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On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives
Journal of Inequalities and Applications volume 2024, Article number: 12 (2024)
Abstract
In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.
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, to obtain 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, [4–8]. In this sense, several interesting topics concerning research for differential equations involving fractional quantum calculus are devoted to the existence and Ulam–Hyers stability of the solutions [9]. Recently, many interesting results concerning the existence and Ulam-type stability of solutions for differential equations with fractional q-calculus were obtained, see [10–16] and the references therein. In [17–19], the existence and uniqueness of solutions were investigated for sequential differential equations with q-fractional calculus.
We have already seen that chaotic behavior can emerge in a system as simple as a logistic map. In that case the “route to chaos” is called period doubling. In practice, one would like to understand the route to chaos in systems described by partial differential equations, such as flow in a randomly stirred fluid. This is, however, very complicated and difficult to treat either analytically or numerically. Here, we consider an intermediate situation where the dynamics is described by a single ordinary differential equation, called the Duffing equation. The Duffing equation is considered with an important type of differential problem that has many applications in chaotic phenomena, for more details, we refer to the articles [20–25]. The classical form of the Duffing problem \((\mathrm{D}\mathbb{P})\) [20], can be displayed by
under initial values \(\mathrm{w} ( 0 ) = d_{1}\), \(\mathrm{D}^{1} \mathrm{w} ( 0 ) = d_{2}\), \(d_{1},d_{2}\in \mathbb{R}\), where φ and g are given continuous functions. Recently, the fractional-type Duffing equation \(( \mathrm{D}\mathbb{E})\) [26] of the form
with initial values \(\mathrm{w} ( 0 ) =d_{1}\), \(\mathrm{D}^{\zeta} \mathrm{w} (0 ) = d_{2}\), \(d_{1},d_{2}\in \mathbb{R}^{\ast}\), for \(\delta >0\), \(1< \varsigma <2\), \(0<\zeta <1, a,b,c,\varrho >0\), where \({}^{\mathrm{C} } \mathrm{D}^{\varkappa }\), \(\varkappa \in \{ \varsigma ,\zeta \}\) is the Caputo fractional derivative, has received considerable interest among scientific researchers, see [27–30] and references therein. In [25], the authors studied the existence and the uniqueness of solutions for sequential \(\mathbb{F}\mathrm{D}\mathbb{P}\) with Caputo-type fractional derivatives of different orders:
for \(0\leq \mu <\epsilon \leq 1\), \(0\leq \varsigma \), \(\zeta \leq 1\), \(1 < \epsilon +\zeta \leq 2\), \(1< \zeta +\varsigma \leq 2\), where \({}^{\mathrm{C} }\mathrm{D}^{\varkappa }, \varkappa \in \{ \varsigma , \zeta ,\varepsilon \} \) is the derivative in the sense of Caputo and \({}_{\mathrm{R.L} }\mathrm{I}^{\eta }\) is the Riemann–Liouville integral of order \(\eta \geq 0\), φ, ψ, g, and h are given continuous functions. Also, the authors discussed the fractional order \(\mathrm{D}\mathbb{P}\) of the form
via initial values \(\mathrm{w} ( \tau _{\circ } ) = \mathrm{w}_{0}\), \(\mathrm{D}^{1} \mathrm{w} ( \tau _{\circ } ) = \mathrm{w}_{1}\), where \({}^{\mathrm{C} }\mathrm{D}^{\varkappa}\) is the Caputo fractional derivative of order \(\varkappa \in \{ \varsigma , \zeta \}\), \(1< \varsigma <2\), \(0<\zeta <1\), and \(\tau _{\circ}\) is an initial value in Ω [29]. Riaz and Zada studied coupled \(\mathbb{FDE}\)s with the help of a Laplace-transform method
where \(\alpha _{i} \in \mathbb{R}\), \(p-1< \theta _{i} < p\), \(q-1< \vartheta _{i}\leq q\), \(p, q \in \mathbb{Z}^{+}\), \(q \leq p\), \(0 < {}_{1}\lambda _{i}\), \({}_{2}\lambda _{i}\leq 1\), \(\digamma _{i} : \Omega \times \mathbb{R}^{2} \to \mathbb{R} \), and \({}_{i}\mathrm{w}_{\kappa}^{\ast }\in \mathbb{R}\), \(i=1,2\) [31]. They studied the following class of implicit \(\mathbb{FDE}\) with implicit integral boundary condition:
for \(\tau \in [0, \ddot{\tau}]\), \(\ddot{\tau}>0\), where the notation \({}^{\mathrm{C} }\mathrm{D}^{\varsigma }\) is used for the Caputo fractional derivative of order \(0 < \varsigma \leq 1\), \(g, \mathfrak{g}, \digamma : [0, \ddot{\tau}] \times \mathbb{R}^{2} \to \mathbb{R}\), δ, σ are real constants greater than zero [32]. Guo et al. investigated the existence, uniqueness, and at least one solution of the coupled system of \(\mathbb{FDE}\)s in the sense of Hadamard derivatives:
for \(1\leq \tau \leq \ddot{\tau}\) under the generalized Hadamard fractional integrodifferential boundary conditions [33].
In this work, we discuss the existence, uniqueness, and Ulam–Hyers and Ulam–Hyers–Rassias stability (\(\mathrm{UH}\mathbb{S}\) & \(\mathrm{UHR}\mathbb{S}\)) of solutions for a sequential fractional Duffing q-differential equation \((\mathbb{F}\mathrm{D}q-\mathbb{DE})\) involving Riemann–Liouville-type and Caputo-type fractional q-derivatives given by
for \(0< \theta , \vartheta , \lambda , \mu , \gamma , q<1\), \(\eta \geq 0\), \(\beta , \alpha _{i} \in \mathbb{R}\), with \(\beta \neq \sum_{i=1}^{2} \alpha _{i}\), where \({}^{\mathrm{R.L} }\mathrm{D}_{ q}^{\theta}\) and \({}^{\mathrm{C} }\mathrm{D}_{ q}^{\upsilon}\), \(\upsilon \in \{ \vartheta ,\lambda \}\) are the Riemann–Liouville and Caputo fractional q-derivatives, respectively, \({}_{\mathrm{R.L} }\mathrm{I}_{ q}^{\eta}\) is the fractional q-derivative of the Riemann–Liouville type, \(\varphi : \Omega \to \mathbb{R}\), and \(g,h : \Omega \times \mathbb{R}^{2} \to \mathbb{R}\) are given continuous functions.
In Sect. 2, we recall some essential definitions of fractional quantum calculus. Sections 3 and 4 contain our main results about the existence, uniqueness, \(\mathrm{UH}\mathbb{S}\), and Ulam–Hyers–Rassias stability \(\mathrm{UHR}\mathbb{S}\) of the \(\mathbb{F}\mathrm{D}q-\mathbb{DE}\) (1), respectively. An application and illustrative example with some needed algorithms for the problem are given in Sect. 5. In Sect. 6, some conclusions are presented.
2 Fractional quantum calculus
We consider the fractional q-calculus on the specific time scale \(\mathbb{T}_{\tau _{0}} = \{0 \} \cup \{ \tau : \tau =\tau _{0}q^{n} \}\), for \(n \in \mathbb{N}\), \(\tau _{0} \in \mathbb{R}\), \(0 < q <1\), and in short we shall denote \(\mathbb{T}_{\tau _{0}}\) by \(\mathbb{T}\). Let \(\kappa \in \mathbb{R}\). Define \({}[ \kappa ]_{q} = \frac{1-q^{\kappa }}{ 1 -q}\) and the q-factorial function \((\tau _{1}-q\tau _{2})^{(n)}\) by
where \(\kappa _{1}, \kappa _{2} \in \mathbb{R}\) [34, 35]. In particular, \(\kappa _{1} =1\), \(\kappa _{2} = q\), we obtain
Algorithm 1 in [36] shows MATLAB lines to obtain the q-gamma function. Also, in [37], one can find that
Obviously, if \(\tau _{2}=0\), then \((\tau _{1} )^{(\sigma )}= \tau _{1}^{\sigma}\). The q-Gamma function is expressed by \(\Gamma _{q} ( \kappa ) = \frac{ ( 1-q )^{ ( \kappa -1 ) }}{ ( 1-q )^{\kappa -1}}\), for \(\kappa \in \mathbb{R} \setminus \{\cdots , -2, -1, 0\}\), and satisfies \(\Gamma _{q} ( \kappa +1 ) = [\kappa ]_{q}\Gamma _{q} ( \kappa )\) [34, 37]. The operator \({}^{\mathrm{R.L} }\mathrm{D}_{ q}^{\theta}\) is the fractional q-derivative of Riemann–Liouville type [38, 39], defined by
where \([\theta ] \) is the smallest integer greater than or equal to θ. The fractional q-derivative of the Caputo type of order υ is given by
while the fractional q-integral of Riemann–Liouville type [38, 39] is defined by
We recall the following lemmas [38, 39].
Lemma 2.1
Let \(\kappa , \rho \geq 0\) and z be a function defined in \([0,1]\). Then,
Lemma 2.2
Suppose that \(\kappa >0\), and ϵ is a positive integer. Then, we have
Lemma 2.3
Let \(\kappa \in \mathbb{R}^{+}\backslash \mathbb{N}\). Then, the following equality is valid
such that n is the smallest integer greater than or equal to κ.
In discrete fractional q-calculus, the fractional Riemann–Liouville-type q-integral of the function w is obtained by [36]
By using [36, Algorithm 2], one can calculate this type of q-integral.
Lemma 2.4
For \(\kappa \in \mathbb{R}_{+}\) and \(\rho >-1\), we have \({}_{\mathrm{R.L} }\mathrm{I}_{ q}^{\kappa } [ \tau ^{ ( \rho ) } ] = \frac{\Gamma _{q} ( \rho +1 ) }{\Gamma _{q} ( \kappa +\rho +1 ) } \tau ^{ ( \kappa + \rho )}\). If \(\rho =0\), we can obtain \({}_{\mathrm{R.L} }\mathrm{I}_{ q}^{\kappa } [ 1] = \frac{1}{\Gamma _{q} ( \kappa +1 ) } \tau ^{ ( \kappa )}\).
In order to study the problem (1), we need the following space \(B = \{ \mathrm{w} : \mathrm{w} \& {}^{\mathrm{C} } \mathrm{D}_{ q}^{\mu} \mathrm{w} \in C ( \Omega ) \}\), endowed with the norm
Then, it is well known that \(( B, \Vert . \Vert _{B} )\) is a Banach space. Now, we consider the Ulam-stability type for the sequential \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1).
Definition 2.5
The \(\mathbb{F}\mathrm{D}q-\mathbb{DE}\) (1) is stable in the
-
UH sense if there exists a real number \(\Sigma _{g^{\ast}, h^{\ast}} >0\) such that for each \(\omega >0\) and for each solution ẃ of the inequality
$$ \bigl\vert {}^{\mathrm{R.L} }\mathrm{D}_{ q}^{\theta} \bigl[ {}^{ \mathrm{C} }\mathrm{D}_{ q}^{\vartheta} \bigl[ {}^{\mathrm{C} } \mathrm{D}_{ q}^{\lambda } \acute{\mathrm{w}} ( \tau ) \bigr] \bigr] - \bigl[ \varphi ( \tau ) - \delta g_{ \acute{\mathrm{w}}}^{\ast } ( \tau ) - h_{ \acute{\mathrm{w}}}^{\ast} ( \tau ) \bigr] \bigr\vert \leq \omega ,\quad \tau \in \Omega , $$(3)there exists a solution w of the \(\mathbb{F}\mathrm{D}q-\mathbb{DE}\) (1) with \(\Vert \acute{\mathrm{w}} - \mathrm{w} \Vert _{B}\leq \Sigma _{g^{\ast}, h^{\ast}} \omega \), where \(g_{\acute{\mathrm{w}}}^{\ast} ( \tau ) = g ( \tau , \acute{\mathrm{w}} ( \tau ), {}^{\mathrm{C} }\mathrm{D}_{ q}^{\mu} \acute{\mathrm{w}} ( \tau ) )\) and \(h_{\acute{\mathrm{w}}}^{\ast} ( \tau ) = h ( \tau , \acute{\mathrm{w}} ( \tau ), {}_{\mathrm{R.L} }\mathrm{I}_{ q}^{ \eta} \acute{\mathrm{w}}( \tau ) )\);
-
UHR sense with respect to \(\mathrm{p}\in C ( \Omega , \mathbb{R}_{+} ) \) if there exists a real number \(\Sigma _{g^{\ast },h^{\ast }}> 0\) such that for each \(\omega >0\) and for each solution ẃ of the inequality
$$ \bigl\vert {}^{\mathrm{R.L} }\mathrm{D}_{ q}^{\theta} \bigl[ {}^{ \mathrm{C} }\mathrm{D}_{ q}^{\vartheta} \bigl[ {}^{\mathrm{C} } \mathrm{D}_{ q}^{\lambda} \acute{\mathrm{w}} ( \tau ) \bigr] \bigr] - \bigl[ \varphi ( \tau ) -\delta g_{ \acute{\mathrm{w}}}^{\ast } ( \tau ) - h_{ \acute{\mathrm{w}}}^{\ast} ( \tau ) \bigr] \bigr\vert \leq \Sigma _{g^{\ast}, h^{\ast}} \omega \mathrm{p} ( \tau ), $$(4)for \(\tau \in \Omega \), there exists a solution w of the \(\mathbb{F}\mathrm{D}q-\mathbb{DE}\) (1) with
$$ \Vert \acute{\mathrm{w}} -\mathrm{w} \Vert _{B} \leq \Sigma _{ g^{ \ast}, h^{\ast }} \omega \mathrm{p}( \tau ). $$
Remark 2.1
A function \(\acute{\mathrm{w}}\in C ( \Omega ) \) is a solution of the inequality (3) iff there exists a function \(\mathrm{p}: \Omega \to \mathbb{R}\) (which depends on ẃ) such that \(\vert \mathrm{p} ( \tau ) \vert \leq \omega \), for each \(\tau \in \Omega \), and
3 Existence of solution for \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\)
In this section, we prove the existence and uniqueness of a solution of problem (1). First, we state the following key lemma.
Lemma 3.1
Suppose that \(\mathrm{v} \in C ( \Omega )\). Then, the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\)
for \(0<\theta ,\vartheta ,\lambda , q<1 \), admits the following solution
such that \(\beta \neq \alpha _{1}+\alpha _{2}\).
Proof
Using Lemma 2.2, we can write
Next, applying Lemma 2.3, we obtain
Now, applying the operator \({}_{\mathrm{R.L} }\mathrm{I}_{ q}^{\lambda }\) to both sides of equation (8), we obtain
here \(b_{i} \in \mathbb{R}\), \(i=1,2,3\). By using the boundary conditions \(\mathrm{w} ( 0 ) =0\) and \([{}^{\mathrm{C} }\mathrm{D}_{ q}^{\vartheta } [ {}^{ \mathrm{C} }\mathrm{D}_{ q}^{\lambda } \mathrm{w} ( 1 ) ] ] =0\), we obtain
Now, by applying conditions (3) and (10), we have
Hence, we obtain Eq. (6). □
Using Lemma 3.1, we introduce an operator \(\mathcal{Z} : B \to B\) as follows
Before stating and proving the main results, we impose the following hypotheses:
- \((H_{1})\):
-
The functions g and h are continuous over \(\Omega \times \mathbb{R}^{2}\) and φ is continuous over Ω.
- \((H_{2})\):
-
There exist constant \(\varpi _{i}>0\) such that for all \(\tau \in \Omega \) and \(\mathrm{w}_{i}, \mathrm{v}_{i}\in \mathbb{R}^{2}\), \(i=1,2\), we have
$$ \bigl\vert g ( \tau , \mathrm{w}_{1}, \mathrm{w}_{2} ) -g ( \tau ,\mathrm{v}_{1}, \mathrm{v}_{2} ) \bigr\vert \leq \varpi _{1} \bigl( \vert \mathrm{w}_{1} - \mathrm{v}_{1} \vert + \vert \mathrm{w}_{2} - \mathrm{v}_{2} \vert \bigr) $$(11)and
$$ \bigl\vert h ( \tau ,\mathrm{w}_{1}, \mathrm{w}_{2} ) -h ( \tau , \mathrm{v}_{1}, \mathrm{v}_{2} ) \bigr\vert \leq \varpi _{2} \bigl( \vert \mathrm{w}_{1} - \mathrm{v}_{1} \vert + \vert \mathrm{w}_{2} - \mathrm{v}_{2} \vert \bigr). $$(12) - \((H_{3})\):
-
There exist positive constants \(N_{i}\), \(i=1,2,3\) such that for all \(\tau \in \Omega \) and \(\mathrm{w}, \mathrm{v}\in \mathbb{R} \),
$$ \begin{aligned} \bigl\vert g ( \tau ,\mathrm{w}, \mathrm{v} ) \bigr\vert \leq N_{1},\qquad \bigl\vert h ( \tau ,\mathrm{w}, \mathrm{v} ) \bigr\vert \leq N_{2}, \qquad \bigl\vert \varphi ( \tau ) \bigr\vert \leq N_{3}. \end{aligned} $$
For the sake of convenience, we introduce the following quantities:
Theorem 3.2
Assume that \(( H_{1} ) \) and \(( H_{2} ) \) hold and that
where \(\aleph _{i}\), \(i=1,2\), are given by (13), then the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) has a unique solution.
Proof
Let us define \(\Lambda =\max \{ \Lambda _{i} : i=1,2,3 \} \), where \(\Lambda _{i}\) are finite numbers given by
Setting real number r such that
we show that \(\mathcal{Z}B_{r}\subset B_{r}\), where \(B_{r} = \{ \mathrm{w}\in B : \Vert \mathrm{w} \Vert _{B}\leq r \}\). For \(\mathrm{w}\in B_{r}\) and by \((H_{1} )\), we obtain
and
By Eqs. (16) and (17), we obtain
which implies that
Also, we have
This implies that
In consequence, we obtain
which means that \(\mathcal{Z} B_{r} \subset B_{r}\). For \(\mathrm{w}, \mathrm{v}\in B_{r}\) and for each \(\tau \in \Omega \), we have
Using \((H_{1})\), we obtain
Hence,
On the other hand, for each \(\tau \in \Omega \), we have
Thanks to \((H_{1})\), we have
Therefore,
Then, thanks to Eqs. (18) and (19), we conclude that
By (14), we see that \(\mathcal{Z}\) is a contractive operator. Consequently, by the Banach fixed-point theorem, \(\mathcal{Z}\) has a fixed point that is a solution of (1). □
Now, we prove the existence of a solution for the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) by applying the following lemma.
Lemma 3.3
([40])
Let S be a Banach space. Assume that \(\mathcal{Z} : S \to S\) is a completely continuous operator and that the set \(B= \{ \mathrm{w}\in S : \mathrm{w} =\zeta \mathcal{Z} ( \mathrm{w} ), \textit{ } 0 < \zeta <1 \}\), is bounded. Then, \(\mathcal{Z}\) has a fixed point in S.
Theorem 3.4
If conditions \((H_{1} )\) and \((H_{3} )\) are satisfied, then the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) has at least one solution.
Proof
By the continuity of functions g, h, and φ, the operator \(\mathcal{Z} \) is continuous. Now, we show that the operator \(\mathcal{Z} \) is completely continuous. First, we show that \(\mathcal{Z}\) maps bounded sets of B into bounded sets of B. Let us take \(\sigma >0\) and \(B_{\sigma } = \{\mathrm{w} \in B : \Vert \mathrm{w} \Vert _{B}\leq \sigma \}\). Then, for all \(\mathrm{w}\in B_{\sigma }\), we have
and
From the above inequalities, it follows that the operator \(\mathcal{Z}\) is uniformly bounded. Next, we show that \(\mathcal{Z}\) is equicontinuous. Let \(\mathrm{w} \in B_{\sigma }\) and \(\tau _{1}, \tau _{2} \in \Omega \), with \(\tau _{1}<\tau _{2} \), we have
and
which imply that \(\Vert \mathcal{Z} \mathrm{w} ( \tau _{2} ) - \mathcal{Z} \mathrm{w} ( \tau _{1} ) \Vert _{B} \to 0\), as \(\tau _{2}\to \tau _{1}\). Combining these results and using the Arzelà–Ascoli theorem, we conclude that \(\mathcal{Z}\) is a completely continuous operator. Finally, it will be verified that the set Π, defined by \(\Pi = \{ \mathrm{w} \in B : \mathrm{w} = \xi \mathcal{Z} ( \mathrm{w} ), \text{ } 0 < \xi < 1 \}\), is bounded. Let \(\mathrm{w}\in \Pi \), then \(\mathrm{w} =\xi \mathcal{Z} ( \mathrm{w} ) \). For all \(\tau \in \Omega \), we have \(\mathrm{w} ( \tau ) = \xi \mathcal{Z} \mathrm{w} ( \tau )\). Then, by \(( H_{3} )\), we obtain
It follows from Inequality (20) that
Consequently, \(\Vert \mathrm{w} \Vert _{B}<\infty \). This shows that the set Π is bounded. Thanks to previous results, and by Lemma 3.3, we deduce that \(\mathcal{Z}\) has at least one fixed point, which is a solution of problem (1). □
4 Ulam-stability of \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\)
In this part, the Ulam stability of the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) will be discussed.
Theorem 4.1
Suppose that conditions \((H_{1})\) and \((H_{2})\) are valid. In addition, it is assumed that
then the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) is UH stable.
Proof
Suppose that \(\acute{\mathrm{w}}\in B\) is a solution (3) and \(\mathrm{w}\in B\) is a unique solution of the problem
Thanks to Lemma 3.1, we have
for \(b_{i}\in \mathbb{R}\), \(i=1,2,3\), where \(\mathrm{v}_{\mathrm{w}} ( \tau ) = \varphi ( \tau ) - \delta g_{\mathrm{w}}^{\ast } ( \tau ) - h_{ \mathrm{w}}^{\ast } ( \tau )\), for \(\tau \in \Omega \). By integration of the inequality (3), we obtain
Then, for any \(\tau \in \Omega \), we obtain
Now, using \((H_{2} ) \), we obtain
Then, we have
Hence, the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) is stable in the UH sense. □
Theorem 4.2
Suppose that conditions \((H_{1} )\), \((H_{2} )\), and Eq. (21) are valid. Suppose there exists \(\chi _{p}>0\) such that
where \(\mathrm{p}\in C ( \Omega ,\mathbb{R}_{+} ) \) is nondecreasing. Then, the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) is UHR stable.
Proof
Let \(\acute{\mathrm{w}}\in B\) be a solution of the inequality (4). Applying Remark 2.1, we can obtain
Let us denote by \(\mathrm{w}\in B\) the unique solution of the problem (22). Then, we have
here \(b_{i}\in \mathbb{R}\), \(i=1,2,3\). Then,
Hence, by \((H_{2} ) \) and (23), we obtain
Then, we have
for \(\tau \in \Omega \). Hence, the \(\mathbb{F} \mathrm{D}q - \mathbb{DP}\) (1) is stable in the UHR sense. □
5 Numerical results
5.1 An illustrative example
Example 5.1
Based on the problem (1), we consider the following \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\)
for \(\tau \in \Omega = [ 0,1 ]\), \(q \in \{\frac{3}{10}, \frac{1}{2}, \frac{8}{9} \} \in (0,1)\), and the following q-fractional inequalities
and
where
Clearly \(\theta = \frac{1}{2}\exp (1) \in (0,1)\), \(\vartheta = \frac{\sqrt{11}}{6} \in (0,1)\), \(\lambda = \frac{3}{5} \in (0,1)\), \(\delta = \frac{1}{25^{2}}\sqrt{\exp (1)} >0\), \(\varphi (\tau ) = \frac{1}{ 3+\tau ^{2}} (\ln \tau +2)\), \(\tau \in \Omega \), \(\mu = \frac{1}{4} \in (0,1)\), \(\eta = \frac{5}{4} \geq 0\), and \(\gamma =\frac{2}{5} \in (0,1)\), \(\beta = \frac{2\exp (1)}{13}\in \mathbb{R}\), \(\alpha _{1} = \frac{\sqrt{7}}{3}\in \mathbb{R}\), \(\alpha _{2} = \frac{\sin 7 }{5}\in \mathbb{R}\), with \(\beta \neq \alpha _{1} + \alpha _{1}\). Condition \((H_{1})\) holds because g, h are continuous over \(\Omega \times \mathbb{R}^{2}\) and φ is continuous over Ω. Also, for \(\tau \in \Omega \) and \(\mathrm{w}_{i}, \mathrm{v}_{i}\in \mathbb{R}^{2}\), \(i=1,2\), we have
and
Hence, by choosing \(\varpi _{1} = \frac{1}{50 \sqrt{2\pi }}\), \(\varpi _{2} = \frac{1}{25(\exp (1))^{2}}\), condition \((H_{2})\), inequalities (11) and (12) hold. Now, we consider three cases for \(q \in \{\frac{3}{10}, \frac{1}{2}, \frac{8}{9} \}\), in \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24). By employing (13), we find that
and
One can see these results in Tables 1, 2, 3, and 4. We can see graphical representation of \(\aleph _{1}\) and \(\aleph _{2}\) for three cases of q in Fig. 1. Curves 1a and 1b show well that as q approaches 1, the values of \(\aleph _{1}\) and \(\aleph _{2}\) decrease harmoniously. Similarly, Tables 1, 2, and 3 show the values of \(\aleph _{1}\) and \(\aleph _{2}\) for three cases of \(q=\frac{3}{10},\frac{1}{2}, \frac{8}{9}\), respectively, and the interesting thing is that when q approaches 1, we reach a constant value up to 4 decimal places in the number of steps higher than n.
Graphical representation of \(\aleph _{1}\) and \(\aleph _{2}\) for \(q\in \{\frac{3}{10}, \frac{1}{2}, \frac{8}{9} \}\) in Example 5.1
It is found that \(\Lambda _{1} =\sup_{\tau \in \Omega } \vert g ( \tau ,0,0 ) \vert =0\), \(\Lambda _{2} =\sup_{\tau \in \Omega } \vert h ( \tau ,0,0 ) \vert =0\),
and so \(\Lambda =\max \{ \Lambda _{i} : i=1,2,3 \}= \frac{1}{2}\). From inequality (14), we remark that
According to Table 4, the calculations performed are consistent with our analysis and analytical proofs. Figure 2 shows this for the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) with three cases of q. Curves 2a, 2b, and 2c show well that as q approaches 1, the value of Δ, Eq. (14) holds, but its rate of change increases with the increase of q. Similarly, Table 4 shows the values of Δ for three cases of \(q=\frac{3}{10},\frac{1}{2}, \frac{8}{9}\), respectively, and the interesting thing is that when q approaches 1, we reach a constant value up to 4 decimal places in the number of steps higher than n. Setting real number r such that \(r\geq 5.8553, 3.3620, 0.5098\), whenever \(q=\frac{3}{10}, \frac{1}{2}, \frac{8}{9}\), respectively. Therefore, the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) satisfies all conditions of Theorem 3.2 and so problem (24) has a unique solution. In the other hand, condition \((H_{3})\) holds, because there exists \(N_{i}\geq 0\), \(i=1,2,3\) such that
and \(|\varphi (\tau )| = |\frac{\ln \tau +2}{3+\tau ^{2}} | \leq \frac{1}{2}=:N_{3}\). Hence Theorem 3.4 implies that the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) has at least one solution. Also, according to the numerical results in Table 5, we have
Hence, the inequality (21) in Theorem 4.1 holds for all \(q \in (0,1)\) and so Theorem 4.1 implies that the \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) is UH stable with
We can see graphical representations of UH stable \(\Sigma _{g^{\ast },h^{\ast }}\) with three cases of q in Fig. 3. Let \(\mathrm{p}( \tau ) = \tau ^{\frac{\sqrt{3}}{2}}\), then
Graphical representation of Δ and \(\frac{1}{\aleph _{1}+\aleph _{2}}\) for \(q\in \{ \frac{3}{10}, \frac{1}{2}, \frac{8}{9} \}\) in Example 5.1
Graphical representation of UH stable \(\Sigma _{g^{\ast },h^{\ast }}\) for \(q \in \{\frac{3}{10} , \frac{1}{2}, \frac{8}{9} \}\) in Example 5.1
Table 6 shows these results. Then, the condition (23) is satisfied with \(\mathrm{p}(\tau )=\tau ^{\frac{\sqrt{3}}{2}}\) and
It follows from Theorem 4.2 that \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) is UHR stable with
One can see graphical representations of UHR stable \(\Sigma _{g^{\ast },h^{\ast }}\) with three cases of q in Fig. 4.
Graphical representation of UHR stable \(\Sigma _{g^{\ast },h^{\ast }}\) for \(q \in \{\frac{3}{10} , \frac{1}{2}, \frac{8}{9} \}\) in Example 5.1
5.2 An application: a Duffing-type oscillator
As is known, the equation of motion of a Duffing oscillator under initial conditions is normally written
\(\mathrm{w}(0) = r_{1}\), \(\mathrm{w}^{\prime }(0) = r_{2}\), where \(\upalpha _{i}\), \(r_{1}\), \(r_{2}\) are real constants, \(\mathrm{w}^{\prime \prime}+ \upalpha _{2} \mathrm{w}\) is a simple harmonic oscillator with angular frequency \(\sqrt{\upalpha _{2}}\), \(\upalpha _{1} \mathrm{w}^{\prime} \) is a small damping, \(\upalpha _{3} \mathrm{w}^{3}\) is a small nonlinearity, and \(F_{0} \cos \omega \tau \) is a small periodic forcing term with angular frequency ω. In this work, the differential equation (27) is rearranged to \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) as follows:
for \(\delta >0\), \(\tau \in \Omega := [ 0,2\pi ]\). Hence, \(\varphi ( \tau )= F_{0} \cos \omega \tau \),
with \(\mu =1\), \(\eta =0\), \(\delta = \upalpha _{1}\), and \(\upalpha _{2}, \upalpha _{3} \neq 0\). It is obvious that Theorems 3.2, 3.4, and 4.1 confirm the existence of the solution and its stability.
6 Conclusion
The \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) 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. \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (1) has been studied on a time scale under some BCs. An application that describes the motion of a particle in the plane has been provided to support our results’ validity and applicability in fields of physics and engineering.
Data availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support in writing this paper.
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Houas, M., Samei, M.E., Sundar Santra, S. et al. On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives. J Inequal Appl 2024, 12 (2024). https://doi.org/10.1186/s13660-024-03093-6
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DOI: https://doi.org/10.1186/s13660-024-03093-6