On a System of ψ-Caputo Hybrid Fractional Differential Equations with Dirichlet Boundary Conditions

Fractional calculus has a long history, going all the way back to Leibniz’s 17th-century explanation of the derivative order in 1965. Mathematicians use fractional calculus to study how derivatives and integrals of noninteger order work and how they change over time. Since then, the new theory has proven to be very appealing to mathematicians, biologists, chemists, economists, engineers, and physicists. Subsequently, the subject attracted the interest of numerous famous mathematicians, including Fourier, Laplace, Abel, Liouville, Riemann, and Letnikov. For current and wide-ranging analyses of fractional derivatives and their applications, we recommend the monographs [1,2,3,4]. In [5], the authors investigated new results of the existence and uniqueness of systems of nonlinear coupled DEs and inclusions involving Caputo-type sequential derivatives of fractional order and new kinds of boundary conditions. In [6], the authors investigated a new type of SFDE and inclusions involving $\psi$-Hilfer fractional derivatives, associated with integral multi-point BCs.

Fractional derivatives have played a very important role in mathematical modeling in many diverse applied sciences, see [7]. In [8], the authors applied a new technique called “local fractional Laplace homotopy perturbation method” (LFLHPM) on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. In [9], the authors present a new analytical method called the “local fractional Laplace variational iteration method” (LFLVIM) for solving the two-dimensional Helmholtz and coupled Helmholtz equations. In [10], the authors find the solution of the LFFPE on the Cantor set. They make a comparison between the RDTM and LFSEM used in LFFPE. For example, the authors in [11] employed the LFLVIM and LFLDM to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within LFDOs. The authors in [12] employed the fractional derivative of the $\psi$-Caputo type in modeling the logistic population equation, through which they were able to show that the model with the fractional derivative led to a better approximation of the variables than the classical model. In addition, the authors in [13] employ the fractional derivative of the $\psi$-Caputo type, and use the kernel Rayleigh, to improve the model again in modeling the logistic population equation. Various research has studied the existence and uniqueness of solutions to initial and boundary value problems utilizing $\psi$-fractional derivatives, see [14,15,16,17,18].

Fractional differential equations have been used to describe a wide variety of occurrences in a number of different engineering and scientific areas. Differential equations of fractional order are suitable for critical aspects in finance, electromagnetics, acoustics, viscoelasticity, biochemistry, and material science, see [19,20,21].

Additionally, it is essential to examine coupled systems through the use of fractional differential equations, as these systems are found in a wide range of applications. A number of scholars have also investigated coupled fractional differential equation systems. Some theoretical work on coupled fractional differential equations is included in this article, see [22,23,24].

The fractional derivatives of an unknown function are included in hybrid differential equations, as is the nonlinearity that relies on them. This class of equations arises in a wide variety of applications and physical science areas, for example, in the redirection of a bent pillar with a constant or variable cross-area, a three-layer shaft, electromagnetic waves, or gravity-driven streams. In the literature, hybrid FDEs have been examined by employing a variety of different forms of fractional derivatives; see [23,25,26]. Some recent results on the existence and uniqueness of initial and boundary value problems and Ulam–Hyers stability can be found in [27,28,29] and the references therein. For recent results from the $\psi$-Caputo hybrid fractional derivatives (CHFDs), we refer to [22,23,30,31] and the references cited therein. Choukri Derbazi et al. recently investigated the existence of extremal solutions to the nonlinear coupled system in [32]. Using the so-called “monotone iterative technique” together with the method of upper and lower solutions, the authors investigate the existence of extremal solutions of the following BVP that involves the $\psi$-Caputo derivative with ICs. where ${}^C{\mathcal{D}}_{a^{+}}^{\upsilon ;\psi }$ denote the $\psi$-Caputo fractional derivatives (CFDs) of order $\upsilon$ and $\mathfrak{f},\mathfrak{g}:[a,b]\times {\mathcal{R}}_e^2\to {\mathcal{R}}_e$ are continuous functions and ${\phi }_a,{\zeta }_b\in {\mathcal{R}}_e$ with ${\phi }_a\le {\zeta }_b$.

Mohamed I Abbas [30] investigated the uniqueness of solutions for the following coupled system of fractional differential equations (CSFDEs). Based on the Leray–Schauder alternative and Banach’s fixed point theorem, the authors investigated the existence and uniqueness of the following BVP associated with four-point BCs. where ${}^C{\mathcal{D}}_{0^{+}}^{\upsilon ;\psi },{}^C{\mathcal{D}}_{0^{+}}^{\beta ;\psi }$ denote the $\psi$-CFDs of order $\upsilon ,\beta$ and $\mathfrak{f},\mathfrak{g}:[0,1]\times {\mathcal{R}}_e^2\to {\mathcal{R}}_e$ are continuous functions.

In 2020, the authors of [33] studied the existence and uniqueness of the following BVP associated with multi-point BCs, with results obtained via topological degree theory and Banach’s contraction principle: where ${}^C{\mathcal{D}}_{a^{+}}^{\alpha ;\psi }$ denotes $\psi$-Caputo fractional derivatives, $h:[a,b]\times {\mathcal{R}}_e\to {\mathcal{R}}_e$ is assumed to be continous and ${\delta }_k\in {\mathcal{R}}_e,k=1,2,\dots ,n.$

In previous works, researchers investigated the existence and uniqueness of linear fractional differential equations involving $\psi$-Caputo.

This work is devoted to investigating the existence and uniqueness of the solutions for the following system of equations with Dirichlet BCs. Adding to this, we show that BVP is stable via the Ulam–Hyers technique. where ${}^C{\mathcal{D}}_{0^{+}}^{{\upsilon }_i,\psi },i=1,2.$ is the $\psi$-CFDs of order ${\upsilon }_i$, and $\mathfrak{f},\mathfrak{g}:[\xi ,\mathcal{T}]\times {\mathcal{R}}_e^2\to {\mathcal{R}}_e$ are continuous functions. To be valuable, the findings of this paper must be novel and generalize several earlier findings that are important to the research. To the best of our knowledge, there are no articles that discuss boundary value problems for systems of fractional differential equations with $\psi$-Caputo and no articles that investigate Ulam–Hyers stability for differential equations that contain $\psi$-Caputo derivatives. This paper is organized as follows. In Section 2, we will briefly recall some basic definitions and some preliminary concepts about fractional calculus and auxiliary results used in the following sections. In Section 3, we establish the existence of solutions to the $\psi$-Caputo fractional hybrid differential equation by using the Leray–Schauder alternative and Banach’s fixed point theorem. In Section 4, the stability of Ulam–Hyers solutions is shown. In Section 5, we finally give an example to illustrate the application of the results obtained and we give our conclusion in Section 6.

There are some basic definitions, lemmas and results of the $\psi$-CFDs with regard to another function ([1,2,3,4]).

Before presenting our main results, the following auxiliary lemma is presented.

Defining the space $\mathcal{B}=\left\{\left(\phi \left(\omega \right),\zeta \left(\omega \right)\right):(\phi ,\zeta )\in \mathcal{C}\left([\xi ,\mathcal{T}],{\mathcal{R}}_e\right)\times \mathcal{C}\left([\xi ,\mathcal{T}],{\mathcal{R}}_e\right)\right\}$, it is obvious that $\mathcal{B}$ is a Banach space. Furthermore, this space is endowed with the norm

By Lemma 2, we define an operator $\Phi :\mathcal{B}\to \mathcal{B}$ as where and

Now, let us assume that the following assumptions hold true:

$\left({\mathcal{A}}_1\right)$$\phi ,\zeta$ are assumed to be continuous and bounded, and there exist ${\partial }_{\mathfrak{f}},{\partial }_{\mathfrak{g}}>0$ such that $\left({\mathcal{A}}_2\right)$ Both ${\mathfrak{H}}_1$ and ${\mathfrak{H}}_2$ are assumed to be continuous and there exist ${\delta }_i,{\epsilon }_i>0,i=1,2$ such that $\forall \omega \in [\xi ,\mathcal{T}],{\phi }_i,{\zeta }_i\in {\mathcal{R}}_e$, $i=1,2$.

$\left({\mathcal{A}}_3\right)$ There exist ${\lambda }_0,{\mu }_0>0$, and ${\lambda }_i,{\mu }_i\le 0,i=1,2$ such that $\left({\mathcal{A}}_4\right)$ Let $\mathcal{S}\subset \mathcal{B}$ be a bounded set, then there exist ${\mathcal{K}}_i>0,i=1,2$ such that

Using ${\mathcal{A}}_4$, observe that $\forall i=1,2.$

For computational convenience, we let

Next, we introduce our main result by setting two theorems with their proofs.