Метод скінченних різниць для еволюційних включень та варіаційних нерівностей
The method of finite-difference approximations, advanced by C. Bardos and H. Brezis for the nonlinear evolutionary equations, is generalized on differential-operational inclusions which are tightly connected to evolutionary variational inequalities in Banach spaces.
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| author | Kasyanov, P. O. Mel’nik, V. S. Toscano, L. |
| author_facet | Kasyanov, P. O. Mel’nik, V. S. Toscano, L. |
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| description | The method of finite-difference approximations, advanced by C. Bardos and H. Brezis for the nonlinear evolutionary equations, is generalized on differential-operational inclusions which are tightly connected to evolutionary variational inequalities in Banach spaces. |
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© P.O. Kasyanov, V.S. Mel’nik, L. Toscano, 2005
106 ISSN 1681–6048 System Research & Information Technologies, 2005, № 4
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
УДК 517.9
METHOD OF APPROXIMATION OF EVOLUTIONARY
INCLUSIONS AND VARIATIONAL INEQUALITIES BY
STATIONARY
P.O. KASYANOV, V.S. MEL’NIK, L. TOSCANO
The method of finite-difference approximations, advanced by C. Bardos and
H. Brezis for the nonlinear evolutionary equations, is generalized on differential-
operational inclusions which are tightly connected to evolutionary variational ine-
qualities in Banach spaces.
INTRODUCTION
At studying of nonlinear evolutionary equations the some spread methods are
used: Faedo-Galerkin, singular perturbations, difference approximations, nonlin-
ear semigroups of operators and others [1, 2]. The dissemination of these ap-
proaches on evolutionary inclusions and variational inequalities encounters a se-
ries of basic difficulties. The method of nonlinear semigroups of operators in
Banach spaces was developed for evolutionary inclusions in works of
A.A. Tolstonogov [3], A.A. Tolstonogov and J.I. Umanskij [4], V. Barbu [2] and
others. A method of singular perturbations H. Brezis [5] and Yu. Dubinskiy [6] on
evolutionary inclusions have disseminated in A.N. Vakulenko’s and V.S. Mel’nik
works [7–9], a method of Galerkin’s approximations in P.O. Kasyanov’s works
[10, 11].
In the present work the attempt to disseminate a method of difference ap-
proximations [1] on evolutionary inclusions and variational inequalities is under-
taken for the first time.
PROBLEM FORMALIZATION
Let Φ be separable locally convex linear topological space; Φ′ be the space
identified to topologically conjugate to Φ space such, that Φ′⊂Φ ; ),( ϕf is the
inner product (canonical pairing) of devices Φ′∈f and Φ∈ϕ .
Let the three spaces HV , and V ′ are given, moreover
Φ′⊂′⊂ΦΦ′⊂⊂ΦΦ′⊂⊂Φ VHV ,, (1)
with continuous and dense embedding;
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H is a Hilbert space (with inner product Hhh ),( 21 and corresponding norm
Hh |||| );
V be reflexive separable Banach space with norm Vv |||| ;
V ′ is the conjugate to V space with dual norm Vf ′|||| .
If Φ∈ψϕ, , that H),(=),( ψϕψϕ is inner product of devices V∈ϕ and
V ′∈ψ .
Let 21= VVV ∩ and
21
||||||=|||||| VVV ′′ ⋅+⋅⋅ , where )||||,(
iViV ⋅ , 1,2=i is re-
flexive separable Banach spaces, embedding Φ′⊂⊂Φ iV and Φ′⊂′⊂Φ iV is
dense and continuous. Spaces )||||,(
iViV ′⋅′ , 1,2=i are topologically conjugate to
)||||,(
iViV ⋅ concerning the bilinear form ),( ⋅⋅ . Then 21= VVV ′+′′ .
Let 11: VVA ′→ , RV →2:ϕ be a functional, Λ is non-bounded operator,
which operates from V to V ′ with definitional domain ),;( VVD ′Λ . The follow-
ing problem on searching of solutions by a method of finite differences is consid-
ered (see [1, chapter 2.7]):
),,;( VVDu ′Λ∈ (2)
,)()( fuuAu ∋∂++Λ ϕ (3)
where Vf ′∈ fixed element; 22: VV ′∂ →
→ϕ is subdifferential from the functional
ϕ (see [13]).
THE BASIC GUESSES
Let us assume, that a set Φ is dense in space
)||||||||,( VV vvVV ′+′∩ . (4)
Remark 1. From (4) it follows, that
.HVV ⊂′∩ (5)
Really, if Φ∈v , that VVH vvv |||||||||||| 2
′≤ whence, due to (4) it follows (5).
Remark 2. If HV ⊂ , it is possible to not introduce Φ and identifying H
and 'H , at once receive the following line-up of embeddings:
.VHV ′⊂⊂ (6)
Definition 1. The family of maps 0)}({ ≥ssG refers to as a continuous semi-
group in a Banach space X , if 0≥∀s );()( XXLsG ∈ , IdG =(0) ,
)()(=)( tGsGtsG + 0, ≥∀ ts , xxtG
w
→)( as +→ 0t Xx∈∀ .
Operator Λ . Let the family of maps 0)}({ ≥ssG be such that 0)}({ ≥ssG is
continuous semigroup on VHV ′,, , that is there are three semigroups, defined in
spaces HV , , and V ′ correspondingly, which coincide on Φ . Each of them we
shall designate as 0)}({ ≥ssG ;
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 108
0)}({ ≥ssG is non-expanding semigroup in H ,
that is 1||)(|| );( ≤HHLsG 0≥∀ s . (7)
Further let Λ− be the infinitesimal generator of a semigroup 0)}({ ≥ssG
with a definitional domain );( VD Λ (accordingly );( HD Λ or );( VD ′Λ ) in V
(accordingly in H or in V ′ ). In virtue of [14, theorem 13.35] such generator ex-
ists, moreover, it is densely defined closed linear operator in space V (accord-
ingly in H or in V ′ ).
Let 0
* )}({ ≥ssG be the semigroup conjugated to )(sG , which operates ac-
cordingly in HV , , and V ′ . Let *Λ− is the infinitesimal generator of a semi-
group 0
* )}({ ≥ssG with definitional domain );( * VD Λ in V , );( * HD Λ in H
and );( * VD ′Λ in V ′ . The operator *Λ in H (accordingly in V or in V ′ ) is con-
jugated in sense of the theory of unlimited operators to the operator Λ in H (ac-
cordingly in V or in V ′ ). It takes place the following.
Lemma 1. The sets VVD ∩);( ′Λ and VVD ∩);( * ′Λ are dense in V .
Proof. Really, Vu∈∀ 0>ε∀ Φ∈∃ϕ : ,<|||| εϕ Vu − =:nϕ
VVD
n
I ∩);(1 1
′Λ∈⎟
⎠
⎞
⎜
⎝
⎛ Λ−=
−
ϕ , ϕϕ →n in V as ∞→n .
The lemma is proved.
Now we define Λ as non-bounded operator, which operates from V to V ′
with definitional domain ),;( VVD ′Λ . Let us put
),(formthe|{=),;( *wvwVvVVD Λ→∈′Λ is continuous on
}spacefrominducedtopology,in);( * VVVD ∩′Λ . (8)
Then there is unique element :Vv ′∈ξ ),(=),( * wwv vξΛ . If ∩);( VDv ′Λ∈
V∩ , that vv Λ=ξ . Thus, generally we can put vv Λξ = , whence
VVDwwvwv ∩);(),(=),( ** ′Λ∈∀ΛΛ . (9)
If we enter on ),;( VVD ′Λ the norm VV vv ′Λ+ |||||||| , we receive a Banach
space. Let us similarly define space ),;( * VVD ′Λ .
Remark 3. If HV ⊂ , then
).;(=),;(and);(=),;( ** VDVVVDVDVVVD ′Λ′Λ′Λ′Λ ∩∩
In case when V does not include in H we assume that
),;(indense);( VVDVDV ′Λ′Λ∩ ,
),;(indense);( ** VVDVDV ′Λ′Λ∩ . (10)
Remark 4. ([1, chapter 2, remark 7.5., 7.6.]).
Method of approximation of evolutionary inclusions and variational inequalities by stationary
Системні дослідження та інформаційні технології, 2005, № 4 109
),;(0),(),,;(0),( ** VVDvvvVVDvvv ′Λ∈∀≥Λ′Λ∈∀≥Λ . (11)
Let us enter some new denotations. Let Y be some reflexive Banach space.
As )(YCv we designate the system of all nonempty convex closed bounded sub-
sets from Y . For nonempty subset YB ⊂ we consider the closed convex hull
of the given set ))(co(cl:=)(co BB Y . With multi-valued map A
it is comparable upper Y
yAd
wdyA >,<sup=]),([
)(∈
+ω and lower =_]),([ ωyA
Y
yAd
wd >,<inf=
)(∈
function of support, where ., Yy ∈ω Properties of the given
maps are considered in works [15–17]. Later on yy
w
n→ in Y will mean, that ny
weakly converges to y in space Y .
THE CLASSES OF MAPS
Let us consider the next classes of maps of pseudomonotone type:
Definition 2. Operator VVA ′→: refers to pseudomonotone, if from
Vy nn ⊂≥ 0}{ , 0yy
w
n → in V , and 0)),((lim 0 ≤−
∞→
yyyA nn
n
it follows, that
11 }{}{ ≥≥ ⊂∃ nnkkn yy :
VwwyyAwyyA
knkn
k
∈∀−≥−
∞→
)),(()),((lim 00 .
Definition 3. The next set:
})()(>,<|'{=)( VuvuvupVpv ∈∀−≤−∈∂ ϕϕϕ
refers to subdifferential map form functional R→V:ϕ in point Vv∈ .
Definition 4. Multi-valued map *: VVA →→ refers to:
1) λ -pseudomonotone, if from Vy nn ⊂≥0}{ , 0yy
w
n → in V and
0),(lim 0 ≤−
∞→
yyd nn
n
, where )(co nn yAd ∈ 1≥∀n it follows, that it is possible
to choose such 0000 }{}{,}{}{ ≥≥≥≥ ⊂⊂ nnkknnnkkn ddyy that
;]),([),(lim _00 wyyAwydVw
knkn
k
−≥−∈∀
∞→
2) bounded, if A translates arbitrary bounded in V set in bounded in *V ;
3) coercive, if +∞→+
− ]),([|||| 1 vvAv V as +∞→Vv |||| ;
4) satisfies condition )(κ if the map R]),([|||| 1 ∈→∋ +
− vvAvvV V is
bounded from below on bounded in 0\V sets, that is
Dvc
v
vvA
cVVD
V
∈∀≥∈∃−⊂∀ +
11 ||||
]),([
:Rin bounded}0{\ .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 110
Remark, that the bounded multi-valued maps and monotone multi-valued
operators, including subdifferential maps, are satisfying condition )(κ .
Definition 5. Multivalued map )(: *VCVA v→ satisfies property )(M , if
from Vy nn ⊂≥0}{ , )( nn yAd ∈ 1≥∀n : 0yy
w
n → in V , 0dd
w
n → in V ′ ,
),(),(lim 00 ydyd nn
n
≤
∞→
it follows, that )( 00 yAd ∈ .
Definition 6. Operator *)(: VVLDL →⊂ refers to maximally monotone, if
it is monotone and from )(0)),(( LDuuvuLw ∈∀≥−− it follows, that
)(LDv∈ and wvL =)( .
Lemma 2. Let V , W be Banach spaces, densely and continuously embed-
ded in locally convex linear topological space Y , VVA ′→→: , WWB ′→→: —
multi-valued λ -pseudomonotone maps and one of them is bound-valued. Then
the multi-valued operator WVWVBAA ′+′→→+ ∩::= is λ -pseudomonotone.
Proof. Let yy
w
n → in WVX ∩=: (that is yy
w
n → in V and yy
w
n → in W )
and the next inequality is holds:
0>,<lim ≤−
∞→
Xnn
n
yyd , (12)
where
)(co)(co=)(co nnnn yByAyAd +∈ . (13)
Let us prove the last equality. It is obvious, that +)(co=)(co nn yAyA
)(co nyB+ and, moreover, )(co)(co)(co nnn yByAyA +⊃ . Let us prove the in-
verse inclusion. Let x is a frontier point of )( nyA . Then =⊂∃ ≥ )(co}{ 1 nmm yAx
)(co)(co= nn yByA + : xx
w
m → in X as ∞→m , because of Mazur theorem
(see [14]), for an arbitrary convex set its weak and the strong closure is coincide.
Hence, 1≥∀m ),( nm yAv ∈∃ )( nm yBw ∈∃ : mmm xwv =+ and, taking into
account bound-valuededness of one of the maps and Banach-Alaoglu theorem, we
obtain, within to a subsequence, vv
w
m → in V , ww
w
m → in W for some
)(co nyAv∈ , )(co nyBw∈ . The statement (13) is proved. Consequently
,= nnn ddd ′′+′ where )(co nn yAd ∈′ , )(co nn yBd ∈′′ . From here, within to a sub-
sequence, we obtain one of two inequalities:
0>,<lim0,>,<lim ≤−′′≤−′
∞→∞→
Wnn
n
Vnn
n
yydyyd . (14)
Without loss of generality, let us consider, that (within to a subse-
quence) 0>,<lim ≤−′
∞→
Vnn
n
yyd . Then, due to λ -pseudomonotony of A ,
1}{}{ ≥⊂∃ nnmm yy :
.]),([>,<lim _ VvvyyAvyd Vmm
m
∈∀−≥−′
∞→
Method of approximation of evolutionary inclusions and variational inequalities by stationary
Системні дослідження та інформаційні технології, 2005, № 4 111
Let us put in last equality yv = , then
0=]),([>,<lim _yyyAyyd Vmm
m
−≥−′
∞→
.
Hence, 0=>,<lim Vmm
m
yyd −′∃
∞→
. Then, due to (12), −′<
∞→
mmn
yd ,lim
0> ≤− Wy . Taking into account (14), λ -pseudomonotony of A and B , we have
,]),([>,<lim _ VvvyyAvyd Vknkn
k
∈∀−≥−′
∞→
.]),([>,<lim _ WwwyyBwyd Wknkn
k
∈∀−≥−′′
∞→
Then from last two relations it follows
≥−′′+−′≥−
∞→∞→∞→
Wknkn
k
Vknkn
k
Xknkn
k
xydxydxyd >,<lim>,<lim>,<lim
.]),([=]),([]),([ ___ WVxxyyAxyyBxyyA ∩∈∀−−+−≥
The lemma is proved.
Lemma 3. Let V , W be Banach spaces, densely and continuously embed-
ded in locally convex linear topological space Y , VVA ′→→: , WWB ′→→: are
multi-valued coercive maps, which satisfies condition )(κ . Then the multi-valued
operator WVWVBAA ′+′→→+ ∩::= is coercive.
Proof. We obtain this statement arguing by contradiction. Let’s assume, that
:}{ 1≥∃ nnx +∞→+ WnVnXn xxx ||||||=|||||| as ∞→n , but <
||||
]),([
sup
1 Xn
nn
n x
xxA +
≥
+∞< .
Case 1. +∞→Vnx |||| as ∞→n , cx Wn ≤|||| 1≥∀n ;
0>,
||||
]),([
inf:=)(,
||||
]),([
inf:=)(
||||||||
r
w
wwB
r
v
vvA
r
WWw
B
VVv
A
+
=
+
= γγ
γγ .
Remark, that +∞→+∞→ )(,)( rr BA γγ as +∞→r . Then 1≥∀n
VnVnAnnVn xxxxAx ||||)||||(]),([|||| 1 γ≥+
− and ×≥+ )||(||
||||
]),([
VnA
Xn
nn x
x
xxA
γ
.||||and||||as
||||
|||| cxx
x
x
WnVn
Xn
Vn ≤+∞→+∞→×
In this case, due to condition )(κ , 1≥∀n
∞→→≥≥+ n
x
x
c
x
x
x
x
xxB
Xn
Wn
Xn
Wn
WnB
Xn
nn at0
||||
||||
||||
||||
)||(||
||||
]),([
1γ ,
where R1 ∈c is the constant from condition )(κ . It is clear, that
∞→+∞→+ +++ n
x
xxB
x
xxA
x
xxA
Xn
nn
Xn
nn
Xn
nn as
||||
]),([
||||
]),([
=
||||
]),([
.
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 112
We have an inconsistency with boundedness of the left part of the given ex-
pression.
Case 2. The case cx Vn ≤|||| 1≥∀ n and ∞→Wnx |||| as ∞→n is investi-
gated similarly.
Case 3. Let us consider the situation, when ∞→Vnx |||| and ∞→Wnx ||||
as ∞→n . Then,
+
+
≥∞+ +
≥ WnVn
Vn
VnA
Xn
nn
n xx
xx
x
xxA
||||||||
||||)||(||
||||
]),([
sup>
1
γ
WnVn
Wn
WnB xx
xx
||||||||
||||)||(||
+
+ γ . (15)
It is obvious, that 1≥∀n 0>
||||
||||
Xn
Vn
x
x
and 0>
||||
||||
Xn
Wn
x
x
. And, if even one of
limits, for example 0
||||
||||
→
Xn
Vn
x
x
, that 1
||||
||||
1=
||||
||||
→−
Xn
Vn
Xn
Wn
x
x
x
x
. We have an
inconsistency with (15).
The lemma is proved.
THE MAIN RESULT
Theorem. Let a) 11: VVA ′→ be bounded pseudomonotone on 1V operator, which
satisfies the following coercive condition:
+∞→+∞→
1
1
||||as
||||
)),((
V
V
u
u
uuA ; (16)
b) functional R: 2 →Vϕ is convex, lower semicontinuous and the following
takes place:
+∞→+∞→
2
2
||||as
||||
)(
V
V
v
v
vϕ ; (17)
c) The operator Λ satisfies all listed above conditions, including conditions
(7) and (10).
Then for every Vf ′∈ there exists such u , that satisfies (2) and (3).
Remark 5. If HV ⊂ , inclusion (2) implies, that );( VDVu ′Λ∈ ∩ .
Proof. The approximate solutions. Natural approximation of inclusion (3) is
inclusion
0)>()()()( hfuuAu
h
hGI
hhh ∋∂++
−
ϕ . (18)
Though, if V does not include in H (18), generally speaking, has no solu-
tions, and it is necessary to modify the given inclusion in appropriate way. We
choose such sequence 1)(0,∈hθ , that
0as0
1
→→
−
h
h
hθ . (19)
Method of approximation of evolutionary inclusions and variational inequalities by stationary
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Let us put 1=hθ when H⊂V . Further, we take
h
hGI h
h
)(
=
θ−
Λ (20)
and also replace (18) with the inclusion
fuuAu hhhh ∋∂++Λ )()( ϕ . (21)
Lemma 4. Inclusion (21) has a solution H∩Vuh ∈ .
Proof. Let us enter the map
11:= VHVHAB h ′+→+Λ ∩ . (22)
We consider the following variation inequality:
HVvuvfuvuvuB hhhh ∩∈∀−≥−+− ),()()()),(( ϕϕ . (23)
Let us prove the existence of such H∩Vuh ∈ , that is a solution of the given
inequality. The given statement follows from [15, theorem 7], if to put
1= VHV ∩ , 2= VW , BA = , ϕϕ = and under condition of realization
Lemma 5. Operator B satisfies to the following conditions:
i) ∞→+∞→
1
1
||||as
||||
)),((
VH
VH
u
u
uuB
∩
∩
; (24)
ii) 1ontonepseudomono VHisB ∩ ; (25)
iii) 1onbounded VHisB ∩ . (26)
Proof. і) As )(sG is non-stretched on H , then Hv∈∀
( )≥−≥−Λ HHhHhh vvsGv
h
vvhGv
h
vv ||||||)(||||||1),)((1=),( 2 θθ
2||||1
H
h v
h
θ−
≥ . (27)
From here it follows the coercive condition and condition )(κ for hΛ on
H . Thus, due to (2), we can use lemma 3 for maps hA Λ= on HV = and
AB = on 1= VW , whence it follows (24), if we prove, that A satisfies condition
)(κ . Really, if it is not true, then 0\}{ 11 Vw nn ⊂∃ ≥ such bounded in W , that
∞−→−
+]),([1||||
1 nnVn wwAw as ∞→n , but in virtue of boundedness of A , we
have
∞−−≥−−
≥
+ >||)(||sup)),((1||||=]),([1||||
1111 Vn
n
nnVnnnVn wAwwAwwwAw .
iіі) The boundedness of B on 1VH ∩ follows from the boundedness of hΛ
on H and A on 1V . The boundedness of hΛ on H immediately follows from
the definition of hΛ and estimation (6).
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 114
іі). Let us prove the pseudomonotony of B on 1VH ∩ . For this purpose we
use lemma 2 with hA Λ= on HV = and AB = on 1= VW . From here, due to
the pseudomonotony and to the property of bound-valuedness of A on 1V , it is
enough to prove pseudomonotony of hΛ on H . Let
0.),(lim,in ≤−Λ→
∞→
yyyHyy nnh
n
n
Then, from estimation (27) we have
0=00),(lim),(lim),(lim +≥−Λ+−Λ−Λ≥−Λ
∞→∞→∞→
yyyyyyyyyy nh
n
nhnh
n
nnh
n
.
Hence 0=),(lim yyy nnh
n
−Λ∃
∞→
. Further, Hu∈∀ , 0>s∀ let +yw =:
)( yus −+ . Then
1),(),(),(),( ≥∀−Λ−−Λ+−Λ−≥−Λ nyuwsyywyyyuyys hnhnnhnh
and
),(),(lim),(),(lim yuwuyyyuwsuyys hnh
n
hnh
n
−Λ−≥−Λ⇔−Λ−≥−Λ
∞→∞→
.
Let +→ 0s then ),(=),(),(lim uyyyuyuyy hhnh
n
−Λ−Λ−≥−Λ
∞→
and
+−Λ≥−Λ
∞→∞→
),(lim),(lim yyyuyy hnh
n
hnh
n
Huuyyuyy hnh
n
∈∀−Λ≥−Λ+
∞→
),(),(lim .
Thus we have the required statement.
The lemma is proved.
To complete the proof of lemma 4 it is necessary to show, that for fixed
1VHuh ∩∈ the variation inequality (23) is equivalent to inclusion (22). If
1VHv ∩∈ is arbitrary, then, by definition of subdifferential map, the inequality
(23) is equivalent to )()( hh uuBf ϕ∂∈− , that in turn, by definition of B , it is
equivalent to (22).
The lemma is proved.
The boundary transition on h . From lemma 4 for every 0>h the exis-
tence of such 1VHuh ∩∈ and )( hh ud ϕ∂∈ , that
fduAu hhhh =)( ++Λ . (28)
is follows. If we put in (23) 0=v , we obtain
)0(),()()),(( ϕϕ +≤+ hhhh ufuuuB . (29)
Let us prove boundedness of 0>}{ hhu in V as h close to zero. For this pur-
pose we use advantage coercive conditions (16) and (24). Let us assume, that
∞→+
21
||||||=|||||| VhVhVh uuu .
Case 1. ∞→
1
|||| Vhu , cu Vh ≤
2
|||| ;
Method of approximation of evolutionary inclusions and variational inequalities by stationary
Системні дослідження та інформаційні технології, 2005, № 4 115
0>,
||||
)(
inf
=
=:)(,
||||
)),((
inf
=
:=)(
22
||||11
||||
r
u
u
r
r
u
uuB
r
r
VVuVVu
B
ϕ
γγ ϕ .
Remark, that +∞→)(rBγ and +∞→)(rϕγ as ∞+→r . Then
11
1
1
||||)||(||)),((|||| VVBhhVh uuuuBu γ≥− and
≥
+
≥
+
≥+←
Vh
hhh
Vh
h
Vh
VV u
uuuB
u
uf
u
ff
||||
)()),((
||||
)0(),(
||||
)0(|||||||| ''
ϕϕϕ
≥+≥
Vh
VhVh
Vh
VhVhB
u
uu
u
uu
||||
||||)||(||
||||
||||)||(||
2211 ϕγγ
∞→+∞→+
+
≥ Vh
Vh
VhVh
Vh
VhVhB
uas
u
uu
cu
uu
||||
||||
||||)||(||
||||
||||)||(||
22
1
11 ϕγγ
.
We have an inconsistency with boundedness of the left part of the given ine-
quality. It is necessary to notice, that last item in a right-side of last inequality
tends to zero. It follows from boundedness from below of ϕ on the bounded sets
(see [13]).
Case 2. The case cu Vh ≤
1
|||| , ∞→
2
|||| Vhu is investigated similarly.
Case 3. Let us consider the situation, when ∞→
1
|||| Vhu , .||||
2
∞→Vhu Then,
21
22
21
11
'' ||||||||
||||)||(||
||||||||
||||)||(||
||||
)0(||||||||
VhVh
VhVh
VhVh
VhVhB
Vh
VV uu
uu
uu
uu
u
ff
+
+
+
≥+←
ϕγγϕ . (30)
It is obvious, that 0>
||||
||||
1
V
V
u
u
and 0>
||||
||||
2
V
V
u
u
. And, if even one of bounda-
ries, for example, 0
||||
||||
1 →
V
V
u
u
, that 1
||||
||||
1=
||||
||||
12 →−
V
V
V
V
u
u
u
u
. We have an incon-
sistency in (30). Thus,
0asinboundedare →hVuh . (31)
Prove, that
0asinboundedare 2 →′ hVdh . (32)
First, from equality (28) we receive:
∞→→+∞⊂∀∞ nhhud nnnhnh
n
as0:)(0,}{<),(sup . (33)
Due to ,Huh ∈ from equality (28), estimation (31) and boundednesses of an
operator A we have
+−+ )),((sup),(sup=),(sup nhnh
nnh
nnhnh
n
uuAufud
+∞+′≤Λ−+ ′ <||||sup||)(||sup||||sup||||),(sup Vnh
n
Vnh
n
Vnh
n
Vnhnhnh
n
uuAufuu .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 116
Now, in virtue of (33), we prove (32). From )(
nhnh yd ϕ∂∈ and from defini-
tion of subdifferential map, 2Vv∈∀
≤−+≤−+≤ )()(),(sup),(sup),(sup),(sup nhnhnh
nnhnh
nnhnh
nnh
n
yvydyvdydvd ϕϕ
∞+−+≤ <)(inf)(),(sup nh
nnhnh
n
yvyd ϕϕ ,
as functional ϕ is bounded from below on bounded sets. From here, under Ba-
nach-Steingauss theorem (32) is follows.
From (31) and boundedness of an operator A on 1V it follows, that
0asinboundedare)( 1 →′ hVuA h . (34)
From equality (28), estimates (31), (32) and (34), under Banach-Alaoglu
theorem, the existence of such subsequences 0>1 }{}{ hhnnh uu ⊂≥ , ⊂≥1}{ nnhd
0>}{ hhd⊂ , 0>1 )}({)}({ hhnnh uAuA ⊂≥ 0)<(0 →nh , which further we will des-
ignate simply as 0>}{ hhu , 0>}{ hhd , 0>)}({ hhuA accordingly, and elements
Vu∈ , 1V∈χ , 2Vd ∈ the next convergences
ddVuAVuu
w
h
w
h
w
h →→→ 'in)(in 1χ
VLuuLV
w
hh ′→′ inin 2 (35)
are follows, in particular,
'in:=)(=: VwdduAv
w
hhh +→+ χ . (36)
Let us enter the following map: )(:)()(=)( VCVvvAvC v ′→∂+ ϕ . Now
prove, that the given map satisfies property )(M . For this purpose it is enough to
show λ -pseudomonotony of C on V . If C is λ -pseudomonotone on V and
Vy nn ⊂≥0}{ , )( nn yCd ∈ 1≥∀n :
),(),(limand'in,in 0000 ydydVddVyy nn
n
w
n
w
n ≤→→
∞→
,
then
0=),(),(),(lim),(lim),(lim 000000 ydydydydyyd n
n
nn
n
nn
n
−≤−+≤−
∞→∞→∞→
.
Hence, due to λ -pseudomonotony of C it follows, that ⊂∃ ≥1}{ kkny
1}{ ≥⊂ nny , 11 }{}{ ≥≥ ⊂ nnkkn dd :
_00 ]),([),(lim wyyCwydVw
knkn
k
−≥−∈∀
∞→
.
From here
≤−≤−≤−
∞→∞→
− ),(lim),(lim]),([ 00 wydwydwyyC nn
nknkn
k
Method of approximation of evolutionary inclusions and variational inequalities by stationary
Системні дослідження та інформаційні технології, 2005, № 4 117
Vwwyd ∈∀−≤ ),( 00 .
Hence )( 00 yCd ∈ . Thus C satisfies condition )(M on V .
In turn, lemma 2, pseudomonotony and bounded-valuedness of A on 1V
provides the last, if to prove λ -pseudomonotony of ϕ∂ on 2V . As it is known,
the last statement follows from [20.ІІІ, lemma 2, remark 2].
We use the fact, that C satisfies property )(M on V . Let us take v from
);( VDV ′Λ∗∩ . From (28) and (36) it follows, that
),(=),(),( * vfvvvu hhh +Λ . (37)
But
vhG
h
I
v
h
hGIv h
h
*
*
* )()(=
θ−
+
−
Λ (38)
and due to (20), vvh
** Λ→Λ in V ′ ; and consequently, as h tends to zero in (37)
we receive:
);(V),(=),(),( * VDvvfvwvu ′Λ∈∀+Λ ∗∩
and (in virtue of (7), (8)) ),,( VVDu ′Λ∈
fwu =+Λ
and we prove the theorem, if we show that
)(uCw∈ . (39)
On the other hand, because of (28) and (36) for H);(V ⊂′Λ∈ VDv ∩ , we have
≤−−Λ−−Λ−−− )),((),(),(=),( vuvuvuvvufvuv hhhhhhhh
),(),( vuvvuf hhh −Λ−−≤ ,
as 0≥Λ h in );( HHΛ . From here
);(V),(),(),(),(suplim VDvvuvvufvwuv hh ′Λ∈∀−Λ−−−≤ ∩ .
But, due to (9), the same inequality is fulfilled ),;( VVDv ′Λ∈∀ , and when
uv = we obtain
),(),(suplim uwuv hh ≤ ,
and also (39), because of C is the operator of type )(M . The theorem is proved.
Example. Let Ω in nR be a bounded region with regular boundary Ω∂ ,
]0,[= TS be finite time interval, )(0;= TQ ×Ω , )(0;= TT ×Ω∂Γ . As operator A
we take ))((=))(( tuAtAu , where
ϕϕϕϕϕ 2
2
1=
=)( −
−
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
−∑ p
i
p
ii
n
i xxx
A (40)
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 118
(see [1, chapter 2.9.5]); V is closed subspace in Sobolev space )(1, ΩpW , 1>p
such, that
)()( 1,1,
0 Ω⊂⊂Ω pp WVW (41)
and
)).(;(0,=)),(;(0,=),;(0,= 222221 ΩΩ LTLVLTLHVTLV p
We consider convex lower semicontinuous coercive functional RR →:ψ
and its subdifferential RR→→Φ : , that satisfies growth condition.
If we put 21= VVV ∩ (from here ))(;(0,);(0,= 22
* Ω+′ LTLVTLV q , where
1=11
qp
+ ), we obtain the situation (6), if 2≥p . At 2<<1 p the common case
takes place, if to take );(0,= VTDΦ (see [1]).
As an operator Λ we take the derivation operator in sense of space of sca-
lar distributions );(0, ** VTD , }|{=:=)',;( VHyHVyWVVD ′+∈′∈Λ ∩
}at0;at)({=:)()( stststtsG ≤≥−ϕϕ .
Due to [1, chapter 2.9.5] and to the theorem, the next problem:
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
−
∂
∂
−
∑
i
p
ii
n
i x
txy
x
txy
xt
txy ),(),(),(
2
1=
Qtxftxytxytxy p ona.e.),()),((),(|),(| 2 ∋Φ++ − , (42)
Ωona.e.0=0),(xy , (43)
T
A
txgtxy
Γ
∂
∂ ona.e.),(=),(
ν
, (44)
has a solution Wy∈ , obtained by finite differences method. Remark, that in
(42)–(44) )(,': 20 Ω∈∈ LyVf are fixed elements.
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Received 22.06.2005
From the Editorial Board: The article corresponds completely to submitted
manuscript.
|
| id | journaliasakpiua-article-165558 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:24:57Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/da/ab8b70260096e7be7e7fdcafdb8002da.pdf |
| spelling | journaliasakpiua-article-1655582019-04-25T16:24:40Z Method of approximation of evolutionary inclusions and variational inequalities bi stationary Метод конечных разностей для эволюционных включений и вариационных неравенств Метод скінченних різниць для еволюційних включень та варіаційних нерівностей Kasyanov, P. O. Mel’nik, V. S. Toscano, L. The method of finite-difference approximations, advanced by C. Bardos and H. Brezis for the nonlinear evolutionary equations, is generalized on differential-operational inclusions which are tightly connected to evolutionary variational inequalities in Banach spaces. Обобщен метод конечно-разностных аппроксимаций, развитый Бардосом и Брезисом для нелинейных эволюционных уравнений, на дифференциально-операторные включения, которые тесно связаны с эволюционными вариационными неравенствами в банаховых пространствах. Узагальнено метод кінцево-різницевих апроксимацій, розвинутий Бардосом і Брезисом для нелінійних еволюційних рівнять, на диференційно-операторні включення, які тісно пов’язані з еволюційними варіаційними нерівностями у банахових просторах. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-04-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/165558 System research and information technologies; No. 4 (2005); 106-119 Системные исследования и информационные технологии; № 4 (2005); 106-119 Системні дослідження та інформаційні технології; № 4 (2005); 106-119 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/165558/164760 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Kasyanov, P. O. Mel’nik, V. S. Toscano, L. Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title | Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title_alt | Method of approximation of evolutionary inclusions and variational inequalities bi stationary Метод конечных разностей для эволюционных включений и вариационных неравенств |
| title_full | Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title_fullStr | Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title_full_unstemmed | Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title_short | Метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| title_sort | метод скінченних різниць для еволюційних включень та варіаційних нерівностей |
| url | https://journal.iasa.kpi.ua/article/view/165558 |
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