On measure differential inclusions 73
3. On measure differential inclusions, weak and strong formulations
Let Q : I →
n
, with I ⊂
closed interval, be a given multifunction with nonempty closed convex values and let µ be a Borel measure on I , of bounded variation.
In [23] Stewart considered the two formulations of measure differential inclusions.
Strong formulation.
S
dµ
a
dλ
t ∈ Qt
λ –a.e. in I
dµ
s
d|µ
s
|
t ∈ Q
∞
t µ
s
–a.e. in I where µ
= µ
a
+ µ
s
be the Lebesgue decomposition of measure µ.
Weak formulation.
W R
I
φ dµ
R
I
φ dλ
∈ cl co [
t ∈I ∩
Supp
φ
Qt for every φ
∈ , where
denotes the set of all continuous functions φ : →
+
, with compact support, such that
R
I
φ dλ
6= 0. Stewart proved that the two formulations are equivalent, under suitable assumptions on Q
see Theorem 2, by means of a transfinite induction process. We provide here a direct proof of the equivalence, under weaker assumption.
Moreover, for our convenience, we introduce also the following local version of weak for- mulation.
Local-weak formulation. Let t
∈ I be fixed. There exists h = ht 0 such that for every 0 h h,
LW R
B
h
φ dµ
R
B
h
φ dλ
∈ cl co [
t ∈B
h
Qt for every φ
∈ such that Supp φ
⊂ B
h
. Of course, if µ satisfies W, then LW holds for every t
∈ I . Rather surprising also the convers hold, as we shall show in the following Theorem 3.
In other words, also this last formulation proves to be equivalent to the previous ones. T
HEOREM
1. Every solution of S is also a solution of W. Proof. Let φ
∈ be given. Note that
R
I
φ dµ
= R
I
φ dµ
a
+ R
I
φ dµ
s
moreover Z
I
φ dµ
a
= Z
I
dµ
a
dλ φ
dλ =
Z
I ∩
Supp
φ
dµ
a
dλ dλ
φ
5 Z
I
φ dµ
s
= Z
I
dµ
s
d |µ
s
| φ
d |µ
s
| = Z
I ∩
Supp
φ
dµ
s
d |µ
s
| dµ
s,φ
6 where λ
φ
and µ
s,φ
are the Borel measures defined respectively by λ
φ
E =
Z
E
φ dλ
µ
s,φ
E =
Z
E
φ d
|µ
s
| E
⊂ I .
74 P. Brandi – A. Salvadori
From 5, in force of the assumption and taking Theorem 1.3 in [1] into account, we get φ
a
: =
R
I
φ dµ
a
R
I
φ dλ
= R
I ∩
Supp
φ dµ
a
dλ
dλ
φ
λ
φ
I ∩ Supp φ
∈ cl co [
t ∈I ∩
Supp
φ
Qt . 7
In the case R
I
φ d
|µ
s
| = 0, then R
I
φ dµ
s
= 0 and the assertion is an immediate consequence of 7.
Let us put 7
′
Q
φ
: = cl co
[
t ∈I ∩
Supp
φ
Qt . Let us assume now that
R
I
φ d
|µ
s
| 6= 0. Then from 6, in force of the assumption we get, as before
R
I
φ dµ
s
R
I
φ d
|µ
s
| =
R
I ∩
Supp
φ dµ
s
d|µ
s
|
dµ
s,φ
µ
s,φ
I ∩ Supp φ
∈ cl co [
t ∈I ∩
Supp
φ
Q
∞
t ⊂
cl co
[
t ∈I ∩
Supp
φ
Qt
∞
= [Q
φ
]
∞
and since the right-hand side is a cone, we deduce φ
s
: =
R
I
φ dµ
s
R
I
φ dλ
= R
I
φ dµ
s
R
I
φ d
|µ
s
| ·
R
I
φ d
|µ
s
| R
I
φ dλ
∈ [Q
φ
]
∞
. 8
From 7 and 8 we have that R
I
φ dµ
R
I
φ dλ
= φ
a
+ φ
s
with φ
a
∈ Q
φ
φ
s
∈ [Q
φ
]
∞
and, by virtue of P
3
, we conclude that R
I
φ dµ
R
I
φ dλ
∈ Q
φ
= cl co [
t ∈
Supp
φ
Qt which proves the assertion.
T
HEOREM
2. Let µ be a solution of LW in t ∈ I .
a If Q has properties Q at t and the derivative
dµ
a
dλ
t exists, then
dµ
a
dλ t
∈ Qt .
b If Q
∞
has properties Q at t and the derivative
dµ
s
d|µ
s
|
t exists, then
dµ
s
d |µ
s
| t
∈ Q
∞
t .
On measure differential inclusions 75
Proof. Let S
µ
denote the set where measure µ
s
is concentrated, i.e. S
µ
= {t ∈ I : µ
s
{t} 6= 0}. Since µ
s
is of bounded variation, then S
µ
is denumerable; let us put S
µ
= {s
n
, n
∈ } . Let us fix a point t
∈ I . The case where t
is an end-point for I is analogous. The proof will proceed into steps.
Step 1. Let us prove first that for every B
h
= Bt ,
h ⊂ I with 0 h ht
and such that ∂
B
h
∩ S
µ
= φ, we have µ
B
h
− S
µ
2h =
µ
a
B
h
2h ∈ cl co
[
t ∈B
h
Qt . 9
Let n ∈
be fixed. For every 1 ≤ i ≤ n, we consider a constant 0 r
i
= r
i
n ≤
1 n2
i
such that Bs
i
, r
i
∩ Bs
j
, r
j
= φ, i 6= j, 1 ≤ i, j ≤ n. Moreover, we put I
n
=
n
[
i=1
B s
i
, r
i
. Fixed a constant 0 η min
{h, r
i
, 1
≤ i ≤ n}, we denote by I
n,η
=
n
[
i=1
B s
i
, r
i
− η and consider the function
φ
n,η
t =
t
∈ I − B
h
∪ I
n,η
1 t
∈ B
h−η
− I
n
linear otherwise
Of course φ
n,η
∈ thus, by virtue of the assumption, we have
R
n,η
: =
R
I
φ
n,η
dµ R
I
φ
n,η
dλ ∈ cl co
[
t ∈B
h
Qt . 10
Note that, put C
n,η
= B
h
− I
n,η
∪ B
h−η
− I
n
, we have R
n,η
= R
B
h
−I
n,η
φ
n,η
dµ R
B
h
−I
n,η
φ
n,η
dλ =
µ B
h−η
− I
n
+ R
C
n,η
φ
n,η
dµ λ
B
h−η
− I
n
+ R
C
n,η
φ
n,η
dλ .
11 If we let η
→ 0, we get B
h−η
− I
n
ր B
h
− I
n
I
n,η
ր I
n
and hence C
n,η
ց ∂ B
h
= {t − h, t
+ h} . As a consequence, we have see e.g. [14]
lim
η→
µ B
h−η
− I
n
= µB
h
− I
n
lim
η→
λ B
h−η
− I
n
= λB
h
− I
n
lim
η→
|µ|C
n,η
= lim
η→
λ C
n,η
= 0 12
76 P. Brandi – A. Salvadori
and hence 12
′
lim
η→
Z
C
n,η
φ
n,η
dµ = lim
η→
Z
C
n,η
φ
n,η
dλ = 0 .
From 11, 12 and 12
′
, we obtain lim
η→
R
n,η
= µ
B
h
− I
n
λ B
h
− I
n
= µ
a
B
h
− I
n
+ µ
s
B
h
− I
n
λ B
h
− I
n
. 13
Note that since λ
I
n
=
n
X
i=1
2r
i
≤ 2
n
n
X
i=1
1 2
i
2 n
we have lim
n→+∞
λ I
n
= lim
n→+∞
µ
a
I
n
= 0 . 14
Moreover |µ
s
B
h
− I
n
| ≤ |µ
s
|B
h
− I
n
≤ |µ
s
|S
µ
− I
n
= X
nn
|µ
s
|{s
n
} and, recalling that µ has bounded variation
14
′
lim
n→+∞
|µ
s
B
h
− I
n
| ≤ lim
n→+∞
X
nn
|µ
s
|{s
n
} = 0 . Finally, from 13, 14 and 14
′
we conclude that lim
n→+∞
lim
η→
R
n,η
= µ
a
B
h
2h that, by virtue of 10, proves 9.
Step 2. Let us prove now part a. We recall that
dµ
a
dλ t
= lim
h→0
µ
a
B
h
2h 15
By virtue of step 1, for every fixed h 0 such that B
h
⊂ I , we have µ
a
B
h
2h ∈ cl co
[
t ∈B
h
Qt ⊂ cl co
[
t ∈B
h
Qt λ
–a.e. 0 h h
and hence, by letting h → 0, and taking 15 into account, we get
dµ
a
dλ t
∈ cl co [
t ∈B
h
Qt . By virtue of the arbitrariness of h 0 and in force of assumption Q, we conclude that
dµ
a
dλ t
∈ \
h0
cl co [
t ∈B
h
Qt = Qt
.
On measure differential inclusions 77
Step 3. For the proof of part b let us note that
dµ
s
d |µ
s
| t
= µ
s
{t }
|µ
s
|{t }
16 since µ
s
{t } =
R
{t }
dµ
s
= R
{t }
dµ
s
d|µ
s
|
d |µ
s
| =
dµ
s
d|µ
s
|
t |µ
s
|{t }.
Let h 0 be fixed in such a way that B
h
= Bt ,
h ⊂ I . For every 0 η h we
consider the continuous function defined by φ
η
t =
1
t ∈ B
η 2
t ∈ I − B
η
linear otherwise.
Note that see e.g. [14] µ
s
{t } = lim
η→
µ B
η 2
= lim
η→
µ B
η
. 17
Moreover we have µ
B
η 2
= Z
B
η
φ
η
dµ =
Z
I
φ
η
dµ −
Z
B
η
−B
η 2
φ
η
dµ =
R
I
φ
η
dµ R
I
φ
η
dλ ·
Z
I
φ
η
dλ −
Z
B
η
−B
η 2
φ
η
dµ . 18
By assumption we know that R
I
φ
η
dµ R
I
φ
η
dλ ∈ cl co
[
t ∈B
η
Qt ⊂ cl co
[
t ∈B
h
Qt let us put
Q
h
: = cl co
[
t ∈B
h
Qt . Since lim
η→
Z
I
φ
η
dµ = 0, by virtue of P
4
we get lim
η→
R
I
φ
η
dµ R
I
φ
η
dλ ·
Z
I
φ
η
dλ ∈ [Q
h
]
∞
. 19
Furthermore, by virtue of 17 we have Z
B
η
−B
η 2
φ
η
dµ ≤ |µ|B
η
− |µ|B
η 2
η→
longr i ght arr ow 0 20
thus, from 18 and taking 17, 19 and 20 into account, we obtain µ
{t } ∈ [Q
h
]
∞
for every h 0such that B
h
= Bt ,
h ⊂ I .
Finally, recalling P
5
we deduce that µ
{t } ∈
\
h0
[Q
h
]
∞
= \
h0
Q
h ∞
= Q
∞
t and taking 16 into account, since Q
∞
t is a cone, the assertion follows.
78 P. Brandi – A. Salvadori
D
EFINITION
4. Let µ be a given measure. We will say that a property P holds λ, µ
s
– a.e. if property P is satisfied for every point t with the exception perhaps of a set N with
λ N
+ µ
s
N = 0.
From Theorem 2 the following result can be deduced. T
HEOREM
3. Assume that i Q has properties Q λ–a.e.
ii Q
∞
has properties Q µ
s
–a.e. Then every measure µ which is a solution of LW λ, µ
s
–a.e. is also a solution of S. As we will observe in Section 6, the present equivalence result [among the three formula-
tions S, W, LW] improves the equivalence between strong and weak formulation proved by Stewart, by means of a transfinite process in [23].
It is easy to see that Theorem 3 admits the following generalization. T
HEOREM
4. Let Q
h
: I →
m
, h ≥ 0 be a net of multifunctions and let µ be a Borel
measure. Assume that i Q
t =
\
h0
Q
h
t λ
–a.e.; ii [Q
]
∞
t =
\
h0
[Q
h
]
∞
t µ
s
–a.e.; iii for λ, µ
s
–a.e. t there exists h
= ht 0 such that for every 0 h h
R
B
h
φ dµ
R
B
h
φ dλ
∈ Q
h
t for every φ
∈ such that Supp φ
⊂ B
h
. Then µ is a solution of S.
Proof. Let t ∈ I be fixed in such a way that all the assumptions hold.
Following the proof of step 1 in Theorem 3, from assumption iii we deduce that µ
a
B
h
2h ∈ Q
h
t and hence from assumption i as in step 2 we get
dµ
a
dλ t
∈ \
h0
Q
h
t = Q
t .
Finally, analogously to the proof of step 3, from asumptions iii and ii we obtain µ
{t } ∈
\
h0
[Q
h
]
∞
t = Q
∞
t
On measure differential inclusions 79
and since Q
∞
t is a cone, we get
dµ
s
d |µ
s
| t
= µ
{t }
|µ|{t }
∈ Q
∞
t .
4. The main closure theorem