Electronic Journal of Differential Equations, Vol. 2005(2005), No. 114, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
CARATHÉODORY PERTURBATION OF A SECOND-ORDER
DIFFERENTIAL INCLUSION WITH CONSTRAINTS
SAÏDA AMINE, RADOUAN MORCHADI, SAÏD SAJID
Abstract. We prove the existence of local solutions for the second-order viability problem
ẍ(t) ∈ f (t, x(t), ẋ(t)) + F (x(t), ẋ(t)),
x(t) ∈ K.
Here K is a closed subset of Rn , F is an upper semicontinuous multifunction
with compact values contained in the subdifferential of a convex proper lower
semicontinuous function V , and f is a Carathéodory function.
1. Introduction
This paper concerns the second-order nonconvex viability problem
ẍ(t) ∈ f (t, x(t), ẋ(t)) + F (x(t), ẋ(t))
(x(0), ẋ(0)) = (x0 , v0 )
(1.1)
x(t) ∈ K.
Where F is a globally upper semicontinuous multifunction, cyclically monotone,
i.e. F (x, y) ⊂ ∂V (y), defined from K × U into the subset of all nonempty compact
subset of Rn ; f is a Carathéodory function from R × K × U to Rn , where K is a
closed subset of Rn , U is an open subset of Rn and ∂V is the subdifferential of a
lower semicontinuous and convex function from U into Rn .
Existence of solutions of differential inclusions with upper semicontinuous and
cyclically monotone right-hand side was first established by Bressan, Cellina and
Colombo [6]. It has been proved the existence of local solutions of a first order
problem without constraints: ẋ(t) ∈ F (x(t)). The approach is based on some technics related to the subdifferential properties applied to approximate solutions. To
avoid the difficulty of the weak convergence of the derivatives of such approximate
solutions, authors rely on the basic relation
d
(V (x(t))) = kẋ(t)k2 .
dt
Regarding the existence of viable solutions of second-order upper semicontinuous
differential inclusions without convexity, we refer to Lupulescu [10, 11], where two
2000 Mathematics Subject Classification. 34A60.
Key words and phrases. Upper semicontinuous multifunction; cyclically monotone;
Carathéodory function.
c
2005
Texas State University - San Marcos.
Submitted June 30, 2005. Published October 21, 2005.
1
2
S. AMINE, R. MORCHADI, S. SAJID
EJDE-2005/114
results are given in this subject: the first deals with the problem (1.1) where f ≡ 0;
while the second studies problem (1.1), but without constraint, i.e. K = Rn .
The first work in the second-order viability problem, was done by Cornet and
Haddad [8], the authors were subject concerned by a problem with upper semicontinuous right-hand side, not cyclically monotone but convex. This research program
was pursued by some works, see [4, 12]. For the nonconvex case, existence results
may be found in [1, 9, 13].
It is known that viability problems require tangential conditions. So far as
we know, for a second-order viability problem, most of the time, authors use the
second-order contingent set
dK (x + hy + 21 h2 z)
=0
AK (x, y) = z ∈ Rn : lim inf
h2
h→0+
2
introduced by Ben-Tal. In the present paper we prove the existence solutions to
(1.1) assuming the tangential condition: For all (t, x, y) ∈ I × K × U , there exists
w ∈ F (x, y) such that
Z t+h
1
h2
lim inf
dK (x + hy + w +
f (τ, x, y)dτ ) = 0.
2
h→0+ h2
t
This condition is also used by Lupulescu [10] with f ≡ 0.
2. Notation and statement of the main result
Let Rn be the n-dimensional Euclidean space with scalar product h ; i and norm
k k. Let K be a closed subset of Rn , U be a nonempty open subset of Rn and
denote Ω = K × U . For each x ∈ Rn we denote by dK (x) the distance from x to
K. For r > 0, B(x, r) stands for the ball centered at x with radius r and B(x, r)
its closure, B is the unit ball of Rn .
Let F be a multifunction from Ω into the set of all nonempty compact subsets
of Rn . Let f be a function from R × Ω into Rn . Assume that F and f satisfy the
following conditions:
(A1) F is upper semicontinuous, i.e. for all (x, y) and for every ε > 0 there exists
δ > 0 such that F (x0 , y 0 ) ⊆ F (x, y) + εB, whenever k(x, y) − (x0 , y 0 )k ≤ δ;
(A2) There exists a convex proper and lower semicontinuous function V : Rn →
Rn such that F (x, y) ⊂ ∂V (y), where ∂V denotes the subdifferential of the
function V ;
(A3) f : R × Ω → Rn is a Carathéodory function, i.e. for each (x, y) ∈ Ω, t →
f (t, x, y) is measurable and for all t ∈ R, (x, y) → f (t, x, y) is continuous;
(A4) There exists m ∈ L2 (R) such that kf (t, x, y)k ≤ m(t) for all (t, x, y) ∈ R×Ω;
(A5) (Tangential condition) For all (t, x, v) ∈ R × Ω, there exists w ∈ F (x, v)
such that
Z t+h
1
h2
lim inf
d
x
+
hv
+
w
+
f
(τ,
x,
v)dτ
= 0.
K
2
h→0+ h2
t
Let (x0 , y0 ) ∈ Ω. Assuming that F and f satisfy (A1)–(A5), we shall prove the
following result.
Theorem 2.1. There exist T > 0 and an absolutely continuous x : [0, T ] → Rn for
which ẋ is also absolutely continuous such that
ẍ(t) ∈ f (t, x(t), ẋ(t)) + F (x(t), ẋ(t))
a.e. t ∈ [0, T ]
EJDE-2005/114
CARATHÉODORY PERTURBATION
3
(x(0), ẋ(0)) = (x0 , y0 )
x(t) ∈ K
∀t ∈ [0, T ].
3. Proof of the main result
We begin by recalling the following result which was proved in [6], and will be
used in the second step of the proof of the main result.
Lemma 3.1. Let V be a convex proper lower semicontinuous function such that
F (x, y) ⊂ ∂V (y), for any (x, y) ∈ Ω. Then there exist r = rx,y > 0 and M =
Mx,y > 0 such that
kF (x, y)k =
sup
kzk ≤ M
on B((x, y), r)
z∈F (x,y)
and V is Lipschitz continuous on B(y, r) with constant M .
Let r and M be the real numbers defined in the Lemma above, and such that
B(y0 , r) ⊂ U . Choose T1 > 0 such that
Z T1
r
(3.1)
(m(s) + M + 1) ds < .
3
0
Set
2r
r
,
.
(3.2)
T2 = min
3(M + 1) 3(ky0 k + r)
In the sequel, we denote by Ω0 the compact set (K × B(y0 , r)) ∩ B((x0 , y0 ), r) and
choose T such that
T ∈]0, min{T1 , T2 }] .
(3.3)
The following result will be used for proving the viability property of the solutions
to (1.1).
Lemma 3.2. Let F and f satisfy assumptions (A1)–(A5). Then for each ε > 0
there exists η ∈]0, ε[ such that for each (t, x, v) in [0, T ] × Ω0 , there exist w in
F (x, v) + Tε B and h in [η, ε]; that is,
Z t+h
h2
x + hv + w +
f (τ, x, v)dτ ∈ K.
2
t
Proof. Let (t, x, v) ∈ [0, T ] × Ω0 , let ε > 0. Since F is upper semicontinuous, then
there exists δ(x,v) > 0 such that
ε
F (y, u) ⊂ F (x, v) + B ∀(y, u) ∈ B((x, v), δ(x,v) ).
(3.4)
T
On the other hand, for all (s, y, u) ∈ [0, T ] × Ω0 , by the tangential condition, there
exist h(s,y,u) ∈]0, ε] and c ∈ F (y, u) such that
Z s+h(s,y,u)
h2(s,y,u)
ε
dK y + h(s,y,u) u +
c+
f (τ, y, u)dτ < h2(s,y,u)
.
2
4T
s
Consider the subset
h2(s,y,u)
N (s, y, u) = (l, a, b) ∈ R × (Rn )2 : dK a + h(s,y,u) b +
c
2
Z l+h(s,y,u)
ε +
f (τ, a, b)dτ < h2(s,y,u)
.
4T
l
4
S. AMINE, R. MORCHADI, S. SAJID
EJDE-2005/114
Since kf (l, a, b)k ≤ m(l) for all (l, a, b) ∈ R×Ω, the dominated convergence theorem
shows that the function
Z l+h(s,y,u)
h2(s,y,u)
(l, a, b) → a + h(s,y,u) b +
c+
f (τ, a, b)dτ
2
l
is continuous. So that, the function
(l, a, b) → dK (a + h(s,y,u) b +
h2(s,y,u)
2
Z
c+
l+h(s,y,u)
f (τ, a, b)dτ )
l
is continuous and consequently the subset N (s, y, u) is open. Moreover, since
(s, y, u) belongs to N (s, y, u), there exists a ball B((s, y, u), η(s,y,u) ) with radius
η(s,y,u) < δ(x,v) and contained in N (s, y, u). Therefore, the compactness of [0, T ] ×
Ω0 implies that it can be covered by q such balls B((si , yi , ui ), η(si ,yi ,ui ) ). For
simplicity, put
h(si ,yi ,ui ) := hi ,
η(si ,yi ,ui ) := ηi ,
η := min hi > 0.
i=1,...,q
Let (t, x, v) ∈ [0, T ] × Ω0 . Since (t, x, v) belongs to one of the balls B((si , yi , ui ), ηi ),
there exist xi ∈ K and ci ∈ F (yi , ui ) such that
Z t+hi
ci − 2 (xi − x − hi v −
f (τ, x, v) dτ )
2
hi
t
Z t+hi
h2i
ε
ε
1
f (τ, x, v) dτ ) +
≤
.
≤ 2 dK (x + hi v + ci +
hi
2
4T
2T
t
Let us set
w=
2
(xi − x − hi v −
h2i
t+hi
Z
f (τ, x, v) dτ ),
t
then
(x + hi v +
h2i
w+
2
Z
t+hi
f (sτ, x, v)dτ ) ∈ K
and kci − wk ≤
t
ε
.
2T
Since (t, x, v) ∈ B((si , yi , ui ), ηi ) and ηi < δ(x,v) , relation (3.4) implies
ε
F (yi , ui ) ⊂ F (x, v) +
B;
2T
so that w ∈ F (x, v) + Tε B. Hence the Lemma is proved.
Now, we are able to prove the main result. Our approach consists of constructing,
in a first step, a sequence of approximate solutions and deduce, in a second step,
from available estimates that a subsequence converges to a solution of (1.1).
Step 1. Construction of approximate solutions. Let (x0 , y0 ) ∈ Ω0 and ε > 0.
By Lemma 3.2, there exist η > 0, h0 in [η, ε] and w0 in F (x0 , y0 ) + Tε B such that
h2
x0 + h0 y0 + 0 w0 +
2
Z
h0
f (τ, x0 , y0 )dτ ∈ K.
0
Put
h2
x1 = x0 + h0 y0 + 0 w0 +
2
Z
h0
f (τ, x0 , y0 )dτ
0
and y1 = y0 + h0 w0 .
EJDE-2005/114
CARATHÉODORY PERTURBATION
5
Since w0 ∈ F (x0 , y0 ) ⊂ B(0, M +1), kf (t, x0 , y0 )k ≤ m(t), by (3.1), (3.2), we obtain
Z h0
h2
kx1 − x0 k = h0 y0 + 0 w0 +
f (τ, x0 , y0 ) dτ 2
0
Z h0
T
f (τ, x0 , y0 ) dτ k
≤ T ky0 k + kw0 k + k
2
0
Z T
(M + 1 + m(τ ))dτ
≤ T ky0 k +
0
r
≤ T ky0 k + ≤ r,
3
and
r
<r
3
and thus (x1 , y1 ) ∈ Ω0 . By induction, for p ≥ 2 and for every i = 1, . . . , p − 1, we
Pp−1
construct (hi , (xi , yi ), wi ) in [η, ε] × Ω0 × Rn such that i=0 hi ≤ T and
Z hi−2 +hi−1
h2
xi = (xi−1 + hi−1 yi−1 + i−1 wi−1 +
f (τ, xi−1 , yi−1 ) dτ ) ∈ K;
2
hi−2
ky1 − y0 k = kh0 w0 k ≤ T kw0 k <
yi = yi−1 + hi−1 wi−1 ;
ε
wi ∈ F (xi , yi ) + B.
T
Since hi ∈]η, ε[ there exists an integer s, such that
s−1
X
hi < T ≤
s
X
hi .
i=0
i=0
In what follows, choose ε small such that
s−1 2
X
h
i
i=0
2
≤
s−1
X
hi < T.
i=0
For all p = 1, . . . , s − 1 define (hp )p ⊂ [η, ε], (xp , yp )p ⊂ Ω0 , and (wp )p as follows
Z hp−2 +hp−1
h2p−1
xp = (xp−1 + hp−1 yp−1 +
wp−1 +
f (τ, xp−1 , yp−1 ) dτ ) ∈ K;
2
hp−2
yp = yp−1 + hp−1 wp−1 ;
ε
wp ∈ F (xp , yp ) + B.
T
Claim: For p = 1, . . . , s − 1, the points (xp , yp ) are in Ω0 .
Indeed, by definition of (xp , yp ), we have
Z h0
p−1 Z Pi
p−1
p−1 2
X
X
X
j=0 hj
hi
f (τ, xi , yi ) dτ ;
xp = x0 +
hi y i +
wi +
f (τ, x0 , y0 ) dτ +
P
i−1
2
0
j=0 hj
i=1
i=0
i=0
yp = yp−1 + hp−1 wp−1 ;
ε
wp ∈ F (xp , yp ) + B.
T
Hence
kyp − y0 k ≤ k
p−1
X
i=0
hi wi k ≤ T (M + 1) ≤
r
≤ r,
3
(3.5)
6
S. AMINE, R. MORCHADI, S. SAJID
EJDE-2005/114
and
kxp − x0 k
p−1
p−1 2
X
X
hi
=
hi y i +
wi +
2
i=0
i=0
Z
h0
f (τ, x0 , y0 ) dτ +
0
p−1
X
p−1 Z
X
i=1
p−1
X
r
h2i
≤ (ky0 k + )
hi + (M + 1)
+
3 i=0
2
i=0
Z
Pi
j=0
Pi−1
j=0
hj
f (τ, xi , yi ) dτ hj
T
m(τ ) dτ.
0
Since
p−1
X
hi ≤ T
p−1 2
X
h
i
and
i=0
i=0
2
≤ T,
by (3.1)–(3.3), we have
r
kxp − x0 k ≤ (ky0 k + )T +
3
Z
T
(M + 1 + m(τ )) dτ,
(3.6)
r
r
kxp − x0 k ≤ T (ky0 k + ) + ≤ r;
3
3
hence (xp , yp )p ⊂ Ω0 which proves the claim.
For any nonzero integer k and for q = 1, . . . , s denote by hkq a real associated to
ε = k1 and (t, x, y) = (hkq−1 , xq , yq ) given by Lemma 3.2. Let the sequence (τkq )k
defined by
τk0 = 0,
0
τks = T,
τkq = hk0 + · · · + hkq−1 ;
and consider the sequence of functions (xk (.))k defined on each interval [τkq−1 , τkq [
by
(t − τkq−1 )2
wq−1
xk (t) = xq−1 + (t − τkq−1 )yq−1 +
2
Z t
+
(t − τ )f (τ, xq−1 , yq−1 )dτ ;
τkq−1
xk (0) = x0 .
Step 2. Convergence of approximate solutions. By the definition of xk , for
all t ∈ [τkq−1 , τkq [ we have
Z t
f (τ, xq−1 , yq−1 )dτ ;
ẋk (t) = yq−1 + (t − τkq−1 )wq−1 +
τkq−1
ẍk (t) = wq−1 + f (t, xq−1 , yq−1 ).
Hence by (3.5) and (3.6) we have the estimates
kẍk (t)k ≤ kwq−1 k + kf (t, xq−1 , yq−1 )k ≤ M + 1 + m(t);
Z
kẋk (t)k = kẋk (τkq−1 ) +
kẋk (t)k ≤ kyq−1 k + t
τkq−1
Z
0
ẍk (τ ))dτ k;
T
(M + 1 + m(τ ))dτ (3.7)
EJDE-2005/114
CARATHÉODORY PERTURBATION
≤ kyq−1 k +
7
r
2r
≤ ky0 k + ;
3
3
and
kxk (t)k = kxk (τkq−1 ) +
Z
Z
t
τkq−1
ẋk (τ ))dτ k
T
2r
)dτ
3
0
2T
≤ kx0 k + T ky0 k + (1 +
)r.
3
≤ kxq−1 k +
(ky0 k +
By (3.7) one has
Z
T
2
T
Z
(M + 1 + m(t))2 dt.
kẍk (t)k dt ≤
0
0
Then the sequence (ẍk (.))k is bounded in L2 ([0, T ], Rn ) and (ẋk (.))k is equiuniformly continuous. Moreover, we see that (xk (.))k is equi-Lipschitzian, hence equiuniformly continuous. Therefore, the sequence (ẍk (.))k is bounded in L2 ([0, T ], Rn ),
(ẋk (.))k and (xk (.))k are bounded in C([0, T ], Rn ) and equiuniformly continuous,
hence, by [3, Theorem 0.3.4] there exist a subsequence, still denoted by (xk (.))k
and an absolutely continuous function x : [0, T ] → Rn such that
(i) xk converges uniformly to x;
(ii) ẋk converges uniformly to ẋ;
(iii) ẍk converges weakly in L2 ([0, T ], Rn ) to ẍ.
The family of approximate solutions xk satisfies the following property.
Proposition 3.3. For every t ∈ [0, T [ there exits q ∈ {1, . . . , s} such that
lim dgrF xk (t), ẋk (t); ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) = 0
k→∞
Proof. Let t ∈ [0, T ]. By construction of τkq , there exists q ∈ 1, . . . s such that
t ∈ [τkq−1 , τkq [ and (τkq )k converges to t. Moreover, for q = 1, . . . s
ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) = wq−1 ∈ F (xk (τkq−1 ), ẋk (τkq−1 )) +
1
B,
kT
then
lim dgrF ((xk (t), ẋk (t)); ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )))
k→∞
≤ lim (kxk (t) − xk (τkq−1 )k + kẋk (t) − ẋk (τkq−1 )k +
k→∞
Since kẍk (t)k ≤ M + 1 + m(t) , kẋk (t)k ≤ ky0 k +
follows that
2r
3
1
).
kT
and (τkq )k converges to t, it
lim kxk (t) − xk (τkq−1 )k = lim kẋk (t) − ẋk (τkq−1 )k = 0,
k→∞
k→∞
hence
lim dgrF (xk (t), ẋk (t)); ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) = 0.
k→∞
This completes the proof.
8
S. AMINE, R. MORCHADI, S. SAJID
EJDE-2005/114
Since xk → x uniformly, ẋk → ẋ uniformly, ẍk → ẍ weakly in L2 ([0, T ], Rn ) and
(f (., xk (τkq−1 ), ẋk (τkq−1 )))k converges to f (., x(.), ẋ(.)) in L2 ([0, T ]; Rn ) and F is
upper semicontinuous, then by [3, Theorem 1.4.1], x is a solution of the convexified
problem
ẍ(t) ∈ f (t, x(t), ẋ(t)) + co(F (x(t), ẋ(t))) a.e. on [0, T ];
x(0) = x0 ,
ẋ(0) = y0 .
Consequently for all t ∈ [0, T ] we have
ẍ(t) − f (t, x(t), ẋ(t)) ∈ ∂V (ẋ(t))
(3.8)
Proposition 3.4. The application x is a solution of (1.1).
Proof. By (3.8) and [7, Lemma 3.3], we obtain
d
(V (ẋ(t))) = hẍ(t), ẍ(t) − f (t, x(t), ẋ(t))i a.e in [0, T ];
dt
therefore,
Z
V (ẋ(T )) − V (y0 ) =
T
kẍ(τ )k2 dτ −
0
Z
T
hẍ(τ ); f (τ, x(τ ), ẋ(τ ))idτ.
(3.9)
0
On the other hand, for q = 1, . . . , s and t ∈ [τkq−1 , τkq [,
1
ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) ∈ F (xk (τkq−1 ), ẋk (τkq−1 )) +
B.
kT
Then
1
B,
ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) ∈ ∂V (ẋk (τkq−1 ) +
kT
hence, there exists bq ∈ B such that
1
q−1
q−1
bq ∈ ∂V (ẋk (τkq−1 ).
ẍk (t) − f (t, xk (τk ), ẋk (τk )) +
kT
(3.10)
Properties of the subdifferential of a convex function imply that for every z in
∂V (ẋk (τkq−1 ), we have
V (ẋk (τkq )) − V (ẋk (τkq−1 )) ≥ hẋk (τkq ) − ẋk (τkq−1 ); zi.
Then by (3.10)
V (ẋk (τkq )) − V (ẋk (τkq−1 ))
≥ hẋk (τkq ) − ẋk (τkq−1 ); ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) +
1
bq i;
kT
thus
V (ẋk (τkq )) − V (ẋk (τkq−1 ))
Z τkq
1
ẍk (τ )dτ ; ẍk (t) − f (t, xk (τkq−1 ), ẋk (τkq−1 )) +
≥h
bq i .
kT
τkq−1
(3.11)
EJDE-2005/114
CARATHÉODORY PERTURBATION
9
Since ẍk is constant in [τkq−1 , τkq [, it follows that
Z τkq
q
q−1
V (ẋk (τk )) − V (ẋk (τk )) ≥
hẍk (τ ); ẍk (τ )idτ
τkq−1
Z
τkq
−
τkq−1
Z
hẍk (τ ); f (t, xk (τkq−1 ), ẋk (τkq−1 ))idτ
τkq
+
τkq−1
hẍk (τ );
1
bq idτ ;
kT
hence we have
V (ẋk (T )) − V (y0 )
Z T
s Z
X
2
≥
kẍk (τ )k dτ −
0
q=1
τkq
τkq−1
hẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))idτ
(3.12)
Z τkq
s
X
1
hẍk (τ ); bq idτ.
+
kT τkq−1
q=1
Ps R τkq
q−1
Claim: The sequence
), ẋk (τkq−1 ))idτ )k converges
q=1 τ q−1 hẍk (τ ); f (τ, xk (τk
k
RT
to 0 hẍ(τ ); f (τ, x(τ ), ẋ(τ ))idτ .
Proof. Since [0, T ] =
s
X
Z
τkq−1
q=1
=k
τkq
τkq−1
q=1
τkq
s Z
X
≤
q=1
q−1 q
, τk ],
q=1 [τk
we have
hẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))idτ −
τkq
s Z
X
Ss
τkq−1
Z
T
hẍ(τ ); f (τ, x(τ ), ẋ(τ ))idτ 0
(hẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))i)dτ k
khẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))ikdτ .
Since
s Z
X
q=1
≤
τkq
τkq−1
s Z
X
q=1
+
s Z
X
s Z
X
q=1
khẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍk (τ ); f (τ, xk (τ ), ẋk (τ ))ikdτ
τkq
τkq−1
s Z
X
q=1
=
τkq
τkq−1
q=1
+
khẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))ikdτ
khẍk (τ ); f (τ, xk (τ ), ẋk (τ ))i − hẍk (τ ); f (τ, x(τ ), ẋ(τ ))ikdτ
τkq
τkq−1
τkq
τkq−1
khẍk (τ ); f (τ, x(τ ), ẋ(τ ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))ikdτ
khẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍk (τ ); f (τ, xk (τ ), ẋk (τ ))ikdτ
10
S. AMINE, R. MORCHADI, S. SAJID
EJDE-2005/114
T
Z
khẍk (τ ); f (τ, xk (τ ), ẋk (τ ))i − hẍk (τ ); f (τ, x(τ ), ẋ(τ ))ikdτ
+
0
T
Z
khẍk (τ ); f (τ, x(τ ), ẋ(τ ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))ikdτ,
+
0
it follows that
Z
s Z τq
k
X
q−1
q−1
hẍk (τ ); f (τ, xk (τk ), ẋk (τk ))idτ −
τkq−1
q=1
=
s Z τq
X
k
τkq−1
q=1
Z
T
hẍ(τ ); f (τ, x(τ ), ẋ(τ ))idτ 0
khẍk (τ ); f (τ, xk (τkq−1 ), ẋk (τkq−1 ))i − hẍk (τ ); f (τ, xk (τ ), ẋk (τ ))ikdτ
T
khẍk (τ ); f (τ, xk (τ ), ẋk (τ ))i − hẍk (τ ); f (τ, x(τ ), ẋ(τ ))ikdτ
+
0
Z
T
khẍk (τ ); f (τ, x(τ ), ẋ(τ ))i − hẍ(τ ); f (τ, x(τ ), ẋ(τ ))ikdτ .
+
0
Since f is a Carathéodory function, xk and ẋk are uniformly lipschitz continuous,
kẍk (s)k ≤ M + 1 + m(s), m ∈ L2 ([0, T ], Rn ), xk → x, ẋk → ẋ uniformly and ẍk
→ ẍ weakly in L2 ([0, T ], Rn ) then the last term converges to 0. Hence the claim is
proved.
Since
Z q
s
X
1 τk
hẍk (τ ); bq idτ = 0,
k→∞
k τkq−1
q=1
lim
by passing to the limit as k → ∞ in (3.12) and using the continuity of the function
V on the ball B(y0 , r), we obtain the estimate
Z T
Z T
V (ẋ(T )) − V (y0 ) ≥ lim sup
kẍk (τ )k2 dτ −
< ẍ(τ ); f (τ, x(τ ), ẋ(τ ) > dτ.
k→∞
0
0
Moreover, by (3.8), we have
kẍk22 ≥ lim sup kẍk k22 ,
k→∞
and by the weak lower semicontinuity of the norm, it follows that
kẍk22 ≤ lim inf kẍk k22 .
k→∞
Hence limk→∞ kẍk k22 = kẍk22 , i.e. ((ẍk ))k converges to ẍ strongly in L2 ([0, T ], Rn ).
So that there exists a subsequence ẍk which converges pointwisely almost every
where to ẍ. In view of Proposition 3.3, we conclude that
dgrF (x(t), ẋ(t), ẍ(t) − f (t, x(t), ẋ(t))) = 0
a.e. t ∈ [0, T ].
Since the graph of F is closed, we have
ẍ(t) ∈ f (t, x(t), ẋ(t)) + F (x(t), ẋ(t)) a.e. t ∈ [0, T ].
Finally, let t ∈ [0, T ]. Recall that there exits (τkq )k such that limk→∞ τkq = t for all
t ∈ [0, T ]. Since lim kx(t) − xk (τkq )k = 0, xk (τkq ) ∈ K and K is closed, by passing
k→∞
to the limit for k → ∞ we obtain x(t) ∈ K. This completes the proof.
Acknowledgements. The authors would like to thank the anonymous referee for
reading carefully the original manuscript and for giving us some suggestions.
EJDE-2005/114
CARATHÉODORY PERTURBATION
11
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Saı̈da Amine
U.F.R Mathematics and Applications, Department of Mathematics, Faculty of Sciences
and Techniques of Mohammedia, BP 146, Mohammedia, Morocco
E-mail address: aminesaida@hotmail.com
Radouan Morchadi
U.F.R Mathematics and Applications, Department of Mathematics, Faculty of Sciences
and Techniques of Mohammedia, BP 146, Mohammedia, Morocco
E-mail address: morchadi@hotmail.com
Saı̈d Sajid
U.F.R Mathematics and Applications, Department of Mathematics, Faculty of Sciences
and Techniques of Mohammedia, BP 146, Mohammedia, Morocco
E-mail address: saidsajid@hotmail.com