PORTUGALIAE MATHEMATICA
Vol. 61 Fasc. 2 – 2004
Nova Série
ON THE EXISTENCE OF MONOTONE SOLUTIONS FOR
SECOND-ORDER NON-CONVEX DIFFERENTIAL INCLUSIONS
IN INFINITE DIMENSIONAL SPACES
A.G. Ibrahim and K.S. Alkulaibi
Abstract: This paper is concerned with the existence of monotone solutions in an
infinite dimensional Hilbert space for a second order differential inclusion and without
the assumption of the convexity.
1 – Introduction
The existence of solutions for either first or second order differential inclustions
or functional differential inclusions has been studied extensively in recent papers.
For instance we refer to [1, 4, 6, 7, 8, 10, 12, 14, 16, 17, 18, 19, 20, 21, 22].
In order to explain our aim let H be an infinite dimensional Hilbert space,
Q = K× Ω be a subset of H×H, F be a set-valued function defined on Q and its
values are not necessary convex subsets of H. Consider the following differential
inclusion:
x00 (t) ∈ F (x(t), x0 (t)),
(∗)
a.e. on [0, T ] ,
(x(t), x0 (t)) = (x◦ , y◦ ) ∈ Q = K × Ω .
By a solution of (∗) we mean an absolutely continuous function x : [0, T ] → K
with absolutely continuous derivative such that (∗) is satisfied. A solution
x : [0, T ] → K of (∗) is called monotone if there is a set-valued function P defined from K to the family of nonempty subsets of K such that (i) x ∈ P (x) for
all x ∈ K, (ii) if y ∈ P (x) then P (y) ⊆ P (x) and (iii) if t ≤ s, t, s ∈ [0, T ]
Received : February 24, 2003; Revised : March 30, 2003.
232
A.G. IBRAHIM and K.S. ALKULAIBI
then x(s) ∈ P (x(t)). In the particular case H = R (the set of real numbers) and
P (x) = [x, ∞) ∩ K the monotoncity of the solution becomes, according to this
definition, as follows if t ≤ s then x(t) ≤ x(s).
The purpose of this paper is to obtain conditions on the data that guarantee
the existence of monotone solution for (∗).
We refer to in a recent paper V. Lupulescu [20] proved the existence of a
solution wich is not necessary monotone and in the case when dimension H is
finite. Also, in the above mentioned papers the monotoncity of the obtained
solutions were not researched.
The paper will organize as follows: in section 2 we will recall briefly some
basic definitions and preliminary facts which will be used throught the sequel.
In section 3 we will establish the main result.
2 – Notations and preliminaries
In this section we give the notations and known facts that we will use throught
the paper.
• H is an infinite dimensional seperable real Hilbert space.
• If x ∈ H and δ > 0, B(x, δ) = {y ∈ H : ky − xk < δ} is the ball centered
at x with radius δ and B(x, δ) its closure.
• If A is a subset of H and x ∈ H, d(x, A) = inf{ky − xk : y ∈ A} is the
distance from x to A.
• If A is a subset of H then |A| = sup{kak : a ∈ A} is the excess of A over
{0} and co A is the convex hull of A.
• A function u : [0, T ] → H is called Lebesgue–Bochner integrable if
t → ku(t)k is Lebesgue integrable and u is strongly measurable, i.e. the
a.e. limit of a sequence of step functions. The Banach space of equivalence class of such u will be denoted by L1 ([0, T ], H). It’s known that if
R
0
w ∈ L1 ([0, T ], H) then ( 0t w(s) ds) = w(t), a.e.
• L2 ([0, T ], H) is the Banach space of all strongly measurable functions
R
u : [0, T ] → H such that 0T ku(t)k2 dt < ∞.
• A function u : [0, T ] → H is absolutely continuous if there is a function
R
v ∈ L1 ([0, T ], H) such that u(t) = u(0) + 0t v(s) ds, for all t ∈ [0, T ].
233
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
• If X and Y are two topological spaces, a set-valued function G : X → Y
is called upper semicontinuous (lower semicontinuous) at x◦ ∈ X if for any
open set U of Y containing G(x◦ ) (G(x◦ ) ∩ U 6= ∅), the set {x ∈ X :
G(x) ⊆ U } ({x ∈ X : G(x) ∩ U 6= ∅}) is a neighbourhood of x◦ . G is upper
semicontinuous (lower semicontinuous) if it’s upper semicontinuous (lower
semicontinuous) at each point in X.
• If E is a topological vector space, f : E → R and x◦ ∈ E, then x0 ∈ E 0 ,
the topological dual of E, is said to be a subgradient of f at x◦ if for every
x ∈ E,
f (x) − f (x◦ ) ≥ hx0 , x − x◦ i .
The set of all subgradients of f at x◦ is called subdifferential and is denoted
by ∂f (x◦ ). It’s known that if E is a Hausdorff locally convex space, then
∂f (x◦ ) is closed and convex. (See for instance [13]).
• If K is a subset of H and x ∈ K, then the Bouligand’s contingent cone of
K at x is defined by:
TK (x) =
(
d(x + hy, K)
y ∈ H : lim inf
=0
+
h
h→0
)
.
It’s known that if x is an interior point in K, then TK (x) = H and if K is
closed and convex then TK (x) = {λ(z − x) : λ ≥ 0, z ∈ K}. (See [3]).
• If K is a subset of H, x ∈ K and y ∈ H then the second order contingent
cone of K at (x, y) is defined by (see [9]):
(2)
TK (x, y)
=
(
z ∈ H : lim inf
d(x + hy +
h→0+
h2
2
h2
2 z,
K)
=0
)
.
(2)
We remark that if TK (x, y) 6= ∅ then y ∈ TK (x).
• If B is a bounded set of a normed space E, then the Kuratowski’s measure of
S
noncompactness of B, α(B), is defined by α(B) = inf{d > 0 : B = m
i=1 Bi
for some m and Bi with diameter less than or equal to d}. In the following
lemma we recall some useful properties for the measure of noncompactness
α. For instance see Prop. 9.1 [15].
Lemma 2.1. Let X be an infinite dimensional real Banach space and D1 ,
D2 be two bounded subsets of X.
(i) α(D1 ) = 0 ⇐⇒ D1 is relatively compact .
234
A.G. IBRAHIM and K.S. ALKULAIBI
(ii) α(λD1 ) = |λ| α(D1 ); λ ∈ R .
(iii) D1 ⊆ D2 =⇒ α(D1 ) ≤ α(D2 ) .
(iv) α(D1 + D2 ) ≤ α(D1 ) + α(D2 ) .
(v) If x◦ ∈ X and r is a positive real number then α(B(x◦ , r)) = 2 r .
For other properties of α we refer to ([5] and [15]), and for more details about
set-valued function we refer to ([2], [3], [13], [15], [19]).
3 – Main result
In this section we give the main result. First we start by the following Lemma
which plays an important role in the sequel. The proof will be based on the same
technique that was used in Lemma 3.1 in [20].
Lemma 3.1. Let K, Ω be two nonempty subsets of H, P be a lower semicontinuous set-valued function from K to the non-empty subsets of K and F be
a set-valued function defined on Q = K × Ω with non-empty subsets of H.
Assume that:
(i) For all x ∈ K, x ∈ P (x) ;
(2)
(ii) For all (x, y) ∈ Q, F (x, y) ∩ TP (x) (x, y) 6= ∅ .
If Q◦ is a compact subset of Q and k is a positive integer, then there is
ηk > 0 such that for all (x◦ , y◦ ) ∈ Q◦ there exist h◦,k ∈ [ηk , k1 ], u◦,k , v◦,k ∈ H and
(xj◦ , yj◦ ) ∈ Q◦ such that:
1. z◦ = x◦ + h◦,k y◦ + 12 h2◦,k u◦,k ∈ P (x◦ ) ;
2. v◦,k ∈ F (xj◦ , yj◦ ) ;
3. d((x◦ , y◦ ), (xj◦ , yj◦ )) <
4. ku◦,k − v◦,k k <
1
k
1
k
;
.
Proof: Let (x, y) be a fixed element in Q = K × Ω. By (ii) there is v =
v(x, y) ∈ F (x, y) such that:
lim inf
h→0+
³
d x + hy +
h2
2 v,
h2
2
P (x)
´
= 0.
235
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
Hence there is hk = hk (x, y) ∈ (0, k1 ] such that
µ
(1)
d x + hk y +
h2
h2k
v, P (x) < k .
2
4k
¶
Since P is lower semicontinuous, Cor. 1.2.1 [3] yields that the function (a, b) →
d(b, P (a)) is upper semicontinuous. Consequently, the function (a, b) →
d(a + hk b + 21 h2k v, P (a)) is upper semicontinuous from H×H to R. Thus the
subset
¾
¶
½
µ
h2
1
N (x, y) = (a, b) : d a + hk b + h2k v, P (a) < k
2
4k
is open. By (1), (x, y) ∈ N (x, y). Then there exists r = r(x, y) ∈ (0, k1 ] such that
B((x, y), r) ⊂ N (x, y).
Now {B((x, y), r) : (x, y) ∈ Q◦ } is an open cover for Q◦ . Since Q◦ is compact,
there exists a finite set {(xi , yi ) ∈ Q◦ : 1 ≤ i ≤ m} such that:
Q◦ ⊆
m
[
B((xi , yi ), ri ) .
i=1
Put ηk = min{hk (xi , yi ) : 1 ≤ i ≤ m}. Since (x◦ , y◦ ) ∈ Q◦ there is j◦ ∈ {1, 2, ..., m}
such that:
(x◦ , y◦ ) ∈ B((xj◦ , yj◦ ), rj◦ ) ⊆ N (xj◦ , yj◦ ) ,
(xj◦ , yj◦ ) ∈ Q◦ .
Denote by h◦,k = hk (xj◦ , yj◦ ), v◦,k = v(xj◦ , yj◦ ) ∈ F (xj◦ , yj◦ ). From the definition
of the distance we can find z◦ ∈ P (x◦ ) such that:
³
h2
´
¶
d x◦ + h◦,k y◦ + ◦,k
h2◦,k
1
1
2 v◦,k , P (x◦ )
v
,
z
d
x
+
h
y
+
≤
+
◦
◦
◦
◦,k
◦,k
2
2
h◦,k
h◦,k
2
2k
2
µ
h2◦,k /4k
1
< 2
+
2k
h◦,k /2
1
1
+
=
2k 2k
1
=
,
k
hence:
°
°
°
°z − x − h y
1
◦
°
° ◦
◦,k ◦
−
v
.
°<
°
◦,k
2
h◦,k
°
°
k
2
2
236
A.G. IBRAHIM and K.S. ALKULAIBI
Let
u◦,k =
z◦ − x◦ − h◦,k y◦
h2◦,k
2
Then
ku◦,k − v◦,k k <
x◦ + h◦,k y◦ +
.
1
,
k
h2◦,k
u◦,k = z◦ ∈ P (x◦ ) ,
2
v◦,k ∈ F (xj◦ , yj◦ ) ,
and
³
´
d (x◦ , y◦ ), (xj◦ , yj◦ ) <
1
.
k
Theorem 3.2. Let K be a subset of H, Ω be an open subset of H such that
Q = K× Ω be a locally compact subset of H× H, F be an upper semicontinuous
set-valued function from Q to the family of non-empty compact subsets of H,
and P be a lower semicontinuous set-valued function from K to the family of
non-empty subsets of K with closed graph.
Assume the following conditions:
(H1) (i) For all x ∈ K, x ∈ P (x) ,
(ii) For all x ∈ K and all y ∈ P (x) we have P (y) ⊆ P (x) .
(2)
(H2) For all (x, y) ∈ Q, F (x, y) ∩ TP (x) (x, y) 6= ∅ .
(H3) There exist a proper convex and lower semicontinuous function
V : H → R such that:
F (x, y) ⊆ ∂V (y) ,
∀(x, y) ∈ Q ,
where ∂V (y) is the subdifferential of V .
Then for all (x◦ , y◦ ) ∈ Q there exists T > 0 and an absolutely continuous
function x : [0, T ] → H with absolutely continuous derivative such that:
x00 (t) ∈ F (x(t), x0 (t))
x(s) ∈ P (x(t))
a.e. on [0, T ] ,
for all t ∈ [0, T ] and all s ∈ [t, T ] ,
x(0) = x◦ ,
x0 (0) = y◦ .
237
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
Proof: Let (x◦ , y◦ ) ∈ Q. Since Q is locally compact, each component K and
Ω is locally compact. Because x◦ ∈ K we can find δ1 > 0 such that B(x◦ , δ1 ) ∩ K
is compact in H. Also, since y◦ ∈ Ω and Ω is open we can find δ2 > 0 such that
B(y◦ , δ2 ) ⊆ Ω is compact in H. Let δ = min(δ1 , δ2 ) and put Q◦ = (B(x◦ , δ)∩K)×
(B(y◦ , δ)). So, Q◦ is a compact subset of Q. Since F is upper semicontinuous,
F (Q◦ ) is compact subset of H. Then we can find M > 0 such that:
n
sup kvk : v ∈ F (Q◦ )
Put


δ
,
T = min
 2(M + 1)
(1)
s
o
≤ M .


δ
δ
,
.
M + 1 2(ky◦ k + 1) 
Let k be a fixed positive integer. We are going to show that there are a positive
real number ηk and a positive integer m(k) such that for each r ∈ {0, 1, ..., m(k)−1}
there exist hr,k ∈ [ηk , k1 ], (xr,k , yr,k ) ∈ Q◦ , ur,k , vr,k ∈ H and (xjr , yjr ) ∈ Q◦ with
the following properties:
(i)
Pm(k)−1
r=0
hr,k ≤ T <
Pm(k)
r=0
hr,k .
(ii) x◦,k = x◦ , y◦,k = y◦ .
(iii) For all r = 0, 1, 2, ..., m(k)−2 we have
1
xr+1,k = xr,k + hr,k yr,k + (hr,k )2 ur,k ∈ P (xr,k )
2
and
(2)
yr+1,k = yr,k + hr,k ur,k .
(iv) For all r = 0, 1, 2, ..., m(k)−1 we have
vr,k ∈ F (xjr , yjr ) ,
and
³
´
d (xr,k , yr,k ), (xjr , yjr ) <
kur,k − vr,k k <
1
k
1
.
k
By Lemma 3.1 there exist ηk > 0, h◦,x ∈ [ηk , k1 ], u◦,k , v◦,k ∈ H and (xj◦ , yj◦ ) ∈ Q◦ ,
such that:
1
x◦ + h◦,k y◦ + (h◦,k )2 u◦,k ∈ P (x◦ ) ⊂ K ,
2
v◦,k ∈ F (xj◦ , yj◦ ) ,
(3)
³
´
d (x◦ , y◦ ), (xj◦ , yj◦ ) <
1
,
k
ku◦,k − v◦,k k <
1
.
k
238
A.G. IBRAHIM and K.S. ALKULAIBI
Define
x1,k = x◦ + h◦,k y◦ +
and
1
(h◦,k )2 u◦,k
2
y1,k = y◦ + h◦,k u◦,k .
Then x1,k ∈ P (x◦ ) and if h◦,k < T we have
1
(h◦,k )2 ku◦,k k
2
³
1´
1
< h◦,k ky◦ k + (h◦,k )2 kv◦,k k +
2
k
1 2
< T ky◦ k + T (M + 1)
2
δ δ
= δ,
< +
2 2
kx1,k − x◦ k ≤ h◦,k ky◦ k +
and
ky1,k − y◦ k ≤ h◦,k ku◦,k k
1´
k
< T (M + 1) < δ .
³
< h◦,k M +
Therefore (x1,k , y1,k ) ∈ Q◦ . Again by Lemma 3.1 there exist h1,k ∈ [ηk , k1 ],
u1,k , v1,k ∈ H and (xj1 , yj1 ) ∈ Q◦ such that
x1,k + h1,k y1,k +
1
(h1,k )2 u1,k ∈ P (x1,k ) ⊆ K ,
2
v1,k ∈ F (xj1 , yj1 ) ,
(4)
³
´
d (x1,k , y1,k ), (xj1 , yj1 ) <
1
,
k
ku1,k − v1,k k <
1
.
k
If h◦,k + h1,k ≥ T we set m(k) = 1, otherwise we define
x2,k = x1,k + h1,k y1,k +
and
1 2
h u1,k
2 1,k
y2,k = y1,k + h1,k u1,k .
By (3) and (4) we obtain x2,k ∈ P (x1,k ) and
kx2,k
°
°
°
°
1 2
1 2
°
− x◦ k = °x◦ + h◦,k y◦ + h◦,k u◦,k + h1,k (y◦ + h◦,k u◦,k ) + h1,k u1,k − x◦ °
°
2
2
≤ (h◦,k + h1,k ) ky◦ k +
1
1 2
h ku◦,k k + h1,k h◦,k ku◦,k k + h21,k ku1,k k <
2 ◦,k
2
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
239
1
1 2
h◦,k (M + 1) + h1,k h◦,k (M + 1) + h21,k (M + 1)
2
2
1
2
= (h◦,k + h1,k ) ky◦ k + (M + 1) (h◦,k + h1,k )
2
1
δ δ
< T ky◦ k + (M + 1) T 2 <
+
= δ.
2
2 2
< (h◦,k + h1,k ) ky◦ k +
Also,
ky2,k − y◦ k ≤ h◦,k ku◦,k k + h1,k ku1,k k
< h◦,k (M + 1) + h1,k (M + 1)
= (h◦,k + h1,k ) (M + 1) < T (M + 1) < δ .
Thus (x2,k , y2,k ) ∈ Q◦ . Invoking to Lemma 3.1, there exist h2,k ∈ [ηk , k1 ],
u2,k , v2,k ∈ H and (xj2 , yj2 ) ∈ Q◦ such that
z2,k = x2,k + h2,k y2,k +
1 2
h u2,k ∈ P (x2,k ) ,
2 2,k
v2,k ∈ F (xj2 , yj2 ) ,
(5)
³
´
d (x2,k , y2,k ), (xj2 , yj2 ) <
1
,
k
ku2,k − v2,k k <
1
.
k
We reiterate this process. Since h◦,k , h1,k , h2,k , ... are in [ηk , k1 ] we are sure that
there exists a positive integer m(k) such that for each r ∈ {0, 1, ..., m(k) − 1}
there exist hr,k ∈ [ηk , k1 ], (xr,k , yr,k ) ∈ Q◦ , ur,k , vr,k ∈ H and (xjr , yjr ) ∈ Q◦ with
properties in (2).
Now let us set t◦k = 0 and trk = h◦,k + h1,k + · · · + hr−1,k ; r ∈ {1, 2, ..., m(k)}.
We remark that for all r ∈ {1, 2, ..., m(k)} we have
trk − tkr−1 <
1
k
and
m(k)−1
tk
m(k)
≤ T < tk
.
We define a function xk : [0, T ] → H as follows. If t ∈ [tkr−1 , trk ], r ∈ {1, 2, ..., m(k)}
we put
1
2
xk (t) = xr−1,k + (t − tkr−1 ) yr−1,k + (t − tkr−1 ) ur−1,k .
2
Then we get
(6)
xk (tkr−1 ) = xr−1,k ∈ P (xr−2,k ) ,
(7)
xk 0 (t) = yr−1,k + (t − tkr−1 ) ur−1,k ,
(8)
xk 00 (t) = ur−1,k ,
r ∈ {1, 2, ..., m(k)} ,
∀ t ∈ [tkr−1 , trk ], r ∈ {1, 2, ..., m(k)} ,
∀ t ∈ [tkr−1 , trk ], r ∈ {1, 2, . . . , m(k)} .
240
A.G. IBRAHIM and K.S. ALKULAIBI
Hence by (2), for all t ∈ [0, T ] we obtain
1
1 1
kxk (t)k ≤ kxr−1,k k + kyr−1,k k +
kur−1,k k
k
2 k2
≤ kxr−1,k − x◦ k + kx◦ k + kyr−1,k − y◦ k
(9)
+ ky◦ k + kur−1,k − vr−1,k k + kvr−1,k k
< 2 δ + kx◦ k + ky◦ k +
1
+M
k
≤ 2 δ + kx◦ k + ky◦ k + 1 + M
and
1
kur−1,k k
k
´
1³
≤ ky◦ k + δ +
kur−1,k − vr−1,k k + kvr−1,k k
k
´
1³1
< ky◦ k + δ +
+M
k k
≤ ky◦ k + δ + M + 1
kxk 0 (t)k ≤ kyr−1,k k +
(10)
and
kxk 00 (t)k = kur−1,k k ≤ kur−1,k − vr−1,k k + kvr−1,k k
(11)
1
+M
k
≤ M +1 .
<
Moreover, let t be a fixed point in [0, T ]. Then there is r ∈ {1, 2, ..., m(k)} such
that t ∈ [tkr−1 , trk ]. We have
1
kxk (t) − xjr−1 k ≤ kxr−1,k − xjr−1 k + kyr−1,k k
k
´
1 1 ³
ku
−
v
k
+
kv
k
+
r−1,k
r−1,k
r−1,k
2 k2
´ 1 1 ³1
´
1 1³
< +
δ + ky◦ k +
+
M
k k
2 k2 k
³
´
1
2 + δ + ky◦ k + M ,
≤
k
´
1³
kxk 0 (t) − yjr−1 k ≤ kyr−1,k − yjr−1 k +
kur−1,k − vr−1,k k + kvr−1,k k
k
´
1 1 ³1
< +
+M
k k k
1
≤ (M + 2)
k
241
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
and
1
.
k
kxk 00 (t) − vr−1,k k = kur−1,k − vr−1,k k <
Since vr−1,k ∈ F (xjr−1 , yjr−1 ), then we obtain
(12)
³
³
´
´
xk (t), xk 0 (t), xk 00 (t) ∈ graph F + εk B(0, 1) × B(0, 1) × B(0, 1) ,
where εk → 0 as k → ∞. Since t is arbitrary point in [0, T ], the relation (12) is
true for all t ∈ [0, T ].
By (10) and (11) the sequences (xk ) and (xk 0 ) are equicontinuous. In order to
apply Ascoli–Arzela theorem we are going to show that for every t ∈ [0, T ] the two
sets Z1 (t) = {xk (t) : k ≥ 1} and Z2 (t) = {xk 0 (t) : k ≥ 1} are relatively compact
in H. So, for every k ≥ 1 let θk : [0, T ] → [0, T ] defined by θk (0) = 0, θk (t) = trk ,
t ∈ ]tkr−1 , trk ]. Also let Q◦,1 = {x : (x, y) ∈ Q◦ for some y}, Q◦,2 = {y : (x, y) ∈
Q◦ for some x}. Hence, each of Q◦,1 and Q◦,2 is compact in H. From the definition of (xk ) and (xk 0 ) we have for all k ≥ 1 and all t ∈ [0, T ], xk (θk (t)) ∈ Q◦,1 ,
xk 0 (θk (t)) ∈ Q◦,2 . Thus for all t ∈ [0, T ] the two sets {xk (θk (t)) : k ≥ 1} and
{xk 0 (θk (t)) : k ≥ 1} are relatively compact in H. Now, for all t ∈ [0, T ]
n
α(Z1 (t)) = α xk (t) : k ≥ 1
n
o
o
= α xk (t) − xk (θk (t)) + xk (θk (t)) : k ≥ 1 .
From (iii) and (iv) of Lemma 2.1 we get
n
α(Z1 (t)) ≤ α xk (t) − xk (θk (t)) : k ≥ 1
o
n
o
+ α xk (θk (t)) : k ≥ 1 .
Since the set {xk (θk (t)) : k ≥ 1} is relatively compact, α{xk (θk (t)) : k ≥ 1} = 0
(Lemma 2.1 (i)). Then
n
α(Z1 (t)) ≤ α xk (t) − xk (θk (t)) : k ≥ 1
= α
½Z
θk (t)
t
0
xk (s) ds : k ≥ 1
¾
o
.
By relation (10) we obtain
à µ
¶!
´
1³
α(Z1 (t)) ≤ α B 0,
ky◦ k + δ + M + 1
k
³
´
2
=
ky◦ k + δ + M + 1 .
(By Lemma 2.1 (v))
k
242
A.G. IBRAHIM and K.S. ALKULAIBI
Since k1 → 0 as k → ∞, α(Z1 (t)) = 0. Hence Z1 (t) is relatively compact. Similarly the set Z2 (t) is relatively compact. By a corollary of Ascoli–Arzela theorem
(see Th. 0.3.4 [3]) the sequence (xk ), k ≥ 1 has a subsequence (again denoted
by (xk )) and absolutely continuous function x : [0, T ] → H with absolutely
continuous derivative x0 such that (xk ) converges uniformly to x on [0, T ],
(xk 0 ) converges uniformly to x0 on [0,T ] and (xk 00 ) converges weakly in L2 ([0,T ], H)
to x00 . Invoking to the convergence theorem (see Th. 1.4.1 [3]) we get that
x00 (t) ∈ co F (x(t), x0 (t))
(13)
a.e. on [0, T ] .
Note that here the values of F are not necessary convex. Now we use condition
(H3) to show that
x00 (t) ∈ F (x(t), x0 (t))
a.e. on [0, T ] .
Since V is proper convex lower semicontinuous then by Lemma 3.3 in [11],
we have
d
2
V (x0 (t)) = kx00 (t)k
a.e. on [0, T ] .
dt
Then
0
0
V (x (T )) − V (x (0)) =
(14)
Z
T
2
kx00 (t)k dt .
0
From (2) for every integer k ≥ 1 and every r ∈ {1, 2, ..., m(k)} there exist
αr−1,k , βr−1,k , γr−1,k ∈ B(0, k1 ) such that
(15)
³
ur−1,k − γr−1,k = vr−1,k ∈ F xr−1,k − αr−1,k , yr−1,k − βr−1,k
⊆ ∂V (yr−1,k − βr−1,k ) .
´
From the definition of the subdifferential ∂V , the last relation gets us
V (yr,k − βr,k ) − V (yr−1,k − βr−1,k ) ≥
≥
=
=
=
D
D
D
ur−1,k − γr−1,k , yr,k − βr,k − (yr−1,k − βr−1,k )
ur−1,k − γr−1,k , yr,k − yr−1,k + βr−1,k − βr,k
E
E
ur−1,k − γr−1,k , xk 0 (trk ) − xk 0 (tkr−1 ) + βr−1,k − βr,k
¿
ur−1,k − γr−1,k ,
Z
trk
00
À
D
E
xk (t) dt + ur−1,k − γr−1,k , βr−1,k − βr,k
r−1
tk
E
=
243
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
D
=
¿
E
D
+ ur−1,k − γr−1,k , βr−1,k − βr,k
(trk
=
Z
ur−1,k , ur−1,k (trk − tkr−1 ) − γr−1,k ,
−
tkr−1 ) kur−1,k k2
¿
− γr−1,k ,
D
E
Z
trk
tr−1
k
trk
x00 (t) dt
r−1
tk
00
x (t) dt
À
À
E
+ ur−1,k − γr−1,k , βr−1,k − βr,k .
Since
D
ur−1,k , ur−1,k (trk − tkr−1 )
E
= (trk − tkr−1 ) hur−1,k , ur−1,k i
= (trk − tkr−1 ) kur−1,k k2 ,
thus, for all positive integer number k and all r ∈ {1, 2, ..., m(k)−1} we have
³
´
³
V x0 (trk ) − βr,k − V x0 (tkr−1 ) − βr−1,k
≥
(16)
Z
trk
´
¿
2
00
tr−1
k
≥
kx (t)k dt −
D
Z
γr−1,k ,
trk
00
tkr−1
x (t) dt
À
E
+ ur−1,k − γr−1,k , βr−1,k − βr,k .
m(k)−1
Also, from (2)–(i) we have tk
one has,
³
´
³
V xk 0 (T ) − V ym(k)−1,k − βm(k)−1,k
≥
=
=
D
D
m(k)
≤ T < tk
´
, then from (15) when r = m(k)
≥
um(k)−1,k − γm(k)−1,k , xk 0 (T ) − ym(k)−1,k + βm(k)−1,k
m(k)−1
um(k)−1,k − γm(k)−1,k , xk 0 (T ) − xk 0 (tk
¿
Z
um(k)−1,k ,
T
00
m(k)−1
tk
xk (t) dt
À
−
E
) + βm(k)−1,k
¿
Z
γm(k)−1,k ,
T
00
m(k)−1
tk
E
xk (t) dt
E
D
+ um(k)−1,k − γm(k)−1,k , βm(k)−1,k .
Then
³
´
³
m(k)−1
V xk 0 (T ) − V xk 0 (tk
(17)
≥
Z
T
m(k)−1
tk
D
) − βm(k)−1,k
00
2
kx (t)k dt −
´
≥
¿
γm(k)−1,k ,
Z
T
m(k)−1
tk
E
+ um(k)−1,k − γm(k)−1,k , βm(k)−1,k .
00
xk (t) dt
À
À
244
A.G. IBRAHIM and K.S. ALKULAIBI
By adding the m(k)−1 inequalities from (16) and inequality from (17) we get
³
´
³
V xk 0 (T ) − V y◦ − β◦,k
³
´
´
=
³
m(k)−1
= V xk 0 (T ) − V x0 (tk
³
m(k)−1
+ V xk 0 (tk
) − βm(k)−1,k
´
m(k)−2
) − βm(k)−1,k − V xk 0 (tk
+ ··· +
(18)
³
´
³
) − βm(k)−2,k
´
+ V xk 0 (tk 1 ) − β1,k − V (y◦ − β◦,k )
≥
where
Z
T
0
2
kxk 00 (t)k dt + ρ(k) ,
m(k)−1¿
ρ(k) = −
+
Z
trk
xk 00 (t) dt
r−1
À
X
γr−1,k ,
X
ur−1,k − γr−1,k , βr−1,k − βr,k
r=1
m(k)−1D
tk
r=1
−
¿
γm(k)−1 ,
Z
T
00
m(k)−1
tk
xk (t) dt
E
À
E
D
+ um(k)−1,k − γm(k)−1,k , βm(k)−1,k .
We have
m(k)−1
|ρ(k)| ≤
X
kγr−1,k k
r=1
Z
trk
tkr−1
kxk 00 (t)k dt
m(k)−1
+
X
kur−1,k − γr−1,k k kβr−1,k − βr,k k
r=1
+ kγm(k)−1 k
Z
T
m(k)−1
tk
kxk 00 (t)k dt
+ kum(k)−1,k − γm(k)−1,k k kβm(k)−1,k k
m(k)−1
≤
X
r=1
+
m(k)−1
X
1
2
kur−1,k k (trk − tkr−1 ) +
kvr−1,k k
k
k
r=1
1
1
m(k)−1
kum(k)−1,k k (T − tk
) + kvr−1,k k
k
k
m(k)−1
≤
X M +1
r=1
k2
m(k)−1
+
X 2M
r=1
k
+
M +1 M
+
,
k2
k
´
245
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
and this yields lim ρ(k) = 0. By (18) we obtain
k→∞
0
lim V (xk (T )) − V (y◦ ) ≥ lim sup
k→∞
k→∞
Z
T
0
2
kxk 00 (t)k dt .
Therefore, by (14) we have
(19)
Z
T
0
2
0
0
kx (t)k dt = V (x (T )) − V (x (0)) ≥ lim sup
k→∞
0
Z
T
0
2
kxk 00 (t)k dt .
Since (xk 00 ) converges weakly to x00 in L2 ([0, T ], H) then relation (19) implies
that (xk 00 ) converges strongly to x00 in L2 ([0, T ], H). Consequently, there is a
subsequence of xk 00 , denoted again by (xk 00 ) converges to x00 almost everywhere
on [0, T ]. From (12) we obtain,
(20)
lim d
k→∞
µ³
0
00
´
xk (t), xk (t), xk (t) , graph F
¶
= 0.
By the assumptions on F and by Prop. 1.1.2 [3] the graph of F is closed. Then
relation (20) yields that
x00 (t) ∈ F (x(t), x0 (t))
a.e. on [0, T ] .
It remains to prove that x(t) ∈ P (x(t)), for all t ∈ [0, T ] and if s > t then
x(s) ∈ P (x(t)).
In order to do this for all k ≥ 1 let δk : [0, T ] → [0, T ] be a function defined
by:
δk (0) = 0 , δk (t) = tkr−1
for all t ∈ (tkr−1 , trk ] and all r ∈ {1, 2, ..., m(k)}. Since
trk − tkr−1 <
1
k
we get
lim xk (θk (t)) = lim xk (δk (t)) = x(t)
k→∞
k→∞
for all t ∈ [0, T ]. Let t ∈ [0, T ] be fixed. For every positive integer k there is
r ∈ {1, 2, ..., m(k)} such that t ∈ (tkr−1 , trk ]. We have
xk (θk (t)) = xk (trk ) ∈ P (xk (tkr−1 )) . = P (xk (δk (t)))
Since the graph of P is closed, we conclude that
x(t) ∈ P (x(t)) .
Now let t, s ∈ [0, T ] be such that s > t. Then for k large enough we can find
q
r, q ∈ {1, 2, ..., m(k) − 1} such that r > q, s ∈ [tkr−1 , trk ] and t ∈ [tq−1
k , tk ].
246
A.G. IBRAHIM and K.S. ALKULAIBI
Assume that r = q + j. Clearly 1 ≤ j < m(k). We have
xk (tkr−1 ) ∈ P (xk (tkr−2 )) .
Since P is transitive,
P (xk (tkr−1 )) ⊂ P (xk (tkr−2 )) .
Similarly,
P (xk (tkr−2 )) ⊂ P (xk (tkr−3 )) .
We continue for j steps, hence we get
P (xk (tkr−1 )) ⊂ P (xk (tqk )) .
But
xk (trk ) ∈ P (xk (tkr−1 )) .
We obtain
xk (trk ) ∈ P (xk (tqk )) .
This means that
xk (θk (s)) ∈ P (xk (θk (t))) .
Since lim θk (t) = t, lim θk (s) = s and the graph of P is closed, we get
k→∞
k→∞
x(s) ∈ P (x(t)) .
If we consider the particular case when P (x) = K for all x ∈ K we obtain the
following Viability Theorem.
Theorem 3.3. Let K be a closed subset of H, Ω be an open subset of H
such that Q = K × Ω be a locally compact subset of H× H, F be an upper
semicontinuous set-valued function from Q to the family of nonempty compact
subsets of H.
Assume that condition (H3) of Theorem 3.2 and the following condition are
satisfied:
(2)
(H4) For all (x, y) ∈ Q, F (x, y) ∩ TK (x, y) 6= ∅ .
Then for all (x◦ , y◦ ) ∈ Q there exists T > 0 and an absolutely continuous
function x : [0, T ] → H with absolutely continuous derivative such that:
x00 (t) ∈ F (x(t), x0 (t))
(x(t), x0 (t)) ∈ Q ,
x(0) = 0 ,
a.e. on [0, T ] ,
∀ t ∈ [0, T ] ,
x0 (0) = y◦ .
ON THE EXISTENCE OF MONOTONE SOLUTIONS...
247
Remark. If we suppose that the dimension of H is finite, in Theorem 3.3,
we obtain Theorem 2.1 of [20].
ACKNOWLEDGEMENT – The authors would like to thank the referee for his useful
remarks.
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A.G. Ibrahim,
Department of Mathematics, Faculty of Science, King Faisal University,
P.O. Box 1759, Hofuf, Al-Ahsa 31982 – SAUDI ARABIA
E-mail: agamal@kfu.edu.sa
and
K.S. Alkulaibi,
Department of Mathematics, Faculty of Science, King Faisal University,
P.O. Box 10714, Hofuf, Al-Ahsa 31982 – SAUDI ARABIA
E-mail: kholod salih@yahoo.com