Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 74, pp. 1–10. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.74 IMPULSIVE REGULAR q-DIRAC SYSTEMS BILENDER P. ALLAHVERDIEV, HÜSEYIN TUNA, HAMLET A. ISAYEV Abstract. This article concerns a regular q-Dirac system under impulsive conditions. We study the existence of solutions, symmetry of the corresponding operator, eigenvalues and eigenfunctions of the system. Also we obtain Green’s function and its basic properties. 1. Introduction Boundary value problems under impulsive conditions have been extensively in- vestigated because they arise as the mathematical modeling of various areas of science. For example in population dynamics, economics, optimal control, and chemotherapy. It is well-known that these equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. For recent studies see [4, 5, 10, 16, 18, 19] and their references. In 2005, Annaby and Mansour [2] applied quantum calculus to classical Sturm- Liouville problems and investigated q-Sturm-Liouville problem −1 q Dq−1Dqy + v(ξ)y = λy, a11y(0) + a12Dq−1y(0) = 0, a21y(a) + a22Dq−1y(a) = 0, where ξ ∈ [0, a], aij (i, j = 1, 2) are real numbers, λ ∈ C, v is a real-valued function defined on [0, a] and continuous at zero. Later on, q-Sturm-Liouville problems were studied by some authors by putting impulsive boundary conditions. Çetinkaya [8] studied discontinuous q-Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Karahan and Mamedov [12, 13, 14] investigated a q-Sturm- Liouville problem with discontinuity conditions. In [15], the author studied the singular q-Sturm-Liouville problem with impulsive conditions. Aygar and Bairamov [6] studied the properties of the scattering function of an impulsive q-difference equation. Bohner and Cebesoy In [7] investigated the locations of the eigenvalues and spectral singularities of an operator corresponding to impulsive q-difference equations. 2020 Mathematics Subject Classification. 39A13, 34A36, 34L40, 34B27. Key words and phrases. Difference equations; q-Dirac system; Green’s function. ©2023. This work is licensed under a CC BY 4.0 license. Submitted August 1, 2023. Published October 27, 2023. 1 2 B. P. ALLAHVERDIEV, H. TUNA, H. A. ISAYEV EJDE-2023/74 In 2017, Allahverdiev and Tuna [1] moved the classical Dirac system to q-calculus and entered the regular q-Dirac system defined as( 0 − 1 qDq−1 Dq 0 )( y1 y2 ) + ( p(ξ) 0 0 r(ξ) )( y1 y2 ) = λ ( y1 y2 ) , a11y1(0) + a12y2(0) = 0, a21y1(a) + a22y2( a q ) = 0, where ξ ∈ [0, a], aij (i, j = 1, 2) are real numbers, λ ∈ C, p and r are real-valued functions defined on [0, a] and continuous at zero and p, r ∈ L1 q(0, a). In this study, the above system is considered under impulsive boundary condi- tions. First, the existence theorem is proved. Then, the symmetry of the impulsive q-Dirac operator, the basic properties of eigenvalues and eigenfunctions are given. Finally, the Green’s function was established and its basic properties were exam- ined. 2. Preliminaries In this section, we give the basic concepts of q-calculus that will be used in this article. Detailed information can be found in [11, 3, 9]. Let q ∈ (0, 1) and let A ⊂ R := (−∞,∞) be a q-geometric set, i.e., if qξ ∈ A for all ξ ∈ A. We begin by defining the operator Dqh(ξ) = { h(qξ)−h(ξ) (q−1)ξ , ξ 6= 0 limn→∞ h(qnς)−h(0) qnς , ξ = 0, where ξ, ς ∈ A. When it is required, q will be replaced by q−1. The following facts, which will be frequently used, can be verified directly from the definition: Dq−1h(ξ) = (Dqh)(q−1ξ), (D2 qh)(q−1ξ) = qDq[Dqh(q−1ξ)] = Dq−1Dqh(ξ). Related to this operator there exists a non-symmetric formula for the q-differentiation of a product Dq[h(ξ)g(ξ)] = g(ξ)Dqh(ξ) + h(qξ)Dqg(ξ). We define the Jackson q-integration by∫ ξ 0 h(γ)dqγ = ξ(1− q) ∞∑ n=0 qnh(qnξ) (ξ ∈ A), provided that the series converges, and∫ b a h(γ)dqγ = ∫ b 0 h(γ)dqγ − ∫ a 0 h(γ)dqγ, where a, b ∈ A. Through the remainder of this article, we deal only with functions q-regular at zero, i.e., functions satisfying lim n→∞ h(ξqn) = h(0), for every ξ ∈ A. The q-trigonometric functions are given by the formulas (see [3]) cos(z; q) = ∞∑ n=0 (−1)n qn 2 (z(1− q))2n (q; q)2n , EJDE-2023/74 IMPULSIVE REGULAR q-DIRAC SYSTEMS 3 sin(z; q) = ∞∑ n=0 (−1)n qn(n+1)(z(1− q))2n+1 (q; q)2n+1 , where (a; q)0 = 1, (a; q)n = n−1∏ k=0 (1− aqk) . 3. Impulsive regular q-Dirac systems We consider the following regular boundary-value problem (BVP) with impulsive conditions l(y) = λy, ξ ∈ I, (3.1) L1(y) := y1(0) + k1y2(0) = 0, (3.2) L2(y) := y1(d−)− k2y1(d+) = 0, (3.3) L3(y) := y2(q−1d−)− k3y2(q−1d+) = 0, (3.4) L4(y) := y1(a) + k4y2(q−1a) = 0, (3.5) where k1, k2, k3, k4 are real numbers, 0 < d < a < ∞, I1 := [0, d), I2 := (d, a], I := I1 ∪ I2, l(y) := { − 1 qDq−1y2 + p(ξ)y1 Dqy1 + r(ξ)y2, y := ( y1 y2 ) , and λ is a complex eigenvalue parameter. Our basic assumptions throughout the paper are the following: (A2) Let q ∈ (0, 1) and k2k3 = α > 0. (A2) p and r are real-valued continuous functions on [0, d) ∪ (d, q−1a] and have finite limits p(d±), r(d±). Let H = L2 q((0, d);C2) · + L2 q((d, a);C2) be a Hilbert space endowed with the inner product 〈h, g〉 := ∫ d 0 (h(1), g(1))C2dqξ + α ∫ a d (h(2), g(2))C2dqξ, where h(ξ) = { h(1)(ξ), ξ ∈ I1 h(2)(ξ), ξ ∈ I2, g(ξ) = { g(1)(ξ), ξ ∈ I1 g(2)(ξ), ξ ∈ I2. Theorem 3.1. For λ ∈ C, the initial-value problem with impulsive conditions l(y) = λy, y1(0) = c1, y2(0) = c2, c1, c2 ∈ R, y1(d−)− k2y1(d+) = 0, y2(q−1d−)− k3y2(q−1d+) = 0, has a unique solution ψ which is an entire function of λ for every ζ ∈ [0, d)∪ (d, a]. Proof. From [1], we infer that the problem l(y(ξ)) = λy(ξ), ξ ∈ (0, d), y1(0) = c1, y2(0) = c2, 4 B. P. ALLAHVERDIEV, H. TUNA, H. A. ISAYEV EJDE-2023/74 has a unique solution ψ1 = ( ψ11 ψ12 ) which is an entire function of λ. � Consider the problem l(y) = λy, ξ ∈ (d, a], (3.6) y1(d+) = 1 k2 ψ11(d−), (3.7) y2(q−1d+) = 1 k3 ψ12(q−1d−). (3.8) Let Un(ξ, λ) = U0(ξ, λ) + q ∫ ξ d ( ϕ2(ξ, λ)ϕT1 (qt, λ) −ϕ1(ξ, λ)ϕT2 (qt, λ) ) M(qt)Un−1(qt, λ)dqt, where U0(ξ, λ) = 1 k2 ψ11(d−, λ) + 1 k3 (ξ − d)ψ12(q−1d−, λ), ξ ∈ I2, ϕ1(ξ, λ) = ( ϕ11(ξ, λ) ϕ12(ξ, λ) ) = ( cos(λξ; q) sin(λξ;q) λ ) , ϕ2(ξ, λ) = ( ϕ21(ξ, λ) ϕ22(ξ, λ) ) = ( − sin(λξ;q) λ cos(λξ; q) ) , M = ( p 0 0 r ) , and the supraindex T means the transpose of a matrix. Here ϕ1 and ϕ2 are the fundamental solutions of (3.6) for M = ( p 0 0 r ) = 0. It is clear that the functions Un are entire functions. Let λ ∈ C be fixed. Then there exist positive numbers N(λ) and A such that |ϕij(ξ, λ)| ≤ √ N(λ) 2 , i, j = 1, 2, ‖M(ξ, λ)‖ ≤ A, ‖U0(ξ, λ)‖ ≤ Ñ(λ), ξ ∈ I2. Then, ‖U1(ξ, λ)− U0(ξ, λ) ≤ ∣∣∣q ∫ ξ d [ cos(λξ; q) sin(λqt; q) λ − sin(λξ; q) λ cos(λqt; q) ] p(qt)U01(qt, λ)dqt − q ∫ ξ d [ cos(λξ; q) cos(λqt; q) + sin(λξ; q) λ sin(λqt; q) λ ] r(qt)U02(qt, λ)dqt ∣∣∣ ≤ 2qN(λ)AÑ(λ) ∣∣ ∫ ξ d dqt ∣∣∣ EJDE-2023/74 IMPULSIVE REGULAR q-DIRAC SYSTEMS 5 ≤ 2qN(λ)AÑ(λ) ξ(1− q) (1− q) . Similarly, we obtain ‖U3(ξ, λ)− U2(ξ, λ)‖ ≤ 22q2Ñ(λ) A2N2(λ)ξ2(1− q)2 (1− q)(1− q2) , which implies ‖Un+1(ξ, λ)− Un(ξ, λ)‖ ≤ 2nqnÑ(λ) (AN(λ)ξ(1− q))n (q; q)n (3.9) for n = 1, 2, 3, . . . . From Weierstrass’s test, we see that the series ∞∑ n=1 2nq n(n+1) 2 Ñ(λ) (AN(λ)ξ(1− q))n (q; q)n is uniformly convergent. Thus the series U1(ξ, λ) + ∞∑ n=1 {Un+1(ξ, λ)− Un(ξ, λ)} (3.10) is uniformly convergent with respect to ξ on (d, a]. Then we have lim n→∞ Un(ξ, λ) = ψ2(ξ, λ), i.e., ψ2(ξ, λ) = U1(ξ, λ) + ∞∑ n=1 {Un+1(ξ, λ)− Un(ξ, λ)}. Now we show that ψ2 of (3.6). DqUn+1(ξ, λ)−DqUn(ξ, λ) = q ∫ ξ d [Dqϕ2(ξ, λ)ϕT1 (qt, λ)−Dqϕ1(ξ, λ)ϕT2 (qt, λ)]M(qt) × [Un(qt, λ)− Un−1(qt, λ)]dqt. It follows from (3.9) that the series ∞∑ n=1 {DqUn+1(ξ, λ)−DqUn(ξ, λ)} is uniformly convergent with respect to ξ on (d, a]. Since Dq cos(λξ; q) = (−r(ξ) + λ) sin(λξ; q) λ , Dq sin(λξ; q) λ = (−r(ξ) + λ) cos(λξ; q), we conclude that Dqψ21(ξ, λ) = ∞∑ n=1 {DqUn+1,1(ξ, λ)−DqUn,1(ξ, λ)} = (−r(ξ) + λ) ∞∑ n=1 {Un,1(ξ, λ)− Un−1,1(ξ, λ)} = (−r(ξ) + λ)ψ22(ξ, λ). 6 B. P. ALLAHVERDIEV, H. TUNA, H. A. ISAYEV EJDE-2023/74 The validity of the other equation in (3.6) is proved similarly. Moreover, ψ2 satisfies (3.7)-(3.8). Consequently, ψ(ξ, λ) = { ψ1(ξ, λ) ξ ∈ I1 ψ2(ξ, λ) ξ ∈ I2 (3.11) satisfies (3.1)-(3.4). Likewise, we can obtain the following theorem. Theorem 3.2. For any λ ∈ C, Equation (3.1) has a solution χ(ζ, λ) = { χ1(ζ, λ), ζ ∈ I1 χ2(ζ, λ), ζ ∈ I2 (3.12) satisfying conditions (3.2)-(3.5) which is an entire function of λ for every ζ ∈ I. Now, we consider the sets Dmax = { y ∈ H : the one-sided limits y1(d±) and y2(q−1d±) exist and are finite, L2(y) = L3(y) = 0, l(y) ∈ H } , Dmin = {y ∈ Dmax : y1(0) = y2(0) = y1(a) = y2(a) = 0}. Then the maximal operator Lmax on Dmax is defined by Lmaxy = l(y). If we restrict the operator Lmax to the set Dmin, then we obtain the minimal operator Lmin. Let y = ( y1 y2 ) , z = ( z1 z2 ) ∈ Dmax. Then the q-Green formula is given by∫ a 0 [(l(y), z)C2 − (y, (l(z))C2 ]dqξ = [y, z](a)− [y, z](d+) + [y, z](d−)− [y, z](0), (3.13) where [y, z](ξ) := Wq(y, z) = y1(ξ)z2(q−1ξ)− z1(ξ)y2(q−1ξ). Let us consider the operator L with domain consisting of vectors y ∈ Dmax, (Ly = l(y)) that satisfy (3.2)-(3.5). Then the following theorem is obtained from (3.13) and conditions (3.2)-(3.5). Theorem 3.3. The operator L is symmetric. Corollary 3.4. (i) All eigenvalues of the BVP (3.1)-(3.5) are real. (ii) If λ1 and λ2 are two different eigenvalues of the BVP (3.1)-(3.5), then the corresponding eigenfunctions u1 and u2 are orthogonal. (iii) All eigenvalues of the BVP (3.1)-(3.5) are simple from the geometric point of view. Now, we shall define the characteristic function of the BVP (3.1)-(3.5). Let us define the following entire functions ω1(λ) = Wq,1(ϕ1, χ1)(ξ), ω2(λ) = Wq,2(ϕ2, χ2)(ξ), because these Wronskians are independent of ξ for ξ ∈ I2 and ξ ∈ I1, respectively. From (3.3)-(3.4), we find that ω1(λ) = αω2(λ). Thus, the characteristic function of the BVP (3.1)-(3.5) is defined as ω(λ) := ω1(λ) = αω2(λ). EJDE-2023/74 IMPULSIVE REGULAR q-DIRAC SYSTEMS 7 Lemma 3.5. Let ∆(λ) := ∣∣∣∣∣∣∣∣ L1ψ1 L1χ1 L1ψ2 L1χ2 L2ψ1 L2χ1 L2ψ2 L2χ2 L3ψ1 L3χ1 L3ψ2 L3χ2 L4χ1 L4χ1 L4ψ2 L4χ2 ∣∣∣∣∣∣∣∣ . Then, every λ ∈ C, we obtain ∆(λ) = − 1 αω 3(λ). Proof. From (3.11) and (3.12), we obtain ∆(λ) = ∣∣∣∣∣∣∣∣ 0 ω1(λ) 0 0 ψ11(d−, λ) χ11(d−, λ) −k2ψ21(d+, λ) −k2χ21(d+, λ) ψ12(q−1d−, λ) χ12(q−1d−, λ) −k3ψ22(q−1d+, λ) −k3χ22(q−1d+, λ) 0 0 −ω2(λ) 0 ∣∣∣∣∣∣∣∣ = ω1(λ)| ∣∣∣∣∣∣ ψ11(d−, λ) −k2ψ21(d+, λ) −k2χ21(d+, λ) ψ12(q−1d−, λ) −k3ψ22(q−1d+, λ) −k3χ22(q−1d+, λ) 0 −ω2(λ) 0 ∣∣∣∣∣∣ = ω1(λ)ω2(λ) ∣∣∣∣ ψ11(d−, λ) −k2χ21(d+, λ) ψ12(q−1d−, λ) −k3χ22(q−1d+, λ) ∣∣∣∣ = −ω1(λ)ω2(λ) ∣∣∣∣ψ11(d−, λ) χ11(d−, λ) ψ12(d−, λ) χ12(d−, λ) ∣∣∣∣ = −ω2 1(λ)ω2(λ) = − 1 k2k3 ω3(λ). � Theorem 3.6. The eigenvalues of (3.1)-(3.5) are the same as the zeros of the entire function ω(λ). Hence the eigenvalues of (3.1)-(3.5) form a finite or infinite sequence without a finite accumulation point. Proof. Let λ(0) be a zero of ω(λ). Then ω2(λ(0)) = Wq,2(ψ2, χ2) = 0, i.e., ψ2 = ζχ2 for some ζ 6= 0. Thus ψ2 satisfies (3.5). Therefore the function ψ(ξ, λ(0)) = { ψ1(ξ, λ(0)), ξ ∈ I1 ψ2(ξ, λ(0)). ξ ∈ I2 satisfies the BVP (3.1)-(3.5), i.e., λ(0) is an eigenvalue. Let λ(0) be an eigenvalue and η(ξ, λ(0)) be any corresponding eigenfunction. We want to show that ω(λ(0)) = 0. Assume that ω(λ(0)) 6= 0. Then we conclude that ω1(λ(0)) 6= 0 and ω2(λ(0)) 6= 0. Thus there exist constants ζi, i = 1, 2, 3, 4, at least one of which is not zero, such that η(ξ, λ(0)) = { ζ1ψ1(ξ, λ(0)) + ζ2χ1(ξ, λ(0)), ξ ∈ I1 ζ3ψ2(ξ, λ(0)) + ζ4χ2(ξ, λ(0)), ξ ∈ I2. Therefore, Liη(ξ, λ(0)) = 0, i = 1, 2, 3, 4, since η(ξ, λ(0)) is the eigenfunction. Hence det ( Liη(ξ, λ(0)) ) = ∆(λ) = 0, because at least one of the constants ξi, i = 1, 2, 3, 4 is not zero. It follows from Lemma 3.5 that ∆(λ) 6= 0, a contradiction. � 8 B. P. ALLAHVERDIEV, H. TUNA, H. A. ISAYEV EJDE-2023/74 4. Green’s function In this section, we construct the Green function of the BVP −1 q Dq−1y2 + {λ+ p(ξ)}y1 = h1, (4.1) Dqy1 + {λ+ r(ξ)}y2 = h2, (4.2) y1(0) + k1y2(0) = 0, (4.3) y1(d−)− k2y1(d+) = 0, (4.4) y2(q−1d−)− k3y2(q−1d+) = 0, (4.5) y1(a) + k4y2(q−1a) = 0, (4.6) where ξ ∈ I and h = ( h1 h2 ) ∈ H Theorem 4.1. Suppose that λ is not an eigenvalue of (3.1)-(3.5). The BVP (4.1)- (4.6) has a unique solution y defined as y(ξ, λ) = ∫ d 0 G(ξ, t, λ)h(t)dqt+ α ∫ a d G(ξ, t, λ)h(t)dqt, (4.7) where G(ξ, t, λ) = 1 ω(λ) { χ(ξ, λ)ψT (t, λ), 0 ≤ t ≤ ξ ≤ a, t 6= d, ξ 6= d, ψ(ξ, λ)χT (t, λ), 0 ≤ ξ ≤ t ≤ a, t 6= d, ξ 6= d. (4.8) Proof. By (4.7), we have y1(ξ, λ) =  q ω(λ)χ11(ξ, λ) ∫ ξ 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)ψ11(ξ, λ) ∫ d ξ (χ11(qt, λ)h1(qt) + χ21(qt, λ)h2(qt))dqt + q ω(λ)ψ11(ξ, λ)α ∫ a d (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I1, q ω(λ)χ12(ξ, λ) ∫ d 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)χ12(ξ, λ)α ∫ ξ d (ψ12(qt, λ)h1(qt) + ψ22(qt, λ)h2(qt))dqt + q ω(λ)ψ12(ξ, λ)α ∫ a ξ (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I2, (4.9) y2(ξ, λ) =  q ω(λ)χ21(ξ, λ) ∫ ξ 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)ψ21(ξ, λ) ∫ d ξ (χ11(qt, λ)h1(qt) + χ21(qt, λ)h2(qt))dqt + q ω(λ)ψ21(ξ, λ)α ∫ a d (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I1, q ω(λ)χ22(ξ, λ) ∫ d 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)χ22(ξ, λ)α ∫ ξ d (ψ12(qt, λ)h1(qt) + ψ22(qt, λ)h2(qt))dqt + q ω(λ)ψ22(ξ, λ)α ∫ a ξ (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I2, (4.10) EJDE-2023/74 IMPULSIVE REGULAR q-DIRAC SYSTEMS 9 From (4.9), we find that Dqy1(ξ, λ) =  q ω(λ)Dqχ11(ξ, λ) ∫ ξ 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)Dqψ11(ξ, λ) ∫ d ξ (χ11(qt, λ)h1(qt) + χ21(qt, λ)h2(qt))dqt + q ω(λ)Dqψ11(ξ, λ)α ∫ a d (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt +Wq,1(χ, ψ)(qξ)h2, ξ ∈ I1, q ω(λ)Dqχ12(ξ, λ) ∫ d 0 (ψ11(qt, λ)h1(qt) + ψ21(qt, λ)h2(qt))dqt + q ω(λ)Dqχ12(ξ, λ)α ∫ ξ d (ψ12(qt, λ)h1(qt) + ψ22(qt, λ)h2(qt))dqt + q ω(λ)Dqψ12(ξ, λ)α ∫ a ξ (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt +αWq,2(χ, ψ)(qξ)h2, ξ ∈ I2 =  q ω(λ){λ− r(ξ)}χ21(ξ, λ) ∫ ξ 0 ( ψ11(qt, λ)h1(qt) +ψ21(qt, λ)h2(qt) ) dqt + q ω(λ){λ− r(ξ)}ψ21(ξ, λ) ∫ d ξ ( χ11(qt, λ)h1(qt) +χ21(qt, λ)h2(qt) ) dqt + q ω(λ){λ− r(ξ)}ψ21(ξ, λ)α ∫ a d ( χ12(qt, λ)h1(qt) +χ22(qt, λ)h2(qt) ) dqt+ h2, ξ ∈ I1, q ω(λ){λ− r(ξ)}χ22(ξ, λ) ∫ d 0 ( ψ11(qt, λ)h1(qt) +ψ21(qt, λ)h2(qt) ) dqt + q ω(λ){λ− r(ξ)}χ22(ξ, λ)α ∫ ξ d ( ψ12(qt, λ)h1(qt) +ψ22(qt, λ)h2(qt) ) dqt + q ω(λ){λ− r(ξ)}ψ22(ξ, λ)α ∫ a ξ ( χ12(qt, λ)h1(qt) +χ22(qt, λ)h2(qt) ) dqt+ h2, ξ ∈ I2, = {λ− r(ξ)}y2(ξ) + h2(ξ). The validity of (4.1) is proved similarly. Hence the function y(ξ, λ) in (4.7) is the solution of system (4.1)-(4.6). It is clear that (4.7) satisfies (4.3)-(4.6). � Lemma 4.2. (i) The Green function is unique. (ii) G(ξ, t, λ) = GT (t, ξ, λ). (iii) G(ξ, t, λ) is continuous at the point (0, 0). The proof of the above lemma is similar to the proof of [2, Theorem 5.2]. We omit it. References [1] B. P. Allahverdiev, H. Tuna; One dimensional q-Dirac equation, Math. Meth. Appl. Sci., 40 (2017), 7287-7306. [2] M. H. Annaby, Z. S. Mansour; Basic Sturm-Liouville problems, J. Phys. A: Math. Gen., 38 (17) (2005), 3775-3797. [3] M. H. Annaby, Z. S. Mansour; q-Fractional calculus and equations, Lecture Notes in Mathe- matics, vol. 2056, Springer, Berlin, 2012. 10 B. P. ALLAHVERDIEV, H. TUNA, H. A. ISAYEV EJDE-2023/74 [4] K. Aydemir, H. Olğar, O. Sh. Mukhtarov; The principal eigenvalue and the principal eigen- function of a boundary-value-transmission problem, Turkish J. Math. Comput. 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Panakhov; Solution of a discontinuous inverse nodal problem on a finite interval, Math. Comput. Model., 44 (1-2) (2006), 204-209. [17] B. M. Levitan, I. S. Sargsjan; Sturm-Liouville and Dirac operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian). [18] A. S. Ozkan, R. Kh. Amirov; An interior inverse problem for the impulsive Dirac operator, Tamkang J. Math., 42 (3) (2011), 259-263. [19] E. Tunç, O. Sh. Muhtarov; Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput., 157 (2) (2004), 347-355. Bilender P. Allahverdiev Department of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan. Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan Email address: bilenderpasaoglu@gmail.com Hüseyin Tuna Department of Mathematics, Mehmet Akif Ersoy University, 15030 Burdur, Turkey Email address: hustuna@gmail.com Hamlet A. Isayev Department of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan Email address: hamlet@khazar.org 1. Introduction 2. Preliminaries 3. Impulsive regular q-Dirac systems 4. Green's function References