Solvability of a ϱ-Hilfer Fractional Snap Dynamic System on Unbounded Domains

Abstract

This paper is devoted to studying the $\varrho$-Hilfer fractional snap dynamic system under the $\varrho$-Riemann–Liouville fractional integral conditions on unbounded domains $[a,\infty ),a\ge 0$, for the first time. The results concerning the existence and uniqueness, along with the Ulam–Hyers, Ulam–Hyers–Rassias, and semi-Ulam–Hyers–Rassias stabilities, are established in an appropriate special Banach space according to fractional calculus, fixed point theory, and nonlinear analysis. At the end, a numerical example is presented for the interpretation of the main results.

1. Introduction

The study of fractional calculus has recently gained momentum and has become an important research area. Fractional derivatives provide excellent tools for characterizing the memory and genetic properties of various materials, processes, and dynamic systems [1,2,3,4,5,6,7,8,9]. In fractional calculus, many operators have appeared; one of these is called the Hilfer fractional derivative with respect to another non-negative increasing function $\varrho$ ($\varrho$-Hilfer fractional derivatives), and this has unified several operators that exist in the literature [10]. So, this definition has attracted the attention of many researchers and has been used to cover a variety of fractional differential problems; for example, see [11,12,13,14,15,16,17,18,19] and the references cited therein.

Fixed point theory is a mathematical discipline that explores the existence and behavior of fixed points of mappings. The study of fixed points has many applications in various fields, such as analysis, geometry, topology, physics, economics, and engineering. For instance, fixed point theory can be used to assess the stability of dynamic systems in physics. In general, fixed point theory provides a powerful tool for finding solutions to problems involving nonlinear equations and systems; we refer the readers to some works that have used fixed point techniques [20,21,22,23,24,25,26,27,28,29,30,31,32,33].

In 2021, Rezapour et al. focused on the existence of solutions and established four classes of the Hyres–Ulam stability of a generalized FBVP with a p-Laplacian operator and with three-point integral boundary conditions given by

$$\left\{\begin{array}{cc}{\displaystyle {}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_1,{\xi }_1}\left({\phi }_p\left({}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_2,{\xi }_2}\Psi \left(u\right)\right)\right)=\mathcal{W}\left(u,\Psi \left(u\right)\right),}\hfill & u\in [u_0,K],u_0\ge 0,\hfill \\ {\displaystyle \Psi \left(u_0\right)+{\mu }_1\Psi \left(K\right)={\sigma }_1{\int }_0^K{\hslash }_1\left(r\right)\mathrm{d}r,}\hfill & {\mu }_1\ne -1,\hfill \\ {\displaystyle {\Psi }^{\prime }\left(u_0\right)+{\mu }_2\Psi \left(K\right)={\sigma }_2{\int }_0^K{\hslash }_2\left(r\right)\mathrm{d}r,}\hfill & {\mu }_2\ne -{\left(\frac{u_0}{K}\right)}^{{\xi }_2-1},\hfill \\ {\displaystyle {}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_2,{\xi }_2}\Psi \left(u_0\right)=0,{}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_2,{\xi }_2}\Psi \left(K\right)=\gamma {}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_2,{\xi }_2}\Psi \left(\eta \right),}\hfill & \eta \in (u_0,K),\hfill \end{array}\right.$$

(1)

so that ${}^{\mathrm{C}}{\mathfrak{D}}_{a^{+}}^{{\nu }_i,{\xi }_i}$ is a generalized derivative in the sense of Caputo and is of the order $0<{\nu }_i<1$ and ${\xi }_i>0$ for $i=1,2$ [34]. The authors of [35] investigated a snap dynamic system via simulations, analysis, and a real circuit in terms of integer derivatives:

$$\left\{\begin{array}{c}{\mathfrak{D}}^1{\Psi }_1={\Psi }_2\left(u\right),{\mathfrak{D}}^1{\Psi }_2={\Psi }_3\left(u\right),{\mathfrak{D}}^1{\Psi }_3={\Psi }_4\left(u\right),\hfill \\ {\mathfrak{D}}^1{\Psi }_4=\mathcal{G}\left({\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right),\hfill \end{array}\right.$$

(2)

where $\mathcal{G}({\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4)=-a{\Psi }_1-{\Psi }_2-{\Psi }_4+b{\Psi }_1{\Psi }_3$, which can be reformulated as

$${\mathfrak{D}}^4{\Psi }_1=\mathcal{G}\left({\Psi }_1,{\mathfrak{D}}^1{\Psi }_1,{\mathfrak{D}}^2{\Psi }_1,{\mathfrak{D}}^3{\Psi }_1\right).$$

(3)

In 2021, Mahmoud et al. ([36]) examined the qualitative properties of the following Caputo fractional snap dynamic system:

$$\left\{\begin{array}{c}{}^{\mathrm{C}}{\mathfrak{D}}^p{\Psi }_1={\Psi }_2\left(u\right),{}^{\mathrm{C}}{\mathfrak{D}}^p{\Psi }_2={\Psi }_3\left(u\right),{}^{\mathrm{C}}{\mathfrak{D}}^p{\Psi }_3={\Psi }_4\left(u\right),\hfill \\ {}^{\mathrm{C}}{\mathfrak{D}}^p{\Psi }_4=-2{\Psi }_1-{\Psi }_2-{\Psi }_4+{\Psi }_1{\Psi }_3,\hfill \end{array}\right.$$

(4)

where $p=0.95$. Recently, in 2021, Samei et al. [37] studied a new structure of the $\varrho$-Caputo fractional snap dynamic system with initial value boundary conditions on bounded domains $[a,b]$:

$$\left\{\begin{array}{cc}{}^{\mathrm{C}}{\mathfrak{D}}^{p_1;\varrho }{\Psi }_1\left(u\right)={\Psi }_2\left(u\right),{\Psi }_1\left(a\right)={\lambda }_1,\hfill & u\in [a,b],\hfill \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_2;\varrho }{\Psi }_2\left(u\right)={\Psi }_3\left(u\right),{\Psi }_2\left(a\right)={\lambda }_2,\hfill & \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_3;\varrho }{\Psi }_3\left(u\right)={\Psi }_4\left(u\right),{\Psi }_3\left(a\right)={\lambda }_3,\hfill & \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_4;\varrho }{\Psi }_4\left(u\right)=\mathcal{G}\left(u,{\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right),\hfill & {\Psi }_4\left(a\right)={\lambda }_4,\hfill \end{array}\right.$$

(5)

where ${}^{\mathrm{C}}{\mathfrak{D}}^{q;\varrho }$ is the $\varrho$-Caputo fractional derivative of the order $q\in (0,1]$ and $q\in \{p_1,p_2,p_3,p_4\}$. Very recently, in 2022 [38], some authors established qualitative theories, such as those of existence, uniqueness, and Ulam–Hyers stability, for a nonlinear fractional snap dynamic system in the sense of $\varrho$-Caputo fractional derivatives under non-periodic boundary conditions on bounded domains $[a,b]$:

$$\left\{\begin{array}{c}{}^{\mathrm{C}}{\mathfrak{D}}^{p_1;\varrho }{\Psi }_1\left(u\right)={\Psi }_2\left(u\right),{\Psi }_1\left(a\right)={\lambda }_1{\Psi }_1\left(b\right),\hfill \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_2;\varrho }{\Psi }_2\left(u\right)={\Psi }_3\left(u\right),{\Psi }_2\left(a\right)={\lambda }_2{\Psi }_2\left(b\right),\hfill \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_3;\varrho }{\Psi }_3\left(u\right)={\Psi }_4\left(u\right),{\Psi }_3\left(a\right)={\lambda }_3{\Psi }_3\left(b\right),\hfill \\ {}^{\mathrm{C}}{\mathfrak{D}}^{p_4;\varrho }{\Psi }_4\left(u\right)=\mathcal{G}\left(u,{\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right),{\Psi }_4\left(a\right)={\lambda }_4{\Psi }_4\left(b\right).\hfill \end{array}\right.$$

(6)

Motivated by these articles and by a generalization of a recently published articles [39,40], our focus is centered on studying the existence and uniqueness, as well as various types of Ulam–Hyers stability, of solutions for the following $\varrho$-Hilfer fractional snap dynamic system on unbounded domains $[a,\infty )$:

$$\left\{\begin{array}{c}{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)={\Psi }_2\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }{\Psi }_2\left(u\right)={\Psi }_3\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }{\Psi }_3\left(u\right)={\Psi }_4\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_4,{\beta }_4;\varrho }{\Psi }_4\left(u\right)=\mathcal{G}\left(u,{\Psi }_1\left(u\right),{\Psi }_2\left(u\right),{\Psi }_3\left(u\right),{\Psi }_4\left(u\right)\right),\hfill \end{array}\right.$$

(7)

for $u\in J:=[a,\infty )$, $a\ge 0$, under the $\varrho$-Riemann–Liouville (R-L) fractional integral conditions:

$$\left\{\begin{array}{cc}{\mathfrak{I}}_{a^{+}}^{1-{\gamma }_1;\varrho }{\Psi }_1\left(a\right)={\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right),\hfill & {\delta }_1\in J,{\theta }_1>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_2;\varrho }{\Psi }_2\left(a\right)={\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right),\hfill & {\delta }_2\in J,{\theta }_2>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_3;\varrho }{\Psi }_3\left(a\right)={\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right),\hfill & {\delta }_3\in J,{\theta }_3>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_4;\varrho }{\Psi }_4\left(a\right)={\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right),\hfill & {\delta }_4\in J,{\theta }_4>0,\hfill \end{array}\right.$$

(8)

where ${\lambda }_i\in \mathbb{R},i=1,\cdots ,4$, ${{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varrho }$ is the $\varrho$-Hilfer fractional derivative of order $\alpha \in (0,1]$ and type $\beta \in [0,1],$ where $\alpha \in \left\{{\alpha }_i\right\}$ and $\beta \in \left\{{\beta }_i\right\}$, respectively, ${\mathfrak{I}}_{a^{+}}^{\theta ;\varrho }$ is the $\varrho$-R-L fractional integral of order $\theta =\{1-{\gamma }_i,{\theta }_i\}$, where ${\alpha }_i\le {\gamma }_i={\alpha }_i+{\beta }_i-{\alpha }_i{\beta }_i,$ and $\mathcal{G}:J\times {\mathrm{\Sigma }}^4\to \mathrm{\Sigma }$ and ${\omega }_i:J\to \mathrm{\Sigma }$ are continuous functions in the real Banach space $\mathrm{\Sigma }$.

It is worth mentioning that the analysis space and approach in this paper are different from those in [36,37,38]. Additionally, the novelty and contributions of this work lie in the study of a more general $\varrho$-Hilfer fractional snap dynamic system under the $\varrho$-R-L fractional integral boundary conditions on unbounded domains $[a,\infty )$, $a\ge 0$ in a specially applicable $\mathrm{\Sigma }$. In particular, it reduces to snap problem (3) when $\varrho \left(u\right)=1$, ${\alpha }_i=1$, ${\beta }_i\to 1,$ and it reduces to snap problem (4) when $\varrho \left(u\right)=1,{\alpha }_i=p,{\beta }_i\to 1;$ it returns to snap problem (5) with boundary conditions at ${\beta }_i\to 1$, ${\theta }_i\to 0$, ${\omega }_i\left({\delta }_i\right)=1$, and it reduces to snap problem (6) with boundary conditions for ${\beta }_i\to 1$, ${\theta }_i\to 0$, ${\omega }_i\left({\delta }_i\right)={\Psi }_i\left(b\right)$. Moreover, we can conclude that there are many new problems that are special cases of our problem.

This paper is arranged as follows: In Section 2, we introduce various definitions and preliminaries. In Section 3, we establish the existence and uniqueness of the solution of the fractional snap system (4). In Section 4, we discuss several kinds of stability, such as Ulam–Hyers ($\mathcal{U}.\mathcal{H}$), Ulam–Hyers–Rassias ($\mathcal{U}.\mathcal{H}.\mathcal{R}$), and semi-Ulam–Hyers–Rassias stability (s-$\mathcal{U}.\mathcal{H}.\mathcal{R}$). Finally, an example is given to illustrate the main results in Section 5.

2. Preliminaries

In this section, we recall some important preliminaries that are related to our analysis. The R-L fractional integral of an integrable function g on $[a,b]$ of the order ${\vartheta }_1>0$ and the R-L fractional derivative of a function g of the order ${\vartheta }_1\in (n-1,n]$ with respect to another function $\varrho$ are given by

$${\mathfrak{I}}_{a^{+}}^{{\vartheta }_1;\varrho }g\left(u\right)=\frac{1}{\mathrm{\Gamma }\left({\vartheta }_1\right)}{\int }_a^u{\varrho }^{\prime }\left(v\right){(\varrho \left(u\right)-\varrho \left(v\right))}^{{\vartheta }_1-1}g\left(v\right)dv,u>a\in \mathbb{R},$$

where $\varrho \in C^1\left([a,b]\right)$ is an increasing function and $\mathrm{\Gamma }(·)$ is the following Euler Gamma function:

$$\begin{array}{cc}\hfill {}^{\mathrm{R}}{\mathfrak{D}}_{a^{+}}^{{\vartheta }_1;\varrho }g\left(u\right)& ={\left(\frac{1}{{\varrho }^{\prime }\left(u\right)}\frac{\mathrm{d}}{\mathrm{d}u}\right)}^n{\int }_a^u\frac{{(\varrho \left(u\right)-\varrho \left(v\right))}^{n-{\vartheta }_1-1}}{\mathrm{\Gamma }(n-{\vartheta }_1)}{\varrho }^{\prime }\left(v\right)g\left(v\right)\mathrm{d}v,u>a,\hfill \end{array}$$

where $\varrho \in C^n\left([a,b]\right)$ is an increasing and integrable function with ${\varrho }^{\prime }\left(u\right)\ne 0$ for each $u\in [a,b]$, and $n=\left[{\vartheta }_1\right]+1$ such that $\left[{\vartheta }_1\right]$ is an integer part of ${\vartheta }_1$ [2].

Lemma 1.

([2]). Let ${\vartheta }_1,{\vartheta }_2,\zeta >0$. Then,

(1)

${\mathfrak{I}}_{a^{+}}^{{\vartheta }_1;\varrho }\left({\mathfrak{I}}_{a^{+}}^{{\vartheta }_2;\varrho }\varrho \left(u\right)\right)={\mathfrak{I}}_{a^{+}}^{{\vartheta }_1+{\vartheta }_2;\varrho }g\left(u\right)$;

(2)

${}^{\mathrm{R}}{\mathfrak{D}}_{a^{+}}^{{\vartheta }_1;\varrho }\left({\mathfrak{I}}_{a^{+}}^{{\vartheta }_1;\varrho }g\left(u\right)\right)=g\left(u\right)$;

(3)

$\left[{\mathfrak{I}}_{a^{+}}^{{\vartheta }_1;\varrho }{(\varrho \left(u\right)-\varrho \left(a\right))}^{\zeta -1}\right]\left(u\right)={\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{\mathrm{\Gamma }(\zeta +{\vartheta }_1)}}{(\varrho \left(u\right)-\varrho \left(a\right))}^{\zeta +{\vartheta }_1-1}$;

(4)

$\left[{}^{\mathrm{R}}{\mathfrak{D}}_{a^{+}}^{\vartheta ;\varrho }{(\varrho \left(u\right)-\varrho \left(a\right))}^{\zeta -1}\right]\left(u\right)={\displaystyle \frac{\mathrm{\Gamma }\left(\zeta \right)}{\mathrm{\Gamma }(\zeta -\vartheta )}}{(\varrho \left(u\right)-\varrho \left(a\right))}^{\zeta -\vartheta -1},0<\vartheta \le 1$.

Definition 1.

([10]). Let $\varrho \in C^n\left([a,b]\right)$ be an increasing and integrable function with ${\varrho }^{\prime }\left(u\right)\ne 0$ for $u\in [a,b]$. Then, the ϱ-Hilfer fractional derivative of a function g of the order ${\vartheta }_1\in (n-1,n]$ and type ${\vartheta }_2\in [0,1]$ with respect to another function ϱ is defined by

$${}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\vartheta }_1,{\vartheta }_2;\varrho }g\left(u\right)={\mathfrak{I}}_{a^{+}}^{{\vartheta }_2(n-{\vartheta }_1);\varrho }\left({\left(\frac{1}{{\varrho }^{\prime }\left(u\right)}\frac{\mathrm{d}}{\mathrm{d}u}\right)}^n{\mathfrak{I}}_{a^{+}}^{(1-{\vartheta }_2)(n-{\vartheta }_1);\varrho }g\left(u\right)\right).$$

Moreover, the operator ${}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\vartheta }_1,{\vartheta }_2;\varrho }$ can be written as

$${}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\vartheta }_1,{\vartheta }_2;\varrho }={\mathfrak{I}}_{a^{+}}^{{\vartheta }_2(n-{\vartheta }_1);\varrho }{}^R{\mathfrak{D}}_{a^{+}}^{\vartheta ;\varrho },\vartheta ={\vartheta }_1+n{\vartheta }_2-{\vartheta }_1{\vartheta }_2.$$

(9)

Lemma 2.

([2,10]). If ${\vartheta }_1\in (0,1]$, ${\vartheta }_2\in [0,1],0\le \vartheta <1,\vartheta ={\vartheta }_1+{\vartheta }_2-{\vartheta }_1{\vartheta }_2$, and $g^1\in C\left([a,b]\right)$, ${\mathfrak{I}}_{a^{+}}^{1-\vartheta ;\varrho }g\in C^1\left([a,b]\right)$, then

$$\begin{array}{cc}\hfill {\mathfrak{I}}_{a^{+}}^{{\vartheta }_1;\varrho }{}^H\left({\mathfrak{D}}_{a^{+}}^{{\vartheta }_1,{\vartheta }_2;\varrho }g\left(u\right)\right)& ={\mathfrak{I}}_{a^{+}}^{\vartheta ;\varrho }{}^R\left({\mathfrak{D}}_{a^{+}}^{\vartheta ;\varrho }g\left(u\right)\right)\hfill \\ \hfill & =g\left(u\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{\vartheta -1}}{\mathrm{\Gamma }\left(\vartheta \right)}}{\mathfrak{I}}_{a^{+}}^{1-\vartheta ;\varrho }g\left(a\right),\forall u\in [a,b].\hfill \end{array}$$

Theorem 1.

([41]). Assume that the generalized complete metric space is denoted by $(\mathfrak{G},\tilde{d})$ and let the operator $\mathrm{\Lambda }:\mathfrak{G}\to \mathfrak{G}$ be contractive with the Lipschitz constant $\ell <1$. If there is a positive integer r, where $\tilde{d}({\mathrm{\Lambda }}^{r+1}u,{\mathrm{\Lambda }}^{r}u)<\infty ,$ for some $u\in \mathfrak{G}$, then the following holds.

(i)

The sequence $\left\{{\mathrm{\Lambda }}^r\right\}$ converges to a fixed point $u_0\in \mathfrak{G}$;

(ii)

$u_0$ is the unique fixed point of Λ in ${\mathfrak{G}}^{*}=\{v\in \mathfrak{G}:\tilde{d}({\mathrm{\Lambda }}^{r}u,v)<\infty \}$;

(iii)

If $v\in {\mathfrak{G}}^{*}$, then $\tilde{d}(v,u_0)\le \frac{1}{1-\ell }\tilde{d}(\mathrm{\Lambda }v,v)$.

Clearly, the $\varrho$-Hilfer fractional snap dynamic system in (7) and (8) can be rewritten as follows:

$$\left\{\begin{array}{cc}{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_4,{\beta }_4;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\right)={\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right),\hfill & \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_1;\varrho }{\Psi }_1\left(a\right)={\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right),\hfill & {\delta }_1\in J,{\theta }_1>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)={\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right),\hfill & {\delta }_2\in J,{\theta }_2>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)\right)={\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right),\hfill & {\delta }_3\in J,{\theta }_3>0,\hfill \\ {\mathfrak{I}}_{a^{+}}^{1-{\gamma }_4;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)\right)\right)={\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right),\hfill & {\delta }_4\in J,{\theta }_4>0,\hfill \end{array}\right.$$

(10)

for $u\in J=[a,\infty )$, $a\ge 0,$ where

$$\begin{array}{cc}\hfill {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)=\mathcal{G}& \left(u,{\Psi }_1\left(u\right),{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right),{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right),\right.\hfill \\ \hfill & \left.{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\right).\hfill \end{array}$$

(11)

Now, we consider the Banach space $\mathcal{C}(J,\mathrm{\Sigma })$, which is endowed by the norm ${\parallel {\Psi }_1\parallel }_{\mathrm{\Sigma }}={sup}_{u\in J}\parallel {\Psi }_1\left(u\right)\parallel .$ For the compatibility of our work and due to inspiration from [42,43,44,45], we introduce the following suitable special Banach space $(\mathrm{\Omega },\parallel ·{\parallel }_{\mathrm{\Omega }})$:

$$\begin{array}{cc}\hfill \mathrm{\Omega }& =\left\{{\Psi }_1\right|{\Psi }_1\left(u\right)\in \mathcal{C}(J,\mathrm{\Sigma }),{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right),{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right),\hfill \\ \hfill & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\in {\mathcal{C}}^1(J,\mathrm{\Sigma }),\hfill \\ \hfill & \underset{u\in J}{sup}\frac{\parallel {\Psi }_1\left(u\right)\parallel }{\xi \left(u\right)}<\infty ,\underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)∥<\infty ,\hfill \\ \hfill & \underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)∥<\infty ,\hfill \\ \hfill & \underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)∥<\infty \},\hfill \end{array}$$

which has the supremum norm

$$\begin{array}{cc}\hfill \parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}& =max\{\underset{u\in J}{sup}\frac{\parallel {\Psi }_1\left(u\right)\parallel }{\xi \left(u\right)},\underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)∥,\hfill \\ \hfill & \underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)∥,\hfill \\ \hfill & \underset{u\in J}{sup}\frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)∥\},\hfill \end{array}$$

where the function $\xi :J\to (0,\infty )$ is non-decreasing and continuous.

3. The Existence and Uniqueness Results

Firstly, we introduce an important corresponding fractional integral equation to the $\varrho$-Hilfer fractional snap dynamic system (10), as given in the next lemma.

Lemma 3.

Consider a continuous function ${\Psi }_1$ that is differentiable. Then, the solution of the ϱ-Hilfer fractional snap dynamic system (10) is given in the following fractional integral form:

$$\begin{array}{cc}\hfill {\Psi }_1\left(u\right)& ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)+{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

(12)

Proof. 

In view of Lemma 2, and by applying ${\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }$ on both sides of (10), we have

$$\begin{array}{cc}\hfill {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }& \left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4-1}}{\mathrm{\Gamma }\left({\gamma }_4\right)}}{\mathfrak{I}}_{a^{+}}^{1-{\gamma }_4;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)\right)\right)\hfill \\ \hfill & +{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

By using the boundary condition

$${\mathfrak{I}}_{a^{+}}^{1-{\gamma }_4;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)\right)\right)={\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right),$$

we get

$$\begin{array}{cc}\hfill {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }& \left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4-1}}{\mathrm{\Gamma }\left({\gamma }_4\right)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

(13)

Now, by applying ${\mathfrak{I}}_{a^{+}}^{{\alpha }_3;\varrho }$ on both sides of (13) and by using Lemma 1 with the substitution of the boundary condition

$${\mathfrak{I}}_{a^{+}}^{1-{\gamma }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)\right)={\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right),$$

we find

$$\begin{array}{cc}\hfill {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)& ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-1}}{\mathrm{\Gamma }\left({\gamma }_3\right)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

(14)

Again, by applying ${\mathfrak{I}}_{a^{+}}^{{\alpha }_2;\varrho }$ on both sides of (14) and by substituting the boundary condition

$${\mathfrak{I}}_{a^{+}}^{1-{\gamma }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(a\right)\right)={\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right),$$

we obtain

$$\begin{array}{cc}\hfill {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)& =\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-1}}{\mathrm{\Gamma }\left({\gamma }_2\right)}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2)}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2)}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

(15)

Similarly, with the boundary condition ${\mathfrak{I}}_{a^{+}}^{1-{\gamma }_1;\varrho }{\Psi }_1\left(a\right)={\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)$ and by applying ${\mathfrak{I}}_{a^{+}}^{{\alpha }_1;\varrho }$ on both sides of (15), we have

$$\begin{array}{cc}\hfill {\Psi }_1\left(u\right)& ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)+{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right).\hfill \end{array}$$

Then, the proof is complete. □

Before establishing the existence and uniqueness results, we need to introduce the following hypotheses.

Hypothesis 1.

Let $\mathcal{G}:I\times {\mathrm{\Omega }}^4\to \mathrm{\Omega }$ be a continuously differentiable function, and we assume that ${\sigma }_i(·)\ge 0,(i=1,\cdots ,4)$ are continuous functions satisfying

$$\begin{array}{cc}\hfill & \parallel \mathcal{G}\left(u,{\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right)-\mathcal{G}\left(u,{\overline{\Psi }}_1,{\overline{\Psi }}_2,{\overline{\Psi }}_3,{\overline{\Psi }}_4\right)\parallel \le \sum _{i=1}^4{\sigma }_i\left(u\right){\displaystyle \frac{\parallel {\Psi }_i\left(u\right)-{\overline{\Psi }}_i\left(u\right)\parallel }{\xi \left(u\right)}},\hfill \end{array}$$

for all ${\Psi }_i,{\overline{\Psi }}_i\in \mathrm{\Omega },(i=1,2,3,4)$ and $u\in J$.

Hypothesis 2.

There are constants $\breve{L},\breve{Q}>0,$ where $\breve{Q}\in (0,1)$, and these satisfy the following restrictions:

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }\left[{\sigma }_1\left(u\right)+{\sigma }_2\left(u\right)+{\sigma }_3\left(u\right)+{\sigma }_4\left(u\right)\right]\right\}\le \breve{Q}<1,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}\right({\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_1-1}}{\mathrm{\Gamma }\left({\eta }_1\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_2-1}}{\mathrm{\Gamma }\left({\eta }_2\right)}}\left({\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_3-1}}{\mathrm{\Gamma }\left({\eta }_3\right)}}\left({\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_4-1}}{\mathrm{\Gamma }\left({\eta }_4\right)}}\left({\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\right)\hfill \\ \hfill & +{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }∥{\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)∥\left)\right\}\le \breve{L}<\infty ,\hfill \end{array}$$

(17)

where ${\omega }_i^{*}={sup}_{u\in [a,{\delta }_i]}\parallel {\omega }_i\left({\delta }_i\right)\parallel$, $(i=1,\cdots ,4)$, and

$$\begin{array}{cc}\hfill \alpha & ={\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1,\mathrm{or}{\alpha }_4+{\alpha }_3+{\alpha }_2,\mathrm{or}{\alpha }_4+{\alpha }_3,\mathrm{or}{\alpha }_4;\hfill \\ \hfill {\eta }_1& ={\gamma }_1\mathrm{or}{\gamma }_1-{\alpha }_1,\mathrm{or}{\gamma }_1-{\alpha }_2-{\alpha }_1,\mathrm{or}{\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1;\hfill \\ \hfill {\eta }_2& ={\gamma }_2+{\alpha }_1,\mathrm{or}{\gamma }_2,\mathrm{or}{\gamma }_2-{\alpha }_2,\mathrm{or}{\gamma }_2-{\alpha }_3-{\alpha }_2;\hfill \\ \hfill {\eta }_3& ={\gamma }_3+{\alpha }_2+{\alpha }_1,\mathrm{or}{\gamma }_3+{\alpha }_2,\mathrm{or}{\gamma }_3,or{\gamma }_3-{\alpha }_3;\hfill \\ \hfill {\eta }_4& ={\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1,\mathrm{or}{\gamma }_4+{\alpha }_3+{\alpha }_2,\mathrm{or}{\gamma }_4+{\alpha }_3,\mathrm{or}{\gamma }_4.\hfill \end{array}$$

(18)

The following lemma is for the study of the existence and uniqueness of solutions for the $\varrho$-Hilfer fractional snap dynamic system (10), and it is used by employing the well-known Banach fixed point theorem.

Theorem 2.

If the hypotheses(HP${}_1$)and(HP${}_2$)hold, then, the ϱ-Hilfer fractional snap dynamic system (10) has one solution on the unbounded domain J.

Proof. 

Based on Lemma 3, let the mapping $T:\mathrm{\Omega }\to \mathrm{\Omega }$ be defined as follows:

$$\begin{array}{cc}\hfill \left(T{\Psi }_1\right)\left(u\right)& ={\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)+{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)+{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right),\hfill \end{array}$$

(19)

where ${\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)$ is defined in Equation (11). For simplicity, let ${\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)=\mathcal{G}(u,0,0,0,0)$. Indeed, T maps $\mathrm{\Omega }$ into $\mathrm{\Omega }$, that is, for any ${\Psi }_1\in \mathrm{\Omega },$ by using (HP${}_1$) and (HP${}_2$), we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel \left(T{\Psi }_1\right)\left(u\right)\parallel }{\xi \left(u\right)}}& \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)\xi \left(u\right)}}\left|{\lambda }_1\right|{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }\parallel {\omega }_1\left({\delta }_1\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)\xi \left(u\right)}}|{\lambda }_2|{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }\parallel {\omega }_2\left({\delta }_2\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}|{\lambda }_3|{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }\parallel {\omega }_3\left({\delta }_3\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}|{\lambda }_4|{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }\parallel {\omega }_4\left({\delta }_4\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\hfill \\ \hfill & +\frac{1}{\xi \left(u\right)}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }[{\sigma }_1\left(u\right){\displaystyle \frac{\parallel {\Psi }_1\left(u\right)\parallel }{\xi \left(u\right)}}+{\sigma }_2\left(u\right){\displaystyle \frac{∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)∥}{\xi \left(u\right)}}\hfill \\ \hfill & +{\sigma }_3\left(u\right){\displaystyle \frac{∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)∥}{\xi \left(u\right)}}\hfill \\ \hfill & +{\sigma }_4\left(u\right){\displaystyle \frac{∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)∥}{\xi \left(u\right)}}]\hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{\parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{L}+\breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \end{array}$$

for $\breve{Q}\in (0,1).$ In addition, due to Equation (9), we have

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)∥& \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_1)\xi \left(u\right)}}|{\lambda }_1|{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }\parallel {\omega }_1\left({\delta }_1\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-1}}{\mathrm{\Gamma }\left({\gamma }_2\right)\xi \left(u\right)}}|{\lambda }_2|{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }\parallel {\omega }_2\left({\delta }_2\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2)\xi \left(u\right)}}|{\lambda }_3|{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }\parallel {\omega }_3\left({\delta }_3\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2)\xi \left(u\right)}}|{\lambda }_4|{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }\parallel {\omega }_4\left({\delta }_4\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-1}}{\mathrm{\Gamma }\left({\gamma }_2\right)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{\parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{L}+\breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_2-{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_2)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-1}}{\mathrm{\Gamma }\left({\gamma }_3\right)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)\parallel \hfill \\ \hfill & +{\displaystyle \frac{\parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{L}+\breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_3-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_3-{\alpha }_2)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_3-{\alpha }_3)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4-1}}{\mathrm{\Gamma }\left({\gamma }_4\right)\xi \left(u\right)}}{\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)\parallel +{\displaystyle \frac{\parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{L}+\breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \end{array}$$

for $\breve{Q}\in (0,1).$ Hence, $\parallel {\Psi }_1{\parallel }_{\mathrm{\Omega }}\le \breve{L}+\breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }}<\infty ,$ which means that $T:\mathrm{\Omega }\to \mathrm{\Omega }$. Now, we will prove that T is contractive on $\mathrm{\Omega }$. By using (HP${}_1$) and (HP${}_2$), for any ${\Psi }_{,}\overline{{\Psi }_1}\in \mathrm{\Omega },$ we find

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel \left(T{\Psi }_1\right)\left(u\right)-\left(T\overline{{\Psi }_1}\right)\left(u\right)\parallel }{\xi \left(u\right)}}& \le {\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }∥{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)-{\breve{\mathcal{G}}}_{\overline{{\Psi }_1}}\left(u\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }[{\sigma }_1\left(u\right){\displaystyle \frac{∥{\Psi }_1\left(u\right)-\overline{{\Psi }_1}\left(u\right)∥}{\xi \left(u\right)}}\hfill \\ \hfill & +{\sigma }_2\left(u\right){\displaystyle \frac{∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)∥}{\xi \left(u\right)}}\hfill \\ \hfill & +{\displaystyle \frac{{\sigma }_3\left(u\right)}{\xi \left(u\right)}}[\parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\hfill \\ \hfill & -{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)\parallel ]\hfill \\ \hfill & +{\displaystyle \frac{{\sigma }_4\left(u\right)}{\xi \left(u\right)}}[\parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)\right)\parallel \left]\right]\hfill \\ \hfill & \le {\displaystyle \frac{\parallel {\Psi }_1-\overline{{\Psi }_1}{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }}.\hfill \end{array}$$

Likewise,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)-{\breve{\mathcal{G}}}_{\overline{{\Psi }_1}}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel {\Psi }_1-\overline{{\Psi }_1}{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \\ \hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\right)\left(u\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\right)\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)-{\breve{\mathcal{G}}}_{\overline{{\Psi }_1}}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel {\Psi }_1-\overline{{\Psi }_1}{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\right)\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\parallel {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)-{\breve{\mathcal{G}}}_{\overline{{\Psi }_1}}\left(u\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel {\Psi }_1-\overline{{\Psi }_1}{\parallel }_{\mathrm{\Omega }}}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}{\parallel {\Psi }_1\parallel }_{\mathrm{\Omega }}.\hfill \end{array}$$

Thus, we conclude that

$$\parallel T{\Psi }_1-T\overline{{\Psi }_1}{\parallel }_{\mathrm{\Omega }}\le \breve{Q}{\parallel {\Psi }_1-\overline{{\Psi }_1}\parallel }_{\mathrm{\Omega }}.$$

Since $\breve{Q}\in (0,1)$, T is a contractive operator. According to the Banach fixed point theorem, the operator T possesses one fixed point ${\Psi }_0$ in $\mathrm{\Omega }$, i.e., $T{\Psi }_0={\Psi }_0$. Hence, the $\varrho$-Hilfer fractional snap dynamic system (10) has one solution on the unbounded domain J. □

4. The Stability

Throughout this section, we will study different types of stability, such as $\mathcal{U}.\mathcal{H}$, $\mathcal{U}.\mathcal{H}.\mathcal{R}$, and s-$\mathcal{U}.\mathcal{H}.\mathcal{R}$. For definitions thereof and more details, we refer readers to [46]. The aim in this section is to define the appropriate metrics ${\mathrm{d}}_1(·)$ and ${\mathrm{d}}_2(·)$ on the Banach space $\mathrm{\Omega }$. The metric ${\mathrm{d}}_1(·)$ is defined by

$$\begin{array}{cc}\hfill {\mathrm{d}}_1({\Psi }_1,\overline{{\Psi }_1})& =\underset{u\in J}{inf}\{N\in J|\frac{\parallel {\Psi }_1\left(u\right)-\overline{{\Psi }_1}\left(u\right)\parallel }{\xi \left(u\right)}\le N\upsilon \left(u\right),\hfill \\ \hfill & \frac{1}{\xi \left(u\right)}∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)∥\le N\upsilon \left(u\right),\hfill \\ \hfill & \frac{1}{\xi \left(u\right)}∥{}^H{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)∥\le N\upsilon \left(u\right),\hfill \\ \hfill & \frac{1}{\xi \left(u\right)}\parallel {}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)\right)\parallel \le N\upsilon \left(u\right)\},\hfill \end{array}$$

for any non-negative, non-decreasing continuous function $\upsilon \left(u\right)$ on J, and the metric ${\mathrm{d}}_2(·)$ is defined for any non-negative, non-increasing continuous function $\upsilon \left(u\right)$ on J as follows:

$$\begin{array}{cc}\hfill {\mathrm{d}}_2({\Psi }_1,\overline{{\Psi }_1})& =\underset{u\in J}{sup}\{N\in J|\frac{\parallel {\Psi }_1\left(u\right)-\overline{{\Psi }_1}\left(u\right)\parallel }{\upsilon \left(u\right)\xi \left(u\right)}\le N,\hfill \\ \hfill & \frac{∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)∥}{\upsilon \left(u\right)\xi \left(u\right)}\le N,\hfill \\ \hfill & \frac{∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)∥}{\upsilon \left(u\right)\xi \left(u\right)}\le N,\hfill \\ \hfill & \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}\parallel {}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }\overline{{\Psi }_1}\left(u\right)\right)\right)\parallel \le N\}.\hfill \end{array}$$

According to [47], it is not difficult to prove that ${\mathrm{d}}_1(·)$ and ${\mathrm{d}}_2(·)$ are metrics on a special Banach space $\mathrm{\Omega }$. Next, we state and prove the $\mathcal{U}.\mathcal{H}$, $\mathcal{U}.\mathcal{H}.\mathcal{R}$, and s-$\mathcal{U}.\mathcal{H}.\mathcal{R}$ stability theorems on the unbounded domains J.

Theorem 3.

Assume that (HP${}_1$) and (HP${}_2$) hold and that ${\Psi }_1:J\to \mathrm{\Omega }$ is a continuously differentiable function, and let $\upsilon \left(u\right)$ be a non-negative, non-decreasing continuous function on J that satisfies

$$\begin{array}{cc}\hfill \parallel {\Psi }_1\left(u\right)& -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)\hfill \\ \hfill & +{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\upsilon \left(u\right),u\in J,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)& -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-1}}{\mathrm{\Gamma }\left({\gamma }_2\right)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)\hfill \\ \hfill & -{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\upsilon \left(u\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \parallel & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_2-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_2)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-1}}{\mathrm{\Gamma }\left({\gamma }_3\right)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)-{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\upsilon \left(u\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \parallel & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_3-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_3-{\alpha }_2)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_3-{\alpha }_3)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4-1}}{\mathrm{\Gamma }\left({\gamma }_4\right)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)-{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\upsilon \left(u\right),\hfill \end{array}$$

(23)

for $u\in J$. Then, there is one solution ${\Psi }_0\in \mathrm{\Omega }$ for all $u\in J$ and $0<\breve{Q}<1$, which satisfies

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{\Psi }_1\left(u\right)-{\Psi }_0\left(u\right)∥\le {\displaystyle \frac{M}{1-\breve{Q}}}\upsilon \left(u\right),\hfill \\ \hfill \frac{1}{\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)∥\le \frac{M}{1-\breve{Q}}\upsilon \left(u\right),\hfill \\ \hfill \frac{1}{\xi \left(u\right)}& \parallel {}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)\right)\parallel \le \frac{M}{1-\breve{Q}}\upsilon \left(u\right),\hfill \\ \hfill \frac{1}{\xi \left(u\right)}& \parallel {}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)\right)\right)\parallel \le \frac{M}{1-\breve{Q}}\upsilon \left(u\right),\hfill \end{array}$$

(24)

where

$$\underset{u\in J}{sup}{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)\xi \left(u\right)}}\le M<\infty ,$$

(25)

which implies that the solution of the ϱ-Hilfer fractional snap dynamic system (10) is $\mathcal{U}.\mathcal{H}.\mathcal{R}$ stable and, accordingly, is $\mathcal{U}.\mathcal{H}$ stable.

Proof. 

At the beginning, $T:\mathrm{\Omega }\to \mathrm{\Omega }$ was as defined in (19). By the metric ${\mathrm{d}}_1(·)$ and hypotheses (HP${}_1$) and (HP${}_2$), for any ${\Psi }_1,\overline{{\Psi }_1}\in \mathrm{\Omega }$, we have

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}& ∥\left(T{\Psi }_1\right)\left(u\right)-\left(T\overline{{\Psi }_1}\right)\left(u\right)∥\le \frac{N\upsilon \left(u\right)}{\xi \left(u\right)}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}N\upsilon \left(u\right),\hfill \\ \hfill \frac{1}{\xi \left(u\right)}& ∥\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\right)\left(u\right)-\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\right)\left(u\right)∥\hfill \\ \hfill & \le \frac{N\upsilon \left(u\right)}{\xi \left(u\right)}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}N\upsilon \left(u\right),\hfill \\ \hfill \frac{1}{\xi \left(u\right)}& ∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\right)∥\hfill \\ \hfill & \le \frac{N\upsilon \left(u\right)}{\xi \left(u\right)}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}N\upsilon \left(u\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}& ∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\right)\right)∥\hfill \\ \hfill & \le \frac{N\upsilon \left(u\right)}{\xi \left(u\right)}{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\left[\sum _{i=1}^4{\sigma }_i\left(u\right)\right]\le \breve{Q}N\upsilon \left(u\right),\hfill \end{array}$$

for $\breve{Q}\in (0,1)$. Hence, we find

$${\mathrm{d}}_1\left(T{\Psi }_1,T\overline{{\Psi }_1}\right)\le \breve{Q}N=\breve{Q}{\mathrm{d}}_1\left({\Psi }_1,\overline{{\Psi }_1}\right),0<\breve{Q}<1.$$

Next, due to inequalities (20)–(23), we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\Psi }_1\right)\left(u\right)-\left(T{\Psi }_1\right)\left(u\right)\parallel }{\xi \left(u\right)}}\le \underset{u\in J}{sup}{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1+1)\xi \left(u\right)}}\upsilon \left(u\right)=M\upsilon \left(u\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}& ∥{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)∥\hfill \\ \hfill & \le \underset{u\in J}{sup}{\displaystyle \frac{\upsilon \left(u\right){(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3+{\alpha }_2}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+{\alpha }_2+1)\xi \left(u\right)}}=M\upsilon \left(u\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}& \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)-{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\parallel \hfill \\ \hfill & \le \underset{u\in J}{sup}{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+1)\xi \left(u\right)}}\upsilon \left(u\right)=M\upsilon \left(u\right),\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill \frac{1}{\xi \left(u\right)}& \parallel {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)\parallel \hfill \\ \hfill & \le \underset{u\in J}{sup}{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4}}{\mathrm{\Gamma }({\alpha }_4+1)\xi \left(u\right)}}\upsilon \left(u\right)=M\upsilon \left(u\right).\hfill \end{array}$$

(29)

Therefore, inequalities (26)–(29) imply that

$${\mathrm{d}}_1({\Psi }_1,T{\Psi }_1)\le M<\infty .$$

Furthermore, according to Theorem 1 (parts (i) and (ii)), T has one fixed point ${\Psi }_0$, i.e., $T{\Psi }_0={\Psi }_0$. Moreover, from Theorem 1 (part (iii)), we have

$${\mathrm{d}}_1({\Psi }_1,{\Psi }_0)\le {\displaystyle \frac{1}{1-\breve{Q}}}{\mathrm{d}}_1(T{\Psi }_1,{\Psi }_1)\le {\displaystyle \frac{M}{1-\breve{Q}}},0<\breve{Q}<1.$$

Thus, the solution of the $\varrho$-Hilfer fractional snap dynamic system (10) is $\mathcal{U}.\mathcal{H}.\mathcal{R}$ stable. Along with this, if $\upsilon \left(u\right)=1$, then the solution of the $\varrho$-Hilfer fractional snap dynamic system (10) is $\mathcal{U}.\mathcal{H}$ stable. □

Theorem 4.

Let (HP${}_1$) and (HP${}_2$) hold, let ${\Psi }_1:J\to \mathrm{\Omega }$ be a continuously differentiable function, and let $\upsilon \left(u\right)$ be a non-negative, non-increasing continuous function on J, all of which satisfy

$$\begin{array}{cc}\hfill \parallel & {\Psi }_1\left(u\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-1}}{\mathrm{\Gamma }\left({\gamma }_1\right)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_2+{\alpha }_1)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)\hfill \\ \hfill & +{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1;\varrho }\rho ,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \parallel & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-1}}{\mathrm{\Gamma }\left({\gamma }_2\right)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_3+{\alpha }_2)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3+{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3+{\alpha }_2)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)\hfill \\ \hfill & -{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3+{\alpha }_2;\varrho }\rho ,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \parallel & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)-{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_2-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_2)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-1}}{\mathrm{\Gamma }\left({\gamma }_3\right)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4+{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_4+{\alpha }_3)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)-{\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4+{\alpha }_3;\varrho }\rho ,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \parallel & {{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1-1}}{\mathrm{\Gamma }({\gamma }_1-{\alpha }_3-{\alpha }_2-{\alpha }_1)}}{\lambda }_1{\mathfrak{I}}_{a^{+}}^{{\theta }_1;\varrho }{\omega }_1\left({\delta }_1\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_2-{\alpha }_3-{\alpha }_2-1}}{\mathrm{\Gamma }({\gamma }_2-{\alpha }_3-{\alpha }_2)}}{\lambda }_2{\mathfrak{I}}_{a^{+}}^{{\theta }_2;\varrho }{\omega }_2\left({\delta }_2\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_3-{\alpha }_3-1}}{\mathrm{\Gamma }({\gamma }_3-{\alpha }_3)}}{\lambda }_3{\mathfrak{I}}_{a^{+}}^{{\theta }_3;\varrho }{\omega }_3\left({\delta }_3\right)\hfill \\ \hfill & -{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\gamma }_4-1}}{\mathrm{\Gamma }\left({\gamma }_4\right)}}{\lambda }_4{\mathfrak{I}}_{a^{+}}^{{\theta }_4;\varrho }{\omega }_4\left({\delta }_4\right)-{\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }{\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)\parallel \le {\mathfrak{I}}_{a^{+}}^{{\alpha }_4;\varrho }\rho ,\hfill \end{array}$$

(33)

for $u\in J$, where $\rho >0$. Then, there is one solution ${\Psi }_0\in \mathrm{\Omega }$ and a constant $\lambda >0$ for all $u\in J$ and $0<\breve{Q}<1$ satisfying

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{\Psi }_1\left(u\right)-{\Psi }_0\left(u\right)∥\le {\displaystyle \frac{\lambda \rho M}{1-\breve{Q}}}\upsilon \left(u\right),\hfill \\ \hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)∥\le {\displaystyle \frac{\lambda \rho M}{1-\breve{Q}}}\upsilon \left(u\right),\hfill \\ \hfill {\displaystyle \frac{}{\xi \left(u\right)}}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)\right)∥\le {\displaystyle \frac{\lambda \rho M}{1-\breve{Q}}}\upsilon \left(u\right),\hfill \\ \hfill {\displaystyle \frac{1}{\xi \left(u\right)}}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_0\left(u\right)\right)\right)∥\le {\displaystyle \frac{\lambda \rho M}{1-\breve{Q}}}\upsilon \left(u\right),\hfill \end{array}$$

where

$$\underset{u\in J}{sup}\frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)\xi \left(u\right)}\le M<\infty ,$$

which implies that the solution of the ϱ-Hilfer fractional snap dynamic system (10) is s-$\mathcal{U}.\mathcal{H}.\mathcal{R}$ stable.

Proof. 

Let the operator $T:\mathrm{\Omega }\to \mathrm{\Omega }$ be defined as in (19); as in Theorem 3, the metric ${\mathrm{d}}_2(·)$ and hypotheses 1 and 2 imply that

$$\begin{array}{cc}\hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥\left(T{\Psi }_1\right)\left(u\right)-\left(T\overline{{\Psi }_1}\right)\left(u\right)∥\le \breve{Q}N,\hfill \\ \hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)∥\le \breve{Q}N,\hfill \\ \hfill {\displaystyle \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\right)∥\le \breve{Q}N,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T\overline{{\Psi }_1}\left(u\right)\right)\right)∥\le \breve{Q}N,\hfill \end{array}$$

for $\breve{Q}\in (0,1)$. Hence, we conclude that

$${\mathrm{d}}_2\left(T{\Psi }_1,T\overline{{\Psi }_1}\right)\le \breve{Q}N=\breve{Q}{\mathrm{d}}_2\left({\Psi }_1,\overline{{\Psi }_1}\right),0<\breve{Q}<1.$$

Now, in view of the continuity and positiveness of the decreasing function $\upsilon \left(u\right),\forall u\in J$, we have $\frac{1}{\upsilon \left(u\right)}\le \lambda$, $\forall u\in J,\lambda >0$. In addition, from inequalities (30)–(33), we find

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\Psi }_1\right)\left(u\right)-\left(T{\Psi }_1\right)\left(u\right)\parallel }{\upsilon \left(u\right)\xi \left(u\right)}}\le \underset{u\in J}{sup}{\displaystyle \frac{\rho {(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1+1)\upsilon \left(u\right)\xi \left(u\right)}}=\lambda \rho M,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)∥\hfill \\ \hfill & \le \underset{u\in J}{sup}\frac{\rho {(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3+{\alpha }_2}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+{\alpha }_2+1)\upsilon \left(u\right)\xi \left(u\right)}=\lambda \rho M,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)-{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)∥\hfill \\ \hfill & \le \underset{u\in J}{sup}\frac{\rho {(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4+{\alpha }_3}}{\mathrm{\Gamma }({\alpha }_4+{\alpha }_3+1)\upsilon \left(u\right)\xi \left(u\right)}=\lambda \rho M,\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}\hfill \frac{1}{\upsilon \left(u\right)\xi \left(u\right)}& ∥{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }{\Psi }_1\left(u\right)\right)\right)\hfill \\ \hfill & -{}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_3,{\beta }_3;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_2,{\beta }_2;\varrho }\left({}^{\mathrm{H}}{\mathfrak{D}}_{a^{+}}^{{\alpha }_1,{\beta }_1;\varrho }T{\Psi }_1\left(u\right)\right)\right)∥\hfill \\ \hfill & \le \underset{u\in J}{sup}{\displaystyle \frac{\rho {(\varrho \left(u\right)-\varrho \left(a\right))}^{{\alpha }_4}}{\mathrm{\Gamma }({\alpha }_4+1)\upsilon \left(u\right)\xi \left(u\right)}}=\lambda \rho M.\hfill \end{array}$$

(37)

Thus, by (34)–(37), we obtain ${\mathrm{d}}_2({\Psi }_1,T{\Psi }_1)\le \lambda \rho M<\infty .$ Based on Theorem 1 (parts (i) and (ii)), there is one fixed point ${\Psi }_0$ i.e., $T{\Psi }_0={\Psi }_0$. Furthermore, according to Theorem 1 (part (iii)), we conclude that

$${\mathrm{d}}_2({\Psi }_1,{\Psi }_0)\le {\displaystyle \frac{1}{1-\breve{Q}}}{\mathrm{d}}_2(T{\Psi }_1,{\Psi }_1)\le {\displaystyle \frac{\lambda \rho M}{1-\breve{Q}}},0<\breve{Q}<1.$$

Hence, from the above conclusions, the solution of the $\varrho$-Hilfer fractional snap dynamic system (10) is s-$\mathcal{U}.\mathcal{H}.\mathcal{R}$ stable, and the proof is complete. □

5. Examples

Herein, we give some examples of the $\varrho$-Hilfer fractional snap dynamic system on unbounded domains $[a,\infty )$ based on numerical simulations to analyze their solutions. In these examples, we consider different cases of the function $\varrho$ to cover the Caputo, Caputo–Hadamard, and Katugampola versions.

Example 1.

Based on system (7), we consider a ϱ-Hilfer fractional snap dynamic system on the unbounded domains $J=[1.05,\infty )$ as

$$\left\{\begin{array}{c}{{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(u\right)={\Psi }_2\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }{\Psi }_2\left(u\right)={\Psi }_3\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{2}{15},\frac{11}{20};\varrho }{\Psi }_3\left(u\right)={\Psi }_4\left(u\right),\hfill \\ {{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{7}{20},\frac{13}{50};\varrho }{\Psi }_4\left(u\right)=\mathcal{G}\left(u,{\Psi }_1\left(u\right),{\Psi }_2\left(u\right),{\Psi }_3\left(u\right),{\Psi }_4\left(u\right)\right),\hfill \end{array}\right.$$

(38)

with

$$\begin{array}{cc}\hfill \mathcal{G}\left(u,{\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right)& ={\displaystyle \frac{\sqrt{u^2+1}}{47(2+\sqrt{u^2+1})}}+\frac{1}{exp(u^2+1)}(\frac{2\sqrt[3]{u+12}\left|{\Psi }_1\left(u\right)\right|}{47(0.5+|{\Psi }_1\left(u\right)\left|\right)}\hfill \\ \hfill & +\frac{3\left|u\right|{sin}^{-1}\left({\Psi }_2\left(u\right)\right)}{94}+\frac{u}{47}\left(\frac{{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}{\sqrt{5}+{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}\right)\hfill \\ \hfill & +\frac{2u}{47}\left(\frac{sin\left({\Psi }_4\left(u\right)\right)}{8+sin\left({\Psi }_4\left(u\right)\right)}\right)),\hfill \end{array}$$

(39)

for $u\in J$, $a=1.05\ge 0$, under the ϱ-Riemann–Liouville (R-L) fractional integral conditions:

$$\left\{\begin{array}{cc}{\displaystyle {\mathfrak{I}}_{a^{+}}^{\frac{167}{961};\varrho }{\Psi }_1\left(a\right)=-\frac{121}{16}{\mathfrak{I}}_{a^{+}}^{\frac{7}{8};\varrho }{\omega }_1\left(3.76\right),}\hfill & \\ {\displaystyle {\mathfrak{I}}_{a^{+}}^{\frac{37}{1000};\varrho }{\Psi }_2\left(a\right)=-\frac{72}{39}{\mathfrak{I}}_{a^{+}}^{\frac{3}{10};\varrho }{\omega }_2\left(2.41\right),}\hfill & \\ {\displaystyle {\mathfrak{I}}_{a^{+}}^{\frac{39}{100};\varrho }{\Psi }_3\left(a\right)=\frac{215}{108}{\mathfrak{I}}_{a^{+}}^{\frac{9}{13};\varrho }{\omega }_3\left(4.58\right),}\hfill & \\ {\displaystyle {\mathfrak{I}}_{a^{+}}^{\frac{481}{1000};\varrho }{\Psi }_4\left(a\right)=-\frac{83}{105}{\mathfrak{I}}_{a^{+}}^{\frac{3}{7};\varrho }{\omega }_4\left(5.93\right),}\hfill \end{array}\right.$$

(40)

where ${\omega }_i\left(u\right)=\frac{\sqrt{2}{u}^2}{5i(u^2+i)}$, $i=1,2,3,4$. One can check that

$${\alpha }_i=\left\{\frac{67}{90},\frac{43}{80},\frac{2}{15},\frac{7}{20}\right\}\subset (0,1],{\beta }_i=\left\{\frac{8}{25},\frac{23}{25},\frac{11}{20},\frac{13}{50}\right\}\subset [0,1],$$

$$\begin{array}{cc}\hfill {\alpha }_i\le {\gamma }_i& ={\alpha }_i+{\beta }_i-{\alpha }_i{\beta }_i=\left\{\begin{array}{cc}\frac{794}{961},\hfill & i=1,\hfill \\ \frac{963}{1000},\hfill & i=2,\hfill \\ \frac{61}{100},\hfill & i=3,\hfill \\ \frac{519}{1000},\hfill & i=4,\hfill \end{array}\right.1-{\gamma }_i=\left\{\begin{array}{cc}\frac{167}{961},\hfill & i=1,\hfill \\ \frac{37}{1000},\hfill & i=2,\hfill \\ \frac{39}{100},\hfill & i=3,\hfill \\ \frac{481}{1000},\hfill & i=4,\hfill \end{array}\right.\hfill \end{array}$$

$${\lambda }_i=\left\{-\frac{121}{16},-\frac{72}{39},\frac{215}{108},-\frac{83}{105}\right\}\subset \mathbb{R},{\theta }_i=\left\{\frac{7}{8},\frac{3}{10},\frac{9}{13},\frac{3}{7}\right\}\subset (0,\infty ),$$

and

$$\delta =\left\{3.76,2.41,4.58,5.93\right\}\subset J.$$

Thus, the ϱ-Hilfer fractional snap dynamic system (38)–(40) can be rewritten as follows:

$$\left\{\begin{array}{c}{\displaystyle {{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{7}{20},\frac{13}{50};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{2}{15},\frac{13}{50};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{8}{25};\varrho }{\Psi }_1\left(u\right)\right)\right)\right)={\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right),}\hfill \\ {\displaystyle {\mathfrak{I}}_{1.05}^{\frac{167}{961};\varrho }{\Psi }_1\left(1.05\right)=-\frac{121}{16}{\mathfrak{I}}_{1.05}^{\frac{7}{8};\varrho }{\omega }_1\left(3.76\right),}\hfill \\ {\mathfrak{I}}_{1.05}^{\frac{37}{1000};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(1.05\right)\right)=-\frac{72}{39}{\mathfrak{I}}_{1.05}^{\frac{3}{10};\varrho }{\omega }_2\left(2.41\right),\hfill \\ {\displaystyle {\mathfrak{I}}_{1.05}^{\frac{39}{100};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(1.05\right)\right)\right)=\frac{215}{108}{\mathfrak{I}}_{1.05}^{\frac{9}{13};\varrho }{\omega }_3\left(4.58\right),}\hfill \\ {\displaystyle {\mathfrak{I}}_{1.05}^{\frac{481}{1000};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{2}{15},\frac{13}{50};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(1.05\right)\right)\right)\right)}\hfill \\ {\displaystyle =-\frac{83}{105}{\mathfrak{I}}_{1.05}^{\frac{3}{7};\varrho }{\omega }_4\left(5.93\right),}\hfill \end{array}\right.$$

(41)

where

$$\begin{array}{cc}\hfill {\breve{\mathcal{G}}}_{{\Psi }_1}\left(u\right)=\mathcal{G}(u,& {\Psi }_1\left(u\right),{{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(u\right),{{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(u\right)\right),\hfill \\ \hfill & {{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{2}{15},\frac{13}{50};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{43}{80},\frac{23}{25};\varrho }\left({{}^{\mathrm{H}}\mathfrak{D}}_{1.05}^{\frac{67}{90},\frac{8}{25};\varrho }{\Psi }_1\left(u\right)\right)\right)).\hfill \end{array}$$

(42)

Now, we have

$$\begin{array}{cc}\hfill \left|\mathcal{G}\right(u,& {\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4)-\mathcal{G}\left(u,{\Psi }_1^{*},{\Psi }_2^{*},{\Psi }_3^{*},{\Psi }_4^{*}\right)|\hfill \\ \hfill & =|{\displaystyle \frac{\sqrt{u^2+1}}{47(2+\sqrt{u^2+1})}}+\frac{1}{exp(u^2+1)}({\displaystyle \frac{2\sqrt[3]{u+12}\left|{\Psi }_1\left(u\right)\right|}{47(0.5+|{\Psi }_1\left(u\right)\left|\right)}}\hfill \\ \hfill & +{\displaystyle \frac{3\left|u\right|{sin}^{-1}\left({\Psi }_2\left(u\right)\right)}{94}}+{\displaystyle \frac{u}{47}}{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}{\sqrt{5}+{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}}+{\displaystyle \frac{2u}{47}}{\displaystyle \frac{sin\left({\Psi }_4\left(u\right)\right)}{8+sin\left({\Psi }_4\left(u\right)\right)}})\hfill \\ \hfill & -[{\displaystyle \frac{\sqrt{u^2+1}}{47(2+\sqrt{u^2+1})}}+\frac{1}{exp(u^2+1)}({\displaystyle \frac{2\sqrt[3]{u+12}\left|{\Psi }_1^{*}\left(u\right)\right|}{47(0.5+|{\Psi }_1^{*}\left(u\right)\left|\right)}}\hfill \\ \hfill & +{\displaystyle \frac{3\left|u\right|{sin}^{-1}\left({\Psi }_2^{*}\left(u\right)\right)}{94}}+{\displaystyle \frac{u}{47}}{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)}{\sqrt{5}+{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)}}+{\displaystyle \frac{2u}{47}}{\displaystyle \frac{sin\left({\Psi }_4^{*}\left(u\right)\right)}{8+sin\left({\Psi }_4^{*}\left(u\right)\right)}}\left)\right]|\hfill \\ \hfill & \le {\displaystyle \frac{2|\sqrt[3]{u+12}|}{47exp(u^2+1)}}|{\displaystyle \frac{|{\Psi }_1\left(u\right)|}{0.5+|{\Psi }_1\left(u\right)|}}-{\displaystyle \frac{|{\Psi }_1^{*}\left(u\right)|}{0.5+|{\Psi }_1^{*}\left(u\right)|}}|\hfill \\ \hfill & +{\displaystyle \frac{3\left|u\right|}{94exp(u^2+1)}}\left|{sin}^{-1}\left({\Psi }_2\left(u\right)\right)-{sin}^{-1}\left({\Psi }_2^{*}\left(u\right)\right)\right|\hfill \\ \hfill & +{\displaystyle \frac{\left|u\right|}{47exp(u^2+1)}}|{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}{\sqrt{5}+{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}}-{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)}{\sqrt{5}+{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)}}|\hfill \\ \hfill & +{\displaystyle \frac{2\left|u\right|}{47exp(u^2+1)}}|{\displaystyle \frac{sin\left({\Psi }_4\left(u\right)\right)}{8+sin\left({\Psi }_4\left(u\right)\right)}}-{\displaystyle \frac{sin\left({\Psi }_4^{*}\left(u\right)\right)}{8+sin\left({\Psi }_4^{*}\left(u\right)\right)}}|\hfill \\ \hfill & \le \frac{1}{exp(u^2+1)}({\displaystyle \frac{2|\sqrt[3]{u+12}|}{47}}\left|{\Psi }_1\left(u\right)-{\Psi }_1^{*}\left(u\right)\right|+{\displaystyle \frac{3\left|u\right|}{94}}\left|{\Psi }_2\left(u\right)-{\Psi }_2^{*}\left(u\right)\right|\hfill \\ \hfill & +{\displaystyle \frac{\left|u\right|}{47}}\left|{\Psi }_3\left(u\right)-{\Psi }_3^{*}\left(u\right)\right|+{\displaystyle \frac{2\left|u\right|}{47}}\left|{\Psi }_4\left(u\right)-{\Psi }_4^{*}\left(u\right)\right|).\hfill \end{array}$$

So, we can choose $\xi \left(u\right)=47exp(u^2+1)$, ${\sigma }_1=2\left|\sqrt[3]{u+12}\right|$, ${\sigma }_2=\frac{3\left|u\right|}{2}$, and ${\sigma }_3=\left|u\right|$, ${\sigma }_4=2\left|u\right|$. Additionally, for $u\in J$,

$$\begin{array}{cc}\hfill {\omega }_i^{*}& =\underset{u\in [a,{\delta }_i]}{sup}\parallel {\omega }_i\left(u\right)\parallel =\left\{\begin{array}{cc}0.264157,\hfill & i=1,\hfill \\ 0.105197,\hfill & i=2,\hfill \\ 0.082484,\hfill & i=3,\hfill \\ 0.063488,\hfill & i=4.\hfill \end{array}\right.\hfill \end{array}$$

(43)

Now, we consider four cases for ϱ:

$${\varrho }_1\left(u\right)=2^u,{\varrho }_2\left(u\right)=u,{\varrho }_3\left(u\right)=ln(u+0.1),{\varrho }_4\left(u\right)=\sqrt{u}.$$

By using Equations (16)–(18), $\alpha ={\alpha }_4+{\alpha }_3+{\alpha }_2+{\alpha }_1\simeq 1.76527$, and by choosing suitable values of $\breve{Q}$ and $\breve{L}$ in the four cases, we have

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }\left[{\sigma }_1\left(u\right)+{\sigma }_2\left(u\right)+{\sigma }_3\left(u\right)+{\sigma }_4\left(u\right)\right]\right\}\hfill \\ \hfill & \simeq \left\{\begin{array}{cc}\hfill 0.004048,& {\varrho }_1\left(u\right)=2^u,\hfill \\ \hfill 0.004703,& {\varrho }_2\left(u\right)=u,\hfill \\ \hfill 0.000975,& {\varrho }_3\left(u\right)=ln(u+0.1),\hfill \\ \hfill 0.000363,& {\varrho }_4\left(u\right)=\sqrt{u},\hfill \end{array}\right\}\le \breve{Q}<1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}\left({\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_1-1}}{\mathrm{\Gamma }\left({\eta }_1\right)}}{\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\right.\right.\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_2-1}}{\mathrm{\Gamma }\left({\eta }_2\right)}}\left({\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_3-1}}{\mathrm{\Gamma }\left({\eta }_3\right)}}\left({\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\right)\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_4-1}}{\mathrm{\Gamma }\left({\eta }_4\right)}}\left({\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\right)+{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }∥{\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)∥\right)\right\}\hfill \\ \hfill & \simeq \left\{\begin{array}{cc}\hfill 0.0000026,& {\varrho }_1\left(u\right)=2^u,\hfill \\ \hfill 0.0000007,& {\varrho }_2\left(u\right)=u,\hfill \\ \hfill 0.0000009,& {\varrho }_3\left(u\right)=ln(u+0.1),\hfill \\ \hfill 0.0000000,& {\varrho }_4\left(u\right)=\sqrt{u},\hfill \end{array}\right.\le \breve{L}<\infty ,\hfill \end{array}$$

for $u\in J$. By considering $\breve{Q}=0.05\in (0,1)$ and $\breve{L}>0.05$, assumptions (HP${}_1$) and (HP${}_2$) hold.

Table 1. Suitable numerical values of $\breve{Q}$ and $\breve{L}$ for four cases of ${\varrho }_i\left(u\right)$.

	${\mathit{\varrho }}_1\left(\mathit{u}\right)=2^{\mathit{u}}$		${\mathit{\varrho }}_2\left(\mathit{u}\right)=\mathit{u}$		${\mathit{\varrho }}_3\left(\mathit{u}\right)=ln(\mathit{u}+0.1)$		${\mathit{\varrho }}_4\left(\mathit{u}\right)=\sqrt{\mathit{u}}$
$\mathit{u}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$
$1.05$	$0.000000$	$NaN$	$0.000000$	$NaN$	$0.000000$	$NaN$	$0.000000$	$NaN$
$1.55$	$0.004048$	$-0.004703$	$0.001550$	$-0.003246$	$0.000975$	$-0.002532$	$0.000363$	$-0.004722$
$2.05$	$0.003531$	$-0.000141$	$0.000950$	$-0.000164$	$0.000450$	$-0.000169$	$0.000192$	$-0.000287$
$2.55$	$0.001146$	$0.000002$	$0.000211$	$-0.000007$	$0.000079$	$-0.000010$	$0.000038$	$-0.000016$
$3.05$	$0.000186$	$0.000001$	$0.000023$	$0.000000$	$0.000007$	$0.000000$	$0.000004$	$-0.000001$
$3.55$	$0.000017$	$0.000000$	$0.000001$	$0.000000$	$0.000000$	$0.000000$	$0.000000$	$0.000000$

shows the numerical results of $\breve{Q}$ and $\breve{L}$ for $u\in [1.05,\infty )$. These values are also shown in Figure 1a,b for the four cases of ϱ. In all four cases of the function ϱ, we saw that all requirements of Theorem 2 were fulfilled. Therefore, this guaranteed that for all four cases of ϱ, the ϱ-Hilfer fractional snap dynamic system (38) admitted one solution on the unbounded domains $J=[1.05,\infty )$.

In the next example, we examine the correctness of the results obtained with Theorems 1 and 3. Of course, we shall consider the case in which $\varrho \left(u\right)=u$ (Caputo type) for three different orders ${\alpha }_1$ and show the obtained results computationally and graphically.

Example 2.

We consider a ϱ-Hilfer fractional snap dynamic system (38) on the domains $J=[1.05,\infty )$, $\varrho \left(u\right)=u$ (Caputo type), where

$$\begin{array}{cc}\hfill \mathcal{G}\left(u,{\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4\right)& ={\displaystyle \frac{|cos(\pi u\left)\right|}{39(\sqrt{5}+|cos\left(\pi u\right)\left|\right)}}+\frac{1}{exp\left(u\right)+\sqrt{3}}[{\displaystyle \frac{4\sqrt[3]{u}}{3+39\sqrt{\left|u\right|}}}\hfill \\ \hfill & \times \left({\displaystyle \frac{|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)|}{1.6+|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)|}}\right)+{\displaystyle \frac{u{\left({\Psi }_2\left(u\right)\right)}^2}{78(9+{\left({\Psi }_2\left(u\right)\right)}^2)}}\hfill \\ \hfill & +{\displaystyle \frac{\left({tan}^{-1}\left({\Psi }_3\left(u\right)\right)\right)}{11(39+2u^2)}}+{\displaystyle \frac{\sqrt{5}u}{39}}\left({\displaystyle \frac{sin\left({\Psi }_4\left(u\right)\right)}{\sqrt{5}+sin\left({\Psi }_4\left(u\right)\right)}}\right)],\hfill \end{array}$$

(44)

for $u\in J$, $a=1.05\ge 0$, and $\alpha \in \{\frac{2}{11},\frac{1}{2},\frac{19}{20}\}$, under the ϱ-Riemann–Liouville (R-L) fractional integral conditions (40), where ${\omega }_1\left(u\right)=\frac{\sqrt{2}}{3}\left|u\right|$, ${\omega }_2\left(u\right)=\frac{1}{5}u$, ${\omega }_3\left(u\right)=\frac{1}{4}\sqrt[3]{u}$ and ${\omega }_4\left(u\right)=\frac{1}{\sqrt{3}}{u}^2$. Clearly,

$$\begin{array}{cc}\hfill {}_k{\alpha }_1\le {\gamma }_1& =\left\{\begin{array}{cc}0.4436,\hfill & k=1,{}_1{\alpha }_1=\frac{2}{11}\hfill \\ 0.6600,\hfill & k=2,{}_2{\alpha }_1=\frac{1}{2},\hfill \\ 0.9660,\hfill & k=3,{}_3{\alpha }_1=\frac{19}{20},\hfill \end{array}\right.1-{\gamma }_1=\left\{\begin{array}{cc}0.5563,\hfill & {}_1{\alpha }_1=\frac{2}{11},\hfill \\ 0.3400,\hfill & {}_2{\alpha }_1=\frac{1}{2},\hfill \\ 0.0340,\hfill & {}_3{\alpha }_1=\frac{19}{20},\hfill \end{array}\right.\hfill \end{array}$$

and for $i=2,3,4$,

$$\begin{array}{cc}\hfill {\alpha }_i\le {\gamma }_i& ={\alpha }_i+{\beta }_i-{\alpha }_i{\beta }_i=\left\{\begin{array}{cc}\frac{963}{1000},\hfill & i=2,\hfill \\ \frac{61}{100},\hfill & i=3,\hfill \\ \frac{519}{1000},\hfill & i=4,\hfill \end{array}\right.1-{\gamma }_i=\left\{\begin{array}{cc}\frac{37}{1000},\hfill & i=2,\hfill \\ \frac{39}{100},\hfill & i=3,\hfill \\ \frac{481}{1000},\hfill & i=4.\hfill \end{array}\right.\hfill \end{array}$$

Now, we have

$$\begin{array}{cc}\hfill \left|\mathcal{G}\right(u,& {\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4)-\mathcal{G}\left(u,{\Psi }_1^{*},{\Psi }_2^{*},{\Psi }_3^{*},{\Psi }_4^{*}\right)|\hfill \\ \hfill & =\left|\frac{1}{exp\left(u\right)+\sqrt{3}}\right[{\displaystyle \frac{4\sqrt[3]{u}\left|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)\right|}{(3+39\sqrt{\left|u\right|})(1.6+|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)\left|\right)}}\hfill \\ \hfill & +{\displaystyle \frac{u{\left({\Psi }_2\left(u\right)\right)}^2}{78(9+{\left({\Psi }_2\left(u\right)\right)}^2)}}+{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3\left(u\right)\right)}{11(39+2u^2)}}\hfill \\ \hfill & +{\displaystyle \frac{\sqrt{5}u}{39}}\left({\displaystyle \frac{sin\left({\Psi }_4\left(u\right)\right)}{\sqrt{5}+sin\left({\Psi }_4\left(u\right)\right)}}\right)]\hfill \\ \hfill & -\frac{1}{exp\left(u\right)+\sqrt{3}}({\displaystyle \frac{4\sqrt[3]{u}\left|{sin}^{-1}\left({\Psi }_1^{*}\left(u\right)\right)\right|}{(3+39\sqrt{\left|u\right|})(1.6+|{sin}^{-1}\left({\Psi }_1^{*}\left(u\right)\right)\left|\right)}}\hfill \\ \hfill & +{\displaystyle \frac{u{\left({\Psi }_2^{*}\left(u\right)\right)}^2}{78(9+{\left({\Psi }_2^{*}\left(u\right)\right)}^2)}}+{\displaystyle \frac{{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)}{11(39+2u^2)}}\hfill \\ \hfill & +{\displaystyle \frac{\sqrt{5}u}{39}}\left({\displaystyle \frac{sin\left({\Psi }_4^{*}\left(u\right)\right)}{\sqrt{5}+{\left(sin\left({\Psi }_4^{*}\left(u\right)\right)\right)}^2}}\right)\left)\right|\hfill \\ \hfill & \le \frac{1}{exp\left(u\right)+\sqrt{3}}\left(\right|{\displaystyle \frac{4\sqrt[3]{u}}{3+39\sqrt{\left|u\right|}}}\left|\right|{\displaystyle \frac{|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)|}{1.6+|{sin}^{-1}\left({\Psi }_1\left(u\right)\right)|}}\hfill \\ \hfill & -{\displaystyle \frac{|{sin}^{-1}\left({\Psi }_1^{*}\left(u\right)\right)|}{1.6+|{sin}^{-1}\left({\Psi }_1^{*}\left(u\right)\right)|}}|\hfill \\ \hfill & +\left|{\displaystyle \frac{u}{78}}\right||{\displaystyle \frac{{\left({\Psi }_2\left(u\right)\right)}^2}{9+{\left({\Psi }_2\left(u\right)\right)}^2}}-{\displaystyle \frac{{\left({\Psi }_2^{*}\left(u\right)\right)}^2}{9+{\left({\Psi }_2^{*}\left(u\right)\right)}^2}}|\hfill \\ \hfill & +\left|{\displaystyle \frac{1}{11(39+2u^2)}}\right|\left|{tan}^{-1}\left({\Psi }_3\left(u\right)\right)-{tan}^{-1}\left({\Psi }_3^{*}\left(u\right)\right)\right|\hfill \\ \hfill & +\left|{\displaystyle \frac{\sqrt{5}u}{39}}\right||{\displaystyle \frac{sin\left({\Psi }_4\left(u\right)\right)}{\sqrt{5}+sin\left({\Psi }_4\left(u\right)\right)}}-{\displaystyle \frac{sin\left({\Psi }_4^{*}\left(u\right)\right)}{\sqrt{5}+sin\left({\Psi }_4^{*}\left(u\right)\right)}}|)\hfill \\ \hfill & \le \left|{\displaystyle \frac{4\frac{1}{exp\left(u\right)+\sqrt{3}}(\sqrt[3]{u}}{3+39\sqrt{\left|u\right|}}}\right|\left|{\Psi }_1\left(u\right)-{\Psi }_1^{*}\left(u\right)\right|+\left|{\displaystyle \frac{u}{78}}\right|\left|{\Psi }_2\left(u\right)-{\Psi }_2^{*}\left(u\right)\right|\hfill \\ \hfill & +\left|{\displaystyle \frac{1}{11(39+2u^2)}}\right|\left|{\Psi }_3\left(u\right)-{\Psi }_3^{*}\left(u\right)\right|+\left|{\displaystyle \frac{\sqrt{5}u}{39}}\right|\left|{\Psi }_4\left(u\right)-{\Psi }_4^{*}\left(u\right)\right|).\hfill \end{array}$$

Indeed, we can consider $\xi \left(u\right)=78(exp\left(u\right)+\sqrt{3})$ and

$${\sigma }_1=\frac{312|\sqrt[3]{u}|}{3+39\sqrt{\left|u\right|}},{\sigma }_2=\left|u\right|,{\sigma }_3=\frac{78}{11(39+2u^2)},{\sigma }_4=2\sqrt{5}\left|u\right|.$$

Additionally, we can consider ${\omega }_i^{*}$, which was defined by (40) in the previous example. Now, we consider three cases: ${\alpha }_1\in \{\frac{2}{11},\frac{1}{2},\frac{19}{20}\}$. By using Equations (16)–(18), we obtain

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }\left[{\sigma }_1\left(u\right)+{\sigma }_2\left(u\right)+{\sigma }_3\left(u\right)+{\sigma }_4\left(u\right)\right]\right\}\hfill \\ \hfill & \simeq \left\{\begin{array}{cc}0.0207,\hfill & {}_1{\alpha }_1=\frac{2}{11},\hfill \\ 0.0204,\hfill & {}_2{\alpha }_1=\frac{1}{2},\hfill \\ 0.0186,\hfill & {}_3{\alpha }_1=\frac{19}{20},\hfill \end{array}\right\}\le \breve{Q}<1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{u\in J}{sup}& \left\{{\displaystyle \frac{1}{\xi \left(u\right)}}\left({\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_1-1}}{\mathrm{\Gamma }\left({\eta }_1\right)}}\left({\displaystyle \frac{|{\lambda }_1|{\omega }_1^{*}{(\varrho \left({\delta }_1\right)-\varrho \left(a\right))}^{{\theta }_1}}{\mathrm{\Gamma }({\theta }_1+1)}}\right)\right.\right.\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_2-1}}{\mathrm{\Gamma }\left({\eta }_2\right)}}\left({\displaystyle \frac{|{\lambda }_2|{\omega }_2^{*}{(\varrho \left({\delta }_2\right)-\varrho \left(a\right))}^{{\theta }_2}}{\mathrm{\Gamma }({\theta }_2+1)}}\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_3-1}}{\mathrm{\Gamma }\left({\eta }_3\right)}}\left({\displaystyle \frac{|{\lambda }_3|{\omega }_3^{*}{(\varrho \left({\delta }_3\right)-\varrho \left(a\right))}^{{\theta }_3}}{\mathrm{\Gamma }({\theta }_3+1)}}\right)\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{{(\varrho \left(u\right)-\varrho \left(a\right))}^{{\eta }_4-1}}{\mathrm{\Gamma }\left({\eta }_4\right)}}\left({\displaystyle \frac{|{\lambda }_4|{\omega }_4^{*}{(\varrho \left({\delta }_4\right)-\varrho \left(a\right))}^{{\theta }_4}}{\mathrm{\Gamma }({\theta }_4+1)}}\right)+{\mathfrak{I}}_{a^{+}}^{\alpha ;\varrho }∥{\breve{\mathcal{G}}}_{{\Psi }_1}\left(0\right)∥\right)\right\}\hfill \\ \hfill & \simeq \left\{\begin{array}{cc}0.000001,\hfill & {}_1{\alpha }_1=\frac{2}{11},\hfill \\ 0.000001,\hfill & {}_2{\alpha }_1=\frac{1}{2},\hfill \\ 0.000011,\hfill & {}_3{\alpha }_1=\frac{19}{20}.\hfill \end{array}\right.\le \breve{L}<\infty .\hfill \end{array}$$

We can consider the value of $\breve{Q}$ to belong to $(0,1)$ whenever $u\in (1,4.05)$, $(1,4.05)$, and $(1,3.55)$ for ${}_i{\alpha }_1=\frac{2}{11},\frac{1}{2},\frac{19}{20}$, respectively. Thus,(HP${}_1$) and (HP${}_2$) hold for the three cases of ${\alpha }_1$.

Table 2. Suitable numerical values of $\breve{Q}$ and $\breve{L}$ for the three case of ${\alpha }_1$.

	${}_1{\mathit{\alpha }}_1=\frac{2}{11}$		${}_2{\mathit{\alpha }}_1=\frac{1}{2}$		${}_3{\mathit{\alpha }}_1=\frac{19}{20}$
$\mathit{u}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$	$\le \breve{\mathit{Q}}$	$\le \breve{\mathit{L}}$
$1.05$	$0.000000$	$NaN$	$0.000000$	$NaN$	$0.000000$	$NaN$
$2.05$	$0.018888$	$-0.001894$	$0.015172$	$-0.002016$	$0.010312$	$-0.001726$
$3.05$	$0.020704$	$-0.000248$	$0.020433$	$-0.000236$	$0.018668$	$-0.000159$
$4.05$	$0.014720$	$-0.000040$	$0.016335$	$-0.000030$	$0.017674$	$-0.000004$
$5.05$	$0.008707$	$-0.000005$	$0.010489$	$0.000000$	$0.012776$	$0.000011$

shows these results. These results are also plotted in Figure 2. In all three cases in the order $q_i$, we see that all requirements of Theorems 1 and 4 are fulfilled. Therefore, this guarantees that for all three different cases, in terms of the order ${\alpha }_1$, the solution of the ϱ-Hilfer fractional snap dynamic system (38)–(40) on the domains $J=[1.05,\infty )$ with (44) is $\mathcal{U}.\mathcal{H}.\mathcal{R}$ stable and, accordingly, is $\mathcal{U}.\mathcal{H}$ stable.

6. Conclusions

In this paper, we defined a new fractional mathematical model of a dynamic system consisting of a $\varrho$-Hilfer fractional snap equation with the $\varrho$-R-L fractional integral conditions in the framework of the generalized sequential $\varrho$-operators. To confirm the existence criterion, we used the valid fixed point theorems. In the end, we designed examples; by assuming different cases for the function $\varrho$ and order ${\alpha }_1$, we obtained numerical results for these two suggested $\varrho$-Hilfer fractional snap dynamic systems. It is notable that, by considering the different types of $\varrho \left(u\right)$, one can discuss the rest of the qualitative properties of such an extended $\varrho$-Hilfer fractional snap dynamic system by regarding other generalized models in future projects.