The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps
We consider the main classes of Wλ0-pseudomonotone multi-valued maps. The main properties of these operators have been investigated. The new classes of these operators have been obtained. Рассматриваются основные классы Wλ0-псевдомонотонных отображений. Исследованы базовые свойства этих операторов....
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2007
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| Cite this: | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps / P.O. Kasyanov, V.S. Mel’nik, L. Toscano // Систем. дослідж. та інформ. технології. — 2007. — № 3. — С. 122-144. — Бібліогр.: 20 назв. — англ. |
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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. |
| citation_txt | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps / P.O. Kasyanov, V.S. Mel’nik, L. Toscano // Систем. дослідж. та інформ. технології. — 2007. — № 3. — С. 122-144. — Бібліогр.: 20 назв. — англ. |
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| description | We consider the main classes of Wλ0-pseudomonotone multi-valued maps. The main properties of these operators have been investigated. The new classes of these operators have been obtained.
Рассматриваются основные классы Wλ0-псевдомонотонных отображений. Исследованы базовые свойства этих операторов. Получены новые классы рассматриваемых отображений.
Розглядаються основні класи Wλ0-псевдомонотонних відображень. Досліджено базові якості цих відображень. Отримано нові класи відображень, що розглядаються.
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| first_indexed | 2025-12-07T18:54:25Z |
| format | Article |
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P.O. Kasyanov, V.S. Mel'nik, L. Toscano, 2007
122 ISSN 1681–6048 System Research & Information Technologies, 2007, № 3
TIДC
НОВІ МЕТОДИ
В СИСТЕМНОМУ АНАЛІЗІ, ІНФОРМАТИЦІ
ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
UDC 517.9
THE CLASSES AND THE MAIN PROPERTIES OF THE MULTI-
VALUED
0λ
W -PSEUDOMONOTONE MAPS
P.O. KASYANOV, V.S. MEL’NIK, L. TOSCANO
We consider the main classes of
0λ
W -pseudomonotone multi-valued maps. The
main properties of these operators have been investigated. The new classes of these
operators have been obtained.
1. INTRODUCTION
One of the most effective approach to investigate nonlinear problems, represented
by partial differential equations, inclusions and evolution inequalities with bound-
ary values, consists in the reduction of them into equations in Banach spaces gov-
erned by nonlinear operators. The given theory was developed by many authors
[1–20]. In particular, the idea of
0λ
W -pseudomonotone maps was introduced in
I.V.Skripnik paper and it was developed in papers [3–10, 14, 17, 19].
Here we investigate the main properties of the given multi-valued operators.
In particular, we will show that the sum of these operators is
0λ
W -
pseudomonotone, it is difficult in the classical definitions. We will also con-
sider the generous pseudomonotone operators and we will prove its
0λ
W -
pseudomonotony.
Finally, we will consider a class of the
0λ
W -pseudomonotone multi-valued
maps and, by using the obtained results and one abstract result for such operators
[7], we obtain the solvability for a class of nonlinear evolutional problems.
2. CLASSES OF MAPS
Let )||||,( YY ⋅ be some Banach space,
*
2: YYA → be a multi-valued map. We
consider its corresponding maps
*
2:co YYA → and
**
2:co YYA → defined by
the relations ))((co=))(co( yAyA and ))((co=))(co(
**
yAyA respectively, where
* is * -weak closure in the space *Y .
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 123
For each multi-valued map A we introduce its upper and lower function of
support:
X
yAd
X
yAd
wdyAwdyA >,<inf=]),([,>,<sup=]),([
)()( ∈
−
∈
+ ωω ,
where Xy ∈ω, . We also consider its upper and lower norms:
.||||inf=||)(||,||||sup=||)(|| *
)(
*
)( XyAdXyAd
dyAdyA
∈
−
∈
+
Proposition 1. Let
*
2:, YYBA → . Then the next relations are valid:
1) +++ +≤+ ]),([]),([]),([ 2121 vyAvyAvvyA ,
−−− +≥+ ]),([]),([]),([ 2121 vyAvyAvvyA ,
−++ +≥+ ]),([]),([]),([ 2121 vyAvyAvvyA ,
−+− +≤+ ]),([]),([]),([ 2121 vyAvyAvvyA y∀ , 1v , Yv ∈2 ;
2) −+ −− ]),([=]),([ vyAvyA ,
)()()( ]),([]),([=]),()([ −+−+−+ ++ vyBvyAvyByA Yvy ∈∀ , ;
3) )(
*
)( ]),(co[=]),([ −+−+ vyAvyA Yvy ∈∀ , ;
4) YvyAvyA ||||||)(||]),([ )()( −+−+ ≤ ,
)()( ||)(||||)(||))(),(( −+−+ −≥ yByAyByAd H ,
−++ −≥− ||)()(||)()(|| yByAyByA ,
where ),( ⋅⋅Hd is the Hausdorff metric;
5) ++ ||)(=||||)(co||
*
yAyA and if the space Y is reflexive then
YyyAyA ∈∀−− ||)(=||||)(co||
*
;
6) the functional ++ →⋅ R)(:|||| *XCv defines the norm on )( *XCv ;
7) the functional +− →⋅ R)(:|||| *XCv satisfies the conditions:
a) 0=||)(||)(0 −⇔∈ yAyA ,
b) XyyAyA ∈∈∀−− ,||)(|||=|||)(|| Rααα ,
c) −−− +≤+ ||)(||||)(||||)()(|| yByAyByA .
Proof. The properties 1), 2), 4), 6), 7) can be proved directly. Property 3) is
well known. Let us consider the property 5). It is obvious that ≥+||)(co||
*
yA
++ ≥≥ ||)(||||)(co|| yAyA , and so we will prove the inverse inequality. For arbi-
trary )(co
*
yAf ∈ there exists the sequence )(co yAfn ∈ such that ff n → * -
weakly in *Y and from the Banach-Steinhaus theorem it follows
** ||||||||lim||)(co||
YYn
n
ffyA ≥≥
∞→
+ .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 124
Since the last inequality is valid for all )(co
*
yAf ∈ then
.||)(co||=||)(co||
*
++ yAyA
Let us prove that ++ ≤ ||)(||||)(co|| yAyA . Let )(co yAf ∈ be arbitrary then
for 1≥n there exist nαα ,...,1 ( 0≥iα , 1=
1=
i
n
i
α∑ ), ngg ,...,1 ( )(yAgi ∈ ) such
that ii
n
i
gf α∑
1=
= . Hence
.||)(||=||)(||||||||||
1=
*
1=
* ++∑∑ ≤≤ yAyAgf i
n
i
Yii
n
i
Y
αα
From here and from the arbitrariness of )(co yAf ∈ we obtain the required ine-
quality which proves the first equality in 5). Let us prove the second one. Let us
introduce the mapping
R→⊂×⊂ )())((: 1
* YBYyAf
defined by the equality Yddf >,=<),( ξξ where 1B is the unitary closed sphere
in the space Y with the center in zero. Let ),(=)( ξξ ⋅⋅ ff then =)(* pfξ
+−= ]),(co[
*
ξpyA and
}<]),(co[|{=dom
*
* +∞−∈ +ξξ pyAYpf .
Notice that )dom(int0 *
1
ξ
ξ
f
B
∈
∈ . In fact, *
0dom0 f∈ , *
1 dom ξfB ⊂ and the
function f satisfies the condition of non-symmetric theorem on minimax [14].
Therefore
),,(infsup=),(supinf
)(11)(
ξξ
ξξ
dfdf
yAdBByAd ∈∈∈∈
from which the required equality follows.
Proposition 2. The inclusion )(co
*
yAd ∈ is fulfilled if and only if
YvvdvyA Y ∈∀≥+ >,<]),([ .
Proof. Let )(co
*
yAd ∈ then Yv∈∀ , from the proposition 1, it follows that
.]),([=]),(co[>,<
*
++≤ vyAvyAvd Y
Now let the inequality
YvvdvyA Y ∈∀≥+ >,<]),([
be valid and nevertheless )(co
*
yAd ∉ . The set )(co
*
yA is convex and closed in
);( * YYσ -topology of the space *Y , therefore from the separability theorem there
exists Yv ∈0 such that
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 125
YvdvyAvyA >,<<]),(co[=]),([ 00
*
0 ++
which contradicts the condition of the proposition.
Proposition 3. Let }{=:),( +∞∪→×⊂⋅⋅ RRYYDa . For each YDy ⊂∈
a functional ),( wyawY ∋ is positive homogeneous convex and lower semi-
continuous if and only if there exists a multi-valued map
*
2: YYA → such that
DAD =)( and
.),(]),([=),( YwADywyAwya ∈∈∀+
Proof. Let
*
2)(: YYADA →⊂ . Then for each )(ADy∈ the functional
+∋ ]),([=),( vyAvyavY is positive homogeneous and semi-additive since the
proposition 1. Hence it is convex. Its lower semicontinuity is obvious.
Now let ),( vyavY ∋ be a positive homogeneous convex and lower
semicontinuous functional for each YDy ⊂∈ . Since 0=0),(ya , it is the point-
wise upper bound of a set of continuous linear functionals. We denote this set by
*)( YyA ⊂ . Thus +]),([=),( vyAvya .
We remind that the multi-valued map
*
2)(: YYADA →⊂ is called mono-
tone if 1y∀ , )(2 ADy ∈ 0>,< 2121 ≥−− Yyydd )( 11 yAd ∈∀ , )( 22 yAd ∈ .
By using the above mentioned introduced brackets it is easy to note that the
multi-valued operator
*
2)(: YYADA →⊂ is monotone if and only if
+− −≥− ]),([]),([ 212211 yyyAyyyA )(, 21 ADyy ∈∀ .
Besides the usual monotony of the multi-valued maps we are interested in:
• N -monotony, i.e.
)(,]),([]),([ 21212211 ADyyyyyAyyyA ∈∀−≥− −− ;
• V -monotony, i.e.
)(,]),([]),([ 21212211 ADyyyyyAyyyA ∈∀−≥− ++ ;
• w -monotony, i.e.
)(,]),([]),([ 21212211 ADyyyyyAyyyA ∈∀−≥− −+ .
Remark 1. Together with the forms +a , −a we consider the forms
|>,<|sup=]]),([[=),(
)(
wdyAya
yAd∈
++ ωω and =−− ]]),([[=),( ωω yAya
>|,|<inf
)(
wd
yAd∈
= Xy ∈∀ ω, . Thus it is obvious that
XyAyAyAyA ||||||)(||]]),([[|]),([|]),([ ωωωω ++++ ≤≤≤ ,
XyAyAyAyA ||||||)(||]]),([[|]),([|]),([ ωωωω −−−− ≤≤≤ .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 126
Remark 2. Further yy
w
n → in Y will mean that ny weakly converges to y
in the space Y . If Y is not reflexive, then yy
w
n → in *Y means that ny
*-weakly converges to y in the space *Y . We denote as )(YCv the family of all
nonempty closed convex bounded subsets of Y .
Definition 1. Let )(AD be some subset. The multi-valued map ⊂)(: ADA
*
2YY →⊂ is called:
• )(−+ -coercive if +∞→−+
−
)(
1 ]),([|||| yyAy Y as +∞→Yy |||| , )(ADy∈ ;
• uniformly )(−+ -coercive if for some 0>c
);(,||||as
||||
||)(||]),([ )()( ADyy
y
yAcyyA
Y
Y
∈+∞→+∞→
− −+−+
• bounded if for any 0>L there exists 0>l such that lyA ≤+||)(||
)(ADy∈∀ Ly Y ≤|||| ;
• locally bounded if for any fixed )(ADy∈ there exist the constants 0>m
and 0>M such that MA ≤+||)(|| ξ when my Y ≤− |||| ξ , )(AD∈ξ ;
• d -closed if from the fact that )()( ADyyAD n ∈→∋ strongly in Y it
follows
YyAyA n
n
∈∀≥ −−
∞→
ϕϕϕ ]),([]),([lim .
Let W be also a normalized space with the norm W|||| ⋅ . We consider
YW ⊂ with continuous embedding.
Definition 2. The multi-valued map
*
2)(: YYADA →⊂ with the convex
definitional domain )(AD is called:
• radial lower semicontinuous if for any fixed y , )(:)( ADyAD ∈−∈ ξξ
−+
+→
≥+ ]),([]),([lim
0
ξξξ yAtyA
t
;
• radial continuous if the real function −+→∋ ]),([][0, ξξε tyAt is con-
tinuous from the right in the point 0=t for any fixed y , )(:)( ADyAD ∈−∈ ξξ ;
• radial continuous from above if the real function
++→∋ ]),([][0, ξξε tyAt
is upper semi-continuous from the right in point 0=t for any fixed y , )(AD∈ξ ;
• an operator with semi-bounded variation on W (with ),( WY -semi-
bounded variation) if )(, 21 ADyy ∈∀ , Ry Y ≤|||| 1 , Ry Y ≤|||| 2
WyyRCyyyAyyyA ||||;(]),([]),([ 21212211 ′−−−≥− +− ;
• an operator with N -semi-bounded variation on W if
)||||;(]),([]),([ 21212211 WyyRCyyyAyyyA ′−−−≥− −− ;
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 127
• an operator with V -semi-bounded variation on W if
;)||||;(]),([]),([ 21212211 WyyRCyyyAyyyA ′−−−≥− ++
• λ -pseudomonotone on W ( λw -pseudomonotone), if for every sequence
)(}{ 0 ADWy nn ∩⊂≥ such that 0yy
w
n → in W , from the inequality
0>,<lim 0 ≤−
∞→
Ynn
n
yyd , (2.1)
where )( nn yAd ∈ 1≥∀n , it follows the existence of 1}{ ≥kkny from 1}{ ≥nny and
1}{ ≥kknd from 1}{ ≥nnd such that
)(]),([>,<lim 00 ADwwyyAwyd Yknkn
k
∈∀−≥− −
∞→
; (2.2)
• 0λ -pseudomonotone on W (
0λ
w -pseudomonotone), if for every se-
quence )(}{ 0 ADWy nn ∩⊂≥ such that 0yy w
n → in W , 0dd
w
n → in *Y , where
)( nn yAd ∈ 1≥∀n , from the inequality (2.1), it follows the existence of
1}{ ≥kkny from 1}{ ≥nny and 1}{ ≥kknd from 1}{ ≥nnd such that (2.2) is true.
The mentioned above multi-valued map satisfies:
• the property )()( −+κ , if for every bounded set D in X there exists Rc∈
such that
}.0{\||||]),([ )( DvvcvvA X ∈∀−≥−+
Here Φ∈C , i.e. RR →⋅ +:);( 1rC is a continuous function for every 01 ≥r
and such that 0);( 21
1 →− rrC ττ for +→ 0τ 0, 21 ≥∀ rr and W|||| ′⋅ is the (semi)
norm on ,Y that is relatively compact on W and relatively continuous on Y !
Remark 3. The idea of the passage to a subsequences in the definition 1 was
adopted by us from Skripnik's work [15].
Now let 21= YYY ∩ , where )||||,(
11 YY ⋅ and )||||,(
22 YY ⋅ are Banach spaces.
Definition 3. The pair of the multi-valued maps
*
1
1 2)(: YYADA →⊂ and
*
2
2 2)(: YYBDB →⊂ is called s -mutually bounded, if for every 0>M there ex-
ists 0>)(MK such that from
)()(,|||| BDADyMy Y ∩∈≤ and Myydyyd YY ≤+
2211 >),(<>),(<
we have
)(||)(||or),(||)(||or *
2
2*
1
1 MKydMKyd
YY
≤≤
for some selectors Ad ∈1 and Bd ∈2 .
Remark 4. Further
*
2: YYA → will mean that A maps Y into ∅\2
*Y ,
i.e. A is a multi-valued map with nonempty bounded values.
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 128
3. THE MAIN PROPERTIES OF THE
0λ
w -PSEUDOMONOTONE MAPS
Lemma 1. Let
*
1
1 2: YYA → and
*
2
2 2: YYB → be some multi-valued )(−+ -
coercive maps that satisfy the condition )()( −+κ . Then the multi-valued operator
*
2::= YYBAC →+ is also )(−+ -coercive.
Proof. We obtain this statement arguing by contradiction. Let ⊂∃ ≥1}{ nny
:Y⊂ +∞→+
21
||||||=|||||| YnYnYn yyy as +∞→n , but
+∞−+
≥
<
||||
]),([
sup )(
1 Yn
nn
n y
yyC
. (3.1)
Case 1. +∞→
1
|||| Yny as ∞→n , cy Yn ≤
2
|||| 1≥∀n .
0.>,
||||
]),([
inf=:)(,
||||
]),([
inf
=
:=)(
2
)(
2
||||1
)(
1
||||
r
w
wwB
r
v
vvA
r
r
YrYw
B
YYvA
−+
=
−+ γγ
We remark that +∞→+∞→ )(,)( rr BA γγ as +∞→r . Then 1≥∀n
11)(1
||||)||(||]),([1|||| YnYnAnnYn yyyyAy γ≥−
−+ and ≥−+
Yn
nn
y
yyA
||||
]),([ )(
+∞→≥
Yn
Yn
YnA y
y
y
||||
||||
)||(|| 1
1
γ as +∞→
1
|||| Yny and cy Yn ≤
2
|||| .
Due to the condition )()( −+κ for every 1≥n
,as0
||||
||||
||||
||||
)||(||
||||
]),([ 2
1
2
2
)( ∞→→−≥≥−+ n
y
y
c
y
y
y
y
yyB
Yn
Yn
Yn
Yn
YnB
Yn
nn γ
where Rc ∈1 is a constant as in the condition )()( −+κ with
{ }cyYyD Y ≤∈
22 ||||= .
It is obvious that
.as
||||
]),([
||||
]),([
=
||||
]),([ )()()( ∞→+∞→+ −+−+−+ n
y
yyB
y
yyA
y
yyC
Yn
nn
Yn
nn
Yn
nn
This is in contradiction with (3.1).
Case 2. The case when cy Yn ≤
1
|||| 1≥∀n and ∞→
2
|||| Yny as +∞→n
can be examined in the same way.
Case 3. Let us consider the situation +∞→
1
|||| Yny and +∞→
2
|||| Yny as
+∞→n . Then
+
+
≥∞+ −+
≥ 21
1
1
)(
1 ||||||||
||||
)||(||
||||
]),([
sup>
YnYn
Yn
YnA
Yn
nn
n yy
y
y
y
yyC
γ
21
2
2 ||||||||
||||
)||(||
YnYn
Yn
YnB yy
y
y
+
+ γ . (3.2)
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 129
It is obvious that 1≥∀n 0>
||||
||||
1
Yn
Yn
y
y
and 0>
||||
||||
2
Yn
Yn
y
y
and moreover if even
one of the boundaries, for example, 0
||||
||||
1 →
Yn
Yn
y
y
, then −=1
||||
||||
2
Yn
Yn
y
y
1
||||
||||
1 →−
Yn
Yn
y
y
. Then we have a contradiction in (3.2).
Lemma 2. Every strict multi-valued operator
*
2: YYA → with
);( WY -semi-bounded variation is bounded-valued, i.e.
*
2: YYA → .
Proof. We remark that for every Yy∈
}{]),([ +∞∈→∋ + Rωω yAY , }{]),([ −∞∈→∋ − Rωω yAY .
So, due to the definition of the semi-bounded variation on ),( WY we obtain
that for all Y∈ω , for some 0>),(= yRR ω
.<)||||;(]),([]),([ +∞′++≤ −+ WA RCyAyA ωωωω
From last, in virtue of Banach-Steinhaus theorem, it follows that <||)(|| +yA
+∞< for every Yy∈ .
Lemma 3. The multi-valued operator
*
2: YYA → with );( WY -semi-
bounded variation is locally bounded.
Proof. We obtain this statement arguing by contradiction. If A is not locally
bounded then for some Yy∈ there exists a sequence Yy nn ⊂≥1}{ such that
yyn → in Y and +∞→+||)(|| nyA as +∞→n . We suppose that
Ynnn yyyA ||||||)(||1= −+ +α
for every 1≥n . Then, due to the proposition 1, Y∈∀ω and some > 0R we
have
≤−++−≤ ++
−
+
− }]),([]),({[]),([ 11
nnnnnnn yyyAyyyAyA ωαωα
{ })||||;(]),([]),([1
WnAnnnn yyRCyyyAyyyA ′−−+−+++−≤ ++
− ωωωα .
Since the sequence }{ 1−
nα is bounded and WWn yy |||||||| ′→′−− ωω (according to
the assumption YW k |||||||| ξξ ≤′ for all Yy∈ ), due to proposition 1, we have
+′−−≤≥∀ −
+
− )||||;({]),([1 11
WnAnnn yyRCyAn ωαωα
,1}||||||)(|| 1NyyyA Yn ≤+−+⋅++ + ωω
where 1N does not depend on 1≥n . Thus,
YyA nn
n
∈∀∞+
−
≥
ωωα <]),([sup 1
1
.
Therefore, since the Banach-Steinhaus theorem, there exists 0>N such that
( ) 1.||||||)(||1=||)(|| ≥∀−⋅+≤ ++ nyyyANNyA Ynnnn α
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 130
By choosing 10 ≥n from the condition 1/2|||| ≤− yyN n 0nn ≥∀ we obtain that
for every 0nn ≥ NyA n 2||)(|| ≤+ , which contradicts the assumption. So, the local
boundedness is proved.
Lemma 4. The multi-valued operator
*
2: YYA → with );( WY -semi-
bounded variation has the property )(Π : if for some 0, 21 >kk and Ad ∈
21 ||||:eachfor>),(< kyYykyyd YY ≤∈≤
then there exists 0>C such that
.||||:allfor||)(|| 2* kyYyCyd YY
≤∈≤
Proof. In virtue of the locally boundedness of A there exist 0>ε and
0>εM such that εξξ ε ≤∀≤+ YMA ||||||)(|| . It means that for some ε≥R
{ }≤+−
≤
≤
≤
+ Y
Y
Y
Y
Y
yydyyAydyd >),(<]),([1sup>),(<1sup=||)(||
||||||||
* ξ
εε
ξ
εε ξξ
≤′−++−
≤
≤ − )}||||;(>),(<]),({[1sup
||||
WAY
Y
yRCyydyA ξξξ
εεξ
≤′−++−⋅
≤
≤ + )}||||;(>),(<||||||)({||1sup
||||
WAYY
Y
yRCyydyA ξξξ
εεξ
ClkMkM =)(1
12 +++≤ εεε
ε
,
where +∞′−
≤≤
<)||||;(supsup=
||||2||||
WA
YYy
yRC
k
l ξ
εξ
, since RR →⋅ +:);(RC is a
continuous function and W|||| ′⋅ is relatively continuous Y|||| ⋅ on Y .
Remark 5. It is obvious that if one of the maps of the pair *:, XXBA →
→ is
bounded, then the pair );( BA is s -mutually bounded. Moreover, if the pair
);( BA is s -mutually bounded and each of them satisfies the condition )(Π , then
the operator *:= XXBAC →
→+ satisfies the property )(Π .
Lemma 5. Let Y be a reflexive Banach space. Then every
λ -pseudomonotone on W map is 0λ -pseudomonotone on W . For bounded
maps the converse implication is true.
Proof. The direct implication is obvious. Let us prove the converse implica-
tion. We consider the 0λ -pseudomonotone on W map *: YYA →
→ , yyn →
weakly in W , the (2.1) holds, where )(co
*
nn yAd ∈ . From the boundedness of
the operator A it immediately follows the boundedness of Aco
*
and so the
boundedness of the sequence }{ nd in *Y . Consequently, there exists a subse-
quence }{}{ nm dd ⊂ and, respectively, }{}{ nm yy ⊂ , such that ddm → weakly
in *Y and at the same time
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 131
0.>,<lim>,<lim ≤−≤−
∞→∞→ YnnnYmmm
vydvyd
However the operator A is 0λ -pseudomonotone on W , therefore there exist
the subsequences }{}{ 1 mkkn yy ⊂≥ and }{}{ 1 mkkn dd ⊂≥ for what (2.2) is true.
This proves our statement.
Remark 6. Let us pay our attention on the fact that for the classical defini-
tions (not passing to the subsequences) this statement is problematically!
In F. Browder and P. Hess work [16] the class of generous pseudomonotone
operators has been introduced.
Definition 4. The operator )(: *YCYA v→ is called generous pseudo-
monotone on W , if for each pair of sequences Wy nn ⊂≥1}{ and *
1}{ Yd nn ⊂≥
such that )( nn yAd ∈ , yyn → weakly in W , ddn → *-weakly in *Y , from the
inequality
YYnnn
ydyd >,<>,<lim ≤
∞→
(3.3)
we have )(yAd ∈ and YYnn ydyd >,<>,< → .
Proposition 4. Every generous pseudomonotone on W operator is 0λ -
pseudomonotone on W .
Proof. Let yyn → weakly in W , ddyA nn →∋)( *-weakly in *Y and
(3.3) holds (we remark that in this case the inequality (2.1) is also true). Then, in
view of the generous pseudomonotony, YYnn ydyd >,<>,< → , )(yAd ∈ , con-
sequently, in virtue of the proposition 2,
[ ] .),(>,=<>,<lim YvvyyAvydvyd YYnn
n
∈∀−≥−− −
∞→
The converse statement in the proposition is not true, but
Proposition 5. Let
*
2: YYA → be a 0λ -pseudomonotone operator. Then
the next property takes place: from yyn → weakly in W , ddyA nn →∋)(co
*
*-
weakly in *Y and from the inequality (2.1) the existence of the subsequences
}{}{ nm yy ⊂ and }{}{ nm dd ⊂ such that YYmm ydyd >,<>,< → , with
)(co
*
yAd ∈ , follows.
Proof. Let }{},{ nn dy be required sequences, consequently, one can choose
such subsequences }{},{ mm dy , that the inequality (2.2) is true. By fixing in the
last relation y=ω , we get
0>,< →− Ymm yyd or YYmm ydyd >,<>,< → ,
[ ] YvvyyAvydvyd Ymm
m
Y ∈∀−≥−− −
∞→
),(>,<lim=>,< .
From here, in virtue of the proposition 2 we obtain that )(co
*
yAd ∈ .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 132
Proposition 6. Let *
10 := YYAAA →
→+ , where *
0 : YYA →
→ is a monotone
map, and the operator *
1 : YYA →
→ has the following properties:
1) there exists a linear normalized space Z in which W is compactly and
densely enclosed and ZY ⊂ with continuous and dense embedding;
2) the operator *
1 : ZZA →
→ univocal and locally polynomial, i.e. 0>R∀
there exists )(= Rnn and a polynomial function α
α
α
λ tRtP
n
R )(=)(
<0
∑
≤
with
continuous factors 0)( ≥Rαλ such that the estimation is valid
( ) 1,2.=,)()( 21
)(
2111
*
iRyyyPyAyA ZiZR
Z ≤∀−≤− +
Then A is the operator with semi-bounded variation on W .
Proposition 7. Let in the previous proposition the operator *
0 : YYA →
→ be
N -monotone, and instead of the condition 2) we make the following one:
2') a map (multi-valued) *
1 : ZZA →
→ is locally polynomial in the sense that
0>R∀ there exists )(= Rnn and a polynomial )(tPR for which
( ) ( ) 1,2=,||)(),(dist 212111 iRyyyPyAyA ZiZR ≤∀−≤ . (3.4)
Then 10= AAA + is the operator with N -semi-bounded variation on W .
Proof. We give the proof in the proposition 7. In the case of the proposition
6 the reasonings are similar. Since for each Yyy ∈21,
( )[ ] ( )[ ]+− −≥− 21202110 ,, yyyAyyyA ,
we must estimate ( )[ ] ( )[ ]−− −−− 21212111 ,, yyyAyyyA .
For any ( ) ( )212111 , yAdyAd ∈∈ we find
≤−−−−−− ZZYY yydyydyydyyd >,<>,=<>,<>,< 211212211212
,21*21 ZZ yydd −−≤
hence
[ ] [ ] ( ) .)(,)(dist),(),( 21211121112121 ZyyyAyAyyyAyyyA −≤−−− −−
From here and from the estimation (3.4) as ( )1,2=iRy Zi ≤ (respectively
Ry Yi
ˆ≤ , )ˆ(= RRR ) we obtain
[ ] [ ] ( )'
2121212111 ||;ˆ,)(,)( WyyRCyyyAyyyA −−−≥− −− ,
where ttPtRC RZW )(=),(,=' ⋅⋅ .
It is easy to check that Φ∈C .
Proposition 8. Let one of two conditions hold:
1) *: YYA →
→ is radially lower semi-continuous operator with semi-bounded
variation on W ;
2) *: YYA →
→ is radially continuous from above operator with N -semi-
bounded variation on W with compact values in *Y .
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 133
Then A is 0λ -pseudomonotone on W map.
Proof. Let yyn → weakly in W, ( ) ddyA nn →∋co
*
* -weakly in *Y and
(2.1) is true. By using the property of semi-bounded variation on W of the opera-
tor A , we conclude that for every Yv∈
( )[ ] ( )[ ] ( )';,,>,< WnnnnYnn vyRCvyvAvyyAvyd −−−≥−≥− +− .
The function ( )[ ]+∋ wvAwX , is convex and semi-continuous from be-
low, and so it is weakly semi-continuous from below, therefore by substituting in
the last inequality yv = and passing to the limit, in view of the properties of the
function C , we obtain 0,>,<lim ≥−
∞→
Ynnn
yyd i.e 0>,< →− Ynn yyd .
For any Yh∈ and [ ]0,1∈τ we shall put ( )yh ττωτ −+ 1= , then
( )[ ] ( )';,>,< WnnYnn yRCyAyd ττττ ωωωω −−−≥− +
or by passing to the limit
( )[ ] ( )';,>,<lim WYn
n
hyRChywAhyd −−−≥− +
∞→
τττ τ .
By dividing the last inequality by τ and by passing to the limit as +→ 0τ , in
view of the radial lower semi-continuity of A and of the properties of the func-
tion C , we obtain that for each Yh∈
≥−
∞→
Yn
n
hyd >,<lim
( )[ ] ( ) ( )[ ] .,;1lim,lim '
00
−+→+
+→
−≥−+−≥ hyyAhyRChyA Wτ
τ
ω
τ
τ
τ
Moreover as 0>,< →− Ynn yyd we get
( )[ ] ,,>,<lim=>,<lim YhhyyAhydhyd Yn
n
Ynn
n
∈∀−≥−− −
∞→∞→
and this proves the first statement of the proposition 8.
Now we stop on the basic distinctive moments of the second statement. Be-
cause of the N -semi-boundedness of the variation for the operator A we con-
clude that
( )[ ] ≥−≥− −
∞→∞→
vyyAvyd nn
n
Ynn
n
,lim>,<lim
( )[ ] ( )';,lim
Wn
n
vyRCvyvA −−−≥ −
∞→
. (3.5)
Let us estimate the first member in the right part of (3.5). Let us prove that
the function ( )[ ]−∋ hvAhX , is weakly lower semi-continuous Yv∈∀ . Let
zzn → weakly in Y , then for each ( )vAn n co1,2,=
*
∈∃ξ such that
( )[ ] .>,<=, Ynnn zzvA ξ−
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 134
From the sequence { }nn z;ξ we take a subsequence { }mm z;ξ such that
( )[ ] Ymm
m
Ynn
n
n
n
zzzvA >,<lim=>,<lim=,lim ξξ
∞→∞→
−
∞→
and by virtue of the compactness of the set )(co
*
vA we find that ξξ →m strongly
in *Y with )(co
*
vA∈ξ . Hence
[ ] [ ]−∞→−
∞→
zvAzzzvA YYmmnn
n
,)(=>,=<>,<lim=,)(lim ξξ ,
and this proves the weak lower semi-continuity of the function [ ]−hvAh ,)( .
So from (3.5) we get
[ ] [ ] ( )';,)(,)(lim>,<lim Wnn
n
Ynn
n
vyRCvyvAvyyAvyd −−−≥−≥− −−
∞→∞→
.
Then by substituting v with y in the last inequality we have
0>,< →− Xnn yyd , therefore
[ ] ( ) YvvyRCwvvAvyd YYnn
n
∈∀−−−≥− −
∞→
';,)(>,<lim .
By substituting in the last inequality v with ( )yttw −+ 1 , where Yw∈ ,
[ ]0,1∈t , then by dividing the result on t and by passing to the limit as 0+→t ,
because of the radial semi-continuity from above we find
[ ] .,)(>,<lim YwwyyAwyd Ynn
n
∈∀−≥− −
∞→
Now let 21= WWW ∩ , where )||||,(
11 WW ⋅ and )||||,(
22 WW ⋅ are Banach
spaces such that ii YW ⊂ with continuous embedding.
Lemma 6. Let 1Y , 2Y be reflexive Banach spaces, )(: *
11 YCYA v→ and
)(: *
22 YCYB v→ be s -mutually bounded 0λ -pseudomonotone respectively
on 1W and on 2W multivalued maps. Then )(:=: *YCYBAC v→+ is
0λ -pseudomonotone on W map.
Remark 7. If the pair ( BA; ) is not s -mutually bounded, then the last
proposition holds only for λ -pseudomonotone (respectively on 1W and on 2W )
maps.
Proof. At first we check that Yy∈∀ )()( *YCyC v∈ . The convexity of
)(yC follows from the same property for )(yA and )(yB . By virtue of the
Mazur theorem, it is enough to prove that the set )(yC is weakly closed. Let c
be a frontier point of )(yC with respect to the topology );(=);( **** YYYY σσ
(the space Y is reflexive). Then
.asinweakly:)(}{ *
1 +∞→→⊂∃ ≥ mYccyCc mmm
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 135
From here, since the maps A and B have bounded values, due to the Banach-
Alaoglu theorem, we can assume that for each 1≥m there exist )(yAvm ∈ and
)(yBwm ∈ such that mmm cwv =+ and by passing (if it is necessary) to the sub-
sequences we obtain:
*
2
*
1 inand YwwYinvv
w
m
w
m →→
for some )(yAv∈ and )(yBw∈ . Hence )(= yCwvc ∈+ . So it is proved that
the set )(yC is weakly closed in *Y .
Now let 0yy
w
n → in W (from here it follows that 0yy
w
n → in 1W and
0yy
w
n → in 2W ), 0)()( dydyC
w
nn →∋ in *Y and the inequality (2.1) be true.
Hence
)(=)()(:)()(and)()( nnBnAnnBnnA ydydydyBydyAyd +∈∈ .
Since the pair ( BA; ) is s -mutually bounded, from the estimation
=+ YnnBnAYnn yydydyyd >),()(=<>),(<
kyydyyd YnnBYnnA ≤+=
21
>),(<>),(<
we have or Cyd
YnA ≤*
1
||)(|| or Cyd
YnB ≤*
2
||)(|| . Then, due to the reflexivity of
1Y and 2Y , by passing (if it is necessary) to a subsequence we get
*
20
*
10 in)(andin)( YdydYdyd
w
nB
w
nA ′′→′→ . (3.6)
From the inequality (2.1) we have
≤−+−
∞→∞→
1020 >),(<lim>),(<lim YnnA
n
YnnB
n
yyydyyyd
0>),(<lim 0 ≤−≤
∞→
Ynnn
yyyd ,
or symmetrically
≤−+−
∞→∞→
2010 >),(<lim>),(<lim YnnB
n
YnnA
n
yyydyyyd
0>),(<lim 0 ≤−≤
∞→
Ynn
n
yyyd .
Let us consider the last inequality. It is obvious that there exists a subse-
quence 1}{}{ ≥⊂ nnmm yy such that
≥−+−≥
∞→∞→ 1020 >),(<lim>),(<lim0 YnnA
n
YnnB
n
yyydyyyd
1020 >),(<lim>),(<lim YmmAmYmmB
m
yyydyyyd −+−≥
∞→∞→
. (3.7)
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 136
From here we obtain:
or 0>),(<limor0,>),(<lim
2010 ≤−≤−
∞→∞→ YmmB
m
YmmAm
yyydyyyd .
Without loss of generality we suppose that
0>),(<lim
10 ≤−
∞→ YmmAm
yyyd .
Then because of (3.6) and of the 0λ -pseudomonotony of A on 1W there ex-
ists a subsequence mmkkm yy }{}{ 1 ⊂≥ such that
1001
]),([>),(<lim YvvyyAvyyd YkmkmAk
∈∀−≥− −
∞→
. (3.8)
By substituting in the last relation v with 0y it results in
+∞→→− kyyyd YkmkmA as0>),(<
10 .
Therefore, taking into account (3.7), we have
0.>),(<lim 20 ≤−
∞→
YkmkmB
k
yyyd
By virtue of the 0λ -pseudomonotony of B on 2W , by passing to a subse-
quence 1}{}{ ≥′
⊂ kkmkm yy we find
2002
]),([>),(<lim YwwyyBwyyd YkmkmB
k
∈∀−≥− −′′
∞→
. (3.9)
So from the relations (3.8) and (3.9) we finally obtain
+−≥−
′′∞→′′
∞→
1
>),(<lim>),(<lim YkmkmA
k
Ykmkm
k
xyydxyyd
+−≥−+ −′′
∞→
]),([>),(<lim 002
xyyAxyyd YkmkmB
k
YxxyyCxyyB ∈∀−−+ −− ]),([=]),([ 0000 .
Proposition 9. Every –-coercive multi-valued map
*
2: YYA → is + -
coercive; every monotone + -coercive multi-valued map is –-coercive, uniformly
–-coercive and uniformly + -coercive.
Proof. The first part of this proposition is the direct corollary of the defini-
tions of +⋅⋅ ],[ and of −⋅⋅ ],[ . Let us check the second one.
Let
*
2: YYA → be a monotone )(−+ -coercive map. Let us prove that A is
uniformly )(−+ -coercive.
From the lemma 3 it follows that there exist a ball
}|||||{= ryYyB Yr ≤∈
and a constant 0>1c such that
rBcA ∈∀≤+ ωω 1||)(|| .
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 137
Hence for each Yy∈
≤+
∈∈∈
+ ]),([sup
1=>),(<1supsup=||)(||
)()(
ωω
ωω
yA
r
yd
r
yA
rB
Y
rByAyd
{ } { +−≤+−≤ +
∈
++
∈
]),([sup1]),([]),([sup1 yA
r
yyAyyA
r rBrB
ωωω
ωω
} { }=]),([)||||(1]),([ 1 ++ ++≤+ yyAyrc
r
yyA Y
;||||]),([1= 1
1 Yy
r
c
cyyA
r
+++
=>),(<1supinf=||)(||
)()(
Y
rByAyd
yd
r
yA ω
ω∈∈
−
{ }≤+−=
∈∈
YY
rByAyd
yydyyd
r
>),(<>),(<1supinf
)()(
ω
ω
{ }≤+−≤
∈∈
YY
rByAyd
yydyd
r
>),(<>),(<1supinf
)()(
ωω
ω
{ } ,||||]),([1=)||||(>),(<inf
1 1
11
)()(
YXY
yAyd
y
r
c
cyyA
r
yrcyd
r
++++≤ −
∈
ω
i.e.
.||||]),([1||)(|| 1
1)()( Yy
r
c
cyyA
r
yAYy ++≤∈∀ −+−+
Thus as 0>
2
= rc the uniform )(−+ -coercivity for A follows from the fol-
lowing estimations:
≥
− −+−+
Yy
yAcyyA
||||
||)(||]),([ )()(
=
−−−
≥
−+−+
Y
Y
y
ycrcyyAyyA
||||
||||
22
]),([
2
1]),([ 11
)()(
≥
−−
−
−
Y
Y
y
y
crc
yyyA
||||
||||
222
1),(
=
11
=
−−
−
≥ +
Y
Y
y
y
crc
yyyA
||||
||||
222
1,
2
1 11
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 138
.||||as
||||22||
2
1||2
2
1,
2
1
= 11 ∞→+∞→−−
−
+
Y
Y
Y
y
y
rcc
y
yyyA
To finish the proof it is enough to show that every monotone + -coercive
map is - -coercive. This follows from the next estimations:
=
||||
]
2
1),
2
1(2[
||||
]
2
1),(2[
=
||||
]),([
YYY y
yyyA
y
yyyA
y
yyA +−
−
−
≥
−
.||||as
||
2
1||
2
1,
2
1
= +∞→+∞→
+
Y
Y
y
y
yyA
Corollary 1. Let RY →:ϕ be a convex lower semicontinuous functional
such that
.||||as
||||
)(
∞→+∞→ Y
Y
y
y
yϕ
Then its subdifferential map
YyYyypYpy Y ∈∅≠∈∀−≤−∈∂ ,})()(>,<|{=)( * ωϕωϕωϕ
is + -coercive, and hence, - -coercive, uniformly - -coercive and uniformly
+ -coercive.
Proof. Due to the monotony of the map
*
2: YY →∂ϕ and to the proposition
9, it is enough to prove only that it is +-coercive. This follows from the next esti-
mations:
+∞→+∞→−≥∂ −−
+
−
YYYY yyyyyyy ||||as(0)||||)(||||]),([|||| 111 ϕϕϕ .
Definition 5. The multi-valued map
*
2: YYA → satisfies the uniform prop-
erty )()( −+κ if for each bounded set D in Y and for each 0>c there exists
0>1c such that
}0{\||||]),([1||)(|| 1
)()( Dvv
c
c
vvA
c
vA Y ∈∀+≤ −+−+ .
Lemma 7. Let *
11: YYA →→ , *
22: YYB →→ be +-coercive maps, which satisfy
the uniform property )()( −+κ . Then the map
*
2:=: YYBAC →+ is uniformly
)(−+ -coercive.
Proof. We obtain this statement arguing by contradiction. Let Yx nn ⊂≥1}{
with 0≠nx and +∞→+
21
||||||=|||||| YnYnYn xxx as +∞→n . Taking into ac-
count that
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 139
+∞
− −+−+
≥
<
||||
||)(||]),([
sup )()(
1 Yn
nCnn
n x
xCcxxC
, (3.10)
where },{min= BAC ccc , 0>, BA cc are the constants as in the uniform +(-)-
coercive condition for A and B respectively. Let
1
)()(
1
|||| ||||
||)(||]),([
inf=:)(
Y
A
rYv
A v
vAcvvA
r −+−+
=
−
γ ,
0>,
||||
||)(||]),([
inf:=)(
2
)()(
2
||||
r
w
wBcwwB
r
Y
B
rYw
B
−+−+
=
−
γ ,
we remark that +∞→+∞→ )(,)( rr BA γγ as +∞→r . In the case +∞→
1
|||| Ynx
as +∞→n and cx Yn ≤
2
|||| 1≥∀n we get
+∞→+∞→≥
− −+−+ n
x
x
x
x
xAcxxA
Yn
Yn
YnA
Yn
nAnn as
||||
||||
)||(||
||||
||)(||]),([ 1
1
)()( γ
and moreover
∞→+→−≥
− −+−+ n
x
x
c
x
wBcxxB
Yn
Yn
Yn
Bnn as0
||||
||||
||||
||)(||]),([ 2
1
)()( ,
where Rc ∈1 is a constant as in the condition )(κ with
}|||||{=
22 cyYyD Y ≤∈ , Bcc = .
Consequently
≥
− −+−+
Yn
nCnn
x
xCcxxC
||||
||)(||]),([ )()(
+
−
≥ −+−+
Yn
nAnn
x
xAcxxA
||||
||)(||]),([ )()(
,as
||||
||)(||]),([ )()( +∞→+∞→
−
+ −+−+ n
x
xBcxxB
Yn
nBnn
and this is in contradiction with (3.10).
If cx Yn ≤
1
|||| 1≥∀n and +∞→
2
|||| Ynx as +∞→n the reasoning is the
same.
When +∞→
1
|||| Ynx and +∞→
2
|||| Ynx as +∞→n , we get the contradic-
tion
+
+
≥
−
∞+ −+−+
≥ 21
1
1
)()(
1 ||||||||
||||
)||(||
||||
||)(||]),([
sup>
YnYn
Yn
YnA
Yn
nCnn
n xx
x
x
x
xCcxxC
γ
{ } +∞→≥
+
+ )||(||),||(||min
||||||||
||||
)||(||
21
21
2
2 YnBYnA
YnYn
Yn
YnB xx
xx
x
x γγγ .
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 140
Proposition 10. If the multi-valued operator
*
2: YYA → satisfies the con-
dition (Π ), then it satisfies the condition +)(κ .
Proof. We prove this proposition arguing by contradiction. Let YD ⊂
be a bounded set such that for each 0>c there exists }0{\Dvc ∈ :
0||||]),([ ≤−≤+ Yccc vcvvA . Then due to the condition )(Π
.<:=||)(||sup
0>
+∞+ dvA c
c
Thus
YcYccccYc vdvvAvvAvc ||||||||||)(||]),([|||| −≥−≥≥− ++
and 0||||)( ≥− Ycvcd for each 0>c . This is a contradiction with 0≠cv .
Proposition 11. Let the functional RY →:ϕ be convex, lower semicon-
tinuous on Y . Then the multi-valued map )(:= *YCYB v→∂ϕ is 0λ -
pseudomonotone on Y and it satisfies the condition (Π ).
Proof. a) Property )(Π . Let 0>k and the bounded set YB ⊂ be arbitrary
fixed. Then By∈∀ and )()( yyd ϕ∂∈∀ kyyyd Y ≤− >),(< 0 is fulfilled. Let
Yu∈ be arbitrary fixed, so
≤+−≤+− kyuyydyuyduyd YYY )()(>),(<>),(=<>),(< ϕϕ
,<const)(inf)( +∞≡+−≤
∈
kyu
By
ϕϕ
since every convex lower semicontinuous functional is lower bounded by every
bounded set. Hence, thanks to the Banach-Steinhaus theorem, there exists
),,(= 0 BkyNN such that Nyd
Y
≤*||)(|| for each By∈ ;
b) 0λ -pseudomonotony on Y . Let 0yy
w
n → in Y , ddy
w
nn →∋∂ )(ϕ in *Y
and the inequality (2.1) true. Then, due to the monotony of ϕ∂ , for each
)( 00 yd ϕ∂∈ and for each 1≥n
YnYnYnnYnn yydyydyyddyyd >,<>,<>,=<>,< 0000000 −≥−+−−− .
Hence
0=>,<lim>,<lim 000 Yn
n
Ynn
n
yydyyd −≥−
+∞→+∞→
.
Because of the last inequality and of the inequality (2.1) it results in
0=>,<lim 0 Ynnn
yyd −
+∞→
.
Thus for each Yw∈
+−≥−
+∞→+∞→
YnnnYnn
n
yydwyd >,<lim>,<lim 0
YYn
n
wydwyd >,=<>,<lim 000 −−+
+∞→
. (3.11)
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 141
From another side we have
−≤−≤−
+∞→
)(>,<lim>,< 00 wywdywd Ynn
n
Y ϕ
)()()(lim 0ywyn
n
ϕϕϕ −≤−
+∞→
. (3.12)
since every convex lower semicontinuous functional is weakly lower semicon-
tinuous. From (3.12) it follows that )( 00 yd ϕ∂∈ . From here, due to the inequal-
ity (3.11), we obtain the inequality (2.2) as ϕ∂=B on Y .
4. EXAMPLE
Now we consider a class of
0λ
w -pseudomonotone maps. Let us consider the
bounded domain nR⊂Ω with rather smooth boundary Ω∂ , ][0,= TS ,
)(0;= TQ ×Ω , )(0;= TT ×Ω∂Γ . Let as 1,2=i Rmi ∈ , iN1 (respectively iN 2 )
the number of the derivatives respect to the variable x of order 1−≤ im (respec-
tively im ) and { }
im
i txA ≤||),,,( αα ξη be a family of real functions defined in
iNiN RRQ 21 ×× . Let
xkuDuDk byationsdifferentithebe}|=|,{= ββ ,
}1...,,,{= uDDuuu im
i
−δ ,
)),(),,(,,(,:),,,( txvDtxutxAtxvDutxA im
i
iim
i
i δδ αα → .
Moreover, let RR →:ψ be a convex, lower semicontinuous coercive real
function and )(: RCR v→Φ be its subdifferential.
Let us assume )(= 2 ΩLH and )(,= 0 Ωiim
i
pWV with (1,2]∈ip such that
HVi ⊂ with continuous embedding, 1=11 −− + ii qp , )(yϕ∂ is the Gateaux sub-
differential of the convex lower semicontinuous coercive functional
∫→∋Ω
Q
dtdxtxyyyLSL )),((=)())(;( 22 ψϕ
in the space ))(;( 22 ΩLSL .
Definition of operators iA . Let ),,,( ξηα txAi , defined in
iNiN RRQ 21 ×× ,
satisfying the conditions
),,,(,mapthe,eachalmostfor ξηξη α txAQtx i→∈
iNiN RR 21oncontinuousis × ; (4.1)
QtxAtx i onmeasurableis),,,(,mapthe,eachfor ξηξη α→ ,
P.O. Kasyanov, V.S. Mel'nik, L. Toscano
ISSN 1681–6048 System Research & Information Technologies, 2007, № 3 142
)(),,,(=:);(0,,eachfor QLuDutxAKVTLvu iqim
i
i
ii
ip ∈∈ δα . (4.2)
Then for each iKu∈ the map
dtdxwDuDutxAwuaw im
i
i
Qim
i
α
α
α
δ ),,,(=),(
||
∫∑
≤
→ ,
is continuous on iK and then
>),(=<),(thatsuch)(existsthere * wuAwuaKuA iiii ∈ . (4.3)
Conditions on .iA Similarly to [20, sections 2.2.5, 2.2.6, 3.2.1] we have
)(),(=),(),,(=)( 21 uAvuAvuAuuAuA iiiii + ,
where
dtdxwDvDutxAwvuA im
i
i
Qim
i
α
α
α
δ ),,,(>=),,(<
|=|
1 ∫∑ ,
dtdxwDuDutxAwuA im
i
i
Qim
i
α
α
α
δ ),,,(
1
>=),(<
||
2 ∫∑
−≤
.
We add the next conditions:
iii KvuvuvuAvuuuA ∈∀≥−−− ,0>),,(<>),,(< 11 , (4.4)
0>),,(),(<ifandin',inif 11
* →−−′→→ uuuuAuuAKuuKuu jjijjii
w
ji
w
j ,
)(in),,,(),,,(then QLuDutxAuDutxA iqimiw
j
im
j
i δδ αα → , (4.5)
coercivity . (4.6)
Remark 8. Similarly to [20, theorem 2.2.8] the sufficient conditions to get
(4.4), (4.5) are:
∞→+∞→−+
∑ ||1||||
1),,,(
|=|
ξ
ξξ
ξξη αα
α
astxA
ip
i
im
for almost each Qtx ∈, and ||η bounded;
***
|=|
0>)))(,,,(),,,(( ξξξξξηξη αααα
α
≠−−∑ astxAtxA ii
im
for almost each Qtx ∈, and η .
The next condition gives the coercivity:
||largeratheras||),,,(
|=|
ξξξξη αα
α
ipi
im
ctxA ≥∑ .
A sufficient condition to get (4.2) (see [20, p. 332]) is the following one:
)(,),(1||1|||),,,(| QLktxkctxA
iq
ipipi ∈
+−+−≤ ξηξηα . (4.7)
Arguing by analogy with the proof of [20, theorem 3.2.1] and of [20, state-
ment 2.2.6] we get the next.
The classes and the main properties of the multi-valued
0λ
W -pseudomonotone maps
Системні дослідження та інформаційні технології, 2007, № 3 143
Proposition 12. ([20], p.337) Let *: iii KKA → ( 1,2=i ) be the operator,
defined in (4.3), satisfying (4.1), (4.2), (4.4), (4.5) and (4.6). Then iA is
λ -pseudomonotone on
}|{= *
2
*
1 KKyKyW ii +∈′∈
(in the classical sense) and coercive. Moreover it is bounded if (4.7) holds.
From the lemma 6 and the lemma 1 we deduce the next corollary:
Corollary 2. Let *: iii KKA → ( 21,=i ) be the operator, defined in (4.3),
satisfying (4.1), (4.2), (4.4)–(4.7). Then →∩+ 21121 =:= KKXAAA
**
2
*
1 = XKK +→ is 0λ -pseudomonotone on
}|{= *
1 XyXyW ∈′∈
and coercive.
Due to the proposition 11, to the proposition 12, to the lemma 6 and the
lemma 1 it is easy to obtain the next
Corollary 3. Let *: iii KKA → ( = 1,2i ) be the operator, defined in (4.3),
satisfying (4.1), (4.2), (4.4)–(4.7); RHSLG →);(=: 2ϕ be the functional satis-
fying the conditions of the proposition 11 and of the corollary 1. Then the multi-
valued map
)(:= *
21 GXCGXAAA v +→∩∂++ ϕ
is )(−+ -reflexive, 0λ -pseudomonotone on
}|{= * GXyGXyW +∈′∩∈
and it satisfies the condition (Π ).
4.1. An application. By virtue of the corollary 3 and of the [7] (theorem 1),
under the conditions of the corollary 3, the problem
+−+
∂
∂ ∑
≤
)),,,((1)(),( 1
1
1||
1||
yDytxAD
t
txy m
m
δα
αα
α
QtxftxyyDytxAD m
m
in),()),(()),,,((1)( 2
2
2||
2||
∋Φ+−+ ∑
≤
δα
αα
α
,
21,=and1||ason0=),( imtxyD iT −≤Γ αα ,
Ωin0=,0)(xy
has a solution in W .
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Received 06.02.2007
From the Editorial Board: the article corresponds completely to submitted
manuscript.
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ, ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
The classes and the main properties of the multi-valued -pseudomonotone maps
P.O. Kasyanov, V.S. Mel’nik, L. Toscano
1. Introduction
2. Classes of maps
3. The main properties of the -pseudomonotone maps
4. Example
|
| id | nasplib_isofts_kiev_ua-123456789-14096 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1681–6048 |
| language | English |
| last_indexed | 2025-12-07T18:54:25Z |
| publishDate | 2007 |
| publisher | Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| record_format | dspace |
| spelling | Kasyanov, P.O. Mel'nik, V.S. Toscano, L. 2010-12-14T11:35:08Z 2010-12-14T11:35:08Z 2007 The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps / P.O. Kasyanov, V.S. Mel’nik, L. Toscano // Систем. дослідж. та інформ. технології. — 2007. — № 3. — С. 122-144. — Бібліогр.: 20 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/14096 517.9 We consider the main classes of Wλ0-pseudomonotone multi-valued maps. The main properties of these operators have been investigated. The new classes of these operators have been obtained. Рассматриваются основные классы Wλ0-псевдомонотонных отображений. Исследованы базовые свойства этих операторов. Получены новые классы рассматриваемых отображений. Розглядаються основні класи Wλ0-псевдомонотонних відображень. Досліджено базові якості цих відображень. Отримано нові класи відображень, що розглядаються. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Нові методи в системному аналізі, інформатиці та теорії прийняття рішень The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps Классы и основные свойства многозначных Wλ0-псевдомонотонных отображений Класи та основні якості багатозначних Wλ0-псевдомонотонних відображень Article published earlier |
| spellingShingle | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps Kasyanov, P.O. Mel'nik, V.S. Toscano, L. Нові методи в системному аналізі, інформатиці та теорії прийняття рішень |
| title | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps |
| title_alt | Классы и основные свойства многозначных Wλ0-псевдомонотонных отображений Класи та основні якості багатозначних Wλ0-псевдомонотонних відображень |
| title_full | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps |
| title_fullStr | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps |
| title_full_unstemmed | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps |
| title_short | The classes and the main properties of the multi-valued Wλ0-pseudomonotone maps |
| title_sort | classes and the main properties of the multi-valued wλ0-pseudomonotone maps |
| topic | Нові методи в системному аналізі, інформатиці та теорії прийняття рішень |
| topic_facet | Нові методи в системному аналізі, інформатиці та теорії прийняття рішень |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/14096 |
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