Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph

Chemical graph theory is a field of mathematics that studies ramifications of chemical network interactions. Using the concept of star graphs, several investigators have looked into the solutions to certain boundary value problems. Their choice to utilize star graphs was based on including a common point connected to other nodes. Our aim is to expand the range of the method by incorporating the graph of hexasilinane compound, which has a chemical formula H12Si6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{H}_{12} \mathrm{Si}_{6}$\end{document}. In this paper, we examine the existence of solutions to fractional boundary value problems on such graphs, where the fractional derivative is in the Caputo sense. Finally, we include an example to support our significant findings.


Introduction
A subfield of mathematics known as chemical graph theory is concerned with the implications of the connectedness of chemical networks. Whether natural or synthetic, almost every chemical system may be represented by a chemical graph (i.e., molecular transformations in a chemical reaction). Moreover, the word "chemical" is used to highlight that, in contrast to graph theory, we may depend on scientific investigation of many ideas and theorems rather than exact mathematical proofs, which is a significant difference.
When it comes to graph theory, Lumer [1] was the first to use differential equation theory on graphs. With the use of ramification spaces and different operator specifications, he explored the solutions of extended evolution equations. Zavgorodnij [2] investigated linear differential equations in 1989 using a geometric network, with suggested boundary value problem solutions arranged at the network interior nodes. On the other hand, Gordeziani et al. [3] utilized the double-sweep method to obtain analytical results for differential equations, which they observed to be more productive on graphs;. However, utilizing fixed point methods, a limited amount of studies on star graphs (see Fig. 1) associated with the solutions of boundary value problems has emerged in the particular research (see, e.g., [4,5]). Mehandiratta et al. [5], extended the work of Graef et al. [4] by proposing the following fractional differential equation: where ∈ (1, 2], ν ∈ (0, -1], S γ : [0,˜ γ ] × R × R → R are continuous functions, and D δ denotes the Caputo fractional derivative of order δ ∈ { , ν}.
Recently, Mophou et al. [6] investigated the solution of the following fractional Sturm-Liouville boundary value problems on a star graph: where D a + and D b -1 γ , γ = 1, 2, . . . , n, are, respectively, the left Riemann-Liouville and right Caputo fractional derivatives of order ∈ (0, 1), I a + is the Riemann-Liouville fractional For the recent research in this area, we refer to [7][8][9] and the references therein.
To extend the work presented in [4][5][6], we use the concept of hexasilinane graphs (see Fig. 2), which are more general than star graphs.
Moreover, the techniques employed in [4][5][6] are inadequate since the hexasilinane graphs contain more junction points than star graphs. As a result, we adopt a different method, in which the graph vertices are labeled by 0 or 1 with edge length |b k | = 1 (see Fig. 3).
Our goal is to prove the existence of solutions to the suggested problem (1.4) by using appropriate fixed point theorems. Finally, we give an example to demonstrate the significance of our findings in light of the existing literature.

Preliminaries
where [ ] is the integer part of .
if and only if z γ is a solution of the fractional integral equation Proof Let z γ be a solution of (2.1), where γ = 1, 2, . . . , 18. Thus there are constants b Using the boundary conditions for (2.1), we have Substituting the values of b 3), we obtain the desired solution (2.2). With regard to the contrary, when z γ is a solution of (2.1), it is self-evident that z γ is a solution of (2.2).
We now present the fixed point theorems of Krasnoselskii and Schaefer.

Theorem 2.3 ([26])
Let V be a closed, bounded, convex, and nonempty subset of a Banach space U, and let S 1 , S 2 : V → U be two operators such that S 1 k + S 2 k ∈ V whenever k, k ∈ V. Suppose that S 1 is compact and continuous and S 2 is a contraction. Then S 1 + S 2 has a fixed point.

Theorem 2.4 ([26])
Let U be a Banach space, and let S : U → U be a completely continuous mapping. If the set {z ∈ U : z = ϑSz for some ϑ ∈ [0, 1]} is bounded, then S has at least one fixed point in U .

Main results
Throughout this paper, Referring to Lemma 2.2, we introduce the operator S : U → U by To facilitate calculations, we use the following notation:
We will show that is bounded. For this, let (z 1 , z 2 , . . . , z 18 ) ∈ . Then we can write and by similar computations we have where I * 0 -I * 3 are given in (3.5)-(3.8). Hence which shows the boundedness of . Now using Theorem 2.4 and Lemma 2.2, we see that S has a fixed point in U . This demonstrates that (1.4) does indeed have a solution.
We will now demonstrate that S 1 is uniformly bounded. For this, we have for all z ∈ V ε γ . Thus which shows that S 1 is uniformly bounded on V ε γ . Now we will prove that S 1 is compact on V ε γ . For this, let s 1 , s 2 ∈ [0, 1] with s 1 < s 2 . Then we have Hence |(S Hence (S 1 z)(s 2 ) -(S 1 z)(s 1 ) U tends to zero as s 1 → s 2 . Thus S 1 is equicontinuous, and therefore S 1 is relatively compact operator on V ε γ . So S 1 is compact on V ε γ by the Arzelà-Ascoli theorem.
Since < 1, S 2 is a contraction on V ε γ . As a result of Theorem 2.3, we infer that S contains a fixed point, which is a solution to problem (1.4).
To illustrate the significance of our results, we provide the following example.