Hyers–Ulam Stability for Quantum Equations of Euler Type

Many applications using discrete dynamics employ either q-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.


Introduction
Recently, there has been much interest in questions of Hyers-Ulam stability for differential equations and h-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. e results in this paper connecting h-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][5][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. e rest of the paper will develop as follows. In Section 2, 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.
Let α ∈ C\ − 1/(q − 1) and λ ∈ C\ − 1/h { } be given, and let I be the identity operator. en, the (factored) quantum equation of the Euler type sD q − αI y(s) � g(s), has a solution y for s ∈ q Z if and only if the (factored) h-difference equation has a solution x for t ∈ hZ, where is a change of variables between s and α to t and λ, while is a change of functions between the variables.
Proof. Let y be a solution of (sD q − αI)y(s) � g(s) for α ∈ C\ − 1/(q − 1) . en, the change of variables (4) converts this equation to where λ ∈ C\ − 1/h { }. Now, make the change of functions (5). en, the function x is a solution of (Δ h − λI)x(t) � f(t). Using (4) and (5), this process is reversible, yielding the converse. □ Definition 1. Assume q > 1 and α ∈ C\ − 1/(q − 1) . e Euler-type quantum equation 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: q Z ⟶ C satisfies for all s ∈ q Z , then there exists a solution y: q Z ⟶ C of (7) such that for all s ∈ q Z . Such a constant K is called an HUS constant for (7) on q Z . Remark 1. If, given an arbitrary ε > 0, there exists a function ψ such that (8) holds for s ∈ q Z , then holds as well, where we have used the change of variable (4) and, similar to (5), the change of function to rewrite (8) as (10).

Second-Order Quantum Equations of Euler Type
Let q > 1, h > 0, and α j ∈ 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 Definition 2. e 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: q Z ⟶ C satisfies for all s ∈ q Z , then there exists a solution y: q Z ⟶ C of (18) such that for all s ∈ q Z . Such a constant K is called an HUS constant for (18) on q Z .

then the second-order quantum equation of Euler type (18) has Hyers-Ulam stability with an HUS constant of
As a result, (18) implies that (sD q − α 2 I)Y 1 (s) � 0 so that where we employ the change of variables en, we have Note that as and this implies Take to match the notation used in [2], eorem 3.4. By [2], eorem 3.4, the second-order h-difference equation (28) has HUS, with an HUS constant on hZ, where K(·, ·) is the constant expressed in (13). Given an arbitrary ε > 0, suppose there exists a function ψ such that en, letting Discrete Dynamics in Nature and Society erefore, by [2], eorem 3.4, there exists a solution x of (28) such that which implies that using (26). It follows that is an HUS constant for (18), for K(·, ·) given in (13). □

then the second-order quantum equation of Euler type (18) has Hyers-Ulam stability with minimum HUS constant
through simplification so that this constant is an HUS constant for (18) on q Z . Invoking [2], eorem 3.4 (i) or [3], Corollary 3.1 and the change of variables to the corresponding h-difference equation, the constant is the minimum HUS constant for the second-order h-difference equation (28) on hZ. e result follows on q Z after a change of variables back.
again through simplification of the expression, making this constant an HUS constant for (18) on q Z . Referring to [2], eorem 3.4 (iii) and proceeding as in case (i) of this proof, and the result follows for (18) on q Z .

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 q > 1, h > 0, α j ∈ C\ − 1/(q − 1) , and λ j ∈ C\ − 1/h { } be given for j ∈ 1, 2, . . . , n { }. In this section, we consider the 4 Discrete Dynamics in Nature and Society nth-order quantum equation of the Euler type given in factored operator form by sD q − α n I sD q − α n− 1 I · · · sD q − α 1 I y(s) � 0, Definition 3. e 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: q Z ⟶ C satisfies for all s ∈ q Z , then there exists a solution y: q Z ⟶ C of (42) such that for all s ∈ q Z . Such a constant K is called an HUS constant for (42) on q Z .

then the higher-order quantum equation of Euler type (42) has Hyers-Ulam stability with an HUS constant of
on q Z , where K(·, ·) is given in (13).
Proof. We proceed by mathematical induction on n ∈ N. For n � 1, equation (42) is simply (7) so that by eorem 1, (7) has Hyers-Ulam stability with minimum HUS constant on q Z . Let n � 2. For an arbitrary ε > 0, suppose there exists a function ψ: q Z ⟶ C that satisfies for all s ∈ q Z . If we let then for all s ∈ q Z . erefore, Hyers-Ulam stability for the firstorder equation implies there exists a solution y 2 of (sD q − α 2 I)y 2 (s) � 0 such that Let Y 2 solve the equation: is is possible by converting the equation using Lemma 1 to the corresponding h-difference equation and using the variation of parameters formula and then converting back. In (50), substitute for Ψ 2 using (48) and for y 2 using (51). en, we can rewrite (50) as so that Again, Hyers-Ulam stability for the first-order equation implies there exists a solution y 0 of (7) such that which implies that Note that making (Y 2 + y 0 ) a solution of (42) with n � 2. By Definition 3, with n � 2, equation (42) has Hyers-Ulam stability with HUS constant K(q − 1, α 1 )K(q − 1, α 2 ). Let n � k for some k ∈ N. Without loss of generality, write (42) with n � k as sD q − α k+1 I sD q − α k I · · · sD q − α 2 I y(s) � 0, (57) where we have reindexed parameters as necessary and make the induction assumption that this equation has Hyers-Ulam stability with HUS constant on q Z . Now, consider (42) with n � k + 1, namely, For an arbitrary ε > 0, suppose there exists a function ψ: q Z ⟶ C that satisfies sD q − α k+1 I sD q − α k I · · · sD q − α 1 I ψ(s) ≤ ε, (60) for all s ∈ q Z . If we let Discrete Dynamics in Nature and Society then sD q − α k+1 I sD q − α k I · · · sD q − α 2 I Ψ k+1 (s) ≤ ε, for all s ∈ q Z . erefore, by the induction assumption for n � k, Hyers-Ulam stability for that equation implies there exists a solution y k+1 of (sD q − α k+1 I)(sD q − α k I) · · · (sD q − α 2 I)y k+1 (s) � 0 such that Let Y k+1 solve the equation Using this result, we can rewrite (63) as so that Again, Hyers-Ulam stability for the first-order equation implies there exists a solution y 0 of (7) such that which implies that Note that sD q − α k+1 I · · · sD q − α 2 I sD q − α 1 I Y k+1 + y 0 (s) � sD q − α k+1 I · · · sD q − α 2 I sD q − α 1 I Y k+1 (s) by the choice of y k+1 , making (Y k+1 + y 0 ) a solution of (42) with n � k + 1. By Definition 3 with n � k + 1, equation (42) has Hyers-Ulam stability with HUS constant k+1 j�1 K(q − 1, α j ). Consequently, by the principle of mathematical induction, the overall result holds. We now use eorem 3 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.

then the higher-order h-difference equation with constant coefficients in factored form given by
has Hyers-Ulam stability with an HUS constant of on hZ, where K(·, ·) is given in (13).
Proof. Let q > 1 and h > 0, and We now consider the change of variables and functions: for j ∈ 1, 2, . . . , n { }. en, the quantum equation of Euler type (42) has a solution y for s ∈ q Z 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.
on q Z .