1 Introduction
In a series of recent papers [7‐9] the authors introduced the study of Ulam stability for linear quantum (q-difference) equations of first order with a complex constant coefficient. See [1, 2, 19] for the literature on related topics. As yet, there are no works in the literature dealing with first-order linear quantum equations with nonautonomous (variable) coefficient functions, which we initiate below.
As one of the stability types of functional equations, Ulam (or Hyers–Ulam) stability has been investigated by many researchers. Since the paper is devoted to a highly active domain with a plethora of interesting and applicable results, we must pay attention to more classical and recent results in the fields of functional equations. For example, for some highly important works, we direct the reader to [3, 4, 20, 22, 24, 27‐29, 31‐33, 42, 46]. We also draw attention to the books [5, 21, 23, 25, 30, 43, 44].
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Since Popa [39, 40] began studying the Ulam stability of linear difference equations (linear recurrences) in 2005, many researchers have investigated this problem; for example, see [6, 10, 15, 16, 37, 38, 41, 45]. For higher-order difference equations, see [13, 14], and for nonlinear difference equations, see [26, 34‐36]. As of yet, the results for variable coefficients are very few, even for difference equations. For the latest studies on the Ulam stability related to variable and periodic coefficients, see [11, 17, 18]. For results on the Ulam stability for a first-order linear difference equation with nonconstant coefficients in Banach spaces, with the best Ulam constant, see [12].
Let \(\mathbb{N}\) be the set of natural numbers, and let \(\mathbb{N}_{0}:=\mathbb{N}\cup \{0\}\). Define the quantum set
for \(q>1\). In this paper, we consider the nonautonomous Cayley quantum equation
where \(\alpha (t)\) is a complex-valued time-varying coefficient, the q-difference operator is
and the Cayley component is
If \(\beta =0\), then the Cayley quantum equation reduces to the mere quantum equation
It is well known that \(D_{q} z(t) \to z'(t)\) as \(q\searrow 1\), so we can say that the quantum equation is an approximate equation of the differential equation \(z'(t) = \alpha (t) z(t)\). Notice that equation (1.1) can be rewritten as
This formula shows that the condition
is necessary to keep the recurrence viable. For this reason, we assume this condition throughout this paper.
$$ q^{\mathbb{N}_{0}}:=\bigl\{ 1,q,q^{2},q^{3},\ldots \bigr\} $$
$$ D_{q} z(t) = \alpha (t) \bigl\langle z(t) \bigr\rangle _{\beta },\quad t\in q^{\mathbb{N}_{0}}, $$
(1.1)
$$ D_{q} z(t):=\frac{z(qt)-z(t)}{(q-1)t},\quad q>1, $$
$$ \bigl\langle z(t) \bigr\rangle _{\beta }:=\beta z(qt)+(1-\beta )z(t), \quad 0 \le \beta \le 1. $$
$$ D_{q} z(t) = \alpha (t) z(t),\quad t\in q^{\mathbb{N}_{0}}. $$
$$ \bigl[1-\beta (q-1)t\alpha (t) \bigr]z(qt) = \bigl[1+(1-\beta ) (q-1)t \alpha (t) \bigr]z(t). $$
$$ 1-\beta (q-1)t\alpha (t)\ne 0 \ne 1+(1-\beta ) (q-1)t\alpha (t) \quad \text{for } t\in q^{\mathbb{N}_{0}} $$
(1.2)
Definition 1.1
Equation (1.1) is Ulam stable on \(q^{\mathbb{N}_{0}}\) if there is a constant \(C>0\) with the following property:
We call such C a Ulam constant for (1.1) on \(q^{\mathbb{N}_{0}}\).For any \(\varepsilon >0\) and for any function ζ satisfyingthere is a solution z of (1.1) such that$$ \bigl\vert D_{q}\zeta (t) - \alpha (t) \bigl\langle \zeta (t) \bigr\rangle _{\beta } \bigr\vert \le \varepsilon \quad \text{for } t \in q^{ \mathbb{N}_{0}}, $$(1.3)$$ \bigl\vert \zeta (t)-z(t) \bigr\vert \le C\varepsilon \quad \text{for } t\in q^{ \mathbb{N}_{0}}. $$
The paper will proceed as follows. In the next section, we highlight the q-difference (quantum) exponential function and its properties and provide details on the solution to the related nonhomogeneous equation. In Sect. 3, we establish our main result, the Ulam stability of (1.1). In Sect. 4, we show some conditions under which (1.1) is Ulam unstable.
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2 Exponential function and its properties
In this section, we introduce the exponential function of equation (1.1). Define
for \(t\in q^{\mathbb{N}_{0}}\); note that for a function f, we define
which is the standard definition. We immediately have the following lemma.
$$ e_{\alpha }(t) := \prod_{j=0}^{\log _{q} t-1} \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} $$
(2.1)
$$ \prod_{j=0}^{-1} f(j):= 1, $$
Lemma 2.1
Let \(\alpha (t)\) satisfy (1.2), and let \(e_{\alpha }(t)\) be given by (2.1). Then \(e_{\alpha }(t)\) is the solution of (1.1) with \(e_{\alpha }(1)=1\). Moreover, \(z(t) = z_{0} e_{\alpha }(t)\) is the solution of (1.1) with \(z(1)=z_{0}\), where \(z_{0}\) is an arbitrary complex constant, that is, \(z(t) = z_{0} e_{\alpha }(t)\) is the general solution of (1.1).
Proof
It is clear that \(e_{\alpha }(1) = 1\). Now we will show that \(e_{\alpha }(t)\) solves (1.1). Substituting it into the left side of (1.1) gives
On the other hand, substituting \(e_{\alpha }(t)\) into the right side, we get
Hence \(e_{\alpha }(t)\) solves equation (1.1).
$$\begin{aligned} D_{q} e_{\alpha }(t) =& \frac{1}{(q-1)t} \biggl[ \frac{1+(1-\beta )(q-1)t\alpha (t)}{1-\beta (q-1)t\alpha (t)}-1 \biggr] \prod_{j=0}^{\log _{q} t-1} \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \\ =& \frac{\alpha (t)}{1-\beta (q-1)t\alpha (t)}e_{\alpha }(t). \end{aligned}$$
$$\begin{aligned} \alpha (t) \bigl\langle e_{\alpha }(t) \bigr\rangle _{\beta } =& \alpha (t) \biggl[ \beta \frac{1+(1-\beta )(q-1)t\alpha (t)}{1-\beta (q-1)t\alpha (t)}+(1- \beta ) \biggr] \\ & {}\times \prod_{j=0}^{\log _{q} t-1} \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \\ =& \frac{\alpha (t)}{1-\beta (q-1)t\alpha (t)}e_{\alpha }(t). \end{aligned}$$
By \(e_{\alpha }(1)=1\) we have \(z(1)=z_{0}\). From the linearity of the solutions of linear equations we can conclude that \(z(t) = z_{0} e_{\alpha }(t)\) is also a solution of (1.1). This completes the proof. □
Needless to say, the function \(e_{\alpha }(t)\) as defined above will play the role of the exponential function in q-difference equations.
Define
The following lemma holds according to the method of variation of parameters.
$$ \gamma (t) := \sum_{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})}. $$
(2.2)
Lemma 2.2
Let \(\alpha (t)\) satisfy (1.2), and let \(e_{\alpha }(t)\) and \(\gamma (t)\) be given by (2.1) and (2.2), respectively. Then the solution of the equation
with \(\zeta (1)=z_{0}\in \mathbb{C}\) is given by \(\zeta (t) = (z_{0}+\gamma (t))e_{\alpha }(t)\) for \(t\in q^{\mathbb{N}_{0}}\), that is, \(\zeta (t) = (z_{0}+\gamma (t))e_{\alpha }(t)\) is the general solution of (2.3).
$$ D_{q} \zeta (t) = \alpha (t) \bigl\langle \zeta (t) \bigr\rangle _{ \beta } + f(t) $$
(2.3)
Proof
Let \(\zeta (t):= \eta (t)e_{\alpha }(t)\) for \(t\in q^{\mathbb{N}_{0}}\), where \(\eta (t)\) is an unclear function here. We assume that \(\zeta (t)\) is a solution of (2.3). Noting that
we have
This implies
for \(t\in q^{\mathbb{N}_{0}}\). Hence the solution of this equation is inductively obtained in the following form:
for \(t\in q^{\mathbb{N}_{0}}\).
$$ e_{\alpha }(qt) = \frac{1+(1-\beta )(q-1)t\alpha (t)}{1-\beta (q-1)t\alpha (t)}e_{ \alpha }(t), $$
$$\begin{aligned} f(t) =& D_{q} \zeta (t) - \alpha (t) \bigl\langle \zeta (t) \bigr\rangle _{\beta } = D_{q} \bigl(\eta (t)e_{\alpha }(t)\bigr) - \alpha (t) \bigl\langle \eta (t)e_{\alpha }(t) \bigr\rangle _{\beta } \\ =& \frac{\eta (qt)e_{\alpha }(qt)-\eta (t)e_{\alpha }(t)}{(q-1)t} - \alpha (t) \bigl[\beta \eta (qt)e_{\alpha }(qt) + (1-\beta )\eta (t)e_{ \alpha }(t) \bigr] \\ =& \bigl[1-\beta (q-1)t\alpha (t)\bigr]\frac{\eta (qt)e_{\alpha }(qt)}{(q-1)t} - \bigl[1+(1-\beta ) (q-1)t\alpha (t)\bigr]\frac{\eta (t)e_{\alpha }(t)}{(q-1)t} \\ =& \biggl\{ \bigl[1-\beta (q-1)t\alpha (t)\bigr] \frac{1+(1-\beta )(q-1)t\alpha (t)}{1-\beta (q-1)t\alpha (t)}\eta (qt) \\ &{} - \bigl[1+(1-\beta ) (q-1)t\alpha (t)\bigr]\eta (t) \biggr\} \frac{e_{\alpha }(t)}{(q-1)t} \\ =& \bigl[1+(1-\beta ) (q-1)t\alpha (t)\bigr]e_{\alpha }(t) D_{q}\eta (t). \end{aligned}$$
$$ D_{q}\eta (t) = \frac{f(t)}{[1+(1-\beta )(q-1)t\alpha (t)]e_{\alpha }(t)} $$
$$ \eta (t) = z_{0}+\sum_{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} $$
Conversely, it satisfies the above equation. Indeed, we can check that
for \(t\in q^{\mathbb{N}_{0}}\). If we go back to the above calculation, we can see that \(\zeta (t)= \eta (t)e_{\alpha }(t)\) is the solution of (2.3) with \(\zeta (1)=z_{0}\). Hence we have \(\eta (t) \equiv z_{0}+\gamma (t)\), and this completes the proof. □
$$\begin{aligned} D_{q}\eta (t) =& \frac{\eta (qt)-\eta (t)}{(q-1)t} = \frac{1}{(q-1)t} \frac{(q-1)tf(t)}{[1+(1-\beta )(q-1)t\alpha (t)]e_{\alpha }(t)} \\ =& \frac{f(t)}{[1+(1-\beta )(q-1)t\alpha (t)]e_{\alpha }(t)} \end{aligned}$$
Proposition 2.3
Let \(\alpha (t)\) satisfy (1.2) and
and let \(e_{\alpha }(t)\) and \(\gamma (t)\) be given by (2.1) and (2.2), respectively. If for any \(\varepsilon >0\), the function \(f(t)\) appearing in \(\gamma (t)\) satisfies
then:
$$ \liminf_{t\to \infty } \bigl\vert \alpha (t) \bigr\vert >0, $$
(2.4)
$$ \bigl\vert f(t) \bigr\vert \le \varepsilon\quad \textit{for } t\in q^{\mathbb{N}_{0}}, $$
(2.5)
(i)
if \(\beta \in [0,\frac{1}{2} )\), then \(\lim_{t\to \infty }|e_{\alpha }(t)|=\infty \), \(\lim_{t\to \infty }\gamma (t)\) exists, and
is bounded above on \(q^{\mathbb{N}_{0}}\);
$$ \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=\log _{q} t}^{\infty } \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert $$
(ii)
if \(\beta \in (\frac{1}{2},1 ]\), then \(\lim_{t\to \infty }|e_{\alpha }(t)|=0\), and
is bounded above on \(q^{\mathbb{N}_{0}}\).
$$ \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=0}^{\log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert $$
Proof
By (2.4) we have \(\lim_{t\to \infty }t|\alpha (t)|=\infty \). It follows that
Let
Then we obtain
so there is a constant \(L>0\) such that
for \(j\in q^{\mathbb{N}_{0}}\).
$$\begin{aligned} \lim_{j\to \infty } \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert &= \lim_{j\to \infty } \left \vert \frac{\frac{1}{q^{j} \vert \alpha (q^{j}) \vert }+(1-\beta )(q-1)\frac{\alpha (q^{j})}{ \vert \alpha (q^{j}) \vert }}{\frac{1}{q^{j} \vert \alpha (q^{j}) \vert }-\beta (q-1) \frac{\alpha (q^{j})}{ \vert \alpha (q^{j}) \vert }} \right \vert \\ & = \textstyle\begin{cases} \infty , & \beta =0, \\ \frac{1-\beta }{\beta } \in (1,\infty ) , & \beta \in (0, \frac{1}{2} ), \\ 1 : & \beta =\frac{1}{2}, \\ \frac{1-\beta }{\beta } \in (0,1) , & \beta \in (\frac{1}{2},1 ), \\ 0 ,& \beta =1. \end{cases}\displaystyle \end{aligned}$$
(2.6)
$$ \underline{\alpha }:=\liminf_{t\to \infty } \bigl\vert \alpha (t) \bigr\vert >0. $$
$$\begin{aligned} \limsup_{j\to \infty } \biggl\vert \frac{(q-1)q^{j}}{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})} \biggr\vert =& \limsup_{j\to \infty } \left \vert \frac{q-1}{\frac{1}{q^{j}}+(1-\beta )(q-1) \alpha (q^{j})} \right \vert = \frac{1}{(1-\beta )\underline{\alpha }}, \end{aligned}$$
$$ \biggl\vert \frac{(q-1)q^{j}}{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})} \biggr\vert \le L $$
(2.7)
First, we consider case (i). Let \(\beta \in [0,\frac{1}{2} )\). By (2.6) there exist constants \(\mu _{1}>1\) and \(k\in \mathbb{N}\) such that
for \(j\ge k\), and thus
for \(t\ge q^{k}\), where
This implies that \(\lim_{t\to \infty }|e_{\alpha }(t)|=\infty \).
$$ \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert \ge \mu _{1} >1 $$
(2.8)
$$ \bigl\vert e_{\alpha }(t) \bigr\vert \ge \Biggl[\prod _{j=0}^{k-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert \Biggr] \Biggl[\prod_{j=k}^{\log _{q} t-1} \mu _{1} \Biggr] = \nu _{1}\mu _{1}^{\log _{q} t-k} $$
(2.9)
$$ \nu _{1}:= \prod_{j=0}^{k-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert . $$
Next, we will show that \(\lim_{t\to \infty }\gamma (t)\) exists. Using (2.5), (2.7), and (2.9), we have
Consequently,
exists.
$$\begin{aligned} \lim_{t\to \infty } \bigl\vert \gamma (t) \bigr\vert \le & \lim _{t\to \infty }\sum_{j=0}^{ \log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \le \varepsilon L\lim_{t\to \infty }\sum _{j=0}^{\log _{q} t-1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert } \\ \le & \varepsilon L \Biggl[\sum_{j=0}^{k-1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert }+\frac{1}{\nu _{1}}\lim_{t\to \infty } \sum _{j=k}^{\log _{q} t-1} \biggl(\frac{1}{\mu _{1}} \biggr)^{j-k} \Biggr] \\ =& \varepsilon L \Biggl[\sum_{j=0}^{k-1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert }+\frac{\mu _{1}}{\nu _{1}(\mu _{1}-1)} \Biggr] < \infty . \end{aligned}$$
$$ \lim_{t\to \infty }\gamma (t) = \sum_{j=0}^{\infty } \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} $$
By (2.7) and (2.8) we obtain
for \(t\ge q^{k}\).
$$\begin{aligned}& \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=\log _{q} t}^{\infty } \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \\& \quad \le L \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=\log _{q} t}^{\infty } \frac{1}{ \vert e_{\alpha }(q^{j}) \vert } = L \bigl\vert e_{\alpha }(t) \bigr\vert \biggl( \frac{1}{ \vert e_{\alpha }(t) \vert }+\frac{1}{ \vert e_{\alpha }(qt) \vert }+ \frac{1}{ \vert e_{\alpha }(q^{2}t) \vert }+\cdots \biggr) \\& \quad = L \biggl(1+ \biggl\vert \frac{1-\beta (q-1)t\alpha (t)}{1+(1-\beta )(q-1)t\alpha (t)} \biggr\vert \\& \qquad {} + \biggl\vert \frac{1-\beta (q-1)t\alpha (t)}{1+(1-\beta )(q-1)t\alpha (t)} \biggr\vert \biggl\vert \frac{1-\beta (q-1)qt\alpha (qt)}{1+(1-\beta )(q-1)qt\alpha (qt)} \biggr\vert +\cdots \biggr) \\& \quad = L\sum_{j=0}^{\infty } \Biggl(\prod _{m=0}^{j-1} \biggl\vert \frac{1-\beta (q-1)q^{m}t\alpha (q^{m}t)}{1+(1-\beta )(q-1)q^{m}t\alpha (q^{m}t)} \biggr\vert \Biggr) \\& \quad \le L\sum_{j=0}^{\infty } \Biggl[\prod _{m=0}^{j-1} \biggl( \frac{1}{\mu _{1}} \biggr) \Biggr] = L\sum_{j=0}^{\infty } \biggl( \frac{1}{\mu _{1}} \biggr)^{j} = \frac{L\mu _{1}}{\mu _{1}-1} < \infty \end{aligned}$$
Next, we consider case (ii). Let \(\beta \in (\frac{1}{2},1 ]\). By (2.6) there exist constants \(0<\mu _{2}<1\) and \(l\in \mathbb{N}\) such that
for \(j\ge l\), and thus
for \(0 \le j \le l-1\) and \(t\ge q^{l+1}\), where
This, with \(j=0\), implies that \(\lim_{t\to \infty }|e_{\alpha }(t)|=0\).
$$ \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert \le \mu _{2} < 1 $$
(2.10)
$$\begin{aligned}& \prod_{m=j}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \\& \quad \le \Biggl[\prod_{m=j}^{l-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \Biggr] \Biggl[\prod _{m=l}^{\log _{q} t-1} \mu _{2} \Biggr] \\& \quad \le \nu _{2}\mu _{2}^{\log _{q} t-l} \end{aligned}$$
(2.11)
$$ \nu _{2}:= \max_{0\le j \le l-1} \Biggl\{ \prod _{m=j}^{l-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \Biggr\} . $$
By (2.7) we have
Moreover, using (2.10) and (2.11), we obtain
for \(t\ge q^{l+1}\). This completes the proof. □
$$\begin{aligned}& \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=0}^{\log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \\& \quad \le L \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=0}^{\log _{q} t-1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert } = L \bigl\vert e_{\alpha }(t) \bigr\vert \biggl(\frac{1}{ \vert e_{\alpha }(1) \vert }+ \frac{1}{ \vert e_{\alpha }(q) \vert }+\frac{1}{ \vert e_{\alpha }(q^{2}) \vert }+\cdots + \frac{1}{ \vert e_{\alpha }(q^{-1}t) \vert } \biggr) \\& \quad = L \Biggl(\prod_{j=1}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert + \prod_{j=2}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1) q^{j}\alpha (q^{j})} \biggr\vert \\& \qquad {}+\cdots + \prod_{j=\log _{q} t-2}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{j}\alpha (q^{j})}{1-\beta (q-1)q^{j}\alpha (q^{j})} \biggr\vert \Biggr) \\& \quad = L\sum_{j=1}^{\log _{q} t-2} \Biggl(\prod _{m=j}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \Biggr) \\& \quad = L\sum_{j=1}^{l-1} \Biggl(\prod _{m=j}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \Biggr) \\& \qquad {} + L\sum_{j=l}^{\log _{q} t-2} \Biggl(\prod _{m=j}^{\log _{q} t-1} \biggl\vert \frac{1+(1-\beta )(q-1)q^{m}\alpha (q^{m})}{1-\beta (q-1)q^{m}\alpha (q^{m})} \biggr\vert \Biggr). \end{aligned}$$
$$\begin{aligned}& \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=0}^{\log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \\& \quad \le L\sum_{j=1}^{l-1}\nu _{2} \mu _{2}^{\log _{q} t-l} + L\sum_{j=l}^{ \log _{q} t-2} \Biggl(\prod_{m=j}^{\log _{q} t-1} \mu _{2} \Biggr) \\& \quad = L(l-1)\nu _{2}\mu _{2}^{\log _{q} t-l} + L\sum _{j=l}^{\log _{q} t-2} \mu _{2}^{\log _{q} t-j} \\& \quad = L(l-1)\nu _{2}\mu _{2}^{\log _{q} t-l} + L \frac{\mu _{2} \left(\mu _{2}-\mu _{2}^{\log _{q} t-l} \right)}{1-\mu _{2}} < L\nu _{2}(l-1) + L\frac{\mu _{2}^{2}}{1-\mu _{2}} < \infty \end{aligned}$$
3 Ulam stability
The main Ulam stability result of this paper is as follows.
Theorem 3.1
Let \(\alpha (t)\) satisfy (1.2) and (2.4), and let \(e_{\alpha }(t)\) be given by (2.1). Let \(\varepsilon >0\) be arbitrary. Suppose that \(\zeta (t)\) satisfies (2.3) with (2.5). Then:
(i)
if \(\beta \in [0,\frac{1}{2} )\), then \(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)}\) exists, the function
uniquely fulfills (1.1), and \(|\zeta (t)-z_{1}(t)| \le C_{1}\varepsilon \) for all \(t\in q^{\mathbb{N}_{0}}\), where
$$ z_{1}(t):= \biggl(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} \biggr)e_{\alpha }(t) $$
$$ C_{1}:= \sup_{t\in q^{\mathbb{N}_{0}}} \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j= \log _{q} t}^{\infty } \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert < \infty ; $$
(ii)
if \(\beta \in (\frac{1}{2},1 ]\), then there is a constant \(z_{0}\in \mathbb{C}\) such that
fulfills (1.1), and \(|\zeta (t)-z_{2}(t)| \le C_{2}\varepsilon \) for all \(t\in q^{\mathbb{N}_{0}}\), where
$$ z_{2}(t):= z_{0} e_{\alpha }(t) $$
$$ C_{2}:= \sup_{t\in q^{\mathbb{N}_{0}}} \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=0}^{ \log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert < \infty . $$
Proof
Let \(\varepsilon >0\). We suppose that \(\alpha (t)\) satisfies (1.2) and (2.4), whereas \(\zeta (t)\) satisfies (2.3) with (2.5). By Lemma 2.2 we can write \(\zeta (t)\) in the form
for some \(z_{0}\in \mathbb{C}\), where \(\gamma (t)\) is given by (2.2).
$$ \zeta (t) = \bigl(z_{0}+\gamma (t) \bigr)e_{\alpha }(t) $$
(3.1)
First, we consider case (i), that is, suppose \(\beta \in [0,\frac{1}{2} )\). From Proposition 2.3 we see that
exists. Using this, define the function
for \(t\in q^{\mathbb{N}_{0}}\). Then from Lemma 2.1 we note that \(z_{1}(t)\) is the solution of (1.1) with \(z_{1}(1)= \big(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} \big)\). Hence by Proposition 2.3 and (2.5) we obtain
for \(t\in q^{\mathbb{N}_{0}}\).
$$ \lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} = z_{0}+\lim _{t \to \infty }\gamma (t) $$
$$ z_{1}(t):= \biggl(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} \biggr)e_{\alpha }(t) $$
$$\begin{aligned} \bigl\vert \zeta (t)-z_{1}(t) \bigr\vert =& \Bigl\vert \bigl(z_{0}+\gamma (t)\bigr)e_{\alpha }(t)- \Bigl(z_{0}+ \lim_{t\to \infty }\gamma (t) \Bigr)e_{\alpha }(t) \Bigr\vert \\ =& \Bigl\vert \Bigl(\gamma (t)-\lim_{t\to \infty }\gamma (t) \Bigr)e_{ \alpha }(t) \Bigr\vert \\ =& \Biggl\vert e_{\alpha }(t)\sum_{j=\log _{q} t}^{\infty } \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \Biggr\vert \\ \le & \varepsilon \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=\log _{q} t}^{\infty } \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \\ \le & \varepsilon \sup_{t\in q^{\mathbb{N}_{0}}} \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=\log _{q} t}^{\infty } \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert < \infty \end{aligned}$$
Next, we will show that \(z_{1}(t)\) satisfies (1.1) and \(|\zeta (t)-z_{1}(t)| \le C_{1}\varepsilon \) uniquely. Consider the function
satisfying \(|\zeta (t)-y(t)| \le C_{1}\varepsilon \) for \(t\in q^{\mathbb{N}_{0}}\), where \(y_{0} \neq \big(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} \big)\). From Lemma 2.1 it follows that \(y(t)\) satisfies (1.1). Hence we obtain
for \(t\in q^{\mathbb{N}_{0}}\). By Proposition 2.3 we know that \(\lim_{t\to \infty }|e_{\alpha }(t)|=\infty \), and so the above inequality derives a contradiction.
$$ y(t):=y_{0}e_{\alpha }(t) $$
$$ \biggl\vert \biggl(\lim_{t\to \infty }\frac{\zeta (t)}{e_{\alpha }(t)} \biggr)-y_{0} \biggr\vert \bigl\vert e_{\alpha }(t) \bigr\vert = \bigl\vert z_{1}(t)-y(t) \bigr\vert \le \bigl\vert \zeta (t)-z_{1}(t) \bigr\vert + \bigl\vert \zeta (t)-y(t) \bigr\vert \le 2 C_{1}\varepsilon $$
Next, we consider case (ii), that is, suppose \(\beta \in (\frac{1}{2},1 ]\). Let
Then by Lemma 2.1\(z_{2}(t)\) is a solution of (1.1). Using Proposition 2.3, we see that
for \(t\in q^{\mathbb{N}_{0}}\). Consequently, the statement in this theorem is true. This completes the proof. □
$$ z_{2}(t):= z_{0} e_{\alpha }(t). $$
$$\begin{aligned} \bigl\vert \zeta (t)-z_{2}(t) \bigr\vert =& \bigl\vert \bigl(z_{0}+\gamma (t)\bigr)e_{\alpha }(t)-z_{0}e_{ \alpha }(t) \bigr\vert = \bigl\vert \gamma (t)e_{\alpha }(t) \bigr\vert \\ =& \Biggl\vert e_{\alpha }(t)\sum_{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}f(q^{j})}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \Biggr\vert \\ \le & \varepsilon \bigl\vert e_{\alpha }(t) \bigr\vert \sum _{j=0}^{\log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert \\ \le & \varepsilon \sup_{t\in q^{\mathbb{N}_{0}}} \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=0}^{\log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert < \infty \end{aligned}$$
Remark 3.2
The results in Theorem 3.1 include and extend the results given in [8, Theorem 2.6] and [9, Theorem 2.4], which deal with the Ulam stability when the coefficient is a complex constant. Let
for \(j\in \mathbb{N}_{0}\). According to the results in [8, Theorem 2.8] and [9, Theorem 2.6], the following facts hold under the assumption that \(\alpha (t)\) satisfies (1.2) and \(\alpha (t)\equiv \alpha \neq 0\):
$$ g(j):= \frac{(q-1)q^{j}}{[1+(1-\beta )(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} $$
(i)
if \(\beta \in [0,\frac{1}{2} )\) and \(\sum_{j=\log _{q} t}^{\infty } \vert g(j) \vert = \big \vert \sum_{j= \log _{q} t}^{\infty } g(j) \big \vert \), then (1.1) is Ulam stable with the best Ulam constant \(C_{1} = \frac{1}{|\alpha |}\).
(ii)
if \(\beta \in (\frac{1}{2},1 ]\) and \(\sum_{j=0}^{\log _{q} t-1} \vert g(j) \vert = \Big \vert \sum_{j=0}^{ \log _{q} t-1} g(j) \Big \vert \) for sufficiently large \(t\in q^{\mathbb{N}_{0}}\), then (1.1) is Ulam stable, and there is \(\delta >0\) such that \(C_{2} = \frac{1}{|\alpha |}+\delta \) is an Ulam constant for sufficiently large \(t\in q^{\mathbb{N}_{0}}\).
A natural follow-up question is what happens if \(\beta =\frac{1}{2}\)? We give partial answers in the following theorem and in the next section on instability.
Theorem 3.3
Proof
In addition to the hypotheses in the statement of this theorem, let \(e_{\alpha }(t)\) be given by (2.1). Note that (3.2) implies (2.4). With \(\beta =\frac{1}{2}\), the exponential function takes the form
and
since α satisfies conditions (1.2) and (2.4). It follows that \(e_{\alpha }(t)\) converges to a two-cycle \(\pm \xi ^{*}\) for some \(\xi ^{*}\in \mathbb{C}\backslash \{0\}\) as \(t\rightarrow \infty \). Let \(\varepsilon >0\) be arbitrary. For \(\gamma (t)\) given by (2.2) with (2.5),
Using the ratio test, we have
by assumption, so that \(\gamma (t)\) converges absolutely. Suppose that \(\zeta (t)\) satisfies (2.3) with (2.5), and suppose that \(z(t)\) satisfies (1.1) with \(z(1)=z_{0}\). Then
where
This completes the proof. □
$$ e_{\alpha }(t) = \prod_{j=0}^{\log _{q} t-1} \frac{1+\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}{1-\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}, \quad t\in q^{\mathbb{N}_{0}}, $$
$$ \lim_{t\rightarrow \infty } \frac{1+\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}{1-\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}=-1, $$
$$ \bigl\vert \gamma (t) \bigr\vert \le \varepsilon \sum _{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}}{ \left \vert 1+\frac{1}{2}(q-1)q^{j}\alpha (q^{j}) \right \vert \cdot \vert e_{\alpha }(q^{j}) \vert }. $$
$$\begin{aligned} \lim_{j\rightarrow \infty } \frac{ \vert q-\frac{1}{2}(q-1)q^{j+1} \alpha (q^{j}) \vert }{ \vert 1+\frac{1}{2}(q-1)q^{j+1}\alpha (q^{j+1}) \vert } =& \lim_{j\rightarrow \infty } \left \vert \frac{\frac{q}{\frac{1}{2}(q-1)q^{j+1} \vert \alpha (q^{j}) \vert }-\frac{\alpha (q^{j})}{ \vert \alpha (q^{j}) \vert }}{\frac{1}{\frac{1}{2}(q-1)q^{j+1} \vert \alpha (q^{j+1}) \vert } +\frac{\alpha (q^{j+1})}{ \vert \alpha (q^{j+1}) \vert }} \right \vert \biggl\vert \frac{\alpha (q^{j})}{\alpha (q^{j+1})} \biggr\vert \\ =& \lim_{j\rightarrow \infty } \biggl\vert \frac{\alpha (q^{j})}{\alpha (q^{j+1})} \biggr\vert < 1 \end{aligned}$$
$$ \bigl\vert \zeta (t)-z(t) \bigr\vert = \bigl\vert \bigl(z_{0}+ \gamma (t)\bigr)e_{\alpha }(t)-z_{0}e_{\alpha }(t) \bigr\vert = \bigl\vert \gamma (t)e_{\alpha }(t) \bigr\vert \le C_{3}\varepsilon , $$
$$ C_{3}:=\sup_{t\in q^{\mathbb{N}_{0}}} \bigl\vert e_{\alpha }(t) \bigr\vert \sum_{j=0}^{ \log _{q} t-1} \biggl\vert \frac{(q-1)q^{j}}{[1+\frac{1}{2}(q-1)q^{j}\alpha (q^{j})]e_{\alpha }(q^{j})} \biggr\vert < \infty . $$
(3.3)
Remark 3.4
The theorems in this section can be summarized as follows.
4 Ulam instability
What happens if the coefficient function α fails to satisfy (2.4)? In the following example, we show an example where (1.1) is unstable in the Ulam sense.
Example 4.1
Let \(q>1\) be fixed, take \(\rho \in (-\infty ,-1)\cup (1,\infty )\), and let
We easily see that this α satisfies (1.2) for any \(\beta \in [0,1]\) but fails to satisfy (2.4). Plugging this α into (2.1), we have the exponential function
$$ \alpha (t)=\frac{1}{(q-1)\rho t}, \quad t\in q^{\mathbb{N}_{0}}. $$
$$ e_{\alpha }(t) = \prod_{j=0}^{\log _{q} t-1} \frac{1+\frac{1}{\rho }(1-\beta )}{1-\frac{1}{\rho }\beta } = \biggl( \frac{1+\rho -\beta }{\rho -\beta } \biggr)^{\log _{q} t}. $$
(4.1)
For arbitrary \(\varepsilon >0\), let \(f(t)\equiv \varepsilon \). Substituting these α, \(e_{\alpha }\), and f into (2.2), we have
It follows that \(\zeta (t)=\gamma (t)e_{\alpha }(t)\) is a solution of (2.3). Let \(z(t)=z_{0}e_{\alpha }(t)\) be any solution of (1.1). There are three cases.
$$ \gamma (t)=\sum_{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}f(q^{j})}{ [1+\frac{1}{\rho }(1-\beta ) ]e_{\alpha }(q^{j})} =\frac{(q-1)\rho \varepsilon }{1+\rho -\beta } \sum_{j=0}^{\log _{q} t-1} \biggl(\frac{(\rho -\beta )q}{1+\rho -\beta } \biggr)^{j} \ge 0. $$
(4.2)
Case i. Let \(\beta \in [0,1]\) and \(\rho \in (-\infty ,-1)\). Clearly, \(e_{\alpha }(t)\) in (4.1) goes to zero as t goes to infinity in \(q^{\mathbb{N}_{0}}\). Moreover, \(\gamma (t)\) in (4.2) and \(\zeta (t)=\gamma (t)e_{\alpha }(t)\) diverge to positive infinity as t goes to positive infinity, so that
for any \(z_{0}\in \mathbb{C}\), and (1.1) is not Ulam stable.
$$ \bigl\vert \zeta (t)-z(t) \bigr\vert = \bigl\vert \gamma (t)e_{\alpha }(t)-z_{0}e_{\alpha }(t) \bigr\vert \rightarrow \infty $$
Case ii. Let \(\beta \in [0,1]\) and \(\rho \in (1,\infty )\). Clearly, \(e_{\alpha }(t)\) in (4.1) goes to infinity as t goes to infinity in \(q^{\mathbb{N}_{0}}\). Suppose \(q\ge \frac{1+\rho -\beta }{\rho -\beta }\). Then \(\gamma (t)\) in (4.2) diverges to positive infinity as t goes to positive infinity, so that
for any \(z_{0}\in \mathbb{C}\), and (1.1) is not Ulam stable.
$$ \bigl\vert \zeta (t)-z(t) \bigr\vert = \bigl\vert \gamma (t)-z_{0} \bigr\vert e_{\alpha }(t)\rightarrow \infty $$
Case iii. Let \(\beta \in [0,1]\) and \(\rho \in (1,\infty )\). As in (ii), \(e_{\alpha }(t)\) in (4.1) goes to infinity as t goes to infinity. Suppose \(1< q<\frac{1+\rho -\beta }{\rho -\beta }\). Then γ in (4.2) is a convergent geometric series. Using this fact, rewrite (4.2) as
Consequently,
If \(z_{0}=0\), then
and (1.1) is not Ulam stable. If \(z_{0}=\frac{(q-1)\rho \varepsilon }{(1+\rho -\beta )-q(\rho -\beta )}\), then
and again (1.1) is not Ulam stable. Any other choice of \(z_{0}\in \mathbb{C}\) leads to a similar conclusion. Therefore (1.1) is Ulam unstable in all cases for this example.
$$\begin{aligned} \gamma (t) =& \frac{(q-1)\rho \varepsilon }{1+\rho -\beta } \sum_{j=0}^{ \infty } \biggl(\frac{(\rho -\beta )q}{1+\rho -\beta } \biggr)^{j} - \frac{(q-1)\rho \varepsilon }{1+\rho -\beta } \sum _{j=\log _{q} t}^{ \infty } \biggl(\frac{(\rho -\beta )q}{1+\rho -\beta } \biggr)^{j} \\ =& \frac{(q-1)\rho \varepsilon }{(1+\rho -\beta )-q(\rho -\beta )} - \frac{(q-1)\rho \varepsilon t \biggl(\frac{\rho -\beta }{1+\rho -\beta } \biggr)^{\log _{q} t}}{(1+\rho -\beta )-q(\rho -\beta )}. \end{aligned}$$
$$\begin{aligned} \bigl\vert \zeta (t)-z(t) \bigr\vert =& \bigl\vert \gamma (t)-z_{0} \bigr\vert e_{\alpha }(t) \\ =& \left \vert \frac{(q-1)\rho \varepsilon }{(1+\rho -\beta )-q(\rho -\beta )} - \frac{(q-1)\rho \varepsilon t \biggl(\frac{\rho -\beta }{1+\rho -\beta } \biggr)^{\log _{q} t}}{(1+\rho -\beta )-q(\rho -\beta )}-z_{0} \right \vert \\ & {}\times \biggl(\frac{1+\rho -\beta }{\rho -\beta } \biggr)^{\log _{q} t}. \end{aligned}$$
$$\begin{aligned}& \lim_{t\rightarrow \infty } \bigl\vert \zeta (t)-z(t) \bigr\vert \\& \quad = \lim_{t\rightarrow \infty } \left \vert \frac{(q-1)\rho \varepsilon }{(1+\rho -\beta )-q(\rho -\beta )} - \frac{(q-1)\rho \varepsilon t \big(\frac{\rho -\beta }{1+\rho -\beta } \big)^{\log _{q} t}}{(1+\rho -\beta )-q(\rho -\beta )} \right \vert \biggl(\frac{1+\rho -\beta }{\rho -\beta } \biggr)^{\log _{q} t} \\& \quad = \frac{(q-1)\rho \varepsilon }{(1+\rho -\beta )-q(\rho -\beta )} \lim_{t\rightarrow \infty } \biggl(\frac{1+\rho -\beta }{\rho -\beta } \biggr)^{\log _{q} t}=\infty , \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow \infty } \bigl\vert \zeta (t)-z(t) \bigr\vert =& \lim _{t\rightarrow \infty } \left \vert - \frac{(q-1)\rho \varepsilon t \big(\frac{\rho -\beta }{1+\rho -\beta } \big)^{\log _{q} t}}{(1+\rho -\beta )-q(\rho -\beta )} \right \vert \biggl( \frac{1+\rho -\beta }{\rho -\beta } \biggr)^{\log _{q} t} \\ =& \lim_{t\rightarrow \infty } \frac{\rho \varepsilon (q-1)t}{(1+\rho -\beta )-q(\rho -\beta )}= \infty , \end{aligned}$$
Theorem 4.2
Let \(\alpha (t)\) satisfy (1.2) and
and let \(e_{\alpha }(t)\) be given by (2.1). If
then (1.1) is unstable in the Ulam sense.
$$ \limsup_{t\to \infty } \bigl\vert \alpha (t) \bigr\vert < \infty , $$
(4.3)
$$ 0 < \liminf_{t\to \infty } \bigl\vert e_{\alpha }(t) \bigr\vert \quad \textit{and}\quad \limsup_{t\to \infty } \bigl\vert e_{\alpha }(t) \bigr\vert < \infty , $$
(4.4)
Proof
For arbitrary \(\varepsilon >0\), let
Substituting this f into (2.2), we have
From Lemma 2.2 it follows that \(\zeta (t) = \gamma (t)e_{\alpha }(t)\) is a solution of (2.3). Let \(z(t)=z_{0}e_{\alpha }(t)\) be any solution of (1.1).
$$ f(t)\equiv \frac{\varepsilon [1+(1-\beta )(q-1)q^{j}\alpha (q^{j}) ]e_{\alpha }(t)}{ \vert [1+\beta (q-1)q^{j}\alpha (q^{j}) ]e_{\alpha }(t) \vert }. $$
$$ \gamma (t) = \varepsilon \sum_{j=0}^{\log _{q} t-1} \frac{(q-1)q^{j}}{ \vert 1+\beta (q-1)q^{j}\alpha (q^{j}) \vert \vert e_{\alpha }(q^{j}) \vert }. $$
(4.5)
From conditions (4.3) and (4.4) we see that there exists a constant \(\mu _{1}>0\) such that
for all \(j\in \mathbb{N}_{0}\), and there exist \(\mu _{2}\), \(\mu _{3}>0\) and \(\nu \in \mathbb{N}_{0}\) such that
for all \(t\ge q^{\nu }\). This, together with (4.5), yields
for \(t\ge q^{\nu }\). Hence we have \(\lim_{t\to \infty }\gamma (t)=\infty \), so
for any \(z_{0}\in \mathbb{C}\), and (1.1) is not Ulam stable. □
$$ \frac{(q-1)q^{j}}{ \vert 1+\beta (q-1)q^{j}\alpha (q^{j}) \vert } = \frac{1}{ \Big\vert \frac{1}{(q-1)q^{j}}+\beta \alpha (q^{j}) \Big\vert } \ge \mu _{1} $$
$$ \mu _{2} \le \bigl\vert e_{\alpha }(t) \bigr\vert \le \mu _{3} $$
$$\begin{aligned} \gamma (t) =& \varepsilon \sum_{j=0}^{\log _{q} t-1} \frac{1}{ \left \vert \frac{1}{(q-1)q^{j}}+\beta \alpha (q^{j}) \right \vert \vert e_{\alpha }(q^{j}) \vert } \ge \varepsilon \mu _{1}\sum _{j=0}^{\log _{q} t-1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert } \\ \ge & \varepsilon \mu _{1} \Biggl(\sum_{j=0}^{\nu -1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert }+\frac{1}{\mu _{3}}\sum_{j=\nu }^{\log _{q} t-1} 1 \Biggr) \\ =& \varepsilon \mu _{1} \Biggl(\sum_{j=0}^{\nu -1} \frac{1}{ \vert e_{\alpha }(q^{j}) \vert }+\frac{\log _{q} t-\nu }{\mu _{3}} \Biggr) \end{aligned}$$
$$ \bigl\vert \zeta (t)-z(t) \bigr\vert = \bigl\vert \gamma (t)-z_{0} \bigr\vert \bigl\vert e_{\alpha }(t) \bigr\vert \ge \mu _{2} \bigl\vert \gamma (t)-z_{0} \bigr\vert \rightarrow \infty $$
Corollary 4.3
Proof
If \(\beta =\frac{1}{2}\), then \(e_{\alpha }(t)\) is given by
Conditions (1.2) and (2.4) imply that \(e_{\alpha }(t)\) converges to a two-cycle \(\pm \xi ^{*}\) for some \(\xi ^{*}\in \mathbb{C}\backslash \{0\}\) as \(t\rightarrow \infty \). Hence (4.4) holds. Then, by Theorem 4.2, (1.1) is not Ulam stable. □
$$ e_{\alpha }(t) = \prod_{j=0}^{\log _{q} t-1} \frac{1+\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}{1-\frac{1}{2}(q-1)q^{j}\alpha (q^{j})}. $$
Remark 4.4
Theorem 4.2 implies the instability result for the constant coefficient case given in [8, Theorem 3.1]. Indeed, consider the case \(\alpha (t)\equiv \alpha \) with (1.2). If \(\alpha \ne 0\), then Corollary 4.3 immediately shows instability. If \(\alpha = 0\), then \(e_{\alpha }(t) \equiv 1\), and (4.4) holds. Hence, by Theorem 4.2, (1.1) is not Ulam stable.
5 Conclusions
Using the properties of the exponential function for nonautonomous Cayley quantum equations, we established sufficient conditions for the Ulam stability of quantum equations with a variable coefficient under the assumptions that the Cayley parameter satisfies \(\beta \neq \frac{1}{2}\) and the absolute value of the variable coefficient does not approach zero. After that, these assumptions are elaborated. The situation is clarified by presenting an example where Ulam stability breaks down if the absolute value of the variable coefficient approaches zero. If the coefficient is a constant, it has already been shown in [8] that \(\beta = \frac{1}{2}\) means the Ulam instability, but with the variable coefficient, something interesting happens, that is, if the absolute value of the variable coefficient increases, the Ulam stability is derived. Therefore it turns out that both Ulam stable and unstable cases may occur for \(\beta =\frac{1}{2}\). In this way, we found in this study that by considering variable coefficients there is a problem of balance between stability and instability, which does not appear in the case of constant coefficients.
Acknowledgements
Both authors would like to thank the referees for their helpful suggestions.
Competing interests
The authors declare that they have no competing interests.
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