On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula $C_{6}H_{12}$ C 6 H 12 and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in Rn

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable x∈Rn and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained.

Discrete fractional boundary value problems and inequalities

In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

Common fixed point theorems for auxiliary functions with applications in fractional differential equation

In this work, we investigate h-ϕ contraction mappings with two metrics endowed with a directed graph which involve auxiliary functions. The achievement allows us to obtain applications for the existence of the solutions for Caputo fractional boundary value problems with the integral boundary condition type. In addition, we also give examples and numerical experiments supporting our main results.

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 $\mathrm{H}_{12} \mathrm{Si}_{6}$ H 12 Si 6 . 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.

Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.

Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.