Abstract
The main purpose of this study is to clarify the Hyers–Ulam stability (HUS) for the Cayley quantum equation. In addition, the result obtained for all parameters is applied to HUS of h-difference equations with a specific variable coefficient using a new transformation.
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1 Introduction
Quantum calculus has been of interest for some time, but really received a boost with the publication of the monograph of the same name, by Kac and Cheung [13]. In that work, both q-difference equations and h-difference equations are dealt with, but no direct transformation is given relating equations of one with the other. We introduce such a direct nexus later on in this work. First, we consider the first-order linear quantum equation
where \(q>1\), \(\gamma \in [0,1]\),
and \(\lambda \in \mathbb {C}\) satisfies the condition
Equation (1.1) is called a Cayley equation, and \(\gamma \in [0,1]\) is called the Cayley parameter, see [11]. Let \(\mathbb {N}\) be the set of natural numbers, and let \(\mathbb {N}_0:=\mathbb {N}\cup \{0\}\) and \(q^{\mathbb {N}_0}:=\{1,q,q^2,q^3,\cdots \}\).
Definition 1.1
The Cayley quantum equation (1.1) has Hyers–Ulam stability (HUS) if and only if there exists a constant \(K>0\) with the following property:
For an arbitrary \(\varepsilon >0\), if a function \(\eta :q^{\mathbb {N}_0}\rightarrow \mathbb {C}\) satisfies
$$\begin{aligned} \left| D_q\eta (s) - \lambda \left\langle \eta (s)\right\rangle _{\gamma }\right| \le \varepsilon \end{aligned}$$(1.3)for all \(s\in q^{\mathbb {N}_0}\), then there exists a solution \(y:q^{\mathbb {N}_0}\rightarrow \mathbb {C}\) of (1.1) such that
$$\begin{aligned} |\eta (s)-y(s)|\le K\varepsilon \end{aligned}$$for all \(s\in q^{\mathbb {N}_0}\).
Such a constant K is called an HUS constant for (1.1) on \(q^{\mathbb {N}_0}\).
Recently, the authors [5] considered the Hyers–Ulam stability of the Cayley quantum equation (1.1) with Cayley parameter \(\gamma \in \left[ 0,\frac{1}{2}\right] \). Under assumption (1.2), they proved the following facts: if \(\lambda =0\), then (1.1) is not HUS on \(q^{\mathbb {N}_0}\) [5, Lemma 2.4]; if \(\gamma =\frac{1}{2}\), and \(\lambda \in \mathbb {C}\) satisfies (1.2), then (1.1) is not HUS on \(q^{\mathbb {N}_0}\) [5, Theorem 3.1]; if \(\gamma \in \left[ 0,\frac{1}{2}\right) \), and \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfies (1.2), then (1.1) is HUS on \(q^{\mathbb {N}_0}\) [5, Corollary 2.7]. Now arises a natural question. What happens in the case where \(\gamma \in \left( \frac{1}{2},1\right] \) and \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfies (1.2)? The first purpose of this study is to consider the Hyers–Ulam stability of the Cayley quantum equation (1.1) with Cayley parameter \(\gamma \in \left( \frac{1}{2},1\right] \). Note that the proof used here is quite different from the previous case.
In 2020, the authors [4] introduced a new, direct connection between HUS for h-difference equations and HUS for quantum equations of Euler type. The second purpose of this study is to establish a novel connection between HUS for Cayley quantum equations and HUS for the h-difference equations with a specific variable coefficient, based on the ideas in this paper.
Hyers–Ulam stability is a burgeoning area of study, encompassing functional equations, differential and difference equations, fractional equations, and the like. Some representative publications include the following. Linear h-difference equations and linear difference equations are explored by [3, 6, 8], and first and second order linear equations in [17, 18]. The Pielou logistic equation is considered by [12], and the various Möbius equations by [14,15,16]. Implicit fractional q-difference equations are treated by [1, 2, 10]. Fractional stability is investigated by [7] and [20], time-dependent and periodic coefficients by [9], and differential equations and HUS driven by measures is the focus of [19].
2 Hyers–Ulam Stability for Cayley Parameter \(\gamma \in \left( \frac{1}{2},1\right] \)
Under the assumption (1.2), we can solve quantum equation (1.1) on \(q^{\mathbb {N}_0}\). The following facts were given in [5].
Lemma 2.1
For any \(\lambda \) satisfying (1.2), the general solution of (1.1) is given by
where \(c\in \mathbb {C}\) is an arbitrary constant.
Lemma 2.2
Fix \(q>1\). Let \(\lambda \) satisfy (1.2). For an arbitrary \(\varepsilon >0\), if a function \(\eta :q^{\mathbb {N}_0}\rightarrow \mathbb {C}\) satisfies the inequality (1.3), then for \(s\in q^{\mathbb {N}_0}\), \(\eta \) has the form \(\eta :=\tau \sigma +c\tau \), where
\(c\in \mathbb {C}\) is an arbitrary constant, and the function P satisfies \(|P(s)|\le \varepsilon \) for all \(s\in q^{\mathbb {N}_0}\).
Lemma 2.3
Let \(q>1\), \(\gamma \in \left( \frac{1}{2},1\right] \), and let \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfy (1.2). Let \(\tau \) be the function given in (2.2). Then, \(\lim _{s\rightarrow \infty }|\tau (s)|=0\), and the function
is bounded above on \(q^{\mathbb {N}_0}\).
Proof
First, we will show that \(\lim _{s\rightarrow \infty }\tau (s)=0\). Since
holds for \(\gamma \in \left( \frac{1}{2},1\right] \) and \(\lambda \ne 0\), we see that there exists a \(s_1\in q^{\mathbb {N}_0}\) such that
for \(s\ge s_1\). Note here that
for \(\gamma \in \left( \frac{1}{2},1\right] \). Using the above-mentioned estimation, we have
for \(s\ge s_1\). Let \(k_1:= \log _q s_1\). Then,
Thus, setting \(s:= q^{k_1+k}\), we have
for \(s\ge q^{k_1+1}=qs_1\). Therefore, we obtain \(\lim _{s\rightarrow \infty }|\tau (s)|=0\).
Next, we will show that the function
is bounded on \(q^{\mathbb {N}_0}\). First, we consider the case \(\gamma \in \left( \frac{1}{2},1\right) \) and \(\lambda \ne 0\). From
we see that there exists a \(k_2\in \mathbb {N}_0\) such that
for \(k\ge k_2\). Let \(m\ge k_2\) with \(m\in \mathbb {N}\). Using the same arguments as in the first part of this proof, we obtain the following:
for all \(m\ge k_2\) and \(l\in \mathbb {N}\). Set \(s:= q^{m+l}\). Then,
for \(k_2 \le m \le \log _q s-1\). From
there exists an \(m_1\ge k_2\) with \(m_1\in \mathbb {N}_0\) such that
for \(m\ge m_1\). Consequently, we have
for \(s\ge q^{m_1+1}\), where
This inequality together with \(\lim _{s\rightarrow \infty }|\tau (s)|=0\) yields
This says that \(\beta (s)\) is bounded above on \(q^{\mathbb {N}_0}\).
Next we consider the case \(\gamma = 1\) and \(\lambda \ne 0\). In this case, \(\tau (s)\) and \(\beta (s)\) are written in the following form:
From
there exists a \(k_3\in \mathbb {N}_0\) such that
for \(k\ge k_3\), and thus,
for \(k\ge k_3\). Since \(\lim _{k\rightarrow \infty }\frac{2}{(q-1)|\lambda |q^k} = 0\) holds, we see that there exists a \(k_4\in \mathbb {N}_0\) such that
for \(k\ge k_4\). Let \(m_2:= \max \{k_3,k_4\}\). Then,
for \(k\ge m_2\). Consequently, we have
for \(m\ge m_2\) with \(m\in \mathbb {N}_0\) and \(l\in \mathbb {N}\). Put \(s:=q^{m+l}\). Then,
for \(m_2\le m \le \log _q s-1\). Using this inequality, we see that
for \(s\ge q^{m_2+1}\), where
Thus, this together with \(\lim _{s\rightarrow \infty }|\tau (s)|=0\) yields the boundedness of \(\beta (s)\) on \(q^{\mathbb {N}_0}\) when \(\gamma =1\) and \(\lambda \ne 0\). This completes the proof. \(\square \)
Theorem 2.4
Let \(q>1\), \(\gamma \in \left( \frac{1}{2},1\right] \), and let \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfy (1.2). Let \(\tau \) be the function defined by (2.2). Then, (1.1) has HUS with HUS constant
on \(q^{\mathbb {N}_0}\).
Proof
Let an arbitrary \(\varepsilon >0\) be given, and \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfy (1.2). Assume that \(\left| D_q\eta (s) - \lambda \left\langle \eta (s)\right\rangle _{\gamma }\right| \le \varepsilon \) for all \(s\in q^{\mathbb {N}_0}\). Now we consider the functions \(\tau \) and \(\sigma \) given in (2.2). Then, \(\eta \) has the form
and satisfies
for all \(s\in q^{\mathbb {N}_0}\) by Lemma 2.2, where \(\eta _0\) is an arbitrary complex constant. Define
Then, y is a solution to (1.1) from Lemma 2.1. Therefore,
for all \(s\in q^{\mathbb {N}_0}\). By Lemma 2.3, the right-hand side is bounded above on \(q^{\mathbb {N}_0}\). Hence, (1.1) has Hyers–Ulam stability with HUS constant
This completes the proof. \(\square \)
By combining Theorem 2.4 with the previous results (already mentioned in the introduction) given in [5], we get the following immediately.
Theorem 2.5
Let \(q>1\), \(\gamma \in [0,1]\), and \(\lambda \in \mathbb {C}\) satisfy (1.2). Then, (1.1) has HUS on \(q^{\mathbb {N}_0}\) if and only if \(\lambda \ne 0\) and \(\gamma \ne \frac{1}{2}\).
Theorem 2.6
Let \(q>1\), and let \(\gamma \in \left( \frac{1}{2},1\right] \) and \(\lambda \in \mathbb {C}\backslash \{0\}\) satisfy (1.2), and
for sufficiently large \(s\in q^{\mathbb {N}_0}\). Then, (1.1) has HUS on \(q^{\mathbb {N}_0}\). Furthermore, for sufficiently large \(s\in q^{\mathbb {N}_0}\), there exists a \(\delta >0\) such that an HUS constant is \(\frac{1}{|\lambda |}+\delta \).
Proof
From the assumptions, (1.1) has HUS with HUS constant K on \(q^{\mathbb {N}_0}\), where K is given in 2.3. Define \(\eta _1(s):=-\frac{1}{\lambda }\). Then, \(\eta _1\) is a member of the solutions to the equation
On the other hand, by Lemma 2.2, we see that the general solution of this equation is written by
where \(c\in \mathbb {C}\) is an arbitrary constant, and \(\tau \) is given in (2.2). Combining these facts, we have
for a suitable constant \(c_0\in \mathbb {C}\). From Lemma 2.3, \(\lim _{s\rightarrow \infty }|\tau (s)|=0\) holds, and so that
This together with the assumption in this theorem yields
This means that for sufficiently large \(s\in q^{\mathbb {N}_0}\), there exists a \(\delta >0\) such that
Therefore, \(\frac{1}{|\lambda |}+\delta \) is an HUS constant for sufficiently large \(s\in q^{\mathbb {N}_0}\). \(\square \)
Example 2.7
We give several examples related to Theorem 2.6. Let \(\gamma =1\), \(q=2\) for the following.
If \(\lambda =5\), then \(\delta =\frac{1}{20}\) for \(s=q^2\), as \(|\eta (2^2)-y(2^2)|\le \frac{1}{4}\varepsilon = \left( \frac{1}{5}+\frac{1}{20}\right) \varepsilon \), and \(|\eta (s)-y(s)|\le |\eta (2^2)-y(2^2)|\) for all \(s\in q^{\mathbb {N}_0}\).
If \(\lambda =-5\), then \(\delta =0\), as \(|\eta (s)-y(s)|\le \frac{1}{5}\varepsilon = \frac{1}{|-5|}\varepsilon \) for all \(s\in q^{\mathbb {N}_0}\).
If \(\lambda =1-i\), then \(\delta =\frac{3}{\sqrt{5}}-\frac{1}{\sqrt{2}}\approx 0.634534\) for \(s=q^2\), as \(|\eta (2^2)-y(2^2)|\le \frac{3}{\sqrt{5}}\varepsilon = \left( \frac{1}{|1-i|}+\frac{3}{\sqrt{5}}-\frac{1}{\sqrt{2}}\right) \varepsilon \), and \(|\eta (s)-y(s)|\le |\eta (2^2)-y(2^2)|\) for all \(s\in q^{\mathbb {N}_0}\).
Remark 2.8
Of course, condition (2.4) does not hold in general. In fact, we know that
as shown in the proof of Theorem 2.6, while numerical evidence indicates that
for any \(q>1\), any \(\lambda \) satisfying (1.2), and any \(\gamma \in \left( \frac{1}{2},1\right] \). It is clear that the two limits are equal for \(\gamma =1\).
3 Application to h-Difference Equations
In [4], it has been shown that there is a suitable transformation between the quantum (q and h difference) equations on two different time scales to guarantee stability for both equations. More specifically, it turns out that if the h-difference equation has HUS, then the corresponding quantum equation of Euler type also has HUS. The reverse is also true. In this section, based on this idea, we will introduce a connection established between the Cayley quantum equation and an h-difference equation with variable coefficient.
Lemma 3.1
Let \(q>1\) and \(h>0\). Set
Let \(\lambda \in \mathbb {C}\) satisfy (1.2), and \(\alpha \in \mathbb {C}\) satisfy
Then, the Cayley quantum equation (1.1) has a solution y for \(s\in q^{\mathbb {N}_0}\) if and only if the Cayley h-difference equation
has a solution x for \(t\in h\mathbb {N}_0\), where
and satisfying the following relationships:
Proof
Let y be a solution of (1.1) for \(s\in q^{\mathbb {N}_0}\). From
and
we find that
Thus, x is a solution of (3.2) for \(t\in h\mathbb {N}_0\). The reverse is clearly true. \(\square \)
Remark 3.2
If \(q=1+h\) and \(\gamma =0\), then (3.2) is reduced to the h-difference equation
We can easily find that \((1+h)^{\frac{t}{h}}\) is a solution of \(\Delta _h x(t) = x(t)\) for \(t\in h\mathbb {N}_0\), and \(\lim _{t\rightarrow 0}(1+h)^{\frac{t}{h}} = e^t\). Hence, we can regard the above h-difference equation as an approximate equation of the differential equation
Definition 3.3
The Cayley h-difference equation (3.2) has Hyers–Ulam stability if and only if there exists a constant \(K>0\) with the following property:
For an arbitrary \(\varepsilon >0\), if a function \(\xi :h\mathbb {N}_0\rightarrow \mathbb {C}\) satisfies
$$\begin{aligned} \left| q^{-\frac{t}{h}} \Delta _h \xi (t)-\alpha [\xi (t)]_{\gamma }\right| \le \varepsilon \end{aligned}$$(3.4)for all \(t\in h\mathbb {N}_0\), then there exists a solution \(x:h\mathbb {N}_0\rightarrow \mathbb {C}\) of (3.2) such that
$$\begin{aligned} |\xi (t)-x(t)|\le K\varepsilon \end{aligned}$$for all \(t\in h\mathbb {N}_0\).
Such a constant K is called an HUS constant for (3.2) on \(h\mathbb {N}_0\).
We establish the following result.
Theorem 3.4
Let \(q>1\), \(h>0\), \(\gamma \in [0,1]\), and \(\alpha \in \mathbb {C}\) satisfy (3.1). Then, (3.2) has HUS on \(h\mathbb {N}_0\) if and only if \(\alpha \ne 0\) and \(\gamma \ne \frac{1}{2}\).
Proof
Suppose that \(\alpha \ne 0\), \(\gamma \ne \frac{1}{2}\), and condition (3.4) holds on \(h\mathbb {N}_0\). Using the transformation
we obtain
and
As a result,
This together with the assumption (3.4) says that
on \(q^{\mathbb {N}_0}\). Since \(\lambda = \frac{h\alpha }{q-1}\), \(\alpha \ne 0\) and restriction (3.1) imply \(\lambda \ne 0\) and condition (1.2) is met. By Theorem 2.5, we see that there exist a \(K>0\) and a solution \(y:q^{\mathbb {N}_0}\rightarrow \mathbb {C}\) of (1.1) such that
for all \(s\in q^{\mathbb {N}_0}\). Let \(x(t) = \frac{h}{q-1}y\left( q^{\frac{t}{h}}\right) \). Then, x is a solution to (3.2) by Lemma 3.1. Moreover, the above inequality implies
for all \(t \in h\mathbb {N}_0\). Therefore, (3.2) has HUS on \(h\mathbb {N}_0\) if \(\alpha \ne 0\) and \(\gamma \ne \frac{1}{2}\).
Conversely, we will show that HUS implies \(\alpha \ne 0\) and \(\gamma \ne \frac{1}{2}\). By way of contradiction, we suppose that \(\alpha = 0\) or \(\gamma = \frac{1}{2}\) holds. Since (3.2) is HUS on \(h\mathbb {N}_0\), we see that if (3.4) holds for \(t\in h\mathbb {N}_0\), then there exist a \(K_0>0\) and a solution \(x:h\mathbb {N}_0\rightarrow \mathbb {C}\) of (3.2) such that
for all \(t \in h\mathbb {N}_0\). Using transformation (3.5) again, we obtain
on \(q^{\mathbb {N}_0}\). On the other hand, if \(y(s) = \frac{q-1}{h}x\left( h\log _q s\right) \), then
for all \(s \in q^{\mathbb {N}_0}\), and y is a solution to (1.1) by Lemma 3.1. That is, (1.1) has HUS. However, by Theorem 2.5, we know that (1.1) is not HUS when \(\alpha = 0\) or \(\gamma = \frac{1}{2}\). This is a contradiction. \(\square \)
Remark 3.5
From inequality (3.6), we conclude that the following holds: if (1.1) has HUS with HUS constant \(K_1\) on \(q^{\mathbb {N}_0}\), then (3.2) has HUS with HUS constant \(\frac{hK_1}{q-1}\) on \(h\mathbb {N}_0\). On the other hand, if (3.2) has HUS with HUS constant \(K_2\) on \(h\mathbb {N}_0\), then (1.1) has HUS with HUS constant \(\frac{(q-1)K_2}{h}\) on \(q^{\mathbb {N}_0}\). Note that if \(q=h+1\), then both HUS constants are the same.
Hence, we can establish the following results using the change of variable connection given in (3.3).
Theorem 3.6
Let \(q>1\), \(h>0\), \(\gamma \in \left( \frac{1}{2},1\right] \), and let \(\alpha \in \mathbb {C}\backslash \{0\}\) satisfy (3.1). Then, (3.2) has HUS with HUS constant
on \(h\mathbb {N}_0\), where
Theorem 3.7
Let \(q>1\), \(h>0\), and let \(\gamma \in \left( \frac{1}{2},1\right] \) and \(\alpha \in \mathbb {C}\backslash \{0\}\) satisfy (3.1), and
for sufficiently large \(t\in h\mathbb {N}_0\). Then, (3.2) has HUS on \(h\mathbb {N}_0\). Furthermore, for sufficiently large \(t\in h\mathbb {N}_0\), there exists a \(\delta >0\) such that an HUS constant is \(\frac{h}{q-1}\left( \frac{1}{|\lambda |}+\delta \right) \).
4 Conclusion
In this work, we have shown for any quantum base \(q>1\), any Cayley parameter \(\gamma \in [0,1]\), and for eigenvalues \(\lambda \in \mathbb {C}\) that satisfy a certain restriction that rules out division by zero, that the Cayley quantum equation with constant complex coefficient \(\lambda \) has Hyers–Ulam stability (HUS) on the quantum time scale \(q^{\mathbb {N}_0}\), if and only if \(\lambda \ne 0\) and \(\gamma \ne \frac{1}{2}\). We have also given precise estimates for the HUS constant of stability, and scenarios where it is the best (minimal) possible. Moreover, an entirely new connection is made between this Cayley quantum equation and a corresponding h-difference equation with variable coefficient. The HUS of one tracks exactly the HUS of the other through a change of variables, and the HUS constants are likewise related.
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The second author was supported by JSPS KAKENHI Grant Number JP20K03668.
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Anderson, D.R., Onitsuka, M. Hyers–Ulam Stability for Cayley Quantum Equations and Its Application to h-Difference Equations. Mediterr. J. Math. 18, 168 (2021). https://doi.org/10.1007/s00009-021-01794-6
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DOI: https://doi.org/10.1007/s00009-021-01794-6