Analytical study of $\mathcal{ABC}$-fractional pantograph implicit differential equation with respect to another function

1.   Introduction

Fractional differential equations (FDEs) are a type of mathematical equation that involves non-integer orders of derivatives. These equations have become increasingly important in recent years, as they provide a powerful tool for modeling systems with long-range dependence and memory. The solutions to FDEs can exhibit unique behaviors, including power-law decay and non-locality, which distinguish them from solutions to traditional differential equations, for more detailed information, see these works ^{[1,2,3,4,5,6,7,8,9,10,11]}. Furthermore, $\mathbb{FPT}$ is a mathematical discipline that explores the existence and behavior of fixed points of the mapping equation. These points are those that remain invariant under the action of the mapping. The study of fixed points has many applications in diverse fields such as analysis, geometry, topology, physics, economics, and engineering. For instance, in physics, $\mathbb{FPT}$ can be used to assess the stability of dynamic systems. In economics, it can help identify equilibrium points of economic models. In general, $\mathbb{FPT}$ provides a powerful tool for finding solutions to problems involving nonlinear equations and systems, we refer the researchers to some works applied $\mathbb{FPT}$ in the literature studies ^{[12,13,14,15,16,17,18,19,20,21]}. In fact, pantograph problems are a type of differential equations that arises in the study of systems with a delayed feedback, particularly in mechanical systems. These equations are characterized by fractional derivatives and a time delay term, making them difficult to solve. Pantograph problems have a wide range of applications in fields such as physics, engineering, and biology ^{[22,23,24,25,26,27]}.

In fractional calculus field, Atangana and Baleanu ($\mathcal{AB}$) ^[28] introduced a new non-singular fractional operator via a function of Mittag-Leffler in the sense of Caputo and Riemann-Liouville. These two operators attracted the attention of many researchers to study various problems and applications ^{[29,30,31,32,33,34,35,36,37]}. Afterwards, Fernandez and Baleanu ^[38] generalized the $\mathcal{AB}$ operator to differintegration with respect to other functions and so called $\varrho$-$\mathcal{ABC}$ in the Caputo sense and $\varrho$-$\mathcal{ABR}$ in the Riemann-Liouville sense, where $\varrho$ is a positive and increasing function of its domains. Very recently, Abdeljawad et al. ^[39] extended these operators to higher-order and derived the Gronwall-type inequality in the framework of an $\varrho$-$\mathcal{AB}$ factional integral.

In 2020, Ali et al. ^[40] studied some essential conditions of qualitative results for a delay problem in the sense of $\mathcal{ABC}$ fractional derivatives, given by:

After that, in 2021 ^[41], Banach and Krasnoselskii used $\mathbb{FPT}s$ to establish the existence and uniqueness theorems for a $\mathcal{ABC}$-fractional implicit problem, given as follows:

Furthermore, the authors of ^[42] investigated an implicit $\mathcal{ABC}$-fractional differential system under integral conditions:

Motivated by the aforementioned research works ^[40,41,42], and as an application for the work in ^[39], in this article, we aim to study the existence, uniqueness and verity types of $\mathbb{UH}$ stability of the solution for a new class of $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit systems supplemented by the integral boundary conditions of the form:

where $0 < \delta < 1, \gamma_1, \gamma_2 \in\mathbb{R},$ $^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }$ denotes the $\varrho$-$\mathcal{ABC}$ fractional derivative of the arbitrary order $\vartheta\in(1, 2]$, and the functions $\mathfrak{g}:\mathrm{J}\times\mathbb{R}^3\rightarrow \mathbb{R}, \mathfrak{f}:\mathrm{J}\times\mathbb{R}\rightarrow \mathbb{R},$ are continuous. Moreover, let $\varrho:[\iota, \mathfrak z]\rightarrow \mathbb{R}^+$ be an increasing and positive function with $\varrho(u)\in \mathcal{C}^1((\iota, \mathfrak z), \mathbb{R}^+)$ and $\varrho ^{\prime }(\upsilon) \neq 0, \, \forall\, \upsilon\in\mathrm{J}.$

Here, we ensure that this research work displays a novelty by considering a new class of pantograph implicit problems in the framework of $\varrho$-$\mathcal{ABC}$ fractional derivatives. Furthermore, the contributions of this work are interesting and more generalized of a new and existing problems in the literature, for instance:

$\textbf{i)}$ If $\varrho(\upsilon) = \upsilon$, then the system (1.4) returns to a system in the sense of $\mathcal{ABC}$-fractional derivatives;

$\textbf{ii)}$ If $\varrho(\upsilon) = \upsilon$, and $\int_{\iota}^{\mathfrak z} \mathfrak{f}(\kappa, \varpi(\kappa)) d\kappa = 1$, then the system (1.4), reduces to the problem (1.1) when the implicit term finishes and reduces to problem (1.2) when the delay term finishes;

$\textbf{iii)}$ The system (1.4), returns to Eq (1.3) in the case where $\varrho(\upsilon) = \upsilon$ and delay term finishes;

$\textbf{iv)}$ Our system (1.4) covers numerous problems by taking any positive and increasing the function $\varrho$ on the interval ${\rm{J}}$.

The paper's remaining sections are organized as follows. In Section 2, some fundamental preliminaries are presented. In Section 3, a corresponding fractional integral equation of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is derived. In Section 4, the existence and uniqueness results are proven by applying the Schaefer and Banach $\mathbb{FPT}s$. In Section 5, $\mathbb{UH}$ and $\mathbb{UHR}$ stability are discussed. In Section 6, some examples are provided to illustrate the main findings.

2.   Preliminaries

In this section, we recall several major preliminaries and definitions. Let $\Omega: = \mathcal{C}(\mathrm{J} = [\iota, \mathfrak z], \mathbb{R})$ denote the Banach space of all the continuous functions $\varpi$ endowed with the supremum norm $\left\| \varpi \right\| = {\sup }_{\upsilon \in [\iota, \mathfrak z]} \left| {\varpi(\upsilon)} \right|.$

Definition 2.1. (^[43]) The $\varrho$-Riemann-Liouville fractional integral of order $\vartheta > 0$ for an integrable function $\varpi:[\iota, \mathfrak z]\rightarrow \mathbb{R}$ is given by the following:

where $\Gamma (\vartheta) = \int_{0}^{+\infty }e^{-\upsilon}\; \upsilon^{\vartheta -1}\, \mathrm{d}\upsilon, \, \vartheta > 0$.

Lemma 2.2. (^[43]) Let $\vartheta, \mu > 0$ and $\varpi:[\iota, \mathfrak z] \rightarrow \mathbb{R}$. Then,

(i) $^{\mathcal{RL}}\mathfrak{I}_{\iota}^{\vartheta, \varrho}[\varrho(\upsilon)-\varrho(\iota)]^{\mu-1} = \frac{\Gamma(\mu)}{\Gamma(\vartheta+\mu)}[\varrho(\upsilon)-\varrho(\iota)]^{\vartheta+\mu-1}$;

(ii) $^{\mathcal{RL}}\mathfrak{I}_{\iota}^{\vartheta, \varrho} \; ^{\mathcal{RL}}\mathfrak{I}_{\iota}^{\mu, \varrho} \varpi(\upsilon) = ^{\mathcal{RL}}\mathfrak{I}_{\iota}^{\vartheta+\mu, \varrho}\varpi(\upsilon)$;

(iii) $\left(\left(\frac{1}{\varrho(\upsilon)}\frac{d}{d\upsilon}\right)^n \; ^{\mathcal{RL}}\mathfrak{I}_\iota^{n, \varrho} \varpi\right) (\upsilon) = \varpi (\upsilon), n\in\mathbb{N}.$

Definition 2.3. (^[38,44]) The $\varrho$-$\mathcal{ABC}$ fractional derivative of order $\vartheta \in (0, 1]\ $ of a function $\varpi \in \mathcal{H}^{1}(\iota, \mathfrak z)$ is given by the following:

where $\varpi_\varrho^{\prime }(\upsilon) = \dfrac{\varpi'(\upsilon)}{\varrho^{\prime }(\upsilon)},$ $\Lambda(\vartheta)$ is the normalization function with $\Lambda(0) = \Lambda(1) = 1$, and $\mathbb{E}_{\vartheta }$ is called the Mittag-Leffler function defined by the following:

Definition 2.4. (^[44]) The $\varrho$-$\mathcal{AB}$ fractional integral of order $\vartheta \in (0, 1]\ $ of an integrable function $\varpi$ is given by the following:

Definition 2.5. (^[39]) The $\varrho$-$\mathcal{ABC}$ fractional derivative of order $\vartheta\in(n, n+1], \nu = \vartheta-n, n = 0, 1, 2, \dots$, of a function $\varpi^{(n+1)}\in \mathcal{H}^{1}(\iota, \mathfrak z)$ is given by the following:

where $\varpi_\varrho^{(n)}(\upsilon) = \left(\dfrac{1}{\varrho'(\upsilon)} \dfrac{d}{d\upsilon}\right)^n \varpi(\upsilon)$ and $\varpi_\varrho^{(0)}(\upsilon) = \varpi(\upsilon)$. If $\vartheta = k\in\mathbb{N}$, then $({}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\varpi)(\upsilon) = \varpi_\varrho^{(k)}(\upsilon).$

Definition 2.6. (^[39]) The $\varrho$-$\mathcal{AB}$ fractional integral of order $\vartheta\in(n, n+1], \nu = \vartheta-n, n = 0, 1, 2, \dots$, of a function $\varpi$ is given by the following:

where ${}^{\mathcal{RL}}\mathfrak{I}_\iota^{n, \varrho}$ is defined as:

Lemma 2.7. (^[39]) Let $\varpi\in\mathcal{C}(\mathrm{J}, \mathbb{R})$ and $\varrho\in\mathcal{C}^n(\mathrm{J}, \mathbb{R})$. For $\vartheta \in (n, n+1], \nu = \vartheta-n, n = 0, 1, 2, \dots,$ the following relation holds:

Lemma 2.8. (^[39]) Let $\varrho\in\mathcal{C}^n(\mathrm{J}, \mathbb{R}^+)$ with $\varrho'(\upsilon)\neq0.$ For $\vartheta \in (n, n+1], \nu = \vartheta-n, n = 0, 1, 2, \dots, \beta\geq{n+1}.$ Then, the following equations hold:

(i) $^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }[\varrho(\upsilon)-\varrho(\mathrm{\iota})]^{\beta} = \dfrac{\Lambda(\vartheta-n)}{(n+1-\vartheta) } \sum_{i = 0}^{\infty} \left(\dfrac{-(\vartheta-n) }{n+1-\vartheta}\right)^i \dfrac{\Gamma(\beta+1)[\varrho(\upsilon)-\varrho(\mathrm{\iota})]^{i(\vartheta-n)+\beta-n}}{\Gamma(i(\vartheta-n)+\beta-n+1)};$

(ii) $^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }[\varrho(\upsilon)-\varrho(\mathrm{\iota})]^{\zeta} = 0, \; \; \zeta = 0, 1, \dots, n;$

(iii) $(^{\mathcal{AB}}\mathfrak{I}_\iota^{\vartheta, \varrho }\; 1) (\upsilon) = \dfrac{(n+1-\vartheta)[\varrho(\upsilon)-\varrho(\mathrm{\iota})]^{n}}{\Lambda(\vartheta -n)\Gamma(n+1)}+ \dfrac{(\vartheta-n)[\varrho(\upsilon)-\varrho(\mathrm{\iota})]^{\vartheta}}{\Lambda(\vartheta-n)\Gamma(\vartheta+1)}.$

Theorem 2.9. (^[45], Banach's $\mathbb {FPT}$) Suppose that $K$ be a Banach space. If $\aleph: K \rightarrow K$ is a contraction, then $\aleph$ has one and only one fixed point in $K$.

Theorem 2.10. (^[46], Schaefer's $\mathbb {FPT}$) Let $K$ be a Banach space and $\aleph :K \to K$ be a completely continuous mapping. If ${\rm{U}} = \{ \zeta\in {\rm{K}}:\varpi = \eta \aleph \varpi$, for some $\eta \in (0, 1)\}$ is bounded set. Then, $\aleph$ possesses fixed points.

3.   The corresponding fractional integral equation

We start this section by deriving the corresponding fractional integral equation of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4). For this, we need to introduce the next lemma.

Lemma 3.1. Consider $\varpi\in \Omega$. Then, the corresponding fractional integral equation of the following $\varrho$-$\mathcal{ABC}$ fractional system,

is given by

Proof. We start our proof by applying ${}^{\mathcal{AB}}\mathfrak{I}_\iota^{\vartheta, \varrho }$ on both sides of Eq (3.1). According to the Definition 2.6 with using Lemma 2.7, we get

Then, by the boundary condition $\varpi(\iota) = \gamma_1$, we deduce that $e_0 = \gamma_1$, which implies that

Again, due to the boundary condition $\varpi(\mathfrak z) = \gamma_2\int_{\iota}^{\mathfrak z} \mathfrak{f}(\kappa, \varpi(\kappa)) d\kappa,$ we conclude that

By substituting the values of $e_0$ and $e_1$ into Eq (3.2), we obtain

Hence, the proof is finished. □

As a consequence of the above lemma, we conclude the following interesting lemma.

Lemma 3.2. Consider $\varpi\in \Omega$. Then, the corresponding fractional integral equation of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4), is given by

4.   Existence and uniqueness

We start this section by investigating the uniqueness criteria of the solution for the system $(1.4)$ via Banach $\mathbb {FPT}$s. Thus, the next assumption is required:

${\rm{H}}_1)$ There are constants $\ell_1, \ell_2, \ell_3 > 0$ such that for all $\upsilon \in [\iota, \mathfrak z]$ and ${\varpi}_i, \hat{\varpi}_i\in \Omega, (i = 1, 2, 3)$ satisfy

and

Theorem 4.1. Let ${\rm{H}}_1$ hold, and if

then there exists a exactly one solution for the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) on $\mathrm{J}$.

Proof. At the beginning, we define the operator $\aleph:\Omega \to \Omega$ as follows:

In order to investigate the uniqueness solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4), we will use the Banach $\mathbb{FPT}$. Regarding this, let $\varpi, \hat{\varpi}\in\Omega$, $\upsilon\in\mathrm{J}$, and by using ${\rm{H}}_1$, we have

In fact, we have ${}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\varpi(\upsilon) = \mathfrak{g}\left(\upsilon, \varpi(\upsilon), \varpi(\delta\upsilon), {}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\varpi(\upsilon)\right),$ thus

Therefore,

In view of Eqs (4.3), (4.4), and Lemma 2.8 part $(iii)$, we obtain

Hence, $\|\aleph\varpi-\aleph\hat\varpi\|\le\beta_1\left\| \varpi-\hat\varpi\right\|$, and by condition (4.1), the operator $\aleph$ is a contraction. Based on the Banach $\mathbb{FPT}$ 2.9, the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) possesses exactly one solution on $\mathrm{J}.$ □

Next, we discuss the existence result of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) via Schaefer's $\mathbb{FPT}$. Thus, we need to present the following hypothesis:

${\rm{H}}_2)$ There are constants $\xi_1, \xi_2, \xi_3, \xi_4, \xi_5 > 0$, such that for all $\upsilon \in [\iota, \mathfrak z]$ and ${\varpi}_i\in \Omega, (i = 1, 2, 3)$ satisfy

and

Theorem 4.2. Let ${\rm{H}}_2$ hold. Then, there exists a solution for the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) provided that

Proof. In order to investigate the conditions of Schaefer's $\mathbb{FPT}$, we define the operator $\aleph:\Omega \to \Omega$ as given in (4.2). Additionally, we set the bounded closed and convex ball $\mho_\varsigma: = \{\varpi\in\Omega: \|\varpi\|\le\varsigma\},$ with radius $\varsigma\geq \dfrac{\beta_2}{1-\beta_3}$ and $\beta_3 < 1$, such that

and $\beta_3$ given in Eq (4.6). Our proof will be divided into two of the procedures.

First: We show that the operator $\aleph$ is completely continuous. For this, let the sequence $\{\varpi_n\}_{n\in\mathbb{N}}$ is convergence to $\varpi$ in the ball $\mho_\varsigma$ as $n\to\infty.$ Therefore, based on the continuity of the functions $\mathfrak{g}$, $\mathfrak{f}$, and by using the dominated convergence theorem in the sense of Lebesgue, we find that

Thus, the operator $\aleph$ is continuous.

Next, we show that $\aleph\mho_\varsigma\subset\mho_\varsigma$. By using ${\rm{H}}_2$, for $\varpi\in\mho_\varsigma$ and $\upsilon\in\mathrm{J},$ we obtain

By the relation 4.5, we find that

Thus,

Now, by substituting Eq (4.8) into Eq (4.7) and by taking the supremum to both sides, and based on Lemma 2.8, part ($iii$), with $\frac{\left(\varrho(\upsilon)-\varrho(\mathrm{\iota})\right)}{\left(\varrho({\mathfrak z})-\varrho(\mathrm{\iota})\right)} < 1$, we obtain

Hence, $\aleph\mho_\varsigma\subset\mho_\varsigma$, concluding that $\aleph: \mho_\varsigma\to \mho_\varsigma.$

In the following, we prove that the operator $\aleph:\mho_\varsigma\to \mho_\varsigma$ is equicontinuos on $\mho_\varsigma$. For each $\varpi\in\mho_\varsigma$ and $\iota\le\upsilon_1\le\upsilon_2\le\mathfrak{z}$, we have the following:

Clearly, $\left| {(\aleph \varpi)({\upsilon_2}) - (\aleph \varpi)({\upsilon_1})} \right|\to 0$ as ${\upsilon_1}\to{\upsilon_2}$ and is independent of $\varpi\in {\mho_\varsigma }$. Therefore, $\aleph$ is equicontinuos on $\mho_\varsigma$. Thus, according to the Arzela Ascoli theorem, the mapping $\aleph :\Omega \to \Omega$ is completely continuous.

Second: We investigate that the set ${\rm{U}} = \{ \varpi\in \Omega: \varpi = \eta \aleph \varpi, 0 < \eta < 1\}$ is bounded. Then, for $\varpi\in {\rm{U}},$ $\upsilon \in [\iota, \mathfrak z]$, and by direct computation as in (4.9), we obtain

Hence, $\left\| \varpi \right\| = \dfrac{\beta_2}{1-\beta_3},$ then ${\rm{U}}$ is bounded set. Therefore, based on Theorem 2.10, the mapping $\aleph$ possesses at least fixed point in $\Omega$, which is a solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4). □

5.   Stability

The $\mathbb{UH}$ stability are a type of stability for functional equations. This stability was first introduced by Ulam in 1940 ^[48]; Hyers ^[49] gave a partial affirmative answer to Ulam's question in 1941, after which, Rassias extended Hyers's theorem in 1978 ^[50]. The $\mathbb{UH}$ and $\mathbb{UHR}$ stabilities have been applied to a variety of problems in mathematics, physics, and economics and are also useful for studying the behavior of solutions to functional equations when the equations are only approximately satisfied.

In this section, we are going to discuss the $\mathbb{UH}$ and $\mathbb{UHR}$ stabilities by using nonlinear analysis themes. Thus, we present the following concepts that are related to $\mathbb{UH}$ stability:

Definition 5.1. (^[51]) The $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit problem (1.4) is defined as $\mathbb{UH}$ stable if there is a constant $\chi_{\mathfrak g} > 0$ such that for each $\varepsilon > 0$, when $\hat\varpi\in \Omega$ is any solution of the inequality

then there is one solution $\varpi\in \Omega$ of the Eq (1.4) satisfied

Definition 5.2. (^[51]) The $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit problem (1.4) is defined as generalized $\mathbb{UH}$ stable if there is a function $\alpha\in C(\mathbb{R}^+, \mathbb{R}^+), \alpha(0) = 0$, such that for when $\hat\varpi\in \Omega$ is any solution of the identity (5.1), then there is one solution $\varpi\in \Omega$, of the Eq (1.4) satisfied by

Remark 5.3. For $\varepsilon > 0,$ a function $\hat\varpi\in \Omega$ is a solution of the inequality (5.1), for all $\upsilon \in [\iota, \mathfrak z]$ iff there is a function $\rho \in \Omega$ such that

i) $\left| {\rho(\upsilon)} \right| \le \varepsilon, \quad \upsilon \in [\iota, \mathfrak z];$

ii) ${}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\hat\varpi(\upsilon) = \mathfrak{g}\left(\upsilon, \hat\varpi(\upsilon), \hat\varpi(\delta\upsilon), {}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\hat\varpi(\upsilon)\right)+ \rho(\upsilon).$

Theorem 5.4. Suppose that ${{\rm H}_1}$ is satisfied and $\beta_1 < 1.$ Then, the unique solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph system (1.4) is $\mathbb{UH}$ stable and generalized $\mathbb{UH}$ stable.

Proof. Consider $\hat\varpi\in \Omega$ which satisfies the inequality (5.1), hence by using the Remark 5.3, we obtain

In view of Eq (3.3), we have

which gives

Now, for $\varpi, \hat\varpi\in \Omega$, by using Eqs (3.3), (5.5) and ${\rm{H}}_1$, we obtain

which further implies

where

Thus, yields that

Therefore, the solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is $\mathbb{UH}$ stable. Moreover, there is an increasing function $\alpha:(0, \infty)\to(0, \infty),$ where $\alpha(\varepsilon) = \varepsilon$ with $\alpha(0) = 0$; thus from (5.6), we obtain

Hence, the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is generalized $\mathbb{UH}$ stable. □

Next, before discussing the $\mathbb{UHR}$ stability, we present the following concepts:

Definition 5.5. (^[52]) The $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit problem (1.4) is called $\mathbb{UHR}$ stable with respect to $\theta \in \mathcal{C}({\rm{J}}, \mathbb{R}),$ if there is a constant $\chi_{\mathfrak{g}, \theta } > 0,$ $\forall\; \varepsilon > 0$, for all $\hat{\varpi}\in \Omega$ which satisfies

then there is one solution $\varpi\in \Omega$ of the Eq (1.4) which satisfies

Definition 5.6. (^[52]) The $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit problem (1.4) is called generalized $\mathbb{UHR}$ stable with respect to $\theta \in \mathcal{C}({\rm{J}}, \mathbb{R}),$ if there is a constant $\chi_{\mathfrak{g}, \theta } > 0,$ for all $\hat{\varpi}\in \Omega$ satisfy (5.7), then there is one solution $\varpi\in \Omega$ of the Eq (1.4) which satisfies

Remark 5.7. For $\varepsilon > 0,$ a function $\hat\varpi\in \Omega$ is a solution of the inequality (5.7), for all $\upsilon \in [\iota, \mathfrak z]$ iff there is a function $\zeta \in \Omega$ such that

i) $\left| {\zeta(\upsilon)} \right| \le \varepsilon\theta(\upsilon), \quad \upsilon \in [\iota, \mathfrak z];$

ii) ${}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\hat\varpi(\upsilon) = \mathfrak{g}\left(\upsilon, \hat\varpi(\upsilon), \hat\varpi(\delta\upsilon), {}^{\mathcal{ABC}}\mathfrak{D}_\iota^{\vartheta, \varrho }\hat\varpi(\upsilon)\right)+ \zeta(\upsilon).$

Remark 5.8. There exists a real number $\lambda_\theta > 0$ and non-decreasing function $\theta(\upsilon)\in \Omega$ such that ${}^{\mathcal{AB}}\mathfrak{I}_\iota^{\vartheta, \varrho} \theta(\upsilon)\leq\lambda_\theta \theta(\upsilon), \forall \upsilon \in [\iota, \mathfrak z].$

Theorem 5.9. If ${{\rm{H}}}_{1}$ holds, subject to $\beta_1 < 1,$ then the unique solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is $\mathbb{UHR}$ stable and is consequently generalized $\mathbb{UHR}$ stable.

Proof. Suppose that $\hat\varpi\in \Omega$ verifies the inequality (5.7). Thus, due to the Remark 5.7, we find

Based on the Eq (3.3), we have

In view of the Remarks 5.7 and 5.8, it is implied that

Now, for $\varpi, \hat\varpi\in \Omega$, by using Eqs (3.3), (5.11) and ${\rm{H}}_1$, we obtain

which yields

Hence, we have

Then, the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is $\mathbb{UHR}$ stable. Additionally, if $\varepsilon = 1$, the solution of the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (1.4) is generalized $\mathbb{UHR}$ stable. □

6.   Examples

Throughout this section, we present two examples for testing the effectiveness of our findings.

Example 6.1. Consider the following $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system:

Here, $\vartheta = 1.5, \iota = 1, \mathfrak z = e, \gamma_1 = 1, \gamma_2 = 0.5, \varrho(\upsilon) = \log(\upsilon), \mathfrak{f}(\upsilon, \varpi) = \dfrac{\varpi(\upsilon)}{\log(\sqrt{\upsilon})+\varpi(\upsilon)},$ and

Hence,

and

Thus, we have $\ell_1 = \frac{1}{9}, \ell_2 = \frac{1}{4}, \text{and} \; \ell_3 = \frac{1}{2}.$ Moreover, for $\Lambda(\vartheta-1) = 1$, by the Mathematica software, we can calculate the following:

Therefore, based on the Theorem 4.1, the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (6.1) possesses exactly one solution on $[1, e].$ Furthermore, in view of Theorem 5.4, the solution is $\mathbb{UH}$ stable with $\chi_{\mathfrak g}\approx 20.263474$, and is consequently generalized $\mathbb{UH}$ stable. Analogously, we can easily draw a conclusion that $\mathbb{UHR}$ and generalized $\mathbb{UHR}$ stability conditions by obtaining a non-decreasing function $\theta(\upsilon) = \upsilon, \forall \upsilon \in[1, e]$.

Example 6.2. Consider the following $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system:

Here, $\vartheta = 1.8, \iota = 0, \mathfrak z = 1, \gamma_1 = 0, \gamma_2 = 0.25, \varrho(\upsilon) = \upsilon^2, \mathfrak{f}(\upsilon, \varpi) = \dfrac{\upsilon}{3}+\dfrac{\cos(\varpi(\upsilon))}{1+\varpi(\upsilon)},$ and

Hence,

and

Then, we obtain $\xi_1 = \frac{1}{10}, \xi_2 = \frac{1}{12}, \xi_3 = \frac{1}{2}, \xi_4 = \frac{1}{3}, \text{and} \; \xi_5 = 1.$ Furthermore, by putting $\Lambda(\vartheta-1) = 1$, and using the Mathematica software, we find the following:

Hence, in view of the Theorem 4.2, the $\varrho$-$\mathcal{ABC}$ fractional pantograph implicit system (6.2) possesses a solution on $[0, 1].$

7.   Conclusions

The current study was devoted to investigate the existence, uniqueness and several types of $\mathbb{UH}$ stability of the solution for a new class of $\varrho$-$\mathcal{ABC}$ factional pantograph implicit systems under integral boundary conditions (1.4). The Scheafer and Banach $\mathbb{FPT}s$ were employed to establish sufficient conditions of the existence and uniqueness results of such system (1.4). The $\mathbb{UH}$, generalized $\mathbb{UH}$, $\mathbb{UHR}$ and generalized $\mathbb{UHR}$ stabilities were discussed by non-linear analysis themes and fractional calculus. Finally, two mathematical examples were enhanced to examine the validity of the main outcomes. In fact, with respect to another function which was used in this study, the $\mathcal{ABC}$ fractional operator is more generalized and supports our system to be more extensive and covers several of new and existing problems in the literature. As a future direction, the studied problem would be interesting if it was studied with an integral term and under impulsive boundary conditions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Pontificia Universidad Católica del Ecuador, Proyecto Título: "Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales" Cod UIO2022.

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.

Conflict of interest

The authors declare that they have no competing interests.