# Hyers–Ulam Stability for Quantum Equations of Euler Type

Authors: Douglas R. Anderson; Masakazu Onitsuka
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Many applications using discrete dynamics employ eitherq-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found.


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## Body

## 1. Introduction

Recently, there has been much interest in questions of Hyers–Ulam stability for differential equations andh-difference equations, but little has been published specifically on q-difference (quantum) equations [1], in particular, on quantum equations of Euler type. In this work, we introduce a new and direct connection between Hyers–Ulam stability results for h-difference equations with constant coefficients, of first, second, and all higher orders, with Hyers–Ulam stability results for quantum equations of Euler type, of all integer orders, through a change of variables. First, we will connect the two types of equations and then introduce Hyers–Ulam stability.The results in this paper connectingh-difference equations and q-difference equations of Euler type are novel. Even if we just consider the higher-order h-difference results independently, they extend first-order and second-order results found in [2, 3] to nth order and are not the same as the results in [4–6], where h=1 and different techniques are used. For a great introduction to quantum calculus, see the monograph [1], which has sections on both q-calculus and h-calculus, but does not show the nexus that we do here.The rest of the paper will develop as follows. In Section2, we establish the connection between q-difference equations of the Euler type and h-difference equations, via a change of variable. We then define Hyers–Ulam stability (HUS) and prove for which the parameter values the first-order q-difference equation of the Euler type has HUS; in the case that it does exhibit HUS, a minimum HUS constant is found. In Section 3, the Hyers–Ulam stability of second-order quantum equations of the Euler type is established from known results for h-difference equations. In Section 4, the stability of both higher-order quantum equations of the Euler type and higher-order h-difference equations with constant coefficients is proven by mathematical induction; these results are new in each context. For some cases, the best HUS constant is found. In Section 5, higher-order perturbed quantum equations of the Euler type and higher-order perturbed h-difference equations with constant (complex) coefficients are analyzed, and HUS with specific HUS constants is established for each setting.


## 2. Connections and First-Order Stability

Lemma 1.
Assumeq>1 and h>0 and set(1)qℤ=…,q−2,q−1,1,q,q2,…,hℤ=…,−2h,−h,0,h,2h,….Letα∈ℂ\−1/q−1 and λ∈ℂ\−1/h be given, and let I be the identity operator. Then, the (factored) quantum equation of the Euler type(2)sDq−αIys=gs,Dqys≔yqs−ysq−1s,has a solution y for s∈qℤ if and only if the (factored) h-difference equation(3)Δh−λIxt=ft,Δhxt≔xt+h−xth,has a solution x for t∈hℤ, where(4)t=hlogqsandλ=αq−1h,equivalentlys=qt/handα=λhq−1,is a change of variables between s and α to t and λ, while(5)xt=hq−1yqt/h&ft=gqt/h,i.e.ys=q−1hxhlogqs&gs=fhlogqs,is a change of functions between the variables.Proof.
Lety be a solution of sDq−αIys=gs for α∈ℂ\−1/q−1. Then, the change of variables (4) converts this equation to(6)hq−1yqt+h/h−yqt/hh−hλq−1yqt/h=gqt/h,where λ∈ℂ\−1/h. Now, make the change of functions (5). Then, the function x is a solution of Δh−λIxt=ft. Using (4) and (5), this process is reversible, yielding the converse.Definition 1.
Assumeq>1 and α∈ℂ\−1/q−1. The Euler-type quantum equation(7)sDq−αIys=0,has Hyers–Ulam stability (HUS) if and only if there exists a constant K>0 with the following property. For an arbitrary ε>0, if a function ψ:qℤ⟶ℂ satisfies(8)sDq−αIψs≤ε,for all s∈qℤ, then there exists a solution y:qℤ⟶ℂ of (7) such that(9)ψs−ys≤Kε,for all s∈qℤ. Such a constant K is called an HUS constant for (7) on qℤ.Remark 1.
If, given an arbitraryε>0, there exists a function ψ such that (8) holds for s∈qℤ, then(10)Δh−λIϕt≤ε,t∈hℤ,holds as well, where we have used the change of variable (4) and, similar to (5), the change of function(11)ϕt=hq−1ψqt/h,to rewrite (8) as (10).Lemma 2.
Assumeq>1 and α∈ℂ\−1/q−1. If αq−1+1=1, then the quantum equation of the Euler type given in (7) is not Hyers–Ulam stable.Proof.
Assumeh>0 and λ∈ℂ\−1/h. Using ([2], Remark 2.1), if λh+1=1, then the h-difference equation,(12)Δh−λIxt=0,is not Hyers–Ulam stable. By the change of variables (4) and change of functions (5) and (11), which connect (7) to (12) and (8) to (10), the result follows.Remark 2.
Throughout the rest of the paper, let(13)Kτ,θ≔τ1−θτ+1,for θτ+1≠1.Theorem 1.
Ifα∈ℂ\−1/q−1 with αq−1+1≠1, then (7) has Hyers–Ulam stability with minimum HUS constant Kq−1,α=q−1/1−αq−1+1 on qℤ.Proof.
Using ([2], Theorem 2.6), if λ∈ℂ\−1/h with λh+1≠1, then (12) has Hyers–Ulam stability with minimum HUS constant Kh,λ=h/1−λh+1 on hℤ, for K⋅,⋅ given in (13). Let an arbitrary ε>0 be given. Suppose ϕ satisfies (10). By ([2], Theorem 2.5) there exists a solution x of (12) such that(14)ϕt−xt≤εKh,λ,t∈hℤ,making the change using (4) and (11), we have(15)hq−1ψs−hq−1ys≤hε1−αq−1+1,s∈qℤ,so that(16)ψs−ys≤q−1ε1−αq−1+1=εKq−1,α,s∈qℤ,where y is a solution of (7). Now from ([2], Lemma 2.3), we know there exists a specific function:(17)ϕt≔cλh+1t/h+εhγ−λh+1γt/h,for γ≔λh+1/λh+1 that satisfies (10); then, γ=1 and γ−λh+1≠0, whereby it is proven that the minimum HUS constant for (12) is at least Kh,λ, for K⋅,⋅ given in (13). By the change of variable (4) and the change of function (5), which connect (7) to (12) and (8) to (10) via (11), there exists a specific function ψ satisfying (8), whereby the minimum HUS constant for (7) is thus Kq−1,α, and the result follows.


## 3. Second-Order Quantum Equations of Euler Type

Letq>1, h>0, and αj∈ℂ\−1/q−1 be given for j∈1,2.Now, consider the second-order quantum equation of the Euler type, written in the factored operator form as(18)sDq−α2IsDq−α1Iys=0,Dqys≔yqs−ysq−1s.Definition 2.
The second-order Euler-type quantum equation (18) has Hyers–Ulam stability (HUS) if and only if there exists a constant K>0 with the following property: For an arbitrary ε>0, if a function ψ:qℤ⟶ℂ satisfies(19)sDq−α2IsDq−α1Iψs≤ε,for all s∈qℤ, then there exists a solution y:qℤ⟶ℂ of (18) such that(20)ψs−ys≤Kε,for all s∈qℤ. Such a constant K is called an HUS constant for (18) on qℤ.Theorem 2.
Ifαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2, then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with an HUS constant of(21)Kq−1,α1Kq−1,α2=q−121−α1q−1+1⋅1−α2q−1+1,on qℤ.Proof.
Let(22)Y1s≔sDq−α1Iys.
As a result, (18) implies that sDq−α2IY1s=0 so that(23)Δh−λ2IX1t=0,ΔhX1t≔X1t+h−X1th,where we employ the change of variables(24)λj=αjq−1h,t=hlogqs,X1t≔hq−1Y1qt/h.
Then, we have(25)X1t=sDq−α1Ihq−1ys.
Let(26)xt≔hq−12yqt/h.
Note that asαj≠−1/q−1, we have λj≠−1/h, for j=1,2. Moreover, αjq−1+1≠1 for j∈1,2 implies that λjh+1≠1 for j∈1,2. Consequently,(27)X1t=Δh−λ1Ixt,and this implies(28)0=Δh−λ2IX1t=Δh−λ2IΔh−λ1Ixt=Δh2xt−λ1+λ2Δhxt+λ1λ2xt.
Take(29)α=−λ1+λ2,β=λ1λ2,to match the notation used in [2], Theorem 3.4.
By [2], Theorem 3.4, the second-order h-difference equation (28) has HUS, with an HUS constant(30)Kh,−α+α2−4β2Kh,−α−α2−4β2=Kh,λ1Kh,λ2,on hℤ, where K⋅,⋅ is the constant expressed in (13). Given an arbitrary ε>0, suppose there exists a function ψ such that(31)sDq−α2IsDq−α1Iψs≤ε,s∈qℤ.
Then, letting(32)ϕt≔hq−12ψqt/h,t=hlogqs,λj=αjq−1h,for j∈1,2, we have(33)Δh−λ2IΔh−λ1Iϕt≤ε,t∈hℤ.
Therefore, by [2], Theorem 3.4, there exists a solution x of (28) such that(34)ϕt−xt≤εKh,−α+α2−4β2Kh,−α−α2−4β2=Kh,λ1Kh,λ2,which implies that(35)hq−12ψqt/h−hq−12yqt/h≤εh1−λ1h+1h1−λ2h+1,ψqt/h−yqt/h≤εq−11−α1q−1+1q−11−α2q−1+1,using (26). It follows that(36)q−11−α1q−1+1q−11−α2q−1+1=Kq−1,α1Kq−1,α2,is an HUS constant for (18), for K⋅,⋅ given in (13).Corollary 1.
Assumeαj∈ℂ\−1/q−1 with αjq−1+1|≠1 for j∈1,2.(i)
Ifαjq−1+1∈0,1∪1,∞ for j∈1,2, then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with minimum HUS constant(37)K=1α1α2,onqℤ.(ii)
Ifαjq−1+1∈−∞,−1∪−1,0 for j∈1,2, then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with minimum HUS constant(38)K=q−12α1q−1+2α2q−1+2,onqℤ.Proof.
Assumeαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2.(i)
Ifαjq−1+1∈0,1∪1,∞ for j∈1,2, then(39)Kq−1,α1Kq−1,α2=1α1α2,through simplification so that this constant is an HUS constant for (18) on qℤ. Invoking [2], Theorem 3.4 (i) or [3], Corollary 3.1 and the change of variables to the corresponding h-difference equation, the constant(40)Kh,λ1Kh,λ2=1λ1λ2,is the minimum HUS constant for the second-orderh-difference equation (28) on hℤ. The result follows on qℤ after a change of variables back.(ii)
Ifαjq−1+1∈−∞,−1∪−1,0 for j∈1,2, then(41)Kq−1,α1Kq−1,α2=q−12α1q−1+2α2q−1+2,again through simplification of the expression, making this constant an HUS constant for (18) on qℤ. Referring to [2], Theorem 3.4 (iii) and proceeding as in case (i) of this proof, and the result follows for (18) on qℤ.


## 4. Higher-Order Quantum Equations of Euler Type

In this section, we extend the results in the previous two sections to higher-order quantum equations of the Euler type.Letq>1, h>0, αj∈ℂ\−1/q−1, and λj∈ℂ\−1/h be given for j∈1,2,…,n. In this section, we consider the nth-order quantum equation of the Euler type given in factored operator form by(42)sDq−αnIsDq−αn−1I⋯sDq−α1Iys=0,Dqys≔yqs−ysq−1s.Definition 3.
The higher-order Euler-type quantum equation (42) has Hyers–Ulam stability (HUS) if and only if there exists a constant K>0 with the following property. For an arbitrary ε>0, if a function ψ:qℤ⟶ℂ satisfies(43)sDq−αnIsDq−αn−1I⋯sDq−α1Iψs≤ε,for all s∈qℤ, then there exists a solution y:qℤ⟶ℂ of (42) such that(44)ψs−ys≤Kε,for all s∈qℤ. Such a constant K is called an HUS constant for (42) on qℤ.Theorem 3.
Ifαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2,…,n, then the higher-order quantum equation of Euler type (42) has Hyers–Ulam stability with an HUS constant of(45)∏j=1nKq−1,αj=q−1n∏j=1n1−αjq−1+1,on qℤ, where K⋅,⋅ is given in (13).Proof.
We proceed by mathematical induction onn∈ℕ. For n=1, equation (42) is simply (7) so that by Theorem 1, (7) has Hyers–Ulam stability with minimum HUS constant(46)Kq−1,α1=q−11−α1q−1+1,on qℤ.
Letn=2. For an arbitrary ε>0, suppose there exists a function ψ:qℤ⟶ℂ that satisfies(47)sDq−α2IsDq−α1Iψs≤ε,for all s∈qℤ. If we let(48)sDq−α1Iψs=Ψ2s,then(49)sDq−α2IΨ2s≤ε,for all s∈qℤ. Therefore, Hyers–Ulam stability for the first-order equation implies there exists a solution y2 of sDq−α2Iy2s=0 such that(50)Ψ2s−y2s≤εKq−1,α2.
LetY2 solve the equation:(51)sDq−α1IY2s=y2s.
This is possible by converting the equation using Lemma1 to the corresponding h-difference equation and using the variation of parameters formula and then converting back. In (50), substitute for Ψ2 using (48) and for y2 using (51). Then, we can rewrite (50) as(52)sDq−α1Iψs−sDq−α1IY2s≤εKq−1,α2,so that(53)sDq−α1Iψ−Y2s≤εKq−1,α2=ε′.
Again, Hyers–Ulam stability for the first-order equation implies there exists a solutiony0 of (7) such that(54)ψ−Y2s−y0s≤ε′Kq−1,α1,which implies that(55)ψs−Y2+y0s≤εKq−1,α1Kq−1,α2.
Note that(56)sDq−α2IsDq−α1IY2+y0s=sDq−α2IsDq−α1IY2s+sDq−α1Iy0s=sDq−α2Iy2s+0=sDq−α2Iy2s=0,making Y2+y0 a solution of (42) with n=2. By Definition 3, with n=2, equation (42) has Hyers–Ulam stability with HUS constant Kq−1,α1Kq−1,α2.
Letn=k for some k∈ℕ. Without loss of generality, write (42) with n=k as(57)sDq−αk+1IsDq−αkI⋯sDq−α2Iys=0,where we have reindexed parameters as necessary and make the induction assumption that this equation has Hyers–Ulam stability with HUS constant(58)∏j=2k+1Kq−1,αj=∏j=2k+1q−11−αjq−1+1,on qℤ.
Now, consider (42) with n=k+1, namely,(59)sDq−αk+1IsDq−αkI⋯sDq−α1Iys=0.
For an arbitraryε>0, suppose there exists a function ψ:qℤ⟶ℂ that satisfies(60)sDq−αk+1IsDq−αkI⋯sDq−α1Iψs≤ε,for all s∈qℤ. If we let(61)sDq−α1Iψs=Ψk+1s,then(62)sDq−αk+1IsDq−αkI⋯sDq−α2IΨk+1s≤ε,for all s∈qℤ. Therefore, by the induction assumption for n=k, Hyers–Ulam stability for that equation implies there exists a solution yk+1 of sDq−αk+1IsDq−αkI⋯sDq−α2Iyk+1s=0 such that(63)Ψk+1s−yk+1s≤ε∏j=2k+1Kq−1,αj.
LetYk+1 solve the equation(64)sDq−α1IYk+1s=yk+1s.
Using this result, we can rewrite (63) as(65)sDq−α1Iψs−sDq−α1IYk+1s≤ε∏j=2k+1Kq−1,αj,so that(66)sDq−α1Iψ−Yk+1s≤ε∏j=2k+1Kq−1,αj=ε′.
Again, Hyers–Ulam stability for the first-order equation implies there exists a solutiony0 of (7) such that(67)ψ−Yk+1s−y0s≤ε′Kq−1,α1,which implies that(68)ψs−Yk+1+y0s≤ε∏j=1k+1Kq−1,αj.
Note that(69)sDq−αk+1I⋯sDq−α2IsDq−α1IYk+1+y0s=sDq−αk+1I⋯sDq−α2IsDq−α1IYk+1s+sDq−α1Iy0s=sDq−αk+1I⋯sDq−α2Iyk+1s+0=sDq−αk+1I⋯sDq−α2Iyk+1s=0,by the choice of yk+1, making Yk+1+y0 a solution of (42) with n=k+1. By Definition 3 with n=k+1, equation (42) has Hyers–Ulam stability with HUS constant ∏j=1k+1Kq−1,αj. Consequently, by the principle of mathematical induction, the overall result holds.
We now use Theorem3 and the connection between q-difference equations of Euler type and h-difference equations with constant coefficients articulated earlier, to extend known results about (28) to general higher-order equations.Corollary 2.
Leth>0 be given. If λj∈ℂ\−1/h with λjh+1≠1 for j∈1,2,…,n, then the higher-order h-difference equation with constant coefficients in factored form given by(70)Δh−λnIΔh−λn−1I⋯Δh−λ1Ixt=0,has Hyers–Ulam stability with an HUS constant of(71)∏j=1nKh,λj=hn∏j=1n1−λjh+1,on hℤ, where K⋅,⋅ is given in (13).Lemma 3.
Assumeαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2,…,n. Then the following hold:(i)
Ifα1, α2, ⋯, αn are distinct, then the general solution to (42) is given by(72)ys=∑j=1najαjq−1+1logqs,whereaj are arbitrary constants.(ii)
Ifα1, α2, ⋯, αr are distinct with multiplicities m1, m2, ⋯, mr with ∑j=1rmj=n, respectively, then the general solution to(73)sDq−αrImrsDq−αr−1Imr−1⋯sDq−α1Im1ys=0,is given by(74)ys=∑j=1rαjq−1+1logqsbj0+bj1logqs+bj2logqs2+⋯+bjmj−1logqsmj−1,wherebjii∈0,1,2,…,mj−1 are arbitrary constants.Proof.
Letq>1 and h>0, and αi∈ℂ\−1/q−1 with αiq−1+1≠1 for i∈1,2,…,n. We now consider the change of variables and functions:(75)t=hlogqs,λj=αjq−1h,xt=hq−1yqt/h=hq−1ys,for j∈1,2,…,n. Then, the quantum equation of Euler type (42) has a solution y for s∈qℤ if and only if the h-difference equation (70) has a solution x for t∈hℤ by using the same argument as in the proof of Lemma 1.
First, we will prove case (i). Suppose thatαjj∈1,2,…,n are distinct. Since(76)Δhλjh+1t/h=1hλjh+1t+h/h−λjh+1t/h=λjλjh+1t/h,holds, we have(77)Δh−λjIxjt=0,where xjt=λjh+1t/h. That is, xjtj∈1,2,…,n are solutions to (70). Since αjj∈1,2,…,n are distinct, λjj∈1,2,…,n are also distinct. Consequently, the general solution to (70) is given by(78)xt=∑j=1ncjxjt=∑j=1ncjλjh+1t/h,where cj are arbitrary constants. Using connect (75), we can find the general solution(79)ys=∑j=1nq−1cjhαjq−1+1logqs=∑j=1najαjq−1+1logqsof (42).
Next, we prove case (ii). Suppose thatαjj∈1,2,…,r are distinct with multiplicities mjj∈1,2,…,r with ∑j=1rmj=n, respectively. From connect (75), we see that λjj∈1,2,…,r are also distinct with multiplicities mjj∈1,2,…,r with ∑j=1rmj=n, respectively. By [7], Corollary 2.24, we can know that(80)xt=∑j=1rλjh+1t/hdj0+dj1t+dj2t2+⋯+djmj−1tmj−1,is the general solution to the h-difference equation:(81)Δh−λrImrΔh−λn−1Imr−1⋯Δh−λ1Im1xt=0.
Using connect (75), we can find the general solution(82)ys=∑j=1rq−1hαjq−1+1logqsdj0+dj1hlogqs+dj2hlogqs2+⋯+djmj−1hlogqsmj−1=∑j=1rαjq−1+1logqsbj0+bj1logqs+bj2logqs2+⋯+bjmj−1logqsmj−1,of (73).
Corollary1 is extended to the following result.Corollary 3.
Assumeαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2,…,n.(i)
Ifαjq−1+1∈0,1∪1,∞ for j∈1,2,…,n, then the nth-order quantum equation of Euler type (42) has Hyers–Ulam stability with minimum HUS constant(83)K=1∏j=1nαj,onqℤ.(ii)
Ifαjq−1+1∈−∞,−1∪−1,0 for j∈1,2,…,n, then the nth-order quantum equation of Euler type (42) has Hyers–Ulam stability with minimum HUS constant(84)K=q−1n∏j=1nαjq−1+2,onqℤ.Proof.
Assumeαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2,…,n.(i)
Ifαjq−1+1∈0,1∪1,∞ for j∈1,2,…,n, then (42) has Hyers–Ulam stability with an HUS constant(85)∏j=1nKq−1,αj=1∏j=1nαj,onqℤ from Theorem 3.We will prove that any HUS constant for (42) is greater than or equal to this constant. Let ε>0 be given. If we suppose that 0<K1<1/∏j=1nαj, and for any function ψ satisfying(86)sDq−αnIsDq−αn−1I⋯sDq−α1Iψs≤ε,there exists a solutiony of (42) such that ψs−ys≤K1ε for all s∈qℤ. Define(87)γ1s≔ε∏j=1nαj.Then,γ1 satisfies(88)sDq−αnIsDq−αn−1I⋯sDq−α1Iγ1s=−αn−αn−1⋯−α1γ1s.This implies that(89)sDq−αnIsDq−αn−1I⋯sDq−α1Iγ1s=ε,for alls∈qℤ. Thus, by the assumption, we can find a solution y1 of (42) such that γ1s−y1s≤K1ε for all s∈qℤ. By Lemma 3, we can rewrite y1 as(90)y1s=∑j=1najαjq−1+1logqsory1s=∑j=1rαjq−1+1logqsbj0+bj1logqs+bj2logqs2+⋯+bjmj−1logqsmj−1,whereaj and bjii∈0,1,2,…,mj−1 are arbitrary constants so that we have(91)γ1s−∑j=1najαjq−1+1logqs=γ1s−y1s≤K1εorγ1s−∑j=1rαjq−1+1logqsbj0+bj1logqs+bj2logqs2+⋯+bjmj−1logqsmj−1≤K1ε.Since(92)lims⟶∞αjq−1+1logqs=∞,ifαjq−1+1∈1,∞orlims⟶+0αjq−1+1logqs=∞,ifαjq−1+1∈0,1,holds for allj∈1,2,…,n, we conclude that aj=0 for all j∈1,2,…,n and bji=0 for all j∈1,2,…,r and i∈0,1,2,…,mj−1. That is, y1s≡0. Hence, we see that(93)K1ε≥γ1s−y1s=γ1s=ε∏j=1nαj>K1ε.This is a contradiction. Thus,1/∏j=1nαj is the minimum HUS constant.(ii)
Ifαjq−1+1∈−∞,−1∪−1,0 for j∈1,2,…,n, then (42) has Hyers–Ulam stability with an HUS constant(94)∏j=1nKq−1,αj=q−1n∏j=1nαjq−1+2,onqℤ from Theorem 3.We will prove that any HUS constant for (42) is greater than or equal to this constant. Let ε>0 be given. If we suppose that 0<K2<q−1n/∏j=1nαjq−1+2, and for any function ψ satisfying(95)sDq−αnIsDq−αn−1I⋯sDq−α1Iψs≤ε,there exists a solutiony of (42) such that ψs−ys≤K2ε for all s∈qℤ. Define(96)γ2s≔q−1nε∏j=1nαjq−1+2−1logqs.Since(97)sDq−αjIγ2s=sq−1nε∏j=1nαjq−1+2−1logqqs−−1logqs/q−1s−αjγ2s=−2/q−1−αjγ2s,holds, we have(98)sDq−αnIsDq−αn−1I⋯sDq−α1Iγ1s=−2q−1−αn−2q−1−αn−1⋯−2q−1−α1γ2s.This implies that(99)sDq−αnIsDq−αn−1I⋯sDq−α1Iγ2s=ε,for alls∈qℤ. Thus, by the assumption, we can find a solution y2 of (42) such that γ2s−y2s≤K2ε for all s∈qℤ. By Lemma 3, we can rewrite y2 as(100)y2s=∑j=1ncjαjq−1+1logqsory2s=∑j=1rαjq−1+1logqsdj0+dj1logqs+dj2logqs2+⋯+djmj−1logqsmj−1,wherecj and djii∈0,1,2,…,mj−1 are arbitrary constants so that we have(101)γ2s−∑j=1ncjαjq−1+1logqs=γ2s−y2s≤K2εorγ2s−∑j=1rαjq−1+1logqsdj0+dj1logqs+dj2logqs2+⋯+djmj−1logqsmj−1≤K2ε.Since(102)lims⟶∞αjq−1+1logqs=∞,ifαjq−1+1∈−∞,−1orlims⟶+0αjq−1+1logqs=∞,ifαjq−1+1∈−1,0,holds for allj∈1,2,…,n, we conclude that cj=0 for all j∈1,2,…,n and dji=0 for all j∈1,2,…,r and i∈0,1,2,…,mj−1. That is, y2s≡0. Hence, we see that(103)K2ε≥γ2s−y2s=γ2s=q−1nε∏j=1nαjq−1+2>K2ε.This is a contradiction. Thus,q−1n/∏j=1nαjq−1+2 is the minimum HUS constant.
Using Corollary3 and connect (75), we can obtain the following result.Corollary 4.
Assumeλj∈ℂ\−1/h with λjh+1≠1 for j∈1,2,…,n.(i)
Ifλjh+1∈0,1∪1,∞ for j∈1,2,…,n, then the nth-order h-difference equation with constant coefficients (70) has Hyers–Ulam stability with minimum HUS constant(104)K=1∏j=1nλj,onqℤ.(ii)
Ifλjh+1∈−∞,−1∪−1,0 for j∈1,2,…,n, then the nth-order h-difference equation with constant coefficients (70) has Hyers–Ulam stability with minimum HUS constant(105)K=hn∏j=1nλjh+2,onqℤ.


## 5. Perturbed Quantum Equations of Euler Type

In this section, we consider thenth-order perturbed quantum equation of the Euler type:(106)sDq−αnIsDq−αn−1I⋯sDq−α1Iys=gs,Dqys≔yqs−ysq−1s,where g:qℤ⟶ℂ is a perturbation. By using Theorem 3, we obtain the following result.Theorem 4.
Ifαj∈ℂ\−1/q−1 with αjq−1+1≠1 for j∈1,2,…,n, then the higher-order perturbed quantum equation of Euler type (106) has Hyers–Ulam stability with an HUS constant of(107)∏j=1nKq−1,αj=q−1n∏j=1n1−αjq−1+1,on qℤ, where K⋅,⋅ is given in (13).Proof.
Letw:qℤ⟶ℂ be a solution of (106), and let ε>0 be given. Suppose that ψ:qℤ⟶ℂ satisfies(108)sDq−αnIsDq−αn−1I⋯sDq−α1Iψs−gs≤ε,s∈qℤ.
Sincew is a solution of (106), we have(109)sDq−αnIsDq−αn−1I⋯sDq−α1Iψs−ws≤ε.
Using Theorem3, we can find a solution z:qℤ⟶ℂ of (42) such that(110)ψs−ws−zs≤q−1nε∏j=1n1−αjq−1+1,s∈qℤ.
Note here thatws+zs is a solution of (106) because(111)sDq−αnIsDq−αn−1I⋯sDq−α1Iws+zs=gs+0=gs,holds. This says that (106) has Hyers–Ulam stability.
Using Theorem4 and connect (75) with ft=gqt/h, we can establish the following result.Theorem 5.
Ifλj∈ℂ\−1/h with λjh+1≠1 for j∈1,2,…,n, then the higher-order perturbed h-difference equation with constant coefficients(112)Δh−λnIΔh−λn−1I⋯Δh−λ1Ixt=ft,has Hyers–Ulam stability with an HUS constant of(113)∏j=1nKh,λj=hn∏j=1n1−λjh+1,on hℤ, where K⋅,⋅ is given in (13).


## 6. Conclusion

New results connectingh-difference equations with complex constant coefficients and q-difference equations of the Euler type are presented, for equations of all integer orders. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found. The key idea introduced here is establishing results for one type of equation and then converting those results over to the other through a change of the variable.



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