, the Hilfer derivative14, the Harmard derivative15 and some other are available in literature. These ideas have recently been used as the foundation for breaking down numerous mathematical problems, we refer to16, and17. Problem related to heterogeneities in materials can not be studied using the above mentioned concepts18. Researchers19 used modified Caputo definition by replacing the singular kernel by non-singular kernel and named the new operator is Caputo-Fabrizio derivative(CFD). Authors18 studied some important properties of the aforesaid operators. Many problems have been analyzed by using mentioned derivative for existence theory and introduced various properties for the mentioned operator. For some recent work on the mentioned topic, we refer20, and21 and references therein. In CFD researchers identified property of locality in the kernel used in the operator. To overcome this difficulty, Atangana and Baleanu22 in 2016 replaced the kernel of exponential by Mittag-Leffler in the CFD. The newly defined definition was called ABC derivative and applied by different authors for the solution of real world problem. For instance, Xu et al.23 discussed HIV-1 model, authors24 explained in details the Klein-Gordan equation using ABC differential operators. Using ABC derivative with fractional order, authors25 computed the numerical results for parabolic partial differential equations. In additional, researchers26 applied ZZ Transform to compute the approximate solution of Fokker Plank equations using fuzzy concepts with ABC derivative.

Furthermore, fractional differential operators, as previously stated, are insufficient to adequately explain problems involving irregular forms and geometry. To describe these kinds of behaviours, the fractal notion has been introduced in the literature. Historically, fractal was mainly used from the \(17^{th}\) century and there is a disagreement among mathematicians on how to define fractals formally. Weierstrass in 1872 presented a function with graph which nowadays called fractal. Different mathematician define fractal in different ways like a fractal is a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. In other words, fractal is a fragmented geometrical shapes that may be split into figments, each figment represent a copy of the whole shape. Because they appear in the geometric representations of the majority of chaotic processes, fractals have thus been noted by authors as being significant to the theory of chaos. Atangana27 has made a significant contribution to this field by bridging the gap between FC and fractal calculus. The aforementioned scholar established numerous definitions that are now widely used and made a substantial contribution to the subject of fractals fractional theory. Newly proposed operators can help to explain fractal behaviour of complicated physical systems and nonlocal phenomena. For instant, Hu and He28 explained fractals space time and dimensions. In addition, authors29. Furthermore, Qureshi and Atangana30 have studied diarrheal illness models using fractal operators. In the same way, researchers31 studied mathematical model of Ebola virus disease using the the mentioned operator. The use of fractal theory to issues including fractal heat exchangers, heat transport in porous media, etc., is growing. Here is where some outstanding accomplishments related to the properties and uses of fractals in porous media should be mentioned. The authors investigated a fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media32. An analytical model for the transverse permeability of the gas diffusion layer in proton exchange membrane fuel cells with electrical double layer effects was carried out by Liang et al.33. Yu, et.al34 investigated the Characterization of the behaviour of water migration during spontaneous imbibition in coal by using fractals fractional model. Recently, Ahmad, et.al35 studied a malaria disease mathematical model using fractals fractional concepts. Author36 investigated a dynamical problem by using generalized fractal-fractional derivative with Mittag-Leffler kernel. Recently, authors37 have used fractals-fractional derivative to study re-infection model of COVID-19. Author38 studied a class of hybrid problems by using confirmable fractal fractional derivative. Researchers39 have studied a mathematical model by using ABC fractals fractional derivative.

Delay differential equations (DDEs) are a unique class of differential equations where the unknown function depends on both its past and present times. The physical phenomenon has a time delay as a result of these issues. DDEs have been used in a number of disciplines, most notably computational physics and electrodynamics. It has also been successfully used to challenges in the domains of economics, chemistry, engineering, and infectious diseases (delay can be added for the time a disease takes to display its symptoms). It is very helpful for modelling issues with time delay systems or memory effects. Examples include population dynamics with time delays in birth or death rates, control systems with communication delays, and chemical reactions with delayed consequences. Many scholars have devoted a tremendous amount of time to finding solutions to delay-type problems. Authors40 explained how DDEs are using in mathematical modeling of life sciences phenomenon. Author41 gave a comprehensive analysis of DDEs. The qualitative analysis of DDEs related to the existence theory of solutions and their properties, we refer to42. Recently, authors43 have published a detailed work on computations of numerical results for fractals fractional DDEs.

Investigating the existence literature, we found that very rare work has been done for the class of hybrid fractals HFDEs. Recently, Shafiullah, et.al44 deduced some computational and theoretical results for a class of fractals HFDEs involving power law kernel by using fixed point analysis and numerical algorithm. But the mentioned area still has not explored very well under various fractals fractional differential operators. Therefore, it was needed to investigate a class of fractals HFDEs under the Mittag-Leffler type kernel. Hence, motivated by the above discussion, in this study a class of fractal HFDEs of linear perturbation type is selected for qualitative analysis and approximate solution. Here we remark that it is possible to model and describe non-homogeneous physical processes that occur in their form using HFDEs. Because the mentioned area incorporate different dynamical systems as specific examples. This category of differential equations comprises the hybrid derivative of an unknown function that depends on nonlinearity. Here a continuous function for fractals fractional derivatives and a piecewise continuous or discrete function needs for fractals fractional integration and such combination make an hybrid problem. For some numerous applications of the said area, we refer to45.

To the best of our knowledge no one has considered the general problem defined as follow:

$$\begin{aligned} \left\{ \begin{aligned}&^{FAB}{_0D}_\theta ^{\sigma ,\xi }[{\varvec{\Phi }(\theta )}-\textbf{f}({\theta },{\varvec{\Phi }(\theta )})]= \textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )}), ~ {\theta }\in I=[0, T], \\&{\varvec{\Phi }(0)}= \varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta )), \end{aligned}\right. \end{aligned}$$
(1)

where \(0<\sigma \le 1\), \(0<\lambda <1\), \(\textbf{f},\textbf{g}\in C[I\times R,R]\). Proportional delay term is shown by the symbol \(\lambda\). Pantograph equations are differential equations that have proportional delay terms. Hence, special kind of functional differential equation with proportional delay is known as the pantograph equation. It appears in a variety of pure and practical mathematics domains, including probability, number theory, quantum physics, electrodynamics, and control systems. Certain kinds of the said equations are using in modeling various process of astrophysic. In the concerned problem (1), if \(\lambda =1\), the said equation will reduce to usual fractals HFDE. If \(\lambda >1\), the said problem (1) become illposed. The detail theory about the pantograph equations and various applications were discussed in46. Here \(\xi\) stands for fractal dimension. Additionally, let \(\xi =1\) then, the fractional differential operator that is typically used becomes the considered operator. In past, hybrid differential equations have been studied by using ordinary or usual fractional derivatives. But investigation of aforementioned problems by using fractals fractional derivatives were found very rare in literature. On the other hand, the concerned area has numerous applications in modelling real world problems. Therefore, it was needed to investigate hybrid problems by using the fractals fractional derivatives. Stability theory is an important aspects of the qualitative theory which play important role in constructing various numerical schemes. Researchers have studied various kinds of stabilities for different problems. In this research work, we are interested to investigate the Ulam-Hyers (U-H) stability theory for our considered problem. The aforementioned stability was introduced by Ulam47 and explained by Hyers48. In addition, further the aforesaid stability was utilized for various functional problems by Rassias49. Recently, the mentioned stability was investigated for the solution of stochastic50, impulsive51,52,53, and piecewise54 fractional differential problems. By using hybrid fixed point theory, we will establish some important results related to existence theory and stability theory to the mentioned problem. Then by using Lagrange’s interpolation method a general numerical scheme will be developed. At the final stage, an examples from biological sciences is given to elaborate our theoretical and numerical results. Here, we remark that numerical analysis under fractals fractional concepts for hybrid problems has not well studied yet.

Preliminaries

Some basic results are recalled from12,22,28,30. Let, \(\textbf{J}=[0,\textbf{T}]\), and \(\Omega =C(\textbf{J},R)\) stands for Banach space endowed with norm \(\Vert \cdot \Vert\) on \(\Omega\) as follow:

$$\Vert \varvec{\Phi }\Vert =\sup _{ \theta \in \textbf{J}}|\varvec{\Phi }(\theta )|.$$

Definition 2.1

The R-L fractional integral of order \(0<\sigma \le 1\) is defined as follows:

$$\begin{aligned} ^{RL}{I}^{{\sigma }}{\varvec{\Phi }}({{\theta }})=\int _{0}^{{\theta }}({{\theta }}-{s})^{{\sigma }-1}{\varvec{\Phi }}({s})\frac{1}{\Gamma ({\sigma })}d{s},\quad \ {{\theta }}\in [0, T]. \end{aligned}$$

Definition 2.2

ABC type arbitrary order integral with order \({\sigma }>0\) is defined as follow:

$$\begin{aligned} ^{ABC}\textbf{I}_{0^+}^{{\sigma }}\varvec{\Phi }({\theta })=\frac{1-{\sigma }}{M({\sigma })}\varvec{\Phi }({\theta })+\frac{{\sigma }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }} ({\theta }-s)^{{\sigma }-1}\varvec{\Phi }(s)dx, \end{aligned}$$

such that the right side exists.

Definition 2.3

ABC type arbitrary order derivative of order \({\sigma } (0<{\sigma }\le 1)\) is defined as follow:

$$\begin{aligned} ^{ABC}\textbf{D}_{0}^{{\sigma }}\varvec{\Phi }({\theta })=\frac{M({\sigma })}{(1-{\sigma })}\int _{0}^{{\theta }} E_{{\sigma }}[-\frac{{\sigma }}{1-{\sigma }}({\theta }-s)^{{\sigma }}]\varvec{\Phi }^{\prime }(s)dx. \end{aligned}$$

The functions \(M({\sigma })\) in the definition is called normalization obeys \(M(0)=M(1)=1,\) and \(E_{{\sigma }}\) is Mittag-Leffler function.

Definition 2.4

Fractal type arbitrary order integral of order \({\sigma }>0\) is defined as follow:

$$\begin{aligned} ^{F}\textbf{I}_{0^+}^{{\sigma }}\varvec{\Phi }({\theta })=\sigma \int _{0}^{\theta }s^{\sigma -1}\varvec{\Phi }(s)ds. \end{aligned}$$
(2)

Definition 2.5

Fractal fractional ABC integral with fractals order \(\xi \in (0, 1],\) and fractional order \(\sigma \in (0, 1]\) is defined as follow:

$$\begin{aligned} ^{FAB}\textbf{I}_{0^+}^{{\sigma ,\xi }}\varvec{\Phi }({\theta })=\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M({\sigma })}\varvec{\Phi }({\theta })+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\varvec{\Phi }(s)ds. \end{aligned}$$

Definition 2.6

Fractal fractional ABC derivative with fractals order \(\xi \in (0, 1],\) and fractional order \(\sigma \in (0, 1]\) is defined as follow:

$$\begin{aligned} ^{FAB}\textbf{D}_{0}^{{\sigma ,\xi }}\varvec{\Phi }({\theta })=\frac{M({\sigma })}{(1-{\sigma })}\int _{0}^{{\theta }} E_{{\sigma }}[-\frac{{\sigma }}{1-{\sigma }}({\theta }-s)^{{\sigma }}]\frac{du(s)}{d s^{\xi }} ds. \end{aligned}$$

Lemma 2.7

If \(x\in L[0, {T}]\) and \(x(0)=0\), then the solution of

$$\begin{aligned} ^{FAB}{D}^{{\sigma ,\xi }}{\varvec{\Phi }}({{\theta }})= & {} {x}({{\theta }}),\ \text {with}\ \ {\sigma }\in (0, 1],\\ {\varvec{\Phi }}(0)= & {} {\varvec{\Phi }}_0, \end{aligned}$$

is described as follows:

$$\begin{aligned} {\varvec{\Phi }}({{\theta }})={\varvec{\Phi }_0}+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M({\sigma })}x({\theta })+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}x(s)ds. \end{aligned}$$

Existence theory

In this part, qualitative theory of problem (1) will be discussed.

Lemma 3.1

Let \(\textbf{f}({\theta },{\varvec{\Phi }(\theta )}),\) and \(\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\) be continuous functions on \(\Omega\), then the solution of fractal-fractional differential equation described by

$$\begin{aligned} \left\{ \begin{aligned}&^{FAB}{_0D}_\theta ^{\sigma ,\xi }[{\varvec{\Phi }(\theta )}-\textbf{f}({\theta },{\varvec{\Phi }(\theta )})]= \textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )}), ~ {\theta }\in I=[0, T], \\ {}&{\varvec{\Phi }(0)}= \varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta )) \end{aligned}\right. \end{aligned}$$
(3)

is given by

$$\begin{aligned} \varvec{\Phi }(\theta )=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}\right. \end{aligned}$$
(4)

Proof

Using Lemma 2.7, we get the solution as follows:

$$\begin{aligned} \varvec{\Phi }(\theta )=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\ {}&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}\right. \end{aligned}$$
(5)

\(\square\)

We define the operator as follows: \(\textbf{Z}:\Omega \rightarrow \Omega\) by:

$$\begin{aligned} \textbf{Z} \varvec{\Phi }(\theta )=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}\right. \end{aligned}$$
(6)

Divide the above operator Eq. (6) into two sub operators as follow:

$$\begin{aligned} \mathbf {Z_1} \varvec{\Phi }(\theta )=\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )}) \end{aligned}$$
(7)

and

$$\begin{aligned} \mathbf {Z_2} \varvec{\Phi }(\theta )=\frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}$$
(8)

From Eq.(7) and Eq.(8), \(\textbf{Z}(\varvec{\Phi })\) can be written as follows:

$$\begin{aligned} \textbf{Z}(\varvec{\Phi })=\mathbf {Z_1} (\varvec{\Phi })+\mathbf {Z_2} (\varvec{\Phi }). \end{aligned}$$
(9)

For additional investigation, the ensuing presumptions are required.

\(\mathbf {A_1}\):

There is a constant \(K_1>0,\) such that

$$\begin{aligned} |\textbf{h}(\theta ,\varvec{\Phi }(\theta ))-\textbf{h}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\le K_1|\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|. \end{aligned}$$
\(\mathbf {A_2}\):

One has a constant \(K_2>0,\) such that

$$\begin{aligned} |\textbf{f}(\theta ,\varvec{\Phi }(\theta ))-\textbf{f}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\le K_2|\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|. \end{aligned}$$
\(\mathbf {A_3}\):

One has a constant \(K_3>0\), such that

$$\begin{aligned} |\textbf{g}(\theta ,\varvec{\Phi }(\theta ))-\textbf{g}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\le K_3|\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|. \end{aligned}$$
\(\mathbf {A_4}\):

Corresponding to function \(\textbf{g}(\theta ,\varvec{\Phi }(\theta )),\ \exists \ K_4>0,\) and \(K_5>0\), which yield

$$\begin{aligned} \Vert \textbf{g}\Vert \le K_4\Vert \varvec{\Phi }\Vert + K_5. \end{aligned}$$

Theorem 3.2

Our suggested problem Eq.(1) has one solution at most under the assumptions \(\mathbf {A_1}- \mathbf {A_4}\), provided that \(\textbf{K}=\left( K_1 +K_2 +\frac{K_3 \xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}\right) <1\).

Proof

Here it is needed to prove that \(\mathbf {Z_1}\) is a contraction and the operator \(\mathbf {Z_2}\) is equi-continuous in order to verify the aforementioned claim. \(\textbf{D}=\{\textbf{U}\in {B}:\ \Vert \varvec{\Phi }\Vert \le \sigma \}\) is a bounded, closed, and convex subset of B for this define. It’s obvious that \(\mathbf {Z_1}\) is continuous. Assuming that \(\varvec{\Phi },\varvec{\bar{\Phi }}\in \textbf{D}\), taking

$$\begin{aligned} \Vert \mathbf {Z_1} (\varvec{\Phi })-\mathbf {Z_1} (\varvec{\bar{\Phi }})\Vert= & {} \sup _{\theta \in [0,T]}\bigg \{\bigg |\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\{} & {} -\left( \varvec{\bar{\Phi }}_0 + \textbf{h}(\theta ,\varvec{\bar{\Phi }}(\theta ))+\textbf{f}({\theta },{\varvec{\bar{\Phi }}(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\bar{\Phi }}(\lambda \theta )})\right) \bigg |\bigg \}, \\\le & {} \sup _{\theta \in [0,T]}\bigg \{|\textbf{h}(\theta ,\varvec{\Phi }(\theta ))-\textbf{h}(\theta ,\varvec{\bar{\Phi }}(\theta ))|+|\textbf{f}(\theta ,\varvec{\Phi }(\theta ))-\textbf{h}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\\{} & {} +\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )} |\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})-\textbf{g}({\theta },{\varvec{\bar{\Phi }}(\lambda \theta )})|\bigg \},\\\le & {} \sup _{\theta \in [0,T]}\bigg \{K_1 |\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|+K_2 |\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|\\{} & {} +\frac{K_3 \xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )} |\varvec{\Phi }(\lambda \theta )-\varvec{\bar{\Phi }}(\lambda \theta )|\bigg \},\\\le & {} \bigg [K_1 +K_2 +\frac{K_3 \xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}\bigg ]\Vert \varvec{\Phi }-\varvec{\bar{\Phi }}\Vert ,\\= & {} \textbf{K}\Vert \varvec{\Phi }-\varvec{\bar{\Phi }}\Vert . \end{aligned}$$

As a result, \(\mathbf {Z_1}\) is contracted. Taking \(\varvec{\Phi }\in \textbf{D}\), we can now demonstrate that \(\mathbf {Z_2}\) is equi-continuous.

$$\begin{aligned} \Vert \mathbf {Z_2}(\varvec{\Phi })\Vert= & {} \sup _{\theta \in [0,T]}\bigg \{\left| \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })} \int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds\right| \bigg \}, \\\le & {} \frac{{\sigma \xi }\Vert \textbf{g}\Vert }{M({\sigma })}\sup _{\theta \in [0,T]}\bigg \{\left| \int _{0}^{{\theta }}\frac{({\theta }-s)^{{\sigma }-1}}{\Gamma ({\sigma })}s^{\xi -1}ds\right| \bigg \}, \\\le & {} \left( \frac{\sigma \Gamma (\xi +1)T^{\sigma +\xi -1}}{M({\sigma })\Gamma (\sigma +\xi )}\right) \Vert \textbf{g}\Vert ,\\< & {} \infty . \end{aligned}$$

As a result, \(\mathbf {Z_2}(\varvec{\Phi })\) has a bound. \(\mathbf {Z_2}(\varvec{\Phi })\) is continuous, just as \(\textbf{f}\). Additionally, if \(\theta _1<\theta _2\in [0, \textbf{T}],\)

$$\begin{aligned} \Vert \mathbf {Z_2}\varvec{\Phi }(\theta _2)-\mathbf {Z_2}\varvec{\Phi }(\theta _1)\Vert= & {} \sup _{\theta _1,\theta _2\in [0, T]}\bigg \{\frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\bigg |\int _{0}^{{\theta _2}}({\theta _2}-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds\\{} & {} -\int _{0}^{{\theta _1}}({\theta _1}-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds\bigg |\bigg \}, \\\le & {} \frac{{\sigma \xi }\Vert \textbf{g}\Vert }{\Gamma ({\sigma })M({\sigma })}\bigg | \int _{0}^{{\theta _2}}({\theta _2}-s)^{{\sigma }-1}s^{\xi -1}ds\\{} & {} -\int _{0}^{{\theta _1}}({\theta _1}-s)^{{\sigma }-1}s^{\xi -1}ds\bigg | \\\le & {} \frac{{\sigma \Gamma (\xi +1)}\Vert \textbf{g}\Vert }{M({\sigma })\Gamma (\sigma +\xi )}\left( \theta _{2}^{\sigma +\xi -1}-\theta _{1}^{\sigma +\xi -1}\right) \rightarrow 0, \text { as } \theta _1 \rightarrow \theta _2. \end{aligned}$$

As a result, \(\mathbf {Z_2}\) is bounded and equi-continuous. Since the aforementioned operator is compact, it can be made reasonably compact and consequently entirely continuous by applying the Arzel\(\acute{a}\)-Ascoli theorem. Schauder’s fixed point theorem thus guarantees that \(\mathbf {Z_2}\) has a minimum of one fixed point. Therefore, there is at least one solution to problem Eq. (1). \(\square\)

Theorem 3.3

Assuming \(\mathbf {A_1}-\mathbf {A_3},\) there exists a unique solution to our problem if the condition

$$\textbf{K}^{\prime }=\left[ K_1 +K_2 +\bigg (\frac{K_3 T^{\xi -1}}{M(\sigma )}\bigg )\bigg (\xi (1-\sigma )+ \frac{{\sigma \Gamma (\xi +1)}T^{\sigma }}{\Gamma ({\sigma +\xi })}\bigg )\right] <1,$$

holds.

Proof

Take \(\varvec{\Phi },\bar{\varvec{\Phi }}\in \textbf{D},\) we have

$$\begin{aligned} \Vert \textbf{Z}(\varvec{\Phi })-\textbf{Z}(\bar{\varvec{\Phi }})\Vert\le & {} \Vert \textbf{Z}_{1}(\varvec{\Phi })-\textbf{Z}_{1}(\bar{\varvec{\Phi }})\Vert + \Vert \textbf{Z}_{2}(\varvec{\Phi })-\textbf{Z}_{2}(\bar{\varvec{\Phi }})\Vert , \\\le & {} \sup _{\theta \in [0,T]}\bigg \{\bigg |\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\{} & {} -\bigg (\bar{\varvec{\Phi }}_0 + \textbf{h}(\theta ,\bar{\varvec{\Phi }}(\theta ))+\textbf{f}({\theta },{\bar{\varvec{\Phi }}(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\bar{\varvec{\Phi }}(\lambda \theta )})\bigg )\bigg |\bigg \} \\{} & {} +\frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\sup _{\theta \in [0,T]}\bigg \{\bigg |\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds\\{} & {} -\bigg (\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\bar{\varvec{\Phi }}(\lambda s)})ds\bigg )\bigg |\bigg \}, \\\le & {} \sup _{\theta \in [0,T]}\bigg [K_{1}|\varvec{\Phi }(\theta )-\bar{\varvec{\Phi }}(\theta )|\Vert +K_{2}|\varvec{\Phi }(\theta )-\bar{\varvec{\Phi }}(\theta )|\\{} & {} +\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}K_{3}|\varvec{\Phi }(\theta )-\bar{\varvec{\Phi }}(\lambda \theta )|\bigg ]\\{} & {} +\frac{{\sigma \xi }K_{3}}{\Gamma ({\sigma })M({\sigma })}\bigg [\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}|\varvec{\Phi }(\theta )-\bar{\varvec{\Phi }}(\lambda \theta )|ds\bigg ],\\\le & {} \left( K_1 +K_2 +\frac{K_3\xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}\right) \Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert \\{} & {} +\frac{{\sigma \Gamma (\xi +1)}K_{3}T^{\sigma +\xi -1}}{M({\sigma })\Gamma ({\sigma +\xi })}\Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert ,\\= & {} \textbf{K}^{\prime }\Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert , \end{aligned}$$

where

$$\textbf{K}^{\prime }=\left[ K_1 +K_2 +\bigg (\frac{K_3 T^{\xi -1}}{M(\sigma )}\bigg )\bigg (\xi (1-\sigma )+ \frac{{\sigma \Gamma (\xi +1)}T^{\sigma }}{\Gamma ({\sigma +\xi })}\bigg )\right] .$$

So, our problem under consideration has a unique solution according to Banach’s contraction principle. \(\square\)

UH stability

In this section, we will establish the conditions required for U-H type stability. The numerical solution of a problem depends on this kind of stability. What conditions allow the approximation of a problem to be almost equal to the exact answer is the fundamental question about an approximation. In the 1940s, Ulam offered a thorough analysis of this important topic (see47). The stability that was previously discussed was extended and generalized by Hyer and Rassias to become generalized U-H and U-H-Rassias stability (see48,49).

Examine the matching perturbation issue of Eq. (1) as

$$\begin{aligned} \left\{ \begin{aligned}&^{FAB}{_0D}_\theta ^{\sigma ,\xi }[{\varvec{\Phi }(\theta )}-\textbf{f}({\theta },{\varvec{\Phi }(\theta )})]= \textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})+\textbf{h}(\theta ), ~ {\theta }\in I=[0, T], \\ {}&{\varvec{\Phi }(0)}= \varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta )). \end{aligned}\right. \end{aligned}$$
(10)

Here, \(\textbf{h}(\theta )\in C([0,T],R),\ni ,\) for \(\epsilon >0\), and \(|\textbf{h}(\theta )|\le \epsilon\). Corresponding to any solution \(\varvec{\Phi }\) of problem Eq. (10), one has

$$\begin{aligned} \textbf{Z}(\varvec{\Phi }(\theta ))=\varvec{\Phi }(\theta )+\textbf{h}(\theta ). \end{aligned}$$
(11)

The solution of Eq.(10) is given as follows:

$$\begin{aligned} \varvec{\Phi }(\theta )=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}[\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})+\textbf{h}(\theta )]\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}[\textbf{g}({s},{\varvec{\Phi }(\lambda s)})+\textbf{h}(s)]ds. \end{aligned}\right. \end{aligned}$$
(12)

Using Eq. (6), and Eq. (12), we have

$$\begin{aligned} |\textbf{Z}(\varvec{\Phi }(\theta ))-\varvec{\Phi }(\theta )|\le & {} \left[ \frac{\xi T^{\xi -1}}{M(\sigma )}\left( (1-\sigma )+\frac{\sigma \Gamma (\xi +1)}{\Gamma (\sigma +\xi )}T^{\sigma }\right) \right] \epsilon , \\\le & {} \Pi \epsilon , \end{aligned}$$

where

$$\Pi =\left[ \frac{\xi T^{\xi -1}}{M(\sigma )}\left( (1-\sigma )+\frac{\sigma \Gamma (\xi +1)}{\Gamma (\sigma +\xi )}T^{\sigma }\right) \right] .$$

Theorem 4.1

System Eq. (1) has a U-H and generalized U-H stable solution if

$$\begin{aligned} \textbf{K}^{\prime }=\left[ K_1 +K_2 +\bigg (\frac{K_3 T^{\xi -1}}{M(\sigma )}\bigg )\bigg (\xi (1-\sigma )+ \frac{{\sigma \Gamma (\xi +1)}T^{\sigma }}{\Gamma ({\sigma +\xi })}\bigg )\right] <1 \end{aligned}$$

holds.

Proof

Given that \(\bar{\varvec{\Phi }}\in \Omega\) represents a unique solution to problem Eq. (10), and \({\varvec{\Phi }}\in \Omega\) is any solution of problem Eq. (1), then

$$\begin{aligned} \begin{aligned} \Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert =\Vert \varvec{\Phi }-\textbf{Z}\bar{\varvec{\Phi }}\Vert&\le \Vert \varvec{\Phi }-\textbf{Z}\varvec{\Phi }\Vert +\Vert \textbf{Z}\varvec{\Phi }-\textbf{Z}\bar{\varvec{\Phi }}\Vert \\&\le \Pi \epsilon +\textbf{K}^{\prime }\Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert \\&\le \frac{\Pi \epsilon }{1-\textbf{K}^{\prime }}. \end{aligned} \end{aligned}$$
(13)

Thus the solution is U-H stable. Further, let there exists a nondecreasing function say \(\varsigma :(0, T)\rightarrow R\), such that \(\varsigma (\epsilon )=\epsilon ,\) then from Eq.(13), we can write

$$\begin{aligned} \Vert \varvec{\Phi }-\bar{\varvec{\Phi }}\Vert \le \frac{\Pi }{1-\textbf{K}^{\prime }} \varsigma (\epsilon ), \end{aligned}$$
(14)

thus, Eq.(14) indicates that the solution is generalized U-H stable. \(\square\)

Numerical approximation

We will use a numerical approach in this section of the manuscript to solve our considered problem. The Lagrange interpolation method will be used to build the numerical scheme, in accordance with the numerical method55. The integral form equivalent to our suggested problem, Eq. (1), is provided by

$$\begin{aligned} \varvec{\Phi }(\theta )=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta ,\varvec{\Phi }(\theta ))+\textbf{f}({\theta },{\varvec{\Phi }(\theta )})+\frac{\xi (1-{\sigma })\theta ^{\xi -1}}{M(\sigma )}\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta }}({\theta }-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}\right. \end{aligned}$$
(15)

Using \(\theta =\theta _{n+1}, \quad n=0,1,2,3,\cdots ,\) Eq. (15) implies

$$\begin{aligned} \varvec{\Phi }(\theta _{n+1})=\left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta _{n},\varvec{\Phi }(\theta _{n}))+\textbf{f}({\theta _{n}},{\varvec{\Phi }(\theta _{n})})+\frac{\xi (1-{\sigma })\theta _{n}^{\xi -1}}{M(\sigma )}\textbf{g}({\theta _{n}},{\varvec{\Phi }(\lambda \theta _{n})})\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\int _{0}^{{\theta _{n}}}({\theta _{n}}-s)^{{\sigma }-1}s^{\xi -1}\textbf{g}({s},{\varvec{\Phi }(\lambda s)})ds. \end{aligned}\right. \end{aligned}$$
(16)

Now approximating function \(\textbf{g}\) on \([\theta _{k-1},\theta _{k}]\) with \(\textbf{h}=\frac{\theta _{k}-\theta _{k-1}}{n}\), and using the following interpolations

$$\begin{aligned} P(\theta )=\frac{(s-\theta _{k})}{\textbf{h}}\textbf{g}({\theta _{k}},{\varvec{\Phi }(\lambda \theta _{k})})-\frac{(s-\theta _{k-1})}{\textbf{h}}\textbf{g}({\theta _{k-1}},{\varvec{\Phi }(\lambda \theta _{k-1})}), \end{aligned}$$
(17)

we obtain

$$\begin{aligned} \varvec{\Phi }(\theta _{n+1})= & {} \left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta _{n},\varvec{\Phi }(\theta _{n}))+\textbf{f}({\theta _{n}},{\varvec{\Phi }(\theta _{n})})+\frac{\xi (1-{\sigma })\theta _{n}^{\xi -1}}{M(\sigma )}\textbf{g}({\theta _{n}},{\varvec{\Phi }(\lambda \theta _{n})})\\&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\sum _{k=0}^{n} \bigg [\int _{\theta _{k-1}}^{{\theta _{k}}}({\theta _{k}}-s)^{{\sigma }-1}s^{\xi -1}\frac{(s-\theta _{k})}{\textbf{h}}\textbf{g}({\theta _{k}},{\varvec{\Phi }(\lambda \theta _{k})})ds\\&-\int _{\theta _{k-1}}^{{\theta _{k}}}({\theta _{k}}-s)^{{\sigma }-1}s^{\xi -1}\frac{(s-\theta _{k-1})}{\textbf{h}}\textbf{g}({\theta _{k-1}},{\varvec{\Phi }(\lambda \theta _{k-1})})ds\bigg ]. \end{aligned}\right. \\ \varvec{\Phi }(\theta _{n+1})= & {} \left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta _{n},\varvec{\Phi }(\theta _{n}))+\textbf{f}({\theta _{n}},{\varvec{\Phi }(\theta _{n})})+\frac{\xi (1-{\sigma })\theta _{n}^{\xi -1}}{M(\sigma )}\textbf{g}({\theta _{n}},{\varvec{\Phi }(\lambda \theta _{n})})\\ {}&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\sum _{k=0}^{n} \bigg [\textbf{g}({\theta _{k}},{\varvec{\Phi }(\lambda \theta _{k})})\int _{\theta _{k-1}}^{{\theta _{k}}}({\theta _{k}}-s)^{{\sigma }-1}s^{\xi -1}\frac{(s-\theta _{k})}{\textbf{h}}ds\\ {}&- \textbf{g}({\theta _{k-1}},{\varvec{\Phi }(\lambda \theta _{k-1})})\int _{\theta _{k-1}}^{{\theta _{k}}}({\theta _{k}}-s)^{{\sigma }-1}s^{\xi -1}\frac{(s-\theta _{k-1})}{\textbf{h}}ds\bigg ]. \end{aligned}\right. \\ \varvec{\Phi }(\theta _{n+1})= & {} \left\{ \begin{aligned}&\varvec{\Phi }_0 + \textbf{h}(\theta _{n},\varvec{\Phi }(\theta _{n}))+\textbf{f}({\theta _{n}},{\varvec{\Phi }(\theta _{n})})+\frac{\xi (1-{\sigma })\theta _{n}^{\xi -1}}{M(\sigma )}\textbf{g}({\theta _{n}},{\varvec{\Phi }(\lambda \theta _{n})})\\ {}&+ \frac{{\sigma \xi }}{\Gamma ({\sigma })M({\sigma })}\sum _{k=0}^{n} \bigg [\theta _{k}^{\xi -1}\textbf{g}({\theta _{k}},{\varvec{\Phi }(\lambda \theta _{k})})(n+1-k)^{\sigma +1}(n+2-k+\sigma )\\ {}&-(-k+n)^{\sigma }(-k+n+2(1+\sigma ))- \theta _{k-1}^{\xi -1}\textbf{g}({\theta _{k-1}},{\varvec{\Phi }(\lambda \theta _{k-1})})(n+1-k)^{\sigma +1}\\ {}&-(-k+n)^{\sigma }(n+1-k+\sigma )\bigg ]. \end{aligned}\right. \end{aligned}$$

Demonstration of our analysis

Example 6.1

To verify our analysis and to demonstrate our numerical scheme, we consider the following problem called Regulation of Haematopoiesis56. Haematopoiesis is the biological process by which precursor stem cells reappear and divide to form mature blood cells. The body’s process of producing blood cells is known as hemopoiesis regulation. Red and white blood cells are produced in the bone marrow and subsequently reach the bloodstream. The primary factor in the production of red blood cells are unique hormones found in the kidney called erythropoiesis. If the blood function fails to supply enough oxygen, renal tubular epithelial cells emit about 90% of the erythropoiesis hormones. Because of this problem, the blood’s oxygen content drops and a chemical is released, which causes the bone marrow to produce more blood cells. A message that blood components deliver to the marrow is what causes haematological illness.

Here, we will apply our chosen findings to a popular hematopoiesis biological model known as \(``{\varvec{Wazewska-Czyzewska}}\,{\varvec{and}}\, {\varvec{Lasota}}\,{\varvec{Model}}''\). The concerned model has numerous applications in real world problems, we refer few as57,58. Time delays have been incorporated into various biological models to illustrate resource regeneration durations, maturation intervals, feeding schedules, reaction times, and so forth. The model in question can be expressed numerically as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{du(\theta )}{dt}= -\mu \varvec{\Phi }(\theta )+p e^{-\gamma \varvec{\Phi }(\theta -\tau )}, \\ {}&{\varvec{\Phi }(0)}= \varvec{\Phi }_0. \end{aligned}\right. \end{aligned}$$
(18)

In this case, \(\varvec{\Phi }(\theta )\) denotes the quantity of red blood cells over time. Red blood cell death rates are indicated by \(\theta\) and \(\mu\), while red blood cell formation rates are indicated by p and \(\gamma\), which are positive constants. Furthermore, \(\tau\) denotes the amount of time needed to produce one red blood cell.

There are some modifications included in the model mentioned above. Here, we added a linear perturbation term to the left side of problem Eq. (18) and substituted the proportional delay term for the discrete delay term. The updated model is provided by

$$\begin{aligned} \left\{ \begin{aligned}&^{FAB}{_0D}_\theta ^{\sigma ,\xi }[{\varvec{\Phi }(\theta )}-\kappa e^{-\varvec{\Phi }(\theta )}]= -\mu \varvec{\Phi }(\lambda \theta )+p e^{-\gamma \varvec{\Phi }(\lambda \theta )}, \\ {}&{\varvec{\Phi }(0)}= 1+\delta \varvec{\Phi }(\theta ). \end{aligned}\right. \end{aligned}$$
(19)

From Eq.(19), we have

$$\textbf{f}(\theta ,\varvec{\Phi }(\theta ))=\kappa e^{-\varvec{\Phi }(\theta )},\ \textbf{g}(\theta ,\varvec{\Phi }(\lambda \theta ))= -\mu \varvec{\Phi }(\lambda \theta )+p e^{-\gamma \varvec{\Phi }(\lambda \theta )},$$

and

$$\textbf{h}(\theta ,\varvec{\Phi }(\theta ))=\delta \varvec{\Phi }(\theta ).$$

Now we deduce the assumptions \((\mathbf {A_1}-\mathbf {A_4})\) by performing the given process

$$|\textbf{h}(\theta ,\varvec{\Phi }(\theta ))-\textbf{h}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\le \delta |\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|,$$
$$|\textbf{f}(\theta ,\varvec{\Phi }(\theta ))-\textbf{f}(\theta ,\varvec{\bar{\Phi }}(\theta ))|\le \kappa |\varvec{\Phi }(\theta )-\varvec{\bar{\Phi }}(\theta )|.$$

Additionally, we compute

$$\begin{aligned} |\textbf{g}(\theta ,\varvec{\Phi }(\lambda \theta ))-\textbf{g}(\theta ,\varvec{\bar{\Phi }}(\lambda \theta ))|= & {} \left| -\mu \varvec{\Phi }(\lambda \theta )+p e^{-\gamma \varvec{\Phi }(\lambda \theta )}-\left( -\mu \varvec{\Phi }(\lambda \theta )+p e^{-\gamma \varvec{\Phi }(\lambda \theta )}\right) \right| \\\le & {} (\mu +p \gamma ) |\varvec{\Phi }(\lambda \theta )-\varvec{\bar{\Phi }}(\lambda \theta )| \end{aligned}$$

and

$$\begin{aligned} \Vert \textbf{g}\Vert =\sup _{\theta \in [0,1]}\left\{ |\textbf{g}(\theta ,\varvec{\Phi }(\lambda \theta ))|\right\}= & {} \sup _{\theta \in [0,1]}\left\{ \left| -\mu \varvec{\Phi }(\lambda \theta )+p e^{-\gamma \varvec{\Phi }(\lambda \theta )}\right| \right\} \\\le & {} (\mu +p \gamma )\Vert \varvec{\Phi }\Vert . \end{aligned}$$

Therefore, \(\textbf{f},\ \textbf{g},\) and \(\textbf{h}\) are functions.fulfil each of the related presumptions. From \((\mathbf {A_1}-\mathbf {A_4}),\) using constants \(K_1=\delta ,\ K_2=\kappa ,\ K_3=\gamma -p,\ K_4=\gamma -p\) and \(K_5=0\).

Moreover \(M(\sigma )=1-\sigma (1-\frac{1}{\Gamma (\sigma +1)}),\ \delta =0.0020,\ T=10.0,\ \kappa =0.10,\ \mu =0.050,\ p=0.0010,\) and \(\gamma =0.250\), we obtain

$$\begin{aligned} \textbf{K}= & {} \left( K_1+\frac{K_3 \xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}+ K_2\right) , \\= & {} \left( \delta +\kappa +\frac{(\mu +p \gamma )\xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}\right) ,\\< & {} 1,\ \text {for all}\ \sigma ,\ \xi \in (0, 1]. \end{aligned}$$

and

$$\begin{aligned} \textbf{K}^{\prime }= & {} \left[ K_1 +\bigg (\frac{K_3 T^{\xi -1}}{M(\sigma )}\bigg )\bigg (\xi (1-\sigma )+ \frac{{\sigma \Gamma (\xi +1)}T^{\sigma }}{\Gamma ({\sigma +\xi })}\bigg )+K_2\right] ,\\= & {} \left[ \delta +\kappa +\bigg (\frac{(\mu +p \gamma )T^{\xi -1}}{M(\sigma )}\bigg )\bigg (\xi (1-\sigma )+ \frac{{\sigma \Gamma (\xi +1)}T^{\sigma }}{\Gamma ({\sigma +\xi })}\bigg )\right] , \\< & {} 1,\ \text {for all}\ \sigma ,\ \xi \in (0, 1]. \end{aligned}$$

In Fig. 1, we display the geometrical behaviours of \(\textbf{K},\ \textbf{K}'\) as:

Figure 1
Figure 1
Full size image

(a). Fractional and fractals behaviour of \(\textbf{K}\). (b). Fractional and fractals behaviour of \(\textbf{K}'\).

As can be seen in Fig. 1, both \(\textbf{K}<1,\ \textbf{K}'<1\). Thus, Theorem 3.2, Theorem 3.3, and Theorem 4.1 hold in their entirety. Consequently, there is at least one solution for Eq. (19). The requirement for the solution’s uniqueness is also met. Additionally, for [0, 10], the solution is U-H and generalized U-H stable. To illustrate the dynamic behaviour, we here provide the approximate solutions of Example 6.1 for different fractals fractional orders in Figs. 2, 3, 4, 5, and 6, respectively.

Figure 2
Figure 2
Full size image

Utilizing first set of different fractional and fractals orders to present the solution graphically of Example 6.1.

Figure 3
Figure 3
Full size image

Utilizing second set of different fractional and fractals orders to present the solution graphically of Example 6.1.

Figure 4
Figure 4
Full size image

Utilizing third set of different fractional and fractals orders to present the solution graphically of Example 6.1.

Figure 5
Figure 5
Full size image

Utilizing fourth set of different fractional and fractals orders to present the solution graphically of Example 6.1.

Figure 6
Figure 6
Full size image

Utilizing fifth set of different fractional and fractals orders to present the solution graphically of Example 6.1.

In Figs. 2, 3, 4, 5, and 6, we have simulated the numerical results graphically for different fractals fractional order values. We see that in start the density of red-blood cells is increasing exponentially and then start to decrease. The decrease in the density of the said amount of red-blood cells is different. Because under different fractals fractional order the decay processes will be different. Usually the decay process is faster on smaller fractional order and greater fractals order and vice versa. In the same way, the growth phenomenon in dynamical system under fractals fractional order is also affected. The said processes is faster at larger fractional order values. Here, we use the fractional order values in (0, 0.25) and fractals values in (0.90, 1.0) and present the results graphically in Fig. 7.

Figure 7
Figure 7
Full size image

Utilizing another set of different fractional and fractals orders to present the solution graphically of Example 6.1.

Here, we use the fractional order values in (0, 0.1) and fractals values in (0.94, 1.0] and present the results graphically in Fig. 8.

Figure 8
Figure 8
Full size image

Utilizing lower values set of different fractional and fractals orders to present the solution graphically of Example 6.1.

From Figs. 7 and 8 we observe that fractional and fractals orders have significant impact on the dynamical behavior of the problem. At lower fractional order values the growth in population for some time is very fast like exponential growth and then becomes stable.

Conclusion

Here, we remark that as non-local operators of differentiation can incorporate increasingly complicated natural phenomena into mathematical equations. The said area has gotten much attention from number of researchers of nearly every field in the sciences, technology, and engineering. The FC has become interested very well for all researchers of science and technology. Since area devoted to fractals HFDEs has not well investigated using non singular fractals fractional differential operators. Therefore, it was needed to provide a sophisticated analysis and numerical results for young researchers to extend their knowledge in this direction. In this research work we have used ABC fractals fractional derivative. Here, we remark that the said operator has all the characteristics which have by other fractional differential operators. This study presents a theoretical and numerical investigation of a class of fractals HFDEs with ABC fractals fractional derivative. The issue under consideration was a hybrid problem involving linear perturbation. Using some fixed point analysis, sufficient requirements were inferred for the existence and uniqueness of the solution. Since stability theory is an important requirement for approximate solutions to nonlinear problems. Because with the help of stability theory we deduce the stable behaviour of solution and methodology we use. Different concepts in this regards were given in literature for stability analysis. U-H concept is one of the powerful procedure to be used to investigate stability results for different problems. Therefore, U-H stability requirements were developed to solve the aforementioned issue. To interpret the results numerically, a potent interpolation-based numerical technique was developed. Our previously mentioned research was applied to an intriguing example Lasota-Wazewska system in order to illustrate our findings. Every theoretical and numerical result was tested with success. We plan to utilize this research in the future for system of fractals HFDEs using more complex dynamical systems addressing real-world problems.