Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Abstract

Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

1. Introduction

Initial and boundary value problems have been extensively used to study natural phenomena in many real-world problems, attracting researchers who employ mathematical techniques and computer simulations to study natural phenomena. It has recently been demonstrated that a wide range of applications in the applied sciences can be described through fractional differential equations (FDEs) [1]. In [2], the authors studied the multiple positive solutions of the boundary value problem for nonlinear FDEs by constructing the Green function using fixed-point theory. Sanjay Bhatter et al. [3] studied a fractional extension of the modified Kawahara equation using the Atangana–Baleanu fractional operator in the Caputo sense to describe plasma waves and capillary–gravity water waves. The inverse problem for a family of multi-term time FDEs with nonlocal boundary conditions is discussed in [4]. The authors of [5] discussed the existence of a multiplicity of solutions to the generalized Bagley–Torvik FDE with the Neumann boundary conditions. Fatma Al-Musalhi et al. [6] considered direct and inverse source problems of a fractional diffusion equation with a regularized Caputo-like hyper-Bessel differential operator, constructing solutions via eigenfunction expansions and establishing their existence and uniqueness. In [7], the authors established Lyapunov-type inequalities for fractional boundary value problems with the Hilfer fractional derivative under multi-point boundary conditions. In [8,9], utilizing the fixed-point theorems and properties of the Mittag–Leffler function, the authors established the existence/uniqueness and stability results for fractional Langevin equations and nonlinear fractional hybrid differential equations, respectively. In [10], the authors presented analytical solutions for some fractional diffusion boundary value problems. Komal Bansal et al. proposed a model dealing with the dynamics of crime transmission using fractional-order differential equations in [11]. An analysis of a human liver model using the Caputo–Fabrizio fractional derivative is discussed in [12]. There are several methods for solving mathematical models based on FDEs, obtaining the solution functions, and then applying certain techniques to analyze the qualitative behavior of solutions under particular boundary conditions. The work in [13] dealt with a class of Hilfer–Hadamard differential equations. The existence and stability of solutions are presented via fixed-point theorems. In [14], the authors investigated the solution of multi-term time–space fractional partial delay differential–algebraic equations with Dirichlet boundary conditions defined on a finite domain using the Laplace transform method. The authors of [15,16,17] discussed the initial/boundary value problem for FDEs in the sense of different fractional derivative operators.

Our goal in this paper is to improve our capacity to predict certain chemical reaction processes by expanding the theoretical features of specific applied concepts in chemistry. If we are able to do so, software developers will be able to create tools that will enable anyone to perform chemical experiments without the need for actual ingredients, and this might be helpful for the environment.

Nowadays, graph theory is an area of interest in which lines interconnect a network of points. Numerous real constructions in our surroundings contain this structure. Stated differently, new descriptive models for studying related processes created by experts in these fields have emerged due to the development and expansion of some dynamic and industrial systems, such as water pipelines, gas transmission lines, computer networks, and the structure of molecules in biology and medicine. The study of mathematical models described by ordinary or fractional differential equations on graphs was considered due to the graph structure of these networks. A boundary value problem on a graph structure is a system of differential equations assigned to each edge with some boundary conditions on each vertex.

With Lumer’s work [18] in the 1980s, the theory of differential equations on graphs came into the picture. He used local operators constructed on ramification spaces to study general evolution equations in those spaces. Nicaise [19] discussed how nerve impulses propagate. This led to many papers on linear eigenvalue problems for metric graphs and Sturm–Liouville-type problems, especially in Von Below’s work [20]. Pokornyi [21] studied the eigenfunctions of a particular problem on graphs with Dirichlet conditions at border nodes, as well as the spectrum and the impact of eigenvalue multiplicity. Using the double-sweep method, Gordeziani et al. [22] provided a numerical way to solve ordinary differential equations on graphs, as well as an investigation of the existence and uniqueness results of these kinds of problems. Currie and Watson [23] provided asymptotic approximations for eigenvalues and examined the spectral structure of second-order boundary value problems on graphs.

Most of the publications listed above involve differential equations on a graph, and computational and numerical techniques are employed to determine their solutions. However, there are only a few studies available in the literature on fractional boundary value problems on graphs where the existence of solutions is demonstrated using certain methods from fixed-point theory. Graef et al. [24] were the first authors to use the concept of fixed-point theory to obtain results on the existence of solutions. The authors assumed a star graph $G=H\cup E$ with three vertices $H=\{h_0,h_1,h_2\}$ and two edges $E=\{e_1=\overrightarrow{h_1{h}_0},e_2=\overrightarrow{h_2{h}_0}\}$, respectively, where $h_0$ is the junction point and $e_i=\overrightarrow{h_i{h}_0}$ is the edge length from $h_i$ to $h_0$ with the length $l_i=\left|\overrightarrow{h_i{h}_0}\right|,i=1,2$. Graef et al. studied the following fractional boundary value problem on this star graph:

$${ }^{RL}{D}^{\rho }{k}_i\left(y\right)=-g_i\left(y\right)r_i(y,k_i\left(y\right)),y\in (0,l_i),\rho \in (1,2],$$

with the boundary conditions

$$k_1\left(0\right)=k_2\left(0\right)=0,k_1\left(l_1\right)=k_2\left(l_2\right),{ }^{RL}{D}^{\sigma }{k}_1\left(l_1\right){+}^{RL}{D}^{\sigma }{k}_2\left(l_2\right)=0,\sigma \in (0,\rho -1),$$

where the functions $g_i:[0,l_i]\to \mathbb{R}$, $r_i:[0,l_i]\times \mathbb{R}\to \mathbb{R},i=1,2$ are continuous with $g_i\left(y\right)\ne 0$. ${ }^{RL}{D}^{\rho }$ and ${ }^{RL}{D}^{\sigma }$ are the fractional derivative operators in the Riemann–Liouville sense. They proved the existence of solutions by using the Banach contraction principle and Schaefer’s fixed-point theorem. In $2019,$ Mehandiratta et al. [25] generalized the work of Graef et al. [24]. They extended the graph from three vertices and two edges to $(n+1)$ vertices and n edges. They considered the generalized star graph $G=H\cup E$ with $(n+1)$ vertices $H=\{h_0,h_1,\dots ,h_n\}$ and n edges $E=\{e_1=\overrightarrow{h_1{h}_0},e_2=\overrightarrow{h_2{h}_0},\dots e_n=\overrightarrow{h_n{h}_0}\}$, respectively, where $h_0$ is the junction point and $e_i=\overrightarrow{h_i{h}_0}$ is the edge length from $h_i$ to $h_0$ with the length $l_i=\left|\overrightarrow{h_i{h}_0}\right|$. The authors discussed the following nonlinear fractional boundary value problem on this generalized star graph:

$${ }^C{D}^{\rho }{k}_i\left(y\right)=r_i(y,k_i\left(y\right){,}^C{D}^{\sigma }{k}_i\left(y\right)),y\in (0,l_i),\rho \in (1,2],\sigma \in (0,1],$$

with the conditions

$$k_i\left(0\right)=0,k_i\left(l_i\right)=k_j\left(l_j\right),i\ne j,\sum _{i=1}^n{k}_i^{\prime }\left(l_i\right)=0,i,j=1,2,\dots ,n,$$

where the functions $r_i:[0,l_i]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},i=1,2,\dots ,n$ are continuous. ${ }^C{D}^{\rho }$ and ${ }^C{D}^{\sigma }$ are the fractional derivative operators in the Caputo sense. By using the appropriate transformation, they converted the given FDE on the generalized star graph to the FDE on the interval $[0,1]$ and proved the existence and uniqueness results. Recently, Wajahat Ali et al. [26] discussed the novel existence result for an FDE on a graph of cyclohexane. For more studies on graphs, one can refer to [27,28,29,30,31,32,33,34,35].

We were motivated to extend these boundary value problems to a new problem on the tetramethylbutane graph. Tetramethylbutane is a hydrocarbon; it is the most heavily branched and most compact of the many octane isomers and has a highly symmetrical structure. The presence of a symmetric bond is related to the number of atoms and bond orbits [36]. The molecule’s symmetrical arrangement leads to significant steric hindrance, influencing its chemical reactivity and resulting in fewer distinct hydrogen environments in spectroscopy. Its compact symmetrical structure allows for efficient packing in the solid state, affecting its melting and boiling points. In graph-theoretical terms, tetramethylbutane can be compared to a generalized graph with multiple junction nodes, distinguishing it from simpler star graphs. Compared to star graphs, it is actually a symmetrical generalized graph with more than one junction node. In this paper, we discuss the following system of nonlinear Caputo fractional-order differential equations on each edge of the tetramethylbutane graph:

$${ }^C{D}^{\rho }{k}_i\left(y\right)=r_i(y,k_i\left(y\right){,}^C{D}^{\sigma }{k}_i\left(y\right),k_i^{\prime }\left(y\right)),i=1,2,\dots ,25,0\le y\le 1,\rho \in [1,2],\sigma \in [0,1],$$

(1)

which satisfies the integral boundary conditions

$$\begin{array}{cc}\hfill m_1{k}_i\left(0\right)+m_2{k}_i\left(1\right)& =m_3{\int }_0^1{k}_i\left(\tau \right)d\tau \hfill \\ \hfill m_1{k}_i^{\prime }\left(0\right)+m_2{k}_i^{\prime }\left(1\right)& =m_3{\int }_0^1{k}_i\left(\tau \right)d\tau \hfill \end{array}$$

(2)

where $r_i:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},i=1,2,\dots ,25$ are continuous, and $m_1,m_2,$$m_3\in \mathbb{R},$ and $m_1+m_2\ne m_3$. ${ }^C{D}^{\rho }$ and ${ }^C{D}^{\sigma }$ are the fractional derivative operators in the Caputo sense. $n=25$ is the number of edges of the tetramethylbutane graph with an edge length of unit 1.

It is noteworthy that there are several practical interpretations of organic chemistry for the fractional boundary value problem’s solutions that are found. The existence and uniqueness of solutions to FDEs in the chemical context ensure that the mathematical models used to describe chemical processes are well defined and reliable [37]. The existence of a solution at any edge of the chemical graph may represent the bond polarity, strength, bond energy, etc. Uniqueness ensures consistent and unambiguous predictions about chemical interactions, such as the distribution of electron density and the stability of bonds. This might have important implications for the theory of chemical processes. Thus, young researchers may find this abstract idea helpful in their future work.

This paper is arranged in the following manner: Section 2 presents some basic preliminary ideas for the graph and some fundamentals of fractional calculus. In Section 3, the main results are proved with well-known fixed-point theorems. The Hyers–Ulam stability of the system is discussed in Section 4. At the end of this manuscript, some illustrative examples are discussed to validate the obtained results.

2. Preliminaries

This section introduces the graph in relation to the tetramethylbutane compound in order to define a new class of fractional boundary value problems on it. First, let us bring readers’ attention to two essential approaches used in [24,25].

(I) The authors of both papers assumed that the graph G is a star graph with one junction node $h_0$, as illustrated in Figure 1 and Figure 2.

However, the graph G might not be a star network and might have more than one junction node. As an illustration, Figure 3 depicts five junction nodes.

(II) The authors of both articles mentioned above treat the length of each edge as the variable value for $|e_i|=l_i,i=1,2,\dots n$, where n is the edge count of the graph G. The length of all edges is then normalized by using a change of variable, and finally, they transform $[0,l_i]$ into a unit interval $[0,1]$. From the start, the length of all edges can be considered a fixed value $|e_i|=1$ without identifying the boundary vertices of every edge as the origin. For this reason, we suggest another method for labeling vertices. In this scenario, each vertex in a graph can be assigned one of two labels: 0 or 1. In other words, each vertex’s label is determined by the orientation of its related edge. When we go along an arbitrary edge, the starting and ending vertex labels are assigned values of 0 and 1, respectively, and vice versa. As a result, several vertices may have both the labels 0 and 1 at the same time. Additionally, every edge’s origin is not fixed; rather, it changes any time the edge’s movement direction changes. We are allowed to choose the origin of each edge to be any one of its two vertices, as there is no need to normalize the length of every edge using the aforementioned transforms. A possible application of labeling in this scenario is illustrated in Figure 4. We start at the yellow vertex and then move along the graph’s edges for labeling (see Figure 4).

In this article, we study a system of FDEs defined on each edge of the tetramethylbutane graph (Figure 5).

The molecular formula for tetramethylbutane is $C_8{H}_{18}$, where C and H represent carbon and hydrogen atoms, respectively. The petrochemical industry uses tetramethylbutane as a precursor molecule. Eight carbon–carbon bonds and six $C{H}_3$ subbranches make up this molecule. Atoms of hydrogen and carbon are viewed as the graph’s vertices because of this structure. The edges of the graph represent the chemical bonds that exist between atoms. Since there are several junction nodes on this molecular graph, the method used in [24,25] to assign the origin at boundary nodes other than the junction node $h_0$ will not apply here because the molecular graph is not a star graph. As a result, we must adopt a new strategy. In this graph, vertices can be labeled with 0 or 1, and the length of each edge is represented by the unit value $l_i=1$ (see Figure 6).

As a result, using the tetramethylbutane graph with vertices labeled 0 or 1, as shown above, we can reach our goals of the existence and uniqueness of solutions for the considered nonlinear FDE defined on it.

We now review some fundamental terms of fractional calculus.

Definition 1.

([38,39]). The Riemann–Liouville fractional integral of order $\rho >0$ of an integrable function $k:[c,d]\to \mathbb{R}$ is defined by

$$I^{\rho }k\left(y\right)=\frac{1}{\Gamma \left(\rho \right)}{\int }_0^{y}k\left(\tau \right){(y-\tau )}^{\rho -1}d\tau ,$$

where Γ denotes the Gamma function.

Definition 2.

([38,39]). The Caputo fractional derivative of order ρ$(m-1<\rho <m)$ of a function $k\in A{C}^m([c,d],\mathbb{R})$ (an m-times differentiable absolutely continuous function) is defined as

$${ }^C{D}^{\rho }k\left(y\right)=\frac{1}{\Gamma (m-\rho )}{\int }_0^y{(y-\tau )}^{m-\rho -1}{k}^{\left(m\right)}\left(\tau \right)d\tau ,$$

where $m=\left[\rho \right]+1$, where $\left[\right]$ denotes the greatest integer less than or equal to ρ.

The general solution of the homogeneous FDE ${ }^C{D}^{\rho }k\left(y\right)=0$ is of the form $k\left(y\right)=s_0+s_{1}y+s_2{y}^2+s_{m-1}{y}^{m-1}$, where $s_0,s_1,\dots ,s_{m-1}\in \mathbb{R}$. Then, we have $I^{\rho }{(}^C{D}^{\rho }k\left(y\right))=k\left(y\right)+s_0+s_{1}y+s_2{y}^2+s_{m-1}{y}^{m-1}$ (see [40]).

Lemma 1.

Assume that function $k_i$ has a fractional derivative and let $q_i:[0,1]\to \mathbb{R},\forall i=1\mathrm{to}25$, be continuous functions; then, the solution $k_i$ of the FDE

$${ }^C{D}^{\rho }{k}_i\left(y\right)=q_i\left(y\right),0\le y\le 1,$$

satisfies the integral boundary conditions

$$m_1{k}_i\left(0\right)+m_2{k}_i\left(1\right)=m_3{\int }_0^1{k}_i\left(\tau \right)d\tau ,$$

$$m_1{k}_i^{\prime }\left(0\right)+m_2{k}_i^{\prime }\left(1\right)=m_3{\int }_0^1{k}_i\left(\tau \right)d\tau ,$$

if and only if it satisfies the following integral equation:

$$\begin{array}{cc}\hfill k_i\left(y\right)& =\frac{1}{\Gamma \left(\rho \right)}{\int }_0^y{(y-\tau )}^{\rho -1}{q}_i\left(\tau \right)d\tau +\frac{A_1-m_2{m}_3(m_1+m_2-m_3)y}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\tau \right)d\tau \hfill \\ \hfill & +\frac{A_2-m_2{(m_1+m_2-m_3)}^{2}y}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -2}}{\Gamma (\rho -1)}{q}_i\left(\tau \right)d\tau \hfill \\ \hfill & +\frac{A_3+m_3(m_1+m_2)(m_1+m_2-m_3)y}{(m_1+m_2-m_3)A}{\int }_0^1{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\mu \right)d\mu d\tau ,\forall i=1,2,\dots ,25,\hfill \end{array}$$

by setting

$$A=m_1^2+m_2^2+2m_1{m}_2-\frac{3m_1{m}_3}{2}-\frac{m_2{m}_3}{2}\ne 0,$$

$$A_1=\frac{3m_2^2{m}_3}{2}-\frac{m_2{m}_3^2}{2}-m_1^2{m}_2-m_2^3-2m_1{m}_2^2+\frac{3m_1{m}_2{m}_3}{2},$$

$$A_2=m_1{m}_2^2+m_2^3-\frac{3m_2^2{m}_3}{2}-\frac{m_1{m}_2{m}_3}{2}+\frac{m_2{m}_3^2}{2},$$

$$A_3=m_1^2{m}_3+m_1{m}_2{m}_3-m_1{m}_3^2.$$

Proof. 

Assume $k_i\left(y\right)$ to be the solution of the following differential equation:

$${ }^C{D}^{\rho }{k}_i\left(y\right)=q_i\left(y\right),0\le y\le 1.$$

Applying the integral on both sides, we get

$$k_i\left(y\right)=\frac{1}{\Gamma \left(\rho \right)}{\int }_0^y{(y-\tau )}^{\rho -1}{q}_i\left(\tau \right)d\tau +s_i^1+s_i^{2}y,\forall i=1,2,\dots ,25,$$

(3)

using boundary conditions

$$m_1{s}_i^1+m_2\left[\frac{1}{\Gamma \left(\rho \right)}{\int }_0^1{(1-\tau )}^{\rho -1}{q}_i\left(\tau \right)d\tau +s_i^1+s_i^2\right]=m_3{\int }_0^1\left[{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\mu \right)d\mu +s_i^1+s_i^2\tau \right]d\tau ,$$

(4)

$$m_1{s}_i^2+m_2\left[\frac{1}{\Gamma (\rho -1)}{\int }_0^1{(1-\tau )}^{\rho -2}{q}_i\left(\tau \right)d\tau +s_i^2\right]=m_3{\int }_0^1\left[{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\mu \right)d\mu +s_i^1+s_i^2\tau \right]d\tau .$$

(5)

On solving $\left(4\right)$ and $\left(5\right)$, we obtain

$$\begin{array}{cc}\hfill s_i^1& =\frac{A_1}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\tau \right)d\tau +\frac{A_2}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -2}}{\Gamma (\rho -1)}{q}_i\left(\tau \right)d\tau \hfill \\ \hfill & +\frac{A_3}{(m_1+m_2-m_3)A}{\int }_0^1{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\mu \right)d\mu d\tau \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill s_i^2& =\frac{-m_2{m}_3}{A}{\int }_0^1\frac{{(1-\tau )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\tau \right)d\tau -\frac{m_2(m_1+m_2-m_3)}{A}{\int }_0^1\frac{{(1-\tau )}^{\rho -2}}{\Gamma (\rho -1)}{q}_i\left(\tau \right)d\tau +\frac{m_3(m_1+m_2)}{A}\hfill \\ \hfill & \times {\int }_0^1{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{q}_i\left(\mu \right)d\mu d\tau .\hfill \end{array}$$

On substituting the values of $s_i^1,s_i^2$ into $\left(3\right)$, we get the required solution $k_i\left(y\right).$

For the converse part, let $k_i\left(y\right)$ be the solution of the integral equation. Obviously, $k_i\left(y\right)$ satisfies the boundary conditions. By applying the Caputo derivative on both sides, we get

$$\begin{array}{cc}\hfill { }^C{D}^{\rho }{k}_i\left(y\right)& {=}^C{D}^{\rho }{I}^{\rho }{q}_i\left(y\right)+0+0+0,\hfill \\ \hfill { }^C{D}^{\rho }{k}_i\left(y\right)& =q_i\left(y\right),y\in [0,1].\hfill \end{array}$$

Hence, these functions having fractional derivatives on $[0,1]$ is the desired solution of the differential equation. □

Theorem 1.

(Schaefer Fixed-Point Theorem [41]) Assume that X is a Banach space and let $T:X\to X$ be a continuous and compact map. Then, either one of the following holds:

(i) The set $\{x\in X:x=\nu Tx\mathrm{for}\mathrm{some}\nu \in (0,1\left)\right\}$ is not bounded.

(ii) T has at least one fixed point.

Theorem 2.

(Banach Contraction Principle [41]) Let X be a Banach space and let $T:X\to X$ be a contraction map. Then, T has a unique fixed point in X.

3. Algorithm

• Vertex Labeling: Label the vertices of the tetramethylbutane graph as 0 or 1 by fixing the edge length to a unit of 1.
• FDEs on Each Edge: On each edge of the graph, define following the FDEs:

  $${ }^C{D}^{\rho }{k}_i\left(y\right)=r_i(y,k_i\left(y\right){,}^C{D}^{\sigma }{k}_i\left(y\right),k_i^{\prime }\left(y\right)),i=1,2,\dots ,25,0\le y\le 1,\rho \in [1,2],\sigma \in [0,1].$$
• Boundary Conditions: Apply the following integral boundary conditions to the problem:

  $$\begin{array}{cc}\hfill m_1{k}_i\left(0\right)+m_2{k}_i\left(1\right)& =m_3{\int }_0^1{k}_i\left(\tau \right)d\tau ,\hfill \\ \hfill m_1{k}_i^{\prime }\left(0\right)+m_2{k}_i^{\prime }\left(1\right)& =m_3{\int }_0^1{k}_i\left(\tau \right)d\tau .\hfill \end{array}$$
• Solution Method: Solve the system of FDEs defined on each edge using fixed-point theory by constructing an operator.

4. Main Results

Consider the space $\Omega =\{k:k\in C\left([0,1]\right),{ }^C{D}^{\sigma }k\in C\left([0,1]\right),k^{\prime }\in C\left([0,1]\right)\}$ with the norm defined by

$${\parallel k\parallel }_{\Omega }={\parallel k\parallel +\parallel }^C{D}^{\sigma }k\parallel +\parallel k^{\prime }\parallel ,$$

where

$$\parallel k\parallel =su{p}_{y\in [0,1]}{\left|k\left(y\right)\right|,\parallel }^C{D}^{\sigma }k\parallel =su{p}_{y\in [0,1]}{|}^C{D}^{\sigma }k\left(y\right)|,\parallel k^{\prime }\parallel =su{p}_{y\in [0,1]}\left|k^{\prime }\left(y\right)\right|.$$

The above-defined space is a Banach space with norm ${\parallel k\parallel }_{\Omega }$ ([42]). Hence, the product space ${\Omega }^{25}=\Omega \times \Omega \times \dots \times \Omega (25-\mathrm{times})$ is also a Banach space ([25]) with the norm

$$\parallel (k_1,k_2,\dots ,k_{25}){\parallel }_{{\Omega }^{25}}=\sum _{i=1}^{25}{\parallel k_i\parallel }_{\Omega },\mathrm{for}(k_1,k_2,\dots ,k_{25})\in {\Omega }^{25}.$$

Using Lemma 1, we define an operator $\Phi :{\Omega }^{25}\to {\Omega }^{25}$ corresponding with the boundary condition (2) by

$$\Phi (k_1,k_2,\dots ,k_{25})\left(y\right)=({\Phi }_1(k_1,k_2,\dots ,k_{25})\left(y\right),\dots ,{\Phi }_{25}(k_1,k_2,\dots ,k_{25})\left(y\right)),$$

where the ith component of $\Phi$ is defined by

$$\begin{array}{cc}\hfill {\Phi }_i(k_1,k_2,\dots ,k_{25})\left(y\right)& =\frac{1}{\Gamma \left(\rho \right)}{\int }_0^y{(y-\tau )}^{\rho -1}{r}_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)d\tau \hfill \\ \hfill & +\frac{A_1-m_2{m}_3(m_1+m_2-m_3)y}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -1}}{\Gamma \left(\rho \right)}{r}_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)d\tau \hfill \\ \hfill & +\frac{A_2-m_2{(m_1+m_2-m_3)}^{2}y}{(m_1+m_2-m_3)A}{\int }_0^1\frac{{(1-\tau )}^{\rho -2}}{\Gamma (\rho -1)}{r}_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)d\tau \hfill \\ \hfill & +\frac{A_3+m_3(m_1+m_2)(m_1+m_2-m_3)y}{(m_1+m_2-m_3)A}\times \hfill \\ \hfill & {\int }_0^1{\int }_0^{\tau }\frac{{(\tau -\mu )}^{\rho -1}}{\Gamma \left(\rho \right)}{r}_i\left(\mu ,k_i\left(\mu \right){,}^C{D}^{\sigma }{k}_i\left(\mu \right),k_i^{\prime }\left(\mu \right)\right)d\mu d\tau ,\hfill \end{array}$$

where $A,A_1,A_2,A_3$ are defined in Lemma 1.

Theorem 3.

Let the functions $r_i:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},i=1,2,\dots ,25$ be continuous and bounded such that $|r_i(y,x,t,z)|<C_i,wherex,t,z\in \mathbb{R}$, and $C_i$ is a positive constant. Then, the nonlinear Caputo fractional boundary value problem (1) has at least a solution satisfying the boundary conditions (2).

Proof. 

We will show the existence result via Schaefer’s fixed-point Theorem 1. The boundary value problem has a solution if $\Phi$ has a fixed point in space ${\Omega }^{25}$. In the beginning, it will be shown that the operator $\Phi :{\Omega }^{25}\to {\Omega }^{25}$ is completely continuous.

Clearly, $\Phi$ is continuous map due to the continuity of the functions $r_i,i=1,2,\dots ,25$. We have to show only the completeness.

Let U be a bounded subset of ${\Omega }^{25}$. Now, for any $k=(k_1,k_2,\dots ,k_{25})\in U,y\in [0,1]$, we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{i}k\parallel & \le \frac{1}{\Gamma \left(\rho \right)}{\int }_0^y\left|{(y-\tau )}^{\rho -1}\right|\left|r_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +\frac{|A_1|+|m_2\left|\right|m_3\left|\right|(m_1+m_2-m_3)\left|\right|y|}{\left|\right(m_1+m_2-m_3\left)\right|\left|A\right|}{\int }_0^1\frac{{|(1-\tau )}^{\rho -1}|}{\Gamma \left(\rho \right)}\left|r_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +\frac{|A_2|+|m_2\left|\right|(m_1+m_2-m_3){|}^2\left|y\right|}{\left|\right(m_1+m_2-m_3\left)\right|\left|A\right|}{\int }_0^1\frac{{|(1-\tau )}^{\rho -2}|}{\Gamma (\rho -1)}\left|r_i\left(\tau ,k_i\left(\tau \right){,}^C{D}^{\sigma }{k}_i\left(\tau \right),k_i^{\prime }\left(\tau \right)\right)\right|d\tau \hfill \\ \hfill & +\frac{|A_3|+|m_3\left|\right|(m_1+m_2)\left|\right|(m_1+m_2-m_3)\left|\right|y|}{\left|\right(m_1+m_2-m_3\left)\right|\left|A\right|}\times \hfill \\ \hfill & {\int }_0^1{\int }_0^{\tau }\frac{{|(\tau -\mu )}^{\rho -1}|}{\Gamma \left(\rho \right)}\left|r_i\left(\mu ,k_i\left(\mu \right){,}^C{D}^{\sigma }{k}_i\left(\mu \right),k_i^{\prime }\left(\mu \right)\right)\right|d\mu d\tau .\hfill \end{array}$$

$$\Rightarrow \parallel {\Phi }_{i}k\parallel \le C_i{M}^{*},i=1,2,\dots ,25,$$

(6)

where

$$\begin{array}{cc}\hfill M^{*}=& \frac{1}{\Gamma (\rho +1)}+\frac{|A_1|+|m_2\left|\right|m_3\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +1)}+\frac{|A_2|+|m_2\left|\right|(m_1+m_2-m_3){|}^2}{|(m_1+m_2-m_3)\left|\right|A|\Gamma \left(\rho \right)}\hfill \\ \hfill & +\frac{|A_3|+|m_3\left|\right|(m_1+m_2)\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +2)}.\hfill \end{array}$$

Similarly,

$${\parallel }^C{D}^{\rho }{\Phi }_{i}k\parallel \le C_i{M}^{**},i=1,2,\dots ,25,$$

(7)

$$\parallel {\Phi }_i^{\prime }k\parallel \le C_i{M}^{***},i=1,2,\dots ,25,$$

(8)

where

$$\begin{array}{c}\hfill M^{**}=\frac{1}{\Gamma (\rho -\sigma +1)}+\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (\rho +1)\Gamma (2-\sigma )}+\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma \left(\rho \right)\Gamma (2-\sigma )}+\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (\rho +2)\Gamma (2-\sigma )},\end{array}$$

$$\begin{array}{c}\hfill M^{***}=\frac{1}{\Gamma \left(\rho \right)}+\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (\rho +1)}+\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma \left(\rho \right)}+\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (\rho +2)}.\end{array}$$

Using $\left(6\right),\left(7\right),\mathrm{and}\left(8\right)$, we obtain

$$\parallel {\Phi }_i{k\parallel }_{\Omega }=\parallel {\Phi }_i{k\parallel +\parallel }^C{D}^{\sigma }{\Phi }_{i}k\parallel +\parallel {\Phi }_i^{\prime }k\parallel \le C_i(M^{*}+M^{**}+M^{***}),i=1,2,\dots ,25.$$

Hence,

$$\begin{array}{c}\hfill {\parallel \Phi k\parallel }_{{\Omega }^{25}}=\sum _{i=1}^{25}{\parallel {\Phi }_{i}k\parallel }_{\Omega }\le \sum _{i=1}^{25}{C}_i(M^{*}+M^{**}+M^{***})<\infty .\end{array}$$

So, $\Phi$ is uniformly bounded.

Now, the equicontinuity of the operator $\Phi$ will be shown. For any $k=(k_1,k_2,\dots ,k_{25})\in U,y_1,y_2\in [0,1],y_1<y_2$, we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{i}k\left(y_2\right)-{\Phi }_{i}k\left(y_1\right)\parallel & \le C_i[\frac{y_2^{\rho }-y_1^{\rho }}{\Gamma \left(\rho \right)}+\frac{|m_2\left|\right|m_3|(y_2-y_1)}{\left|A\right|\Gamma (\rho +1)}+\frac{|m_2\left|\right|(m_1+m_2-m_3)|(y_2-y_1)}{\left|A\right|\Gamma \left(\rho \right)}\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|(y_2-y_1)}{\left|A\right|\Gamma (\rho +2)}].\hfill \end{array}$$

(9)

We also have

$$\begin{array}{cc}\hfill {\parallel }^C{D}^{\sigma }{\Phi }_{i}k\left(y_2\right){-}^C{D}^{\sigma }{\Phi }_{i}k\left(y_1\right)\parallel & \le C_i[\frac{y_2^{\rho -\sigma }-y_1^{\rho -\sigma }}{\Gamma \left(\rho \right)}+\frac{|m_2\left|\right|m_3|(y_2^{1-\sigma }-y_1^{1-\sigma })}{\left|A\right|\Gamma (\rho +1)\Gamma (2-\sigma )}\hfill \\ \hfill & +\frac{|m_2\left|\right|(m_1+m_2-m_3)|(y_2^{1-\sigma }-y_1^{1-\sigma })}{\left|A\right|\Gamma \left(\rho \right)\Gamma (2-\sigma )}\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|(y_2^{1-\sigma }-y_1^{1-\sigma })}{\left|A\right|\Gamma (\rho +2)\Gamma (2-\sigma )}].\hfill \end{array}$$

(10)

Next,

$$\parallel {\Phi }_i^{\prime }k\left(y_2\right)-{\Phi }_i^{\prime }k\left(y_1\right)\parallel \le C_i\left[\frac{y_2^{\rho -1}-y_1^{\rho -1}}{\Gamma (\rho -1)}\right].$$

(11)

So,

$$\begin{array}{cc}\hfill \parallel {\Phi }_{i}k\left(y_2\right)-{\Phi }_{i}k\left(y_1\right){\parallel }_{\Omega }& =\parallel {\Phi }_{i}k\left(y_2\right)-{\Phi }_{i}k\left(y_1\right){\parallel +\parallel }^C{D}^{\sigma }{\Phi }_{i}k\left(y_2\right){-}^C{D}^{\sigma }{\Phi }_{i}k\left(y_1\right)\parallel \hfill \\ \hfill & +\parallel {\Phi }_i^{\prime }k\left(y_2\right)-{\Phi }_i^{\prime }k\left(y_1\right)\parallel .\hfill \end{array}$$

From $\left(9\right),\left(10\right),$ and $\left(11\right)$,

$$\parallel {\Phi }_{i}k\left(y_2\right)-{\Phi }_{i}k\left(y_1\right){\parallel }_{\Omega }\to 0\mathrm{as}{y}_1\to y_2.$$

So,

$$\parallel \Phi k\left(y_2\right)-\Phi k\left(y_1\right){\parallel }_{{\Omega }^{25}}\to 0\mathrm{as}{y}_1\to y_2.$$

Therefore, the operator $\Phi$ is equicontinous on ${\Omega }^{25}$. Applying the Arzela Ascoli theorem, we conclude that $\Phi$ is complete.

Furthermore, we have to prove that the set $P=\{(k_1,k_2,\dots ,k_{25})\in {\Omega }^{25}:(k_1,k_2,\dots ,$$k_{25})=\nu \Phi (k_1,k_2,\dots ,k_{25}),\nu \in (0,1)\}$ is bounded.

Assume that $(k_1,k_2,\dots ,k_{25})\in P$; then, $(k_1,k_2,\dots ,k_{25})=\nu \Phi (k_1,k_2,\dots ,k_{25})$. So,

$$k_i\left(y\right)=\nu {\Phi }_i(k_1,k_2,\dots ,k_{25}),\mathrm{for}\mathrm{every}0\le y\le 1\mathrm{and}i=1,2,\dots 25.$$

Therefore,

$$\parallel k_i\parallel \le \nu C_i{M}^{*},i=1,2,\dots ,25.$$

(12)

$${\parallel }^C{D}^{\sigma }{k}_i\parallel \le \nu C_i{M}^{**},i=1,2,\dots ,25.$$

(13)

$$\parallel k_i^{\prime }\parallel \le \nu C_i{M}^{***},i=1,2,\dots ,25.$$

(14)

Using $\left(12\right),\left(13\right),\mathrm{and}\left(14\right)$, we obtain

$$\parallel k_i{\parallel }_{\Omega }=\parallel k_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i\parallel +\parallel k_i^{\prime }\parallel \le \nu C_i(M^{*}+M^{**}+M^{***}),i=1,2,\dots ,25.$$

Hence,

$$\begin{array}{c}\hfill \parallel k=(k_1,k_2,\dots ,k_{25}){\parallel }_{{\Omega }^{25}}=\sum _{i=1}^{25}{\parallel k_i\parallel }_{\Omega }\le \nu \sum _{i=1}^{25}{C}_i(M^{*}+M^{**}+M^{***})<\infty .\end{array}$$

This proves that set P is bounded. Hence, according to Schaefer’s fixed-point theorem, $\Phi$ has a fixed point in ${\Omega }^{25}$, which ensures at least one solution of the nonlinear FDEs (1) on $[0,1]$. □

Theorem 4.

Let the functions $r_i:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},i=1,2,\dots 25$ be continuous. Assume that $b_i:[0,1]\to {\mathbb{R}}^{+}$ are non-negative continuous functions on $[0,1],i=1,2,\dots ,25$ satisfying the condition

$$\begin{array}{c}\hfill |r_i(y,x,t,z)-r_i(y,x_1,t_1,z_1)|\le b_i\left(y\right)\left(\right|x-x_1|+|t-t_1|+|z-z_1\left|\right),\\ \hfill where\parallel b_i\left(y\right)\parallel =su{p}_{y\in [0,1]}\left|b_i\left(y\right)\right|andx,t,z,x_1,t_1,z_1\in \mathbb{R},\end{array}$$

and

$$\left(\sum _{i=1}^{25}\parallel b_i\parallel (M^{*}+M^{**}+M^{***})\right)<1,$$

where $M^{*},M^{**},M^{***}$ are constants given in Theorem 3;

then, the nonlinear FDEs (1) has a unique solution on $[0,1]$ satisfying the boundary conditions (2).

Proof. 

We will apply the Banach contraction principle 2 to prove the uniqueness of the solution. We will show that $\Phi$ is a contraction mapping.

Assume that $k=(k_1,k_2,\dots ,k_{25})\in {\Omega }^{25},w=(w_1,w_2,\dots ,w_{25})\in {\Omega }^{25},y\in [0,1]$.

Then, we have

$$\begin{array}{cc}\hfill \parallel {\Phi }_{i}k\left(y\right)-{\Phi }_{i}w\left(y\right)\parallel & \le \frac{1}{\Gamma (\rho +1)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|A_1|+|m_2\left|\right|m_3\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +1)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|A_2|+|m_2\left|\right|(m_1+m_2-m_3){|}^2}{|(m_1+m_2-m_3)\left|\right|A|\Gamma \left(\rho \right)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|A_3|+|m_3\left|\right|(m_1+m_2)\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +2)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right).\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\parallel }^C{D}^{\sigma }{\Phi }_{i}k\left(y\right){-}^C{D}^{\sigma }{\Phi }_{i}w\left(y\right)\parallel & \le \frac{1}{\Gamma (\rho -\sigma +1)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +1)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma (2-\sigma )\Gamma \left(\rho \right)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +2)}\times \hfill \\ \hfill & \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel {\Phi }_i^{\prime }k\left(y\right)-{\Phi }_i^{\prime }w\left(y\right)\parallel & \le \frac{1}{\Gamma \left(\rho \right)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (\rho +1)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma \left(\rho \right)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right)\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (\rho +2)}\parallel b_i\parallel \left(\parallel k_i-w_i{\parallel +\parallel }^C{D}^{\sigma }{k}_i{-}^C{D}^{\sigma }{w}_i\parallel +\parallel k_i^{\prime }-w_i^{\prime }\parallel \right).\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill \parallel {\Phi }_{i}k\left(y\right)-{\Phi }_i{w\left(y\right)\parallel }_{\Omega }& \le \parallel b_i\parallel \left(\parallel k_i-w_i{\parallel }_{\Omega }\right)[\frac{1}{\Gamma (\rho +1)}\hfill \\ \hfill & +\frac{|A_1|+|m_2\left|\right|m_3\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +1)}+\frac{|A_2|+|m_2\left|\right|(m_1+m_2-m_3){|}^2}{|(m_1+m_2-m_3)\left|\right|A|\Gamma \left(\rho \right)}\hfill \\ \hfill & +\frac{|A_3|+|m_3\left|\right|(m_1+m_2)\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +2)}+\frac{1}{\Gamma (\rho -\sigma +1)}\hfill \\ \hfill & +\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +1)}+\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma (2-\sigma )\Gamma \left(\rho \right)}+\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +2)}\hfill \\ \hfill & +\frac{1}{\Gamma \left(\rho \right)}+\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (\rho +1)}+\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma \left(\rho \right)}+\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (\rho +2)}].\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\parallel \Phi k\left(y\right)-\Phi w\left(y\right)\parallel }_{{\Omega }^{25}}& =\sum _{i=1}^{25}{\parallel {\Phi }_{i}k\left(y\right)-{\Phi }_{i}w\left(y\right)\parallel }_{\Omega }\hfill \\ \hfill & \le \left(\sum _{i=1}^{25}\parallel b_i\parallel (M^{*}+M^{**}+M^{***})\right)\left(\sum _{i=1}^{25}{\parallel k_i-w_i\parallel }_{\Omega }\right).\hfill \end{array}$$

$$\begin{array}{c}\hfill {\parallel \Phi k\left(y\right)-\Phi w\left(y\right)\parallel }_{{\Omega }^{25}}\le \left(\sum _{i=1}^{25}\parallel b_i\parallel (M^{*}+M^{**}+M^{***})\right){\parallel k_i-w_i\parallel }_{{\Omega }^{25}}.\end{array}$$

Hence, $\Phi$ is a contraction mapping. Hence, according to the Banach contraction principle, the nonlinear FDEs has a unique solution on $[0,1].$ □

5. Ulam–Hyers Stability

Because the concept of stability is so essential in defining the solutions to many dynamic systems, we present the Ulam–Hyers stability in connection to the FDE on the tetramethylbutane graph. Readers should refer to [43,44,45] for more information. Consistently with [45], we have the following definitions.

Definition 3.

A system of FDEs defined on each edge of the tetramethylbutane graph is Ulam–Hyers stable if there exists $c_i>0$ such that for every ${ϵ}_i>0$ and the solutions $k_i^{*}$ of the inequalities

$$|{ }^C{D}^{\rho }{k}_i^{*}\left(y\right)-r_i\left(y,k_i^{*}\left(y\right){,}^C{D}^{\sigma }{k}_i^{*}\left(y\right),k_i^{{*}^{\prime }}\left(y\right)\right)|<{ϵ}_i,y\in [0,1],$$

(15)

there exists a unique solution $k_i$ of the system satisfying $\parallel k_i^{*}\left(y\right)-k_i\left(y\right)\parallel <c_i{ϵ}_i,i=1,2,\dots 25$.

Definition 4.

The function $k_i^{*}$ is a solution of the inequality $\left(15\right)$ if there exists a function $p_i:[0,1]\to \mathbb{R}$ such that $|p_i\left(y\right)|\le {ϵ}_i$ and

$${ }^C{D}^{\rho }{k}_i^{*}\left(y\right)=r_i\left(y,k_i^{*}\left(y\right){,}^C{D}^{\sigma }{k}_i^{*}\left(y\right),k_i^{{*}^{\prime }}\left(y\right)\right)+p_i\left(y\right),y\in [0,1].$$

Theorem 5.

Under the assumptions of Theorem 4, the system of FDEs defined on the tetramethylbutane graph is Ulam–Hyers stable.

Proof. 

Assume $k_i^{*}$ to be the solution satisfying the inequalities in $\left(15\right)$, and let $k_i$ be the unique solution of problem (1). Now,

$$\begin{array}{cc}\hfill \parallel k_i^{*}\left(y\right)-k_i\left(y\right)\parallel & \le \frac{1}{\Gamma (\rho +1)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|A_1|+|m_2\left|\right|m_3\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +1)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|A_2|+|m_2\left|\right|(m_1+m_2-m_3){|}^2}{|(m_1+m_2-m_3)\left|\right|A|\Gamma \left(\rho \right)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|A_3|+|m_3\left|\right|(m_1+m_2)\left|\right|(m_1+m_2-m_3)|}{|(m_1+m_2-m_3)\left|\right|A|\Gamma (\rho +2)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right].\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\parallel }^C{D}^{\sigma }{k}_i^{*}\left(y\right){-}^C{D}^{\sigma }{k}_i\left(y\right)\parallel & \le \frac{1}{\Gamma (\rho -\sigma +1)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +1)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma (2-\sigma )\Gamma \left(\rho \right)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (2-\sigma )\Gamma (\rho +2)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right].\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel k_i^{{*}^{\prime }}\left(y\right)-k_i^{\prime }\left(y\right)\parallel & \le \frac{1}{\Gamma \left(\rho \right)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]+\frac{|m_2\left|\right|m_3|}{\left|A\right|\Gamma (\rho +1)}[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)\hfill \\ \hfill & +{ϵ}_i]+\frac{|m_2\left|\right|(m_1+m_2-m_3)|}{\left|A\right|\Gamma \left(\rho \right)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right]\hfill \\ \hfill & +\frac{|m_3\left|\right|(m_1+m_2)|}{\left|A\right|\Gamma (\rho +2)}\left[\parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)+{ϵ}_i\right].\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill \parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }& \le \parallel b_i\parallel \left(\parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\right)(M^{*}+M^{**}+M^{***})+{ϵ}_i(M^{*}+M^{**}+M^{***}),\hfill \end{array}$$

$$\begin{array}{c}\hfill \parallel k_i^{*}\left(y\right)-k_i{\left(y\right)\parallel }_{\Omega }\le c_i{ϵ}_i,\mathrm{where}{c}_i=\frac{(M^{*}+M^{**}+M^{***})}{1-\parallel b_i\parallel (M^{*}+M^{**}+M^{***})}>0,i=1,2,\dots 25.\end{array}$$

This completes the proof. □

6. Examples

Example 1.

Consider the system of FDEs

$$\left\{\begin{array}{c}{ }^C{D}^{1.43}{k}_1\left(y\right)=0.007ysin\left(k_1\left(y\right)\right)+\frac{{490y|}^C{D}^{0.32}{k}_1{\left(y\right)|}^2}{{\mathrm{70,000}+\mathrm{70,000}|}^C{D}^{0.32}{k}_1{\left(y\right)|}^2}+\frac{98y\left(ta{n}^{-1}\left(k_1^{\prime }\left(y\right)\right)\right)}{\mathrm{14,000}},\hfill \\ { }^C{D}^{1.43}{k}_2\left(y\right)=\frac{1}{40{(y^4+4)}^2}\left(sin\left(k_2\left(y\right)\right)+{|}^C{D}^{0.32}{k}_2\left(y\right)|\right)+cosh(1+y^2)+\frac{si{n}^{-1}\left(k_2^{\prime }\left(y\right)\right)}{40{(y^4+2)}^2},\hfill \\ { }^C{D}^{1.43}{k}_3\left(y\right)=0.00003y\left|sin\left(k_3\left(y\right)\right)\right|+\frac{0.00003y|k_3^{\prime }{\left(y\right)|}^3}{2+|k_3^{\prime }{\left(y\right)|}^3}+\frac{0.00003y\left|sin\right(2\pi { }^C{D}^{0.32}{k}_3\left(y\right)\left)\right|}{2\pi },\hfill \\ { }^C{D}^{1.43}{k}_4\left(y\right)=\frac{y(k_4\left(y\right)+6)}{500}+\frac{y{ }^C{D}^{0.32}{k}_4\left(y\right))}{500}+\frac{y(k_4^{\prime }\left(y\right)+24)}{500},\hfill \end{array}\right.$$

(16)

with the integral boundary conditions

$$\left\{\begin{array}{cc}0.3k_1\left(0\right)+0.5k_1\left(1\right)\hfill & =0.23{\int }_0^1{k}_1\left(\tau \right)d\tau ,\hfill \\ 0.3k_1^{\prime }\left(0\right)+0.5k_1^{\prime }\left(1\right)\hfill & =0.23{\int }_0^1{k}_1\left(\tau \right)d\tau ,\hfill \\ 0.3k_2\left(0\right)+0.5k_2\left(1\right)\hfill & =0.23{\int }_0^1{k}_2\left(\tau \right)d\tau ,\hfill \\ 0.3k_2^{\prime }\left(0\right)+0.5k_2^{\prime }\left(1\right)\hfill & =0.23{\int }_0^1{k}_2\left(\tau \right)d\tau ,\hfill \\ 0.3k_3\left(0\right)+0.5k_3\left(1\right)\hfill & =0.23{\int }_0^1{k}_3\left(\tau \right)d\tau ,\hfill \\ 0.3k_3^{\prime }\left(0\right)+0.5k_3^{\prime }\left(1\right)\hfill & =0.23{\int }_0^1{k}_3\left(\tau \right)d\tau ,\hfill \\ 0.3k_4\left(0\right)+0.5k_4\left(1\right)\hfill & =0.23{\int }_0^1{k}_4\left(\tau \right)d\tau ,\hfill \\ 0.3k_4^{\prime }\left(0\right)+0.5k_4^{\prime }\left(1\right)\hfill & =0.23{\int }_0^1{k}_4\left(\tau \right)d\tau ,\hfill \end{array}\right.$$

(17)

where $\rho =1.43,\sigma =0.32,m_1=0.3,m_2=0.5,m_3=0.23,n=4$. ${ }^C{D}^{1.43}$ and ${ }^C{D}^{0.32}$ are the fractional derivative operators of order $1.43$ and $0.32$, respectively, in the Caputo sense.

Now, we define the continuous functions $r_i:[0,1]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ using

$$\left\{\begin{array}{c}{r}_1(y,x,t,z)=0.007ysin\left(x\right)+\frac{{490y\left|t\right|}^2}{\mathrm{70,000}+{\mathrm{70,000}\left|t\right|}^2}+\frac{98y\left(ta{n}^{-1}\left(z\right)\right)}{\mathrm{14,000}},\hfill \\ r_2(y,x,t,z)=\frac{1}{40{(y^4+2)}^2}\left(sin\left(x\right)+\left|t\right|\right)+cosh(1+y^2)+\frac{si{n}^{-1}\left(z\right)}{40{(y^4+2)}^2},\hfill \\ r_3(y,x,t,z)=0.00003y\left|sin\left(x\right)\right|+\frac{{0.00003y\left|z\right|}^3}{2+{\left|z\right|}^3}+\frac{0.00003y\left|sin\right(2\pi t\left)\right|}{2\pi },\hfill \\ r_4(y,x,t,z)=\frac{y(x+6)}{500}+\frac{yt}{500}+\frac{y(z+24)}{500}.\hfill \end{array}\right.$$

Then, we have

$$|r_1(y,x,t,z)-r_1(y,x_1,t_1,z_1)|\le 0.007y(|x-x_1|+|t-t_1|+|z-z_1\left|\right),$$

$$|r_2(y,x,t,z)-r_2(y,x_1,t_1,z_1)|\le \frac{1}{40{(y^4+2)}^2}\left(\right|x-x_1|+|t-t_1|+|z-z_1\left|\right),$$

$$|r_3(y,x,t,z)-r_3(y,x_1,t_1,z_1)|\le 0.00003y(|x-x_1|+|t-t_1|+|z-z_1\left|\right),$$

$$|r_4(y,x,t,z)-r_4(y,x_1,t_1,z_1)|\le \frac{y}{500}\left(\right|x-x_1|+|t-t_1|+|z-z_1\left|\right).$$

So,

$b_1\left(y\right)=0.007y,b_2\left(y\right)=\frac{1}{40{(y^4+2)}^2},b_3\left(y\right)=0.00003y,b_4\left(y\right)=\frac{y}{500}$,

and we get

$\parallel b_1\parallel =0.007,\parallel b_2\parallel =\frac{1}{160},\parallel b_3\parallel =0.00003,\parallel b_4\parallel =\frac{1}{500}$.

After manipulation, we obtain $M^{*}=12.52594,M^{**}=2.0399,M^{***}=2.11477$

and

$$(\parallel b_1\parallel +\parallel b_2\parallel +\parallel b_3\parallel +\parallel b_4\parallel )(M^{*}+M^{**}+M^{***})=0.117+0.1042+0.005+0.0336=0.253<1.$$

Hence, according to Theorem 4, the problem $\left(16\right)$ satisfying the conditions $\left(17\right)$ has unique solution on $[0,1]$. Moreover, this problem is Ulam–Hyers stable.

Example 2.

We solve the following FDEs analytically in this example:

$${ }^C{D}^{\rho }{k}_i\left(y\right)=r_i\left(y\right),0\le y\le 1,1\le \rho \le 2,$$

satisfying the integral boundary conditions with $r_i\left(y\right)=y^i,i=1,2,\dots ,25$

$$0.3k_i\left(0\right)+0.5k_i\left(1\right)=0.23{\int }_0^1{k}_i\left(\tau \right)d\tau ,$$

$$0.3k_i^{\prime }\left(0\right)+0.5k_i^{\prime }\left(1\right)=0.25{\int }_0^1{k}_i\left(\tau \right)d\tau ,$$

where $r_i\left(y\right)=y^i,i=1,2,\dots ,25$ are continuous and bounded functions. Hence, by applying Theorem 3, this problem has a solution on $[0,1]$.

With the above operator defined in Lemma 1, we get

$$\begin{array}{cc}\hfill k_i\left(y\right)& =\frac{\Gamma (i+1)}{\Gamma (i+\rho +1)}{y}^{i+\rho }+\frac{\Gamma (i+1)}{\Gamma (i+\rho +1)}(0.1492+0.2430y)+\frac{\Gamma (i+1)}{\Gamma (i+\rho )}(0.5949y-0.4019)\hfill \\ \hfill & -\frac{\Gamma (i+1)}{\Gamma (i+\rho +2)}(0.1441+0.1049y),\forall i=1,2,\dots ,25.\hfill \end{array}$$

7. Conclusions

Since the methodology used in [24,25] can only be used for star graphs, it will not work here. The chemical structure of the tetramethylbutane compound can be viewed as a graph that is a generalization of a star graph. The tetramethylbutane graph is actually a non-star graph comprising eight junction nodes. So, we chose another method for labeling the vertices with 0 or 1 to fix the edge length to a unit of 1. We examined FDEs in the sense of a Caputo operator on each edge of the tetramethylbutane graph with the boundary conditions on each vertex of the graph. Employing the Schaefer and Banach fixed-point theorems, we attained the conditions for proving the existence and uniqueness of solutions for the FDEs system (1) satisfying the integral boundary conditions (2). Additionally, we conducted a stability analysis of the system with the Hyers–Ulam method and provided two illustrative examples to support our findings. Due to the significance of differential equations in chemical graph theory, we discussed the system of FDEs (1) on the molecular graph representation of tetramethylbutane, which is an illustration of a non-star graph. This study can be important in the fractional mathematical modeling of various chemical compounds of molecules within complex non-star graph frameworks. Our approach is simple to use and can be used for many different types of graphs, such as chordal bipartite graphs, which are widely utilized in biology and computer networking.