Figures
Abstract
A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method’s range by introducing the concept of an octane graph, which is an essential organic compound having the formula C_{8}H_{18}. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph’s edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study.
Citation: Sintunavarat W, Turab A (2022) A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory. PLoS ONE 17(8): e0270148. https://doi.org/10.1371/journal.pone.0270148
Editor: Mohammed S. Abdo, Hodeidah University, YEMEN
Received: February 24, 2022; Accepted: June 3, 2022; Published: August 12, 2022
Copyright: © 2022 Sintunavarat, Turab. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: Data sharing is not applicable to this article. All relevant data are within the manuscript.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Chemical graph theory is concerned with all elements of graph theory’s application to chemistry. In contrast to graph theory, the term chemical emphasizes that one may rely on the intuitive understanding of several concepts and theorems in chemical graph theory rather than precise mathematical proofs. On the other hand, graph theory is used to mathematically portray the structural properties of chemical compounds to understand them. A substance’s physical properties, such as its boiling point, are related to its geometric structure.
The concept of chemical indices is one of the most fundamental ideas in chemical graph theory. This is done by associating a numerical value with a graph structure that frequently has some relationship with the characteristics of the relevant molecules. As a result, these chemical indices are often presented as identifiers of chemical components. From a graph-theoretical standpoint, investigating such a chemical index often entails researching its behavior in various graphs, particularly minima and maxima, as well as upper and lower limits in terms of various graph characteristics.
Graph theory is closely connected to topology (in fact, it is one-dimensional topology [1]), probability, group theory, matrix theory, set theory, numerical analysis, and combinatorics. It has been used in a wide range of subjects, including psychology [2] and nuclear physics [3], economics [4] and theoretical physics [5], biomathematics [6] and linguistics [7], technology [8] and anthropology [9], sociology [10] and zoology [11], biology [12] and engineering [13], computer science [14] and geography [15], and so on.
Chemical graph theory has grown significantly in popularity in recent years (for the detail, see [16–18]). Numerous factors contribute to graph theory’s growing prominence in chemistry (see [19–21]). First, few concepts in the natural sciences are more closely related to the concept of a graph than the institutional formula of a chemical compound (see [22]). Thus, it would seem that (chemical) graph theory provides the natural language of chemistry through which scientists interact. Second, graph theory enables researchers to make many intuitive assumptions about the composition and reactivity of diverse substances using simple principles. Thirdly, graph theory may describe, classify, and categorize a vast range of chemical interactions (for the detail, see [23–25]). Lastly, graphs provide practical tools for the computer-assisted synthesis design (see [26, 27]).
In [28], Lumer modified the specified local operators on ramification spaces and investigated the solutions of evolution equations on graphs. After that, some researchers examined the solutions of differential equations on graphs by using different methods (for the detail, see [29, 30]).
However, there are just a few research on boundary value problems with graphs in which particular fixed point methods have shown the existence of solutions (see [31, 32]). In such studies, the authors utilized the concept of a star graph, which has only one junction node (see Fig 1). Since then, various authors have used notable methods to extend the problem in different directions see [33–38] and the references within.
The methods described in [31, 32] for determining the origin at edges other than the junction node are inadequate since graphs might contain several junction nodes in general (for examples, see Figs 2 and 3).
Additionally, the authors of [31, 32] treated the length of each edge as a variable, but the length of all edges may be considered constant from the start. Here, we use a novel approach in which we assign a value of 0 or 1 to the vertices of the proposed graph with for all k = 1, 2, …, 25 (see Fig 4).
By utilizing the ideas mentioned above, here, we investigate the existence of solutions to the boundary value problem, which is stated for each k = 1, 2, …, 25 by (1.1) where is an unknown function, with μ_{3} ≠ μ_{1}, and represent the Caputo fractional derivative of orders p ∈ (1, 2] and q ∈ (0, 1), respectively. Also, is a continuously differentiable function for k = 1, 2, …, 25.
In this way, the orientation of the linked edge determines the label given to each vertex. When we proceed along a random edge, the starting and ending vertex labels are interpreted as 0 and 1, and vice versa. As a consequence, some vertices may have the labels 0 and 1, and the origin of each edge is not constant; it fluctuates depending on the path of motion along the border. We are not obliged to normalize the length of each edge using the provided adjustment, and we may also pick one of the associated edge’s two vertices as the origin using such procedures.
There are two points on each edge where unknown functions’ boundary values and their q−derivatives are linearly combined. This study shows that the anonymous functions’ integral is a multiple of these combinations. Additionally, it is worth noting that the solutions derived for the proposed boundary value problem (1.1) can be applied in various chemical graph theory applications. As a result, we assert that this generic concept may be beneficial to future work by young scholars.
On the other hand, numerous advanced fractional modeling techniques are discussed in the literature, notably (but not limited to) the well-known Caputo and Riemann–Liouville operators (for the detail, see [39–45]). This decade has seen the introduction of several novel modifications of the Hadamard, Caputo–Hadamard, and Hilfer operators and numerous simulation efforts using these new operators (for the detail, see [46–50]). Fabrizio and Caputo suggested a new formulation of a fractional framework without singularity six years ago (see [51]). Shortly after this work, Nieto and Losada concentrated on significant computational aspects (see [52]). The inclusion of nonsingular operators resulted in many research publications on fractional modeling (for example, see [53–55]).
This study aims to establish the existence of solutions to the specified boundary value problem (1.1) by using well-known fixed point techniques. Finally, two examples are presented to emphasize the significance of our results in this field of study.
2 Preliminaries
The succeeding results will be needed in the following sections.
Definition 2.1 ([51]). Let p > 0. The Caputo fractional derivative of order p for a function is defined by
For p > 0, the general solution of is given as where k = 0, 1, …, n − 1 (n − 1 < p < n, n = [p] + 1).
Lemma 2.2. Suppose that . Then is a solution of (2.1) if and only if y^{⋆} is a solution of the integral equations stated below (2.2)
Proof. Let is a solution of (2.1). Also, there are constants such that (2.3)
Using the boundary conditions for (2.1), we have
Substituting the values of z_{0} and z_{1} in (2.3), we get the solution (2.2). On the converse part, it is clear that y^{⋆} can be consider as a solution for (2.1) if y^{⋆} is a solution of (2.3).
We now present the Krasnoselskii and Schaefer fixed point theorems, respectively.
Theorem 2.3 ([56]). Let be a closed, bounded, convex, and nonempty subset of a Banach space and are two operators satisfying the following conditions:
- for all ;
- is compact and continuous on ;
- is a contraction mapping on , that is, there is an such that for all .
Then has a fixed point.
Theorem 2.4 ([56]). Let be a Banach space. If is a completely continuous function, that is, is continuous and totally bounded, then either the set is unbounded or has at least one fixed point in .
3 Main results
We define the Banach space having the norm
Furthermore, it is obvious that is a Banach space with
Also, by addressing Lemma 2.2, we can define an operator for each by (3.1) where for each k = 1, 2, …, 25, is defined for each by (3.2) for all s ∈ [0, 1].
For the ease of calculations, we use the following abbreviations: (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9)
Theorem 3.1 Consider the fractional boundary value problem (1.1). Assume that are continuous functions and there are constants , for all k = 1, 2, …, 25 with for all , s ∈ [0, 1]. Then (1.1) has a solution.
Proof. The fixed points of given in (3.1) exist if and only if (1.1) has a solution, as shown by the consequence of (3.2). To demonstrate this, we must first prove that is completely continuous.
As are continuous, therefore is continuous too. Let be a bounded set and , so for each s ∈ [0, 1], we have where is given in (3.3). Also, and for all s ∈ [0, 1], where and are defined in (3.4) and (3.5), respectively. Similarly, for all s ∈ [0, 1], where is given in (3.6). Therefore
Hence, which reveals that is uniformly bounded.
To prove the equicontinuity of the operator , we let and s_{1}, s_{2} ∈ [0, 1] with s_{1} < s_{2}. Then we have
It is clear that if s_{1} → s_{2} then, independently, the right-hand side of the above equation converges to zero. Also and
Hence, we deduce that the operators are equicontinuous, which implies that is equicontinuous. The Arzela–Ascoli theorem now entails the complete continuity of the operator.
Further, we define a set on . Now, we will prove that Υ is bounded. For this, let (y_{1}, y_{2}, …, y_{25})∈Υ. Then, we can write and so for all s ∈ [0, 1] and k = 1, 2, …, 25. Thus, and by similar computations, we have where are given in (3.3)–(3.6). Hence, which demonstrates the boundedness of the operator Υ. Now, using Lemma 2.2 and Theorem 2.4, it is clear that the operator has a fixed point. Consequently, (1.1) does indeed have a solution.
We shall now investigate the solution of (1.1) by applying various conditions.
Theorem 3.2 Consider the fractional boundary value problem (1.1). Suppose that are continuous functions and there are bounded continuous functions , and nondecreasing continuous functions σ_{1}, …, σ_{25}: [0, 1] → [0, ∞] such that and for all s ∈ [0, 1], and k = 1, 2, …, 25. If then (1.1) has a solution, where and the constants – are given in (3.7)–(3.9), respectively.
Proof. For each k = 1, 2, …25, let and for suitable constants , we have (3.10) where are given in (3.3)–(3.6). We define a set where is defined in (3.10). It is obvious that be a closed, nonempty, bounded, and convex subset of . Now, we have and which are define on as where (3.11) and (3.12) for all s ∈ [0, 1] and y =
Let Now, for every we have and
By using similar computations, we have also
This yields that and so Additionally, continuity of entails the continuity of .
We now need to demonstrate that is uniformly bounded. This is why, we have for all y ∈ Also, and for all y∈ Thus,
It shows that the operator is uniformly bounded on . Here, we need to prove the compactness of on . For this, let s_{1}, s_{2} ∈ [0, 1] with s_{1} < s_{2}, we have
Hence, as s_{1} → s_{2}. Also, we have and
Hence, we deduce that the operators are equicontinuous, which implies that is equicontinuous. The Arzela-Ascoli theorem indeed reveals the compactness of the operator on .
Lastly, we need to prove that is a contraction mapping. For this, let Thus, we have for each k = 1, 2, …, 25, where is given in (3.7). Also, by the similar computations, we have where and are given in (3.8) and (3.9), respectively. Thus, we have and so
As which means that is a contraction on We deduce that possesses a fixed point that is a solution to the fractional boundary value problem (1.1) as a consequence of Theorem 2.3.
4 Examples
In this section, we present the following two examples to illustrate the relevance of our key findings.
Example 4.1 Consider the system of differential equations given below (4.1) with boundary conditions (4.2) where p = 1.5, q = 0.08, μ_{1} = 2, μ_{2} = 8 and μ_{3} = 10.
Let are continuous functions given by for all s ∈ [0, 1], whereas are zero functions. For each s ∈ [0, 1], we have
Here, and where and . Let are identity functions. Then we obtain for all s ∈ [0, 1], Also, and for all s ∈ [0, 1],
Moreover, the continuous functions are defined by
It can be seen that all the conditions of Theorem 3.2 are satisfied, therefore, the proposed problem (4.1)–(4.2) has a solution.
Example 4.2 Consider the system of fractional differential equations given below (4.3) with boundary conditions (4.4) where and . Let are continuous functions given by for all s ∈ [0, 1], whereas are zero functions. For each s ∈ [0, 1], we have
Here, and where and . Let be identity functions. Then we obtain
Furthermore, the continuous functions are defined by
It can be seen that all the conditions of Theorem 3.2 are fulfilled, hence, the boundary value problem (4.1)–(4.2) has a solution.
5 Conclusion and open problems
Chemical graph theory is a branch of mathematics in which graphs represent the molecular structures of chemical compounds, and specific mathematical challenges are studied using theoretical and analytical methodologies. In recent decades, the fast growth of this subject has resulted in the development of various ground-breaking and novel ideas and techniques for conducting such research. Several researchers have used the structure of star graphs to investigate the solutions of fractional differential equations. They used star graphs because their approach requires a center node with interconnections to nearby vertices but no node-to-node connections. Since, in general, the graphs can have several junction nodes, therefore in this article, we introduced the idea of an octane graph. We have analyzed a graph with vertices labeled by 0 or 1, which is inspired by a graph representation of the octane compound, and formulated fractional differential equations on each of its edges. The existence of solutions results to the suggested fractional differential equation have been investigated by utilizing the Krasnoselskii and Schaefer fixed point theorems. In the end, we presented two examples to demonstrate the importance of our findings.
Here, we give the following open problems for the interested readers.
- Problem 1: Can we extend this idea to the circular ring type graphs?
- Problem 2: Can we use another method that can guarantee the conclusion of the proposed results?
We also pose the stability of the proposed fractional differential equation (1.1) as an open problem.
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