On the solvability of relaxed one-sided Lipschitz inclusions in Hilbert spaces

0.25 MB·English

by  Janosch Rieger

On the solvability of relaxed one-sided
5
1 Lipschitz inclusions in Hilbert spaces
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2

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Janosch Rieger and Tobias Weth
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J

2 January 5, 2015

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Abstract
.
h
t We prove solvability theorems for relaxed one-sided Lipschitz mul-
a
tivalued mappings in Hilbert spaces and for composed mappings in
m
the Gelfand triple setting. From these theorems, we deduce proper-
[
ties of the inverses of such mappings and convergence properties of a

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numerical scheme for the solution of algebraic inclusions.
v
0
8 Key words: relaxed one-sided Lipschitz property, algebraic inclusion,
3
root-finding method, set-valued analysis.
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0 AMS(MOS) subject classifications: 47H04, 65K10, 49J53.
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5
1
1 Introduction
:
v
i
X
The relaxed one-sided Lipschitz property (see Definition 4 below) was first
r
a considered in [9], where it was identified as an important stability criterion
for time-dependent differential inclusions. The behavior of general multival-
ued mappings with negative relaxed one-sided Lipschitz constants was later
studied in [10, 11]. In particular, surjectivity of the mappings and therefore
solvability of the corresponding algebraic inclusions was shown by consider-
ing the flow of the differential inclusions from [9]. However, no information
on the localization of the solutions was given in these papers.
For relaxed one-sided Lipschitz mappings in finite-dimensional spaces,
the solvability theorem given in [3] specifies a ball in which a solution of the
inclusion is contained. The radius of this ball depends on the norm of the
1
residualoftheinclusionatthecenterpoint. Thistheoremguaranteesthatthe
implicit Euler scheme for stiff ordinary differential inclusions is well-defined
and convergent on the infinite time interval, and it has recently been applied
in [13] to obtain a numerical method for the solution of the generalized Bolza
problem. Arefinedsolvabilityresult presented in[5]andrestatedasTheorem
10 below immediately gives rise to a numerical algorithm for the solution of
algebraic inclusions.
These solvability theorems are relevant for the following reason. For non-
scalar mappings, it is currently unclear whether continuous and relaxed one-
sidedLipschitzmultivaluedmappingspossessparameterizationsthatarecon-
tinuous and one-sided Lipschitz with the same one-sided Lipschitz constant.
Moreover, simple examples show that selections generated by metric projec-
tion such as the minimal selection are not one-sided Lipschitz with the same
constantasthemultimap. Itisthereforeimpossibletoobtainprecisesolvabil-
ity results by applying standard tools like topological fixed point theorems
to selections or parameterizations of one-sided Lipschitz multifunctions.
In the present paper we generalize the finite-dimensional solvability re-
sult from [5] to infinite-dimensional Hilbert spaces, and we discuss implica-
tions both in an abstract framework as well as in the context of a special
class of systems of elliptic differential inclusions. After collecting definitions
and preliminary tools in Section 2, we prove an abstract solvability result
in an infinite-dimensional Hilbert space in Section 3 via an approach based
on Galerkin approximations. The approach avoids strong compactness as-
sumptions which are not satisfied in many applications. In Section 4, we
reformulate the main result in the context of Gelfand triples and composed
multivalued operators. In this setting, special care was taken to obtain op-
timal estimates by considering a mixed scalar product adapted to the prop-
erties of the individual operators. As a byproduct, the main result reveals
certain aspects of the behavior of the inverses of relaxed one-sided Lipschitz
mappings as detailed in Section 5. As in the finite-dimensional context, the
solvability theorem gives rise to a numerical algorithm for the solution of re-
laxed one-sided Lipschitz algebraic inclusions, which is analyzed in Section 6.
In Section 7, we discuss a system of elliptic differential inclusions where the
assumptions of the Gelfand triple version of our main result are verified for
suitable right-hand sides. Moreover, we test the numerical algorithm from
Section 6 in the context of this system.
2
2 Preliminaries
Inthissection, wecollectthenecessarydefinitionsandsomeelementaryfacts.
Let (X, ) be any real normed vector space, and let , : X∗ X
X
k·k h· ·i × → R
denote the dual pairing.
Definition 1. For x X and nonempty subsets M,M′ X, we set
∈ ⊂
dist (x,M′) := inf x x′ ,
X X
x′∈M′k − k
e (M,M′) := supdist (x,M′),
X X
x∈M
M := e (M, 0 ),
X X
k k { }
Proj (x,M) := x′ M : x x′ = dist (x,M) ,
X { ∈ k − kX X }
B (M,R) := y X : dist (y,M) R ,
X X
{ ∈ ≤ }
B (x,R) := y V : y x R .
X X
{ ∈ k − k ≤ }
The nonempty closed, bounded and convex subsets of X are denoted (X),
CBC
and the nonempty convex and compact subsets of X are denoted (X).
CC
Definition 2. a) The support function σ∗ : X (X∗) is defined
X ×CBC → R
by
σ∗ (x,A) := sup ϕ,x x X,A X∗.
X h i ∀ ∈ ⊂
ϕ∈A
b) If X is a real Hilbert space with scalar product ( , ) , then we define
X
· ·
σ : X (X) by
X
×CBC → R
σ (x,A) := sup(y,x) x X,A X.
X
∀ ∈ ⊂
y∈A
Definition 3. Let (M,d) be a metric space and Y a further normed vector
space.
a) A set-valued mapping F : M (Y∗) is called upper hemicontinu-
→ CBC
ous (uhc) at x M if for any sequence (x ) M with x x we
k k∈ k
∈ N ⊂ →
have
limsupσ∗(v,F(x )) σ∗(v,F(x)) for all v Y. (1)
Y k ≤ Y ∈
k→∞
It is called uhc if it is uhc at any x M.
∈
If M is weakly sequentially closed, then F is called compactly upper
hemicontinuous(c-uhc)if condition (1)holds foranysequence(x )
k k∈
N ⊂
M with x ⇀ x M.
k
∈
3
b) If Y is a Hilbert space, then a set-valued mapping F : M (Y)
→ CBC
is called upper hemicontinuous (uhc) at x M if for any sequence
∈
(x ) M with x x we have
k k∈ k
N ⊂ →
limsupσ (v,F(x )) σ (v,F(x)) for all v Y. (2)
X k X
≤ ∈
k→∞
It is called uhc if it is uhc at any x M.
∈
If M is weakly sequentially closed, then F is called compactly upper
hemicontinuous(c-uhc)if condition (2)holds foranysequence(x )
k k∈
N ⊂
M with x ⇀ x M.
k
∈
c) A set-valued mapping F : M (Y) is called upper semicontinuous
→ CC
(usc) at x M if for any sequence (x ) M with x x M we
k k∈ k
∈ N ⊂ → ∈
have
e (F (x ),F (x)) 0 as k . (3)
V N k N
→ → ∞
It is called usc if it is usc at any x M.
∈
The following one-sided property is the central object of investigation in
the present paper.
Definition 4. A mapping F : X ⇒ X∗ is called l-relaxed one-sided Lipschitz
with constant l (or l-ROSL) if for any x,x′ X and y F(x) there
∈ R ∈ ∈
exists y′ F(x′) such that
∈
y′ y,x′ x l x′ x 2 .
h − − i ≤ k − kX
In Theorems 9, 10 and 11, the relaxed one-sided Lipschitz property will
only be required relative to one point in the graph of F. In Sections 5 and
6, however, we will deal with mappings that are relaxed one-sided Lipschitz
in the sense of Definition 4.
Remark 5. The definition of an ROSL mapping with constant l 0 is for-
≥
mallysimilarto thatof a monotonemapping. Nevertheless, ROSLand mono-
tone mappings have fundamentally different properties. Monotone mappings
on Hilbert spaces are single-valued outside a set of first Baire category (see
[12]), and the operator I+αT is onto and possesses a single-valued inverse for
any monotone T and any α > 0, which is the theoretical basis for the proximal
point algorithm (see [15]). In contrast, the 1-ROSL mapping F : ( )
R → CC R
given by F(x) = x + [ 1,+1] is set-valued on the whole space, and for all
−
4
α > 0 the inverse of I+αF, given by (I+αF)(x) = (1 α)x+[ α,α], is set
− −
valued as well. Similarly, ROSL mappings with constants l 0, which are
≤
mainly considered in the present paper, look formally similar to mappings T
with T monotone but have fundamentally different properties.
−
We also recall the following facts which are well known and easy to see.
Lemma 6. Let X be a Hilbert space, and let x X.
∈
a) If M X is closed and convex, then Proj(x,M) is a single point.
⊂
b) If M X is weakly sequentially closed, then Proj(x,M) is nonempty.
⊂
c) If M X is closed, and if (x ) X and x¯ X satisfy x x¯ 0
n n n X
⊂ ⊂ ∈ k − k →
and dist (x ,M) 0 as n , then x¯ M.
X n
→ → ∞ ∈
The following standard observations will also be used later on.
Lemma 7. Let Y be a reflexive Banach space, Z a normed vector space
and T (Y,Z). Then for every A (Y) we have T(A) (Z).
∈ L ∈ CBC ⊂ CBC
Consequently, every map F : M (Y) defined on an arbitrary set M
→ CBC
gives rise to a map T F : M (Z)
◦ → CBC
Proof. Let A (Y). Since T is linear and continuous, T(A) is convex
∈ CBC
and bounded. To see that T(A) is closed, we consider a sequence (z ) in
k k
T(A)suchthatz z Z ask . Choosingy AsuchthatT(y ) = z
k k k k
→ ∈ → ∞ ∈
for k , we obtain a bounded sequence (y ) in A. Since Y is reflexive,
k k
∈ N
we may pass to a subsequence such that y ⇀ y Y as ′ k , and
k
∈ N ∋ → ∞
y A by Mazur’s Theorem. Moreover,
∈
T(y) = w-lim Ty = w-lim z = z
k→∞ k k→∞ k
and thus z T(A). Hence T(A) is closed.
∈
Lemma 8. Let X be a reflexive Banach space, and let M′,M′′ (X).
∈ CBC
Then
M′ +M′′ := m′ +m′′ : m′ M,m′′ M′′ (X).
{ ∈ ∈ } ∈ CBC
Proof. It is easy to check that M′ + M′′ is bounded and convex. We show
that M′ + M′′ is closed. Let (m ) M′ + M′′ be any sequence with
n n
⊂
lim m = m X. Then there exist (m′ ) M′ and (m′′) M′′ such
n→∞ n ∈ n n ⊂ n n ⊂
thatm = m′ +m′′ foralln. BytheBanach-Alaoglutheorem, wehavem′ ⇀
n n n n
m′ M′ along a subsequence, so that m′′ = m m′ ⇀ m m′. By Mazur’s
∈ n n− n −
lemma, m′′ := m m′ M′′. Therefore, m = m′ +m′′ M′ +M′′.
− ∈ ∈
5
3 An infinite-dimensional solvability theorem
Let V be a separable Hilbert space with scalar product ( , ) , associated
V
· ·
norm . The main result of this section is the following solvability
V
k · k
theorem.
Theorem 9. Let x˜,y¯ V, R > 0 and l < 0, and let F : B (x˜,R) (V)
V
∈ → CBC
be a multivalued mapping.
a) Let F be bounded and c-uhc. If there exists some y˜ F(x˜) such that
∈
y¯ y˜ lR and
V
k − k ≤ −
x B (x˜,R) y F(x) : (y y˜,x x˜) l x x˜ 2,
∀ ∈ V ∃ ∈ − − V ≤ k − kV
then there exists some
x¯ B (x , 1 y˜ y¯ ) with x = x˜+ 1(y¯ y˜),
∈ V c −2lk − kV c 2l −
satisfying y¯ F(x¯).
∈
b) If F admits a modulus of continuity relative to x˜ in the sense that
e (F(x),F(x˜)) ω( x x˜ )
V V
≤ k − k
for all x B (x˜,r) and some r minω−1(dist (y¯,F(x˜))), then
V V
∈ ≤
x x˜ r for all x F−1(y¯) B (x˜,R).
V V
k − k ≥ ∈ ∩
A variant of part a) for finite-dimensional Hilbert spaces has been proved
in [5]. This variant will be used in the proof of Theorem 9 together with
a Galerkin approximation. For this we let (w )∞ denote an orthonor-
k k=1
mal basis of V, and for N we consider the finite-dimensional subspace
∈ N
V := span w ,...,w V. The orthogonal projection from V onto V
N 1 N N
{ } ⊂
is denoted P . We will then use the following reformulation of [5, Theorem
N
3.1].
Theorem 10. Let x˜,y¯ V , R > 0 and l < 0, let F : B (x˜,R) (V )
∈ N N VN → CC N
be a multivalued mapping, and let y˜ F (x˜). If F is usc, if y¯ y˜ lR
N N V
∈ k − k ≤ −
and if for every x B (x˜,R) there exists some y F (x) such that
∈ VN ∈ N
(y y˜,x x˜) l x x˜ 2,
− − V ≤ k − kV
then there exists some x¯ B (x , 1 y¯ y˜ ) with x := x˜+ 1(y¯ y˜) and
∈ VN c −2lk − kV c 2l −
y¯ F (x¯).
N
∈
6
The remainder of this section is devoted to the
Proof of Theorem 9.
Statement a), special case: We begin with the case x˜ = 0, y¯ = 0, in which
y˜ lR, and define y˜ := P y˜. Clearly, B (0,R) B (0,R)
k kV ≤ − N N VN ⊂ V ⊂
V and y˜ lR, and the mapping F : B (0,R) ⇒ V given by
k NkV ≤ − N VN N
F (x) := P F(x) is well-defined with y˜ F (0). Moreover, F satisfies
N N N N N
∈
the assumptions of Theorem 10 on B (0,R) with data x˜ = 0, y¯ = 0 and
VN N N
y˜ :
N
i) The mapping F is bounded on B (0,R) with convex and compact
N VN
values: As V is finite-dimensional and P (V,V ), this follows
N N N
∈ L
from Lemma 7.
ii) The mapping F satisfies an ROSL-type condition: Let x B (0,R).
N ∈ VN
Byassumption, thereexistssomey F(x)suchthat(y y˜,x) l x 2 ,
∈ − ≤ k kV
which implies that
(P y y˜ ,x) = (y y˜,P x) = (y y˜,x) l x 2 .
N − N V − N V − V ≤ k kV
iii) The mapping F is usc: Assume that F is not usc at x B (0,R).
N N ∈ VN
Then there exist ε > 0 and two sequences (x ) B (0,R) and
k k∈N ⊂ VN
(y ) V with lim x = x and y F (x ) such that
k k∈ N k→∞ k k N k
N ⊂ ∈
dist (y ,F (x)) ε.
V k N
≥
As a consequence of [14, Theorem 13.1] there exist (v ) V such
k k∈ N
N ⊂
that v = 1 and
k V
k k
(v ,y ) σ (v ,B (F (x),ε)) = σ (v ,F (x))+ε.
k k V ≥ VN k VN N VN k N
As dimV < , there exists v V such that v v 0 along a
N N k V
∞ ∈ k − k →
subsequence. Hence
(v,y ) = (v v ,y ) +(v ,y )
k V k k V k k V
−
sup F (x′) v v +σ (v ,F (x))+ε
≥ −x′∈BVN(0,R)k N kVk − kkV VN k N (4)
σ (v,F (x))+ε
→ VN N
7
as k , because sup F (x′) < according to a). On
→ ∞ x′∈BVN(0,R)k N kV ∞
the other hand, since F is c-uhc, F is uhc, and for any v V , we have
N
∈
limsupσ (v,F (x )) = limsup sup (P y,v) = limsup sup (y,v)
VN N k N V V
k→∞ k→∞ y∈F(xk) k→∞ y∈F(xk)
= limsupσ (v,F(x )) σ (v,F(x))
V k V
≤
k→∞
= sup (y,v) = sup (P y,v) = σ (v,F (x)).
V N V VN N
y∈F(x) y∈F(x)
This contradicts (4), and hence F is uhc.
N
Therefore, Theorem10yieldsforeveryN somex¯ B ( 1y˜ , 1 y˜ )
∈ N N ∈ VN −2l N −2lk NkV
with 0 F (x¯ ). Rewrite x¯ = 1(y˜ +v ) with v y˜ y˜ .
∈ N N N −2l N N k NkV ≤ k NkV ≤ k kV
Then there exists some v¯ V with v¯ y˜ and such that v ⇀ v¯along
V V N
∈ k k ≤ k k
a subsequence ′ . Since moreover y˜ y˜ as N , we infer that
N
N ⊂ N → → ∞
x¯ ⇀ x¯ := 1(y˜+v¯) B ( 1y˜, 1 y˜ ) as ′ N .
N −2l ∈ V −2l −2lk kV N ∋ → ∞
Furthermore, since 0 F (x¯ ), there exist elements ϕ F(x¯ ) with
N N N N
∈ ∈
P ϕ = 0 for N , which implies that (ϕ ,w ) 0 as N for every
N N N k
∈ N → → ∞
k . Since (w ) is an orthonormal basis of X and the sequence, (ϕ ) is
k k N N
∈ N
bounded as F is bounded, it follows that ϕ ⇀ 0 as N . For arbitrary
N
→ ∞
v V, we thus find that
∈
0 = lim ϕ ,v limsupσ (v,F(x¯ )) σ (v,F(x¯)),
N V N V
N→∞h i ≤ N→∞ ≤
because F is c-uhc. This implies 0 F(x¯).
∈
Statement a), general case: Consider x˜ V, y¯ V and the map G :
∈ ∈
B (0,R) (V) given by
V
→ CBC
G(z) := F(z +x˜) y¯,
−
Clearly y := y˜ y¯ F(x˜) y¯ = G(0). For any x B (x˜,R), there exists
0 V
− ∈ − ∈
z B (0,R) such that z + x˜ = x, and by assumption, there exists some
V
∈
y F(x) such that
∈
(y y˜,x x˜) l x x˜ 2 = l z 2 .
− − V ≤ k − kV k kV
But then y′ := y y¯ F(x) y¯= G(z) satisfies
− ∈ −
(y′ y ,z) = (y y¯) (y˜ y¯),x x˜ = (y y˜,x x˜) l z 2 ,
− 0 V − − − − V − − V ≤ k kV
(cid:0) (cid:1)
8
so that G satisfies all assumptions of Step 1, which guarantees the existence
of some z B (0,R) with 0 G(z ) and
0 V 0
∈ ∈
z + 1y 1 y .
k 0 2l 0kV ≤ −2lk 0kV
Setting x¯ := x˜+z we obtain y¯ F(x¯) and
0
∈
x¯ x = x¯ x˜ 1(y¯ y˜) = z + 1y 1 y = 1 y˜ y¯ .
k − ckV k − − 2l − kV k 0 2l 0kV ≤ −2lk 0kV −2lk − kV
Statement b): Assume that y¯ F(x) for some x B (x˜,r). Then
V
∈ ∈
dist (y¯,F(x˜)) e (F(x),F(x˜)) ω( x x˜ )
V V V
≤ ≤ k − k
implies
x x˜ minω−1(dist (y¯,F(x˜))) r.
V V
k − k ≥ ≥
4 A reformulation for Gelfand triples
The aim of this section is to adapt the above solvability theorem to a situ-
ation in which the multivalued operator consists of two parts with different
properties. Theorem 11 improves the approach presented in [4] by consider-
ing the problem in a space that is adapted to the composed operator. We
postpone a comparison of both results to Remark 13 at the end of this sec-
tion. The most prominent setting, in which such a splitting occurs, will be
discussed in the extended example in Section 7.
Let (V, ,( , ) ) and (H, ,( , ) ) be separable Hilbert spaces
V V H H
k · k · · k · k · ·
such that V is densely and continuously embedded into H with embedding
constant c > 0. Identifying H with its dual H∗, we then have embeddings
VH
V ֒i H ֒i∗ V∗.
→ →
Here i∗ denotes the dual of i, and this map is injective due to the density of
i(V) in H. As usual, we regard V as a subspace of H and H as a subspace
of V∗, writing simply v H instead of i(v) and w V∗ instead of i∗(w) for
∈ ∈
v V, w H. With these simplifications, we have
∈ ∈
y,x = (y,x) x V, y H.
H
h i ∀ ∈ ∈
9
In the following, we fix constants
l < 0 and l < l /c2 . (5)
V H − V VH
Then the bilinear form
(x ,x ) (x ,x ) := l (x ,x ) l (x ,x ) , x ,x V
1 2 1 2 W V 1 2 V H 1 2 H 1 2
7→ − − ∈
is a scalar product which induces an equivalent norm on V. In the
W
k · k
following, B (x,R) denotes the ball with radius R > 0 w.r.t. centered
W W
k·k
at x V. Moreover, for y V∗, we denote by y W∗ the dual norm induced
∈ ∈ k k
by k · kW, i.e., kykW∗ = sup khxyk,xWi for y ∈ V∗. We also denote by JW :
x∈V\{0}
V∗ V the corresponding canonical isometric isomorphism given by
→
(J ϕ,v) = ϕ,v v V, ϕ V∗.
W W
h i ∀ ∈ ∈
The following theorem is a variant of Theorem 9 for composite operators.
Theorem 11. Suppose that l ,l satisfy (5), and let x˜ V, y¯ V∗
V H
∈ R ∈ ∈
and R > 0. Moreover, let
F : B (x˜,R) V (V∗) and F : B (x˜,R) V (H)
V W H W
⊂ → CBC ⊂ → CBC
be bounded and c-uhc, and let F : B (x˜,R) ⇒ V∗ be given by F = F +F ,
W V H
i.e.
F(v) := y +y : y F (v),y F (v) for v V.
V H V V H H
{ ∈ ∈ } ∈
Suppose furthermore that there exists y˜ F (x˜), y˜ F (x˜) such that
V V H H
∈ ∈
y¯ y˜ W∗ R
k − k ≤
with y˜:= y˜ +y˜ and
V H
x B (x˜,R) y y˜ ,x x˜ l x x˜ 2 ;
∀ ∈ W with h V − V − i ≤ Vk − kV (6)
y F (x),y F (x) (y y˜ ,x x˜) l x x˜ 2 .
∃ V ∈ V H ∈ H ) ( H − H − H ≤ Hk − kH
Finally, let x := x˜ 1J (y¯ y˜). Then there exists some
c − 2 W −
x¯ ∈ BW(xc, 21ky˜−y¯kW∗) satisfying y¯∈ F(x¯). (7)
10