Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 77, pp. 1–15. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.77 ULAM TYPE STABILITY FOR NONLINEAR HAHN DIFFERENCE EQUATIONS WITH DELAY KAI CHEN, JINRONG WANG Abstract. In this article, we study the Ulam type stability of nonlinear Hahn difference equations with delay over a finite interval. First, we use the Banach fixed point theorem to prove the existence and uniqueness of a solution. Then we establish the Ulam stability for first and second order nonlinear Hahn differ- ence equations with delay. We also extend our analysis to n-th order nonlinear Hahn difference equations with delay. To illustrate our theoretical findings, we provide three examples. 1. Introduction Hahn [10] developed a difference operator, by drawing from two well-known dif- ference operators: the forward difference operator [4] and the Jackson q-difference operator [3, 5, 6, 27]. Subsequently, Annaby et al. [2] extended the concept by in- troducing the q, ω-integral a function, which encompasses both Nörlund sums and Jackson q-integrals. Hamaz et al. [11, 16] explored the existence and uniqueness of solutions to Hahn difference equations using the method of successive approxi- mations and examined the stability of first-order Hahn difference equations. Ab- delkhaliq et al. [1] investigated the stability of Hahn difference equations within Banach spaces. Additional results on the Hahn difference operator can be found in references [12, 14, 15, 17, 18, 22, 24]. Ulam stability originated from a query about stability addressed in [29], and was later termed Ulam-Hyers stability by Hyers [19]. Rassias [25] further developed this concept into Ulam-Hyers-Rassias stability by incorporating additional variables in the form of functions. Following this, numerous studies have explored the Ulam stability of various equations [8, 9, 20, 21, 26]. For instance, Rus [28] examined Ulam stability in ordinary differential equations, Otrocol et al. [23] looked into the Ulam stability of delay differential equations, and Hamaz et al. [13] studied the Ulam stability of first-order linear quantum difference equations. Inspired by [15, 28, 23], we consider the equation Dq,ωx(s) = F (t, x(s), x(Θ(s))), s ∈ I1 = [ω0, b], x(s) = y(s), s ∈ I2 = [ω0 − h0, ω0], (1.1) 2020 Mathematics Subject Classification. 39A05, 39A30. Key words and phrases. Nonlinear Hahn difference equation; delay; Ulam-Hyers stability; Ulam-Hyers-Rassias stability. ©2024. This work is licensed under a CC BY 4.0 license. Submitted September 9, 2024. Published November 30, 2024. 1 2 K. CHEN, J. WANG EJDE-2024/77 where F : I1 × R2 → R and Θ : I1 → I3, I3 = I1 ∪ I2, are continuous at s = ω0, Θ(s) ≤ s, h0 > 0 and y : I2 → R is the initial value condition. We demonstrate both the existence and uniqueness of the solution to equation (1.1) on I3 using the Banach fixed point theorem. Additionally, we explore the Ulam stability of equation (1.1) on I3. Unlike [11], where the method of successive approximations was used and the function f needed to be continuous on the plane I1 × R, our approach requires f to be continuous specifically at s = ω0. Secondly, we examine the equation D2 q,ωx(s) = F (s, x(s),Dq,ωx(s), x(Θ(s))), s ∈ I1, x(s) = y(s), Dq,ωx(s) = Dq,ωy(s), s ∈ I2, (1.2) where F : I1 × R3 → R is continuous at s = ω0. We analyze the existence and uniqueness of the solution to equation (1.2) on I3 using the Banach fixed point theorem. Subsequently, we establish the Ulam stability of equation (1.2) on I3 employing Gronwall’s inequality. Finally, we analyze the equation Dn q,ωx(s) = F (s, x(s),Dq,ωx(s), . . . ,D n−1 q,ω x(s), x(Θ(s))), s ∈ I1, x(s) = y(s), Dj q,ωx(s) = Dj q,ωy(s), s ∈ I2, i = 0, 1, . . . , n− 1, (1.3) where F : I1 ×Rn → R is continuous at s = ω0. We extend the results of the Ulam stability to equation (1.3) on I3. The remainder of this article is organized as follows: In Section 2, we present notations and relevant preliminaries for the paper. Section 3 is dedicated to the study of the Ulam stability of equation (1.1) on interval I3. In Section 4, we establish the Ulam stability of equation (1.2) on interval I3, and provides direct results on the Ulam stability of equation (1.3). Finally, Section 5 includes examples to illustrate these theoretical findings. 2. Preliminaries Throughout the article, R is the set of real numbers, R+ signifies the set of non- negative real numbers, N+ refers to the set of positive integers, and I0 represents any interval of R that includes ω0. We define these function spaces S(I3,R) = {f : I3 → R : f(s) is continuous at s = ω0 and bounded}, S(I3,R+) = {f : I3 → R+ : f(s) is continuous at s = ω0 and bounded}. Let S(I3,R+) have a subspace S1(I3,R+) in which all functions are increasing. Obviously, S(I3,R) ⊇ S(I3,R+) ⊇ S1(I3,R+). For S(I3,R), S(I3,R+) and S1(I3,R+), let the metric ρ be defined by ρ(v1, v2) = max s∈I3 |v1(s)− v2(s)|. Then it is obvious that S(I3,R), S(I3,R+) and S1(I3,R+) are complete metric spaces. Definition 2.1. [10] Assume function f : I0 → R is continuous at s = ω0. Then Hahn difference operator is defined by Dq,ωf(s) = { f(qs+ω)−f(t) s(q−1)+ω , t ̸= ω0, f′(ω0), t = ω0, . EJDE-2024/77 ULAM TYPE STABILITY 3 where 0 < q < 1 and ω > 0 are constants, ω0 = ω 1−q . Definition 2.2. [2] Assume function g : I0 → R is continuous at s = ω0 and let [a1, a2] ⊂ I0. Then the Hahn integral of g from a1 to a2 has the form∫ a2 a1 g(s1)dq,ωs1 = ∫ a2 ω0 g(s1)dq,ωs1 − ∫ a1 ω0 g(s1)dq,ωs1, where∫ x ω0 g(s1)dq,ωs1 = (x(1− q)− ω) ∞∑ j=0 qjg(σj(x)) = ∞∑ j=0 (σj(x)− σj+1(x))g(σk(x)) for x ∈ I0, and σj(x) = qjx+ ω[j]q, x ∈ I0, [j]q = 1− qj 1− q , and the series (x(1− q)− ω) ∑∞ k=0 q kg(σk(x)) converges at x = a1 and x = a2. We can noted that | ∫ a2 a1 g(s1)dq,ωs1| ≤ ∫ a2 a1 |g(s1)|dq,ωs1, ∀a1, a2 ∈ I0, a1 < a2, is not necessarily true [2]. However, for a1 = ω0, we can obtain | ∫ a2 ω0 g(s1)dq,ωs1| ≤ ∫ a2 ω0 |g(s1)|dq,ωs1, ∀a2 ∈ I0, a2 > ω0. Additionally, we can obtain that∫ a1 ω0 |g(s1)|dq,ωs1 ≤ ∫ a2 ω0 |g(s1)|dq,ωs1, ∀a1, a2 ∈ I0, ω0 < a1 < a2, (2.1) is not necessarily true. If function |g| is increasing on I0, inequality (2.1) holds. Definition 2.3. [2] Assume function ζ : I0 → R is continuous at s = ω0 and 1 − ζ(s)(s − σ(s)) ̸= 0, ∀ s ∈ I0. Then exponential functions eζ(s) and Eζ(s) are given by eζ(s) = 1∏∞ j=0(1− ζ(σj(s))qj(s− σ(s))) , (2.2) Eζ(s) = ∞∏ j=0 (1 + ζ(σj(s))qj(s− σ(s))). (2.3) It is obvious that (2.2) and (2.3) are convergent since ∑∞ j=0 | ζ(σj(s)) | qj(s − σ(s)) is convergent. For ζ(s) = a0 ∈ R for all s ∈ I0, we have ea0 (s) = 1∏∞ j=0(1− a0qj(s− σ(s))) = ∞∑ j=0 (a0(s− σ(s)))j (q : q)j , |s− ω0| < 1 |a0(1− q)| , (2.4) and Ea0 (s) = ∞∏ j=0 (1 + a0q j(s− σ(s))) = ∞∑ j=0 q 1 2 j(j−1)(a0(s− σ(s)))j (q : q)j , s ∈ R, (2.5) 4 K. CHEN, J. WANG EJDE-2024/77 where (a : q)n = {∏n j=1(1− aqj−1), n ∈ N+, 1, n = 0. The proofs of (2.4) and (2.5) can be found in [7]. Lemma 2.4 ([2]). Assume f, g : I0 → R are continuous at s = ω0. Then∫ b a g(s)Dq,ω(f(s))dq,ωs+ ∫ b a Dq,ω(g(s))f(σ(s))dq,ωs = f(s)g(s) ∣∣b a , a, b ∈ I0. Lemma 2.5 (Gronwall’s inequality). Assume f, g : I0 → R are continuous at s = ω0 and ζ : I0 → R+ is continuous at s = ω0. Let 1− ζ(s)(s− σ(s)) > 0 for all s ∈ I0. If f(s) ≤ g(s) + ∫ s ω0 ζ(s1)f(s1)dq,ωs1, ∀s ∈ I0, then f(s) ≤ g(s) + eζ(s) ∫ s ω0 ζ(s1)E−s1(σ(s1))g(s1)dq,ωs1. (2.6) Let ζ(s) = a0 ∈ R+, for all s ∈ I0. If f(s) ≤ g(s) + ∫ s ω0 a0f(s1)dq,ωs1, s ∈ [ω0, ω0 + 1 a0(1− q) ], then f(s) ≤ g(s) + a0ea0 (s) ∫ s ω0 E−a0 (σ(s1))g(s1)dq,ωs1. Lemma 2.6 ([23]). Assume (Y, d,≤) is an ordered metric space. V : Y → Y is an increasing Picard operator (FV = {y∗V } denotes the fixed point set of operator V ). Then, for y ∈ Y , we have (i) if y ≤ V (y), then y ≤ y∗V ; (ii) if y ≥ V (y), then y ≥ y∗V . 3. Ulam stability of equation (1.1) Definition 3.1 ([28]). Assuming there is a real number c > 0, for for all ε > 0 and for all y satisfy |Dq,ωy(s)− F (s, y(s), y(Θ(s)))| ≤ ε, s ∈ I1, (3.1) equation (1.1) has a solution x with |y(s)− x(s)| ≤ cε, ∀s ∈ I3. Then (1.1) has Ulam-Hyers stability on I3. Definition 3.2 ([28]). Assuming there is a function θ : R+ → R+ and θ(0) = 0, for each solution y of inequality (3.1), equation (1.1) has a solution x with |y(s)− x(s)| ≤ θ(ε), ∀s ∈ I3. Then (1.1) has generalized Ulam-Hyers stability on I3. EJDE-2024/77 ULAM TYPE STABILITY 5 Definition 3.3 ([28]). Assuming there is c > 0, for all y satisfy |Dq,ωy(s)− F (s, y(s), y(Θ(s)))| ≤ εφ(s), s ∈ I1, (3.2) equation (1.1) has a solution x with |y(s)− x(s)| ≤ cεφ(s), ∀s ∈ I3. Then (1.1) has Ulam-Hyers-Rassias stability with respect to φ on I3. Definition 3.4 ([28]). Assuming there is c > 0, for all y satisfy |Dq,ωy(s)− F (s, y(s), y(Θ(s)))| ≤ φ(s), s ∈ I1, (3.3) equation (1.1) has a solution x with |y(s)− x(s)| ≤ cφ(s), ∀s ∈ I3. Then (1.1) has generalized Ulam-Hyers-Rassias stability with respect to φ on I3. Remark 3.5. A function y satisfies inequality (3.1) if and only if there is a function β : R → R such that (i) |β(s)| ≤ ε for all s ∈ I1; (ii) Dq,ωy(s) = F (s, y(s), y(Θ(s))) + β(s) for all s ∈ I1. The same statements apply to inequalities (3.2) and (3.3). In this article, we use the following assumptions: (A1) there is a real number LF > 0 such that for all s ∈ I1, xj , yj ∈ R, j = 1, 2, |F (s, x1, x2)− F (s, y1, y2)| ≤ LF 2∑ j=1 |xj − yj |. (A2) b− ω0 < 1 2LF . (A3) φ : I1 → R is increasing and continuous at s = ω0. Theorem 3.6. Under assumptions (A1), (A2), Equaton (1.1) has (i) a unique solution on I3, and (ii) Ulam-Hyers stability on I3. Proof. (i) Equation (1.1) is equivalent to the Hahn integral equation x(s) = { y(s), s ∈ I2, y(ω0) + ∫ s ω0 F (s1, x(s1), x(Θ(s1)))dq,ωs1, s ∈ I1. (3.4) We consider the mapping G : S(I3,R) → S(I3,R) as (Gx)(s) = { y(s), s ∈ I2, y(ω0) + ∫ s ω0 F (s1, x(s1), x(Θ(s1)))dq,ωs1, s ∈ I1. For all v ∈ S(I3,R), we have |(Gu)(s)− (Gv)(s)| ≤ ∫ s ω0 |F (s1, u(s1), u(Θ(s1)))− F (s1, v(s1), v(Θ(s1)))|dq,ωs1 ≤ ∫ s ω0 2LF max s∈I3 |u(s)− v(s)|dq,ωs1 ≤ 2LF (b− ω0)max s∈I3 |u(s)− v(s)|. By (A2) and the Banach fixed point theorem, equation (1.1) has a unique solution on I3. 6 K. CHEN, J. WANG EJDE-2024/77 (ii) Let y satisfy (3.1) and x represent the unique solution of (1.1). Then we can get (3.4). Consequently, by Remark 3.5, we obtain |y(s)− x(s)| ≤ ∫ s ω0 |β(s1)|dq,ωs1 + | ∫ s ω0 F (s1, y(s1), y(Θ(s1)))− F (s1, x(s1), x(Θ(s1)))dq,ωs1| ≤ ε(s− ω0) + ∫ s ω0 LF (|y(s1)− x(s1)|+ |y(Θ(s1))− x(Θ(s1))|)dq,ωs1. (3.5) Let us define V : S(I3,R+) → S(I3,R+) by (V u)(s) = { 0, s ∈ I2, ε(s− ω0) + ∫ s ω0 LF (u(s1) + u(Θ(s1)))dq,ωs1, s ∈ I1. For all u, v ∈ S(I3,R+), we have |(V u)(s)− (V v)(s)| ≤ ∫ s ω0 LF (|u(s1)− v(s1)|+ |u(Θ(s1))− v(Θ(s1))|)dq,ωs1 ≤ 2LF (b− ω0)max s∈I3 |u(s)− v(s)|. Then V is a contraction mapping in S(I3,R+). For all u1, v1 ∈ S1(I3,R+), we can also obtain |(V u1)(s)− (V v1)(s)| ≤ 2LF (b− ω0)max s∈I3 |u1(s)− v1(s)|. Then V is also a contraction mapping in S1(I3,R+). Thus, according to Banach fixed theorem, V has the unique fixed point u∗ ∈ S1(I3,R+) in S(I3,R+). We obtain u∗(s) = ε(s− ω0) + ∫ s ω0 LF (u ∗(s1) + u∗(Θ(s1)))dq,ωs1, s ∈ I1. Since u∗ ∈ S1(I3,R+) is increasing, we have u∗(s) ≤ ε(s− ω0) + ∫ s ω0 2LFu ∗(s1)dq,ωs1, s ∈ I1. (3.6) By using Lemma 2.5 (Gronwall’s inequality), from (3.6) it follows that u∗(s) ≤ ε(s− ω0) + e2LF (s)2LF ∫ s ω0 E−2LF (σ(s1))ε(s1 − ω0)dq,ωs1. Therefore, based on Lemma 2.4, we get u∗(s) ≤ ε(s− ω0)− e2LF (s) ∫ s ω0 Dq,ω(E−2LF (s1))ε(s1 − ω0)dq,ωs1 ≤ ε(e2LF (b)− 1) 2LF . Let u = |y − x|. According to (3.5), we have u ≤ V (u). By using Lemma 2.6, we obtain u(s) ≤ u∗(s), s ∈ I1. EJDE-2024/77 ULAM TYPE STABILITY 7 Then |y(s)− x(s)| ≤ ε(e2LF (b)− 1) 2LF , s ∈ I1. Thus equation (1.1) has Ulam-Hyers stability on I3. □ Corollary 3.7. Under assumptions (A1)–(A3), Equation (1.1) has generalized Ulam-Hyers stability on I3. Theorem 3.8. Under assumptions (A1)–(A3), Equation (1.1) has Ulam-Hyers- Rassias stability with respect to φ on I3. Proof. By Theorem 3.6 (i), equation (1.1) has the unique solution on I3. Let y satisfy (3.2). We can obtain (3.4). Thus, by using Remark 3.5, we have |y(s)− x(s)| ≤ ε(b− ω0)φ(s) + | ∫ s ω0 F (s1, y(s1), y(Θ(s1)))− F (s1, x(s1), x(Θ(s1)))dq,ωs1| ≤ ε(b− ω0)φ(s) + ∫ s ω0 LF (|y(s1)− x(s1)|+ |y(Θ(s1u))− x(Θ(s1))|)dq,ωs1. (3.7) As in the proof of Theorem 3.6 (ii), let the operator V : S(I3,R+) → S(I3,R+) be defined by (V u)(s) = { 0, s ∈ I2, ε(b− ω0)φ(s) + ∫ s ω0 LF ((V u)(s1) + (V u)(Θ(s1)))dq,ωs1, s ∈ I1. Then, V has a unique fixed point u∗ ∈ S(I3,R+) such that u∗ is increasing and u∗(s) ≤ ε(b− ω0)φ(s) + ∫ s ω0 2LFu ∗(s1)dq,ωs1. (3.8) By Lemma 2.5, from (3.8) it follows that u∗(s) ≤ ε(b− ω0)φ(s) + e2LF (s)2LF (b− ω0) ∫ s ω0 E−2LF (σ(s1))εφ(s1)dq,ωs1 ≤ (b− ω0)e2LF (b)εφ(s). Then according to Lemma 2.6, we obtain |y(s)− x(s)| ≤ (b− ω0)e2LF (b)εφ(s). Thus, equation (1.1) has Ulam-Hyers-Rassias stability with respect to φ on I3. □ Corollary 3.9. Under assumptions (A1)–(A3), Equation (1.1) has generalized Ulam-Hyers-Rassias stability with respect to φ on I3. 4. Ulam stability of equation (1.2) and (1.3) 4.1. Ulam stability of equation (1.2). In this section, let S(I3,R) be a Banach space in which all u ∈ S(I3,R) with the norm ∥u∥q,ω = max { max s∈I3 |u(s)|, max s∈I3 |Dq,ωu(s)| } . 8 K. CHEN, J. WANG EJDE-2024/77 Lemma 4.1. Equation (1.2) has a solution x : I3 → R in the form x(s) =  y(s), s ∈ I2, y(ω0) + (s− ω0)Dq,ωy(ω0) + ∫ s ω0 (s− σ(s1))F (s1, x(s1),Dq,ωx(s1), x(Θ(s1)))dq,ωs1, s ∈ I1. (4.1) Proof. Equation (1.2) is equivalent to the integral equation x(s) =  y(s), s ∈ I2, y(ω0) + (s− ω0)Dq,ωy(ω0) + ∫ s ω0 ∫ s2 ω0 F (s1, x(s1),Dq,ωx(s1), x(Θ(s1)))dq,ωs1dq,ωs2, s ∈ I1. (4.2) Then, we have∫ s ω0 ∫ s2 ω0 F (s1, x(s1),Dq,ωx(s1), x(Θ(s1)))dq,ωs1dq,ωs2 = ∫ s ω0 ∞∑ j=0 (σj(s2)− σj+1(s2)) × F (σj(s2), x(σ j(s2)),Dq,ωx(σ j(s2)), x(Θ(σj(s2))))dq,ωs2 = ∞∑ j=0 ∞∑ i=0 (σi(s)− σi+1(s))(σj+i(s)− σj+1+i(s)) × F (σj+i(s), x(σj+i(s)),Dq,ωx(σ j+i(s)), x(Θ(σj+i(s)))) = ∞∑ j=0 ∞∑ i=0 qi(s− σ(s))qj+i(s− σ(s)) × F (σj+i(s), x(σj+i(s)),Dq,ωx(σ j+i(s)), x(Θ(σj+i(s)))) = (s− σ(s))2 [ F (s, x(s),Dq,ωx(s), x(Θ(s))) + q(1 + q)F (σ(s), x(σ(s)),Dq,ωx(σ(s)), x(Θ(σ(s)))) + q2(1 + q + q2)F (σ2(s), x(σ2(s)),Dq,ωx(σ 2(s)), x(Θ(σ2(s)))) + q3(1 + q + q2 + q3)F (σ3(s),Dq,ωx(σ 3(s)), x(σ3(s)), x(Θ(σ3(s)))) + . . . ] = (s− σ(s))2 ∞∑ j=0 qj(1− qj+1)F (σj(s), x(σj(s)),Dq,ωx(σ j(s)), x(Θ(σj(s)))) 1− q = ∞∑ j=0 (σj(s)− σj+1(s))(s− σj+1(s))F (σj(s), x(σj(s)),Dq,ωx(σ j(s)), x(Θ(σj(s)))) = ∫ s ω0 (s− σ(s1))F (τ, x(s1),Dq,ωx(s1), x(Θ(s1)))dq,ωs1. Thus, we obtain (4.1). □ Now we introduce two more assumptions: (A4) there is a real number LF > 0 such that for all s ∈ I1, xj , yj ∈ R, j = 1, 2, 3, |F (s, x1, x2, x3)− F (s, y1, y2, y3)| ≤ LF 3∑ j=1 |xj − yj |. EJDE-2024/77 ULAM TYPE STABILITY 9 (A5) (b− ω0) 2 < 1+q 3LF . Theorem 4.2. Assume (A4) and (A5) hold. Then (1.2) has the unique solution on I3. Proof. By Lemma 4.1, equation (1.2) has a solution x : I3 → R in the form of (4.1). Let the mapping V : S(I3,R) → S(I3,R) be define by V (x)(s) =  y(s), s ∈ I2, y(ω0) + (s− ω0)Dq,ωy(ω0) + ∫ s ω0 (s− σ(s1))F (s1, x(s1),Dq,ωx(s1), x(Θ(s1)))dq,ωs1, s ∈ I1. For all x, y ∈ S(I3,R), we obtain |V (x)(s)− V (y)(s)| ≤ ∫ s ω0 (s− σ(s1))LF (|x(s1)− y(s1)|+ |Dq,ωx(s1)−Dq,ωy(s1)| + |x(Θ(s1))− y(Θ(s1))|)dq,ωs1 ≤ ∫ s ω0 3LF (s− σ(s1))∥x− y∥q,ωdq,ωs1 ≤ 3LF (s− ω0) 2 (1 + q) ∥x− y∥q,ω. Then, we can get ∥V (x)− V (y)∥q,ω ≤ 3LF (b− ω0) 2 (1 + q) ∥x− y∥q,ω. By (A5) and the Banach fixed theorem, (1.2) has the unique solution on I3. □ Lemma 4.3. Assume (A5) holds and η(s) = 3LF (s− ω0), s ∈ R. Then eη(s) > 0 is increasing on I1 and 1− η(s)(s− σ(s)) > 0, for all s ∈ I1. Proof. According to the definition of Hahn integral, by calculation, we obtain∫ s ω0 η(s)dq,ωs = 3LF (s− ω0) 2 (1 + q) . From condition (A5), for s ∈ I1, we have ∞∑ k=0 η(σk(s))(σk(s)− σk+1(s)) < 1. Then, 1− η(σk(s))(σk(s)−σk+1(s)) ∈ (0, 1) for all k ∈ N0, s ∈ I1. Thus, eη(s) > 0 is increasing on I1 and 1− η(s)(s− σ(s)) > 0 for all s ∈ I1. □ Theorem 4.4. Assume (A4) and (A5) hold. Then equation (1.2) has Ulam-Hyers stability on I3. Proof. According to Theorem 4.2, we can know equation (1.2) has the unique so- lution on I3. Let y satisfy the inequality |D2 q,ωy(s)− F (s, y(s),Dq,ωy(s), y(Θ(s)))| ≤ ε, s ∈ I1. Let x represent the unique solution to (1.2). Then, we obtain (4.1). 10 K. CHEN, J. WANG EJDE-2024/77 For s ∈ I1, according to Remark 3.5, we obtain |y(s)− x(s)| ≤ ∫ s ω0 ∫ s2 ω0 ε+ LF (|y(s1)− x(s1)| + |Dq,ωy(s1)−Dq,ωx(s1)|+ |y(Θ(s1))− x(Θ(s1))|)dq,ωs1dq,ωs2. (4.3) Let ϕ(s) = max { max s1∈[ω0−h0,s] |y(s1)− x(s1)|, max s1∈[ω0−h0,s] |Dq,ωy(s1)−Dq,ωx(s1)| } . Then ϕ is increasing and |y(s1)− x(s1)| ≤ ϕ(s1), |y(Θ(s1))− x(Θ(s1))| ≤ ϕ(s1), |Dq,ωy(s1)−Dq,ωx(s1)| ≤ ϕ(s1). Consequently, |y(s)− x(s)| ≤ ∫ s ω0 ∫ s2 ω0 ε+ 3LFϕ(s1)dq,ωs1dq,ωs2 ≤ ∫ s ω0 (ε+ 3LFϕ(s2))(s2 − ω0)dq,ωs2 = ε(s− ω0) 2 1 + q + ∫ s ω0 3LFϕ(s2)(s2 − ω0)dq,ωs2. For all s1 ∈ [ω0, s], we have |y(s1)− x(s1)| ≤ ε(s1 − ω0) 2 1 + q + ∫ s1 ω0 3LFϕ(s2)(s2 − ω0)dq,ωs2. Then ϕ(s) ≤ ε(s− ω0) 2 1 + q + ∫ s ω0 3LFϕ(s2)(s2 − ω0)dq,ωs2. (4.4) Let η(s) = 3LF (s − ω0), s ∈ R. According to Lemma 2.5 and Lemma 4.3, from (4.4) it follows that ϕ(s) ≤ ε(s− ω0) 2 1 + q + eη(s) ∫ s ω0 η(s2)E−η(σ(s2)) ε(s2 − ω0) 2 1 + q dq,ωs2 ≤ ε(b− ω0) 2 1 + q eη(b). Thus, |y(s)− x(s)| ≤ ε(b− ω0) 2 1 + q eη(b). Then (1.2) has Ulam-Hyers stability on I3. □ Corollary 4.5. Under assumptions (A4), (A5), Equation (1.2) has generalized Ulam-Hyers stability on I3. Theorem 4.6. Assume (A3)–(A5) hold. Then (1.2) has Ulam-Hyers-Rassias sta- bility with respect to φ on I3. EJDE-2024/77 ULAM TYPE STABILITY 11 Proof. According to Theorem 4.2, equation (1.2) has the unique solution on I3. Let y satisfy the inequality |D2 q,ωy(s)− F (s, y(s),Dq,ωy(s), y(Θ(s)))| ≤ εφ(s), s ∈ I1. Let x represent unique solution to (1.2). According to Remark 3.5, we have |y(s)− x(s)| ≤ ∫ s ω0 ∫ s2 ω0 εφ(s1) + LF (|y(s1)− x(s1)|+ |Dq,ωy(s1)−Dq,ωx(s1)| + |y(Θ(s1))− x(Θ(s1))|)dq,ωs1dq,ωs2. Let ϕ(s) = max { max s1∈[ω0−h0,s] |y(s1)− x(s1)|, max s1∈[ω0−h0,s] |Dq,ωy(s1)−Dq,ωx(s1)|}. As in the proof of Theorem 4.4, we have |y(s)− x(s)| ≤ ∫ s ω0 ∫ s2 ω0 εφ(s1) + 3LFϕ(s1)dq,ωs1dq,ωs2 ≤ ∫ s ω0 (εφ(s2) + 3LFϕ(s2))(s2 − ω0)dq,ωs2 ≤ εφ(s)(s− ω0) 2 1 + q + ∫ s ω0 3LFϕ(s2)(s2 − ω0)dq,ωs2. Then one has ϕ(s) ≤ εφ(s)(s− ω0) 2 1 + q + ∫ s ω0 3LFϕ(s2)(s2 − ω0)dq,ωs2. (4.5) By using Lemma 2.5, from (4.5) it follows that ϕ(s) ≤ εφ(s)(s− ω0) 2 1 + q + ep(s) ∫ s ω0 p(s2)E−η(σ(s2)) εφ(s2)(s2 − ω0) 2 1 + q dq,ωs2 ≤ (b− ω0) 2eη(b)εφ(s) 1 + q . Then |y(s)− x(s)| ≤ (b− ω0) 2eη(b)εφ(s) 1 + q . Thus, (1.2) has Ulam-Hyers-Rassias stability with respect to φ on I3. □ Corollary 4.7. Assume (A3)–(A5) hold. Then (1.2) has generalized Ulam-Hyers- Rassias stability with respect to φ on I3. 4.2. Ulam stability of equation (1.3). Based on the definitions of the Hahn difference and q, ω-integral, it is clear that x : I3 → R satisfies (1.3) if and only if x satisfies the corresponding integral equation x(s) =  y(s), s ∈ I2,∑n−1 k=0 (1−q)k(s−ω0) k (q:q)k Dk q,ωy(ω0) + ∫ s ω0 ∫ sn ω0 . . . ∫ s2 ω0 F (s1, x(s1), Dq,ωx(s1), . . . ,D n−1 q,ω x(s1), x(Θ(s1)))dq,ωs1 . . . dq,ωsn−1dq,ωsn, s ∈ I1. Now we introduce the next assumptions 12 K. CHEN, J. WANG EJDE-2024/77 (A6) there is a real number L > 0 such that for all s ∈ I1, xj , yj ∈ R, j = 1, 2, . . . , n, |F (s, x1, x2, . . . , xn)− F (s, y1, y2, . . . , yn)| ≤ L n∑ j=1 |xj − yj |. (A7) (b− ω0) n < (q:q)n (n+1)L(1−q)n . Consequently, analogous approaches can be used to establish Ulam stability for equation (1.3) on I3. We now easily present these results without proofs. Theorem 4.8. Assume (A6) and (A7) hold. Then (1.3) has the unique solution on I3. Theorem 4.9. Assume (A6) and (A7) hold. Then (1.3) has Ulam-Hyers stability on I3 with c = (1− q)n(b− ω0) neη(b) (q : q)n , where η(s) = (n+1)L(1−q)n−1(s−ω0) n−1 (q:q)n−1 , and s ∈ R. Theorem 4.10. Assume (A6) and (A7) hold. Then equation (1.3) has generalized Ulam-Hyers stability on I3. Theorem 4.11. Assume (A3), (A6), (A7) hold. Then (1.3) has Ulam-Hyers- Rassias stability and generalized Ulam-Hyers-Rassias stability with respect to φ on I3. 5. Examples Example 5.1. We consider the equation D 1 3 ,6 x(s) = 2e−|x(s)| + sin(x(s− 10)) 264 , s ∈ [9, b], x(s) = s2, s ∈ [−1, 9], (5.1) and inequalities |D 1 3 ,6 y(s)− 2e−|y(s)| + sin(y(s− 10)) 264 | ≤ ε, s ∈ [9, b], |D 1 3 ,6 y(s)− 2e−|y(s)| + sin(y(s− 10)) 264 | ≤ εe 1 45 (s), s ∈ [9, b]. When 9 < b < 75, equation (5.1) has the unique solution on [−1, b]. Obviously, equation (5.1) has Ulam-Hyers stability on [8, 75) with c = 66(e1/66(75)− 1). Equation (5.1) has Ulam-Hyers-Rassias stability with respect to e 1 45 (s) on [−1, 75) with c = 66e1/66(75). Example 5.2. We consider the equation D2 1 2 , 1 2 x(s) = 5 cos(x(s)) + 8D 1 2 , 1 2 x(s) + 8 sin(x(s− 5)) 10240 + sin(x(s)) 16806 , s ∈ [1, b], x(s) = s, D 1 2 , 1 2 x(s) = 1, s ∈ [−4, 1], (5.2) EJDE-2024/77 ULAM TYPE STABILITY 13 and inequalities∣∣D2 1 2 , 1 2 y(s)− 5 cos(y(s)) + 8D 1 2 , 1 2 y(s) + 8 sin(y(s− 5)) 10240 − sin(y(s)) 16806 ∣∣ ≤ ε, for s ∈ [1, b], and∣∣D2 1 2 , 1 2 y(s)− 5 cos(y(s)) + 8D 1 2 , 1 2 y(s) + 8 sin(y(s− 5)) 10240 − sin(y(s)) 16806 ∣∣ ≤ εe 1 16 (s), for s ∈ [1, b]. When 1 < b < 1 + 8 √ 10, equation (5.2) has the unique solution on [−4, b]. Then equation (5.2) has Ulam-Hyers stability on [−4, 1 + 8 √ 10) and Ulam-Hyers-Rassias stability with respect to e 1 16 (s) on [−4, 1 + 8 √ 10) with c = 1280 3 ep(1+8 √ 10)(1 + 8 √ 10), where p(s) = 3(s−1) 1280 , s ∈ R. Example 5.3. We consider the equation Dn 1 5 ,8 x(s) = 3Dn−1x(s) + sin( ∑n−2 i=1 Di q,ωx(s)) + 2e−|x(s)| + cos(x(s− 1)) 885 , s ∈ [10, b], x(t) = s, D 1 5 ,8 x(s) = 1, Di 1 5 ,8 x(s) = 0, i = 2, 3, . . . , n− 1, s ∈ [9, 10], (5.3) and inequalities∣∣Dn 1 5 ,8 y(s)− 3Dn−1y(s) + sin( ∑n−2 i=1 Di q,ωy(s)) + 2e−|y(s)| + cos(y(s− 1)) 885 ∣∣ ≤ ε, for s ∈ [10, b], and∣∣Dn 1 5 ,8 y(s)− 3Dn−1y(s) + sin( ∑n−2 i=1 Di q,ωy(s)) + 2e−|y(s)| + cos(y(s− 1)) 885 ∣∣ ≤ εs, for s ∈ [10, b]. When 10 < b < n √ 295[n]q ! n+1 +10, equation (5.3) has the unique solution on [9, b]. Therefore, (5.3) has Ulam-Hyers stability on [9, b] and Ulam-Hyers-Rassias stability with respect to φ(s) = s on [9, b]. Acknowledgments. This work is partially supported by the Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020] 5016). The authors are grateful to the referees for their careful reading of the manuscript and their valuable comments. The authors also want to thank the editor for the help provided. References [1] M. M. Abdelkhaliq, A. E. Hamza; Stability of Hahn difference equations in Banach algebras, Communications of the Korean Mathematical Society, 33(2018), 1141-1158. [2] M. H. Annaby, A. E. Hamza, K. A. Aldwoah; Hahn difference operator and associated Jackson-Nörlund integrals, Journal of Optimization Theory and Application, 154(2012), 133- 153. [3] M. H. Annaby, Z. S. Mansour; q-Taylor and interpolation series for Jackson q-difference operator, Journal of Mathematical Analysis and Applications, 344(2008), 472-483. [4] G. D. Birkhoff; General theory of linear difference equations, Transactions of the American Mathematical Society, 12(1911), 243-284. 14 K. CHEN, J. WANG EJDE-2024/77 [5] R. D. Carmichael; Linear difference equations and their analytic solutions, Transactions of the American Mathematical Society, 12(1911), 99-134. [6] R. D. Carmichael; On the theory of linear difference equations, American Journal of Mathe- matics, 35(1913), 163-182. [7] G. Gasper, M. Rahman; Basic Hypergeometric Series, Cambridge University Press, 2004. [8] P. Gavruta; A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184(1994), 431-436. [9] R. Ger, P. Semrl; The stability of the exponential equation, Proceedings of the American Mathematical Society, 124(1996) 779-787. [10] W. Hahn; Über orthogonalpolynome, die q-differenzenlgleichungen genügen, Mathematische Nachrichten, 2(1949), 4-34. [11] A. E. Hamza, S. M. Ahmed; Existence and uniqueness of solutions of Hahn difference equa- tions, Advances in Difference Equations, 2013(2013), 316. [12] A. E. Hamza, S. M. Ahmed; Theory of linear Hahn difference equations, Journal of Advances in Mathematics, 4(2013), 441-461. [13] A. E. Hamza, M. A. Alghamdi, S. A. Alasmi; Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear quantum difference equations, Journal of Mathematics and Computer Science, 35(2024), 336-347. [14] A. E. Hamza, S. D. Makharesh; Leibniz’ rule and Fubinis theorem associated with Hahn difference operator, Journal of Advances in Mathematics, 12(2016), 6335-6345. [15] A. E. Hamza, E. M. Shehata; Existence and uniqueness of solutions of general quantum difference equations, Advances in Dynamical Systems and Applications, 11(2016), 45-58. [16] A. E. Hamza, A. S. Zaghrout, S. M. Ahmed; Characterization of stability of first order Hahn difference equations, Journal of Advances in Mathematics, 5(2013), 678-687. [17] F. Hıra; Hahn Laplace transform and its applications, Demonstratio Mathematica, 56(2023), 20230259. [18] F. Hıra; On q, ω-differential transform method, Journal of Physics A: Mathematical and Theoretical, 56(2023), 325202. [19] D. H. Hyers; On the stability of the linear functional equation, Proceedings of the National Academy of Sciences, 27(1941), 222-224. [20] D. H. Hyers, T. M. Rassias; Approximate homomorphisms, Aequationes Mathematicae, 44(1992), 125-153. [21] S. M. Jung; Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, 2001. [22] K. Oraby, A. Hamza; Taylor theory associated with Hahn difference operator, Journal of Inequalities and Applications, 2020(2020), 124. [23] D. Otrocol, V. Ilea; Ulam stability for a delay differential equation, Open Mathematics, 11(2013), 1296-1303. [24] L. M. Quarrie, N. Saad, M. S. Islam; Asymptotic iteration method for solving Hahn difference equations, Advances in Difference Equations, 2021(2021), 354. [25] T. M. Rassias; On the stability of linear mappings in Banach spaces, Proceedings of the American Mathematical Society, 72(1978), 297-300. [26] T. M. Rassias; On the stability of functional equations and a problem of Ulam, Acta Appli- candae Mathematica, 62(2000), 23-130. [27] M. H. A. Risha, M. H. Annaby, Z. S. Mansour, et al.; Linear q-difference equations, Zeitschrift für Analysis und ihre Anwendungen, 26(2007), 481-494. [28] I. A. Rus; Ulam stability of ordinary differential equations, Studia Universitatis Babes-Bolyai Mathematica, 54(2009), 125-133. [29] S. M. Ulam; A Collection of Mathematical Problems, Interscience Publishers, New York, 1960. Kai Chen School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China Email address: kaichen589@163.com EJDE-2024/77 ULAM TYPE STABILITY 15 Jin Rong Wang (corresponding author) School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China Email address: jrwang@gzu.edu.cn 1. Introduction 2. Preliminaries 3. Ulam stability of equation (??) 4. Ulam stability of equation (??) and (??) 4.1. Ulam stability of equation (??) 4.2. Ulam stability of equation (??) 5. Examples Acknowledgments References