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20
Математика
MSC: 34L10, 39A13, 47B25, 94A20 DOI: 10.14529/mmph210401
ON A q-BOUNDARY VALUE PROBLEM WITH DISCONTINUITY CONDITIONS
D. Karahan1, K.R. Mamedov2
1 Harran University, Sanlurfa, Turkey E-mail: [email protected]
2 Mersin University, Mersin, Turkey E-mail: [email protected]
In this paper, we studied ^-analogue of Sturm-Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the ^-Sturm-Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of ^-Sturm-Liouville boundary value problem. We shown that eigenfunctions of ^-Sturm-Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson's type.
Keywords: q-Sturm-Liouville operator; self-adjoint operator; completeness of eigenfunctions; sampling theory.
Introduction
Boundary value problems with discontinuity conditions on the interval often appear in mathematics and other branches of sciences. Quantum calculus was initiated at the beginning of the 19th century and in recent years, many papers subject to the boundary value problems consisting a q -Jackson derivative in the classical Sturm-Lioville problem have occured [1]. In [2, 3], q -Sturm-Liouville problems are investigated and a space of boundary values of the minimal operator and describe all maximal dissipative, self-adjoint, maximal accretive and other extensions of q -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.
Also, there are a lot of physical models involving q -difference and their related problems in [4, 5]. In [6], the construction of expansions in q -Fourier series was followed by the derivation of the q -sampling theorems. In [7], a q -version of the sampling theorem was derived using the q -Hankel transform. The sampling theory associated with q -type of Sturm-Liouville equations is conceived (see [8, 9]).
In [10], it is proved that the regular symmetric q -Sturm-Liouville operator is semi-bounded and investigated the continuous spectrum of this operator. In [11], authors established a Parseval equality and an expansion formula in eigenfunctions for a singular q -Sturm-Liouville operator.
In this paper, q -analogue of Sturm-Liouville boundary value problems with discontinuity conditions in an interior point ([12]) are discussed.
Let us consider the boundary value problem L for the equation:
l(y) := -1D _1 Dqy(x) + v(x)y(x) = iy(x), (1)
q q
on the interval x e (0,T) with the boundary conditions
U ( y):= y(0) - gy(0) = 0,
q
together with the jump conditions at a point a e (0,T)
U(y) := D -y(0) - gy(0) = 0, V(y) := D -y(T) + Gy(T) = 0, (2)
q q
y(a + 0) = a1 y(a - 0), D -1 y(a + 0) = a11D -1 y(a - 0) + a2y(a - 0). (3)
q q
2
Here v(x) e Lq (0,T) is a real-valued function, a1, a2, g and r are real numbers; a1 > 0 .
1. Preliminaries on q -calculus
In this section, we give some of the q -notations and we will use these q -notations throughout the paper. These standard notations are founded in [13].
Let q be a positive number with 0 < q <1. Let h be a real or complex valued function on A (A is
q -geometric set (see [2])). The q -difference operator Dq (the Jackson q -derivative) is defined as
h( x) - h(qx), x ^ o.
q x(1 - q) '
When required we will replace q by q-1. We can demonstrate the correctness of the following facts using the definition and will use often
D -ih(x) = (Dqh)(q"1x), Dq2h(q-1x) = qDq[Dqh(q-1x)] = D -1 Dqh(x).
q q
Let h and g be defined on a q -geometric set A such that the q -derivatives of h and g exist for all x e A . Then, there is a non-symmetric formula for the q -differentiation of a product
Dq [h( x) g (x)] = h(qx)Dqg (x) + g (x) Dqh( x). (4)
The q -integral usually associated with the name of Jackson is defined in the interval (0,T), as
T
JqA( x)dqx = (1 - q)Yh(Tq" )Tqn.
n=0
Let l}q (0,T) be the space of all complex-valued functions defined on (0,T), such that
1
2 J . „) 2
И = (f I h(x) I2 dqXj
<.
q
The space L2q (0,T) is a separable Hilbert space (see [6]) with the inner product
q
T _
?(x) g(xX-^qs
<h, g) = J"0 h( x) g (x)dqx. If h and g are both q -regular at zero, there is a rule of q -integration by parts given by
rT rT
J0 g(x)Dqh(x)dqx = (hg)(T) - (hg)(0) - J0 Dqg(x)h(qx)dqx. (5)
The q appearing in the argument of h in the right-hand side integrand is another manifestation of the symmetry that is everywhere present in q -calculus. As an important special case, we have
rT
J0 Dqh(x)dqx = (h)(T) - (h)(0). (6)
Lemma 1. (see [2]) Let h(.), g(.) in L\ (0,T) be defined on [0, q T]. Then, for x e (0,T] we have
(Dqh)(xq"1) = D -1 h(x), (7)
<?
<Dqh,g) = h(T)g(Tq~l) - Hm h(Tqn)g(Tqn-1) + <h,--D -1 g), (8)
n®¥ q q
<-1 D -1h,g) = lim h(Tqn-1)g(Tqn) - h(Tq"1)g(T) + <h,Dqg). (9)
q q n®¥
2. Properties of the spectral characteristics
Let h(x) and g(x) be continuously differentiable functions on [0,a] and [a,T]. Denote
Wq (h, g)(x) = <h, g) := h(x)Dqg(x) - g(x)Dqh(x).
Here Wq (h, g) is defined as the q -Wronskian of two function h and g . If h(x) and g(x) satisfy the jump conditions (3), then
Karahan D., On a q-Boundary Value Problem
Mamedov K.R. with Discontinuity Conditions
(h g>\x=a+0 = (h g>|x=a-0, (10)
i. e. the function (h,g> is continuous on [0,T]. Applying formula (4), we obtain
DqWq(h,g)(x) = Dq (h(x)Dqg(x) - g(x)Dqh(x)) = h(qx)D2qg(x) - g(qx)D2qh(x). (11)
(12)
q" q v л /V"/ ^ q y "w-'-'qô v"/ ô w^q" v ..VI», Jqg(x) - g(qx)Dq
On the other hand,
/Vq_1x) = h( x) Dq2 g (q-1x) - g ( x)Dq2
DqW^q (h, g)(q-\) = h(x)Dq2g(q-1x) - g(x)Dqh(q A) = qh( x)[v( x) g ( x) - ng ( x)] - qg ( x)[v( x)h( x) - nh( x)] = 0.
As a result,
so, for x Ф 0.
1 Wq (h, g )(q x) - Wq (h, g )( x)
0 = DqWq (h, g )(q -1x) = qK'*™i f-, (13)
q 1 x(1 q)
Wq (h, g)(x) = Wq (h, g)(q-1x), (14)
"qV
i. e. the q -Wronskian Wq(h, g)(x) does not depend on x.
Let h(x,v) and X(x,v) be the solution of equation (1) under the boundary conditions
h(0,v) = %(T,v) = 1, D_ih(0,v) = g, D _iX(T,v) = -Г. (15)
q q
and under the jump conditions (3). Then
U (h) = v (X) = o. (16)
Since the q -Wronskian is independent of x, we can evaluate
Wq (h,X)(v) := Wq (v) = -V (h) = U (X). (17)
Wq (v) is called the characteristic function of L.
Lemma 2. The eigenvalues (v„ }„>0 of the boundary value problem L coincide with zeros of the characteristic function. The functions h( x,vn) and X( x,vn) are eigenfunctions and
X(x,v„) = bnh(x,vn), b Ф 0. (18)
Denote
"n J0'' v-'-^q
The set {vn ,an }n>0 is called the spectral date of L. Lemma 3. The following relation holds
ban = Wq (vn), (20)
where Wq (v) = DqWq (v) (respect to v).
The proof of Lemma 2 and Lemma 3 can be done similar to [12].
Theorem 1. The q -Sturm-Liouville eigenvalue problem (1)-(3) is self-adjoint on Cq2(0) П LLq (0,T).
Proof. We first prove that h(.), g(.) in j}q (0,T), we have the following q -Lagrange's identity
JT 2
0h ( x,n„ )dqx. (19)
J0
where
q
|T(l(h(x))g(x) - h(x)l(g(x)))dqx = [h,g](T) - lim [h,g](Tq"), (21)
[h, g](x) := h(x)D -1 g(x) - D -1h(x)g(x). (22)
Applying (9) with h(x) = Dqh(x) and g(x) = g(x), we obtain
< -1D -1 Dqh(x), g(x) > qq
= lim (Dqh)(Tqn-1)g(Tqn)-(Dqh)(Tq4)g(T)+ < Dqh,Dqg >
= lim D _xh(Tqn )g(Tqn) - D _h(T)g(T)+ < Dqh, Dqg >. (23)
Applying (8) with h(x) = h(x), g(x) = Dqg(x), we obtain
< Dqh, Dqg >= h(T ) Dqg (Tq"1) - lim h(Tqn ) Dqg (Tqn_> < h, - - D Dqg >
n®¥ q q
1
(24)
= h(T)D -! g(T) - limh(Tqn)D g(Tqn)+ < h, — D Dqg >.
q n®~ q q q
Therefore,
<-1D -1 Dqh(x),g(x) >= [h,g](T)- lim [h,g](Tqn)+ <h,-1D -1 Dqg >. (25)
qq n®~ q q
2
Lagrange's identity (21) results from (25) and the reality of v(x). Letting h(.),g(.) in CqX0) and assuming the that they satisfy (2)-(3), we obtain
D-1h(0) - gh(0) = 0, D -1 g(0) - gg(0) = 0. (26)
q q
The continuity of h(.),g(.) at zero implies that limn®¥[h,g](Tqn) = [h,g](0). Then (25) will be
<-1D -1 Dqh,g >=[h,g](T)-[h,g](0)+ <h,-1D -1 Dqg >.
q q q q q q
From (26), we have Similarly, from (2) we obtain
[h, g](0) = h(0)D _g(0) - D _h(0)g(0) = 0.
q q
[h,g](T) = h(T)D _1 g(T) _D _-h(T)g(T) = 0.
q q
Since v( x) is real-valued function, then
< l(h), g > = < _-D _1 Dqh(x) + v(x)h(x), g(x) > = < _-D _1 Dqh(x), g(x) > + < v(x)h(x), g(x) >
q q q q q q
= < h,_-D _iDqg > + < h(x),v(x)g(x) >=< h,l(g) >,
i.e. l is a self-adjoint operator.
Lemma 4. The eigenvalues {nn} of the boundary value problem (1)-(3) are real. Eigenfunctions
related to different eigenvalues are orthogonal in L2q (0,T ). All zeros of Wq(n) are simple, i. e. W(Vn)^0.
Proof. Let V0 be an eigenvalue with an eigenfunction h0( ). Then,
< l(h0),h0>=<h0, l(h))>. (27)
Since l(h0) = n0h0 , then
(V0 _ V0)JT I h0 (x)|2 dqx. (28)
Since h0( ) is non-trivial then V0 = V0 . So the eigenvalues are real.
Let V, m be two distinct eigenvalues with corresponding eigenfunctions h( ), X( ), respectively. Then,
T — (V_m)J0h( x)X( x)dqx = 0.
Since V ° m, then h( ) and X( ) are orthogonal.
Since h(x,Vn ) and X(x,V) are solutions of the boundary value problem (1)-(3), we obtain
Dqh(x,Vn),X(x,V)> = (Vn _ V)h(x,Vn)X(x,V). (29)
Karahan D., On a q-Boundary Value Problem
Mamedov K.R. with Discontinuity Conditions
Integrating equation (29) from 0 to T and using the conditions (2), we obtain
T, ^ (y„) - Wq (n)
J0 h( x,n„ ),X( x,v)dqx-
J0 ■ ~ ' n......q v -v
v n v
Since X(x,n„) = pnh(x,n„) as n ®nn , we obtain
Wq (nn )= Pn«n •
Thus it follows that Wq (nn) * 0.
3. Completeness of Eigenfunctions
Theorem 2. The system of eigenfunctions {h(x,nn)}n>0 of the boundary value problem (1)-(3) is
2
complete in Lq (0,T).
Proof. Consider the function
Y (x,n) =
Wq (n)
It is easy to verify that
X( x,v)^n(t,v)h(t )dqt + h( x,v)^ %(t,v)h(t)dqt
-1D 1D Y(x) + {—n + v(x)}Y(x) = h(x), xe [0,T], ne C, (30)
q q
Furthermore, taking into account (19), from (18) and (20) we get
Resw Y (x,v) = 1
Wq (vn )
X( x, vn) j0Xh(t, vn )h(t )dqt + h( x, vn) jxrX(t, vn )h(t )dqt
b T 1 T
Hn h( x,vn )J0h(t,vn )h(t)dqt = —h( xvn )\Qh(t,v„ )h(t)dqt. (31)
Wq (v)'- — * —n
J (
2
Let the function h(x) e Lq (0,T) be such that
rT
I r](t,nn)h(t)d t = 0, n = 0,1,2,... (32)
J0
Then in view of (31), Resn=n Y(x,n) = 0 and consequently for each fixed xe [0,T], the function
n
* 2
Y(x,n) is entire in n . Furthermore, for pe Gs = {p :\ p — pn |> d} and | p |> p where n = p , pn are the zeros of the function
W0 (p) = p(b1 sinpT — b2 sinp(2a — T)), where b2=^^, d is a fixed positive number, p is rather large, the inequality
q
and consequently the inequality
|Wq(v)|> Q | p\etT, pe Cs, p = s + it
Cx
\ Y(x,v) |<P, pe Gs,
\ P|
is obtained (see [12]). Using the maximum principle and Liouville's theorem we conclude that Y(x,n) ° 0 . From this and (30) it follows that h(x) = 0 a. e. on (0,T). Thus the theorem is proved.
4. The q -sampling theory
Theorem 3. Let h(x,n) and X(x,n) be the solutions of (1) selected as above. Then every function h of the form
h(n) = JT*x)h(x,n)dqx, ve LLq(0,T), (33)
can be written as the Lagrange-type sampling expansion
^ Wq (n)
h(n)=Yhivn —T ' (34)
Wq (nn )(n-nn )
where Wq (n) is the q -Wronskian of the functions h(x,n) and X(x,n) .
Proof. We multiply equation (1) with h(x,nn). Then we consider again equation (1), but replace n by nn and multiply this last equation by h(x,n). Subtracting the two results yields
(n-n )h( x,n)h( x,nn ) = D2qh(q -1x,n )h( x,n) - D2^(q _1x,n)h( x,nn). From the rule for the q -differentiation of product (4), we can write
(n-nn)h(x,n)h(x,nn) = Dq Dqh(q-1x,n)h(x,n) - Dqh(q_1x,n)h(x,nn)J If we apply a q -integration by means of (6) we obtain
(n-nn)iorh(x,n)h(x,nn)dqx = jToDq [Dqh(q~lx,nn)h(x,n)-Dqh(q_1x,n)h(x,nn)]dqx
= Dqh(q~lT ,n„ )h(T, n) - Dqh(q ~lT ,n)h(T ,n„)-(Dqh(q~l0,n„ )h(0,n) - Dqh(q_10,n)h(0,n„)) From the condition (2), we have
Dqh(q"10,nn)h(0,n)-Dqh(q"X0,n)h(0,nn) = D-h(0,n„)h(0,n)-D -h(0,n)h(0,n„)
y y q q
= D-1h(0,nn) -h(0,n„ ) = tf (h) = 0.
q
Multiply (17) by h(T ,nn) to obtain
Wq(n)h(T,nn) = -D -1h(T,n)h(T,nn)-Gh(T,n)h(T,nn)
q
= —D —h(T,V)h(TV) + D —h(T,v„)h(T,v).
q q
Then, we get
as a result,
(n - Vn) J0 h( xn)h( x, V„ )dqx = Wq (n)h(T ,v„). f, ,, ,_Wq (n)h(T Vn)
J0 x,n)h(x,Vn)dqX :
nq
Jo y V — V
n
and taking the limit as n ®nn gives
JOT I h( x,nn )|2 dqx = Wq (nn )h(T ,n„ ).
We can therefore apply Kramer's lemma (see [14]) and write an integral transform of the form (33)
as
^ Wq (n)
h(n) = Vh(nn)--q—-. (35)
n=o W (nn )(n - n n) ' '
References
1. Jackson F.H. q-Difference Equations. Am. J. Math., 1910, Vol. 32, no. 4, pp. 305-314.
2. Annaby M.H., Mansour Z.S. q-Difference Equations. In: q -Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin, Heidelberg, 2012. DOI: 10.1007/978-3-642-30898-7_2
3. Annaby M.H., Mansour Z. S. Basic Sturm-Liouville problems. J. Phys. A: Math. Gen, 2005, Vol. 38, pp. 3775-3797.
4. Chung K., Chung W., Nam S., Kang, H. New q-Derivative and q-Logarithm. Int. J. Theor. Phys., 1994, Vol. 33, Iss. 10, pp. 2019-2029. DOI: 10.1007/BF00675167
5. Floreanini R., LeTourneux J., Vinet L. More on the q-Oscillator Algebra and q-Orthogonal Polynomials. Journal of Physics A: Mathematical and General, Vol. 28, no. 10, pp. L287-L293. DOI: 10.1088/0305-4470/28/10/002
Karahan D., Mamedov K.R.
On a q-Boundary Value Problem with Discontinuity Conditions
6. Annaby M.H. q-Type Sampling Theorems. Result. Math., 2003, Vol. 44, Iss. 3, pp. 214-225. DOI: 10.1007/BF03322983
7. Abrue L.D. A q-Sampling Theorem Related to the q-Hankel Transform. Proc. Am. Math. Soc., 2005, Vol. 133, no. 4, pp. 1197-1203. DOI: 10.2307/4097680
8. Abreu L.D. Sampling theory associated with q-difference equations of the Sturm-Liouville type. J Phys. A: Math. Gen., 2005, Vol. 38(48), pp. 10311-10319. DOI: 10.1088/0305-4470/38/48/005
9. Karahan D., Mamedov Kh.R. Sampling Theory Associated with q-Sturm-Liouville Operator with Discontinuity Conditions. Journal of Contemporary Applied Mathematics, 2020, Vol. 10, no. 2, pp.40-48.
10. Allahverdiev B.P., Tuna H. Qualitative Spectral Analysis of Singular q-Sturm-Liouville Operators. Bulletin of the Malaysian Mathematical Sciences Society, 2020, Vol. 43, Iss. 2, pp. 1391-1402. DOI: 10.1007/s40840-019-00747-3
11. Allahverdiev B.P., Tuna H. Eigenfunction Expansion in the Singular Case for q-Sturm-Liouville Operators. CJMS, 2019, Vol. 8, Iss. 2, pp. 91-102. DOI: 10.22080/CJMS.2018.13943.1339
12. Yurko, V. Integral Transforms Connected with Discontinuous Boundary Value Problems. Integral Transforms and Special Functions, 2000, Vol. 10, Iss. 2, pp. 141-164. DOI: 10.1080/10652460008819282
13. Gasper G., Rahman M. Basic Hypergeometric Series. Cambridge; New York: Cambridge University Press, 1990, 287 p.
14. Kramer H.P. A Generalized Sampling Theorem. Journal of Mathematics and Physics, 1959, Vol. 38, Iss.1-4, pp. 68-72. DOI:10.1002/SAPM195938168
Received October 13, 2021
Information about the authors
Karahan Done, Mathematics Department, Science and Letter Faculty, Harran University, Sanlurfa, Turkey, e-mail: [email protected]
Mamedov Khanlar Rashid, Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey, ORCID iD: https://orcid.org/0000-0002-3283-9535, e-mail: [email protected]
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2021, vol. 13, no. 4, pp. 5-12
УДК 515.162.8 DOI: 10.14529/mmph210401
0 q-ГРАНИЧНОЙ ЗАДАЧЕ С РАЗРЫВНЫМИ УСЛОВИЯМИ
Д. Карахан1, K.P. Мамедов2
1 Университет Харран, Шанлыурфа, Турция E-mail: [email protected]
2 Мерсинский университет, Мерсин, Турция E-mail: [email protected]
Изучается q-аналог граничной задачи Штурма-Лиувилля на конечном интервале, имеющем разрыв во внутренней точке. Доказывается, что q-граничная задача Штурма-Лиувилля является само-сопряженной в модифицированном Гильбертовом пространстве. Исследуются спектральные свойства собственных значений и собственных функций q-граничной задача Штурма-Лиувилля. Показано, что собственные функции q-граничной задача Штурма-Лиувилля представимы в виде полной системы. Наконец, доказывается теорема о дискретном представлении для интегральных преобразований, чьи ядра являются базисными функциями, а интеграл имеет тип Джексона.
Ключевые слова: q-оператор Штурма-Лиувилля; самосопряженный оператор; полнота собственных функций; теорема о дискретном представлении.
Литература
1. Jackson, F.H. q-Difference Equations / F.H. Jackson // Am. J. Math. - 1910. - Vol. 32, no. 4. -P. 305-314.
2. Annaby, M.H. q-Difference Equations / M.H. Annaby, Z.S. Mansour // q -Fractional Calculus and Equations. Lecture Notes in Mathematics. - Springer, Berlin, Heidelberg, 2012. - Vol 2056.
3. Annaby, M.H. Basic Sturm-Liouville problems / M.H. Annaby, Z.S. Mansour // J. Phys. A: Math. Gen. - 2005. - Vol. 38. - P. 3775-3797.
4. New q-Derivative and q-Logarithm / K. Chung, W. Chung, S. Nam, H. Kang // Int. J. Theor. Phys. - 1994. - Vol. 33, Iss. 10. - P. 2019-2029.
5. Floreanini, R. More on the q-Oscillator Algebra and q-Orthogonal Polynomials / R. Floreanini, J. LeTourneux, L. Vinet // Journal of Physics A: Mathematical and General. - 1995. - Vol. 28, no. 10. -P. L287-L293.
6. Annaby, M.H. q-Type Sampling Theorems / M.H. Annaby // Result. Math. - 2003. - Vol. 44, Iss. 3. - P. 214-225.
7. Abrue, L.D. A q-Sampling Theorem Related to the q-Hankel Transform / L.D. Abrue // Proc. Am. Math. Soc. - 2005. - Vol. 133. - P. 1197-1203.
8. Abreu, L.D. Sampling theory associated with q-difference equations of the Sturm-Liouville type / L.D. Abreu // J, Phys. A: Math. Gen. - 2005. - Vol. 38(48). - P. 10311-10319.
9. Karahan, D. Sampling Theory Associated with q-Sturm-Liouville Operator with Discontinuity Conditions / D. Karahan, Kh.R. Mamedov // Journal of Contemporary Applied Mathematics. - 2020. -Vol. 10, no. 2. - P. 40-48.
10. Allahverdiev, B.P. Qualitative Spectral Analysis of Singular q-Sturm-Liouville Operators / B.P. Allahverdiev, H. Tuna // Bulletin of the Malaysian Mathematical Sciences Society. - 2020. -Vol. 43, Iss. 2. - P. 1391-1402.
11. Allahverdiev, B.P. Eigenfunction Expansion in the Singular Case for q-Sturm-Liouville Operators / B.P. Allahverdiev, H. Tuna // Caspian Journal of Mathematical Sciences (CJMS). - 2019. - Vol. 8, Iss. 2. - P. 91-102.
12. Yurko, V. Integral Transforms Connected with Discontinuous Boundary Value Problems / V. Yurko // Integral Transforms and Special Functions. - 2000. - Vol. 10, Iss. 2. - P. 141-164.
13. Gasper, G. Basic Hypergeometric Series / G. Gasper, M. Rahman. - Cambridge; New York: Cambridge University Press, 1990. - 287 p.
14. Kramer, H.P. A Generalized Sampling Theorem / H.P. Kramer // Journal of Mathematics and Physics. - 1959. - Vol. 38, Iss.1-4. - P. 68-72.
Поступила в редакцию 13 октября 2021 г.
Сведения об авторах
Карахан Доне, кафедра математики, факультет естественных наук и литературы, Университет Харран, Шанлыурфа, Турция, e-mail: [email protected]
Мамедов Ханлар Рашид, кафедра математики, факультет естественных наук и литературы, Мерсинский университет, Мерсин, Турция, ORCID iD: https://orcid.org/0000-0002-3283-9535, e-mail: [email protected]
