Inequalities for eigenvalues of a system of higher-order differential equations
We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we present some sharper estimates for the upper bound of the (k +1)th eigenvalue and the gaps of its consecutive eigenvalues.
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Інститут математики НАН України
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nasplib_isofts_kiev_ua-123456789-1659842025-02-23T18:17:42Z Inequalities for eigenvalues of a system of higher-order differential equations Нергоності для власних значень системи диференціальних рівнянь вищого порядку Sun, H.J. Статті We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we present some sharper estimates for the upper bound of the (k +1)th eigenvalue and the gaps of its consecutive eigenvalues. Встановлено дєякі 6ільш точні оцінки для власних значень системи диференціальних рівнянь вищого порядку. Ми також наводимо деякі більш точні оцінки для оцінки зверху (k +1)-го власного значення i щілин між будь-якими двома послідовними власними значеннями. This paper was supported by the National Natural Science Foundation of China (Grant No.1100113 2014 Article Inequalities for eigenvalues of a system of higher-order differential equations / H.J. Sun // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 394–403. — Бібліогр.: 10 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165984 517.9 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Sun, H.J. Inequalities for eigenvalues of a system of higher-order differential equations Український математичний журнал |
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We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we present some sharper estimates for the upper bound of the (k +1)th eigenvalue and the gaps of its consecutive eigenvalues. |
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Article |
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Sun, H.J. |
| author_facet |
Sun, H.J. |
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Sun, H.J. |
| title |
Inequalities for eigenvalues of a system of higher-order differential equations |
| title_short |
Inequalities for eigenvalues of a system of higher-order differential equations |
| title_full |
Inequalities for eigenvalues of a system of higher-order differential equations |
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Inequalities for eigenvalues of a system of higher-order differential equations |
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Inequalities for eigenvalues of a system of higher-order differential equations |
| title_sort |
inequalities for eigenvalues of a system of higher-order differential equations |
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Інститут математики НАН України |
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2014 |
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Статті |
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https://nasplib.isofts.kiev.ua/handle/123456789/165984 |
| citation_txt |
Inequalities for eigenvalues of a system of higher-order differential equations / H.J. Sun // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 394–403. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT sunhj inequalitiesforeigenvaluesofasystemofhigherorderdifferentialequations AT sunhj nergonostídlâvlasnihznačenʹsistemidiferencíalʹnihrívnânʹviŝogoporâdku |
| first_indexed |
2025-11-24T06:40:52Z |
| last_indexed |
2025-11-24T06:40:52Z |
| _version_ |
1849652886802268160 |
| fulltext |
UDC 517.9
H.-J. Sun (Nanjing Univ. Sci. and Technology, China)
INEQUALITIES FOR EIGENVALUES OF A SYSTEM
OF HIGHER-ORDER DIFFERENTIAL EQUATIONS*
НЕРIВНОСТI ДЛЯ ВЛАСНИХ ЗНАЧЕНЬ СИСТЕМИ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ВИЩОГО ПОРЯДКУ
We establish some sharper inequalities for eigenvalues of a system of higher-order differential equations. Moreover, we
give some sharper estimates for the upper bound of the (k + 1)th eigenvalue and the gaps of its consecutive eigenvalues.
Встановлено деякi бiльш точнi оцiнки для власних значень системи диференцiальних рiвнянь вищого порядку. Ми
також наводимо деякi бiльш точнi оцiнки для оцiнки зверху (k + 1)-го власного значення i щiлин мiж будь-якими
двома послiдовними власними значеннями.
1. Introduction. Let [a, b] ⊂ R and l ≥ 1. We investigate the following Dirichlet eigenvalue problem
of a system of higher-order differential equations:
(−1)lDl
(
a11(x)Dlu1(x) + . . .+ a1n(x)Dlun(x)
)
= λρ(x)u1(x),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(−1)lDl
(
an1(x)Dlu1(x) + . . .+ ann(x)Dlun(x)
)
= λρ(x)un(x),
Dtuq(a) = Dtuq(b) = 0, for t = 0, 1, . . . , l − 1 and q = 1, . . . , n,
(1.1)
where D =
d
dx
, ρ(x) is a positive function on [a, b] and 0 ≤ aij(x) = aji(x) ∈ C l[a, b] for
i, j = 1, 2, . . . , n. Assume that the coefficients aij(x), i, j = 1, 2, . . . , n, satisfy the uniform ellipticity
assumption: for any column vector ξ =
(
ξ1, . . . , ξn
)T
, there is a positive constant ς such that
n∑
i,j=1
aij(x)ξiξj ≥ ς|ξ|2 ∀x ∈ [a, b], (1.2)
where |ξ| = (ξ21 + . . . + ξ2n)1/2 denotes the Euclidean norm of ξ. Moreover, we assume that the
coefficients aij(x), i, j = 1, 2, . . . , n, satisfy
aij(x) ≤ ζ ∀x ∈ [a, b], (1.3)
where ζ is a positive constant. Suppose that the weight function satisfies:
σ−1 ≤ ρ(x) ≤ τ−1 ∀x ∈ [a, b], (1.4)
where σ and τ are two positive constants. This problem is significant in physics, mechanics and
engineering. Moreover, the weight function ρ denotes the density. Weighted estimates for eigenvalues
have many applications (see [6, 7]).
* This paper was supported by the National Natural Science Foundation of China (Grant No.11001130).
c© H.-J. SUN, 2014
394 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INEQUALITIES FOR EIGENVALUES OF A SYSTEM OF HIGHER-ORDER . . . 395
There have been some results for some eigenvalues problems of systems of equations. Mokeichev
[8] derive some expressions for eigenvalues of a system of equations in Hilbert space. Kostenko [5]
investigated the spectrum of a system of second-order differential equations. In 2011, Jia, Huang and
Liu [4] established the following inequalities for eigenvalues of problem (1.1):
k∑
r=1
λ
1/l
r
λk+1 − λr
≥ ςτ2k2
4l(2l − 1)ζσ2
( k∑
r=1
λ1−1/l
r
)−1
(1.5)
and
λk+1 ≤
[
1 +
4l(2l − 1)ζσ2
ςτ2
]
λk. (1.6)
The goal of this paper is to give some sharper estimates for eigenvalues of problem (1.1). We
first establish the following inequality.
Theorem 1.1. Let λr be the rth eigenvalue of problem (1.1). Suppose that the coefficients aij(x),
i, j = 1, 2, . . . , n, and the weight function ρ(x) satisfy (1.2), (1.3) and (1.4). Then we have
k∑
r=1
(λk+1 − λr)2 ≤
4l(2l − 1)ζσ2
ςτ2
k∑
r=1
(λk+1 − λr)λr. (1.7)
Furthermore, making a modification in the proof of Theorem 1.1, we can obtain another inequal-
ity.
Theorem 1.2. Under the same assumptions as Theorem 1.1, we have
k∑
r=1
(λk+1 − λr)2 ≤
≤ 2
[
l(2l − 1)ζσ2
ςτ2
]1/2[ k∑
r=1
(λk+1 − λr)λ1/lr
]1/2[ k∑
r=1
(λk+1 − λr)2λ1−1/l
r
]1/2
. (1.8)
Remark 1.1. When l = 1, (1.7) and (1.8) all become
k∑
r=1
(λk+1 − λr)2 ≤
4ζσ2
ςτ2
k∑
r=1
(λk+1 − λr)λr.
When l ≥ 2, inequality (1.7) is better than (1.8). In fact, inequality (1.7) is always Yang-type (see
[1, 9, 10]) for any l. Using Chebyshev’s inequality, it is not difficult to get (1.5) from (1.8). That is
to say, (1.8) and (1.7) is sharper than (1.5).
Using (1.7), we can derive some explicit estimates for upper bounds of eigenvalues and gaps of
consecutive eigenvalues. In fact, noting that (1.7) is a quadratic inequality of λk+1, we can obtain
the following two corollaries.
Corollary 1.1. Under the same assumptions of Theorem 1.1, we have
λk+1 ≤
[
1 +
2l(2l − 1)ζσ2
ςτ2
]
1
k
k∑
r=1
λr+
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
396 H.-J. SUN
+
{[
2l(2l − 1)ζσ2
ςτ2
1
k
k∑
r=1
λr
]2
−
[
1 +
4l(2l − 1)ζσ2
ςτ2
]
1
k
k∑
s=1
(λs −
1
k
k∑
r=1
λr)
2
}1/2
. (1.9)
Using the Cauchy – Schwarz inequality, we can get the following inequality from (1.9).
Corollary 1.2. Under the same assumptions of Theorem 1.1, we have
λk+1 ≤
[
1 +
4l(2l − 1)ζσ2
ςτ2
]
1
k
k∑
r=1
λr. (1.10)
Remark 1.2. Inequalities (1.9) and (1.10) give some estimates for upper bounds of λk+1 in terms
of the first k eigenvalues of problem (1.1). Moreover, it is easy to find that (1.10) implies (1.6).
At the same time, an explicit estimate for the gaps of any two consecutive eigenvalues of problem
(1.1) can be obtained from (1.9).
Corollary 1.3. Under the same assumptions of Theorem 1.1, we have
λk+1 − λk ≤ 2
{[
2l(2l − 1)ζσ2
ςτ2
1
k
k∑
r=1
λr
]2
−
−
[
1 +
4l(2l − 1)ζσ2
ςτ2
]
1
k
k∑
s=1
(λs −
1
k
k∑
r=1
λr)
2
}1/2
. (1.11)
2. Proofs of the main results. Let
u =
u1(x)
...
un(x)
, Dlu =
D
lu1(x)
...
Dlun(x)
, A(x) =
a11(x) . . . a1n(x)
. . . . . . . . .
an1(x) . . . ann(x)
.
Then we can rewrite problem (1.1) as the following simpler form:
(−1)lDl
(
A(x)Dlu
)
= λρu, on [a, b],
Dtu(a) = Dtu(b) = 0, for t = 0, 1, . . . , l − 1.
(2.1)
In order to prove the main theorems, we first establish a general inequality for eigenvalues of
problem (2.1).
Lemma 2.1. Let ur =
(
ur1(x), ur2(x), . . . , urn(x)
)T
be the weighted orthonormal eigenvector
corresponding to the rth eigenvalue λr of problem (1.1) for r = 1, 2, . . . , k. Then we have
k∑
r=1
(λk+1 − λr)2
b∫
a
|ur|2dx ≤
≤
k∑
r=1
δr(λk+1 − λr)2ωr +
k∑
r=1
1
δr
(λk+1 − λr)
b∫
a
1
ρ
|Dur|2dx, (2.2)
where the positive constants δr, r = 1, . . . , k, . . . , construct a nonincreasing sequence and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INEQUALITIES FOR EIGENVALUES OF A SYSTEM OF HIGHER-ORDER . . . 397
ωr = l2
b∫
a
Dl−1uT
r A(x)Dl−1urdx− l(l − 1)
b∫
a
DluT
r A(x)Dl−2urdx.
Proof. According to (2.1) and the assumptions of Lemma 1, the weighted orthonormal eigen-
vector ur satisfies
(−1)lDl
(
A(x)Dlur
)
= λrρur,
Dtur(a) = Dtur(b) = 0,
b∫
a
ρuT
r usdx = δrs,
(2.3)
for r, s = 1, . . . , k and t = 0, 1, . . . , l − 1. We define the trial vectors Φr by
Φr = xur −
k∑
s=1
brsus, (2.4)
where
brs =
b∫
a
ρxuT
r usdx = bsr. (2.5)
Then we know that Φr is weighted orthogonal with us. That is to say
b∫
a
ρΦT
r usdx = 0, for r, s = 1, . . . , k. (2.6)
It yields
b∫
a
ρΦT
r xurdx =
b∫
a
ρ|Φr|2dx. (2.7)
Since
Dl
[
aij(x)Dl
(
xurj(x)
)]
=
= lDl
(
aij(x)Dl−1urj(x)
)
+Dl
(
aij(x)xDlurj(x)
)
=
= lDl
(
aij(x)Dl−1urj(x)
)
+ lDl−1
(
aij(x)Dlurj(x)
)
+ xDl
(
aij(x)Dlurj(x)
)
,
we can deduce
Dl
(
A(x)DlΦr
)
= Dl
[
A(x)Dl(xur)
]
−
k∑
s=1
brsD
l
(
A(x)Dlus
)
=
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
398 H.-J. SUN
= lDl
(
A(x)Dl−1ur
)
+ lDl−1
(
A(x)Dlur
)
+ xDl
(
A(x)Dlur
)
−
−
k∑
s=1
brsD
l
(
A(x)Dlus
)
=
= lDl
(
A(x)Dl−1ur
)
+ lDl−1
(
A(x)Dlur
)
+ (−1)lλrρxur − (−1)l
k∑
s=1
brsλsρus. (2.8)
Using (2.6), (2.7), and (2.8), we have
b∫
a
ΦT
r D
l
(
A(x)DlΦr
)
dx =
=
b∫
a
ΦT
r
[
Dl
(
A(x)Dl−1ur
)
+Dl−1
(
A(x)Dlur
)]
dx+ (−1)lλr
b∫
a
ρ|Φr|2dx. (2.9)
Substituting (2.9) into the Rayleigh – Ritz inequality
λk+1 ≤
(−1)l
∫ b
a
ΦT
r D
l
(
A(x)DlΦr
)
dx∫ b
a
ρ|Φr|2dx
,
we obtain
(λk+1 − λr)
b∫
a
ρΦ2
rdx ≤
≤ (−1)ll
b∫
a
ΦT
r
[
Dl
(
A(x)Dl−1ur
)
+Dl−1
(
A(x)Dlur
)]
dx =
= (−1)ll
b∫
a
xuT
r
[
Dl
(
A(x)Dl−1ur
)
+Dl−1
(
A(x)Dlur
)]
dx+
k∑
s=1
brscrs, (2.10)
where
crs = (−1)l+1l
b∫
a
uT
s
[
Dl
(
A(x)Dl−1ur
)
+Dl−1
(
A(x)Dlur
)]
dx.
It is easy to find that crs = −csr.
Using integration by parts and making use of (2.3), we have
λrbrs =
b∫
a
xuT
s
[
(−1)lDl
(
A(x)Dlur
)]
dx = (−1)l
b∫
a
uT
r D
l
[
A(x)Dl
(
xus
)]
dx =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INEQUALITIES FOR EIGENVALUES OF A SYSTEM OF HIGHER-ORDER . . . 399
= (−1)ll
b∫
a
uT
r
[
Dl
(
A(x)Dl−1us
)
+Dl−1
(
A(x)Dlus
)]
dx+
+(−1)l
b∫
a
xuT
r D
l
(
A(x)Dlus
)
dx = λsbrs − csr.
It yields
crs = (λr − λs)brs. (2.11)
At the same time, we get
(−1)ll
b∫
a
xuT
r
[
Dl
(
A(x)Dl−1ur
)
+Dl−1
(
A(x)Dlur
)]
dx =
= l
b∫
a
Dl−1uT
r A(x)Dl
(
xur
)
dx− l
b∫
a
DluT
r A(x)Dl−1
(
xur
)
dx =
= l2
b∫
a
Dl−1uT
r A(x)Dl−1urdx+ l
b∫
a
xDl−1uT
r A(x)Dlurdx−
−l(l − 1)
b∫
a
DluT
r A(x)Dl−2urdx− l
b∫
a
xDluT
r A(x)Dl−1urdx = ωr, (2.12)
where
ωr = l2
b∫
a
Dl−1uT
r A(x)Dl−1urdx− l(l − 1)
b∫
a
DluT
r A(x)Dl−2urdx.
Substituting (2.11) and (2.12) into (2.10), we derive
(λk+1 − λr)
b∫
a
ρ|Φr|2dx ≤ ωr +
k∑
s=1
(λr − λs)b2rs. (2.13)
It is not hard to find
b∫
a
xuT
r Durdx = −1
2
b∫
a
|ur|2dx. (2.14)
Hence, using integration by parts again, we obtain
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
400 H.-J. SUN
−2
b∫
a
ΦT
r Durdx = −2
b∫
a
xuT
r Durdx+ 2
k∑
s=1
brsdrs =
b∫
a
|ur|2dx+ 2
k∑
s=1
brsdrs, (2.15)
where
drs =
b∫
a
uT
sDurdx = −dsr.
On the other hand, we have
−2(λk+1 − λr)2
b∫
a
ΦT
r Durdx =
= −2(λk+1 − λr)2
b∫
a
√
ρΦT
r
(
1
√
ρ
Dur −
√
ρ
k∑
s=1
drsus
)
dx ≤
≤ δr(λk+1 − λr)3
b∫
a
ρ|Φr|2dx+
λk+1 − λr
δr
b∫
a
(
1
√
ρ
Dur −
√
ρ
k∑
s=1
drsus
)2
dx =
=
λk+1 − λr
δr
b∫
a
1
ρ
|Dur|2dx− 2
k∑
s=1
drs
b∫
a
uT
sDurdx+
k∑
s,q=1
drsdrq
b∫
a
ρuT
s uqdx
+
+δr(λk+1 − λr)3
b∫
a
ρ|Φr|2dx =
= δr(λk+1 − λr)3
b∫
a
ρ|Φr|2dx+
λk+1 − λr
δr
b∫
a
1
ρ
|Dur|2dx−
k∑
s=1
d2rs
. (2.16)
Substituting (2.13) into (2.16), we obtain
−2(λk+1 − λr)2
b∫
a
ΦT
r Durdx ≤
≤ δr(λk+1 − λr)2
[
ωr +
k∑
s=1
(λr − λs)b2rs
]
+
λk+1 − λr
δr
b∫
a
1
ρ
|Dur|2dx−
k∑
s=1
d2rs
. (2.17)
Combining (2.15) and (2.17), and taking sum on r from 1 to k, we derive
k∑
r=1
(λk+1 − λr)2
b∫
a
|ur|2dx+ 2
k∑
r,s=1
(λk+1 − λr)2brsdrs ≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INEQUALITIES FOR EIGENVALUES OF A SYSTEM OF HIGHER-ORDER . . . 401
≤
k∑
r=1
δr(λk+1 − λr)2ωr +
k∑
r,s=1
δr(λk+1 − λr)2(λr − λs)b2rs+
+
k∑
r=1
1
δr
(λk+1 − λr)
b∫
a
1
ρ
|Dur|2dx−
k∑
r,s=1
1
δr
(λk+1 − λr)d2rs. (2.18)
Since the sequence {δr} is nonincreasing, one can get
k∑
r,s=1
δr(λk+1 − λr)2(λr − λs)b2rs ≤ −
k∑
r,s=1
δr(λk+1 − λr)(λr − λs)2b2rs. (2.19)
Then we can eliminate the unwanted terms in both sides of (2.18) by using (2.19) and
k∑
r,s=1
(λk+1 − λr)2brsdrs = −
k∑
r,s=1
(λk+1 − λr)(λr − λs)brsdrs.
Thereby, (2.2) is true.
Lemma 2.1 is proved.
Now we give the proof of Theorem 1.1.
Proof of Thereom 1.1. It is not hard to find
0 < τ ≤
b∫
a
|ur|2dx =
b∫
a
1
ρ
ρ|ur|2dx ≤ σ. (2.20)
By virtue of (1.2), we have
λr =
b∫
a
uT
r
[
(−1)lDl
(
A(x)Dlur
)]
dx =
b∫
a
DluT
r A(x)Dlurdx =
=
b∫
a
n∑
i,j=1
aij(x)Dluri(x)Dlurj(x)dx ≥ ς
b∫
a
|Dlur|2dx.
It yields
b∫
a
|Dlur|2dx ≤
1
ς
λr. (2.21)
Using (2.21) and the following inequality (see Lemma 2.1 of [4])
b∫
a
|Dtur|2dx ≤ σ1−t/l
(
λr
ς
)t/l
,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
402 H.-J. SUN
we can deduce
b∫
a
1
ρ
|Dur|2dx ≤ σ2−1/l
(
λr
ς
)1/l
, (2.22)
b∫
a
Dl−1uT
r A(x)Dl−1urdx ≤ ζ
b∫
a
|Dl−1ur|2dx ≤ ζσ1/l
(
λr
ς
)1−1/l
(2.23)
and
−
b∫
a
DluT
r A(x)Dl−2urdx ≤ ζ
b∫
a
|Dl−2ur|2dx
1/2 b∫
a
|Dlur|2dx
1/2
≤
≤ ζσ1/l
(
λr
ς
)1−1/l
. (2.24)
Using (2.23) and (2.24), we obtain
ωr = l2
b∫
a
Dl−1uT
r A(x)Dl−1urdx− l(l − 1)
b∫
a
DluT
r A(x)Dl−2urdx ≤
≤ l(2l − 1)ζσ1/l
(
λr
ς
)1−1/l
. (2.25)
Therefore, substituting (2.20), (2.22), and (2.25) into (2.2), we get
τ
k∑
r=1
(λk+1 − λr)2 ≤
≤ l(2l − 1)ζσ1/l
k∑
r=1
δr(λk+1 − λr)2
(
λr
ς
)1−1/l
+ σ2−1/l
k∑
r=1
1
δr
(λk+1 − λr)
(
λr
ς
)1/l
. (2.26)
Putting
δr =
δ
l(2l − 1)ζσ1/lλ
1−1/l
r ς−1+1/l
in (2.26), we have
τ
k∑
r=1
(λk+1 − λr)2 ≤ δ
k∑
r=1
(λk+1 − λr)2 +
1
δ
l(2l − 1)ζσ2ς−1
k∑
r=1
(λk+1 − λr)λr. (2.27)
Then, putting
δ =
[
l(2l − 1)ζ
]1/2
σς−1/2
[
k∑
r=1
(λk+1 − λr)λr
]1/2 [ k∑
r=1
(λk+1 − λr)2
]−1/2
in (2.27), we obtain
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INEQUALITIES FOR EIGENVALUES OF A SYSTEM OF HIGHER-ORDER . . . 403
τ
k∑
r=1
(λk+1 − λr)2 ≤ 2
[
l(2l − 1)ζ
]1/2
σς−1/2
[
k∑
r=1
(λk+1 − λr)2
k∑
r=1
(λk+1 − λr)λr
]1/2
. (2.28)
This yields (1.7).
Theorem 1.1 is proved.
Proof of Thereom 1.2. Since the sequence {δr} is nonincreasing, we can obtain
τ
k∑
r=1
(λk+1 − λr)2 ≤
≤ l(2l − 1)ζσ1/lδ
k∑
r=1
(λk+1 − λr)2
(
λr
ς
)1−1/l
+ σ2−1/l 1
δ
k∑
r=1
(λk+1 − λr)
(
λr
ς
)1/l
(2.29)
by taking δr = δ in (2.26) for r = 1, . . . , k. Putting
δ =
[
l(2l − 1)ζ
]−1/2
σ1−1/lς1/2−1/l
[
k∑
r=1
(λk+1 − λr)λ1/lr
]1/2 [ k∑
r=1
(λk+1 − λr)2λ1−1/l
r
]−1/2
in (2.29), we can derive (1.8).
Theorem 1.2 is proved.
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Received 22.11.11
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