Existence and stability results of pantograph equation with three sequential fractional derivatives

1.   Introduction

Differential equations (DEs) involving fractional operators of different orders have recently been studied by a number of scientific researchers because of the fact that they are valuable tools in the modeling of numerous problems in sciences and engineering such as biology, chemistry, physics, economics, signal and control theory, etc. The readers might refer to ^[13,14,18] for more information and the references therein. By utilizing diverse nonlinear analysis techniques, many researchers have identified the uniqueness and existence of solutions for numerous classes of DEs of fractional order. For more details, see ^{[3,6,8,9,12,16,17,19]}. Stability analysis, on the other hand, is usually one of the most popular essential concerns in the concept and utilization of fractional differential equations (FDEs). Several authors have recently become interested in the Ulam and MLU-stability concerns, we refer to the papers ^{[1,2,7,8,21,22,23,24]} and the references therein. Following the concerns raised in this work, we will focus on a significant area of differential problems widely used in engineering and other scientific fields. The equation is referred to as the pantograph equation. For additional details and uses of the pantograph equation, we recommend reading works ^{[10,11,20,25,26,27,28,29,30,31,32,33]}. It's worth mentioning that the standard expression of the pantograph equation is

Some scholars have studied several versions of the above pantograph equation. For example, the researchers examined the following multi-pantograph equation in ^[20,25]

In ^[20], the authors investigated the non-linear neutral pantograph equation

Recently, in ^[4] the author evaluated the pantograph type's following difficulty

where $^{C}D^{q}$ denote Caputo fractional derivative. The following sequential fractional pantograph equation is the subject of the current work, which examines the uniqueness, existence, and MLU-stability of solutions.

where $^{RL}D^{\delta }, ^{C}D^{q}, q\in \left\{ \vartheta, \theta \right\}$ stand for the fractional derivatives of Riemann-Liouville (RL) and Caputo, $\phi, \psi :[0, 1]\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ and $\varphi :C\left([0, 1], \mathbb{R} \right) \rightarrow \mathbb{R}$ are continuous functions. $^{RL}D^{\delta }$ ^[15,18], is defined by

where $\Gamma \left(.\right)$ represents Euler gamma function. The operator, $^{C}D^{q}$^[15,18], is defined by

and the RL fractional integral ^[15,18] of order $\alpha > 0,$ stated as

Following are several lemmas that we notice ^[13,18].

Lemma 1. Let $q, p > 0$ and $y\in L^{1}([0, 1])$. Then $I^{q}I^{p}y(\tau) = I^{q+p}y(\tau)$ and $D^{q}I^{q}y(\tau) = y\left(\tau\right).$

Lemma 2. Let $q > p > 0$ and $y\in L^{1}([0, 1])$. Then $D^{p}I^{q}y(\tau) = I^{q-p}y(\tau).$

Also, we recall the observing lemmas.

Lemma 3. ^[13] Let $\delta > 0.$ Then for $w\in C\left(0, 1\right) \cap L^{1}\left(0, 1\right)$ and $^{RL}D^{\delta }w\in C\left(0, 1\right) \cap L^{1}\left(0, 1\right),$ we have

where $c_{i}\in \mathbb{R},$ $i = 1, 2, ..., n, n = \left[ \delta \right] +1$.

Lemma 4. ^[13] Let $q > 0.$ Then

for some $c_{i}\in \mathbb{R},$ $i = 0, 1, 2, ..., n-1, n = \left[ q\right] +1.$

In what follow, we require an important singular type Gronwall inequality.

Theorem 5. ^[15] For each $\tau\in \left[ 0, 1\right).$ If

where all of the functions are continuous and non-negative. The costants $\varepsilon _{i} > 0,$ $k_{i}$ $(i = 1, 2, ..., n)$ are monotonic increasing and bounded functions on $[0, 1)$, then

Remark 6. For $n = 2,$ if $\varepsilon _{1}, \varepsilon _{2} > 0, k_{1}, k_{2}\geq 0, p\left(\tau\right)$ is locally integrable and non-negative on $[0, 1)$ and $u\left(\tau\right)$ is locally integrable and non-negative on $[0, 1)$ with

then

Remark 7. Let $p\left(\tau\right)$ be a non-decreasing function on $[0, 1)$ under the criteria of Remark 6. Then we have

where the Mittag-Leffler function, $E_{\varepsilon }$^[24] defined by: $E_{\varepsilon }\left[ x\right] = \sum_{j = 1}^{\infty }\frac{ x^{\varepsilon }}{\Gamma \left(j\varepsilon +1\right) }, x\in \mathbb{C}.$

The subsequent auxiliary result is also necessary.

Lemma 8. Suppose that $y\left(\tau\right) \in C\left(\left[ 0, 1\right], \mathbb{R} \right)$ and let's examine the fractional problem

with the condition

Then, we have

Proof. Utilizing Lemma 3 and the RL fractional integral of order $\delta$ on both sides of Eq (1.2), we obtain

where $c_{1}\in \mathbb{R}.$ Next, using Lemma 4 and the RL fractional integral of order $\vartheta$ on both sides of Eq (1.5), we obtain

where $c_{2}\in \mathbb{R}.$ When both sides of Eq (1.6) are solved using the RL fractional integral of order $\theta$, we obtain

Using (1.3), we obtain

and

inserting the values of $c_{0}, c_{1}$ and $c_{2}$ in (1.7) provides the solution (1.4).

2.   Existence results for pantograph problem (1.1)

Under this section, let's look at the sequential fractional pantograph problem using fixed point theory.

So, we need to introduce the following space.

Let $Z = C\left(\left[ 0, 1\right], \mathbb{R} \right)$, the space of continuous Banach functions from $\left[ 0, 1 \right]$ into $\mathbb{R}$ with the norm, $\left\Vert w\right\Vert$ where $\left\Vert w\right\Vert = \sup \left\{ \left\vert w\left(\tau\right) \right\vert :\tau\in \left[ 0, 1\right] \right\}.$ We construct operator $O:Z\rightarrow Z$ in the context of Lemma 8 by

For convenience, we consider the following hypotheses:

$\left(H_{1}\right)$ Continuous functions: $\psi, \phi :\left[ 0, 1\right] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ and $\exists$ constants $\mu _{i} > 0, i = 1, 2$ such that for each $\tau\in \left[ 0, 1\right]$ and $z_{j}, w_{j}\in \mathbb{R}, j = 1, 2$

$\left(H_{2}\right)$ $\varphi :C\left(\left[ 0, 1\right], \mathbb{R} \right) \rightarrow \mathbb{R}$ is continuous function with $\varphi \left(0\right) = 0$ and $\exists$ a constant $\sigma > 0$ such that

We also introduce the notations shown below:

The first result of our existence is based on Banach's principle of contraction.

Theorem 9. If $\left(H_{i}\right), i = 1, 2,$ are satisfied and also

where $\mu = \max \left\{ \mu _{i}, i = 1, 2\right\}$ and $\Delta _{i}, i = 1, 2,$ are given by (2.2), then problem (1.1) has a unique solution on $\left[ 0, 1\right]$.

Proof. Let's define $\Lambda = \max \left\{ \Lambda _{i}:i = 1, 2\right\},$ where $\Lambda _{i}$ are finite numbers given by

Setting

we demonstrate that $OB_{r}\subset B_{r}$, where $B_{r} = \left\{ w\in Z:\left\Vert w\right\Vert \leq r\right\}.$

For $w\in B_{r}$ and for each $\tau\in \lbrack 0, 1]$, by using the hypothesis $(H_{i}), i = 1, 2,$ we can write

and

Using these estimates, we obtain

$\Rightarrow$ $OB_{r}\subset B_{r}.$ Now, for $w, z\in B_{r},$ we obtain

in context of condition $2\mu \left(\Delta _{1}+\Delta _{2}\right) +\frac{\sigma }{\left\vert \lambda _{1}-\lambda _{2}\eta ^{\delta +\vartheta +\theta -1\;\;\;\;}\right\vert } < 1$, this demonstrates that $O$ is a contraction. The operator $O$ possesses an unique fixed point that corresponds to an unique solution to the problem according to Banach's fixed point theorem (1.1). The Leray-Schauder alternative yielded the second main result.

Lemma 10. ^[5] Let $Q:X\rightarrow X$ be a completely continuous operator. Let $G(Q) = \{u\in X:u = \rho Q(u)$ for some $0 < \rho < 1\}$. Then either the set $G(Q)$ is unbounded, or $Q$ has at least one fixed point (Leray-Schauder alternative).

For the forthcoming result, we suppose that

$\left(H_{3}\right)$ $\psi, \phi :\left[ 0, 1\right] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous and $\exists$ real constants $\pi _{i}, \gamma _{i}\geq 0, i = 1, 2$ and $\pi _{0} > 0, \gamma _{0} > 0$ such that for any $w_{i}\in \mathbb{R}, i = 1, 2,$ we have

and

$\left(H_{4}\right)$ $\varphi :C\left(\left[ 0, 1\right], \mathbb{R} \right) \longrightarrow \mathbb{R}$ is continuous function with $\varphi \left(0\right) = 0$ and $\exists$ constant $\epsilon > 0$ such that

Theorem 11. If $\left(H_{i}\right), i = 3, 4,$ are satisfied and also

then on $[0, 1]$ the problem (1.1) has at least one solution.

Proof. We demonstrate that the operator $O:Z\longrightarrow Z$ is completely continuous in the first step. Since the functions $\psi, \phi$ and $\varphi$ are continuous, the operator $O$ is also continuous.

Let $\Theta \subset W$ be bounded. Then $\exists$ positive constants $M_{i}, \left(i = 1, 2\right)$ such that

for each $w, z\in \Theta$ and constants $N$ such that $\left\vert \psi \left(w\right) \right\vert \leq N$ for all $z\in C\left(\left[ 0, 1\right], \mathbb{R} \right)$. Then for any $w\in \Theta$ and by $\left(H_{i}\right), i = 3, 4,$ we have

which implies that

The operator $O$ is uniformly bounded, as shown in the above Eq (2.1).

As a follow-up, we demonstrate that $O$ is equicontinuous sets of $Z$. Let $\tau_{1}, \tau_{2}\in \lbrack 0, 1]$ with $\tau_{1} < \tau_{2}$. Next, we obtain

which does not depend on $w$ and tends to $0$ as $\tau_{2}-\tau_{1}\rightarrow 0$. Thus, $O$ is equicontinuous. Thus, by using the Arzelá-Ascoli theorem, $O:Z\rightarrow Z$ is completely continuous.

Finally, we demonstrate that the set, $\chi = \left\{ w\in Z:w = \varrho O\left(w\right), \text{ }0 < \varrho < 1\right\},$ is bounded. Let $w\in \chi,$ then $w = \varrho O\left(w\right).$For each $\tau\in \left[ 0, 1\right],$ we have

Then,

which implies

Consequently,

This implies that $\chi$ is bounded. According to Lemma 10, this indicates that the operator $O$ contains at least one fixed point. The problem (1.1) on $\left[ 0, 1\right]$ has hence at least one solution in this case. Hence, the proof is completed.

3.   MLU-stability (MLUS) of pantograph problem (1.1)

The MLUS of the sequential fractional problem (1.1) will be defined and studied in this section. For $\tau\in \left[ 0, 1\right],$ we provide the following fractional inequalities:

and

where $\kappa \in \mathbb{R}^{+}$ and $m:\left[ 0, 1\right] \rightarrow \mathbb{R} ^{+}$ is continuous function.

Definition 12. The problem (1.1) is MLU-Hyers stable, with respect to $E_{\delta +\vartheta +\theta }$ if $\exists$ a real number $\upsilon$ such that for each $\kappa > 0$ and for each solution $z\in Z$ of the inequality (3.1), $\exists$ a solution $w\in Z$ of the problem (1.1) with

Definition 13. The problem (1.1) is MLU-Hyers-Rassias stable, with respect to $mE_{\delta +\vartheta +\theta }$ if $\exists$ a real number $\upsilon _{m} > 0$ such that for each $\kappa > 0$ and for each solution $z\in Z$ of the inequality (3.2), $\exists$ a solution $w\in Z$ of problem (1.1) with

Remark 14. A function $z\in Z$ is a solution of the inequality (3.1) if and only if $\exists$ a function $f\in C\left(\left[ 0, 1\right], \mathbb{R} \right)$ (which depend on $z$) such that

and

Theorem 15. If hypotheses $(H_{i}), i = 1, 2,$ are satisfied, then the problem (1.1) is MLU-Hyers stable.

Proof. Let $z\in Z$ represent the inequality's (3.1) solution and let $w\in Z$ represents the unique solution of the problem

According to Lemma 8, we have

By integrating the inequality (3.1), we obtain

where

Latter, if $w\left(0\right) = z\left(0\right), w\left(\omega \right) = z\left(\omega \right)$ and $^{C}D^{\theta }w\left(0\right) =$ $^{C}D^{\theta }z\left(0\right)$, then $c_{1} = a_{1}, c_{2} = a_{2}$ and $c_{3} = a_{3}.$

For each $\tau\in \left[ 0, 1\right],$ we have

then

Using $\left(H_{1}\right)$, we get

By using (3.3) and (3.4), we have got

According to Remarks 6 and 7, we have got

As a result, the problem (1.1) is MLU-Hyers stable.

Theorem 16. If hypotheses $(H_{i}), i = 1, 2,$ are satisfied$.$ Suppose there exists $\upsilon _{m} > 0$ such that

where $m\in C(\left[ 0, 1\right], \mathbb{R}_{+})$ is increasing. Then the problem (1.1) is Mittag-Leffler-Ulam-Hyers-Rassias stable with respect to $mE_{\delta +\vartheta +\theta }$.

Proof. By integrating the inequality (3.2), we have

where the inequality (3.2) has $z\in Z$ as a solution. Let's call the unique solution to the problem $w\in Z$

We have

Then

From (3.5), as it can be observed,

Now, by Remarks 6 and 7, we have got

The problem (1.1) is therefore MLU-Hyers-Rassias stable.

4.   Application

Take the fractional pantograph problem below into consideration:

and the following fractional inequalities

and

where

and

For each $\tau\in \left[ 0, 1\right]$ and $w_{j}, z_{j}\in \mathbb{R}, j = 1, 2,$ we have

and

hence conditions $\left(H_{1}\right)$ and $\left(H_{2}\right)$ hold with $\mu _{1} = \frac{1}{45\pi e^{2}}, \mu _{2} = \frac{1}{40e^{2}}$ and $\sigma = \frac{1}{45}$ respectively.

With the given data, it is found that

Thus condition

is satisfied. Therefore, the problem (4.1) has a unique solution on $[0, 1]$, according to Theorem 9, and is MLU-Hyers stable with

Let $m \left(\tau \right) = \tau^{2},$ then

Thus condition (3.4) is satisfied with $m\left(\tau\right) = \tau^{2}$ and $\upsilon _{m} = \frac{2}{\Gamma \left(\frac{149}{30}\right) }.$ Thus, Theorem 16 demonstrates that problem (4.1) is MLU-Hyers-Rassias stable with

5.   Conclusions

In this work, we considered pantograph equations with three sequential fractional derivatives. The existence and Mittag-Leffler-Ulam stability of solutions have been discussed. The existence results of the solutions for the mentioned problem were investigated by the contraction mapping principle and Leray-Schauder's alternative. The Mittag-Leffler-Ulam-Hyers stability and Mittag-Leffler-Ulam-Hyers-Rassias stability results have been proved by applying generalized singular Gronwall's inequality. The main results were illustrated with the aid of an example.

Acknowledgments

Authors would like to express gratitude to Central university of Punjab, India-151401 to encourage us for the research work. The authors Aziz Khan and Thabet Abdeljawad would like to thanks Prince Sultan University for paying the APC and the support through TAS research lab.

Conflict of interest

The publication of this paper, according to the author, is free from any conflicts of interest.