Stability analysis and solutions of fractional boundary value problem on the cyclopentasilane graph

Abstract

The study is being applied to a model involving silane and on cyclopentasilane graph. We consider a graph with labeled vertices by 0 or 1 inspired by the molecular structure of cyclopentasilane. In this paper, we first study the existence of solutions to fractional conformable boundary value problem on the cyclopentasilane graph by applying Scheafer and Krasnoselskii fixed point theorems. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stable, generalized Ulam-Hyers stable, Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable for the given problem. Finally, we give an example to support our important results.

1. Introduction

Silane is the general term for a series of compounds, namely silicon and hydrogen compounds. As a gas source providing silicon components, it can be used to make a range of silicon-containing substances such as monocrystalline silicon, polycrystalline silicon, metal silicide, silicon oxide and other silicon-containing substances. Because silane can achieve the highest purity, the most refined control, and the most flexible chemical reactions, it has become an important special gas that many other silicon sources cannot replace. Silane is the only intermediate product in the world to produce massively granular high purity silicon, which is widely used in microelectronics and optoelectronics industry [1], [2], specifically it is used for the manufacture of solar cells, flat-panel displays and so on [3], [4], [5], [6], [7]. Silane is also constantly appearing in high-tech applications, so silane has become the object of our attention.

As we all know, cyclopentasilane is a compound of silicon (Si) and hydrogen (H) with the chemical formula ${\mathrm{H}}_{\mathrm{10}}\mathrm{S}{\mathrm{i}}_{\mathrm{5}}$ (see Fig. 1). Cyclopentasilane can be used as a liquid silicon ink for printing silicon structures on integrated circuits or solar cells [8]. In 2016, Li et al. [9] examined the impact of ring conformation on the charge transport characteristics of cyclicpentasilane structures bound to gold electrodes in single molecule junctions. In 2021, Gerwig et al. [10] studied as an liquid precursor for the deposition of thin silicon films for printed electronics and related applications. The molecular structure of cyclopentasilane is a ring composed of five silicon atoms, which connect to two hydrogen atoms. In this paper, we research a class of integral-differential system involving fractional conformable derivatives on cyclopentasilane graph, which is a more common graph than a star graph.

Figure 1.

The cyclopentasilane graph is composed of 15 points and 15 edges.

For the past few years, fractional calculus has been widely put into use in chemistry, physics, biology, control, economics, aerodynamics and other fields. It is very suitable for describing materials and processes with memory and genetic characteristics, which has attracted increasing interest from researchers. Up to now, it has derived several results in many areas [11], [12], [13], [14], [15], [16], [17], [18]. In 2017, Jarad et al. [19] came up with a new fractional derivative called the fractional conformable derivative, which is given in (2.1). Some recent studies can be read in [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].

One integral part of the qualitative theory of linear and nonlinear differential equations is the stability analysis proposed by Ulam [30] and Hyers [31], known as Ulam-Hyers stability. Ulam-Hyers stability can be defined as an exact solution near the approximate solution for a fractional differential equation that minimizes the error. Furthermore, the analysis of Ulam-Hyers and Ulam-Hyers-Rassias stability for nonlinear fractional differential equations has become a hot research topic, making it one of the most significant subjects in mathematical analysis. The development of the stability theory for Ulam-Hyers and Ulam-Hyers-Rassias of fractional differential equations can be found in [32], [33], [34], [35], [36], [37], [38], [39], [40], [41].

On the other side, chemical graph theory [42], [43], [44] is a mathematical branch of studying the interaction of chemical networks, which can indicate any abstract or practical chemical system (i.e., molecular transformation in chemical reactions). In other words, chemical graph theory pays attention to every element of graph theory applied in chemistry. However, we concentrate on networks composed of line connection points. This network structure can be found in many real buildings in our surroundings. Given the graph structure of these networks, many researchers have studied the mathematical models on graphs, which are depicted by ordinary or fractional differential equations. Recently, there have been many theoretical and practical developments in fractional differential equations on graphs [45], [46], [47], [48], [49], [50], [51], [52].

In 1980, Lumer [53] first put forward the differential equation theory on the graph, and used local operators defined on branching spaces to study general evolution equations. Nicaise [54] investigated the propagation of nerve impulses by using a similar construct. In 1989, the boundary value problem of linear differential equations on geometric graph was studied by Zavgorodnii et al. [55], and the solutions to the problem on above graph were coordinated over internal vertices. They yielded a self-adjoint criterion by constructing an adjoint boundary value problem. In 2008, Gordeziani et al. [56] adopted a double-scan method to obtain the existence and uniqueness of solutions to ordinary differential equations on graph and proposed a numerical method. In 2014, Graef et al. [45] first explored the existence and uniqueness results on the three-vertex-star graph (see Fig. 2) by using fixed point theory. The system of nonlinear differential equations defined on any edge $m_j=\overrightarrow{n_j{n}_0}$ is as follows:

$$\{\begin{array}{c}-{\mathfrak{D}}_{0+}^{\gamma }{v}_j(t)={\mathfrak{p}}_j(t){\mathfrak{g}}_j(t,v_j(t)),0<t<l_j,j=1,2,\hfill \\ v_1(0)=v_2(0)=0,\hfill \\ v_1(l_1)=v_2(l_2),{\mathfrak{D}}_{0+}^{\alpha }{v}_1(l_1)+{\mathfrak{D}}_{0+}^{\alpha }{v}_2(l_2)=0,\hfill \end{array}$$

where $1<\gamma <2$ and $0<\alpha <\gamma$, ${\mathfrak{p}}_j\in C[0,l_j]$ with ${\mathfrak{p}}_j(t)\ne 0$ on $[0,l_j]$ and ${\mathfrak{g}}_j\in C([0,l_j]\times \mathbb{R},\mathbb{R})$, $j=1,2$. ${\mathfrak{D}}_{0+}^{\gamma }$ and ${\mathfrak{D}}_{0+}^{\alpha }$ denote Riemann-Liouville fractional derivatives. In 2020, Etemad et al. [46] studied the fractional boundary value problem on the ethane graph and obtained the existence of the solution. The equation system is as follows:

$$\{\begin{array}{c}{}^{\mathbf{C}}\mathcal{D}_0^{\alpha }{f}_j(x)=m_j(x,f_j(x),f_j^{\prime }(x)),x\in [0,1],j=1,2,\cdot \cdot \cdot ,7,\hfill \\ {\xi }_1{f}_j(0)+{\xi }_2{f}_j(1)={\xi }_3{\int }_0^1{f}_j(t)dt,{\xi }_1{f}_j^{\prime }(0)+{\xi }_2{f}_j^{\prime }(1)={\xi }_3{\int }_0^1{f}_j(t)dt,\hfill \end{array}$$

where $\alpha \in (1,2],{\xi }_1,{\xi }_2,{\xi }_3\in \mathbb{R}$, ${}^{\mathbf{C}}\mathcal{D}_0^{\alpha }$ is the Caputo fractional derivative and functions $m_j\in C([0,1]\times {\mathbb{R}}^2,\mathbb{R})$, $j=1,2,\cdot \cdot \cdot ,7$. Also they used 0 or 1 to mark the vertices of the ethane graph. The graph representation of the ethane compound has 7 edges with $\left|e_j\right|=1$.

Figure 2.

Star graph consisting of two edges and three points.

In 2022, Rezapour et al. [47] discussed the fractional multi-dimensional system of boundary value problems by using fixed point technique on the methylpropane graph and got the existence results:

$$\{\begin{array}{c}{}^{\mathbf{C}}\mathcal{D}_0^{\gamma }{u}_w(y)=f_w(y,u_w(y),u_w^{\prime }(y))+g_w(y,u_w(y),u_w^{\prime }(y)),y\in [0,1],\hfill \\ u_w(0){+}^{\mathbf{C}}{\mathcal{D}}_0^{\alpha }{u}_w(0)+u_w^{\prime }(1)=0,\hfill \\ {\int }_0^1[u_w(\theta ){+}^{\mathbf{C}}{\mathcal{D}}_0^{\alpha }{u}_w(\theta )+u_w^{\prime }(\theta )]d\theta =0,\hfill \end{array}$$

where ${}^{\mathbf{C}}\mathcal{D}_0^{\gamma }$ and ${}^{\mathbf{C}}\mathcal{D}_0^{\alpha }$ are the Caputo fractional derivatives of orders $1<\gamma <2$ and $0<\alpha <1$, respectively. $f_w,g_w\in C([0,1]\times {\mathbb{R}}^2,\mathbb{R})$, $w=1,2,\cdot \cdot \cdot ,13$, in which $w=13$ represents the total number of edges of methylpropane graph with $\left|l_w\right|=1$. Further, they also studied the Ulam-Hyers type stability for a special case of this multi-dimensional system.

The fixed point theorem is a very classic and excellent tool. When studying the qualitative theoretical problem of differential equations, the fixed point theorem is often used to study the existence and uniqueness of solutions. The method is a popular and effective tool, so we apply the fixed point theorem for our study. Furthermore, the research on graphs is mostly based on Riemann-Liouville and Caputo derivatives, whereas fractional conformable derivative can contain both of these derivatives. Inspired by this, we apply fractional conformable derivative to graphs, which will provide a new perspective for studying fractional differential equations on graphs. We investigate the stability and existence results of the following fractional integral-differential boundary value problem:

$$\{\begin{array}{c}{}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i(t)=h_i(t,u_i(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(t),u_j(t)),t\in [0,1],\hfill \\ {}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(1)=\lambda {\int }_0^1{u}_i(s)ds,\hfill \\ t^{\alpha (1-\beta )}{u}_i(t){|}_{t=0}=k,\hfill \end{array}$$ (1.1)

in which $1<\beta \le 2$, $1<\alpha \le 2$, $\lambda ,k\in \mathbb{R}$, $i=1,2,\cdot \cdot \cdot ,15$, $j=i+1$, $h_i\in C([0,1]\times {\mathbb{R}}^3,\mathbb{R})$. When $j=16$, $u_{16}=u_1$. ${}_0{}^{\beta }\mathcal{D}^{\alpha }$ and ${}_0{}^{\beta -1}\mathcal{D}^{\alpha }$ are the fractional conformable derivatives of Riemann-Liouville type of order β and $\beta -1$, α represents a free index. The cyclopentasilane graph has 15 edges with $\left|s_i\right|=1$. The integral boundary condition can be interpreted as the continuous distribution of the unknown function value on [0,1], and the fractional conformable derivative value of the unknown function at 1 is proportional to the integral boundary condition. It should be noted that the solutions of fractional boundary value problem (1.1) can be explained by diverse actual meanings of inorganic chemistry. That is, each solution $u_i(t)$ on any edge $s_i$ can represent the specific bond energy, bond polarity and bond strength etc. Our conclusions can provide theoretical support in terms of chemical reaction theory.

The rest of the article is arranged as follows. In Section 2, we first present graph representation of the cyclopentasilane molecule and mark its vertices by 0 or 1. Furthermore, some relevant knowledge on fractional calculus is reviewed. In Section 3, we give the existence results to fractional boundary value problem (1.1) by applying known Scheafer and Krasnoselskii fixed point theorems. In Section 4, we consider the Ulam stability for the system (1.1). In Section 5, we give an example to illustrate the validity of our results.

2. Preliminaries

Here, we introduce a new fractional differential system model on the cyclopentasilane graph. From the molecular structure graph of cyclopentasilane, the silicon and hydrogen atoms are taken as the vertices of this graph and the edges of this graph are the chemical bonds between the atoms. In addition, the cyclopentasilane graph contains more knots and has a more general structure than the star graph. To investigate the stability and existence results of fractional boundary value problem (1.1), we think the length of each edge as unit length and label vertices of the cyclopentasilane graph with 0 or 1. On this condition, we establish a local coordinate system on this cyclopentasilane graph, so that the interval between each edge of the graph is the unit length. The label of each vertex is obtained according to the orientation of corresponding edge. The start vertex is marked as 0 and the end vertex is marked as 1, when we move along the edge, and vice versa. So, a vertex may have two labels 0 and 1 at the same time. The origin of each edge is not certain, and it changes correspondingly when the orientation of moving along the edge changes.

According to this rule, we are free to determine any one of two vertices of the corresponding edge as the origin. Fig. 3 is a possible labeling case of cyclopentasilane graph. Now, we recall some related knowledge needed in our proofs later.

Definition 2.1

[19] The fractional conformable integral of Riemann-Liouville type of order $\beta \in \mathbb{C}$, $Re(\beta )>0$ is defined by

$${}_a{}^{\beta }\mathcal{J}^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{a}{\overset{t}{\int }}{\left(\frac{{(t-a)}^{\alpha }-{(x-a)}^{\alpha }}{\alpha }\right)}^{\beta -1}f(x)\frac{dx}{{(x-a)}^{1-\alpha }}.$$

Definition 2.2

[19] The fractional conformable derivative of Riemann-Liouville type of order $\beta \in \mathbb{C}$, $Re(\beta )>0$ is defined by

$${}_a{}^{\beta }\mathcal{D}^{\alpha }f(t)={}_a{}^{n}T^{\alpha }({}_a{}^{n-\beta }\mathcal{J}^{\alpha })f(t)=\frac{{}_a{}^{n}T^{\alpha }}{\mathrm{\Gamma }(n-\beta )}\underset{a}{\overset{t}{\int }}{\left(\frac{{(t-a)}^{\alpha }-{(x-a)}^{\alpha }}{\alpha }\right)}^{n-\beta -1}f(x)\frac{dx}{{(x-a)}^{1-\alpha }},$$ (2.1)

where $n=[Re(\beta )]+1$, ${}_a{}^{n}T^{\alpha }=\underset{ntimes}{\underbrace{{}_{a}T^{\alpha }{}_a{T}^{\alpha }{\cdots }_a{T}^{\alpha }}}$ and ${}_{a}T^{\alpha }$ is the conformable differential operator with ${}_{a}T^{\alpha }f(t)={(t-a)}^{1-\alpha }{f}^{\prime }(t)$ (see [57]).

Lemma 2.1

[19]Let$Re(\beta )>0$,$n=-[-Re(\beta )]$,$f\in L(a,b)$. Then

$${}_a{}^{\beta }\mathcal{J}^{\alpha }({}_a{}^{\beta }\mathcal{D}^{\alpha }f(t))=f(t)-\sum _{j=1}^n\frac{{}_a{}^{\beta -j}\mathcal{D}^{\alpha }f(a)}{{\alpha }^{\beta -j}\mathrm{\Gamma }(\beta -j+1)}{(t-a)}^{\alpha \beta -\alpha j}.$$

Lemma 2.2

[19]For$Re(n-\alpha )>0$, we have

$$\left[{}_a{}^{\beta }\mathcal{D}^{\alpha }{(x-a)}^{\alpha v-\alpha }\right](t)={\alpha }^{\beta }\frac{\mathrm{\Gamma }(v)}{\mathrm{\Gamma }(v-\beta )}{(t-a)}^{\alpha (v-\beta -1)}.$$

Lemma 2.3

[58]Assume that M is a Banach space and $\mathrm{\Psi }:\mathrm{M}\to \mathrm{M}$ is a completely continuous operator. Then either the set $\{v\in \mathrm{M}:v=\lambda \mathrm{\Psi }v,\lambda \in (0,1)\}$ is unbounded or the operator Ψ has a fixed point in M.

Lemma 2.4

[58]Assume that$\mathcal{A}$is a closed, bounded, convex, and nonempty subset of a Banach space M. Suppose ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ are two operators and satisfy ${\mathrm{\Psi }}_{1}u+{\mathrm{\Psi }}_{2}v\in \mathcal{A}$ whenever $u,v\in \mathcal{A}$, ${\mathrm{\Psi }}_1$ is compact and continuous and ${\mathrm{\Psi }}_2$ is a contraction map. Then there exists $w\in \mathcal{A}$ such that $w={\mathrm{\Psi }}_{1}w+{\mathrm{\Psi }}_{2}w$.

Lemma 2.5

[59]Let$\mathrm{\Psi }\subset C([0,1],\mathbb{R})$. Ψ is relatively compact if and only if Ψ is uniformly bounded and equicontinuous.

Figure 3.

The cyclopentasilane graph labeled with 0 or 1.

3. Existence

In this part, we demonstrate the existence results on the cyclopentasilane graph. Therefore, we take into account the Banach spaces ${\bar{\mathcal{X}}}_i=\{u_i:u_i,{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i\in C[0,1]\}$ with the norm ${\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i}=\underset{t\in [0,1]}{\sup }|u_i(t)|+\underset{t\in [0,1]}{\sup }{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|+\underset{t\in [0,1]}{\sup }|u_j(t)|$ for $i=1,2,\cdot \cdot \cdot ,15$, $j=i+1$. It should be noted that the product space $\bar{\mathcal{X}}=({\bar{\mathcal{X}}}_1,{\bar{\mathcal{X}}}_2,...,{\bar{\mathcal{X}}}_{15})$ equipped with the norm ${\Vert u=(u_1,u_2,...,u_{15})\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i}$ is a Banach space.

Theorem 3.1

Suppose that${\phi }_i\in C[0,1]$. Then$u_i^{⁎}$satisfy the boundary value problem

$$\{\begin{array}{c}{}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i(t)={\phi }_i(t),(t\in [0,1]),\hfill \\ {}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(1)=\lambda {\int }_0^1{u}_i(s)ds,\hfill \\ t^{\alpha (1-\beta )}{u}_i(t){|}_{t=0}=k,\hfill \end{array}$$ (3.1)

if and only if$u_i^{⁎}$satisfy the following fractional integral system:

$$u_i(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}ds,$$ (3.2)

where

$\mathrm{\Lambda }=\lambda -{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )[\alpha (\beta -1)+1]\ne 0$,

${\mathrm{\Lambda }}_1=\alpha (\beta -1)+1$,

${\mathrm{\Lambda }}_2=\lambda [\alpha (\beta -1)+1]$.

Proof

Assume that $u_i^{⁎}$ $(i=1,2,\cdot \cdot \cdot ,15)$ satisfy the above problem (3.1). By Lemma 2.1,

$$u_i^{⁎}(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}+a_1^{(i)}{t}^{\alpha (\beta -1)}+a_2^{(i)}{t}^{\alpha (\beta -2)}.$$ (3.3)

According to $t^{\alpha (1-\beta )}{u}_i^{⁎}(t){|}_{t=0}=k$, we get $a_2^{(i)}=0$. Combining (3.3) and Lemma 2.2, we obtain

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=\underset{0}{\overset{t}{\int }}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}+a_1^{(i)}{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta ).$$

Based on ${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(1)=\lambda {\int }_0^1{u}_i^{⁎}(s)ds$, we get

$$a_1^{(i)}=\frac{{\mathrm{\Lambda }}_1}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}ds.$$

Substituting the values of $a_1^{(i)}$ and $a_2^{(i)}$ into (3.3), we get the solution (3.2). On the contrary, assume that $u_i^{⁎}$ satisfy the equation (3.2), we obtain ${}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i^{⁎}(t)={\phi }_i(t)$ according to Lemma 2.2. By calculating, we get

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(1)=\frac{\lambda }{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}-\frac{{\alpha }^{(\beta -1)}{\mathrm{\Lambda }}_2}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\phi }_i(x)\frac{dx}{x^{1-\alpha }}ds=\lambda \underset{0}{\overset{1}{\int }}{u}_i^{⁎}(s)ds$$

and

$$u_i^{⁎}(t){|}_{t=0}=k{t}^{\alpha (\beta -1)}{|}_{t=0}.$$

So we can get that $u_i^{⁎}$ satisfy the problem (3.1). □

Based on Theorem 3.1, we define the operator $\mathrm{\Psi }:\bar{\mathcal{X}}\to \bar{\mathcal{X}}$ as follows:

$$\mathrm{\Psi }(u_1,u_2,...,u_{15})(t)=({\mathrm{\Psi }}_1(u_1,u_2,...,u_{15})(t),{\mathrm{\Psi }}_2(u_1,u_2,...,u_{15})(t),...,{\mathrm{\Psi }}_{15}(u_1,u_2,...,u_{15})(t)),$$

where

$${\mathrm{\Psi }}_i(u_1,u_2,...,u_{15})(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds,$$

for all $u_i\in {\bar{\mathcal{X}}}_i$ $(i=1,2,\cdot \cdot \cdot ,15)$ and $t\in [0,1]$. Put

$${\mathit{\eta }}_0^{⁎}=\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)},{\mathit{\eta }}_1^{⁎}=\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|},{\xi }_0^{⁎}=\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)},{\xi }_1^{⁎}=\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}.$$ (3.4)

Theorem 3.2

Suppose that$h_i\in C([0,1]\times {\mathbb{R}}^3,\mathbb{R})$satisfy the condition$|h_i(t,u_1,u_2,u_3)|\le L_i(i=1,2,\cdot \cdot \cdot ,15)$, where$L_i$are positive constants for all$t\in [0,1]$and$u_1,u_2,u_3\in \mathbb{R}$. Then the fractional boundary value problem(1.1)has a solution on the cyclopentasilane graph.

Proof

According to definition of the operator Ψ, we will prove that the fractional boundary value problem (1.1) has a solution if and only if Ψ has a fixed point on the product space $\bar{\mathcal{X}}={\bar{\mathcal{X}}}_1\times {\bar{\mathcal{X}}}_2\times \cdot \cdot \cdot \times {\bar{\mathcal{X}}}_{15}$.

Firstly, we demonstrate that the operator Ψ is completely continuous. Because of the continuity of the functions $h_i$, we know that the operator $\mathrm{\Psi }:\bar{\mathcal{X}}\to \bar{\mathcal{X}}$ is continuous.

Assume that $\mathcal{A}$ is a bounded subset of $\bar{\mathcal{X}}$ and $u=(u_1,u_2,...,u_{15})\in \mathcal{A}$. Then we have

$$|({\mathrm{\Psi }}_{i}u)(t)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{L}_i\frac{dx}{x^{1-\alpha }}ds\le L_i[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]=L_i{\mathit{\eta }}_0^{⁎},$$

for each $t\in [0,1]$, where ${\mathit{\eta }}_0^{⁎}$ is defined in (3.4). Also, we have

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_{i}u)(t)|\le \underset{0}{\overset{t}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \underset{0}{\overset{t}{\int }}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{L}_i\frac{dx}{x^{1-\alpha }}ds\le L_i[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]=L_i{\mathit{\eta }}_1^{⁎},$$

for each $t\in [0,1]$, in which ${\mathit{\eta }}_1^{⁎}$ is defined in (3.4). This means that ${\Vert {\mathrm{\Psi }}_{i}u\Vert }_{{\bar{\mathcal{X}}}_i}\le L_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})$. Therefore, ${\Vert \mathrm{\Psi }u\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert {\mathrm{\Psi }}_{i}u\Vert }_{{\bar{\mathcal{X}}}_i}\le \sum _{i=1}^{15}{L}_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})<\infty$, which indicates that the operator Ψ is uniformly bounded. Now, we demonstrate that the operator Ψ is equicontinuous.

Assume that $u=(u_1,u_2,...,u_{15})\in \mathcal{A}$ and $t_1,t_2\in [0,1]$ with $t_1<t_2$. Then we have

$$|({\mathrm{\Psi }}_{i}u)(t_2)-({\mathrm{\Psi }}_{i}u)(t_1)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{t_1}{\overset{t_2}{\int }}{\left(\frac{{t_2}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t_1}{\int }}[{\left(\frac{{t_2}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}-{\left(\frac{{t_1}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}]|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{[{t_2}^{\alpha (\beta -1)}-{t_1}^{\alpha (\beta -1)}]\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{[{t_2}^{\alpha (\beta -1)}-{t_1}^{\alpha (\beta -1)}]\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds.$$

Letting $t_1\to t_2$, the right-hand side of the inequality converges to 0.

Analogously, we have

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_{i}u)(t_2)-({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_{i}u)(t_1)|=|\underset{t_1}{\overset{t_2}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}|\le \underset{t_1}{\overset{t_2}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}.$$

Again, letting $t_1\to t_2$, the right-hand side of the inequality converges to 0. As a result, ${\Vert \mathrm{\Psi }u(t_2)-\mathrm{\Psi }u(t_1)\Vert }_{\bar{\mathcal{X}}}\to 0$ as $t_1\to t_2$, which implies that Ψ is an equicontinuous operator on the product space $\bar{\mathcal{X}}$. Because of Lemma 2.5, we deduce that Ψ is a completely continuous operator.

Here, thinking about the subset

$$\mathrm{\Omega }=\{(u_1,u_2,...,u_{15})\in \bar{\mathcal{X}}:(u_1,u_2,...,u_{15})=\mu \mathrm{\Psi }(u_1,u_2,...,u_{15}),\mu \in [0,1]\}$$

of $\bar{\mathcal{X}}$. We show that Ω is a bounded set. Assume that $(u_1,u_2,...,u_{15})\in \mathrm{\Omega }$, then $(u_1,u_2,...,u_{15})=\mu \mathrm{\Psi }(u_1,u_2,...,u_{15})$ and $u_i(t)=\mu {\mathrm{\Psi }}_i(u_1,u_2,...,u_{15})$ for all $t\in [0,1]$ and $i=1,2,\cdot \cdot \cdot ,15$. Therefore,

$$\begin{gathered} |u_i(t)|\le \mu [\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }} \\ +\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }} \\ +\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds]\le \mu [\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{L}_i\frac{dx}{x^{1-\alpha }} \\ +\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}\frac{dx}{x^{1-\alpha }}ds]\le \mu L_i[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]=\mu L_i{\mathit{\eta }}_0^{⁎} \end{gathered}$$

and

$$\begin{gathered} {|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|\le \mu [\underset{0}{\overset{t}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }} \\ +\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }} \\ +\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds]\le \mu [\underset{0}{\overset{t}{\int }}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{L}_i\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{L}_i\frac{dx}{x^{1-\alpha }}ds]\le \mu L_i[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]=\mu L_i{\mathit{\eta }}_1^{⁎}. \end{gathered}$$

This means that ${\Vert u\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le \mu \sum _{i=1}^{15}{L}_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})<\infty$, so Ω is bounded. Now, based on Lemma 2.3 and Theorem 3.1, the operator Ψ has a fixed point in $\bar{\mathcal{X}}$, which is a solution to the fractional boundary value problem (1.1) on the cyclopentasilane graph. □

Next, we recall the fractional boundary value problem (1.1) with diverse conditions.

Theorem 3.3

Suppose that$h_i\in C([0,1]\times {\mathbb{R}}^3,\mathbb{R})$, and there exist functions${\sigma }_i\in C([0,1],\mathbb{R}),{\delta }_i\in C([0,1],{\mathbb{R}}^{+})$and nondecreasing functions${\varphi }_i\in C([0,1],{\mathbb{R}}^{+})$, such that

$$|h_i(t,u_1,u_2,u_3)-h_i(t,v_1,v_2,v_3)|\le {\sigma }_i(t)(|u_1-v_1|+|u_2-v_2|+|u_3-v_3|)$$

and

$$|h_i(t,u_1,u_2,u_3)|\le {\delta }_i(t){\varphi }_i(\left|u_1\right|+\left|u_2\right|+\left|u_3\right|),$$

for all$u_1,u_2,u_3,v_1,v_2,v_3\in \mathbb{R}$,$t\in [0,1]$,$i=1,2,\cdot \cdot \cdot ,15$. If$k:=(2{\xi }_0^{⁎}+{\xi }_1^{⁎})\sum _{i=1}^{15}\Vert {\sigma }_i\Vert <1$, then the fractional boundary value problem(1.1)has a solution on the cyclopentasilane graph, in which$\Vert {\sigma }_i\Vert =\underset{t\in [0,1]}{\sup }|{\sigma }_i(t)|$and${\xi }_w^{⁎}(w=0,1)$are defined in equation(3.4).

Proof

Suppose $\Vert {\delta }_i\Vert =\underset{t\in [0,1]}{\sup }|{\delta }_i(t)|$ and take an appropriate constant ${\rho }_i$ that satisfy the condition

$${\rho }_i\ge \sum _{i=1}^{15}{\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})\Vert {\delta }_i\Vert \{2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎}\},$$ (3.5)

where ${\mathit{\eta }}_w^{⁎}(w=0,1)$ are given in (3.4). Take into account the set ${\mathcal{C}}_{{\rho }_i}:=\{u=(u_1,u_2,...,u_{15})\in \bar{\mathcal{X}}:{\Vert u\Vert }_{\bar{\mathcal{X}}}\le {\rho }_i\}$, where ${\rho }_i$ are defined in equation (3.5). It can be found that ${\mathcal{C}}_{{\rho }_i}$ is closed, bounded, convex and nonempty subset on the product Banach space $\bar{\mathcal{X}}$. Next, define the operators ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ on ${\mathcal{C}}_{{\rho }_i}$ by

$${\mathrm{\Psi }}_1(u_1,u_2,...,u_{15})(t):=({\mathrm{\Psi }}_1^{(1)}(u_1,u_2,...,u_{15})(t),{\mathrm{\Psi }}_1^{(2)}(u_1,u_2,...,u_{15})(t),...,{\mathrm{\Psi }}_1^{(15)}(u_1,u_2,...,u_{15})(t)),{\mathrm{\Psi }}_2(u_1,u_2,...,u_{15})(t):=({\mathrm{\Psi }}_2^{(1)}(u_1,u_2,...,u_{15})(t),{\mathrm{\Psi }}_2^{(2)}(u_1,u_2,...,u_{15})(t),...,{\mathrm{\Psi }}_2^{(15)}(u_1,u_2,...,u_{15})(t)),$$

where

$$({\mathrm{\Psi }}_1^{(i)}u)(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}$$

and

$$({\mathrm{\Psi }}_2^{(i)}u)(t)=\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds,$$

for all $u=(u_1,u_2,...,u_{15})\in {\mathcal{C}}_{{\rho }_i}$ and $t\in [0,1]$.

Let ${\varphi }_i^{⁎}=\underset{u_i\in {\bar{\mathcal{X}}}_i}{\sup }{\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})$, then for any $u=(u_1,u_2,...,u_{15})$, $v=(v_1,v_2,...,v_{15})\in {\mathcal{C}}_{{\rho }_i}$, we have

$$|({\mathrm{\Psi }}_1^{(i)}u+{\mathrm{\Psi }}_2^{(i)}v)(t)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\delta }_i(x){\varphi }_i(|u_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+|u_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{\delta }_i(x){\varphi }_i(|v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|v_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\delta }_i(x){\varphi }_i(|v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|v_j(x)|)\frac{dx}{x^{1-\alpha }}ds\le \Vert {\delta }_i\Vert {\varphi }_i^{⁎}[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]=\Vert {\delta }_i\Vert {\varphi }_i^{⁎}{\mathit{\eta }}_0^{⁎}$$

and

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_1^{(i)}u{+}_0^{\beta -1}{\mathcal{D}}^{\alpha }{\mathrm{\Psi }}_2^{(i)}v)(t)|\le \underset{0}{\overset{t}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \underset{0}{\overset{t}{\int }}{\delta }_i(x){\varphi }_i(|u_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+|u_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{\delta }_i(x){\varphi }_i(|v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|v_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\delta }_i(x){\varphi }_i(|v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|v_j(x)|)\frac{dx}{x^{1-\alpha }}ds\le \Vert {\delta }_i\Vert {\varphi }_i^{⁎}[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]=\Vert {\delta }_i\Vert {\varphi }_i^{⁎}{\mathit{\eta }}_1^{⁎}.$$

This means ${\Vert {\mathrm{\Psi }}_{1}u+{\mathrm{\Psi }}_{2}v\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert {\mathrm{\Psi }}_1^{(i)}u+{\mathrm{\Psi }}_2^{(i)}v\Vert }_{{\bar{\mathcal{X}}}_i}\le \sum _{i=1}^{15}\Vert {\delta }_i\Vert {\varphi }_i^{⁎}(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\le {\rho }_i$, so ${\Vert {\mathrm{\Psi }}_{1}u+{\mathrm{\Psi }}_{2}v\Vert }_{\bar{\mathcal{X}}}\le {\rho }_i$ and ${\mathrm{\Psi }}_{1}u+{\mathrm{\Psi }}_{2}v\in {\mathcal{C}}_{{\rho }_i}$. Furthermore, because of the continuity of functions $h_i$, we can get the continuity of ${\mathrm{\Psi }}_1$.

Now, we prove that the operator ${\mathrm{\Psi }}_1$ is uniformly bounded. Therefore, it should be noted that

$$|({\mathrm{\Psi }}_1^{(i)}u)(t)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\delta }_i(x){\varphi }_i(|u_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+|u_j(x)|)\frac{dx}{x^{1-\alpha }}\le \frac{1}{\mathrm{\Gamma }(\beta +1)}\Vert {\delta }_i\Vert {\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})$$

and

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_1^{(i)}u)(t)|\le \underset{0}{\overset{t}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}\le \underset{0}{\overset{t}{\int }}{\delta }_i(x){\varphi }_i(|u_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+|u_j(x)|)\frac{dx}{x^{1-\alpha }}\le \frac{1}{\alpha }\Vert {\delta }_i\Vert {\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i}),$$

for all u in ${\mathcal{C}}_{{\rho }_i}$. So, ${\Vert {\mathrm{\Psi }}_{1}u\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert {\mathrm{\Psi }}_1^{(i)}u\Vert }_{{\bar{\mathcal{X}}}_i}\le \{\frac{2}{\mathrm{\Gamma }(\beta +1)}+\frac{1}{\alpha }\}\sum _{i=1}^{15}\Vert {\delta }_i\Vert {\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})$. This shows that the operator ${\mathrm{\Psi }}_1$ is uniformly bounded on ${\mathcal{C}}_{{\rho }_i}$.

Now, we demonstrate that the operator ${\mathrm{\Psi }}_1$ is compact on ${\mathcal{C}}_{{\rho }_i}$. Assume that $t_1,t_2\in [0,1]$ with $t_1<t_2$. Then we have

$$|({\mathrm{\Psi }}_1^{(i)}u)(t_2)-({\mathrm{\Psi }}_1^{(i)}u)(t_1)|=|\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t_2}{\int }}{\left(\frac{{t_2}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t_1}{\int }}{\left(\frac{{t_1}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{t_1}{\overset{t_2}{\int }}{\left(\frac{{t_2}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t_1}{\int }}[{\left(\frac{{t_2}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}-{\left(\frac{{t_1}^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}]|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}\le \frac{1}{\mathrm{\Gamma }(\beta +1)}\Vert {\delta }_i\Vert {\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})[{\left(\frac{{t_2}^{\alpha }}{\alpha }\right)}^{\beta }-{\left(\frac{{t_1}^{\alpha }}{\alpha }\right)}^{\beta }].$$

Hence, letting $t_1\to t_2$, the right-hand side of the inequality converges to 0. And we have

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_1^{(i)}u)(t_2)-({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_1^{(i)}u)(t_1)|=|\underset{0}{\overset{t_2}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\underset{0}{\overset{t_1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}|\le \underset{t_1}{\overset{t_2}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}\le \Vert {\delta }_i\Vert {\varphi }_i({\Vert u_i\Vert }_{{\bar{\mathcal{X}}}_i})\left(\frac{{t_2}^{\alpha }-{t_1}^{\alpha }}{\alpha }\right),$$

and letting $t_1\to t_2$, the right-hand side of the inequality converges to 0. As a result, ${\Vert ({\mathrm{\Psi }}_{1}u)(t_2)-({\mathrm{\Psi }}_{1}u)(t_1)\Vert }_{\bar{\mathcal{X}}}$ tends to zero as $t_1\to t_2$. Then ${\mathrm{\Psi }}_1$ is equicontinuous. Further we get that ${\mathrm{\Psi }}_1$ is a relatively compact operator on ${\mathcal{C}}_{{\rho }_i}$. Because of Lemma 2.5, we deduce that the operator ${\mathrm{\Psi }}_1$ is compact on ${\mathcal{C}}_{{\rho }_i}$.

Next, we show that the operator ${\mathrm{\Psi }}_2$ is a contraction. Assume that $u,v\in {\mathcal{C}}_{{\rho }_i}$, then we have

$$|({\mathrm{\Psi }}_2^{(i)}u)(t)-({\mathrm{\Psi }}_2^{(i)}v)(t)|\le \frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))-h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))-h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{\sigma }_i(x)(|u_i(x)-v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|u_j(x)-v_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\sigma }_i(x)(|u_i(x)-v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|u_j(x)-v_j(x)|)\frac{dx}{x^{1-\alpha }}ds\le \Vert {\sigma }_i\Vert {\Vert u_i-v_i\Vert }_{{\bar{\mathcal{X}}}_i}[\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]=\Vert {\sigma }_i\Vert {\Vert u_i-v_i\Vert }_{{\bar{\mathcal{X}}}_i}{\xi }_0^{⁎}$$

and

$$|({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_2^{(i)}u)(t)-({}_0{}^{\beta -1}\mathcal{D}^{\alpha }{\mathrm{\Psi }}_2^{(i)}v)(t)|\le \frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))-h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))-h_i(x,v_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{v}_i(x),v_j(x))|\frac{dx}{x^{1-\alpha }}ds\le \frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}{\sigma }_i(x)(|u_i(x)-v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|u_j(x)-v_j(x)|)\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{\sigma }_i(x)(|u_i(x)-v_i(x)|+{|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{v}_i(x)|+|u_j(x)-v_j(x)|)\frac{dx}{x^{1-\alpha }}ds\le \Vert {\sigma }_i\Vert {\Vert u_i-v_i\Vert }_{{\bar{\mathcal{X}}}_i}\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}=\Vert {\sigma }_i\Vert {\Vert u_i-v_i\Vert }_{{\bar{\mathcal{X}}}_i}{\xi }_1^{⁎}.$$

Thus, we get ${\Vert {\mathrm{\Psi }}_{2}u-{\mathrm{\Psi }}_{2}v\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert {\mathrm{\Psi }}_2^{(i)}u-{\mathrm{\Psi }}_2^{(i)}v\Vert }_{{\bar{\mathcal{X}}}_i}\le (2{\xi }_0^{⁎}+{\xi }_1^{⁎})\sum _{i=1}^{15}\Vert {\sigma }_i\Vert {\Vert u_i-v_i\Vert }_{{\bar{\mathcal{X}}}_i}$, and ${\Vert {\mathrm{\Psi }}_{2}u-{\mathrm{\Psi }}_{2}v\Vert }_{\bar{\mathcal{X}}}\le k{\Vert u-v\Vert }_{\bar{\mathcal{X}}}$. Since $k<1$, ${\mathrm{\Psi }}_2$ is a contraction on ${\mathcal{C}}_{{\rho }_i}$. Now, based on Lemma 2.4, we deduce that the operator Ψ has a fixed point, which is a solution of the fractional boundary value problem (1.1) on the cyclopentasilane graph. □

4. Ulam stability

In this part, we discuss the Ulam stability of system (1.1) on the cyclopentasilane graph, namely Ulam-Hyers stable, generalized Ulam-Hyers stable, Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable.

Definition 4.1

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the condition

$${|}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)-h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))|<{\epsilon }_i,$$ (4.1)

and $u=(u_1,u_2,...,u_{15})$ is the solution of (1.1). If there is a constant $C\in {\mathbb{R}}^{+}$ such that

$$|u^{⁎}(t)-u(t)|\le C\epsilon ,t\in [0,1],$$

for each $\epsilon =\epsilon ({\epsilon }_1,{\epsilon }_2,...,{\epsilon }_{15})>0$, then the system (1.1) is called as Ulam-Hyers stable.

Definition 4.2

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the condition (4.1), and $u=(u_1,u_2,...,u_{15})$ is the solution of (1.1). If there is a function $\mathrm{\Psi }\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ with $\mathrm{\Psi }(0)=0$, such that

$$|u^{⁎}(t)-u(t)|\le \mathrm{\Psi }(\epsilon ),t\in [0,1],$$

for each $\epsilon =\epsilon ({\epsilon }_1,{\epsilon }_2,...,{\epsilon }_{15})>0$, then the system (1.1) is called as generalized Ulam-Hyers stable.

Definition 4.3

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the condition

$${|}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)-h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))|<{\varphi }_i(t){\epsilon }_i,$$ (4.2)

and $u=(u_1,u_2,...,u_{15})$ is the solution of (1.1). If there is a constant $C_{\varphi }\in {\mathbb{R}}^{+}$ such that

$$|u^{⁎}(t)-u(t)|\le C_{\varphi }\varphi (t)\epsilon ,t\in [0,1],$$

for each $\epsilon =\epsilon ({\epsilon }_1,{\epsilon }_2,...,{\epsilon }_{15})>0$, then the system (1.1) is called as Ulam-Hyers-Rassias stable with respect to $\varphi =\varphi ({\varphi }_1,{\varphi }_2,...,{\varphi }_{15})\in C([0,1],{\mathbb{R}}^{+})$.

Definition 4.4

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the condition

$${|}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)-h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))|<{\mathrm{\Phi }}_i(t),$$ (4.3)

and $u=(u_1,u_2,...,u_{15})$ is the solution of (1.1). If there is a constant $C_{\mathrm{\Phi }}\in {\mathbb{R}}^{+}$ such that

$$|u^{⁎}(t)-u(t)|\le C_{\mathrm{\Phi }}\mathrm{\Phi }(t),t\in [0,1],$$

then the system (1.1) is called as generalized Ulam-Hyers-Rassias stable with respect to $\mathrm{\Phi }=\mathrm{\Phi }({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,...,{\mathrm{\Phi }}_{15})\in C([0,1],{\mathbb{R}}^{+})$.

Remark 4.1

We call that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ is the solution of (4.1), if there are functions $M_i\in C([0,1],\mathbb{R})$, which depend on $u_i^{⁎}(t)(i=1,2,\cdot \cdot \cdot ,15)$, such that

$$\text{(1) }|M_i(t)|\le {\epsilon }_i,t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,{\text{(2) }}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+M_i(t),t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,j=i+1.$$

Remark 4.2

We call that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ is the solution of (4.2), if there are functions $N_i\in C([0,1],\mathbb{R})$, which depend on $u_i^{⁎}(t)(i=1,2,\cdot \cdot \cdot ,15)$, such that

$$\text{(1) }|N_i(t)|\le {\varphi }_i(t){\epsilon }_i,t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,{\text{(2) }}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+N_i(t),t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,j=i+1.$$

Remark 4.3

We call that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ is the solution of (4.3), if there are functions $P_i\in C([0,1],\mathbb{R})$, which depend on $u_i^{⁎}(t)(i=1,2,\cdot \cdot \cdot ,15)$, such that

$$\text{(1) }|P_i(t)|\le {\mathrm{\Phi }}_i(t),t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,{\text{(2) }}_0^{\beta }{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+P_i(t),t\in [0,1],i=1,2,\cdot \cdot \cdot ,15,j=i+1.$$

We now analyze the Ulam stability for the fractional boundary value problem (1.1) on the cyclopentasilane graph.

Theorem 4.1

Assume that$h_i\in C([0,1]\times {\mathbb{R}}^3,\mathbb{R})$,$u^{⁎}$is the solution of(4.1), and there are nonnegative bounded integrable functions${\mathcal{L}}_i,{\mathcal{G}}_i,{\mathcal{P}}_i$, such that

$$|h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))-h_i(t,u_i(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(t),u_j(t))|\le {\mathcal{L}}_i(t)|u_i^{⁎}(t)-u_i(t)|+{\mathcal{G}}_i(t){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|+{\mathcal{P}}_i(t)|u_j^{⁎}(t)-u_j(t)|.$$

Then, on the cyclopentasilane graph, the fractional boundary value problem(1.1)is Ulam-Hyers stable and generalized Ulam-Hyers stable if

$$({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})<1,i=1,2,\cdot \cdot \cdot ,15,$$

in which${\mathcal{L}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{L}}_i(t)|$,${\mathcal{G}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{G}}_i(t)|$,${\mathcal{P}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{P}}_i(t)|$and${\mathit{\eta }}_w^{⁎}(w=0,1)$are defined in equation(3.4).

Proof

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the inequality (4.1). According to Remark 4.1, there exist $M_i(t)\in C([0,1],\mathbb{R})$ such that

$${}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+M_i(t),$$

with

$$|M_i(t)|\le {\epsilon }_i.$$

Then we consider the following problem:

$$\{\begin{array}{c}{}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+M_i(t),t\in [0,1],\hfill \\ {}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(1)=\lambda {\int }_0^1{u}_i^{⁎}(s)ds,\hfill \\ t^{\alpha (1-\beta )}{u}_i^{⁎}(t){|}_{t=0}=k.\hfill \end{array}$$

Based on Theorem 3.1, we can obtain

$$u_i^{⁎}(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}ds$$

and

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=\underset{0}{\overset{t}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{\alpha }^{(\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+M_i(x)]\frac{dx}{x^{1-\alpha }}ds.$$

Combining (1.1) and Theorem 3.1, we can get

$$u_i(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds$$

and

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(t)=\underset{0}{\overset{t}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{\alpha }^{(\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds.$$

Then we have

$$|u_i^{⁎}(t)-u_i(t)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}ds\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}ds+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}ds\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]+{\epsilon }_i[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}{\mathit{\eta }}_0^{⁎}+{\epsilon }_i{\mathit{\eta }}_0^{⁎}$$

and

$${|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|\le \underset{0}{\overset{t}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds+\underset{0}{\overset{t}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}ds\le \underset{0}{\overset{t}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_j^{⁎}(x)-u_j(x)|]\frac{dx}{x^{1-\alpha }}ds+\underset{0}{\overset{t}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|M_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|M_i(x)|\frac{dx}{x^{1-\alpha }}ds\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]+{\epsilon }_i[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]\le ({\mathcal{L}}_i^{⁎}+{\mathcal{P}}_i^{⁎}+{\mathcal{G}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}{\mathit{\eta }}_1^{⁎}+{\epsilon }_i{\mathit{\eta }}_1^{⁎}.$$

Hence,

$${\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})+{\epsilon }_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎}).$$

From above, we obtain

$${\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le \frac{{\epsilon }_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}.$$

Now, by taking $\epsilon =\max \{{\epsilon }_1,{\epsilon }_2,...,{\epsilon }_{15}\}$, we get

$${\Vert u^{⁎}-u\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le \sum _{i=1}^{15}\frac{\epsilon (2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}.$$

Thus, we take

$$C=\sum _{i=1}^{15}\frac{(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{G}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})},$$

then, on the cyclopentasilane graph, the fractional boundary value problem (1.1) is Ulam-Hyers stable by Definition 4.1. Further, we take

$$\mathrm{\Psi }(\epsilon )=\sum _{i=1}^{15}\frac{\epsilon (2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})},$$

with

$$\mathrm{\Psi }(0)=0,$$

then, on the cyclopentasilane graph, we conclude that the fractional boundary value problem (1.1) is generalized Ulam-Hyers stable by Definition 4.2. □

Theorem 4.2

Assume that$h_i\in C([0,1]\times {\mathbb{R}}^3,\mathbb{R})$,$u^{⁎}$is the solution of(4.2), and there are nonnegative bounded integrable functions${\mathcal{L}}_i,{\mathcal{G}}_i,{\mathcal{P}}_i$, such that

$$|h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))-h_i(t,u_i(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(t),u_j(t))|\le {\mathcal{L}}_i(t)|u_i^{⁎}(t)-u_i(t)|+{\mathcal{G}}_i(t){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|+{\mathcal{P}}_i(t)|u_j^{⁎}(t)-u_j(t)|.$$

Then, on the cyclopentasilane graph, the fractional boundary value problem(1.1)is Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable with respect to ϕ if

$$({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})<1,i=1,2,\cdot \cdot \cdot ,15,$$

in which${\mathcal{L}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{L}}_i(t)|$,${\mathcal{G}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{G}}_i(t)|$,${\mathcal{P}}_i^{⁎}=\underset{t\in [0,1]}{\sup }|{\mathcal{P}}_i(t)|$and${\mathit{\eta }}_w^{⁎}(w=0,1)$are defined in equation(3.4).

Proof

Suppose that $u^{⁎}=(u_1^{⁎},u_2^{⁎},...,u_{15}^{⁎})$ satisfies the inequality (4.2). According to Remark 4.2, there exist $N_i(t)\in C([0,1],\mathbb{R})$ such that

$${}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+N_i(t),$$

with

$$|N_i(t)|\le {\varphi }_i(t){\epsilon }_i.$$

Then we consider the following problem:

$$\{\begin{array}{c}{}_0{}^{\beta }\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=h_i(t,u_i^{⁎}(t),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t),u_j^{⁎}(t))+N_i(t),t\in [0,1],\hfill \\ {}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(1)=\lambda {\int }_0^1{u}_i^{⁎}(s)ds,\hfill \\ t^{\alpha (1-\beta )}{u}_i^{⁎}(t){|}_{t=0}=k.\hfill \end{array}$$

Based on Theorem 3.1, we can obtain

$$u_i^{⁎}(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}ds$$

and

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(t)=\underset{0}{\overset{t}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{\alpha }^{(\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))+N_i(x)]\frac{dx}{x^{1-\alpha }}ds.$$

Combining (1.1) and Theorem 3.1, we can get

$$u_i(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{t}^{\alpha (\beta -1)}}{\mathrm{\Lambda }\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds$$

and

$${}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(t)=\underset{0}{\overset{t}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}+\frac{{\mathrm{\Lambda }}_1{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}-\frac{{\mathrm{\Lambda }}_2{\alpha }^{(\beta -1)}}{\mathrm{\Lambda }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}{h}_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))\frac{dx}{x^{1-\alpha }}ds.$$

Then we have

$$|u_i^{⁎}(t)-u_i(t)|\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}ds\le \frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}ds+\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{\left(\frac{t^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|t^{\alpha (\beta -1)}}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta )}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}ds\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]+{\varphi }_i(t){\epsilon }_i[\frac{1}{\mathrm{\Gamma }(\beta +1)}+\frac{\left|{\mathrm{\Lambda }}_1\right|}{\left|\mathrm{\Lambda }\right|}+\frac{\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|\mathrm{\Gamma }(\beta +1)}]\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}{\mathit{\eta }}_0^{⁎}+{\varphi }_i(t){\epsilon }_i{\mathit{\eta }}_0^{⁎}$$

and

$${|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(t){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(t)|\le \underset{0}{\overset{t}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|h_i(x,u_i^{⁎}(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i^{⁎}(x),u_j^{⁎}(x))-h_i(x,u_i(x),{}_0{}^{\beta -1}\mathcal{D}^{\alpha }{u}_i(x),u_j(x))|\frac{dx}{x^{1-\alpha }}ds+\underset{0}{\overset{t}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}ds\le \underset{0}{\overset{t}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}[{\mathcal{L}}_i(x)|u_i^{⁎}(x)-u_i(x)|+{\mathcal{G}}_i(x){|}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i^{⁎}(x){-}_0^{\beta -1}{\mathcal{D}}^{\alpha }{u}_i(x)|+{\mathcal{P}}_i(x)|u_i^{⁎}(x)-u_i(x)|]\frac{dx}{x^{1-\alpha }}ds+\underset{0}{\overset{t}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_1\right|{\alpha }^{(\beta -1)}\mathrm{\Gamma }(\beta )}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}|N_i(x)|\frac{dx}{x^{1-\alpha }}+\frac{\left|{\mathrm{\Lambda }}_2\right|{\alpha }^{(\beta -1)}}{\left|\mathrm{\Lambda }\right|}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{s}{\int }}{\left(\frac{s^{\alpha }-x^{\alpha }}{\alpha }\right)}^{\beta -1}|N_i(x)|\frac{dx}{x^{1-\alpha }}ds\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]+{\varphi }_i(t){\epsilon }_i[\frac{1}{\alpha }+\frac{\left|{\mathrm{\Lambda }}_1\right|\mathrm{\Gamma }(\beta )+\left|{\mathrm{\Lambda }}_2\right|}{\left|\mathrm{\Lambda }\right|}]\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}{\mathit{\eta }}_1^{⁎}+{\varphi }_i(t){\epsilon }_i{\mathit{\eta }}_1^{⁎}.$$

Hence,

$${\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le ({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎}){\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})+{\varphi }_i(t){\epsilon }_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎}).$$

From above, we obtain

$${\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le \frac{{\varphi }_i(t){\epsilon }_i(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}.$$

Now, by taking $\epsilon =\max \{{\epsilon }_1,{\epsilon }_2,...,{\epsilon }_{15}\},\varphi (t)=\max \{{\varphi }_1(t),{\varphi }_2(t),...,{\varphi }_{15}(t)\}$, we get

$${\Vert u^{⁎}-u\Vert }_{\bar{\mathcal{X}}}=\sum _{i=1}^{15}{\Vert u_i^{⁎}-u_i\Vert }_{{\bar{\mathcal{X}}}_i}\le \sum _{i=1}^{15}\frac{\varphi (t)\epsilon (2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}.$$

Thus, we take

$$C_{\varphi }=\sum _{i=1}^{15}\frac{(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})}{1-({\mathcal{L}}_i^{⁎}+{\mathcal{G}}_i^{⁎}+{\mathcal{P}}_i^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})},$$

then, on the cyclopentasilane graph, the fractional boundary value problem (1.1) is Ulam-Hyers-Rassias stable with respect to ϕ by Definition 4.3. Moreover, we take $\mathrm{\Phi }(t)=\varphi (t)\epsilon$, then, on the cyclopentasilane graph, we conclude that the fractional boundary value problem (1.1) is generalized Ulam-Hyers-Rassias stable with respect to ϕ by Definition 4.4. □

5. Examples

In this part, we give the following example, which shows the significance of our main results.

Example 5.1

Study the following fractional differential system1

$$\{\begin{array}{c}{}_0{}^{1.3}\mathcal{D}^{1.5}{u}_1(t)=\frac{t|u_1(t)|}{1250(1+|u_1(t)|)}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_1(t)|}{5000}+\frac{3t|\arcsin u_2(t)|}{1250},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_2(t)=\frac{3e^t|\arctan u_2(t)|}{10000}+\frac{e^t\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_2(t)\left)\right|}{2500}+\frac{7e^t|u_3(t)|}{5000},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_3(t)=\frac{21t|\sin u_3(t)|}{10000}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_3(t)|}{10000(1+{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_3(t)|)}+\frac{7t|\arcsin u_4(t)|}{10000},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_4(t)=\frac{t|u_4(t)|}{1250}+\frac{t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_4(t)|}{2000}+\frac{t|\arcsin u_5(t)|}{1250},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_5(t)=\frac{e^t|\arctan u_5(t)|}{10000}+\frac{e^t\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_5(t)\left)\right|}{2500}+\frac{e^t|u_6(t)|}{5000(1+|u_6(t)|},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_6(t)=\frac{7t|\sin u_6(t)|}{10000}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_6(t)|}{5000}+\frac{11t|\arcsin u_7(t)|}{10000},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_7(t)=\frac{t|u_7(t)|}{5000(1+|u_7(t)|)}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_7(t)|}{5000(1+{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_7(t)|)}+\frac{13t|\arcsin u_8(t)|}{1250},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_8(t)=\frac{3e^t|\arctan u_8(t)|}{10000(1+|\arctan u_8(t)|)}+\frac{e^t\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_8(t)\left)\right|}{5000}+\frac{e^t|u_9(t)|}{1250},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_9(t)=\frac{3t|\sin u_9(t)|}{5000(1+|\sin u_9(t)|)}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_9(t)|}{10000}+\frac{t|\arcsin u_{10}(t)|}{1250},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{10}(t)=\frac{t|u_{10}(t)|}{1250}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_{10}(t)|}{5000}+\frac{7t|\arcsin u_{11}(t)|}{10000(1+|\arcsin u_{11}(t)|)},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{11}(t)=\frac{e^t|\arctan u_{11}(t)|}{20000}+\frac{e^t\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_{11}(t)\left)\right|}{20000(1+\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_{11}(t)\left)\right|)}+\frac{e^t|u_{12}(t)|}{10000},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{12}(t)=\frac{7t|\sin u_{12}(t)|}{10000}+\frac{t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_{12}(t)|}{1000}+\frac{7t|\arcsin u_{13}(t)|}{10000(1+|\arcsin u_{13}(t)|)},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{13}(t)=\frac{t|u_{13}(t)|}{1250(1+|u_{13}(t)|)}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_{13}(t)|}{10000}+\frac{3t|\arcsin u_{14}(t)|}{1000(1+|\arcsin u_{14}(t)|)},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{14}(t)=\frac{e^t|\arctan u_{14}(t)|}{10000(1+|\arctan u_{14}(t)|)}+\frac{e^t\left|\sin \right({}_0{}^{0.3}\mathcal{D}^{1.5}{u}_{14}(t)\left)\right|}{10000}+\frac{e^t|u_{15}(t)|}{5000(1+|u_{15}(t)|)},\hfill \\ {}_0{}^{1.3}\mathcal{D}^{1.5}{u}_{15}(t)=\frac{t|\sin u_{15}(t)|}{1000(1+|\sin u_{15}(t)|)}+\frac{3t{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_{15}(t)|}{1000(1+{|}_0^{0.3}{\mathcal{D}}^{1.5}{u}_{15}(t)|)}+\frac{7t|\arcsin u_1(t)|}{2000},\hfill \end{array}$$ (5.1)

with boundary conditions

$${}_0{}^{0.3}\mathcal{D}^{1.5}{u}_i(1)=\frac{9}{4}\underset{0}{\overset{1}{\int }}{u}_i(s)ds,t^{1.5\times 0.3}{u}_i(t){|}_{t=0}=\frac{7}{6},$$ (5.2)

where $\alpha =1.5$, $\beta =1.3$, $\lambda =\frac{9}{4}$, $k=\frac{7}{6}$, $i=1,2,...,15$, $t\in [0,1]$ and ${}_0{}^{1.3}\mathcal{D}^{1.5}$ and ${}_0{}^{0.3}\mathcal{D}^{1.5}$ are the fractional conformable derivatives of Riemann-Liouville type of order 1.3 and 0.3, respectively.

From (5.1), we note

$$\{\begin{array}{c}{\mathcal{H}}_1(t,u(t),v(t),w(t))=\frac{t|u(t)|}{1250(1+|u(t)|)}+\frac{3t|v(t)|}{5000}+\frac{3t|\arcsin w(t)|}{1250},\hfill \\ {\mathcal{H}}_2(t,u(t),v(t),w(t))=\frac{3e^t|\arctan u(t)|}{10000}+\frac{e^t|\sin v(t)|}{2500}+\frac{7e^t|w(t)|}{5000},\hfill \\ {\mathcal{H}}_3(t,u(t),v(t),w(t))=\frac{21t|\sin u(t)|}{10000}+\frac{3t|v(t)|}{10000(1+|v(t)|)}+\frac{7t|\arcsin w(t)|}{10000},\hfill \\ {\mathcal{H}}_4(t,u(t),v(t),w(t))=\frac{t|u(t)|}{1250}+\frac{t|v(t)|}{2000}+\frac{t|\arcsin w(t)|}{1250},\hfill \\ {\mathcal{H}}_5(t,u(t),v(t),w(t))=\frac{e^t|\arctan u(t)|}{10000}+\frac{e^t|\sin v(t)|}{2500}+\frac{e^t|w(t)|}{5000(1+|w(t)|},\hfill \\ {\mathcal{H}}_6(t,u(t),v(t),w(t))=\frac{7t|\sin u(t)|}{10000}+\frac{3t|v(t)|}{5000}+\frac{11t|\arcsin w(t)|}{10000},\hfill \\ {\mathcal{H}}_7(t,u(t),v(t),w(t))=\frac{t|u(t)|}{5000(1+|u(t)|)}+\frac{3t|v(t)|}{5000(1+|v(t)|)}+\frac{13t|\arcsin w(t)|}{1250},\hfill \\ {\mathcal{H}}_8(t,u(t),v(t),w(t))=\frac{3e^t|\arctan u(t)|}{10000(1+|\arctan u(t)|)}+\frac{e^t|\sin v(t)|}{5000}+\frac{e^t|w(t)|}{1250},\hfill \\ {\mathcal{H}}_9(t,u(t),v(t),w(t))=\frac{3t|\sin u(t)|}{5000(1+|\sin u(t)|)}+\frac{3t|v(t)|}{10000}+\frac{t|\arcsin w(t)|}{1250},\hfill \\ {\mathcal{H}}_{10}(t,u(t),v(t),w(t))=\frac{t|u(t)|}{1250}+\frac{3t|v(t)|}{5000}+\frac{7t|\arcsin w(t)|}{10000(1+|\arcsin w(t)|)},\hfill \\ {\mathcal{H}}_{11}(t,u(t),v(t),w(t))=\frac{e^t|\arctan u(t)|}{20000}+\frac{e^t|\sin v(t|}{20000(1+|\sin v(t)|)}+\frac{e^t|w(t)|}{10000},\hfill \\ {\mathcal{H}}_{12}(t,u(t),v(t),w(t))=\frac{7t|\sin u(t)|}{10000}+\frac{t|v(t)|}{1000}+\frac{7t|\arcsin w(t)|}{10000(1+|\arcsin w(t)|)},\hfill \\ {\mathcal{H}}_{13}(t,u(t),v(t),w(t))=\frac{t|u(t)|}{1250(1+|u(t)|)}+\frac{3t|v(t)|}{10000}+\frac{3t|\arcsin w(t)|}{1000(1+|\arcsin w(t)|)},\hfill \\ {\mathcal{H}}_{14}(t,u(t),v(t),w(t))=\frac{e^t|\arctan u(t)|}{10000(1+|\arctan u(t)|)}+\frac{e^t|\sin v(t)|}{10000}+\frac{e^t|w(t)|}{5000(1+|w(t)|)},\hfill \\ {\mathcal{H}}_{15}(t,u(t),v(t),w(t))=\frac{t|\sin u(t)|}{1000(1+|\sin u(t)|)}+\frac{3t|v(t)|}{1000(1+|v(t)|)}+\frac{7t|\arcsin w(t)|}{2000}.\hfill \end{array}$$ (5.3)

And we have

$$|{\mathcal{H}}_1(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_1(t,u_2(t),v_2(t),w_2(t))|\le \frac{3t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_2(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_2(t,u_2(t),v_2(t),w_2(t))|\le \frac{7e^t}{5000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_3(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_3(t,u_2(t),v_2(t),w_2(t))|\le \frac{21t}{10000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_4(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_4(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_5(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_5(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{2500}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_6(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_6(t,u_2(t),v_2(t),w_2(t))|\le \frac{11t}{10000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_7(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_7(t,u_2(t),v_2(t),w_2(t))|\le \frac{13t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_8(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_8(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_9(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_9(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{10}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{10}(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{11}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{11}(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{10000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{12}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{12}(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{13}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{13}(t,u_2(t),v_2(t),w_2(t))|\le \frac{3t}{1000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{14}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{14}(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{5000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),|{\mathcal{H}}_{15}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{15}(t,u_2(t),v_2(t),w_2(t))|\le \frac{7t}{2000}(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|w_1(t)-w_2(t)|),$$

where

$${\sigma }_1(t)=\frac{3t}{1250},{\sigma }_2(t)=\frac{7e^t}{5000},{\sigma }_3(t)=\frac{21t}{10000},{\sigma }_4(t)=\frac{t}{1250},{\sigma }_5(t)=\frac{e^t}{2500},{\sigma }_6(t)=\frac{11t}{10000},{\sigma }_7(t)=\frac{13t}{1250},{\sigma }_8(t)=\frac{e^t}{1250},{\sigma }_9(t)=\frac{t}{1250},{\sigma }_{10}(t)=\frac{t}{1250},{\sigma }_{11}(t)=\frac{e^t}{10000},{\sigma }_{12}(t)=\frac{t}{1000},{\sigma }_{13}(t)=\frac{3t}{1000},{\sigma }_{14}(t)=\frac{e^t}{5000},{\sigma }_{15}(t)=\frac{7t}{2000},\Vert {\sigma }_1\Vert =\frac{3}{1250},\Vert {\sigma }_2\Vert =\frac{7e}{5000},\Vert {\sigma }_3\Vert =\frac{21}{10000},\Vert {\sigma }_4\Vert =\frac{1}{1250},\Vert {\sigma }_5\Vert =\frac{e}{2500},\Vert {\sigma }_6\Vert =\frac{11}{10000},\Vert {\sigma }_7\Vert =\frac{13}{1250},\Vert {\sigma }_8\Vert =\frac{e}{1250},\Vert {\sigma }_9\Vert =\frac{1}{1250},\Vert {\sigma }_{10}\Vert =\frac{1}{1250},\Vert {\sigma }_{11}\Vert =\frac{e}{10000},\Vert {\sigma }_{12}\Vert =\frac{1}{1000},\Vert {\sigma }_{13}\Vert =\frac{3}{1000},\Vert {\sigma }_{14}\Vert =\frac{e}{5000},\Vert {\sigma }_{15}\Vert =\frac{7}{2000}.$$

Simultaneously, we get

$$|{\mathcal{H}}_1(t,u(t),v(t),w(t))|\le \frac{3t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{3t}{1250}{\varphi }_1(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_2(t,u(t),v(t),w(t))|\le \frac{7e^t}{5000}(|u(t)|+|v(t)|+|w(t)|)=\frac{7e^t}{5000}{\varphi }_2(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_3(t,u(t),v(t),w(t))|\le \frac{21t}{10000}(|u(t)|+|v(t)|+|w(t)|)=\frac{21t}{10000}{\varphi }_3(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_4(t,u(t),v(t),w(t))|\le \frac{t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{t}{1250}{\varphi }_4(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_5(t,u(t),v(t),w(t))|\le \frac{e^t}{2500}(|u(t)|+|v(t)|+|w(t)|)=\frac{e^t}{2500}{\varphi }_5(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_6(t,u(t),v(t),w(t))|\le \frac{11t}{10000}(|u(t)|+|v(t)|+|w(t)|)=\frac{11t}{10000}{\varphi }_6(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_7(t,u(t),v(t),w(t))|\le \frac{13t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{13t}{1250}{\varphi }_7(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_8(t,u(t),v(t),w(t))|\le \frac{e^t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{e^t}{1250}{\varphi }_8(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_9(t,u(t),v(t),w(t))|\le \frac{t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{t}{1250}{\varphi }_9(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{10}(t,u(t),v(t),w(t))|\le \frac{t}{1250}(|u(t)|+|v(t)|+|w(t)|)=\frac{t}{1250}{\varphi }_{10}(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{11}(t,u(t),v(t),w(t))|\le \frac{e^t}{10000}(|u(t)|+|v(t)|+|w(t)|)=\frac{e^t}{10000}{\varphi }_{11}(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{12}(t,u(t),v(t),w(t))|\le \frac{t}{1000}(|u(t)|+|v(t)|+|w(t)|)=\frac{t}{1000}{\varphi }_{12}(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{13}(t,u(t),v(t),w(t))|\le \frac{3t}{1000}(|u(t)|+|v(t)|+|w(t)|)=\frac{3t}{1000}{\varphi }_{13}(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{14}(t,u(t),v(t),w(t))|\le \frac{e^t}{5000}(|u(t)|+|v(t)|+|w(t)|)=\frac{e^t}{5000}{\varphi }_{14}(|u(t)|+|v(t)|+|w(t)|),|{\mathcal{H}}_{15}(t,u(t),v(t),w(t))|\le \frac{7t}{2000}(|u(t)|+|v(t)|+|w(t)|)=\frac{7t}{2000}{\varphi }_{15}(|u(t)|+|v(t)|+|w(t)|),$$

where

$${\delta }_1(t)=\frac{3t}{1250},{\delta }_2(t)=\frac{7e^t}{5000},{\delta }_3(t)=\frac{21t}{10000},{\delta }_4(t)=\frac{t}{1250},{\delta }_5(t)=\frac{e^t}{2500},{\delta }_6(t)=\frac{11t}{10000},{\delta }_7(t)=\frac{13t}{1250},{\delta }_8(t)=\frac{e^t}{1250},{\delta }_9(t)=\frac{t}{1250},{\delta }_{10}(t)=\frac{t}{1250},{\delta }_{11}(t)=\frac{e^t}{10000},{\delta }_{12}(t)=\frac{t}{1000},{\delta }_{13}(t)=\frac{3t}{1000},{\delta }_{14}(t)=\frac{e^t}{5000},{\delta }_{15}(t)=\frac{7t}{2000}.$$

According to the obtained values, we get ${\mathit{\xi }}_0^{⁎}\approx 5.4416$, ${\mathit{\xi }}_1^{⁎}\approx 5.8485$, so $2{\mathit{\xi }}_0^{⁎}+{\mathit{\xi }}_1^{⁎}\approx 16.7317$. Hence, $k=(2{\mathit{\xi }}_0^{⁎}+{\mathit{\xi }}_1^{⁎})\sum _{i=1}^{15}\Vert {\sigma }_i\Vert \approx 0.5556\le 1$. Based on Theorem 3.3, the fractional boundary value system (5.1)-(5.2) has a solution.

From (5.3), we also get

$$|{\mathcal{H}}_1(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_1(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}|u_1(t)-u_2(t)|+\frac{3t}{5000}|v_1(t)-v_2(t)|+\frac{3t}{1250}|w_1(t)-w_2(t)|,|{\mathcal{H}}_2(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_2(t,u_2(t),v_2(t),w_2(t))|\le \frac{3e^t}{10000}|u_1(t)-u_2(t)|+\frac{e^t}{2500}|v_1(t)-v_2(t)|+\frac{7e^t}{5000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_3(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_3(t,u_2(t),v_2(t),w_2(t))|\le \frac{21t}{10000}|u_1(t)-u_2(t)|+\frac{3t}{10000}|v_1(t)-v_2(t)|+\frac{7t}{10000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_4(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_4(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}|u_1(t)-u_2(t)|+\frac{t}{2000}|v_1(t)-v_2(t)|+\frac{t}{1250}|w_1(t)-w_2(t)|,|{\mathcal{H}}_5(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_5(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{10000}|u_1(t)-u_2(t)|+\frac{e^t}{2500}|v_1(t)-v_2(t)|+\frac{e^t}{5000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_6(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_6(t,u_2(t),v_2(t),w_2(t))|\le \frac{7t}{10000}|u_1(t)-u_2(t)|+\frac{3t}{5000}|v_1(t)-v_2(t)|+\frac{11t}{10000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_7(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_7(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{5000}|u_1(t)-u_2(t)|+\frac{3t}{5000}|v_1(t)-v_2(t)|+\frac{13t}{1250}|w_1(t)-w_2(t)|,|{\mathcal{H}}_8(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_8(t,u_2(t),v_2(t),w_2(t))|\le \frac{3e^t}{10000}|u_1(t)-u_2(t)|+\frac{e^t}{5000}|v_1(t)-v_2(t)|+\frac{e^t}{1250}|w_1(t)-w_2(t)|,|{\mathcal{H}}_9(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_9(t,u_2(t),v_2(t),w_2(t))|\le \frac{3t}{5000}|u_1(t)-u_2(t)|+\frac{3t}{10000}|v_1(t)-v_2(t)|+\frac{t}{1250}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{10}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{10}(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}|u_1(t)-u_2(t)|+\frac{3t}{5000}|v_1(t)-v_2(t)|+\frac{7t}{10000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{11}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{11}(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{20000}|u_1(t)-u_2(t)|+\frac{e^t}{20000}|v_1(t)-v_2(t)|+\frac{e^t}{10000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{12}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{12}(t,u_2(t),v_2(t),w_2(t))|\le \frac{7t}{10000}|u_1(t)-u_2(t)|+\frac{t}{1000}|v_1(t)-v_2(t)|+\frac{7t}{10000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{13}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{13}(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1250}|u_1(t)-u_2(t)|+\frac{3t}{10000}|v_1(t)-v_2(t)|+\frac{3t}{1000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{14}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{14}(t,u_2(t),v_2(t),w_2(t))|\le \frac{e^t}{10000}|u_1(t)-u_2(t)|+\frac{e^t}{10000}|v_1(t)-v_2(t)|+\frac{e^t}{5000}|w_1(t)-w_2(t)|,|{\mathcal{H}}_{15}(t,u_1(t),v_1(t),w_1(t))-{\mathcal{H}}_{15}(t,u_2(t),v_2(t),w_2(t))|\le \frac{t}{1000}|u_1(t)-u_2(t)|+\frac{3t}{1000}|v_1(t)-v_2(t)|+\frac{7t}{2000}|w_1(t)-w_2(t)|,$$

where

$${\mathcal{L}}_1(t)=\frac{t}{1250},{\mathcal{L}}_2(t)=\frac{3e^t}{10000},{\mathcal{L}}_3(t)=\frac{21t}{10000},{\mathcal{L}}_4(t)=\frac{t}{1250},{\mathcal{L}}_5(t)=\frac{e^t}{10000},{\mathcal{L}}_6(t)=\frac{7t}{10000},{\mathcal{L}}_7(t)=\frac{t}{5000},{\mathcal{L}}_8(t)=\frac{3e^t}{10000},{\mathcal{L}}_9(t)=\frac{3t}{5000},{\mathcal{L}}_{10}(t)=\frac{t}{1250},{\mathcal{L}}_{11}(t)=\frac{e^t}{20000},{\mathcal{L}}_{12}(t)=\frac{7t}{10000},{\mathcal{L}}_{13}(t)=\frac{t}{1250},{\mathcal{L}}_{14}(t)=\frac{e^t}{10000},{\mathcal{L}}_{15}(t)=\frac{t}{1000},{\mathcal{G}}_1(t)=\frac{3t}{5000},{\mathcal{G}}_2(t)=\frac{e^t}{2500},{\mathcal{G}}_3(t)=\frac{3t}{10000},{\mathcal{G}}_4(t)=\frac{t}{2000},{\mathcal{G}}_5(t)=\frac{e^t}{2500},{\mathcal{G}}_6(t)=\frac{3t}{5000},{\mathcal{G}}_7(t)=\frac{3t}{5000},{\mathcal{G}}_8(t)=\frac{e^t}{5000},{\mathcal{G}}_9(t)=\frac{3t}{10000},{\mathcal{G}}_{10}(t)=\frac{3t}{5000},{\mathcal{G}}_{11}(t)=\frac{e^t}{20000},{\mathcal{G}}_{12}(t)=\frac{t}{1000},{\mathcal{G}}_{13}(t)=\frac{3t}{10000},{\mathcal{G}}_{14}(t)=\frac{e^t}{10000},{\mathcal{G}}_{15}(t)=\frac{3t}{1000},{\mathcal{P}}_1(t)=\frac{3t}{1250},{\mathcal{P}}_2(t)=\frac{7e^t}{5000},{\mathcal{P}}_3(t)=\frac{7t}{10000},{\mathcal{P}}_4(t)=\frac{t}{1250},{\mathcal{P}}_5(t)=\frac{e^t}{5000},{\mathcal{P}}_6(t)=\frac{11t}{10000},{\mathcal{P}}_7(t)=\frac{13t}{1250},{\mathcal{P}}_8(t)=\frac{e^t}{1250},{\mathcal{P}}_9(t)=\frac{t}{1250},{\mathcal{P}}_{10}(t)=\frac{7t}{10000},{\mathcal{P}}_{11}(t)=\frac{e^t}{10000},{\mathcal{P}}_{12}(t)=\frac{7t}{10000},{\mathcal{P}}_{13}(t)=\frac{3t}{1000},{\mathcal{P}}_{14}(t)=\frac{e^t}{5000},{\mathcal{P}}_{15}(t)=\frac{7t}{2000},{\mathcal{L}}_1^{⁎}=\frac{1}{1250},{\mathcal{L}}_2^{⁎}=\frac{3e}{10000},{\mathcal{L}}_3^{⁎}=\frac{21}{10000},{\mathcal{L}}_4^{⁎}=\frac{1}{1250},{\mathcal{L}}_5^{⁎}=\frac{e}{10000},{\mathcal{L}}_6^{⁎}=\frac{7}{10000},{\mathcal{L}}_7^{⁎}=\frac{1}{5000},{\mathcal{L}}_8^{⁎}=\frac{3e}{10000},{\mathcal{L}}_9^{⁎}=\frac{3}{5000},{\mathcal{L}}_{10}^{⁎}=\frac{1}{1250},{\mathcal{L}}_{11}^{⁎}=\frac{e}{20000},{\mathcal{L}}_{12}^{⁎}=\frac{7}{10000},{\mathcal{L}}_{13}^{⁎}=\frac{1}{1250},{\mathcal{L}}_{14}^{⁎}=\frac{e}{10000},{\mathcal{L}}_{15}^{⁎}=\frac{1}{1000},{\mathcal{G}}_1^{⁎}=\frac{3}{5000},{\mathcal{G}}_2^{⁎}=\frac{e}{2500},{\mathcal{G}}_3^{⁎}=\frac{3}{10000},{\mathcal{G}}_4^{⁎}=\frac{1}{2000},{\mathcal{G}}_5^{⁎}=\frac{e}{2500},{\mathcal{G}}_6^{⁎}=\frac{3}{5000},{\mathcal{G}}_7^{⁎}=\frac{3}{5000},{\mathcal{G}}_8^{⁎}=\frac{e}{5000},{\mathcal{G}}_9^{⁎}=\frac{3}{10000},{\mathcal{G}}_{10}^{⁎}=\frac{3}{5000},{\mathcal{G}}_{11}^{⁎}=\frac{e}{20000},{\mathcal{G}}_{12}^{⁎}=\frac{1}{1000},{\mathcal{G}}_{13}^{⁎}=\frac{3}{10000},{\mathcal{G}}_{14}^{⁎}=\frac{e}{10000},{\mathcal{G}}_{15}^{⁎}=\frac{3}{1000},{\mathcal{P}}_1^{⁎}=\frac{3}{1250},{\mathcal{P}}_2^{⁎}=\frac{7e}{5000},{\mathcal{P}}_3^{⁎}=\frac{7}{10000},{\mathcal{P}}_4^{⁎}=\frac{1}{1250},{\mathcal{P}}_5^{⁎}=\frac{e}{5000},{\mathcal{P}}_6^{⁎}=\frac{11}{10000},{\mathcal{P}}_7^{⁎}=\frac{13}{1250},{\mathcal{P}}_8^{⁎}=\frac{e}{1250},{\mathcal{P}}_9^{⁎}=\frac{1}{1250},{\mathcal{P}}_{10}^{⁎}=\frac{7}{10000},{\mathcal{P}}_{11}^{⁎}=\frac{e}{10000},{\mathcal{P}}_{12}^{⁎}=\frac{7}{10000},{\mathcal{P}}_{13}^{⁎}=\frac{3}{1000},{\mathcal{P}}_{14}^{⁎}=\frac{e}{5000},{\mathcal{P}}_{15}^{⁎}=\frac{7}{2000}.$$

According to the obtained values, we get ${\mathit{\eta }}_0^{⁎}\approx 6.2987$, ${\mathit{\eta }}_1^{⁎}\approx 6.5152$, so $2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎}\approx 19.1126$. Since

$$({\mathcal{L}}_1^{⁎}+{\mathcal{G}}_1^{⁎}+{\mathcal{P}}_1^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0726<1,({\mathcal{L}}_2^{⁎}+{\mathcal{G}}_2^{⁎}+{\mathcal{P}}_2^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.1089<1,({\mathcal{L}}_3^{⁎}+{\mathcal{G}}_3^{⁎}+{\mathcal{P}}_3^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0592<1,({\mathcal{L}}_4^{⁎}+{\mathcal{G}}_4^{⁎}+{\mathcal{P}}_4^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0401<1,({\mathcal{L}}_5^{⁎}+{\mathcal{G}}_5^{⁎}+{\mathcal{P}}_5^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0363<1,({\mathcal{L}}_6^{⁎}+{\mathcal{G}}_6^{⁎}+{\mathcal{P}}_6^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0459<1,({\mathcal{L}}_7^{⁎}+{\mathcal{G}}_7^{⁎}+{\mathcal{P}}_7^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.2141<1,({\mathcal{L}}_8^{⁎}+{\mathcal{G}}_8^{⁎}+{\mathcal{P}}_8^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0669<1,({\mathcal{L}}_9^{⁎}+{\mathcal{G}}_9^{⁎}+{\mathcal{P}}_9^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0325<1,({\mathcal{L}}_{10}^{⁎}+{\mathcal{G}}_{10}^{⁎}+{\mathcal{P}}_{10}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0401<1,({\mathcal{L}}_{11}^{⁎}+{\mathcal{G}}_{11}^{⁎}+{\mathcal{P}}_{11}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0096<1,({\mathcal{L}}_{12}^{⁎}+{\mathcal{G}}_{12}^{⁎}+{\mathcal{P}}_{12}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0459<1,({\mathcal{L}}_{13}^{⁎}+{\mathcal{G}}_{13}^{⁎}+{\mathcal{P}}_{13}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0784<1,({\mathcal{L}}_{14}^{⁎}+{\mathcal{G}}_{14}^{⁎}+{\mathcal{P}}_{14}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.0211<1,({\mathcal{L}}_{15}^{⁎}+{\mathcal{G}}_{15}^{⁎}+{\mathcal{P}}_{15}^{⁎})(2{\mathit{\eta }}_0^{⁎}+{\mathit{\eta }}_1^{⁎})\approx 0.1433<1,$$

the conclusion of Theorem 4.1 illustrates that the system (5.1)-(5.2) is Ulam-Hyers stable and generalized Ulam-Hyers stable.

6. Conclusion

Chemical graph theory is a multidisciplinary field that applies theoretical and practical techniques to analyze the molecular structure graphs of chemical substances, while considering specific mathematical problems. Over the past few decades, significant advancements in this field have provided us with numerous groundbreaking and unique ideas and methods. In this paper, we employ the structural graph of cyclopentasilane to define boundary value problem in the sense of Riemann-Liouville fractional conformable derivative on its edges. We utilize Scheafer and Krasnoselskii fixed point theorems to prove the existence of solutions to the proposed boundary value problem. Additionally, stability analysis of different types on cyclopentasilane graph is investigated. To showcase the significance of our results, we provide an example. Our approach is easy to implement and can be applied to a wide range of graphs, particularly directed graphs, which are commonly used in medical technologies for protein networks. The paper mainly focuses on theoretical research, which leads to the lack of numerical analysis in the research. In the future, we will consider using numerical methods to solve more problems on graphs with different molecular structures.

Funding statement

This work was supported by the Natural Science Foundation of Shanxi Province, China (No. 20210302123339) and the Graduate Research Innovation Program of Shanxi, China (No. 2022Y497).

CRediT authorship contribution statement

Guotao Wang: Writing – review & editing, Supervision, Conceptualization. Hualei Yuan: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. Dumitru Baleanu: Writing – review & editing, Supervision, Methodology, Conceptualization.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Guotao Wang reports article publishing charges was provided by Elsevier Inc., Cell Press. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

No data was used for the research described in the article.