Hyers–Ulam Stability for Cayley Quantum Equations and Its Application to h-Difference Equations

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. Mathematics Subject Classification. Primary 39A06; Secondary 39A13, 39A30, 34N05.


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 and λ ∈ C satisfies the condition (1.2) Equation (1.1) is called a Cayley equation, and γ ∈ [0, 1] is called the Cayley parameter, see [11]. Let N be the set of natural numbers, and let N 0 := N∪{0} and q N0 := {1, q, q 2 , q 3 , · · · }. D. R. Anderson For an arbitrary ε > 0, if a function η : q N0 → C satisfies for all s ∈ q N0 , then there exists a solution y : q N0 → C of (1. 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].
where c ∈ C is an arbitrary constant.

2)
c ∈ C is an arbitrary constant, and the function P satisfies |P (s)| ≤ ε for all s ∈ q N0 . Lemma 2.3. Let q > 1, γ ∈ 1 2 , 1 , and let λ ∈ C\{0} satisfy (1.2). Let τ be the function given in (2.2). Then, lim s→∞ |τ (s)| = 0, and the function Proof. First, we will show that lim s→∞ τ (s) = 0. Since holds for γ ∈ 1 2 , 1 and λ = 0, we see that there exists a s 1 ∈ q N0 such that for γ ∈ 1 2 , 1 . Using the above-mentioned estimation, we have for s ≥ s 1 . Let k 1 := log q s 1 . Then, Thus, setting s := q k1+k , we have for s ≥ q k1+1 = qs 1 . Therefore, we obtain lim s→∞ |τ (s)| = 0. Next, we will show that the function is bounded on q N0 . First, we consider the case γ ∈ 1 2 , 1 and λ = 0. From we see that there exists a k 2 ∈ N 0 such that Using the same arguments as in the first part of this proof, we obtain the following: for all m ≥ k 2 and l ∈ N. Set s := q m+l . Then, .
This inequality together with lim s→∞ |τ (s)| = 0 yields This says that β(s) is bounded above on q N0 . Next we consider the case γ = 1 and λ = 0. In this case, τ (s) and β(s) are written in the following form: there exists a k 3 ∈ N 0 such that 1 for k ≥ m 2 . Consequently, we have for m ≥ m 2 with m ∈ N 0 and l ∈ N. Put s := q m+l . Then, Using this inequality, we see that Thus, this together with lim s→∞ |τ (s)| = 0 yields the boundedness of β(s) on q N0 when γ = 1 and λ = 0. This completes the proof.
Proof. Let an arbitrary ε > 0 be given, and λ ∈ C\{0} satisfy (1.2). Assume that D q η(s) − λ η(s) γ ≤ ε for all s ∈ q N0 . Now we consider the functions τ and σ given in ( (q − 1)q m P (q m ) for all s ∈ q N0 . By Lemma 2.3, the right-hand side is bounded above on q N0 . Hence, (1.1) has Hyers-Ulam stability with HUS constant This completes the proof.
By combining Theorem 2.4 with the previous results (already mentioned in the introduction) given in [5], we get the following immediately.
for sufficiently large s ∈ q N0 . Then, (1.1) has HUS on q N0 . Furthermore, for sufficiently large s ∈ q N0 , there exists a δ > 0 such that an HUS constant is Proof. From the assumptions, (1.1) has HUS with HUS constant K on q N0 , where K is given in 2.3. Define η 1 (s) := − 1 λ . Then, η 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 ∈ C is an arbitrary constant, and τ is given in (2.2). Combining these facts, we have for a suitable constant c 0 ∈ C. From Lemma 2.3, lim s→∞ |τ (s)| = 0 holds, and so that lim s→∞ τ (s) This together with the assumption in this theorem yields This means that for sufficiently large s ∈ q N0 , there exists a δ > 0 such that Therefore, 1 |λ| + δ is an HUS constant for sufficiently large s ∈ q N0 .
Example 2.7. We give several examples related to Theorem 2.6. Let γ = 1, q = 2 for the following. If λ = 5, then δ = 1 20 for s = q 2 , as |η( Remark 2.8. Of course, condition (2.4) does not hold in general. In fact, we know that lim s→∞ τ (s) as shown in the proof of Theorem 2.6, while numerical evidence indicates that for any q > 1, any λ satisfying (1.2), and any γ ∈ 1 2 , 1 . It is clear that the two limits are equal for γ = 1.

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.
Let λ ∈ C satisfy (1.2), and α ∈ C satisfy (3.1) has a solution x for t ∈ hN 0 , where and satisfying the following relationships: Proof. Let y be a solution of (1.1) for s ∈ q N0 . From and we find that Thus, x is a solution of (3.2) for t ∈ hN 0 . The reverse is clearly true.
Remark 3.2. If q = 1 + h and γ = 0, then (3.2) is reduced to the h-difference equation We can easily find that (1+h) t h is a solution of Δ h x(t) = x(t) for t ∈ hN 0 , and lim t→0 (1 + h) t h = e t . Hence, we can regard the above h-difference equation as an approximate equation of the differential equation For an arbitrary ε > 0, if a function ξ : hN 0 → C satisfies for all t ∈ hN 0 , then there exists a solution x : hN 0 → C of (3.2) such that Proof. Suppose that α = 0, γ = 1 2 , and condition (3.4) holds on hN 0 . Using the transformation we obtain As a result, This together with the assumption (3.4) says that D q η(s) − λ η(s) γ ≤ ε on q N0 . Since λ = hα q−1 , α = 0 and restriction (3.1) imply λ = 0 and condition (1.2) is met. By Theorem 2.5, we see that there exist a K > 0 and a solution y : q N0 → C of (1.1) such that |η(s) − y(s)| ≤ Kε for all s ∈ q N0 . Let x(t) = h q−1 y q t h . Then, x is a solution to (3.2) by Lemma 3.1. Moreover, the above inequality implies for all t ∈ hN 0 . Therefore, (3.2) has HUS on hN 0 if α = 0 and γ = 1 2 . Conversely, we will show that HUS implies α = 0 and γ = 1 2 . By way of contradiction, we suppose that α = 0 or γ = 1 2 holds. Since (3.2) is HUS on hN 0 , we see that if (3.4) holds for t ∈ hN 0 , then there exist a K 0 > 0 and a solution x : hN 0 → C of (3.2) such that |ξ(t) − x(t)| ≤ K 0 ε for all t ∈ hN 0 . Using transformation (3.5) again, we obtain