Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability

Abbreviations

CFD

MLU-Caputo fractional derivative

DE

MLU-differential equation

FP

MLU-fixed point

FDE

fractional differential equation

MLUH

MLU-Hyers

MLUHR

MLU-Hyers-Rassias

MLU

Mittag-Leffler-Ulam

RL

MLU-Riemann-Liouville

SFPE

MLU-sequential fractional pantograph equation

1 Introduction

Research on the analysis of differential equations (DEs) of fractional order has recently gained prominence and interest. Fractional derivatives are an excellent tool for studying memory and the inherited properties of diverse materials and processes. Fractional derivatives are frequently used to replicate the dynamics of real-world processes and phenomena, where previous behavior impacts the current state. Significant efforts have been made to advance the theoretical and application aspects of fractional differential equations (FDEs) in a variety of disciplines, including physics, chemistry, biology, and engineering sciences. Many contributions have been made by researchers about the existence and uniqueness of solutions to many classes of DEs. We urge the reader refer to previous studies [1–13] for more details.

In recent years, there has been a surge of interest in the application of novel analysis techniques to a variety of mathematical models, including delay FDEs. These equations are critical to understanding dynamic systems with delays and memory effects. One important component is the investigation of mild solutions, which provide a powerful framework for understanding the long-term behavior of delay FDEs. These solutions take into consideration both the initial conditions and the influence of previous states. For instance, the fractional Langevin equation that appears in the classical Brownian motion theory is modeled as fractional integro-differential equations [14]. For more details, see [15–17].

In 1960, Kalman proposed the fundamental concept of mathematical control theory, controllability. In general, controllability refers to the ability to employ a correct control function to steer the state of a control dynamical equation from an arbitrary initial state to the desirable terminal state. Many writers [18–26] have explored the controllability results. Furthermore, the investigation of controllability results on temporal scales is a new topic with minimal evidence [27,28].

The existence and uniqueness problems of FDEs are studied using fixed point (FP) theory. Over the course of a century, FP theory began to take shape and quickly advanced. Because of its applications, FP theory is highly respected and is still being investigated. Furthermore, this theory applies to a broad variety of spaces, such as Sobolev spaces, metric spaces, and abstract spaces. Because of this quality, FP theory is extremely beneficial for comprehending a wide variety of practical science problems represented by fractional ordinary, partial, and DEs [29–33].

On the other hand, stability theory is frequently applied to dynamical issues. Lyapunov and Mittag-Leffler stabilities have been established with great success for ordinary classical fractional calculus problems as well as exponential varieties of stabilities. The required attention has recently been given to the Hyer-Ulam [34] stability. Some stability and existence results from the FP approach were looked into in [35]. Furthermore, many writers discussed the stability analysis and existence theory for FDEs. For more details, see [36–40] and references therein, as several authors have recently developed an interest in the Ulam and Mittag-Leffler-Ulam (MLU) stability concerns.

The pantograph equation is a form of functional DE with a proportional delay. It occurs in a variety of pure and applied mathematics domains, including electrodynamics, control systems, number theory, probability, and quantum mechanics. Because of its significance, several academics have recently concentrated on the fractional pantograph equation and produced valuable contributions in this area. Balachandran et al. [41] investigated the existence of solutions to nonlinear fractional pantograph equations. Vivek et al. [42] established the existence and uniqueness of results for nonlinear neutral pantograph equations with generalized fractional derivatives. Several noteworthy findings on this subject have been reached using various types of fractional operators [43–49].

Now, the pantograph equation takes the following analogous form:

where P , Q , and υ are the real constants such that Q ≠ 0 and υ ∈ ( 0 , 1 ) . Academics [48,49] have researched several variations of the aforementioned pantograph as well as the multi-pantograph equation in the following:

under the condition θ ( 0 ) = κ , where P , κ ∈ C ( C is the set of complex numbers ) , δ j ( ℓ ) , and k ( ℓ ) are the analytical functions, and υ j ∈ ( 0 , 1 ) .

The authors of [48] investigated the non-linear neutral pantograph equation as follows:

The author recently assessed the following difficulty for the pantograph type in [41]:

where D ϰ C refers to the Caputo fractional derivative (CFD).

Continuing with the aforementioned contributions, in this manuscript, we suggest the following model, the so-called sequential fractional pantograph equations (SFPEs):

where D ω RL , and D a C , a ∈ { ϰ , ϱ } represent the fractional derivatives of Riemann-Liouville (RL) and Caputo, respectively. Also, ϕ , ψ : U × R 3 → R , σ : C ( U , R ) → R are the continuous functions.

Due to its importance in many applied fields, it is interesting to study the fractional model of the pantograph equations. Such a model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. In this article, we apply the FP technique to study the existence and uniqueness of solutions to the SFPEs (1.1). Furthermore, stability results for the considerable problem in the sense of the MLU have been investigated. Finally, an illustrative example has been highlighted in order to reinforce the theoretical results and suggest applications for this article.

2 Preliminary work

We need some basic definitions and properties of fractional calculus that are used in this article. So, this section is devoted to presenting some definitions and earlier works that will help the reader understand our text and will assist us in overcoming the challenges that we will experience in proofreading.

We need a singular type for the following Gronwall inequality:

The following auxiliary lemma is also necessary:

3 Existence results

In this section, we use the FP approach to present the necessary and sufficient conditions for studying the existence and uniqueness of a solution to the SFPEs (1.1). Therefore, we need to define a space as follows: Assume that Θ = C ( U , R ) is the space of all continuous functions on the interval U . Obviously, this space is a Banach space under the norm ∥ θ ∥ = sup ℓ ∈ U { ∣ θ ( ℓ ) ∣ } . Moreover, the product space Θ = Θ 1 × Θ 1 is also a Banach space with the norm ∥ ( ϖ , θ ) ∥ Θ = ∥ ( ϖ ) ∥ Θ 1 + ∥ ( θ ) ∥ Θ 2 .

According to Lemma 2.10, define the operator ℧ : Θ → Θ by

where υ 1 , υ 2 ∈ ( 0 , 1 ) . For convenience, we consider the following assumptions:

1. For the continuous functions ϕ , ψ : U × R 3 → R , there exist constants G ϕ , G ψ > 0 such that for ℓ ∈ U , θ i , ϖ i ∈ R , i = 1 , 2 , 3 , we have

   ∣ ϕ ( ℓ , θ 1 , θ 2 , θ 3 ) − ϕ ( ℓ , ϖ 1 , ϖ 2 , ϖ 3 ) ∣ ≤ G ϕ 2 ( ∣ θ 1 − ϖ 1 ∣ + ∣ θ 2 − ϖ 2 ∣ + ∣ θ 3 − ϖ 3 ∣ ) , ∣ ψ ( ℓ , θ 1 , θ 2 , θ 3 ) − ψ ( ℓ , ϖ 1 , ϖ 2 , ϖ 3 ) ∣ ≤ G ψ 2 ( ∣ θ 1 − ϖ 1 ∣ + ∣ θ 2 − ϖ 2 ∣ + ∣ θ 3 − ϖ 3 ∣ ) .

2. The function σ : C ( U , R ) → R is continuous with σ ( 0 ) = 0 , and there exists a positive constant ξ such that

   ∣ σ ( θ ) − σ ( ϖ ) ∣ ≤ ξ ∣ θ − ϖ ∣ , θ , ϖ ∈ Θ .

Also, we consider

Now, we can present the first main result of this part as follows: