Electronic Journal of Differential Equations, Vol. 2002(2002), No. 76, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
A viability result for second-order differential
inclusions ∗
Vasile Lupulescu
Abstract
We prove a viability result for the second-order differential inclusion
x00 ∈ F (x, x0 ),
(x(0), x0 (0)) = (x0 , y0 ) ∈ Q := K × Ω,
where K is a closed and Ω is an open subsets of Rm , and is an upper
semicontinuous set-valued map with compact values, such that F (x, y) ⊂
∂V (y), for some convex proper lower semicontinuous function V .
1
Introduction
Bressan, Cellina and Colombo [6] proved the existence of local solutions to the
Cauchy problem
x0 ∈ F (x), x(0) = ξ ∈ K,
where F is an upper semicontinuous, cyclically monotone, and compact valued
multifunction. While Rossi [15] proved a viability result for this problem. On
the other hand, for the second order differential inclusion
x00 ∈ F (x, x0 ),
x(0) = x0 ,
x0 (0) = y0 ,
existence results were obtained by many authors [1, 4, 9, 10, 13, 16]). In [12],
existence results are proven for the case when F (., .) is an upper semicontinuous
set-valued map with compact values, such that F (x, y) ⊂ ∂V (y) for some convex
proper lower semicontinuous function V .
The aim of this paper is to prove a viability result for the second-order
differential inclusion
x00 ∈ F (x, x0 ),
(x(0), x0 (0)) = (x0 , y0 ) ∈ Q := K × Ω,
where K is a closed and Ω is an open subsets of Rm , and F : Q ⊂ R2m →
m
2R is an upper semicontinuous set-valued map with compact values, such that
F (x, y) ⊂ ∂V (y), for some convex proper lower semicontinuous function V .
∗ Mathematics Subject Classifications: 34G20, 47H20.
Key words: second-order contingent set, subdifferential, viable solution.
c
2002
Southwest Texas State University.
Submitted December 9, 2001. Revised March 26, 2002. Published August 20, 2002.
1
2
2
A viability result
EJDE–2002/76
Preliminaries and statement of main result
Let Rm be the m-dimensional Euclidean space with scalar product h., .i and
norm k.k. For x ∈ Rm and ε > 0 let
Bε (x) = {y ∈ Rm : kx − yk < ε}
be the open ball, centered at x with radius ε, and let B ε (x) be its closure.
Denote by B the open unit ball B = {x ∈ Rm : kxk < 1}.
For x ∈ Rm and for a closed subsets A ⊂ Rm we denote by d(x, A) the
distance from x to A given by
d(x, A) = inf{kx − yk : y ∈ A}.
Let V : Rm → R be a proper lower semicontinuous convex function. The
m
multifunction ∂V : Rm → 2R defined by
∂V (x) = {ξ ∈ Rm : V (y) − V (x) > hξ, y − xi, ∀y ∈ Rm }
is called subdifferential (in the sense of convex analysis) of the function V .
m
We say that a multifunction F : Rm → 2R is upper semicontinuous if for
every x ∈ Rm and every ε > 0 there exists δ > 0 such that
F (y) ⊂ F (x) + Bε (0),
∀y ∈ Bδ (x).
This definition of the upper semicontinuous multifunction is less restrictive
than the usual (see Definition 1.1.1 in [3] or Definition 1.1 in [11]). Actually such
a property is called (ε, δ)-upper semicontinuity (see Definition 1.2 in [11]) and it
is only equivalent to the upper semicontinuity for compact-valued multifunctions
(see Proposition 1.1 in [11]).
For K ⊂ Rm and x ∈ K denote by TK (x) the Bouligand’s contingent cone
of K at x, defined by
d(x + hv, K)
TK (x) = v ∈ Rm : lim inf
=0 .
h→0+
h
(2)
For K ⊂ Rm and (x, y) ∈ K × Rm we denote by TK (x, y) the second-order
contingent set of K at (x, y) introduced by Ben-Tal [5] and defined by
2
d(x + hy + h2 v, K)
(2)
TK (x, y) = v ∈ Rm : lim inf
=0 .
2
h→0+
h /2
(2)
We remark that if TK (x, y) is non-empty then, necessarily, y ∈ TK (x).
Moreover (see [4], [10], [13]), if F is upper semicontinuous with compact
convex values and if x : [0, T ] → Rm is a solution of the Cauchy problem
x00 ∈ F (x, x0 ),
x(0) = x0 ,
x0 (0) = y0 ,
EJDE–2002/76
Vasile Lupulescu
3
such that x(t) ∈ K, ∀t ∈ [0, T ], then
(x(t), x0 (t)) ∈ graph(TK ),
∀t ∈ [0, T ),
hence, in particular, (x0 , y0 ) ∈ graph(TK ).
m
For a multifunction F : Q := K × Ω ⊂ R2m → 2R and for any (x0 , y0 ) ∈
graph(TK ) we consider the Cauchy problem
x00 ∈ F (x, x0 ),
(x(0), x0 (0)) = (x0 , y0 ) ∈ Q
(2.1)
under the following assumptions:
(H1) K is a closed and Ω and open subset of Rm , such that
Q := K × Ω ⊂ graph(TK )
(H2) F is an upper semicontinuous compact valued multifunction such that
(2)
F (x, y) ∩ TK (x, y) 6= ∅,
∀(x, y) ∈ Q;
(H3) There exists a proper convex and lower semicontinuous function V : Rm →
R such that
F (x, y) ⊂ ∂V (y), ∀(x, y) ∈ Q.
Remark. A convex function V : Rm → R is continuous in the whole space
Rm (Corollary 10.1.1 in [14]) and almost everywhere differentiable (Theorem
25.5 in [14]). Therefore, (H3) strongly restricts the multivaluedness of F .
Definition. By viable solution of the problem (2.1) we mean any absolutely
continuous function x : [0, T ] → Rm with absolutely continuous derivative x0
such that x(0) = x0 , x(0) = y0 ,
x00 (t) ∈ F (x(t), x0 (t)) a.e. on [0, T ],
(x(t), x0 (t)) ∈ Q ∀t ∈ [0, T ].
Our main result is the following:
m
Theorem 2.1 If F : Q ⊂ R2m → 2R and V : Rm → R satisfy assumptions
(H1)–(H3), then then for every (x0 , y0 ) ∈ Q there exist T > 0 and x : [0, T ] →
Rm , a viable solution of the problem (2.1).
3
Proof of the main result
We start this section with the following technical result, which will be used to
prove the main result.
m
Lemma 3.1 Assume Q = K × Ω ⊂ R2m satisfies (H1), F : Q → 2R satisfies
(H2), Q0 ⊂ Q is a compact subset and (x0 , y0 ) ∈ Q0 . Then for every k ∈ N∗
there exist h0k ∈ (0, k1 ] and u0k ∈ Rm such that
x0 + h0k y0 +
(h0k )2 0
1
uk ∈ K, (x0 , y0 , u0k ) ∈ graph(F ) + (B × B × B).
2
k
4
A viability result
EJDE–2002/76
(2)
Proof. Let (x, y) ∈ Q be fixed. Since by (H2), F (x, y) ∩ TK (x, y) 6= ∅, there
exists v = v(x,y) ∈ F (x, y) such that
2
lim inf
h→0+
d(x + hy + h2 v, K)
= 0.
h2 /2
∗
Hence, for every k ∈ N there exists hk = hk (x, y) ∈ (0, k1 ] such that
d(x + hk y +
h2k
h2
v, K) < k .
2
4k
By the continuity of the map (a, b) → d(a + hk b +
h2k
2 v, K)
(3.1)
it follows that
h2
h2 N (x, y) = (a, b) : d(a + hk b + k v, K) < k
2
4k
is an open set and, by (3.1), it contains (x, y). Then there exists r := r(x, y) ∈
(0, k1 ) such that Br (x, y) ⊂ N (x, y). Since Q0 is compact there exists a finite
subset {(xj , yj ) ∈ Q : 1 6 j 6 m} such that
Q0 ⊂
m
[
Brj (xj , yj ).
j=1
We set
h0 (k) := min{hk (xj , yj ) : j ∈ {1, . . . , m}}.
Since (x0 , y0 ) ∈ Q0 , there exists j0 ∈ {1, 2, . . . m} such that
(x0 , y0 ) ∈ Brj0 (xj0 , yj0 ) ⊂ N (xj0 , yj0 ).
(3.2)
Denote by h0k := hk (xj0 , yj0 ) and remark that, by (3.1) and (3.2), one has
h0k ∈ [h0 (k), k1 ] and there exists z0 ∈ K such that we have that
(h0 )2
(h0 )2
d(x0 + h0k y0 + 2k v0 , z0 )
d(x0 + hk y0 + 2k v0 , K)
1
1
6
+
< ,
0
0
2
2
(hk ) /2
(hk ) /2
2k
k
hence
k
z0 − x0 − h0k y0
1
− v0 k < .
(h0k )2 /2
k
Let
u0k :=
(3.3)
z0 − x0 − h0k y0
.
(h0k )2 /2
Then
x0 + h0k y0 +
(h0k )2 0
uk ∈ K.
2
By (3.3) and (3.2) we get successively:
kuk − v0 k <
1
,
k
d((x0 , y0 ), (xj0 , yj0 )) 6 rj0 <
hence (x0 , y0 , u0k ) ∈ graph(F ) + k1 (B × B × B).
1
,
k
EJDE–2002/76
Vasile Lupulescu
5
Proof of Theorem 2.1 Let (x0 , y0 ) ∈ Q ⊂ graph(TK ). Since Ω ⊂ Rm is an
open subset, there exist r > 0 such that B r (y0 ) ⊂ Ω.
We set Q0 := B r (x0 , y0 ) ∩ (K × B r (y0 )). Since Q0 is a compact set, by the
upper semicontinuity of F and Proposition 1.1.3 in [3], we have that
[
F (Q0 ) :=
F (x, y)
(x,y)∈Q0
is a compact set, hence there exists M > 0 such that:
sup{kvk : v ∈ F (x, y),
Let
T = min
r
,
2(M + 1)
r
(x, y) ∈ Q0 } 6 M.
r
r
,
.
M + 1 2(ky0 k + 1)
(3.4)
We shall prove the existence of a viable solution of the problem (2.1) defined
on the interval [0, T ]. Since (x0 , y0 ) ∈ Q0 then, by Lemma 3.1, there exist
h0k ∈ [h0 (k), k1 ] and u0k ∈ Rm such that
1
x0 + h0k y0 + (h0k )2 u0k ∈ K
2
and (x0 , y0 , u0k ) ∈ graph(F ) + k1 (B × B × B). Define
1
x1k :=x0 + h0k y0 + (h0k )2 u0k ;
2
yk1 :=y0 + h0k u0k .
(3.5)
We remark that if h0k < T then
1
1
kx1k − x0 k 6 h0k ky0 k + (h0k )2 ku0k k < h0k ky0 k + (h0k )2 (M + 1),
2
2
kyk1 − y0 k = h0k ku0k k < h0k (M + 1),
and by the choice of T we get
kx1k − x0 k < r,
kyk1 − y0 k < r.
Therefore (x1k , yk1 ) ∈ Q0 and by Lemma 3.1, there exist h1k ∈ [h0 (k), k1 ] and
u1k ∈ Rm such that
1
x1k + h1k yk1 + (h1k )2 u1k ∈ K,
2
1
1
1
1
(xk , yk , uk ) ∈ graph(F ) + (B × B × B).
k
We claim that, for each k ∈ N∗ , there exist m(k) ∈ N∗ and hpk , xpk , ykp , upk , such
that for every p ∈ {2, . . . , m(k) − 1}, we have that:
6
A viability result
(i)
Pm(k)−1
j=0
hjk 6 T <
Pm(k)
j=0
EJDE–2002/76
hjk
(ii)
xpk =x0k +
ykp =yk0 +
p−1
p−2 p−1
1X i 2 i X X i j i
(hk ) uk +
hk hk u k ,
hik y0 +
2 i=0
i=0 j=i+1
i=0
p−1
X
p−1
X
hik uik ;
i=0
(iii) (xpk , ykp ) ∈ Q0
(iv) (xpk , ykp , upk ) ∈ graph(F ) + k1 (B × B × B).
If h0k + h1k ≥ T then we set m(k) = 1. Assume that h0k + h1k < T and define
1
x2k :=x1k + h1k yk1 + (h1k )2 u1k ,
2
yk2 :=yk1 + h1k u1k .
(3.6)
Then by (3.5) and (3.6) we have that
1
x2k :=x0k + (h0k + h1k )yk0 + (h0k )2 u1k + (h1k )2 u1k + h0k h1k u0k ,
2
yk2 :=yk0 + h0k u0k + h1k u1k
and since h0k + h1k ≤ T and
1
1
kx2k − x0 k 6(h0k + h1k )kyk0 k + (h0k )2 ku0k k + (h1k )2 ku1k k + h0k h1k ku0k k
2
2
1
<(h0k + h1k )kyk0 k + (h0k + h1k )2 (M + 1),
2
it follows
kx2k − x0 k < r, kyk2 − y0 k < r,
hence (x2k , yk2 ) ∈ Q0 .
Assume that hqk xqk , ykq uqk , have been constructed for q 6 p satisfying (ii)–
p+1
p+1
(iv) and that we construct hp+1
, up+1
satisfying such properties.
k , xk , yk
k
p
p
Since (xk , yk ) ∈ Q0 , by lemma 2, there exist hpk ∈ [h0 (k), k1 ] and upk ∈ Rm such
that
1
xpk + hpk ykp + (hpk )2 upk ∈ K,
2
1
(xpk , ykp , upk ) ∈ graph(F ) + (B × B × B).
k
If h0k +h1k +· · ·+hpk > T then we set m(k) = p. Assume that h0k +h1k +· · ·+hpk < T
and define
1
xp+1
:=xpk + hpk ykp + (hpk )2 upk ,
k
2
(3.7)
ykp+1 :=ykp + hpk upk .
EJDE–2002/76
Vasile Lupulescu
7
Then, by the above equations and (ii), we obtain that
p−1
p−1
X
1
1X i 2 i
= xpk + hpk ykp + (hpk )2 upk = x0k +
hik y0 +
(h ) u
2
2 i=0 k k
i=0
xp+1
k
+
p−1
p−2 p−1
p−1
X
1X i 2 i X X i j i
1
(hk ) uk +
hk hk uk + hpk
hik uik + (hpk )2 upk
2 i=0
2
i=0 j=i+1
i=0
= x0k +
p
X
p
p−1
p
1X i 2 i X X i j i
hik y0 +
(hk ) uk +
hk hk u k
2 i=0
i=0
i=0 j=i+1
and
ykp+1 := ykp + hpk upk = yk0 +
p−1
X
hik uik + hpk upk = yk0 +
i=0
p
X
hik uik .
i=0
Therefore,
kxp+1
− x0 k 6
k
p
X
p
p−1 X
p
X
1X i 2 i
hik ky0 k +
(hk ) kuk k +
hik hjk kuik k
2
i=0
i=0
i=0 j=i+1
6
p
X
hik
i=0
p
M + 1 X i 2
ky0 k +
hk
2
i=0
and
kykp+1 − x0 k 6
p
X
hik kui0 k 6 (M + 1)
i=0
Since
Pp
i
i=0 hk
p
X
i=0
hik .
< T one obtains that
kxp+1
− x0 k < r,
k
kykp+1 − x0 k < r,
p+1
hence (xp+1
) ∈ Q0 .
k , yk
We remark that this iterative process is finite because hpk ∈ [h0 (k), k1 ], implies
the existence of an integer m(k) such that
m(k)−1
h0k + h1k + · · · + hk
m(k)−1
6 T < h0k + h1k + · · · + hk
m(k)
+ hk
.
By (iv), for every k ∈ N∗ and every p ∈ {0, 1, . . . , m(k)} there exists
∈ graph(F ) such that
(apk , bpk , vkp )
1
,
k
1
;
k
(3.8)
1
+ kx0 k 6 1 + kx0 k,
k
1
kykp k 6 kykp − y0 k + ky0 k 6 + ky0 k 6 1 + ky0 k,
k
1
kupk k 6 kupk − vkp k + kvkp k 6 + M 6 1 + M.
k
(3.9)
kxpk − apk k <
kykp − bpk k <
1
,
k
kupk − vkp k <
hence,
kxpk k 6 kxpk − x0 k + kx0 k 6
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A viability result
EJDE–2002/76
Let us set
tpk = h0k + h1k + · · · + hp−1
,
k
t0k = 0.
We remark that for all k ∈ N∗ and all p ∈ {1, . . . , m(k)}, we have
tpk − tp−1
<
k
1
k
m(k)−1
and tk
m(k)
6 T < tk
.
(3.10)
For each k > 1 and for p ∈ {1, . . . , m(k)} we set Ikp = [tp−1
, tpk ] and for t ∈ Ikp
k
we define
1
xk (t) = xp−1
+ (t − tp−1
)ykp−1 + (t − tp−1
)2 up−1
.
(3.11)
k
k
k
k
2
Then
x0k (t) = ykp−1 + (t − tp−1
)up−1
, ∀t ∈ Ikp ,
k
k
(3.12)
x00k (t)
=
up−1
,
k
∀t ∈
Ikp ,
hence, by (3.9), for all t ∈ [0, T ], we obtain
kx00k (t)k 6kup−1
k<M +1
k
kx0k (t)k 6kykp−1 k + (t − tpk )kupk k < ky0 k + M + 2
1
)2 kup−1
k
kxk (t)k 6kxp−1
k + (t − tp−1
)kykp−1 k + (t − tp−1
k
k
k
k
2
6kx0 k + ky0 k + M + 3.
(3.13)
Moreover, for all t ∈ [0, T ] we have that
(xk (t), x0k (t), x00k (t)) ∈ (xpk , ykp , upk ) +
ky0 k + M + 2
M +1
B×
B × {0};
k
k
hence, by (iv), we have
(xk (t), x0k (t), x00k (t)) ∈ graph(F ) + ε(k)(B × B × {0}),
(3.14)
where ε(k) → 0 when k → ∞. Then, by (3.11), (3.12) and (3.13), we obtain that
(x00k )k is bounded in L2 ([0, T ], Rm ), (x0k )k and (xk )k are bounded in C([0, T ], Rm )
and equi-Lipschitzian, hence, by Theorem 0.3.4 in [3] there exist a subsequence
(again denoted by (xk )k ) and an absolutely continuous function x : [0, T ] → Rm
such that
(a) (xk )k converge uniformly to x
(b) (x0k )k converge uniformly to x0
(c) (x00k )k converge weakly in L2 ([0, T ], Rm ) to x00 .
By (H3) and Theorem 1.4.1 in [3] we get that
x00 (t) ∈ co F (x(t), x0 (t)) ⊂ ∂V (x0 (t)), a.e. on [0, T ],
EJDE–2002/76
Vasile Lupulescu
9
where co stands for the closed convex hull; hence, by Lemma 3.3 in [7], we
obtain that
d
V (x0 (t)) = kx00 (t)k2 , a.e. on [0, T ];
dt
hence
Z
T
V (x0 (T )) − V (x0 (0)) =
kx00 (t)k2 dt.
(3.15)
0
On the other hand, since x00k (t) = up−1
, ∀t ∈ Ikp , by (iv), there exist ap−1
, bp−1
,
k
k
k
p−1
1
zk ∈ k B, such that
up−1
− zkp−1 ∈ F (xkp−1 − ap−1
, ykp−1 − bp−1
) ⊂ ∂V (ykp−1 − bp−1
), ∀k ∈ N∗ (3.16)
k
k
k
k
and so the properties of the subdifferential of a convex function imply that, for
every p < m(k), and for every k ∈ N∗ we have
V (x0k (tpk ) − bpk ) − V (x0k (tp−1
) − bp−1
)>
k
k
>hup−1
− zkp−1 , x0k (tpk ) − x0k (tp−1
) + bp−1
− bpk i =
k
k
k
Z tpk
p−1
=hup−1
−
z
,
x00k (t)dti + hup−1
− zkp−1 , bp−1
− bpk i =
k
k
k
k
tp−1
k
=
Z
tp
k
tp−1
k
kx00k (t)k2 dt − hzkp−1 ,
Z
tp
k
tp−1
k
x00k (t)dti + hup−1
− zkp−1 , bp−1
− bpk i;
k
k
hence
V (x0k (tpk ) − bpk ) − V (x0k (tp−1
) − bp−1
)
k
k
Z tpk
Z tpk
>
kx00k (t)k2 dt − hzkp−1 ,
x00k (t)dti + hup−1
− zkp−1 , bp−1
− bpk i. (3.17)
k
k
tp−1
k
tp−1
k
m(k)
Analogously if T ∈ Ik
, then by (3.10) we have
m(k)−1
V (x0k (T )) − V (x0k (tk
m(k)−1
>huk
=
Z
m(k)−1
− zk
,
m(k)−1
tk
T
m(k)−1
kx00k (t)k2 dt
tk
m(k)−1
+ huk
m(k)−1
) − bk
Z T
−
m(k)−1
x00k (t)dt + bk
m(k)−1
hzk
,
m(k)−1
− zk
m(k)−1
, bk
)
Z
i
(3.18)
T
m(k)−1
x00k (t)dti
tk
i.
By adding the m(k) − 1 inequalities from (3.17) and the inequality from (3.18),
we get
Z T
0
0
V (xk (T )) − V (y0 − bk ) >
kx00k (t)k2 dt + α(k),
(3.19)
0
10
A viability result
EJDE–2002/76
where
m(k)−1
α(k) = −
X
hzkp−1 ,
tp−1
k
p=1
m(k)−1
− hzk
,
Z
m(k)−1
tp
k
Z
X
x00k (t)dti +
− zkp−1 , bp−1
− bpk i
hup−1
k
k
p=1
T
m(k)−1
m(k)−1
tk
x00k (t)dti + huk
m(k)−1
− zk
m(k)−1
, bk
i.
Since
m(k)−1
|α(k)| 6
X
|hzkp−1 ,
Z
tp−1
k
p=1
m(k)
+ |hzk
,
Z
X
kzkp−1 kk
m(k)
6
kk
Z
|hup−1
− zkp−1 , bp−1
− bpk i|+
k
k
p=1
m(k)−1
m(k)−1
tk
p=1
+ kzk
X
x00k (t)dti| +
T
m(k)−1
6
m(k)−1
tp
k
Z
x00k (t)dti| + |huk
m(k)−1
, bk
i|
m(k)−1
tp
k
tp−1
k
x00k (t)dtk +
X
kup−1
− zkp−1 kkbp−1
− bpk k
k
k
p=1
T
m(k)−1
m(k)−1
tk
m(k)−1
− zk
x00k (t)dtk + kuk
m(k)−1
− zk
m(k)−1
kkbk
k
(M + 2)(3m(k) − 1)
k
it following that α(k) → 0 when k → ∞; hence, by (3.19), we passing to the
limit for k → ∞, we obtain
Z T
V (x0 (T )) − V (y0 ) > lim sup
kx00k (t)k2 dt.
(3.20)
k→∞
0
Therefore, by (3.15) and (3.20),
Z T
Z
00
2
kx (t)k dt > lim sup
0
k→∞
T
kx00k (t)k2 dt
0
and, since (x00 )k converges weakly in L2 ([0, T ], Rm ) to x00 , by applying Proposition III.30 in [8], we obtain that (x00 )k converge strongly in L2 ([0, T ], Rm ) to
x00 , hence a subsequence again denoted by (x00 )k converge poinwise a.e. to x00 .
Since by (3.14)
lim d((xk (t), x0k (t), x00k (t)), graph(F )) = 0,
k→∞
and since by (H2) the graph of F is closed ([3], Proposition 1.1.2), we have that
x00 (t) ∈ F (x(t), x0 (t)) a.e. on [0, T ].
It remains to prove that (x(t), x0 (t)) ∈ Q, ∀t ∈ [0, T ]. Indeed, by (3.11),
(3.12), and (3.13), we have that
kxk (t) − xpk k <
ky0 k + M + 2
,
k
kx0k (t) − ykp k <
M +1
,
k
EJDE–2002/76
Vasile Lupulescu
11
hence
lim d((xk (t), x0k (t)), (xpk , ykp )) = 0.
k→∞
Since, (xpk , ykp ) ∈ Q0 , ∀k ∈ N∗ , by (a) and (b) we have that
lim d((x(t), x0 (t)), (xk (t), x0k (t))) = 0.
k→∞
On the other hand
d((x(t), x0 (t)), Q0 )
6 d((x(t), x0 (t)), (xk (t), x0k (t))) + d((xk (t), x0k (t)), (xpk , ykp )) + d((xpk , ykp ), Q0 );
(3.21)
hence, by passing to the limit we obtain that
d((x(t), x0 (t)), Q0 ) = 0,
∀t ∈ [0, T ].
Since Q0 is closed, we obtain that (x(t), x0 (t)) ∈ Q0 , for all t ∈ [0, T ], which
completes the proof.
Acknowledgements. This research was done while the author was visiting
the Research unit “Mathematics and Applications” (Control Theory Group) of
the Department of Mathematics of Aveiro University, where the author was
supported by a post-doctoral fellowship. The author wishes to thank Vasile
Staicu for having introduced him to the subject of the paper; The author also
wants to thanks the referee for his/her remarks that improved the results and
their presentation.
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Vasile Lupulescu
Universitatea “Constantin Brâncusi” of Târgu-Jiu
Bulevardul Republicii, Nr.1
1400 Târgu-Jiu, Romania
e-mail: vasile@utgjiu.ro