Dirac System Associated with Hahn Difference Operator
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Dirac System Associated with Hahn Difference Operator Fatma Hıra1 Received: 12 July 2018 / Revised: 21 December 2018 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we introduce q, ω-Dirac system. We investigate the existence and uniqueness of solutions for this system and obtain some spectral properties based on the Hahn difference operator. Also we give two examples, which indicate that asymptotic formulas for eigenvalues. Keywords Hahn difference operator · q, ω-Dirac system · Eigenvalue problems Mathematics Subject Classification Primary 39A13 · 34B09; Secondary 35Q41 · 33D15
1 Introduction and Preliminaries In [1,2], Hahn introduced the difference operator Dq,ω which is defined by ⎧ ⎨ f (qt + ω) − f (t) , t = ω0 , Dq,ω f (t) := (qt + ω) − t ⎩ f (ω ) , t = ω0 , 0
(1.1)
where q ∈ (0, 1), ω > 0 are fixed and ω0 := ω/ (1 − q). This is valid if f is differentiable at ω0 . In this case, we call Dq,ω f , the q, ω-derivative of f . This operator extends the forward difference operator ω f (t) :=
f (t + ω) − f (t) , (t + ω) − t
(1.2)
Communicated by V. Ravichandran.
B 1
Fatma Hıra [email protected]; [email protected] Department of Mathematics, Arts and Science Faculty, Ondokuz Mayis University, 55139 Samsun, Turkey
123
F. Hıra
where ω > 0 is fixed (see [3–6]) as well as Jackson q-difference operator Dq f (t) :=
f (qt) − f (t) t (q − 1)
(1.3)
where q ∈ (0, 1) is fixed (see [7–12]). In [13], the authors gave a rigorous analysis of Hahn’s difference operator and the associated calculus. The existence and uniqueness theorems for general first-order q, ω-initial value problems and the theory of linear Hahn difference equations were studied in [14,15], respectively. Recently, a q, ω-Sturm–Liouville theory has been established in [16], sampling theorems associated with q, ω-Sturm–Liouville problems in the regular setting have been derived in [17] and fractional Hahn calculus have been investigated in [18–20]. In [21], the authors presented the q-analog of the one-dimensional Dirac system: ⎧ ⎨
1 − Dq −1 y2 + p (x) y1 = λy1 , q ⎩ D y + r (x) y = λy , q 1 2 2 k11 y1 (0) + k12 y2 (0) = 0, k21 y1 (a) + k22 y2 aq −1 = 0,
(1.4) (1.5) (1.6)
y1 (x) , 0 ≤ x ≤ a < ∞. They where ki j (i, j = 1, 2) are real numbers, y (x) = y2 (x) also gave the existence and uniqueness of the solution and some spectral properties of this system. For the same q-Dirac system (1.4)–(1.6), asymptotic formulas for the eigenvalues and the eigenfunctions were investigated in [22] and sampling theory was derived in [23]. In this paper, we introduce a q, ω-version of q-Dirac system (1.4)–(1.6). When the q-difference operator Dq is replaced by the Hahn difference operator Dq,ω , we obtain the following q, ω−Dirac system. Namely, we obtain the system which consists of the q, ω−Dirac equations
⎧ ⎨
1 − D 1 , −ω y2 + p (t) y1 = λy1 , q q q ⎩ D y + r (t) y = λy , q,ω 1 2 2
(1.7)
and the boundary conditions B1 (y) := k11 y1 (ω0 ) + k12 y2 (ω0 ) = 0, B2 (y) := k21
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