Abstract
In this present manuscript, by applying fractional quantum calculus, we study a nonlinear fractional pantograph q-difference equation with nonlocal boundary conditions. We prove the existence and uniqueness results by using the well-known fixed-point theorems of Schaefer and Banach. We also discuss the Ulam–Hyers stability of the mentioned pantograph q-difference problem. Lastly, the paper includes pertinent examples to support our theoretical analysis and justify the validity of the results.
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1 Introduction
It is known that the difference equations involving quantum calculus play an important role in modeling many problems in engineering, physics, and mathematics, for further information the reader can address the following works [1–3]. In recent years, differential equations with fractional quantum calculus have been extensively studied by several scientific researchers, see for instance [4–7]. In this sense, several interesting topics concerning research for differential equations involving fractional quantum calculus have been devoted to the existence and the Ulam–Hyers stability of the solutions. Recently, many interesting results concerning the existence and Ulam-type stability of solutions for differential equations with fractional q-calculus have been obtained, see [8–11] and the references therein. In [12, 13], the existence and uniqueness of solutions were investigated for sequential differential equations with q-fractional calculus.
In the 1960s, British Railways wanted to make the electric locomotive faster and to develop a new type of electric locomotive. The goal was to make the trains faster. An important component for the new high-speed electric locomotive was the pantograph. The purpose of the pantograph is to collect current from an overhead wire, which is necessary for the locomotive to be able to move; see Fig. 1.
To make sure that the electric locomotive can move smoothly with high speed, it is necessary that there are no interruptions in the current-collection system. Therefore, the pantograph should stay in contact with the overhead wire for the whole time, particularly when the pantograph passes the supports of the overhead wire, which is a critical passage. Therefore, Ockendon and Tayler studied the motion of the pantograph head on an electric locomotive in [14] and developed a special delay differential equation of the form
for \(\rho >0\), where \(\gamma _{i}\) is a real constant and \(0< \lambda <1\) for \(\lambda \in \mathbb{R}\). In [15] the authors described different classes of exact solutions to nonlinear pantograph-type reaction–diffusion equations of the form
where \(\upvarphi = \upvarphi ( y, \rho )\) and \(\acute{\upvarphi}= \upvarphi ( py, q \rho )\), \(p,q >0\) such that p and q cannot be equal to 1 at the same time.
The authors in [16] considered the following initial value problem for the fractional pantograph equation in quantum form
where \(\theta , \lambda \in (0,1)\), \(\mathfrak{g}: [ 0,\delta ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) is a continuous function. Abdo et al. in [17] investigated two AB-Caputo-type implicit fractional differential equations with nonlinear integral conditions described by
and
for \(\rho \in [a,\top ]\), where \({}^{ABC }\mathcal{D}_{ a^{+}}^{\theta}\) is the AB-Caputo-type fractional differential of order θ, while \(\mathfrak{g} \in C([a,\top ] \times \mathbb{R}^{2}) \) and \(\mathfrak{u} \in C([a,\top ] \times \mathbb{R})\). In 2021, Ali et al. studied the given class of fractional order of pantograph differential equations under multipoint boundary conditions
where \(\mathcal{D}_{ 0^{+}}^{\theta}\) represents the Riemann–Liouville derivative with arbitrary order \(( m-1, m ]\), \(m \geq 2\) and \(0 < \upgamma _{0i}\), \(\upgamma _{1i} \in < 1\) with \(\sum_{i=1}^{m-2} \upgamma _{0i} \upgamma _{1i} <1\) and \(\mathfrak{g}: [ 0,1 ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) is a continuous function [18]. Also, Alzabut et al. investigated the following nonlinear discrete fractional pantograph equation
where \({}^{C }\mathcal{D}^{(\cdot)}\) is the Caputo fractional derivative, \(\mathbb{N}_{1-\beta} = \{\rho , \rho +1, \rho +2, \dots \}\), \(\lambda\in ( 0,1 )\), \(p : C( [0, \infty ), \mathbb{C} )\to \mathbb{R}\), and \(\mathfrak{g}: [ 0,\delta ] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) is a Lipschitz continuous function with respect to φ [19]. Derbazi et al. determined the existence criteria of extremal solutions for the following θ-Caputo-type fractional differential equations in a Caputo sense with nonlinear boundary conditions
where \({}^{C }\mathcal{D}_{ a^{+}}^{\nu , \theta}\) is the θ-fractional operator of order \(0< \nu \leq 1\) in the Caputo sense and this was investigated and \(\mathfrak{g}\in C([a, b]\times \mathbb{R})\), \(\mathfrak{u}\in C(\mathbb{R}^{2})\) [20]. For more information related to this topics see [21–27].
Motivated by the aforementioned works, we investigated the existence and Ulam-stability analysis for the following class of fractional pantograph q-difference equation \((\mathbb{F}\mathrm{P}q{-}\mathbb{DE})\) with nonlocal boundary conditions
where \({}^{C }\mathcal{D}_{ q}^{(\cdot)}\), \(\mathcal{I}_{ q}^{(\cdot)}\) are the Caputo fractional q-derivative and Riemann–Liouville fractional q-integral, respectively, \(q\in (0,1 )\), \(\rho \in [ 0, \delta ] \), \(m\in \mathbb{N}\), \(\lambda\in ( 0,1 )\),
and \(\mathfrak{g}: [ 0,\delta ] \times \mathbb{R}\times \mathbb{R} \to \mathbb{R} \) is a given continuous function.
In Sect. 2, we recall some essential definitions of fractional quantum calculus. Section 3 contains our main results in this work, while an example is presented to support the validity of our obtained results. An application, together with some needed algorithms for the problems, are given in Sect. 4. In Sect. 5, some conclusions are presented.
2 Preliminary notions
We recall some basic definitions and necessary lemmas related to fractional q-calculus and nonlinear analysis that will be used in the following.
Let \(\Sigma = [ 0,\delta ]\), and consider the Banach spaces \(C ( \Sigma ,\mathbb{R} ) \) and \(L^{1} (\Sigma ,\mathbb{R} ) \) of Lebesgue integrable functions \(\upvarphi : \Sigma \to \mathbb{R}\) with the norm \(\Vert \upvarphi \Vert = \sup \{ \vert \upvarphi ( \rho ) \vert : \rho \in \Sigma \}\), and
respectively. Let \(q\in ( 0,1 ) \). Then, the q-number is defined by
The q-analog of the power \((p-r )^{m}\) is
The q-gamma function is defined by [28]
Note that the q-gamma function satisfies \(\Gamma _{q} ( 1+b ) = [ b ]_{q} \Gamma _{q} ( b ) \). The 1st-q-derivative of a function \(\upvarphi :\mathrm{J} \to \mathbb{R}\) is given by
and for the higher orders, it becomes \(\mathcal{D}_{ q}^{0}\upvarphi ( \rho ) = \upvarphi ( \rho )\), \(\mathcal{D}_{ q}^{m} \upvarphi ( \rho ) = \mathcal{D}_{ q} \mathcal{D}_{ q}^{m-1} \upvarphi ( \rho ) \), for \(\text{ }\rho \in \Sigma \), \(\text{ } m\in \{ 1,2,\dots \}\). Set \(\Sigma _{\rho }:= \{ \rho q^{m}:n\in \mathbb{N} \} \cup \{ 0 \} \) [28]. The 1st-q-integral of a function \(\upvarphi : \Sigma _{\rho} \to \mathbb{R} \) is defined by
provided that the series absolutely converges [28]. We note that \(( \mathcal{D}_{ q} \mathcal{I}_{ q} \upvarphi ) ( \rho ) = \upvarphi ( \rho )\), while φ is continuous at 0, then \(( \mathcal{I}_{ q} \mathcal{D}_{ q} \upvarphi ) ( \rho ) =\upvarphi ( \rho ) -\upvarphi ( 0 )\). The ϑth Riemann–Liouville fractional q-integral of a function \(\upvarphi: \Sigma \to \mathbb{R}\) is defined by [29], \(( \mathcal{I}_{ q}^{0} \upvarphi ) ( \rho ) =\upvarphi ( \rho )\) and
The θth Caputo fractional q-derivative of a function \(\upvarphi: \Sigma \to \mathbb{R}\) is given by [30],
For more details about the fractional q-operators, see [22].
Lemma 2.1
Let \(\theta _{1}, \theta _{2} \geq 0\). Then, we have the following relations
-
(i)
\(\mathcal{I}_{ q}^{\theta _{1}} \mathcal{I}_{ q}^{\theta _{2}} \upvarphi ( \rho ) = \mathcal{I}_{ q}^{\theta _{1}+ \theta _{2}}\upvarphi ( \rho ) \);
-
(ii)
\({}^{C }\mathcal{D}_{ q}^{\theta _{1}} \mathcal{I}_{ q}^{\theta _{1}} \upvarphi ( \rho ) =\upvarphi ( \rho )\);
-
(iii)
\(\mathcal{I}_{ q}^{\theta _{1}} s^{\rho } = \frac{\Gamma _{q} ( 1+\rho ) }{ \Gamma _{q} ( 1+\rho +\theta ) }s^{ \rho +\theta }\), \(\rho \in ( -1,\infty )\), \(s >0\).
Lemma 2.2
([30])
Let \(\theta \in ( m-1,m ) \). Then, the following equality holds
In view of Lemma 2.2, the general series solution of the following equation \(\mathcal{I}_{ q}^{\theta} {}^{C }\mathcal{D}_{ q}^{\theta} \upvarphi ( \rho ) =0\) is
Hence, we have
Theorem 2.3
(Banach fixed-point theorem [31])
Let \(\Omega \neq \emptyset \) be a closed subset of a Banach space \((\mathcal{X}, \Vert \cdot \Vert )\). If \(\mathcal{Z}:\Omega \to \Omega \) is a contraction mapping, then Φ admits a unique fixed point.
Theorem 2.4
(Schaefer fixed-point theorem [31])
Let \(\mathcal{Z} \) be a continuous compact operator of a Banach space \(\mathcal{X}\) into itself, such that the set
is bounded. Then, \(\mathcal{Z}\) has a fixed point.
3 Existence and uniqueness results
In what follows, we apply some fixed-point theorems to demonstrate the existence and uniqueness results for problem (1). To obtain the existence results for problem (1), the following auxiliary lemma is needed.
Lemma 3.1
For any \(\omega \in C(\Sigma , \mathbb{R})\), the \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) with nonlocal boundary conditions
for \(q\in (0,1 )\), \(\rho \in [ 0,\delta ]\), \(m\in \mathbb{N}\),
has a unique solution given by
where
Proof
Assume φ satisfies problem (1). First, we write this equation as
In view of Lemma 2.2, we have
Applying the BCs, we obtain
By substituting (6) into (5), we obtain
Applying the integrator operator \(\mathcal{I}_{ q}^{ \vartheta _{i}}\) to both sides of equation (7), we obtain
Using the condition
we have
By solving (8), we find that
By inserting (9) into (7), we obtain (3). □
To obtain our findings, we need the following assumptions.
-
(As1)
There is a constant \(l_{\mathfrak{g}}>0\) such that
$$ \bigl\vert \mathfrak{g} ( \rho , \upvarphi, \upvarphi ) - \mathfrak{g} ( \rho , \tilde{\upvarphi}, \tilde{\upvarphi} ) \bigr\vert \leq l_{\mathfrak{g}} \bigl( \vert \upvarphi-\tilde{\upvarphi} \vert + \vert \upvarphi - \tilde{ \upvarphi} \vert \bigr),$$for \(\rho \in \Sigma \) and \(\upvarphi ,\tilde{\upvarphi}\in \mathbb{R}\).
-
(As2)
There exist constants \(\mathrm{D}, \mathrm{h}_{\mathfrak{g}}^{ ( 1 )}, \mathrm{h}_{\mathfrak{g}}^{ ( 2 ) }>0\) such that
$$ \bigl\vert \mathfrak{g} \bigl( \rho , \upvarphi ( \rho ), \upvarphi ( \lambda \rho ) \bigr) \bigr\vert \leq \mathrm{D}+ \mathrm{h}_{\mathfrak{g}}^{ ( 1 ) } \bigl\vert \upvarphi ( \rho ) \bigr\vert + \mathrm{h}_{\mathfrak{g}}^{ (2 ) } \bigl\vert \upvarphi ( \lambda \rho ) \bigr\vert , \quad \forall ( \rho , \upvarphi ) \in \Sigma \times \mathbb{R}. $$
3.1 Existence and uniqueness results via Banach’s fixed-point theorem
Theorem 3.2
Let (As1) be valid, then \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) (1) has a unique mild solution on Σ, whenever
where
Proof
We switch problem (1) into a fixed-point problem and we consider the operator \(\tilde{\mathcal{Z}} : C(\Sigma , \mathbb{R})\to C(\Sigma , \mathbb{R})\) as
Clearly, the solution of (1) is as a fixed point of the operator \(\tilde{\mathcal{Z}}\). By (As1), for any \(\upvarphi ,\tilde{\upvarphi}\in C(\Sigma , \mathbb{R})\) and \(\rho \in \Sigma \), we obtain
Thus,
From (10), \(\tilde{\mathcal{Z}}\) is a contraction. As an outcome of Banach’s FPT, \(\tilde{\mathcal{Z}}\) has a unique fixed point that is a unique mild solution of (1) on Σ. □
3.2 Existence results via Schaefer’s fixed-point theorem
Theorem 3.3
Suppose that the hypothesis (As2) is satisfied. Then, \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) (1) has at least one solution on Σ, whenever \(\mathrm{N}_{1}<1\), where
Proof
We shall use Schaefer’s fixed-point theorem to demonstrate that \(\tilde{\mathcal{Z}}\) defined in (12) has a fixed point. The proof will be given in the following steps.
Step 1. \(\tilde{\mathcal{Z}}\) is continuous. Let a sequence \(\upvarphi _{n} \to \upvarphi \) in \(C(\Sigma , \mathbb{R})\). Since \(\mathfrak{g}\) is continuous, we have
as \(n\to \infty \). Thus, for any \(\rho\in \Sigma \), we write
Hence, we obtain
Consequently, \(\tilde{\mathcal{Z}}\) is continuous.
Step 2. The image of a bounded set under \(\tilde{\mathcal{Z}}\) is bounded in \(C(\Sigma , \mathbb{R})\). Indeed, it is enough to show that for any \(\omega >0\), there exists a positive constant ζ such that for each
we have \(\Vert \tilde{\mathcal{Z}} \upvarphi \Vert \leq \zeta \). In fact, we have
and consequently
where \(\mathrm{h}_{\mathfrak{g}} = \mathrm{h}_{\mathfrak{g}}^{ ( 1 ) } + \mathrm{h}_{\mathfrak{g}}^{ ( 2 ) }\).
Step 3. \(\tilde{\mathcal{Z}}\) sends bounded sets of \(C(\Sigma , \mathbb{R})\) into equicontinuous sets. For \(\rho_{1}, \rho _{2}\in \Sigma \), \(\rho _{1} < \rho _{2}\) and for \(\upvarphi \in \Omega _{\omega }\), we have
As \(\rho_{1} \rightarrow \rho _{2}\), we obtain
Consequently, \(\tilde{\mathcal{Z}} ( \Omega _{\omega } ) \) is equicontinuous. From the previous steps, together with the Arzelá–Ascoli theorem, we deduce that \(\tilde{\mathcal{Z}}\) is completely continuous.
Step 4. A priori bounds. Now, it remains to prove that the set
is bounded. Let \(\upvarphi \in \mathcal{U}\) and for each \(\rho \in \Sigma \), we have
From inequality (14), we obtain \(\Vert \upvarphi \Vert \leq \frac{\mathrm{N}_{2}}{ ( 1- \mathrm{N}_{1} )}\), where
This means that the set \(\mathcal{U}\) is bounded. We infer from Schaefer’s fixed-point theorem that \(\tilde{\mathcal{Z}}\) possesses at least one fixed point. Consequently, there is at least one solution to the problem (1). □
4 Ulam–Hyers stability
In this section, we discuss two types of Ulam stability for solutions of problem (1).
Theorem 4.1
Suppose that the hypothesis (As1) and condition (10) are satisfied. Then, the problem (1) is Ulam–Hyers stable. Moreover, it is also generalized Ulam–Hyers stable.
Proof
Let \(\varepsilon >0\). Let \(\hat{\upvarphi}\in C(\Sigma , \mathbb{R})\) be any solution of the inequality
Then, there exists \(\mathrm{Q}\in C(\Sigma , \mathbb{R})\) such that
and \(\vert \mathrm{Q} ( \rho ) \vert \leq \varepsilon \), \(\rho \in \Sigma \). This gives
On the other hand, let \(\upvarphi \in C(\Sigma , \mathbb{R})\) be a unique solution of the \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) (1). From Lemma 3.1, we have
From (17), (18), and assumption (As1), we obtain
Hence,
where \(\mathrm{k}^{\ast }\) is defined in (11). Consequently,
Consequently, the problem (1) is Ulam–Hyers stable. By setting
we obtain
Clearly in (20), \(\phi ( \varepsilon ) =0\). Therefore, the problem (1) is also generalized Ulam–Hyers stable. □
5 Illustrative examples with a numerical approach
In this section, we provide two examples to validate the obtained results.
Example 5.1
Consider the following nonlocal boundary value problem of \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) as
for \(\rho \in \Sigma = [0, 1]\) with \(\delta =1>0\). Here, \(\theta =\frac{7}{2} \in (3, 4]\) with \(m=4 \geq 2\), \(\lambda = \frac{1}{2} \in (0,1)\), \(i=0,1,2\), \(k=2\), \(\upgamma _{01} = \frac{1}{3} \in \mathbb{R}\), \(\upgamma _{02} = \frac{1}{4} \in \mathbb{R}\), \(\vartheta _{1} = \frac{1}{3} >0\), \(\vartheta _{2} = \frac{1}{4}>0\), \(\upgamma _{11} = \frac{1}{2} \in (0,1)\), \(\upgamma _{12} = \frac{1}{4} \in (0,1)\), and \(\upgamma _{21} = \frac{1}{6}\in (0,1)\), \(\upgamma _{22}=\frac{1}{8} \in (0,1)\). We consider three cases
for the problem. With Eq. (4) and these data we find that
One can see these results in Table 1 and the graphical representation of Λ for three cases of q in Fig. 2. Consider the function
Now, for every \(\upvarphi , \tilde{\upvarphi} \in \mathbb{R}\) and \(\rho \in \Sigma \), one has
thus, the assumption (As1) is satisfied for \(l_{\mathfrak{g}}=\frac{1}{6}\) and by using Eq. (11) we obtain
Table 2 shows these results and the graphical representation of \(\mathrm{k}^{\ast}\) for three cases of q can be seen in Fig. 3. Hence, inequality (10) holds. Hence, all hypotheses of Theorem 3.2 hold, and so the problem (21) has at most one solution on Σ. Further, it follows from Theorem 4.1 that the problem (21) is Ulam–Hyers stable and consequently it is also generalized Ulam–Hyers stable.
Example 5.2
Consider problem (21) with the function
Now, for any \(\upvarphi \in \mathbb{R}\) and \(\rho \in \Sigma \), we have
thus, the assumption (As2) is satisfied for \(\mathrm{D}=2\), \(\mathrm{h}_{\mathfrak{g}}^{ ( 1 ) }= \frac{1}{4}\) and \(\mathrm{h}_{\mathfrak{g} }^{ ( 2 ) }=\frac{1}{7}\) and
Table 3 shows these results and the graphical representation of \(\mathrm{N}_{1}\) for three cases of q can be seen in Fig. 4. Hence, all items of Theorem 3.3 are satisfied. Hence, the problem (21) possesses at least one solution on Σ.
6 Conclusion
The \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) has been investigated in this work in detail. The investigation of this particular equation provides us with a powerful tool in modeling most scientific phenomena without the need to remove most parameters that have an essential role in the physical interpretation of the studied phenomena. For the first time, we have described the existence and uniqueness of solutions of various classes of nonlinear pantograph-type \(\mathbb{F}\mathrm{P}q{-}\mathbb{DE}\) (1) on a time scale under some BCs. Also, the two types of Ulam stability of the problem (1) are considered. We presented a few examples of \(\mathbb{F} \mathrm{P}q-\mathbb{DE}\) (1) that describe our outcomes.
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AL: Actualization, methodology, formal analysis, validation, investigation, initial draft and was a major contributor in writing the manuscript. MES: Actualization, methodology, formal analysis, validation, investigation, software, simulation, initial draft and was a major contributor in writing the manuscript. AA: Actualization, methodology, formal analysis, validation, investigation and initial draft. All authors read and approved the final manuscript.
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Appendix
Appendix
Algorithm 1
(MATLAB function for calculation q-gamma function)
Algorithm 2
(MATLAB function for calculation the fractional q-integral of the Riemann–Liouville type)
Algorithm 3
(MATLAB lines for calculation of all variables in Example 5.1)
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Lachouri, A., Samei, M.E. & Ardjouni, A. Existence and stability analysis for a class of fractional pantograph q-difference equations with nonlocal boundary conditions. Bound Value Probl 2023, 2 (2023). https://doi.org/10.1186/s13661-022-01691-1
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DOI: https://doi.org/10.1186/s13661-022-01691-1