On solvability of the variational inequality with +-coercive multivalued mappings
In this work the theory of variational inequalities with multivalued operators is extended for the wider of multivalued maps from the re°exive Banach space to its dual one. In particular, we relax the restriction on boundness of operators and on same properties of convergence. This operator class co...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
1999
|
| Назва видання: | Нелинейные граничные задачи |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/169283 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On solvability of the variational inequality with +-coercive multivalued mappings / O.V. Solonoukha // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 126-129. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169283 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1692832020-06-10T01:26:18Z On solvability of the variational inequality with +-coercive multivalued mappings Solonoukha, O.V. In this work the theory of variational inequalities with multivalued operators is extended for the wider of multivalued maps from the re°exive Banach space to its dual one. In particular, we relax the restriction on boundness of operators and on same properties of convergence. This operator class contains the bounded pseudomonotone maps, the maximal monotone maps on interior of domain, the s{weakly locally bounded generalized pseudomonotone maps and other. 1999 Article On solvability of the variational inequality with +-coercive multivalued mappings / O.V. Solonoukha // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 126-129. — Бібліогр.: 10 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169283 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this work the theory of variational inequalities with multivalued operators is extended for the wider of multivalued maps from the re°exive Banach space to its dual one. In particular, we relax the restriction on boundness of operators and on same properties of convergence. This operator class contains the bounded pseudomonotone maps, the maximal monotone maps on interior of domain, the s{weakly locally bounded generalized pseudomonotone maps and other. |
| format |
Article |
| author |
Solonoukha, O.V. |
| spellingShingle |
Solonoukha, O.V. On solvability of the variational inequality with +-coercive multivalued mappings Нелинейные граничные задачи |
| author_facet |
Solonoukha, O.V. |
| author_sort |
Solonoukha, O.V. |
| title |
On solvability of the variational inequality with +-coercive multivalued mappings |
| title_short |
On solvability of the variational inequality with +-coercive multivalued mappings |
| title_full |
On solvability of the variational inequality with +-coercive multivalued mappings |
| title_fullStr |
On solvability of the variational inequality with +-coercive multivalued mappings |
| title_full_unstemmed |
On solvability of the variational inequality with +-coercive multivalued mappings |
| title_sort |
on solvability of the variational inequality with +-coercive multivalued mappings |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169283 |
| citation_txt |
On solvability of the variational inequality with +-coercive multivalued mappings / O.V. Solonoukha // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 126-129. — Бібліогр.: 10 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT solonoukhaov onsolvabilityofthevariationalinequalitywithcoercivemultivaluedmappings |
| first_indexed |
2025-07-15T04:02:26Z |
| last_indexed |
2025-07-15T04:02:26Z |
| _version_ |
1837684119899734016 |
| fulltext |
ON SOLVABILITY OF THE VARIATIONAL INEQUALITY
WITH + –COERCIVE MULTIVALUED MAPPINGS
c© Solonoukha O.V.
In this work the theory of variational inequalities with multivalued operators is ex-
tended for the wider of multivalued maps from the reflexive Banach space to its dual
one. In particular, we relax the restriction on boundness of operators and on same
properties of convergence. This operator class contains the bounded pseudomonotone
maps ([6–8]), the maximal monotone maps on interior of domain ([2,5]), the s–weakly
locally bounded generalized pseudomonotone maps ([3,10]) and other.
Let X be a reflexive Banach space, X∗ be its topological dual space, 〈·, ·〉 : X×X∗ →
R be the duality pairing, A : X → 2X∗
be a multivalued mapping (2X∗
=
⋃
V⊂X∗
V ).
We define Dom(A) = {y ∈ X : A(y) 6= ∅}. We associated with A the lower and
upper support functions [A(y), ξ]− = inf
d∈A(y)
〈d, ξ〉 and [A(y), ξ]+ = sup
d∈A(y)
〈d, ξ〉, where
y, ξ ∈ X, ‖A(y)‖+ = sup
d∈A(y)
‖d‖X∗ , ‖A(y)‖− = inf
d∈A(y)
‖d‖X∗ . If y /∈ Dom(A) then
[A(y), ξ]− = +∞, [A(y), ξ]+ = −∞ for each ξ ∈ X and ‖A(y)‖− = ‖A(y)‖+ = 0,
−∞+∞ = +∞.
Definition 1. The mapping A : Dom(A) ⊂ X → 2X∗
is s–weakly locally bounded,
if for arbitrary y ∈ Dom(A) and {yn} ⊂ Dom(A), where yn → y weakly in X, there
exist the subsequence {ynk
} and N > 0 such that ‖A(ynk
)‖+ ≤ N .
Definition 2. The mapping A : Dom(A) → 2X∗
has the property (M), if for
arbitrary {(yn, wn)} ⊂ graph co A, such that yn → y weakly in X, wn → w weakly in
X∗ and lim
n→∞
〈wn, yn − y〉 ≤ 0, we have that w ∈ coA(y).
Definition 3. The mapping A : Dom(A) → 2X∗
is monotone, if
[A(y1), y1 − y2]− ≥ [A(y2), y1 − y2]+ ∀yi ∈ Dom(A).
A is maximal monotone if it is monotone and for each monotone operator C
(graphA ⊂ graph C) ⇒ (C = A).
Definition 4. The mapping A : Dom(A) → 2X∗
is + –coercive on K if there exist
y0 ∈ K such that
‖y‖−1
X [A(y), y − y0]+ → +∞ as ‖y‖X → +∞, y ∈ K.
This research is based on properties of multivalued maps which are perturbed by
maximal monotone operators. In particular, we use the properties of maximal monotone
operator ([2,3,5]), of operators with the properties (M) ([6–8] and its References) and
of normal cone as the maximal monotone operator ([4]).
Proposition. Let A : Dom(A) ⊂ X → 2X∗
is s–weakly locally bounded and has the
property (M), B : Dom(B) → 2X∗
is maximal monotone, then A + B has the property
(M) on Dom(A) ∩Dom(B).
Proof. Let yn → y weakly in X, (A+B)(yn) 3 wn → w weakly in X∗ and lim
n→∞
〈wn, yn−
y〉 ≤ 0. Since operator A is s–weakly locally bounded, there exists the subsequence
{ym} ⊂ {yn} such that ‖A(ym)‖+ ≤ N , i.e. there exist {dm ∈ co A(ym)} such that
‖dm‖X∗ ≤ N and ŵm = wm − dm ∈ B(ym). Without restricting the generality, we
can assume that dm → d weakly in X∗. Thus, B(ym) 3 ŵm → ŵ weakly in X∗ and
ŵ + d = w. Then or lim
m→∞
〈ŵm, ym − y〉 ≤ 0 or lim
m→∞
〈dm, ym − y〉 ≤ 0 or both estimate
hold. In first case ŵ ∈ B(y) and 〈ŵm, ym〉 → 〈ŵ, y〉 (see [2]). Thus, lim
m→∞
〈dm, ym−y〉 =
lim
m→∞
〈wm − ŵm, ym − y〉 = lim
m→∞
〈wm, ym − y〉 ≤ 0. I.e. d ∈ A(y), w ∈ (A + B)(y).
If lim
n→∞
〈dn, yn − y〉 ≤ 0 and lim
n→∞
〈ŵn, yn − y〉 > 0, then d ∈ A(y) by property (M)
and ∃ {ym} such that 〈ŵm, ym − y〉 > 〈ŵ, ym − y〉. But B is maximal monotone, i.e.
ŵ ∈ B(y). ¥
For proofing of based result we need the auxiliary property of multivalued operator
on some closed convex set Dr ⊂ Br = {y ∈ X : ‖y‖X ≤ r}, where 0 ∈ Dr, ∂Dr is
the boundary of Dr. Without restricting the generality the following property will be
proved for convex– closed–valued operators.
Theorem 1. Let A : Dr → 2X∗
be a convex– closed–valued s–weakly locally bounded
operator which has the property (M), A = coA, B : Dr → 2X∗
be maximal monotone
and [(A+B)(y), y]+ ≥ 0 for each y ∈ ∂Dr. Then the inclusion 0 ∈ co(A+B)(ξ), ξ ∈ Dr
has nonempty weakly compact set of solutions.
Sketch. Let F (X) be a totality of finite-dimensional subspaces F ⊂ X. For arbitrary
F ∈ F (X) we introduce the projector IF : F → X (‖IF yF ‖X = ‖yF ‖F ∀yF ∈ F ),
I∗F : X∗ → F ∗ is a dual operator, DrF = Dr ∩ F , AF = A|F : F → 2X∗
. And we
introduce the auxiliaries operators ∀y ∈ DrF ≡ Dr ∩ F
I∗F AF (y) =
⋃
d∈AF
{ ∑
{hi}
〈d(y), hi〉hi
}
I∗F BF (y) =
⋃
w∈BF
{ ∑
{hi}
〈w(y), hi〉hi
}
I∗F (A + B)F (y) =
⋃
d∈AF ,w∈BF
{ ∑
{hi}
〈w(y) + d(y), hi〉hi
}
where {hi} is the basis of F .
Using the properties of maximal monotone operators (see [2,5]) and s–weakly locally
bounded operator with the properties (M) (see [7,8,10]), we can prove that the operator
I∗F (A + B)F is closed. Then by Remark 6.4.2 ([1]) for each F ∈ F (X) there exists
yF ∈ DrF such that 0 ∈ I∗F (A + B)F (yF ). We can construct the system with the finite
intersection property {Gw
F }, where G
w
F is the weak closure of GF0 =
⋃
F⊃F0
{
yF ∈ DrF :
0 ∈ I∗F (A + B)F (yF )
}
. Since X is reflexive, then ∃ y ∈ ⋂
F∈F (X)
{Gw
F }. And by property
(M) 0 ∈ (A + B)(y). ¥
For the bounded demiclosed operator with the property α) this theorem had been
considered in [9].
Theorem 2. The variational inequality (VIMO)
[A(y), ξ − y]+ ≥ 〈f, ξ − y〉 ∀ξ ∈ K
has at least one bounded solution if the mapping A is +-coercive, s–weakly locally bounded
and A satisfies the property (M) on K. Moreover, there exists the solution of VIMO y
which is bounded by constant R (‖y‖X ≤ R) where [A(ξ), ξ − y0]+ ≥ 0 for each ξ ∈ K
such that ‖ξ‖X = R.
Sketch. It suffices to show that the inclusion coA(y) + ∂IK(y) 3 f, where IK is the
characteristic map of K (IK(y) = 0 as y ∈ K and IK(y) = +∞ as y /∈ K). the
subdifferentional ∂IK is normal cone (see [4]), i.e. it is maximal monotone (see [2]). By
the definition ∂IK(y) = {w ∈ X∗ : 〈w, ξ〉 ≤ 0 ∀ξ ∈ 1
h
⋃
h>0
(K − y)}, thus, 0 ∈ ∂IK(ξ)
for each ξ ∈ K and [∂IK(y), y − y0]+ = +∞ as y ∈ ∂K (∂K is the boundary of K).
Since [A(y)− f +∂IK(y), y− y0]+ = [A(y)− f, y− y0]+ +[∂IK(y), y− y0]+, at least one
of summed tends to infinity as ‖y‖X → ∞, i.e. there exists the constant R > 0 such
that [A(y) + ∂IK(y), y − y0]+ ≥ 0 if y ∈ ∂(K ∩ BR(y0)). Thus, by Theorem 1 there
exists ŷ such that 0 ∈ coA(ŷ) + ∂IK(ŷ).
Let {yn} be subset of solution set, yn → y weakly in X, 0 ∈ coA(yn) + ∂IK(yn).
Then lim〈0, yn − y〉 = 0. By the property (M) 0 ∈ coA(y) + ∂IK(y), consequently, the
set of solution is nonempty and weakly compact. ¥
Collorary. Let A be s–weakly locally bounded and satisfies the property (M) on K,
K be closed convex and bounded. Then VIMO has nonempty weakly compact set of
solutions.
Example. Let Ω be a bounded set from Rn, Γ be the boundary of Ω. We consider
such Ω that Γ is smooth. And we study the free-boundary problem on Sobolev space
W 1
p (Ω), where p ≥ 2, p−1 + q−1 = 1, the track y|Γ belongs to W
1/q
p (Γ). Let ν(x) be the
normal in x ∈ Γ. Let us consider the free-boundary problem
A(y) = −
n∑
i=1
∂
∂xi
(
ai(x, y)
∂y
∂xi
)
+ a0(x, y)y = f,
y|Γ ≥ 0,
∂y
∂νA
+ γ(y) ∩ [0,∞) 6= ∅, y
( ∂y
∂νA
+ γ(y)
) 3 0,
where ∂y
∂νA
=
n∑
i=1
ai(x, y) ∂y
∂xi
cos(νi, x), γ : W
1/q
p (Γ) → 2W−1/q
q (Γ) is a s-weakly locally
bounded operator which has the property (M). And let ai satisfy to Caratheodori
conditions and following ones:
|ai(x, y)| ≤ g(x) + C1i|y|p−1,
where g ∈ Lq‘(Ω), q‘ = p
p−1 , if p > 2, and g ∈ C∞(Ω), if p = 2;
ai(x, y) > C2i|y|p−2, for sufficiently large |y| >> R (C2i > 0).
Then we can consider the integral forms, i.e. we can construct the variational inequality
n∑
i=1
∫
Ω
ai(x, y)
∂y
∂xi
∂(ξ − y)
∂xi
dx +
∫
Ω
a0(x, y)y(ξ − y)dx+
+ [γ(y), ξ − y]+ ≥
∫
Ω
f(ξ − y)dx ∀ξ ∈ K,
where [γ(y), ξ − y]+ = sup
v∈γ(y)
∫
Γ
v(ξ − y)dΓ, K = {y ∈ W 1
p (Ω) : y|Γ ≥ 0}. By Theorem 2
this VIMO has at least one solution for each f ∈ W−1
q (Ω), this statement holds if some
values of γ are not convex and closed. Also this statement holds if for some (not all)
selector d ∈ γ d(y) → −∞ as ‖y‖X →∞.
In [2,8] the problem had been considered for the operator with ai(x, y) ≡ ai(x) and
maximal monotone operator γ. In [10] this problem had been considered for generalized
pseudomonotone operator γ, the proving of the properties is analogous.
References
1. Aubin J.–P., Ekland I., Applied Nonlinear Analysis, In: J.Wiley and Sons., Inc. (1984).
2. Barbu V., Analysis and control of nonlinear infinite dimensional systems, In: Acad. Press, Inc.
(1995).
3. Browder F.E., Hess P., Nonlinear Mappings of Monotone Type in Banach Spaces, J. Func. Anal.
11 (1972), no. 2, pp. 251-294.
4. Clarce F., Optimization and Nonsmooth analysis, In: J.Wiley and Sons., Inc. (1983).
5. Z.Guan and A.G.Kartsatos., Ranges of Perturbed Maximal Monotone and m-Accretive Operators
in Banach Spaces, Trans.of AMS 347 (1995), 2403–2435.
6. Ivanenko V.I. and Mel’nik V.S., Variational Methods in Control Problems for Distributed Systems,
In: Kiev: Naukova dumka (in Russian) (1988).
7. Mel’nik V.S., Solonoukha O.V., On the Stationary Variational Inequalities with the Multivalued
Operators, Cybernetics and System Analysis 3 (1997), 74–89(in Russian).
8. Mel’nik V.S., Solonoukha O.V., On the Variational Inequalities with the Multivalued Operators,
Dokl.NAN of Ukraine 5 (1997), 33–39(in Russian).
9. Mel’nik V.S., Vakulenko A.N., On the same Class of Operator Inclusions in Banach Spaces,
Dokl.NAN of Ukraine (1997), (in print) (in Russian).
10. Solonoukha O.V., On the Stationary Variational Inequalities with the Generalized Pseudomonotone
Operators, Methods of Functional Analysis and Topology (1997), (in print).
Address Kiev-056, pr. Pobedy 37,
315-7, NTUU”KPI”, FMM
E-mail ssolesya@adam.kiev.ua
|