Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition
We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish the isometry of its wave operators and determine the initial domain...
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2016
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507793955487744 |
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| author | Acikgoz, M. Araci, S. Şen, E. Ацікгоц, М. Аракі, С. Сен, Е. |
| author_facet | Acikgoz, M. Araci, S. Şen, E. Ацікгоц, М. Аракі, С. Сен, Е. |
| author_sort | Acikgoz, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:14Z |
| description | We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form
by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish the isometry
of its wave operators and determine the initial domains of these operators. |
| first_indexed | 2026-03-24T02:14:58Z |
| format | Article |
| fulltext |
UDC 517.9
E. Şen (Namik Kemal Univ., Tekirdağ, Turkey),
M. Acikgoz (Gaziantep Univ., Turkey),
S. Araci (Hasan Kalyoncu Univ., Gaziantep, Turkey)
SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR
WITH RETARDED ARGUMENT WHICH CONTAINS
A SPECTRAL PARAMETER IN BOUNDARY CONDITION
СПЕКТРАЛЬНА ЗАДАЧА ДЛЯ OПEРАТОРА ШТУРМА – ЛIУВIЛЛЯ
З АРГУМЕНТОМ, ЩО ЗАПIЗНЮЄТЬСЯ,
ТА СПЕКТРАЛЬНИМ ПАРАМЕТРОМ У ГРАНИЧНIЙ УМОВI
We consider a discontinuous Sturm – Liouville problem with retarded argument that contains a spectral parameter in the
boundary condition. First, we investigate the simplicity of eigenvalues and then prove the existence theorem. As a result,
we obtain the asymptotic formulas for eigenvalues and eigenfunctions.
Розглянуто розривну задачу Штурма – Лiувiлля з аргументом, що запiзнюється, та спектральним параметром у
граничнiй умовi. Спочатку ми вивчаємо простоту власних значень, а потiм доводимо теорему про iснування. Як
результат, отримано асимптотичнi формули для власних значень i власних функцiй.
1. Preliminaries. Boundary-value problems for differential equations of the second order with
retarded argument were studied in [1 – 9], and various physical applications of such problems can be
found in [2]. The asymptotic formulas for the eigenvalues and eigenfunctions of boundary problem
of Sturm – Liouville type for second order differential equation with retarded argument were obtained
in [1, 2, 5 – 9]. The asymptotic formulas for the eigenvalues and eigenfunctions of classical Sturm –
Liouville problem with the spectral parameter in the boundary condition were obtained in [10 – 13].
In this paper we study the eigenvalues and eigenfunctions of discontinuous boundary-value
problem with retarded argument and a spectral parameter in the boundary condition. That is, we
consider the boundary-value problem for the differential equation
p(x)y\prime \prime (x) + q(x)y(x - \Delta (x)) + \lambda y(x) = 0 (1.1)
on [0, r1) \cup (r1, r2) \cup (r2, \pi ] , with boundary conditions
y\prime (0) = 0, (1.2)
y\prime (\pi ) + \lambda y(\pi ) = 0, (1.3)
and jump conditions
\gamma 1y(r1 - 0) = \delta 1y(r1 + 0), (1.4)
\gamma 2y
\prime (r1 - 0) = \delta 2y
\prime (r1 + 0), (1.5)
\theta 1y(r2 - 0) = \eta 1y(r2 + 0), (1.6)
\theta 2y
\prime (r2 - 0) = \eta 2y
\prime (r2 + 0), (1.7)
c\bigcirc E. ŞEN, M. ACIKGOZ, S. ARACI, 2016
1102 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1103
where p(x) = p21, if x \in [0, r1), p(x) = p22, if x \in (r1, r2), and p(x) = p23, if x \in (r2, \pi ] , the
real-valued function q(x) is continuous in [0, r1) \cup (r1, r2) \cup (r2, \pi ]; and has finite limits q(r1 \pm
\pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r1\pm 0 q(x), q(r2\pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r2\pm 0 q(x); the real valued function \Delta (x) \geq 0 continuous
in [0, r1) \cup (r1, r2) \cup (r2, \pi ] and has finite limits \Delta (r1 \pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r1\pm 0\Delta (x), \Delta (r2 \pm 0) =
= \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r2\pm 0\Delta (x), x - \Delta (x) \geq 0, if x \in
\Bigl[
0,
\pi
2
\Bigr)
;x - \Delta (x) \geq \pi
2
, if x \in
\Bigl( \pi
2
, \pi
\Bigr]
; \lambda is a real spectral
parameter; p1, p2, p3, \gamma 1, \gamma 2, \delta 1, \delta 2, \theta 1, \theta 2, \eta 1, \eta 2 are arbitrary real numbers; | \gamma i| + | \delta i| \not = 0 and
| \theta i| + | \eta i| \not = 0 for i = 1, 2. Also \gamma 1\delta 2p1 = \gamma 2\delta 1p2 and \theta 1\eta 2p2 = \theta 2\eta 1p3 hold.
It must be noted that some problems with jump conditions which arise in mechanics (thermal
condition problem for a thin laminated plate) were studied in [14].
Let w1(x, \lambda ) be a solution of Eq. (1.1) on [0, r1] , satisfying the initial conditions
w1 (0, \lambda ) = 1, w\prime
1 (0, \lambda ) = 0. (1.8)
Conditions (1.8) define a unique solution of Eq. (1.1) on [0, r1] [2, p. 12].
After defining above solution we shall define the solution w2 (x, \lambda ) of Eq. (1.1) on [r1, r2] by
means of the solution w1 (x, \lambda ) by the initial conditions
w2 (r1, \lambda ) = \gamma 1\delta
- 1
1 w1 (r1, \lambda ) , w\prime
2(r1, \lambda ) = \gamma 2\delta
- 1
2 \omega \prime
1(r1, \lambda ). (1.9)
Conditions (1.9) are defined as a unique solution of Eq. (1.1) on [r1, r2] .
After defining above solution we shall define the solution w3 (x, \lambda ) of Eq. (1.1) on [r2, \pi ] by
means of the solution w2 (x, \lambda ) by the initial conditions
w3 (r2, \lambda ) = \theta 1\eta
- 1
1 w2 (r2, \lambda ) , w\prime
3(r2, \lambda ) = \theta 2\eta
- 1
2 \omega \prime
2(r2, \lambda ). (1.10)
Conditions (1.10) are defined as a unique solution of Eq. (1.1) on [r2, \pi ] .
Consequently, the function w (x, \lambda ) is defined on [0, r1) \cup (r1, r2) \cup (r2, \pi ] by the equality
w(x, \lambda ) =
\left\{
w1(x, \lambda ), x \in [0, r1),
w2(x, \lambda ), x \in (r1, r2),
w3(x, \lambda ), x \in (r2, \pi ],
is a such solution of Eq. (1.1) on [0, r1) \cup (r1, r2) \cup (r2, \pi ]; which satisfies one of the boundary
conditions and both transmission conditions.
Lemma 1.1. Let w (x, \lambda ) be a solution of Eq. (1.1) and \lambda > 0. Then the following integral
equations hold:
w1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
x - 1
s
x\int
0
q(\tau )
p1
\mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(x - \tau )w1 (\tau - \Delta (\tau ), \lambda ) d\tau , s =
\surd
\lambda , \lambda > 0, (1.11)
w2(x, \lambda ) =
\gamma 1
\delta 1
w1 (r1, \lambda ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(x - r1) +
\gamma 2p2w
\prime
1 (r1, \lambda )
s\delta 2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(x - r1) -
- 1
s
x\int
r1
q(\tau )
p2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(x - \tau )w2 (\tau - \Delta (\tau ) , \lambda ) d\tau , s =
\surd
\lambda , \lambda > 0, (1.12)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1104 E. ŞEN, M. ACIKGOZ, S. ARACI
w3(x, \lambda ) =
\theta 1
\eta 1
w2 (r2, \lambda ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(x - r2) +
\theta 2p3w
\prime
2 (r2, \lambda )
s\eta 2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(x - r2) -
- 1
s
x\int
r2
q(\tau )
p3
\mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(x - \tau )w3 (\tau - \Delta (\tau ) , \lambda ) d\tau , s =
\surd
\lambda , \lambda > 0. (1.13)
Proof. To prove this, it is enough to substitute
- s2
p21
w1(\tau , \lambda ) - w\prime \prime
1(\tau , \lambda ), - s2
p22
w2(\tau , \lambda ) - w\prime \prime
2(\tau , \lambda )
and
- s2
p23
w3(\tau , \lambda ) - w\prime \prime
3(\tau , \lambda )
instead of
- q(\tau )
p21
w1(\tau - \Delta (\tau ), \lambda ), - q(\tau )
p22
w2(\tau - \Delta (\tau ), \lambda ) and - q(\tau )
p23
w3(\tau - \Delta (\tau ), \lambda )
in the integrals in (1.11), (1.12) and (1.13) respectively and integrate by parts twice.
Theorem 1.1. Problem (1.1) – (1.7) can have only simple eigenvalues.
Proof. Let \widetilde \lambda be an eigenvalue of problem (1.1) – (1.7) and
\widetilde u(x, \widetilde \lambda ) =
\left\{
\widetilde u1(x, \widetilde \lambda ), x \in [0, r1),
\widetilde u2(x, \widetilde \lambda ), x \in (r1, r2),
\widetilde u3(x, \widetilde \lambda ), x \in (r2, \pi ],
be a corresponding eigenfunction. Then from (1.2) and (1.8) the determinant
W
\Bigl[ \widetilde u1(0, \widetilde \lambda ), w1(0, \widetilde \lambda )\Bigr] =
\bigm| \bigm| \bigm| \bigm| \bigm| \widetilde u1(0, \widetilde \lambda ) 1
\widetilde u\prime 1(0, \widetilde \lambda ) 0
\bigm| \bigm| \bigm| \bigm| \bigm| = 0,
and by Theorem 2.2.2 in [2] the functions \widetilde u1(x, \widetilde \lambda ) and w1(x, \widetilde \lambda ) are linearly dependent on [0, r1].
We can also prove that the functions \widetilde u2(x, \widetilde \lambda ) and w2(x, \widetilde \lambda ) are linearly dependent on [r1, r2] and the
functions \widetilde u3(x, \widetilde \lambda ) and w3(x, \widetilde \lambda ) are linearly dependent on [r2, \pi ]. Hence
\widetilde ui(x, \widetilde \lambda ) = Kiwi(x, \widetilde \lambda ), i = 1, 2, 3, (1.14)
for some K1 \not = 0, K2 \not = 0 and K3 \not = 0. We first show that K2 = K3 . Suppose that K2 \not = K3 .
From equalities (1.6) and (1.14), we have
\theta 1\widetilde u(r2 - 0, \widetilde \lambda ) - \eta 1\widetilde u(r2 + 0, \widetilde \lambda ) = \theta 1\widetilde u2(r2, \widetilde \lambda ) - \eta 1\widetilde u3(r2, \widetilde \lambda ) =
= \theta 1K2w2(r2, \widetilde \lambda ) - \eta 1K3w3(r2, \widetilde \lambda ) =
= \theta 1K2\eta 1\theta
- 1
1 w3(r2, \widetilde \lambda ) - \eta 1K3w3(r2, \widetilde \lambda ) =
= \eta 1 (K2 - K3)w3(r2, \widetilde \lambda ) = 0.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1105
Since \eta 1 (K2 - K3) \not = 0, we obtain
w3
\Bigl(
r2, \widetilde \lambda \Bigr) = 0. (1.15)
By the same procedure arising from (1.7), we see that
w\prime
3
\Bigl(
r2, \widetilde \lambda \Bigr) = 0. (1.16)
From the fact that w3(x, \widetilde \lambda ) is a solution of the differential Eq. (1.1) on [r2, \pi ] and satisfies the initial
conditions (1.15) and (1.16), w3(x, \widetilde \lambda ) = 0 identically on [r2, \pi ] (cf. [2, p. 12], Theorem 1.2.1).
By using this procedure, we may also find
w1
\Bigl(
r1, \widetilde \lambda \Bigr) = w\prime
1
\Bigl(
r1, \widetilde \lambda \Bigr) = w2
\Bigl(
r2, \widetilde \lambda \Bigr) = w\prime
2
\Bigl(
r2, \widetilde \lambda \Bigr) = 0.
Thus, we have w2(x, \widetilde \lambda ) = 0 and w1(x, \widetilde \lambda ) = 0 identically on [0, r1) \cup (r1, r2) \cup (r2, \pi ]. But this
contradicts (1.8), thus completing the proof.
2. An existence theorem. The function w(x, \lambda ) defined in Section 1 is a nontrivial solution
of Eq. (1.1) satisfying conditions (1.2), (1.4), (1.5) and (1.6). Putting w(x, \lambda ) into (1.3), we get the
characteristic equation
F (\lambda ) \equiv w\prime (\pi , \lambda ) + \lambda w(\pi , \lambda ) = 0. (2.1)
By Theorem 1.1, the set of eigenvalues of boundary-value problem (1.1) – (1.7) coincides with
the set of real roots of Eq. (2.1). Let
q1 =
1
p1
r1\int
0
| q(\tau )| d\tau , q2 =
1
p2
r2\int
r1
| q(\tau )| d\tau and q3 =
1
p3
\pi \int
r2
| q(\tau )| d\tau .
Lemma 2.1. (1) Let \lambda \geq 4q21 . Then for the solution w1 (x, \lambda ) of Eq. (1.11), the following
inequality holds:
| w1 (x, \lambda )| \leq 2, x \in [0, r1] . (2.2)
(2) Let \lambda \geq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
4q21, 4q
2
2
\bigr\}
. Then for the solution w2 (x, \lambda ) of Eq. (1.12), the following
inequality holds:
| w2 (x, \lambda )| \leq 4
\biggl( \bigm| \bigm| \bigm| \bigm| \gamma 1\delta 1
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| p2\gamma 2p1\delta 2
\bigm| \bigm| \bigm| \bigm| \biggr) , x \in [r1, r2] . (2.3)
(3) Let \lambda \geq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
4q21, 4q
2
2, 4q
2
3
\bigr\}
. Then for the solution w2 (x, \lambda ) of Eq. (1.13), the following
inequality holds:
| w3 (x, \lambda )| \leq
8\theta 1p2 + 4\theta 2p3\eta 1
\eta 1p2\eta 2
\biggl( \bigm| \bigm| \bigm| \bigm| \gamma 1\delta 1
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| p2\gamma 2p1\delta 2
\bigm| \bigm| \bigm| \bigm| \biggr) +
\theta 2p3
\eta 2
\bigm| \bigm| \bigm| \bigm| 4\gamma 1\delta 2q1 + \gamma 2p2\delta 1
2p2\delta 1\delta 2q1
\bigm| \bigm| \bigm| \bigm| , x \in [r2, \pi ]. (2.4)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1106 E. ŞEN, M. ACIKGOZ, S. ARACI
Proof. Let B1\lambda = \mathrm{m}\mathrm{a}\mathrm{x}[0,r1] | w1 (x, \lambda )| . Then from (1.11), for any \lambda > 0, the following
inequality holds:
B1\lambda \leq 1 +
1
s
B1\lambda q1.
If s \geq 2q1 we get (2.2). Differentiating (1.11) with respect to x, we have
w\prime
1(x, \lambda ) = - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
s
p1
x - 1
p21
x\int
0
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(x - \tau )w1(\tau - \Delta (\tau ), \lambda ) d\tau . (2.5)
Taking into account (2.5) and (2.2), for s \geq 2q1, the following inequality holds:
| w\prime
1(x, \lambda )|
s
\leq 2
p1
. (2.6)
Let B2\lambda = \mathrm{m}\mathrm{a}\mathrm{x}[r1,r2] | w2 (x, \lambda )| . Then from (1.12), (2.2) and (2.6), for s \geq 2q1, the following
inequality holds:
B2\lambda \leq 4
\biggl\{ \bigm| \bigm| \bigm| \bigm| \gamma 1\delta 1
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| p2\gamma 2p1\delta 2
\bigm| \bigm| \bigm| \bigm| \biggr\} .
Hence if \lambda \geq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
4q21, 4q
2
2
\bigr\}
we get (2.3).
Differentiating (1.12) with respect to x, we obtain
w\prime
2(x, \lambda ) = - s\gamma 1
p2\delta 1
w1 (r1, \lambda ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(x - r1) +
\gamma 2w
\prime
1 (r1, \lambda )
\delta 2
\mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(x - r1) -
- 1
p22
x\int
r1
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(x - \tau )w2(\tau - \Delta (\tau ), \lambda ) d\tau . (2.7)
By virtue of (2.7) and (2.3), for s \geq 2q2, the following inequality holds true:
| w\prime
2(x, \lambda )|
s
\leq 2\gamma 1
p2\delta 1
+
\gamma 2
2\delta 2q1
+
2
p2
\biggl\{ \bigm| \bigm| \bigm| \bigm| \gamma 1\delta 1
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| p2\gamma 2p1\delta 2
\bigm| \bigm| \bigm| \bigm| \biggr\} . (2.8)
Let B3\lambda = \mathrm{m}\mathrm{a}\mathrm{x}[r2,\pi ] | w3 (x, \lambda )| . Then from (1.13), (2.2), (2.3) and (2.8), for s \geq 2q3, the following
inequality holds:
B3\lambda \leq 8\theta 1p2 + 4\theta 2p3\eta 1
\eta 1p2\eta 2
\biggl( \bigm| \bigm| \bigm| \bigm| \gamma 1\delta 1
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| p2\gamma 2p1\delta 2
\bigm| \bigm| \bigm| \bigm| \biggr) +
\theta 2p3
\eta 2
\bigm| \bigm| \bigm| \bigm| 4\gamma 1\delta 2q1 + \gamma 2p2\delta 1
2p2\delta 1\delta 2q1
\bigm| \bigm| \bigm| \bigm| .
Hence, if \lambda \geq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
4q21, 4q
2
2, 4q
2
3
\bigr\}
, then we arrive at Eq. (2.4).
Theorem 2.1. Problem (1.1) – (1.7) has an infinite set of positive eigenvalues.
Proof. Differentiating (1.13) with respect to x, we have
w\prime
3(x, \lambda ) = - s\theta 1
p3\eta 1
w2 (r2, \lambda ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(x - r2) +
\theta 2w
\prime
2 (r2, \lambda )
\eta 2
\mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(x - r2) -
- 1
p23
x\int
r2
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(x - \tau )w3(\tau - \Delta (\tau ), \lambda )d\tau . (2.9)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1107
From (1.11) – (1.13), (2.1), (2.5), (2.7) and (2.9), we get
- s\theta 1
p3\eta 1
\left[ \gamma 1
\delta 1
\left( \mathrm{c}\mathrm{o}\mathrm{s}
sr1
p1
- 1
sp1
r1\int
0
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - r1)+
+
\gamma 2p2
s\delta 2
\left( - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
sr1
p1
- 1
p21
r1\int
0
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - r1) -
- 1
sp2
r2\int
r1
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - \tau )w2(\tau - \Delta (\tau ), \lambda )d\tau
\right] \mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(\pi - r2)+
+
\theta 2
\eta 2
\left[ - s\gamma 1
p2\delta 1
\left( \mathrm{c}\mathrm{o}\mathrm{s}
sr1
p1
- 1
sp1
r1\int
0
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - r1)+
+
\gamma 2
\delta 2
\left( - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
sr1
p1
- 1
p21
r1\int
0
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - r1) -
- 1
p22
r2\int
r1
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - \tau )w2(\tau - \Delta (\tau ), \lambda )d\tau
\right] \mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(\pi - r2) -
- 1
p23
\pi \int
r2
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(\pi - \tau )w3 (\tau - \Delta (\tau ) , \lambda ) d\tau +
+\lambda
\left\{ \theta 1
\eta 1
\left[ \gamma 1
\delta 1
\left( \mathrm{c}\mathrm{o}\mathrm{s}
sr1
p1
- 1
sp1
r1\int
0
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - r1)+
+
\gamma 2p2
s\delta 2
\left( - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
sr1
p1
- 1
p21
r1\int
0
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - r1) -
- 1
sp2
r2\int
r1
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - \tau )w2(\tau - \Delta (\tau ), \lambda )d\tau
\right] \mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(\pi - r2)+
+
\theta 2p3
s\eta 2
\left[ - s\gamma 1
p2\delta 1
\left( \mathrm{c}\mathrm{o}\mathrm{s}
sr1
p1
- 1
sp1
r1\int
0
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(r2 - r1)+
+
\gamma 2
\delta 2
\left( - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
sr1
p1
- 1
p21
r1\int
0
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(r1 - \tau )w1(\tau - \Delta (\tau ), \lambda )d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - r1) -
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1108 E. ŞEN, M. ACIKGOZ, S. ARACI
- 1
p22
r2\int
r1
q(\tau ) \mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(r2 - \tau )w2(\tau - \Delta (\tau ), \lambda )d\tau
\right] \mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(\pi - r2) -
- 1
sp3
\pi \int
r2
q(\tau ) \mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(\pi - \tau )w3 (\tau - \Delta (\tau ) , \lambda ) d\tau
\right\} = 0. (2.10)
Let \lambda be sufficiently large. Then, by (2.2) – (2.4), Eq. (2.10) may be rewritten in the form
s \mathrm{c}\mathrm{o}\mathrm{s} s
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
+O(1) = 0. (2.11)
Obviously, for large s Eq. (2.11) has an infinite set of roots. Thus, we arrive at the desired result.
3. Asymptotic formulas for eigenvalues and eigenfunctions. Now we begin to study asymp-
totic properties of eigenvalues and eigenfunctions. In the following we shall assume that s is
sufficiently large. From (1.11) and (2.2), we get
w1(x, \lambda ) = O(1). (3.1)
From expressions of (1.12) and (2.3), we see that
w2(x, \lambda ) = O(1). (3.2)
By virtue of (1.13) and (2.4), we procure the following equation:
w3(x, \lambda ) = O(1). (3.3)
The existence and continuity of the derivatives w\prime
1s(x, \lambda ) for 0 \leq x \leq r1, | \lambda | < \infty , w\prime
2s(x, \lambda )
for r1 \leq x \leq r2, | \lambda | < \infty and w\prime
3s(x, \lambda ) for r2 \leq x \leq \pi , | \lambda | < \infty follows from Theorem 1.4.1
in [2]:
w\prime
1s(x, \lambda ) = O(1), x \in [0, r1],
w\prime
2s(x, \lambda ) = O(1), x \in [r1, r2],
w\prime
3s(x, \lambda ) = O(1), x \in [r2, \pi ].
(3.4)
Theorem 3.1. Let n be a natural number. For each sufficiently large n, there is exactly one
eigenvalue of problem (1.1) – (1.7) near
(n+ 1/2)2 \pi 2\bigl(
r1/p1 + (r2 - r1)/p2 + (\pi - r2)/p3
\bigr) 2 .
Proof. We consider the expression which is denoted by O(1) in Eq. (2.11). If formulas (3.1) –
(3.4) are taken into consideration, it can be shown by differentiation with respect to s that for large s
this expression has bounded derivative. We shall show that, for large n, only one root of (2.11) lies
near to each
(n+ 1/2)2 \pi 2\bigl(
r1/p1 + (r2 - r1)/p2 + (\pi - r2)/p3
\bigr) 2 . Let us consider the function
\phi (s) = s \mathrm{c}\mathrm{o}\mathrm{s} s
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
+O(1).
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SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1109
Its derivative, which has the form
\phi \prime (s) = \mathrm{c}\mathrm{o}\mathrm{s} s
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
-
- s
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
\mathrm{s}\mathrm{i}\mathrm{n} s
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
+O(1),
does not vanish for s close to sufficiently large n. Thus our assertion follows by Rolle’s theorem.
Let n be sufficiently large. In what follows we shall denote by \lambda n = s2n the eigenvalue of
problem (1.1) – (1.7) situated near
(n+ 1/2)2 \pi 2\bigl(
r1/p1 + (r2 - r1)/p2 + (\pi - r2)/p3
\bigr) 2 . We set
sn =
\biggl(
n+
1
2
\biggr)
\pi \biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr) + \delta n.
From (2.11) \delta n = O
\biggl(
1
n
\biggr)
. Consequently, we procure
sn =
\biggl(
n+
1
2
\biggr)
\pi \biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr) +O
\biggl(
1
n
\biggr)
. (3.5)
Formula (3.5) make it possible to obtain asymptotic expressions for eigenfunction of problem
(1.1) – (1.7). By (1.11), (2.5) and (3.1), we have
w1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
sx
p1
+O
\biggl(
1
s
\biggr)
, (3.6)
w\prime
1(x, \lambda ) = - s
p1
\mathrm{s}\mathrm{i}\mathrm{n}
sx
p1
+O (1) . (3.7)
By means of (1.12), (3.2), (3.6) and (3.7), we acquire
w2(x, \lambda ) =
\gamma 1
\delta 1
\mathrm{c}\mathrm{o}\mathrm{s}
s
p2
\biggl(
r1 (p2 - p1)
p1
+ x
\biggr)
+O
\biggl(
1
s
\biggr)
, (3.8)
w\prime
2(x, \lambda ) = - s\gamma 1
\delta 1p2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p2
\biggl(
r1 (p2 - p1)
p1
+ x
\biggr)
+O (1) . (3.9)
In view of (1.13), (3.3), (3.8) and (3.9), we attain the following:
w3(x, \lambda ) =
\theta 1\gamma 1
\eta 1\delta 1
\mathrm{c}\mathrm{o}\mathrm{s}
s
p3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr)
+O
\biggl(
1
s
\biggr)
. (3.10)
Putting (3.5) into (3.6), (3.8) and (3.10), we readily derive
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1110 E. ŞEN, M. ACIKGOZ, S. ARACI
u1n(x) = \mathrm{c}\mathrm{o}\mathrm{s}
\left(
\biggl(
n+
1
2
\biggr)
\pi x
p1
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr)
\right) +O
\biggl(
1
n
\biggr)
,
u2n(x) =
\gamma 1
\delta 1
\mathrm{c}\mathrm{o}\mathrm{s}
\left(
\biggl(
n+
1
2
\biggr)
\pi
p2
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr) \biggl(
r1 (p2 - p1)
p1
+ x
\biggr) \right) +O
\biggl(
1
n
\biggr)
,
u3n(x) =
\theta 1\gamma 1
\eta 1\delta 1
\times
\times \mathrm{c}\mathrm{o}\mathrm{s}
\left(
\biggl(
n+
1
2
\biggr)
\pi
p3
\biggl(
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
\biggr) \biggl(
p3(r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr) \right) +O
\biggl(
1
n
\biggr)
.
Hence the eigenfunctions un(x) have the following asymptotic representation:
un(x) =
\left\{
u1n(x) = w1 (x, \lambda n) , x \in [0, r1),
u2n(x) = w2 (x, \lambda n) , x \in (r1, r2) ,
u3n(x) = w3 (x, \lambda n) , x \in (r2, \pi ].
Under some additional conditions the more exact asymptotic formulas which depend upon the
retardation may be obtained. Let us assume that the following conditions are fulfilled:
(a) the derivatives q\prime (x) and \Delta \prime \prime (x) exist and are bounded in [0, r1)\cup (r1, r2)\cup (r2, \pi ] and have
finite limits
q\prime (r1 \pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow r1\pm 0
q\prime (x), q\prime (r2 \pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow r2\pm 0
q\prime (x), \Delta \prime \prime (r1 \pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow r1\pm 0
\Delta \prime \prime (x)
and
\Delta \prime \prime (r2 \pm 0) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow r2\pm 0
\Delta \prime \prime (x),
respectively;
(b) \Delta \prime (x) \leq 1 in [0, r1)\cup (r1, r2)\cup (r2, \pi ], \Delta (0) = 0, \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r1+0\Delta (x) = 0 and \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow r2+0\Delta (x) =
= 0.
By using (b), we have
x - \Delta (x) \geq 0, if x \in [0, r1),
x - \Delta (x) \geq r1, if x \in (r1, r2),
x - \Delta (x) \geq r2, if x \in (r2, \pi ].
(3.11)
From (3.6), (3.8), (3.10) and (3.11), we have
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SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1111
w1 (\tau - \Delta (\tau ), \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
s (\tau - \Delta (\tau ))
p1
+O
\biggl(
1
s
\biggr)
,
w2 (\tau - \Delta (\tau ), \lambda ) =
\gamma 1
\delta 1
\gamma 1
\delta 1
\mathrm{c}\mathrm{o}\mathrm{s}
s
p2
\biggl(
r1 (p2 - p1)
p1
+ \tau - \Delta (\tau )
\biggr)
+O
\biggl(
1
s
\biggr)
,
w3 (\tau - \Delta (\tau ), \lambda ) =
\theta 1\gamma 1
\eta 1\delta 1
\mathrm{c}\mathrm{o}\mathrm{s}
s
p3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ \tau - \Delta (\tau )
\biggr)
+O
\biggl(
1
s
\biggr)
.
(3.12)
Under the conditions (a) and (b), the following formulas:
O
\biggl(
1
s
\biggr)
=
\left\{
\int r1
0
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p1
(2\tau - \Delta (\tau )) d\tau ,\int r1
0
q(\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s
p1
(2\tau - \Delta (\tau )) d\tau ,\int r2
r1
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p2
(2\tau - \Delta (\tau )) d\tau ,\int r2
r1
q(\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s
p2
(2\tau - \Delta (\tau )) d\tau ,\int \pi
r2
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s
p3
(2\tau - \Delta (\tau )) d\tau ,\int \pi
r2
q(\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s
p3
(2\tau - \Delta (\tau )) d\tau
(3.13)
can be proved by the same technique in Lemma 3.3.3 in [2]. Using the abbreviations
A(x) =
x\int
0
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s\Delta (\tau )
p1
d\tau , B(x) =
x\int
0
q (\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s\Delta (\tau )
p1
d\tau ,
C(x) =
x\int
r1
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s\Delta (\tau )
p2
d\tau , D(x) =
x\int
r1
q(\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s\Delta (\tau )
p2
d\tau ,
E(x) =
x\int
r2
q(\tau )
2
\mathrm{s}\mathrm{i}\mathrm{n}
s\Delta (\tau )
p3
d\tau , F (x) =
x\int
r2
q(\tau )
2
\mathrm{c}\mathrm{o}\mathrm{s}
s\Delta (\tau )
p3
d\tau ,
Zr
p =
r1
p1
+
r2 - r1
p2
+
\pi - r2
p3
, \Delta r
p =
1
p3
+
B(r1)
p1
+
D(r2)
p2
+
F (\pi )
p3
and putting expressions (3.13) into (2.10), and then using sn =
(n+ 1/2)\pi
Zr
p
+ \delta n we get \delta n =
= -
\Delta r
p
(n+ 1/2)\pi
+O
\biggl(
1
n2
\biggr)
and finally
sn =
\biggl(
n+
1
2
\biggr)
\pi
Zr
p
-
\Delta r
p\biggl(
n+
1
2
\biggr)
\pi
+O
\biggl(
1
n2
\biggr)
. (3.14)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1112 E. ŞEN, M. ACIKGOZ, S. ARACI
Thus, we proved the following theorem.
Theorem 3.2. If conditions (a) and (b) are satisfied, then the positive eigenvalues \lambda n = s2n of
problem (1.1) – (1.7) have (3.14) asymptotic representation for n \rightarrow \infty .
We now may obtain a more accurate asymptotic formula for the eigenfunctions. From (1.11) and
(3.12)
w1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
sx
p1
\biggl[
1 +
A(x)
sp1
\biggr]
-
B(x) \mathrm{s}\mathrm{i}\mathrm{n}
sx
p1
sp1
+O
\biggl(
1
s2
\biggr)
. (3.15)
Replacing s by sn and using (3.14) we have
u1n(x) = \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
n+
1
2
\biggr)
\pi x
p1Zr
p
\left[ 1 + A(x)Zr
p\biggl(
n+
1
2
\biggr)
\pi p1
\right] +
\left[ x\Delta r
p\biggl(
n+
1
2
\biggr)
\pi p1
\right] \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
n+
1
2
\biggr)
\pi x
p1Zr
p
+O
\biggl(
1
n2
\biggr)
.
(3.16)
From (1.12), (2.5), (3.12), (3.13) and (3.15) we obtain
w2 (x, \lambda ) =
\gamma 1
\delta 1
\Biggl\{ \biggl[
1 +
1
s
\biggl(
A(r1)
p1
+
C (x)
p2
\biggr) \biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s
p2
\biggl(
r1 (p2 - p1)
2p1
+ x
\biggr) \biggr)
-
- (D(x)/p2 +B(r1)/p1)
s
\mathrm{s}\mathrm{i}\mathrm{n}
s
p2
\biggl(
r1 (p2 - p1)
2p1
+ x
\biggr) \Biggr\}
+O
\biggl(
1
s2
\biggr)
. (3.17)
Now, replacing s by sn and using (3.14), we get
u2n(x) =
\gamma 1
\delta 1
\left\{
\left[ 1 + Zr
p
\biggl(
A(r1)
p1
+
C(x)
p2
\biggr)
\biggl(
n+
1
2
\biggr)
\pi
\right] \mathrm{c}\mathrm{o}\mathrm{s}
\left(
\biggl(
n+
1
2
\biggr)
\pi
Zr
pp2
\biggl(
r1 (p2 - p1)
2p1
+ x
\biggr) \right) +
+
Zr
p\Delta
r
p
\biggl(
D(x)
p2
+
B(r1)
p1
\biggr) \biggl(
r1 (p2 - p1)
2p1
+ x
\biggr)
p2
\biggl(
n+
1
2
\biggr) 2
\pi 2
\times
\times \mathrm{s}\mathrm{i}\mathrm{n}
\left(
\biggl(
n+
1
2
\biggr)
\pi
Zr
pp2
\biggl(
r1 (p2 - p1)
2p1
+ x
\biggr) \right)
\right\} +O
\biggl(
1
n2
\biggr)
. (3.18)
From (1.13), (2.7), (3.12), (3.13) and (3.17) we have
w3(x, \lambda ) =
\theta 1\gamma 1
\eta 1\delta 1
\left\{
\left[ 1 +
\biggl(
A(r1)
p1
+
C(r2)
p2
+
E(x)
p3
\biggr)
s
\right] \times
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
SPECTRAL PROBLEM FOR STURM – LIOUVILLE OPERATOR WITH RETARDED ARGUMENT . . . 1113
\times \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s
p3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr) \biggr)
-
- 1
s
\biggl(
B(r1)
p1
+
D(r2)
p2
+
F (x)
p3
\biggr)
\times
\times \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s
p3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr) \biggr) \right\} +O
\biggl(
1
s2
\biggr)
.
Now, replacing s by sn and using (3.14), we obtain
u3n(x) =
\theta 1\gamma 1
\eta 1\delta 1
\left\{
\left[ 1 + Zr
p
\biggl(
A(r1)
p1
+
C(r2)
p2
+
E(x)
p3
\biggr)
\biggl(
n+
1
2
\biggr)
\pi
\right] \times
\times \mathrm{c}\mathrm{o}\mathrm{s}
\left(
\biggl(
n+
1
2
\biggr)
\pi
Zr
pp3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr) \right) +
+
Zr
p\Delta
r
p
\biggl(
B(r1)
p1
+
D(r2)
p2
+
F (x)
p3
\biggr)
p3
\biggl(
n+
1
2
\biggr) 2
\pi 2
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr)
\times
\times \mathrm{s}\mathrm{i}\mathrm{n}
\left(
\biggl(
n+
1
2
\biggr)
\pi
Zr
pp3
\biggl(
p3 (r1 (p2 - p1) + p1r2) - r2p1p2
p1p2
+ x
\biggr) \right)
\right\} +O
\biggl(
1
n2
\biggr)
. (3.19)
Thus, we have proven the following theorem.
Theorem 3.3. If conditions (a) and (b) are satisfied, then the eigenfunctions un(x) of prob-
lem (1.1) – (1.7) have the following asymptotic representation for n \rightarrow \infty :
un(x) =
\left\{
u1n(x), x \in [0, r1),
u2n(x), x \in (r1, r2),
u3n(x), x \in (r2, \pi ],
where u1n(x), u2n(x) and u3n(x) defined as in (3.16), (3.18) and (3.19), respectively.
References
1. Norkin S. B. On boundary problem of Sturm – Liouville type for second-order differential equation with retarded
argument // Izv. Vysś. Ućebn. Zaved. Matematika. – 1958. – 6, № 7. – P. 203 – 214.
2. Norkin S. B. Differential equations of the second order with retarded argument // Transl. Math. Monogr. – Providence,
RI: AMS, 1972.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1114 E. ŞEN, M. ACIKGOZ, S. ARACI
3. Bellman R., Cook K. L. Differential-difference equations. – New York; London: Acad. Press, 1963.
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Received 14.01.13,
after revision — 27.05.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
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| id | umjimathkievua-article-1905 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:58Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4f/5b1dc82a7c9d516b04bfb6533ee1564f.pdf |
| spelling | umjimathkievua-article-19052019-12-05T09:31:14Z Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition Спектральна задача для oпeратора Штурма – Лiувiлля з аргументом, що запiзнюється, та спектральним параметром у граничнiй умовi Acikgoz, M. Araci, S. Şen, E. Ацікгоц, М. Аракі, С. Сен, Е. We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish the isometry of its wave operators and determine the initial domains of these operators. Методом Кука доведено iснування хвильових операторiв для багатовимiрного електромагнiтного оператора Шредiнгера у дивергентнiй формi. Крiм того, при певних умовах на коефiцiєнти даного оператора встановлено iзометричнiсть його хвильових операторiв. При цьому знайдено початковi областi цих операторiв. Institute of Mathematics, NAS of Ukraine 2016-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1905 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 8 (2016); 1102-1114 Український математичний журнал; Том 68 № 8 (2016); 1102-1114 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1905/887 Copyright (c) 2016 Acikgoz M.; Araci S.; Şen E. |
| spellingShingle | Acikgoz, M. Araci, S. Şen, E. Ацікгоц, М. Аракі, С. Сен, Е. Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title | Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title_alt | Спектральна задача для oпeратора Штурма – Лiувiлля з аргументом, що запiзнюється, та спектральним параметром у граничнiй умовi |
| title_full | Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title_fullStr | Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title_full_unstemmed | Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title_short | Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| title_sort | spectral problem for sturm – liouville operator with retarded argument which contains a spectral parameter in boundary condition |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1905 |
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