Abstract

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

1. Introduction

Recently, there has been much interest in questions of Hyers–Ulam stability for differential equations and -difference equations, but little has been published specifically on -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 -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 connecting -difference equations and -difference equations of Euler type are novel. Even if we just consider the higher-order -difference results independently, they extend first-order and second-order results found in [2, 3] to th order and are not the same as the results in [4–6], where and different techniques are used. For a great introduction to quantum calculus, see the monograph [1], which has sections on both -calculus and -calculus, but does not show the nexus that we do here.

The rest of the paper will develop as follows. In Section 2, we establish the connection between -difference equations of the Euler type and -difference equations, via a change of variable. We then define Hyers–Ulam stability (HUS) and prove for which the parameter values the first-order -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 -difference equations. In Section 4, the stability of both higher-order quantum equations of the Euler type and higher-order -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 -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. Assume and and set

Let and be given, and let be the identity operator. Then, the (factored) quantum equation of the Euler typehas a solution for if and only if the (factored) -difference equationhas a solution for , whereis a change of variables between and to and , whileis a change of functions between the variables.

Proof. Let be a solution of for . Then, the change of variables (4) converts this equation towhere . Now, make the change of functions (5). Then, the function is a solution of . Using (4) and (5), this process is reversible, yielding the converse.

Definition 1. Assume and . The Euler-type quantum equationhas Hyers–Ulam stability (HUS) if and only if there exists a constant with the following property. For an arbitrary , if a function satisfiesfor all , then there exists a solution of (7) such thatfor all . Such a constant is called an HUS constant for (7) on .

Remark 1. If, given an arbitrary , there exists a function such that (8) holds for , thenholds as well, where we have used the change of variable (4) and, similar to (5), the change of functionto rewrite (8) as (10).

Lemma 2. Assume and . If , then the quantum equation of the Euler type given in (7) is not Hyers–Ulam stable.

Proof. Assume and . Using ([2], Remark 2.1), if , then the -difference equation,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, letfor .

Theorem 1. If with , then (7) has Hyers–Ulam stability with minimum HUS constant on .

Proof. Using ([2], Theorem 2.6), if with , then (12) has Hyers–Ulam stability with minimum HUS constant on , for given in (13). Let an arbitrary be given. Suppose satisfies (10). By ([2], Theorem 2.5) there exists a solution of (12) such thatmaking the change using (4) and (11), we haveso thatwhere is a solution of (7). Now from ([2], Lemma 2.3), we know there exists a specific function:for that satisfies (10); then, and , whereby it is proven that the minimum HUS constant for (12) is at least , for 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 satisfying (8), whereby the minimum HUS constant for (7) is thus , and the result follows.

3. Second-Order Quantum Equations of Euler Type

Let , , and be given for .

Now, consider the second-order quantum equation of the Euler type, written in the factored operator form as

Definition 2. The second-order Euler-type quantum equation (18) has Hyers–Ulam stability (HUS) if and only if there exists a constant with the following property: For an arbitrary , if a function satisfiesfor all , then there exists a solution of (18) such thatfor all . Such a constant is called an HUS constant for (18) on .

Theorem 2. If with for , then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with an HUS constant ofon .

Proof. LetAs a result, (18) implies that so thatwhere we employ the change of variablesThen, we haveLetNote that as , we have , for . Moreover, for implies that for . Consequently,and this impliesTaketo match the notation used in [2], Theorem 3.4.
By [2], Theorem 3.4, the second-order -difference equation (28) has HUS, with an HUS constanton , where is the constant expressed in (13). Given an arbitrary , suppose there exists a function such thatThen, lettingfor , we haveTherefore, by [2], Theorem 3.4, there exists a solution of (28) such thatwhich implies thatusing (26). It follows thatis an HUS constant for (18), for given in (13).

Corollary 1. Assume with for .(i)If for , then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with minimum HUS constant on .(ii)If for , then the second-order quantum equation of Euler type (18) has Hyers–Ulam stability with minimum HUS constant on .

Proof. Assume with for .(i) If for , then through simplification so that this constant is an HUS constant for (18) on . Invoking [2], Theorem 3.4 (i) or [3], Corollary 3.1 and the change of variables to the corresponding -difference equation, the constant is the minimum HUS constant for the second-order -difference equation (28) on . The result follows on after a change of variables back.(ii)If for , then again through simplification of the expression, making this constant an HUS constant for (18) on . Referring to [2], Theorem 3.4 (iii) and proceeding as in case (i) of this proof, and the result follows for (18) on .

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.

Let , , , and be given for . In this section, we consider the th-order quantum equation of the Euler type given in factored operator form by

Definition 3. The higher-order Euler-type quantum equation (42) has Hyers–Ulam stability (HUS) if and only if there exists a constant with the following property. For an arbitrary , if a function satisfiesfor all , then there exists a solution of (42) such thatfor all . Such a constant is called an HUS constant for (42) on .

Theorem 3. If with for , then the higher-order quantum equation of Euler type (42) has Hyers–Ulam stability with an HUS constant ofon , where is given in (13).

Proof. We proceed by mathematical induction on . For , equation (42) is simply (7) so that by Theorem 1, (7) has Hyers–Ulam stability with minimum HUS constanton .
Let . For an arbitrary , suppose there exists a function that satisfiesfor all . If we letthenfor all . Therefore, Hyers–Ulam stability for the first-order equation implies there exists a solution of such thatLet solve the equation:This is possible by converting the equation using Lemma 1 to the corresponding -difference equation and using the variation of parameters formula and then converting back. In (50), substitute for using (48) and for using (51). Then, we can rewrite (50) asso thatAgain, Hyers–Ulam stability for the first-order equation implies there exists a solution of (7) such thatwhich implies thatNote thatmaking a solution of (42) with . By Definition 3, with , equation (42) has Hyers–Ulam stability with HUS constant .
Let for some . Without loss of generality, write (42) with aswhere we have reindexed parameters as necessary and make the induction assumption that this equation has Hyers–Ulam stability with HUS constanton .
Now, consider (42) with , namely,For an arbitrary , suppose there exists a function that satisfiesfor all . If we letthenfor all . Therefore, by the induction assumption for , Hyers–Ulam stability for that equation implies there exists a solution of such thatLet solve the equationUsing this result, we can rewrite (63) asso thatAgain, Hyers–Ulam stability for the first-order equation implies there exists a solution of (7) such thatwhich implies thatNote thatby the choice of , making a solution of (42) with . By Definition 3 with , equation (42) has Hyers–Ulam stability with HUS constant . Consequently, by the principle of mathematical induction, the overall result holds.
We now use Theorem 3 and the connection between -difference equations of Euler type and -difference equations with constant coefficients articulated earlier, to extend known results about (28) to general higher-order equations.

Corollary 2. Let be given. If with for , then the higher-order -difference equation with constant coefficients in factored form given byhas Hyers–Ulam stability with an HUS constant ofon , where is given in (13).

Lemma 3. Assume with for . Then the following hold:(i)If , , , are distinct, then the general solution to (42) is given by where are arbitrary constants.(ii)If , , , are distinct with multiplicities , , , with , respectively, then the general solution to is given by where are arbitrary constants.

Proof. Let and , and with for . We now consider the change of variables and functions:for . Then, the quantum equation of Euler type (42) has a solution for if and only if the -difference equation (70) has a solution for by using the same argument as in the proof of Lemma 1.
First, we will prove case (i). Suppose that are distinct. Sinceholds, we havewhere . That is, are solutions to (70). Since are distinct, are also distinct. Consequently, the general solution to (70) is given bywhere are arbitrary constants. Using connect (75), we can find the general solutionof (42).
Next, we prove case (ii). Suppose that are distinct with multiplicities with , respectively. From connect (75), we see that are also distinct with multiplicities with , respectively. By [7], Corollary 2.24, we can know thatis the general solution to the -difference equation:Using connect (75), we can find the general solutionof (73).
Corollary 1 is extended to the following result.

Corollary 3. Assume with for .(i)If for , then the th-order quantum equation of Euler type (42) has Hyers–Ulam stability with minimum HUS constant on .(ii)If for , then the th-order quantum equation of Euler type (42) has Hyers–Ulam stability with minimum HUS constant on .

Proof. Assume with for .(i)If for , then (42) has Hyers–Ulam stability with an HUS constant on from Theorem 3. We will prove that any HUS constant for (42) is greater than or equal to this constant. Let be given. If we suppose that , and for any function satisfying there exists a solution of (42) such that for all . Define Then, satisfies This implies that for all . Thus, by the assumption, we can find a solution of (42) such that for all . By Lemma 3, we can rewrite as where and are arbitrary constants so that we have Since holds for all , we conclude that for all and for all and . That is, . Hence, we see that This is a contradiction. Thus, is the minimum HUS constant.(ii)If for , then (42) has Hyers–Ulam stability with an HUS constant on from Theorem 3. We will prove that any HUS constant for (42) is greater than or equal to this constant. Let be given. If we suppose that , and for any function satisfying there exists a solution of (42) such that for all . Define Since holds, we have This implies that for all . Thus, by the assumption, we can find a solution of (42) such that for all . By Lemma 3, we can rewrite as where and are arbitrary constants so that we have Since holds for all , we conclude that for all and for all and . That is, . Hence, we see that This is a contradiction. Thus, is the minimum HUS constant.Using Corollary 3 and connect (75), we can obtain the following result.

Corollary 4. Assume with for .(i)If for , then the th-order -difference equation with constant coefficients (70) has Hyers–Ulam stability with minimum HUS constant on .(ii)If for , then the th-order -difference equation with constant coefficients (70) has Hyers–Ulam stability with minimum HUS constant on .

5. Perturbed Quantum Equations of Euler Type

In this section, we consider the th-order perturbed quantum equation of the Euler type:where is a perturbation. By using Theorem 3, we obtain the following result.

Theorem 4. If with for , then the higher-order perturbed quantum equation of Euler type (106) has Hyers–Ulam stability with an HUS constant ofon , where is given in (13).

Proof. Let be a solution of (106), and let be given. Suppose that satisfiesSince is a solution of (106), we haveUsing Theorem 3, we can find a solution of (42) such thatNote here that is a solution of (106) becauseholds. This says that (106) has Hyers–Ulam stability.
Using Theorem 4 and connect (75) with , we can establish the following result.

Theorem 5. If with for , then the higher-order perturbed -difference equation with constant coefficientshas Hyers–Ulam stability with an HUS constant ofon , where is given in (13).

6. Conclusion

New results connecting -difference equations with complex constant coefficients and -difference equations of the Euler type are presented, for equations of all integer orders. For equation orders greater than two, the -difference results extend first-order and second-order results found in the literature, and the Euler-type -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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Professor Onitsuka was supported by JSPS KAKENHI Grant no. JP20K03668. This work was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.