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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki
[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 3, pp. 407-418 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1934
MSC: 34L10, 39A13, 47B25, 94A20
On a ^-analogue of the Sturm—Liouville operator with discontinuity conditions
D. Karahan
Harran Universitesi, Sanliurfa, Turkey.
Abstract
In this paper, a ^-analogue of the Sturm-Liouville problem with discontinuity condition on a finite interval is studied. It is proved that the ^-Sturm-Liouville problem with discontinuity conditions is self-adjoint in L'2(0,n). The completeness theorem and the sampling theorem are proved.
Keywords: ^-Sturm-Liouville operator, completeness of eigenfunctions, self-adjoint operator.
Received: 3rd June, 2021 / Revised: 1st September, 2021 / Accepted: 13th September, 2022 / First online: 27th September, 2022
1. Introduction
Let us consider a ^-analogue of the Sturm-Liouville equation in the form
l(y) := - X-Dq-i Dq y(t) + u(t)y(t) = vy(t), 0 <t<v, v e C, (1)
together with the discontinuity conditions at a point a e (0, n)
y(a + 0) = ay(a - 0), Dq-iy(a + 0) = a-lDq-iy(a - 0), (2)
and boundary conditions
y(0) = y(n) = 0, (3)
where 0 < q < 1, u(t) e L2q(0, n) is a real function, a is real; a = 1, a> 0.
In [1], it is worth mentioning that this work is based on the ^-difference operator, which is attributed to Jackson. In recent years, many papers have been
Differential Equations and Mathematical Physics Research Article
© Authors, 2022
© Samara State Technical University, 2022 (Compilation, Design, and Layout) Q ©® The content is published under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/) Please cite this article in press as:
Karahan D. On a q-analogue of the Sturm-Liouville operator with discontinuity conditions, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 3, pp. 407-418. EDN: ZNKLPD. DOI: 10.14498/vsgtu1934. Author's Details:
Done Karahan © https://orcid.org/0000-0001-6644-5596
Mathematics Department, Science and Letter Faculty; e-mail: dkarahan@harran.edu.tr
published subject to boundary value problems consisting of a g-Jackson derivative in the classical Sturm-Liouville problem. In [2-4], the ^-analogues of Sturm-Liouville problems are investigated, and a space of boundary values of the minimal operator is created, and all maximal dissipative, self-adjoint, maximal accretive operators are described, and other extensions of the ^-analogue of Sturm-Liouville operators in terms of boundary conditions are raised. A theorem on completeness of the system of eigenfunctions and associated functions of dissipative operators are proved by using the Lidskii's theorem.
In the current decade, many authors have investigated the ^-sampling theory of signal analysis. [6-8] are the first studies in this subject. In these studies, the construction of expansions in the ^-Fourier series [5] was followed by the derivation of the ^-sampling theorems. The sampling theory associated with q-type of Sturm-Liouville equations is conceived (see [9]). In [10], M. Annaby and Z. Mansour obtained asymptotic formulae for eigenvalues and eigenfunctions of g-type of Sturm-Liouville problems.
In [11-13], B. Allahverdiev and H. Tuna investigated the continuous spectrum of the singular g-Sturm-Liouville operators and established some criteria under which the g-Sturm-Liouville equation is of limit-point case at infinity. In [14], authors established a Parseval equality and an expansion formula in eigenfunctions for a singular g-Sturm-Liouville operator on the entire line. (Also, B. Allahverdiev and H. Tuna investigated the resolvent operator of a singular g-Dirac system (see [15])). In [16], the spectral properties of the eigenvalues and the eigenfunctions of the g-Sturm-Liouville boundary value problem are investigated.
Also, there are many physical models involving ^-difference and their related problems in [21-23]. In these studies, several physical models involving q-functions, ^-derivatives, ^-integrals and their related problems are investigated. However, to our knowledge, there is no study of the general problem as we do in the present setting. At this point, it is worth mentioning that this work is based on the ^-difference operator, which is attributed to Jackson and a similar study of the Stum-Liouville systems generated by the Askey-Wilson derivative.
In [24] and [25], the Sturm-Liouville problems are generalized by a fractional derivative of order a, 0 < a ^ 1. The numerical solutions of fractional Sturm-Liouville problems were examined.
2. Preliminaries
In this section, we give some of the ^-notations and we use these ^-notations throughout the paper. These standard notations are based on [18].
If q £ R is fixed, a subset A of C is called ^-geometric if qt £ A whenever t £ A. Let h be a function, real or complex valued, defined on a ^-geometric set A, q = 1. Let q be a positive number with 0 < q < 1. The ^-difference operator Dq is defined as
Dqtm = Wt-M V £ A \{0}.
The q-1 -derivative at zero is defined by
Dq-ih(t)= lim ^Z h(0) = Dih(0).
ra^-TO tq r
When required, we will replace q by q l. We can demonstrate the correctness of the following facts using the definition and will often use it
Dq-1 h(t) = (Dq h)(q-11), D2qh(q-lt) = qDq [Dq h(q-lt)] = Dq-i Dq h(t).
Associated with this operator, there is a non-symmetric formula for the ^-differentiation of a product of the functions h and g defined in the ^-geometric set.
Dq [h(t)g(t)] = h(qt)Dgg(t) + g(t)Dq h(t). (4)
The ^-integral called Jackson integral is given by
/ h{t)dq t = (1 - q)Y\ h(nqn)nqn. J 0
n=0
L2q(0,K) is the space of all complex-valued functions defined in (0,n) with the norm
/г-ж \ i/q
HhH = miqdqtj <
and it is a separable Hilbert space (see [6]) with the inner product
рк _
{h,g} = h(t]g(f)dq t. J 0
The space C^(0) is the space of all continuous functions with the ^-derivative first order at the point zero.
Definition. A function f that is defined on a ^-geometric set A, 0 e A, is said to be ^-regular at zero if
lim f (tqn) = f (0), yt e A.
If h and g are both ^-regular at zero, there is a rule of ^-integration by parts given by
fK fK
/ g(t)Dqh(t)dgt = (hg)(n) - (hg)(0) - Dgg(t)h(qt)dgt.
00
An important special case, we have
fK
/ Dqh(t)dgt = (h)(n) - (h)(0). (5)
0
The g-Wronskian of two functions h and g is defined as
Wq(h,g)(t) = h(t)Dqg(t) - g(t)Dqh(t).
Lemma 1 (see [2]). Let h( ■ ), g( ■ ) in L"^(0,-k) be defined on [0,g-1^]. Then, for t e (0, ж] we have
(Dqh,g) = h(n)g(nq-1) - lim h(nqn)g(nqn-1) + (h, -1D— g), (6)
n^— \ q /
(--D— h, g) = lim h(irqn-1)g(irqn) - h(irq-1)g(ir) + (h, Dqg) . (7)
\q / n^x
3. The self-adjoint problem
Theorem 1. The q-analogue of the Sturm-Liouville eigenvalue problem (1)-(3) is self-adjoint on Cq(0) n L^(0,ir).
Proof. We first prove that k( ■ ), a( ■ ) in L'^(0,tt), we have the following g-Lagrange's identity
fK ___
(iK(t)â(t) - K(t)la(t))dqt = [n, a](n) - lim [n, cr](irqn), (8)
J 0
where
[n,a](t) := n(t)Dq-i<r(t) - Dq-in(t)a(t).
Applying (7) with h(t) = Dqn(t) and g(t) = a(t), we obtain the following: -
(--Dq-1 Dq K(t),a(t)) =
= -(DqK)(vq-1 )a(ir) + lim (Dqn)(^qn-1)a(^qn) + (Dqn,Dqa) =
n^-x
= -Dq-ih(tt) a(ir) + lim Dq-iK(^qn)a(^qn) + (Dqn,Dqa).
H n^X H
Applying (6) to h(t) = n(t), g(t) = Dqa(t), we obtain
(Dq K, Dq a) =
= MDq (r(nq-1) - lim n(nqn)Dq a(^qn-1) + ( k, - 1 Dq-i Dq a) =
n^-x \q /
1
= n(n)Dq-ia(n) - lim n(nqn)Dq-ia(nqn) + (k, — Dq-iDqa\
n^-x \ q /
Therefore,
/-1DiD K(t),a(t)\ = [n,a](ir) - lim [n, a](irqn) + (k, -1 Dq-iDqA (9)
\q / n—x \ q H /
The Lagrange's identity (8) is the result of (9) and the reality of u(t). Letting k( ■), a( ■) in C2(0) and assuming that they satisfy (2), (3), we obtain the following:
k(0) = 0, ct(0) = 0. (10)
The continuity of k( ■), a( ■) at zero implies that
lim [ n,a](TTqn) = [k,ct](0).
n—^^o
Then (9) will be
1Dq-iDqK,a^ = [K,a](7t) - [«,a](0) + (k, -1Dq-iDqa^
From (10), we have
[«,^1(0) = K(0)Dq-i<r(0) — Dq-ik(0)<t(0) = 0.
Similarly,
[k,O"](^) = n(n)Dq-i a(n) — Dq~i n(n)a(n) = 0. Since u(t) is real valued,
K),a) = (-1 Dn-i Dn n(t) + u(t)K(t),a(t)) =
(1(k), a) = ( —1 Dq-iDgK(t) + u(t)n(t),a(t)^
( — -Dg-i Dg K(t),a(t)) + (u(t)K(t),a(t))
= (k, — 1Dg-i Dg a^ + (K(t),u(t)a(t)) = (k, 1(a)),
i.e. I is a self-adjoint operator. □
4. Completeness of eigenfunctions
Lemma 2. Let h and g be q-regular at zero. The Wronskian Wq(h,g)(t) of the q-analogue of the Sturm-Liouville problem (1) does not defend on t.
Proof. The proof can be done similarly to [9]. □
Let ^(t,v) be the solution of equation (1) with discontinuity conditions (2) and initial conditions
rj(0,v)=0, Dg-ir)(0, v) = -, (11)
and ((t,u) be the solution of (1) with discontinuity conditions (2) and
£(^)=0, Dg-i £(*,") = -■
Since the g-Wronskian is independent of t, we can evaluate it at t = 0 and use the above conditions at £ in order to write
wg (v,0(v) = Wg (u) = —t(0,v). (12)
It follows from condition (3) that Wq(v) =0 if and only if v is an eigenvalue of the ^-analogue of the Sturm-Liouville problem (1).
Denote by vn the eigenvalues and by an the normalized numbers of problem (1), (2):
®n = V2(t, Vn)dqt. (13)
J 0
The numbers {vn,an} are said to be the spectral date of the problem (1)-(3). X°n and a°n are eigenvalues and normalized numbers, respectively, in the case of u(t) = 0 in the equation (1), where u(t) is a potential function. Then there exists a sequence such that
1) ((t, vn) = vn), = 0,
2) ftn&n = —Wq(vn), where Wq(v) = DqWq(v) (respect to v),
a 1 fK
= . /,,0+ - + » , e h,an = _ fimt, Xn)dqt, (see [20]),
^un an Jo
4) an = an + ^, {U e h-
Lemma 3. The eigenvalues and eigenfunctions of the q-analogue of the Sturm-Liouville problem (1)-(3) have the following properties:
i) the eigenvalues are real;
ii) eigenfunctions that belong to different eigenvalues are orthogonal;
iii) all eigenvalues are simple.
Proof.
i) Let u0 be an eigenvalue with an eigenfunction ]o( ■). Then
{l(]o), r]o) = {]o, Klo)) -
Since l(]o) = uo]o then
f-K
( Vo - m) \]o{x)\2(dqX. Jo
Since ]o( ■) is non-trivial then vo = To, which proves i). ii) Let v, i be two distinct eigenvalues with the corresponding eigenfunctions ](■), i( ■), respectively. Then
f-K
(V -¡) ](t)W)dqt = 0. o
Since v = i then ](■) and ((■) are orthogonal. iii) Let uo be an eigenvalue with two eigenfunctions ]1() and ]2(:). From [2, Corollary 2.15] we can prove that the functions {]1( ■ ), ]2( ■ )} are linearly dependent by proving that their g-Wronskian vanishes at t = 0. Indeed,
Wq (]l, ]2)(0) = ll(0)Dq ]2(0) - ]2(0)Dq ]l(0) =
= ]1 (0) Dq-1 ]2(0) - l]2(0) Dq-1 ]l(0) = 0,
since both ]1 and ]2 satisfy (3).
□
Theorem 1. The system of eigenfunctions {](t, vn)}n^o of the problem (1)-(3) is complete in L^(0,n).
Proof. Denote
G(, v =__^ ii(t,v)at,u), t^ T,
(, , ) Wq (V)\]( T,V )i(t, V), T,
and consider the function
rK
Y (t ,v )= G(t ,t,v )h(r)dqT = o
W9 (u)
ft fK
) Tj(T,V )h(T )dq T + V(t,v) &T,V )h(r )dq T
J 0 Jt
The function G(t, t, v) is called the g-type Green function for the ^-analogue of the Sturm-Liouville problem (1)-(3). G(t,r,v) is the kernel of the inverse operator for the ^-analogue of the Sturm-Liouville problem, i.e. Y(t, v) is the solution of the problem
-1 Dq-iDqY(t) + {-v + u(t)}Y(t) = h(t), t e M, v e C, (14)
satisfies the discontinuity condition (2) and the boundary condition (3). Furthermore, taking into account (13), we get the following:
Res Y(t, v) =
1
ft fK
£(t,vn) v(T,un)h(r )dq t + v(t,Vn) t,(T,vn)h(T )dq t 0
Wq (Un)
• n N v(t,Vn) ri(T,Vn)h(T)dgT =— ij(t,Vn) ri(T,Vn)h(T)dqt. (15) Wg (u) Jo an Jo
Let the function h(t) e L2q(0,K) be such that
K
/ rj(r,un)h(T)dqt = 0, n = 1, 2,... 0
Then in view of (15), Res Y(t,v) = 0 and consequently for each fixed t e [0,^],
the function Y(t, v) is entire in v. Furthermore, for p e G$ = {p : \p — pk,0\ ^ S, k = ±1, ±2,...} and \p\ ^ p* for sufficiently large p* = p*(S), where v = pq, p^,0 are the zeros of the function
W°(p)= a++ a
A + sin pm - sin p(2a — n) P
.± _ i,
5 is a fixed positive number, a± = 1 (a ± 1) (see [17]), p* is rather large, the inequality
Cs \P\'
\WQ(V)\ ^ TSe'lmplK,
and consequently the inequality
c'
\Y(t,v)\ < , P e Gs, \p\ > p*, \p\
are fulfilled (see [17]). Using the maximum principle and Liouville's theorem, we conclude that Y(t,v) = 0. From this and (14) it follows that h(t) = 0 a.e. on (0,n). Thus, the theorem is proved. □
1
n
Kv) = h(
5. The qr-sampling theory
Theorem 2. Let ](t ,v) and £(t, v) be the solutions of (1) selected as above. Then all functions h of the form
K
h(v) = u(t)](t, u)dqt, u eL2(0,n), (16)
o
can be written as the Lagrange-type sampling expansion:
" Wq (v)
n=o n'Wq(Vn)(V - Vn) ,
where Wq(v) is the q-Wronskian of the functions ](t, v) and £(t, v).
Proof. We multiply equation (1) with ](t, vn). Then we again consider equation (1), but replace v by vn and multiply this last equation by ](t, v). Subtracting the two results yields
(V - Vn)](t, v)](t, Vn) = D2](q-1t, Un)](t, v) - D"2](q-1t, v)](t, Vn).
From the rule for the ^-differentiation of a product (4), we can write
(is - Un)](t, v)](t, Vn) = Dq [Dq](q-1t, un)](t, is) - Dq](q-1t, v)](t, Vn)] .
If we apply a ^-integration by means of (5) we obtain
K
fK
( V - Vn) n{t, v)rj(t, Vn)dqt =
Jo
rK
= Dg [Dqr(q-11, Vn)r(t, v) - Dqr(Q-11, v)r(t, vn)] dqt = Jo
= Dqr(q- V, Vn)r(^, v) - Dqr(q - V v)r(n, vn) -
- (Dqr(q-10, vn)r(0,v) -Dqr(q-10,v)r(0, Vn)).
From conditions (3) and (11) we get the following:
K
(V - Vn) ](t, V)](t, Vn)dqt =
o
= Dq](q-1 7, Vn)](lT, v) - Dq](q-17, v)](7T, Vn) =
= ](7, is)Dq-1 ](7, Vn) - ](7, Vn)Dq-1 ](n, V).
From (12), we have the following:
K
(v - Vn) ](t, v)](t, Vn)dq X = Wq (is) Dq-1 ](n, Vn) - Wq (vn)Dq-1 ](n, V).
o
From vn eigenvalues being zeros of the characteristic function Wq(v) of the (/-analogue of the Sturm-Liouville problem (1)-(3), we obtain Wq(vn) = 0. Then, we have
K
(v - Vn) ](t,v)](t, Vn)dqt = Wq(is) Dq-1 ](7,Vn).
o
As a result,
Wq (v)Dq-i r](ir,vn)
fK
n{t ,V )rj(t, Vn)dgt = ,
.J0 \v — vn)
and taking the limit a v ^ vn gives
K
/ I nit, Vn)l2 dgt = Wq (Vn)Dq-i r](i, Vn).
Jo
Therefore, we can apply Kramer's lemma (see [19]) and write an integral transform of the form (16) as
Ui \ ^Uf \ Wq (v)
n=0 n W1(Vn)(V — Vn)
Example. Consider the following g-Sturm-Liouville problem:
- lDq-iDqy(t) =vy(t), 0 <t<i, ve C, together with the discontinuity conditions at a point a e (0, i)
y( a + 0) = ay (a — 0), Dq-i y(a + 0) = a-1Dq-i y(a — 0), and boundary conditions
y(0) = y(i) = 0,
□
where 0 < q < 1, a is real; a = 1, a > 0. The system of functions {r]o(t, v°n)}£= where v = p
2
sin pt
c-— , 0 <t < a,
^o(t, ^ = \ + pin pt sin p(2a — t)
' a+-— +a-——--, a<t ^i,
p
where a± = q(a ± , is complete in the space L2q(0,i).
6. Conclusion
In this paper, a (/-analogue of the Sturm-Liouville problem with discontinuity condition on a finite interval is studied. It is shown that the eigenfunctions of this problem are in the form of a complete system. A sampling theorem is proved for integral transforms whose kernels are basic functions and the integral is of Jackson's type. Finally, it is proved that the (/-analogue of the Sturm-Liouville problem with discontinuity conditions is self-adjoint in L"^(0,i).
In future studies, the main equation for the (/-analogue of the Sturm-Liouville problem can be obtained. The Weyl solution and the Weyl function can be defined for the ^-analogue of the Sturm-Liouville problem. Uniqueness theorems for the solution of the inverse problem according to the Weyl function and spectral date can be proved.
Competing interests. I declare that I have no competing interests.
Authors' contributions and responsibilities. I take full responsibility for submit
the final manuscript to print. I approved the final version of the manuscript.
Funding. Not applicable.
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Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2022. Т. 26, № 3. С. 407-418
ISSN: 2310-7081 (online), 1991-8615 (print) EDN: ZNKLPD
d https://doi.org/10.14498/vsgtu1934
УДК 517.927.21
Об ^-аналоге оператора Штурма—Лиувилля с условиями разрыва
Д. Карахан
Университет Харран, Шанлыурфа, Турция.
Аннотация
Исследуется (/-аналог задачи Штурма-Лиувилля с условием разрыва на конечном интервале. Доказано, что (/-задача Штурма-Лиувилля с условиями разрыва является самосопряженной в Ь'^(0,п). Доказаны теорема о полноте и теорема о выборке. Приводится пример, иллюстрирующий полученные результаты.
Ключевые слова: (/-оператор Штурма-Лиувилля, полнота собственных функций, самосопряженный оператор.
Получение: 3 июня 2021 г. / Исправление: 1 сентября 2021 г. / Принятие: 13 сентября 2022 г. / Публикация онлайн: 27 сентября 2022 г.
Конкурирующие интересы. Я заявляю, что конкурирующих интересов не имею.
Авторская ответственность. Я несу полную ответственность за предоставление окончательной версии рукописи в печать. Окончательная версия рукописи мною одобрена.
Финансирование. Исследование выполнялось без финансирования.
Дифференциальные уравнения и математическая физика Научная статья
© Коллектив авторов, 2022 © СамГТУ, 2022 (составление, дизайн, макет)
<В ©® Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru) Образец для цитирования
Karahan D. On a q-analogue of the Sturm-Liouville operator with discontinuity conditions, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 3, pp. 407-418. EDN: ZNKLPD. DOI: 10.14498/vsgtu1934.
Сведения об авторе
Доне Карахан© https://orcid.org/0000-0001-6644-5596 кафедра математики, факультет естественных наук и письма; e-mail: dkarahan@harran.edu.tr
