On topological degree to some class of multivalued mappings and its applications
In this work the elements of topological degree theory has been developed for the class of multivalued maps from the re°exive Banach space to its dual one. In particular, this class contains the maps which generated by the inclusions with partial derivatives, by variational inequalities etc.
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Інститут прикладної математики і механіки НАН України
1999
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| Назва видання: | Нелинейные граничные задачи |
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| Цитувати: | On topological degree to some class of multivalued mappings and its applications / V.S. Mel'nik // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 120-125. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1692822020-06-10T01:26:14Z On topological degree to some class of multivalued mappings and its applications Mel'nik, V.S. In this work the elements of topological degree theory has been developed for the class of multivalued maps from the re°exive Banach space to its dual one. In particular, this class contains the maps which generated by the inclusions with partial derivatives, by variational inequalities etc. 1999 Article On topological degree to some class of multivalued mappings and its applications / V.S. Mel'nik // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 120-125. — Бібліогр.: 6 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169282 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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In this work the elements of topological degree theory has been developed for the class of multivalued maps from the re°exive Banach space to its dual one. In particular, this class contains the maps which generated by the inclusions with partial derivatives, by variational inequalities etc. |
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Article |
| author |
Mel'nik, V.S. |
| spellingShingle |
Mel'nik, V.S. On topological degree to some class of multivalued mappings and its applications Нелинейные граничные задачи |
| author_facet |
Mel'nik, V.S. |
| author_sort |
Mel'nik, V.S. |
| title |
On topological degree to some class of multivalued mappings and its applications |
| title_short |
On topological degree to some class of multivalued mappings and its applications |
| title_full |
On topological degree to some class of multivalued mappings and its applications |
| title_fullStr |
On topological degree to some class of multivalued mappings and its applications |
| title_full_unstemmed |
On topological degree to some class of multivalued mappings and its applications |
| title_sort |
on topological degree to some class of multivalued mappings and its applications |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169282 |
| citation_txt |
On topological degree to some class of multivalued mappings and its applications / V.S. Mel'nik // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 120-125. — Бібліогр.: 6 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT melnikvs ontopologicaldegreetosomeclassofmultivaluedmappingsanditsapplications |
| first_indexed |
2025-07-15T04:02:23Z |
| last_indexed |
2025-07-15T04:02:23Z |
| _version_ |
1837684116524367872 |
| fulltext |
ON TOPOLOGICAL DEGREE TO SOME CLASS OF
MULTIVALUED MAPPINGS AND ITS APPLICATIONS
c© Mel’nik V.S.
In this work the elements of topological degree theory has been developed for the
class of multivalued maps from the reflexive Banach space to its dual one. In particular,
this class contains the maps which generated by the inclusions with partial derivatives,
by variational inequalities etc. This research is based on the results of [1–3].
1. The multivalued mappings. 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∗
is the totality of all subset of the space X∗). We define
Dom(A) = {y ∈ X|A(y) 6= ∅}, and the map A is called the strong if Dom(A) = X.
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. If y /∈ Dom(A) then [A(y), ξ]− = +∞,
[A(y), ξ]+ = −∞ for each ξ ∈ X. Moreover, ‖A(y)‖+ = sup
d∈A(y)
‖d‖X∗ , ‖A(y)‖− =
inf
d∈A(y)
‖d‖X∗ and ‖∅‖− = ‖∅‖+ = 0.
Lemma 1.1. Let A,A1, A2 : X → 2X∗
. The following statements hold for each y ∈
Dom(A), ξ1, ξ2 ∈ X:
[A(y), ξ1 + ξ2]+ ≥ [A(y), ξ1]+ + [A(y), ξ2]−;
[A(y), ξ1 + ξ2]− ≤ [A(y), ξ1]+ + [A(y), ξ2]−;
‖A1(y) + A2(y)‖+ ≥
{ |‖A1(y)‖+ − ‖A2(y)‖+|
|‖A1(y)‖+ − ‖A2(y)‖−|
.
Lemma 1.2. For each y ∈ Dom(A) the following equalities hold:
‖coA(y)‖+ = ‖A(y)‖+, ‖coA(y)‖− = ‖A(y)‖−.
Definition 1.1. The mapping A : G ⊂ X → 2X∗
satisfies the condition α0(G) if from
G 3 yn → y weakly on X and
lim
n→∞
[A(yn), yn − y]− ≤ 0 (1)
it follows that yn → y strongly on X.
Let D be an arbitrary open bounded set from X, ∂D be its boundary, D = D ∪ ∂D.
Definition 1.2. The mapping A : D → 2X∗
satisfies the condition α(G), where D ⊃ G,
if from Dom(A)∩G 3 yn → y weakly on X and from (1) it follows that yn → y strongly
on X.
Remark 1.1. For single-valued mappings the conditions α0(G) and α(G) had been in-
troduced by I.V.Skrypnik [2]. Like to them condition (S)+ had been introduced by
F.Browder [4].
We say that the continuous function C : R+ × R+ → R belong to the class Φ if
τ−1C(r1, τr2) = 0 for each r1, r2 > 0 as τ → +0.
Definition 1.3. The mapping A : Dom(A) ⊂ X → 2X∗
is called the operator of
semibounded variation (s.b.v), if for any R > 0 and each y1, y2 ∈ X such that ‖yi‖X ≤ R
(i = 1, 2) the following inequality holds:
[A(y1), y1 − y2]− ≥ [A(y2), y1 − y2]+ − C(R; ‖y1 − y2‖′X),
where C belong to Φ, and ‖ · ‖′X is compact seminorm with respect to the norm ‖ · ‖X .
Remark 1.2. Let A = A1 + A2, where A1 : X → 2X∗
is a monotone map, A1 : Y → Y ∗
is a locally Lipschitz operator, Y is a Banach space, moreover, the embedding X ⊂ Y
is compact. Then A is the map of semibounded variation.
Proposition 1.1. Let the mapping A0 : G ⊂ X → 2X∗
satisfies the condition α(G),
A1 : X → 2X∗
be a map of s.b.v. Then A = A0 + A1 satisfies the condition α(G) if
Dom(A0) ∩Dom(A1) = Dom(A) 6= ∅.
Definition 1.4. The mapping A : Dom(A) ⊂ X → 2X∗
is called (G; F )–pseudomono-
tone, where F ⊂ X, if from Dom(A) ∩ G 3 yn → y ∈ Dom(A) weakly in X and (1) it
follows that
lim
n→∞
[A(yn), yn − ξ]− ≥ [A(y), y − ξ]− ∀ξ ∈ F.
If G = F = X the map A is called pseudomonotone.
Remark 1.3. The multivalued pseudomonotone mappings were studied also in [3,6].
Proposition 1.2. Let the mapping A0 satisfies the conditions from Proposition 1.1,
A1 : X → 2X∗
be a (G; F )–pseudomonotone mapping, moreover, Dom(A0)∩Dom(A1) 6=
∅. Then A = A0 + A1 satisfies the condition α(G).
Proposition 1.3. Finite union of maps {Ai : G ⊂ X → 2X∗}n
i=1 which satisfy the
condition α(G), is closed w.r.t. the positive multiplication and w.r.t. the intersection.
If, moreover,
n⋂
i=1
Dom(Ai) 6= ∅ then the map
n∑
i=1
Ai satisfies the condition α(G).
Definition 1.5. The mapping A : X → 2X∗
is called
a) demiclosed if graph A is closed in X ×X∗ with respect to the strong convergence
on X and weak one on X∗;
b) radially upper semicontinuous at point y0 ∈ Dom(A) if for any ξ, h ∈ X the
function [0, 1] 3 t 7→ [A(y0 + th), ξ]+ is upper semicontinuous at point 0. The mapping
A is radially upper semicontinuous (r.u.c.) if it is r.u.c. at each point of Dom(A).
Remark 1.4. R.u.c. multivalued mappings are the generalization of upper hemicontinu-
ous mappings (see [5]). Moreover, each r.u.c. mapping of s.b.v. is upper hemicontinuous.
Proposition 1.4. Let A : X → 2X∗
be a closed-convex-valued maps and one of follow-
ing conditions holds:
a) A is strong, r.u.c. map of s.b.v.;
b) A is pseudomonotone map and Dom(A) is closed.
Then A is demiclosed mapping.
We recall that the map A : X → 2X∗
is bounded if for each bounded B ⊂ X there
exists k > 0 such that ‖A(y)‖+ ≤ k ∀y ∈ B.
Definition 1.6. We call that the mapping A : D ⊂ X → 2X∗
belong to class U0(D;G)
(respectively, U(D;G)) if it is bounded, demiclosed and satisfies the condition α0(G)
(respectively, α(G)). We will write U0(D) and U(D) in place of U0(D;D) and U(D;D)
(i.e. as G = D).
2. The degree of the mapping A. First we consider the case when X is a reflexive
separable Banach space. Let Cv(X∗) be a totality of nonempty closed convex subset
from X∗, D be an arbitrary bounded open subset in X with the boundary ∂D. We
assume that
2a) A : D ⊂ X → Cv(X∗);
2b) A ∈ U0(D; ∂D);
2c) A(y) 63 0 for each y ∈ ∂D.
Let {hi}∞i=1 be an arbitrary complete system of X, moreover, for any n the elements
h1, . . . , hn are linear independent. Let Xn be the span of {h1, . . . , hn}, Jn : Xn → X
be the inclusion map, J∗n : X∗ → X∗
n be its dual. Under each n we define the finite-
dimensional multivalued mapping An associated with A by formula
An(y) =
⋃
d∈A
{ n∑
i=1
〈d(y), hi〉hi
}
= J∗nA(Jny)
∀y ∈ Dn = D ∩Xn. (2)
The symbol d ∈ A denote that d is a selector of multivalued map A.
Theorem 2.1. Let the multivalued mapping A satisfies conditions 2a)–2c). Then there
exists N such that as n ≥ N the following statements hold:
1) the inclusion An(y) 3 0 has not any solution on ∂Dn;
2) the degree deg(An, Dn, 0) of the map An set Dn with respect to 0 ∈ Xn is defined
and is not depended on n ≥ N .
The proof is based on following statements.
Lemma 2.1. The mapping An : Dn → Cv(X∗
n) is upper continuous and compact.
Let us consider auxiliary mapping Ãn(y) = An−1(y) + 〈ξn, y〉hn where the element
ξn ∈ X∗ such that 〈ξn, hi〉 = 0 for each i < n and 〈ξn, hn〉 = 1.
Lemma 2.2. There exists N such that for each n ≥ N
a) deg(An−1, Dn−1, 0) = deg(Ãn, Dn, 0);
b) deg(An, Dn, 0) = deg(Ãn, Dn, 0).
Remark 2.1. From Theorem 2.1. it follows that there exists the limit
lim
n→∞
deg(An, Dn, 0) = D({hi}).
Theorem 2.2. Under conditions 2a)–2c) the limit D({hi}) is independent on the choice
of {hi}.
From Theorems 2.1. and 2.2. it follows the naturalism of the following definition.
Definition 2.2. For the mapping A : X → 2X∗
which satisfies conditions 2a)–2c) the
number
Deg(A,D, 0) = lim
n→∞
deg(An, Dn, 0)
is called its degree of the set D with respect to the point 0 ∈ X∗, where An and Dn are
defined by (2).
Above we assume that X is separable, let us show that this requirement is not
necessary. Let now X be a reflexive Banach space, F (X) be the totality of its finite-
dimensional subspaces, F ∈ F (X) and h1, . . . , hν be some basis in F . We define the
finite-dimensional mapping
AF (y) =
⋃
d∈A
{ ν∑
i=1
〈d(y), hi〉hi
} ∀y ∈ DF = D ∩ F. (3)
Theorem 2.3. Let the mapping A : D ⊂ X → 2X∗
satisfies conditions 2a),2c) and
A ∈ U(D; ∂D). Then there exists F0 ∈ F (X) such that for each F ∈ F (X) (F0 ⊂ F )
the following properties are:
1) the inclusion AF (y) 3 0 has not any solution on ∂DF ;
2) deg(AF , DF , 0) = deg(AF0 , DF0 , 0),
where deg is the degree of the finite-dimensional map, AF and DF are defined by (3).
Lemma 2.3. There exists F0 ∈ F (X) such that for each F ∈ F (X) (F0 ⊂ F ) the set
ZF
F0
=
⋃
d∈A
ZF
F0
(d) = ∅,
where ZF
F0
(d) =
{
y ∈ ∂DF |〈d(y), ξ〉 = 0 ∀ξ ∈ F0
}
.
Definition 2.2. For the mapping A : X → 2X∗
which satisfies conditions of Theorem
2.3. the number
Deg(A, D, 0) = deg(AF0 , DF0 , 0)
is called its degree of the set D with respect to 0 ∈ X∗, where AF0 and DF0 are defined
by (3) and F0 ∈ F (X) is choiced by Theorem 2.3.
3. The basis properties of degree. In this section we are restricted to the case
when X is separable.
Definition 3.1. The mapping A : [0, 1] × (D ⊂ X) → 2X∗
satisfies the condition
α0,t(∂D) if for arbitrary subsequences {yn} ⊂ ∂D, {tn} ⊂ [0, 1] from yn → y0 weakly
on X, A(tn, yn) 3 dn → d0 weakly on X∗ and
lim
n→∞
[A(tn, yn), yn − y0]− ≤ 0
it follows that yn → y strongly on X.
Definition 3.2. The mapping A0, A1 : D ⊂ X → 2X∗
of class U(D; ∂D) which satisfy
the condition 2c) is called gomotopic in D if there exists the bounded map A : [0, 1]×
D → 2X∗
which satisfy the following conditions:
1) A(0, ·) = A0, A(1, ·) = A1;
2) A satisfies the condition α0,t(∂D);
3) A(t, y) 63 0 for each t ∈ [0, 1] and for each y ∈ ∂D;
4) A is demiclosed, i.e. if tn → t0, yn → y0 strongly in X and A(tn, yn) 3 dn → d0
weakly on X∗, then d0 ∈ A(t0, y0).
Theorem 3.1. Let A0 and A1 be multivalued mappings of the class U(D; ∂D) which
satisfy conditions 2a), 2c). If, in additional,A0 and A1 are gomotopic on D, then
Deg(A0, D, 0) = Deg(A1, D, 0).
Theorem 3.2. Let A : D ⊂ X → 2X∗
be the map of class U0(D), A(y) 63 0 for each
y ∈ D and 2a) is. Then Deg(A, D, 0) = 0.
Corollary 3.1. Let A : D ⊂ X → 2X∗
be the map of class U0(D) and it satisfies
properties 2a), 2b). For the inclusion A(y) 3 0 has at least one solution on D it is
sufficiently that Deg(A, D, 0) 6= 0.
Theorem 3.3. Let A : D ⊂ X → Cv(X∗) be the map of class U0(D; ∂D), 0 ∈ D \ ∂D
and
[A(y), y]− ≥ 0 ∀y ∈ ∂D.
Then Deg(A,D, 0) = 1.
Theorem 3.4. Let D be a symmetric bounded neighborhood of zero, A : D ⊂ X →
Cv(X∗) be the map of class U(D; ∂D) and 0 6∈ A(∂D). In additional let
A(y) ∩ λA(−y) 6= ∅ ∀y ∈ ∂D andλ ∈ [0, 1].
Then Deg(A,D, 0) is odd number.
Theorem 3.5. Let D1 and D2 be a nonintersecting open subset on D, in additional,
A(y) 63 0 ∀y ∈ D \ (D1 ∪D2),
where A ∈ U0(D; ∂D) and 2a) holds.
Then Deg(A, D, 0) = Deg(A,D1, 0) + Deg(A,D2, 0).
Remark 3.1. The statements of this section allow the natural extension in the case of
nonseparable spaces.
4. The degree for pseudomonotone maps. Let D be some bounded open
subset on X, A0 : D ⊂ X → Cv(X∗) be the map of class U(D; ∂D). We assume that
A : D ⊂ X → Cv(X∗) is (∂D; X)–pseudomonotone, demiclosed, bounded mapping and
0 6∈ A(∂D). In additional, there exists δ0 > 0 such that ‖A(y)‖− ≥ δ0 for each y ∈ ∂D.
We consider the map Aε = εA0 + A : D ⊂ X → Cv(X∗). Let M = sup
y∈D
‖A0(y)‖+.
Obviously, M < ∞ and since Proposition 1.2. for each ε > 0 Aε ∈ U(D; ∂D). Moreover,
Aε(y) 63 0 for each y ∈ ∂D. In fact, ‖Aε(y)‖− ≥ ‖A(y)‖− − ε‖A0(y)‖+ ≥ δ0 − εM . If
0 < ε < δ0M
−1, then ‖Aε(y)‖− > 0 for each y ∈ ∂D. Thus, the degree Deg(Aε, D, 0)
is defined as 0 < ε < δ0M
−1. Let us show that the defined degree is independent on ε.
Let 0 < εi < δ0M
−1, i = 1, 2 and we consider corresponding Aεi . Let us assume that
A(t, y) = (tε2 + (1 − t)ε1)A0(y) + A(y). Obviously, A(0, y) = Aε1(y), A(1, y) = Aε2(y)
and in additional A(t, y) 63 0 for each t ∈ [0, 1] and for each y ∈ ∂D.
Lemma 4.1. The mapping A : [0, 1]×D → 2X∗
satisfies the condition α0,t(∂D).
The conditions of Theorem 3.1 are satisfied, thus,
Deg(Aε1 , D, 0) = Deg(Aε2 , D, 0).
Hence, there exists the limit lim
ε→0
Deg(Aε, D, 0) which we will call the degree Deg(A, D, 0)
of mapping A and set D with respect to the point 0 ∈ X∗.
Using the constructions given above we can prove that this limit is not depended on
the mapping A0, i.e. the degree of pseudomonotone map is correct.
References
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of Setvalued Maps. I. Topological Characteristics and Solvability of the Operator Inclusions, Itogi
nauki i techniki. VINITI. Math.analis 25 (1987), 123-137 (in Russian).
2. I.V.Skrypnik, Methods of Investigation of Nonlinear Elliptic Boundary Value Problems, Moscow,
Nauka, 442pp. (in Russian) (1990).
3. Melnik V.S., On Critical Points for Some Classes of Multivalued Mappings, Cybernetics and Sys-
tem Analysis (Cybernetics) 2 (1997), 87-98 (in Russian).
4. Browder F.E., The Degree of Mapping and Its Generalization, Contemp. Math. 31 (1983), 15-40.
5. Aubin J.–P., Ekland I., Applied Nonlinear Analysis, Moscow: ”Mir”, 587pp. (in Russian) (1988).
6. Ivanenko V.I. and Melnik V.S., Variational Methods in Control Problems for Distributed Systems,
In: Kiev: Naukova dumka, 288pp. (in Russian) (1988).
Department of the Mathematical
Simulation of Economical Systems,
Kiev Polytechnic Institute, FMM,
pr.Pobedy 37, Kiev-56, 252056, Ukraine;
E-mail: melnik@consy.ms.kiev.ua
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