Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2
We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. A compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained.
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| Дата: | 2008 |
|---|---|
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2008
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System research and information technologies| _version_ | 1867334318826717184 |
|---|---|
| author | Kasyanov, P. Mel'nik, V. |
| author_facet | Kasyanov, P. Mel'nik, V. |
| author_institution_txt_mv | [
{
"author": "P. Kasyanov",
"institution": null
},
{
"author": "V. Mel'nik",
"institution": null
}
] |
| author_sort | Kasyanov, P. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-04-11T11:07:52Z |
| description | We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. A compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained. |
| first_indexed | 2025-07-17T10:22:59Z |
| format | Article |
| fulltext |
© P. Kasyanov, V. Mel'nik , 2008
88 ISSN 1681–6048 System Research & Information Technologies, 2008, № 3
TIДC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 517.9
ON SOME TOPOLOGICAL PROPERTIES FOR SPECIAL
CLASSES OF BANACH SPACES. PART 2
P. KASYANOV, V. MEL'NIK
We consider some classes of infinite-dimensional Banach spaces with integrable
derivatives. A compactness lemma for nonreflexive spaces is obtained. However
some main topological properties for the given spaces are obtained.
This work is continuation of [1].
Theorem 1. );(*
0 HSCW ⊂ with continuous embedding. Moreover, for
every *
0, Wy ∈ξ and ,s t S∈ the next formula of integration by parts takes place
.))}(),(())(),({(=))(),(())(),(( ττξττξτξξ dyyssytty
t
s
′+′− ∫ (1)
In particular, when ξ=y we have:
τττ dyysyty
t
s
HH ))(),((=))()((
2
1 22 ′− ∫ .
Proof. To simplify the proof we consider ],[ baS = for some
.<<< +∞∞− ba
The validity of formula (1) for );(, 1 VSCy ∈ξ is checked by direct
calculation. Now let )(1 SC∈ϕ be such fixed that 0=)(aϕ and 1=)(bϕ .
Moreover, for );(1 VSCy∈ let yϕξ = and yy ϕη −= . Then, due to (1):
dssysyssysystyt
t
a
))}(),()((2))(),()(({=))(),(( ′+′∫ ϕϕξ ,
dssysyssysystyt
b
t
))}(),())(((12))(),()(({=))(),(( ′−+′−− ∫ ϕϕη ,
from here for );( *
iiqi VSL∈ξ and );( HSL
iri ′
∈η ( 1,2=i ) such that =′y
2121 ηηξξ +++= it follows:
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 89
≤′−′+′ ∫∫ dssysydssysyssysysty
b
t
b
t
H ))(),((2))}(),()((2))(),()(({=)( 2 ϕϕ
≤′−+⋅⋅′≤ ∫∈
dssysysVSyys
S
LVSCSs
))(),(1)()((2);()|)(|max
1*;(
ϕϕ
+′≤
∈ );()|)(|max
1*;( VSyys LVSCSs
ϕ
⎜⎜
⎝
⎛
++−+
∈ );();();();(|1)(|max2
22
*
22
211
*
11
1 VSyVSVSyVSs
pLqLpLqLSs
ξξϕ
≤⎟
⎠
⎞++
′′
);();();();(
22
2
11
1 HSyHSHSyHS rLrLrLrL ηη
+⎟
⎠
⎞
⎜
⎝
⎛ +′≤
∈
21/
22
11/
11
*;(
)(mes);()(mes);()|)(|max q
pL
q
pLVSCSs
SVSySVSyysϕ
×⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+++−+
′′∈
);();();();(|1)(|max2
2
2
1
1*
22
2*
11
1 HSHSVSVSs
rLrLqLqLSs
ηηξξϕ
⎟
⎠
⎞
⎜
⎝
⎛ +++× 21/
);(
11/
);(2211
)(mes)(mes);();(
r
HSC
r
HSCpLpL SySyVSyVSy .
Hence, due to [1, theorem 3], definition of X⋅ , if we take in last inequal-
ity
ab
att
−
−=)(ϕ for all St∈ we obtain
,2
);(*
0
3*
0
2
2
);( HSCWWHSC yyCyCy +≤ (2)
where 1C is the constant from inequality *
0
1*;( ) WVSC
yCy ≤ for every
*
0Wy∈ ,
{ }.,1},)(mesmax2=,
})(mes,)(mes{min
2= 21{min1/
321/11/
1
2
rSC
SS
C
C r
pp+
Remark that 0=1
∞+
, 0>, 32 CC . From (2) it obviously follows that
),;(allfor 1
*
0
4);( VSCyyCy
WHSC ∈≤ (3)
where
2
4
= 2
2
33
4
CCC
C
++
does not depend on y .
Now let us apply [1, theorem 4]. For arbitrary *
0Wy∈ let 1}{ ≥nny be a
sequence of elements from 1( ; )C S V converging to y in *
0W . Then in virtue of
relation (3) we have
0*
0
4);( →−≤−
WknHSCkn yyCyy ,
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 90
therefore, the sequence 1}{ ≥nny converges in );( HSC and it has only limit
);( HSC∈χ such that for a.e. St∈ )(=)( tytχ . So, we have );( HSCy∈ and
now the embedding );(*
0 HSCW ⊂ is proved. If we pass to limit in (3) with
nyy = as ∞→n we obtain the validity of the given estimation *
0Wy∈∀ . It
proves the continuity of the embedding *W into );( HSC .
Now let us prove formula (1). For every *
0, Wy ∈ξ and for corresponding
approximating sequences );(},{ 1
1 VSCy nnn ⊂≥ξ we pass to the limit in (1) with
nyy = , nξξ = as ∞→n . In virtue of Lebesgue's theorem and );( **
0 VSCW ⊂
with continuous embedding formula (1) is true for every *
0Wy ∈ .
The theorem is proved.
In virtue of * *
0W W⊂ with continuous embedding and due to the latter
theorem the next statement is true.
Corollary 1. * ( ; )W C S H⊂ with continuous embedding. Moreover, for
every *,y Wξ ∈ and ,s t S∈ formula (1) takes place.
For every 1≥n let us define the Banach space { }nnn XyXyW ∈′∈ |= **
with the norm
,** nX
nXnW
yyy ′+=
where the derivative y′ is considered in sense of scalar distributions space
);(*
nHSD . As far as
);(=));(());((=);( **** VSVSHSHS nn DDLDLD ω⊂
it is possible to consider the derivative of an element *
nXy∈ in the sense of
);( ** VSD . Remark that for every 1≥n **
1
* WWW nn ⊂⊂ + .
Proposition 1. For every *Xy∈ and 1≥n )(= ′′ yPyP nn , where derivative
of element *x X∈ is in the sense of the scalar distributions space );( ** VSD .
Remark 1. We pay our attention that in virtue of the previous assumptions
the derivatives of an element *
nXx∈ in the sense of );( *VSD and in the sense of
);( nHSD coincide.
Proof. It is sufficient to show that for every )(SD∈ϕ )()(=)( ϕϕ ′′ yPyP nn .
In virtue of definition of derivative in sense of );( ** VSD we have
=′−=′−=′∈∀ ∫ ττϕτϕϕϕ dyPyPyPS
S
nnn )()()()()(D
).()(=)()(=)()(= ϕϕττϕτ ′′−′−∫ yPyPdyP nnn
S
The proposition is proved.
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 91
Due to [1, propositions 3, 4] it follows the next
Proposition 2. For every 1≥n ** = WPW nn , i.e.
}.)(|)({= ** WyyPW nn ∈⋅⋅
Moreover, if the triple ( )HVH jii ;;}{ 1≥ , 1,2=j satisfies condition (γ ) with
jCC = . Then for every *y W∈ and 1n ≥
.)(},{max)( *21* WWn yCCyP ⋅≤⋅
Theorem 2. Let the triple ( )HVH jii ;;}{ 1≥ , 1,2=j satisfy condition (γ )
with jCC = . We consider bounded in *X set *XD ⊂ and XE ⊂ that is
bounded in X . For every 1n ≥ let us consider
{ } .and|=: **
nnnnnnn WEPyDyXyD ⊂∈′∈∈
Then
,and1allfor* nnWn DynECDy ∈≥+≤ ++ (4)
where },{max= 21 CCC , *sup=
XDy
yD
∈
+ and X
Ef
fE
∈
+ sup= .
Remark 2. Due to proposition 2 nD is well-defined and *
nn WD ⊂ is true.
Remark 3. A priori estimates (like (4)) appear at studying of solvability of
differential–operator equations, inclusions and evolutional variational inequalities
in Banach spaces with maps of λw -pseudomonotone type by using Faedo–
Galerkin method (see [2, 3]) at boundary transition, when it is necessary obtain a
priori estimates of approximate solutions ny in *X and of its derivatives ny′
in X .
Proof. Due to proposition 2 for every 1≥n and nn Dy ∈
.},{max= 21** ++++ +≤+≤′+ ECCDEPDyyy nXnXnWn
The theorem is proved.
Further, let 0B , 1B , 2B be some Banach spaces such, that
20 , BB are reflexive 10 BB ⊂ with compacting embedding (5)
210 BBB ⊂⊂ with compacting embedding. (6)
Lemma 1. ([4] lemma 1.5.1, p.71) Under the assumptions (5), (6) for an
arbitrary 0>η there exists 0>ηC such that
.0201
BxxCxx BBB ∈∀+≤ ηη
Corollary 2. Let the assumptions (5), (6) for the Banach spaces 0B , 1B and
2B are verified, ][1;1 +∞∈p , ][0,= TS and the set );( 01
BSLK p⊂ such that
a) K is precompact set in );( 21
BSLp ;
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 92
b) K is bounded set in );( 01
BSLp .
Then K is precompact set in );( 11
BSLp .
Proof. Due to lemma 1 and to the norm definition in );(
1 ip BSL , 0,2=i it
follows that for an arbitrary 0>η there exists such 0>ηC that
);();(2);(2);( 01210111
BSLyBSyCBSyBSy ppLpLpL ∈∀+≤ ηη (7)
Let us check inequality (7), when )[0,1 +∞∈p (the case +∞=1p is direct
corollary of lemma 1):
≤+≤ ∫∫ dttyCtydtptyp
BSy p
BB
S
B
S
pL
1
20
1
1
1
11
])()([)(=);( ηη
=)()(12 1
2
11
0
11
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+−≤ ∫∫ dtptyCdtpty B
S
p
B
S
pp
ηη
≤⎥
⎦
⎤
⎢
⎣
⎡
+− 1
21
11
01
11
);();(
12= p
BSyCp
BSy
pL
p
pL
pp
ηη
);();();(2 01
1
2101
1 BSLyBSyCBSy p
p
pLpL
p ∈∀⎥⎦
⎤
⎢⎣
⎡ +≤ ηη .
The last inequality follows from
( ) 0,12)(
2
1111
11
≥∀+−≤+≤
+ babababa pppp
pp
.
Now let 1}{ ≥nny be an arbitrary sequence from K . Then by the conditions
of the given statement there exists 11 }{}{ ≥≥ ⊂ nnkkn yy that is a Cauchy
subsequence in the space );( 21
BSLp . So, thanks to inequality (7) for every
1, ≥mk
+−≤− );(2);( 0111
BSyyBSyy
pLmnknpLmnkn η
);(2);(2
2121
BSyyCCBSyyC
pLmnknpLmnkn −+≤−+ ηη η ,
where 0>C is a constant that does not depend on η,, km . Therefore, for every
0>ε we can choose 0>η and 1≥N such that
NkmBSyyCC
pLmnkn ≥∀− ,/2<);(2and/2<
21
εεη η
Thus,
NkmBSyyN
pLmnkn ≥∀−≥∃∀ ,<);(:10>
11
εε .
This fact means, that 1}{ ≥kkny converges in );( 11
BSLp . The corollary is
proved.
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 93
Theorem 3. Let conditions (5), (6) for 210 ,, BBB are satisfied, ∈10 , pp
)[1;+∞∈ , S be a finite time interval and );( 01
BSLK p⊂ be such, that
a) K is bounded in );( 01
BSL p ;
b) for every 0>ε there exists such 0>δ that from δ<<0 h it results in
Kudphuu B
S
∈∀+−∫ ετττ <)()( 0
2
. (8)
Then K is precompact in );(}; 110{min BSpL p .
Furthermore, if for some 1>q K is bounded in );( 1BSLq , then K is
precompact in );( 1BSLp for every )[1,qp∈ .
Remark 4. Further we consider that every element )( iBSx →∈ is equal to
0 out of the interval S .
Proof. At the beginning we consider the first case. For our goal it is enough
to show, that it is possible to choose a Cauchy subsequence from every sequence
Ky nn ⊂≥1}{ in );(}; 110{min BSpL p . Due to corollary 2 it is sufficient to prove
this statement for );(}; 210{min BSpL p .
For every Kx∈ 0>h∀ St∈∀ we put
ττ dx
h
tx
ht
t
h )(1=:)( ∫
+
,
where the integral is regarded in the sense of Bochner integral. We point out that
0>h∀ );();( 20 BSCBSCxh ⊂∈ .
Fixing a positive number ε , we construct for a set
);();( 2000
BSLBSLK pp ⊂⊂
a final ε -web in );( 20
BSLp . For 0>ε we choose 0>δ from (8). Then for
every fixed h ( δ<<0 h ) we have:
=−=−+ ∫∫
+++
+
22
)()(1)()( B
ht
t
hut
ut
Bhh dxdx
h
txutx ττττ
τττττττ dxux
h
dxdux
h B
ht
t
B
ht
t
ht
t
22
)()(1)()(1
−+≤−+= ∫∫∫
+++
.
Moreover, from the H o lder inequality we obtain
≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+⎟
⎠
⎞
⎜
⎝
⎛≤−+ ∫∫
++ 0
1
0
2
0
1
2
)()(1)()(1 p
B
ht
t
p
B
ht
t
dpxux
h
dxux
h
ττττττ
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 94
StuKx
h
dpxux
h
pp
B
T
p
∈∀∀∈∀⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+⎟
⎠
⎞
⎜
⎝
⎛≤ ∫ ,<<0,<)()(1 0
1
0
1
0
2
0
0
1
δετττ .
Therefore the family of functions Kxhx ∈}{ is equicontinuous.
Since Kx∈∀ St∈∀ it results in
≤≤ ∫∫
++
ττττ dx
h
dx
h
tx B
ht
t
B
ht
t
Bh 222
)(1)(1=)(
,)(1)(1 1
1
1
1
1
2
0
1
1
1
1
1
2
1
1
pp
B
T
pp
B
ht
t
p
h
Cdpx
h
dpx
h
⎟
⎠
⎞
⎜
⎝
⎛≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛≤ ∫∫
+
ττττ
the family of functions Kxhx ∈}{ is uniformly bounded, because of the constant
0≥C does not depend on Kx∈ . Hence, δ<<0: hh∀ the family of functions
Kxhx ∈}{ is precompact in );( 2BSC , so in );(}, 210{min BSpL p too.
On the other hand, δ<<0 h∀ , Kx∈∀ , St∈∀
≤−≤− ∫
+
ττ dxtx
h
txtx B
ht
t
Bh 22
)()(1)()(
.)()(1)()(1 0
1
0
2
0
0
1
2
0
p
B
h
p
B
h
dptxtx
h
dtxtx
h ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−⎟
⎠
⎞
⎜
⎝
⎛≤+−≤ ∫∫ ττττ
From here, taking into account inequality (8) we receive:
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− ∫∫∫
0
1
0
2
00
0
1
0
2
0
)()(1)()(
p
B
hTp
Bh
T
dtdptxtx
h
dtptxtx ττ
.=1<)()(1 0
1
0
1
0
0
1
0
2
00
pphp
B
Th
d
h
dtdptxtx
h
ετεττ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−= ∫∫∫
Hence, by virtue of the precompactness of system Kxhx ∈}{ in
);(}, 210{min BSpL p δ<<0 h∀ we have that K is a precompact set in
);(}, 210{min BSpL p .
Let us consider the second case. Assume that for some 1>q the set K is
bounded in );( 1BSLq . Similarly to the previous case, it is enough to show that
for every )[1;qp∈ and Ky nn ⊂≥1}{ there exists a subsequence ⊂≥1}{ kkny
1}{ ≥⊂ nny and );( 1BSLy p∈ so that
∞→→ kBSLyy pkn as);(in 1 .
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 95
Because of yyn → in );(}, 110{min BSpL p , up to a subsequence, as ∞→n , we
have 11 }{}{ ≥≥ ⊂∃ nnkkn yy such that 0)( →
knBλ as ∞→k , where =:nB
1})()(|{=:
1
≥−∈ Bn tytySt for every 1≥n , λ is the Lebesgue measure on S .
Then for every 1≥k
+−− ∫∫ dspsysydspsysy Bkn
knA
Bkn
S
11
)()(=)()(
+−≤−+ ∫∫ dspsysydspsysy Bkn
knA
Bkn
knB
11
)()()()(
( ) ,:=)()()(
1 knknq
pq
kn
q
p
Bkn
S
JIBdsqsysy +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+
−
∫ λ
where nn BSA \= for every 1≥n .
It is clear that 0→
knJ as ∞→k . Let us consider
knI . Since 1}{ ≥kkny is
precompact in );(}, 110{min BSpL p , there exists such 11 }{}{ ≥≥ ⊂ kknkkm yy that
)()( tyty
km → in 1B as ∞→k almost everywhere in S . Setting
⎪⎩
⎪
⎨
⎧ ∈−
∈∀≥∀
otherwice,,0
,,)()(
=:)(,1 1 nBkm
km
Atptyty
tStk ϕ
using definition of
kmA , sequence 1}{ ≥kkmϕ satisfies the conditions of the
Lebesgue theorem with the integrable majorant 1≡φ . So 0→
kmϕ in )(1 SL as
∞→k . Thus, within to a subsequence, yyn → in );( 1BSLq .
The theorem is proved.
Let Banach spaces 210 ,, BBB satisfy all assumptions (5), (6),
)[1;, 10 +∞∈pp be arbitrary numbers. We consider the set with the natural
operations
)},;(|);({= 2100
BSLvBSLvW pp ∈′∈
where the derivative v′ of an element );( 00
BSLv p∈ is considered in the sense
of the scalar distribution space );( 2BSD . It is clear, that
);( 00
BSLW p⊂ .
Theorem 4. The set W with the natural operations and the graph norm
);();(=
2100
BSvBSvv
pLpLW ′+
is a Banach space.
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 96
Proof. The executing of the norm properties for W⋅ immediately follows
from its definition. Now we consider the completeness of W referring to just
defined norm. Let 1}{ ≥nnv be a Cauchy sequence in W . Hence, due to the
completeness of );( 00
BSLp and );( 21
BSLp it follows that for some
);( 00
BSLy p∈ and );( 21
BSLv p∈
+∞→→′→ nBSLvyBSLyy pnpn as);(inand);(in 2100
.
Due to [5, lemma IV.1.10] and in virtue of continuous dependence of the
derivative by the distribution in *
2( ; )S BD (see [5, p. 169) it follows, that
);(= 21
BSLvy p∈′ .
The theorem is proved.
Theorem 5. Under conditions (5), (6) );( 2BSCW ⊂ with the continuous
embedding.
Proof. For a fixed Wy∈ let us show that );( 2BSCy∈ . Let us put
Sttdy
t
t
t
∈∀′∫ ,)(=)( 0
0
ττξ .
The integral is well-defined because );( 21 BSLy ∈′ . On the other hand, from the
inequality [5, p. 153]
Sstsdyst B
s
t
B ∈≥∀′≤− ∫ ,)()()(
22
ττξξ
it follows that );( 2BSC∈ξ . Due to [5] (lemma IV.1.8) y′′ =ξ , so from
[5] (lemma IV.1.9) it follows that
Stztty ∈+ a.e.for)(=)( ξ .
for some fixed 2Bz∈ .
Thus the function y also lies in );( 2BSC .
In virtue of the continuous embedding of );( 21
BSLp in );( 21 BSL we have
that for some constant 0>k , which does not depend on y ,
StBSykdyt
pLB
S
B ∈∀′≤′≤ ∫ );()()(
2122
ττξ .
From here, due to the continuous embedding 20 BB ⊂ , we have
≤−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∫ );())(mes(
21
11/
1
2
11/
2 BSydspzSz
pL
p
B
S
p
B ξ
⎟
⎠
⎞
⎜
⎝
⎛ ′+≤⎟
⎠
⎞
⎜
⎝
⎛ +≤ );();());( 2100
22;(21
1 BSyBSykBSyk
pLpLBSCpL ξ ,
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 97
where )(mes S is the “length”' (the measure) of S , 0>2k is a constant that does
not depend on Wy∈ . Therefore, from the last two relations there exists 03 ≥k
such that
Wyyky WBSC ∈∀≤ 32;( ) .
The theorem is proved.
The next result represents a generalization of the compactness lemma [4,
theorem 1.5.1, p. 70] into the case )[1;, 10 +∞∈pp .
Theorem 6. Under conditions (5), (6), for all )[1;, 10 +∞∈pp the Banach
space W is compactly embedded in );( 10
BSLp .
Proof. At the beginning we prove the compact embedding of W in
);( 21 BSL .
For every Wy∈ and R∈h let us take
⎜⎜
⎝
⎛ ∈++
.otherwice,0
,if),(
=)(
Shthty
tyh
In virtue of theorem 5 the given definition is correct.
Lemma 2. For every Wy∈ and R∈h
);();( 2121 BSyhBSyy LLh ′≤− . (9)
Proof. Let Wy∈ be fixed. Then
dtdydttyhtyBSyy B
ht
tS
B
S
Lh 2221
)(=)()(=);( ττ′−+− ∫∫∫
+
.
Let us put 1,2,)()()(=)( =∈∀−+=′∫
+
iSttyhtydytg
ht
t
y ττ . Due to
theorem 5 the element );( 2BSCg y ∈ . So, as S is a compact set, we have that
);( 21 BSLg y ∈ . Therefore, due to proposition [6, p.191] with );(= 21 BSLX and
to [1, theorem 2] it follows the existence of **
2 );( XBSLhy ≡∈ ∞ such that
dttgthdttg Byy
S
By
S
22
)(),()( 〉〈= ∫∫ and 1);( *
2
=
∞ BSh
Ly
Hence,
==′ ∫∫∫∫
+
dttgthdttgdtdy
Byy
S
By
SB
ht
tS 22
2
)(),(=)()( ττ
=′=′ ∫∫∫∫
++
dtdythdtdyth
By
ht
tSB
ht
t
y
S
ττττ
2
2
)(),()(),(=
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 98
≤′=′ ∫∫∫∫
−−
ττττ
τ
τ
τ
τ
dydtthdtdyth
B
y
hS
By
hS 2
2
)(,)()(),(=
);()()(supess
212*
2 BSyhdyhth LB
S
By
St
′≤′≤ ∫
∈
ττ .
So, we have obtained necessary estimation (9).
The lemma is proved.
Let us continue the proof of the given theorem. Let WK ⊂ be an arbitrary
bounded set. Then for some 0>C
KyCBSyCBSy
pLpL ∈∀≤′≤ );(,);( 2100
. (10)
In order to prove the precompactness of K in );( 11 BSL let us apply
theorem 4 with 00 = BB , 11 = BB , 22 = BB , 1=0p , 11 = pp . Due to estimates
(9) and (10) the all conditions of the given theorem hold. So, the set K is
precompact in );( 11 BSL and hence in );( 21 BSL . In virtue of theorem 5 and the
Lebesgue theorem it follows that the set K is precompact in );( 00
BSLp . Hence,
due to corollary 2 we obtain the necessary statement.
The theorem is proved.
Proposition 3. Let Banach spaces 210 ,, BBB satisfy conditions (5), (6),
)[1;, 10 +∞∈pp , );(}{ 01
BSLu pIhh ⊂∈ , where +⊂ R)(0,= δI , ],[ baS = such
that
a) Ihhu ∈}{ is bounded in );( 01
BSLp ;
b) there exists such +→ RIc : that 0=
2
lim ⎟
⎠
⎞
⎜
⎝
⎛ −
∞→
nn
abc and
00
2
)()()( p
Bhh
S
hhcdtphtutuIh ≤+−∈∀ ∫ .
Then there exists Ih nn ⊂≥1}{ ( +0nh as ∞→n ) so that 1}{ ≥nnhu
converges in );(}, 110{min BSpL p .
Remark 5. We assume 0=)(tuh when bt > .
Remark 6. Without loss of generality let us consider [0,1]=S .
Proof. At first we prove this statement for );( 20
BSLp . In virtue of
Minkowski inequality for every Ih N ∈
2
1= and 1≥k
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−≤
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
− ∫∫
0
1
0
2
1
0
0
1
0
2
2
1
0
)()()()(
p
Bhh
p
B
k
hh dtphtutudtptutu
On some topological properties for special classes of Banach spaces. Part 2
Системні дослідження та інформаційні технології, 2008, № 3 99
≤
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−++
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+−++ ∫∫
0
1
0
2
22
1
0
0
1
0
2
2
1
0
)()()()(
p
B
k
h
k
h
p
B
k
hh dtptuhtudtphtuhtu
⎜
⎜
⎜
⎝
⎛
−⎟
⎠
⎞
⎜
⎝
⎛ +
+
−
+
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−+≤ ∫∑∫ hitudtptutuhhc k
k
h
k
i
p
B
k
hh
h
p
2
11
)()()(
2
1
0
2
0=
0
1
0
2
2
1
0
1
++≤
⎟⎟
⎟
⎟
⎠
⎞
⎟
⎠
⎞
⎜
⎝
⎛ +− )/2(
2
2)(
2
0
1
0
10
1
0
22
kp
k
kp
p
B
k
k
h hchhhcdt
p
hitu
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+≤
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−+ ∫ )/2()()()( 0
1
0
1
0
1
0
2
2
1
kpp
p
B
k
hh
h
hchchdtptutu
+
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+−++
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−+ ∫∫
0
1
0
2
2
10
1
0
2
1
)()()()(
p
B
k
hh
h
p
Bhh
h
dtphtuhtudtphtutu
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+≤≤
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−++ ∫ )/2()(2...)()( 0
1
0
1
0
1
0
2
22
1
kpp
p
B
k
h
k
h
h
hchchdtptuhtu
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+≤≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−+ ∫ )/2()(2...)()( 0
1
0
1
0
1
0
2
1
2
kppN
p
Bhh
h
hchchdtphtutu
)/2()( 0
1
0
1
kpp hchc += .
So, for every 1≥N and 1≥k it results in
⎟
⎠
⎞
⎜
⎝
⎛+⎟
⎠
⎞
⎜
⎝
⎛≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
++∫ kN
p
N
pp
BkNN ccdtptutu
2
1
2
1)()( 0
1
0
1
0
1
0
21/21/2
1
0
.
In virtue of assumption b) we can choose Ih
m
mnn ∩
1
1
2
1}{
≥
≥
⎭
⎬
⎫
⎩
⎨
⎧⊂ such that
0)( →nhc as ∞→n . So, the sequence 1}{ ≥nnhu is fundamental in );( 20
BSLp .
Because of 10 BB ⊂ with compact embedding, the sequence 1}{ ≥nnhu is bounded
in );(}, 010{min BSpL p ; due to corollary 2 it follows that 1}{ ≥nnhu is fundamental
in );(}, 110{min BSpL p .
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 100
The proposition is proved.
Now we combine all results to obtain the necessary a priori estimate.
Theorem 7. Let all conditions of theorem 2 are satisfied and HV ⊂ with
compact embedding. Then (4) be true and the set
);(inprecompactand);(inboundedis
1
HSLHSCD pn
n
∪
≥
for every 1≥p .
Proof. Estimation (4) follows from theorem 2. Now we use compactness
theorem 6 with VB =0 , HB =1 , *
2 VB = , 1=0p , 11 =p . Remark that
*
1( ; )X L S V⊂ and *
1( ; )X L S V⊂ with continuous embedding. Hence, the set
);(inprecompactis 1
1
HSLDn
n
∪
≥
.
In virtue of (4) and theorem 1 on continuous embedding of *W in );( HSC ,
it follows that the set
).;(inboundedis
1
HSCDn
n
∪
≥
Further, by using standard conclusions and the Lebesgue theorem we obtain
the necessary statement.
The theorem is proved.
Partially Supported by State Fund of Fundamntal Investigations Grant
№ Ф25.1/029-2008
REFERENCES
1. Kasyanov P., Mel'nik V.S. On some topological properties for special classes of Ba-
nach spaces. Part 1 // System Research & Information Technologies. — 2008. —
№ 1. — P. 127–143.
2. Kasyanov P.O. Galerkin's method for one class differential-operator inclusions //
Dopovidi Natcional'noi Academii Nauk Ukraini. — 2005. — № 9. — P. 20–24.
3. Kasyanov P.O., Mel'nik V.S. Faedo-Galerkin method for differential-operator inclu-
sions in Banach spaces with maps of
0
wλ -pseudomonotone type // Zbirnik prats
institutu mathematiki Nacional'noy Akademiy nauk Ukrainy. Part 2. — 2005. —
№ 1. — P. 82–105.
4. Lions J.L. Quelques m e′ thodes de r e′ solution des problem e′ s aux limites non
lineaires. — Paris: DUNOD GAUTHIER-VILLARS, 1969. — 587 p.
5. Gaevsky H., Greger K., Zaharias K. Nonlinear The operator equations and the opera-
tor-differential equations. — M.:Myr, 1977. — 337 p. (Russian translation).
6. Aubin J.P. Ekeland I. Applied Nonlinear Analysis. — Moscow: Mir, 1988. — p. 510.
Received 05.07.2007
From the Editorial Board: the article corresponds completely to submitted
manuscript.
|
| id | journaliasakpiua-article-108911 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:22:59Z |
| publishDate | 2008 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/9d/605621e3f666cbe2e59c55a486c6ea9d.pdf |
| spelling | journaliasakpiua-article-1089112018-04-11T11:07:52Z On some topological properties for special classes of Banach spaces. Part 2 О некоторых топологических свойствах специальных классов банаховых пространств. Часть 2 Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 Kasyanov, P. Mel'nik, V. We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. A compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained. Рассмотрены некоторые классы бесконечномерных банаховых пространств с интегрируемыми производными. Для нерефлексивных пространств получены лемма про компактность и основные топологические свойства данных пространств. Розглянуто деякі класи нескінченновимірних банахових просторів з інтегрованими похідними. Для нерефлексивних просторів одержано лемму про компактність та основні топологічні властивості даних просторів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2008-09-22 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/108911 System research and information technologies; No. 3 (2008); 88-100 Системные исследования и информационные технологии; № 3 (2008); 88-100 Системні дослідження та інформаційні технології; № 3 (2008); 88-100 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/108911/103828 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Kasyanov, P. Mel'nik, V. Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title_alt | On some topological properties for special classes of Banach spaces. Part 2 О некоторых топологических свойствах специальных классов банаховых пространств. Часть 2 |
| title_full | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title_fullStr | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title_full_unstemmed | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title_short | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 2 |
| title_sort | про деякі топологічні властивості спеціальних класів банахових просторів. частина 2 |
| url | https://journal.iasa.kpi.ua/article/view/108911 |
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