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
Let I ⊂
be a closed interval and let Q
k
: I →
m
, k ≥ 0, be a sequence of multifunctions.
We introduce first the following definition. D
EFINITION
5. We will say that Q
k k≥0
satisfies condition QK at a point t ∈ E pro-
vided QK
Q t
= \
h0
\
n∈N
cl [
k≥n
cl co [
t ∈B
h
Q
k
t . We are able now to state and prove our main closure result.
T
HEOREM
5. Let Q
k
: I →
m
, k ≥ 0 be a sequence of multifunctions and let µ
k k≥0
be a sequence of Borel measures such that i Q
k k≥0
satisfies QK condition λ, µ
0,s
–a.e.; ii µ
k
w
∗
–converges to µ ;
iii
dµ
k,a
dλ
t ∈ Q
k
t λ
–a.e.
dµ
k,s
d|µ
k,s
|
t ∈ [Q
k
]
∞
t µ
k,s
–a.e. Then the following inclusion holds
dµ
0,a
dλ
t ∈ Q
t λ
–a.e.
dµ
0,s
d|µ
0,s
|
t ∈ [Q
]
∞
t µ
0,s
–a.e. Proof. We prove this result as an application of Theorem 4 to the net
Q
h
t =
\
n∈N
cl [
k≥n
cl co \
τ ∈ Bt,h
Q
k
τ . By virtue of P
5
assumption i assures that both assumptions i and ii in Theorem 4 hold. Now, let t
∈ I be fixed in such a way that assumption iii holds and let φ ∈ be given
with Supp φ ⊂ B
h
∩ I . From Theorem 1 we deduce
R Supp
φ
φ dµ
k
R Supp
φ
φ dλ
∈ cl co [
t ∈
Supp
φ
Q
k
t k
∈ 21
and from assumption ii we get R
Supp
φ
φ dµ
R Supp
φ
φ dλ
= lim
k→+∞
R Supp
φ
φ dµ
k
R Supp
φ
φ dλ
∈ Q
h
t 22
which gives assumption iii in Theorem 4.
80 P. Brandi – A. Salvadori
5. Further closure theorems for measure differential inclusions
We present here some applications of the main result to remarkable classes of measure differen- tial inclusions.
According to standard notations, we denote by L
1
the space of summable functions u : I →
m
and by B V the space of the functions u ∈ L
1
which are of bounded variation in the sense of Cesari [7], i.e. V u
+∞. Let u
k
: I →
m
, k ≥ 0, be a given sequence in L
1
and let Q : I × A ⊂
n+1
→
m
be a given multifunction. D
EFINITION
6. We say that the sequence u
k k≥0
satisfies the property of local equi- oscillation at a point t
∈ I provided LEO
lim
h→0
lim sup
k→∞
sup
t ∈B
h
|u
k
t − u
t | = 0 .
It is easy to see that the following result holds. P
ROPOSITION
2. If u
k
converges uniformly to a continuous function u , then condition
LEO holds everywhere in I . In [10] an other sufficient condition for property LEO can be found see the proof of
Theorem 1. P
ROPOSITION
3. If u
k k≥0
is a sequence of B V functions such that i u
k
converges to u λ
–a.e. in I; ii sup
k∈
V u
k
+∞. Then a subsequence u
s
k
k≥0
satisfies condition LEO λ–a.e. in I. Let us prove now a sufficient condition for property QK.
T
HEOREM
6. Assume that the following conditions are satisfied at a point t ∈ I
i Q satisfies property Q; ii u
k k≥0
satisfies condition LEO. Then the sequence of multifunctions Q
k
: I →
m
, k ≥ 0, defined by
Q
k
t = Qt, u
k
t k
≥ 0 satisfies property QK at t
. Proof. By virtue of assumption ii , fixed ε 0 a number 0 h
ε
ε exists such that for every
0 h h
ε
an integer k
h
exists with the property that for every k ≥ k
h
t ∈ B
h
t H⇒ |u
t − u
k
t | ε .
Then for every k ≥ k
h
cl co [
t ∈B
h
Qt, u
k
t ⊂ cl co
[
|t −t |≤ε,|x−u
t |≤ε
Qt, x = Q
ε
.
On measure differential inclusions 81
Fixed n ≥ k
h
cl [
k≥n
cl co [
t ∈B
h
Qt, u
k
t ⊂ Q
ε
and hence \
n∈
cl [
k≥n
cl co [
t ∈B
h
Qt, u
k
t ⊂ Q
ε
. Finally, by virtue of assumption i , we have
\
ε
\
n∈
cl [
k≥n
cl co [
t ∈B
h
Qt, u
k
t ⊂
\
ε
Q
ε
= Qt ,
u t
which proves the assertion. In force of this result, the following closure Theorem 5 can be deduced as an application of
the main theorem. T
HEOREM
7. Let Q : I × A ⊂
n+1
→
m
be a multifunction, let µ
k k≥0
be a sequence of Borel measures of bounded variations and let u
k
: I → A, k ≥ 0 be a sequence of
B V functions which satisfy the conditions i Q has properties Q at every point t, x with the exception of a set of points whose t -
coordinate lie on a set of λ, µ
0,s
–null measure; ii
dµ
k,a
dλ
t ∈ Qt, u
k
t λ
–a.e.
dµ
k,s
d|µ
k,s
|
t ∈ Q
∞
t, u
k
t µ
k,s
–a.e. iii µ
k
w
∗
–converges to µ ;
i v sup
k∈
V u
k
+∞; v
u
k
converges to u pointwise λ–a.e. and satisfies condition LEO at µ
0,s
–a.e. Then the following inclusion holds
dµ
0,a
dλ
t ∈ Qt, u
t λ
–a.e.
dµ
0,s
d|µ
0,s
|
t ∈ Q
∞
t, u t
µ
0,s
–a.e. R
EMARK
1. We recall that the distributional derivative of a B V function u is a Borel mea- sure of bounded variation [17] that we will denote by µ
u
. Moreover u admits an “essential derivative” u
′
i.e. computed by usual incremental quo- tients disregarding the values taken by u on a suitable Lebesgue null set which coincides with
dµ
u,a
dλ
[25]. Note that Theorem 7 is an extension and a generalization of the main closure theorem in
[10] Theorem 1 given for a differential inclusion of the type u
′
t ∈ Qt, ut
λ –a.e. in I .
To this purpose, we recall that if u
k k≥0
, is a sequence of equi–B V functions, then a subse- quence of distributional derivatives w
∗
–converges. The following closure theorem can be considered as a particular case of Theorem 7.
82 P. Brandi – A. Salvadori
T
HEOREM
8. Let Q : I × E →
m
, with E subset of a Banach space, be a multifunc- tion, let µ
k k≥0
be a sequence of Borel measures of bounded variations and let a
k k≥0
be a sequence in E . Assume that the following conditions are satisfied
i Q has properties Q at every point t, x with the exception of a set of points whose t - coordinate lie on a set of λ, µ
0,s
–null measure; ii
dµ
k,a
dλ
t ∈ Qt, a
k
λ –a.e.
dµ
k,s
d|µ
k,s
|
t ∈ Q
∞
t, a
k
µ
k,s
–a.e. iii µ
k
w
∗
–converges to µ ;
i v a
k k
converges to a .
Then the following inclusion holds
dµ
0,a
dλ
t ∈ Qt, a
λ –a.e.
dµ
0,s
d|µ
0,s
|
t ∈ Q
∞
t, a µ
0,s
–a.e. As we will prove in Section 6, this last result is an extension of closure Theorem 3 in [10].
As an application of Theorem 7 also the following result can be proved. T
HEOREM
9. Let Q : I ×
n
×
p
→
m
, be a multifunction, let f : I ×
n
×
q
→
n
be a function and let u
k
, v
k
: I →
n
×
q
, k ≥ 0, be a sequence of functions.
Assume that i Q satisfies property Q at every point t, x, y with the exception of a set of points whose
t -coordinate lie on a set of λ, µ
v ,
s
–null measure; ii f is a Carath´eodory function and
| f t, u, v| ≤ ψ
1
t + ψ
2
t |u| + ψ
3
t |v| with ψ
i
∈ L
1
i = 1, 2, 3;
iii v
′ k
t ∈ Qt, u
k
t − f t, u
k
t , v
k
t λ
–a.e.
dµ
vk ,s
d|µ
vk ,s
|
t ∈ Q
∞
t, u
k
t µ
v
k
, s
–a.e. i v sup
k∈
V v
k
+∞ and v
k k
converges to v λ
–a.e.; v
u
k k
converges uniformly to a continuous function u .
Then the following inclusion holds v
′
t ∈ Qt, u
t − f t, u
t , v t
λ –a.e.
dµ
v0,s
d|µ
v0,s
|
t ∈ Q
∞
t, u µ
v ,
s
–a.e. Proof. If we consider the sequence of Borel measures defined by
ν
k
[a, b] =
Z
b a
[v
′ k
t + f t, u
k
t , v
k
t ] dλ [a, b]
⊂ I k
≥ 0 it is easy to see that
dν
k,s
= dµ
v
k
, s
dν
k,a
dλ t
= v
′ k
t + f t, u
k
t , v
k
t λ
–a.e. It is easy to verify that assumptions assure that ν
k k≥0
is a sequence of B V measure which w
∗
–converges and the result is an immediate application of Theorem 7.
On measure differential inclusions 83
R
EMARK
2. Differential incusions of this type are adopted as a model for rigid body dy- namics see [20] for details. As we will observe in Section 6 the previous result improves the
analogous theorem proved in [23] Theorem 4.
6. On comparison with Stewart’s assumptions
