EXISTENCE OF SOLUTIONS FOR A CLASS OF DIFFERENTIAL INCLUSIONS GOVERNED
BY A SWEEPING PROCESS
AURELIAN CERNEA
We prove the existence of local solutions for a class of evolution differential inclu- sions defined by a sweeping process and by an upper semicontinuous set-valued map contained in the Clarke subdifferential of an uniformly regular function.
AMS 2000 Subject Classification: 34A60.
Key words: sweeping process, normal cone, uniform regular function, Clarke sub- differential.
1. INTRODUCTION
The existence of local solutions to evolution equations governed by dif- ferential inclusions and by sweeping processes has been the subject of many papers in the last two decades. Convex sweeping process were introduced by Moreau ([9]). We refer to [5] and [8] for a complete bibliography on this topic.
In this note we establish an existence result for nonconvex perturbations of the sweeping process associated with a closed convex locally compact setC of a Hilbert space, namely,
(1.1) x(t)∈ −NC(x(t)) +F(x(t)) +g(t) a.e.([0, T]), x(0) =x0, where NC(x(t)) denotes the Clarke normal cone to C at x(t), F(·) is a set- valued map that is upper semicontinuous onH, with nonempty compact values satisfying F(x) ⊂ ∂CV(x), ∀x ∈ H, where ∂CV(·) is Clarke’s subdifferen- tial (generalized gradient) of a uniformly regular function V(·) and g(·) is a bounded measurable function onH.
Our result is an improvement of a previous result of Syam [10], where the set-valued map F(·) is assumed to satisfy F(x) ⊂ ∂V(x), ∀x ∈ H, with V(·) a convex function and ∂V(·) the subdifferential (in the sense of Convex Analysis). Note that an alternative improvement of the result in [10] is pro- vided in [7]. Namely, the convexity of the function V(·) in [10] is relaxed in [7] in the sense thatV(·) is assumed to be aφ-convex of order two function.
The proof of our main result follows the general ideas in [7] and [10].
MATH. REPORTS9(59),4 (2007), 335–341
The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel while in Section 3 we prove our main result.
2. PRELIMINARIES
Let H be a real Hilbert space with norm · and scalar product·,·.
Denote byB the closed unit ball inH. For x∈H and a closed subsetA⊂X, denote by d(x, A) the distance from x to A given by d(x, A) := inf{y−x; y ∈ A}. By co(A) we denote the convex hull of A and by co(A) the closed convex hull ofA
IfK ⊂H is a closed set andx ∈K,Clarke’s tangent cone to K at x is defined by
CK(x) =
v∈H; lim
s→0+, x→Kx
d(x+sv, K)
s = 0
,
where→K denotes the convergence inK. The negative polar of Clarke’s tan- gent cone NK(x) := CK(x)− is also called the normal cone to the set K at x∈K.
Consider a locally Lipschitz continuous function V : H → R. For ev- ery direction v ∈ H, its Clarke directional derivative at x in direction v is defined by
DCV(x;v) = lim sup
y→x, t→0+
V(y+tv)−V(y)
t .
TheClarke subdifferential (generalized gradient) of V at x is defined by
∂CV(x) ={q ∈H, q, v ≤DCV(x;v)∀v∈H}. We recall that theproximal subdifferential ofV(·) is defined by
∂PV(x) ={q ∈H, ∃δ, σ >0 such that V(y)−V(x) +σy−x2≥ q, y−x ∀y∈B(x, δ)}.
Definition 2.1 ([2]). Let V : H → R∪ {∞} be a lower semicontinuous function and letO ⊂dom(V) be a nonempty open subset. We say that V is uniformly regularover O if there exists a positive numberβ >0 such that
ξ, y−x ≤V(y)−V(x) +βy−x2 ∀y ∈O
for all x ∈ O and all ξ ∈ ∂PV(x). We say that V is uniformly regular over a closed set K if there exists an open set O containing K such that V is uniformly regular over O.
In [2] there are several examples and properties of such maps. For ex- ample, according to [2], any lower semicontinuous proper convex function V is uniformly regular over any nonempty compact subset of its domain with
β= 0; any lower-C2 functionV is uniformly regular over any nonempty con- vex compact subset of its domain.
The next results will be used in the proof of our main result.
Lemma2.2 ([3]). Let V :H →R be a locally Lipschitz function and let
∅ =K⊂dom(V) a closed set. If V is uniformly regular over K then i) ∂CV(x) =∂PV(x) ∀x∈K;
ii) if z(·) : [0, τ] → H is absolutely continuous, g(·) : [0, τ] → H is measurable and g(t)∈∂CV(z(t)) a.e. ([0, τ]), then
(V ◦z)(t) =z(t)2 a.e.([0, τ]).
Proposition 2.3 ([10]). Consider a nonempty convex set C ⊂ H and x0 ∈ C. Then for any g(·) ∈ L1([0, τ], H) there exists a unique absolutely continuous functionxg(·); [0, τ]→H solution to
x(t)∈ −NC(x(t)) +g(t) a.e. ([0, τ]), x(0) =x0.
Moreover,g(t)−xg(t), xg(t)= 0 a.e. ([0, τ]), xg(t) ≤2g(t) a.e. ([0, τ]).
3. THE MAIN RESULT
We are now able to prove the main result of this paper.
Theorem 3.1. Consider a bounded measurable function g(·) : [0,∞)→ H and a convex locally compact setC ⊂H. LetF(·) :H → P(H)be an upper semicontinuous set-valued map with nonempty compact values such that there exists a locally Lipschitz function V(·) : H → R uniformly regular over C, withF(x)⊂∂CV(x), ∀x∈C. Then for anyx0∈C there exist T >0 and an absolutely continuous functionx(·) : [0, T]→H solution to problem(1.1).
Proof. Letr, L >0 be such thatV(·) isL-Lipschitz on x0+rB. By the properties of the Fr´echet subdifferential, ∂CV(x) ⊂ LB ∀x ∈ rB. Without any loss of generality we can assume thatg(t) ≤1 ∀t∈[0,∞).
TakeT >0 such thatT < 2(Lr+1), denoteI = [0, T] and definetin= 2inT, i = 0,1, . . . ,2n. Next, take y0 ∈ F(x0) and define f1n(·) : [0, tn1] → H by f1n(t)≡y0. Obviously, f1n(·)∈L2([0, tn1], H).
Proposition 2.3 ensures the existence of an unique absolutely continuous functionxn1(·) : [0, tn1]→H solution to
x(t)∈ −NC(x(t)) +f1n(t) +g(t) a.e.([0, tn1]), x(0) =x0. Since for any t∈[0, tn1] we have f1n(t) ≤L, we deduce that
(xn1)(t) ≤2f1n(t) +g(t)<2(L+ 1) a.e.([0, tn1]),
thus xn1(t)−x0 ≤ 2−n2(L+ 1)T < r ∀t ∈ [0, tn1], i.e., xn1(t) ∈ x0+rB
∀t∈[0, tn1].
Repeating the same construction for any k∈ {2,3, . . . ,2n}, take ynk−1∈ F(xnk−1(tnk−1)) with xnk−1(tnk−1) ∈ x0+rB and define fkn(·) : (tnk−1, tnk] → H by fkn(t) ≡ ykn−1. By Proposition 2.3, consider the unique solution xnk(·) : [tnk−1, tnk]→H to
x(t)∈ −NC(x(t)) +fkn(t) +g(t) a.e.([tnk−1, tnk]), x(tnk−1) =xnk−1(tnk−1).
One has (xnk)(t) ≤2fkn(t) +g(t) <2(L+ 1) and, therefore, for any t∈[tnk−1, tnk] we have
xnk(t)−xnk(tnk−1) ≤2(L+ 1)(t−tnk−1)≤2−n2(L+ 1)T < r.
Define
θn(t) =tnk−1 ∀t∈[tnk−1, tnk), k= 1,2, . . . ,2n, θn(T) =T, xn(t) = 2
n
k=1
xnk(t)χ[tn
k−1,tnk](t), fn(t) = 2
n
k=1
fkn(t)χ(tn
k−1,tnk](t),
whereχAis the characteristic function of the setA. Then, for anyt∈[tnk−1, tnk], xn(t)−x0≤xn(t)−xn(tnk−1)+xn(tnk−1)−xn(tnk−2)+· · ·+xn(tn1)−x0≤
≤2k(L+ 1)2−nT <2n·2(L+ 1)2−nT = 2(L+ 1)T < r, i.e.,xn(t)∈x0+rB ∀t∈I. Therefore, we have
(3.1) xn(t)∈C∩(x0+rB) ∀t∈I, (3.2) xn(t)∈ −NC(xn(t)) +fn(t) +g(t) a.e.(I), (3.3) fn(t)∈F(xn(θn(t)))⊂∂CV(xn(θn(t)))⊂LB a.e.(I) and, via Proposition 2.3,
(3.4) xn(t) ≤2(g(t)+fn(t))≤2(L+ 1) a.e.(I), (3.5) xn(t), xn(t)=xn(t), g(t) +fn(t) a.e.(I).
We next prove (3.6)
T
0 fn(t), xn(t)dt≤V(xn(T))−V(x0) +βT 2n
T
0 xn(t)2dt, whereβ >0 is the constant appearing in Definition 2.1. Using the properties of the functionV(·), we have
V(xn(tnk))≥V(xn(tnk−1)) +ynk−1, xnk−xnk−1 −βxnk−xnk−12.
So,
ykn−1, tn
k
tnk−1xn(t)dt
≤V(xn(tnk))−V(xn(tnk−1)) +β tn
k
tnk−1xn(t)dt2. We deduce that
tnk
tnk−1fn(t), xn(t)dt≤V(xn(tnk))−V(xn(tnk−1)) +βT 2n
tnk
tnk−1xn(t)2dt, hence (3.6) holds.
From (3.4) and Theorem III. 27 in [4] there follows the existence of a subsequence also denotedxn(·) which converges weaklyL1(I, H) to a function y(·)∈L1(I, H). In particular, lim
n→∞(x0+t
0xn(s)ds) =x0+t
0y(s)ds∀t∈I. On the other hand, from (3.1) and the fact that the setC∩(x0+rB) is convex we deduce thatxn(·) converges uniformly tox(·), wherex(t) :=x0+t
0 y(s)ds
∀t∈I.
The continuity ofV(·), (3.4) and (3.6) yield
(3.7) lim sup
n→∞
T
0 fn(t), xn(t)dt≤V(x(T))−V(x0).
At the same time, by (3.3) and Theorem III. 27 in [4] there exists a subsequence also denoted fn(·) which converges weakly L1(I, H) to a func- tion f(·) ∈ L1(I, H). Thus, since xn(θn(·)) converges uniformly to x(·) and coF(·) is upper semicontinuous with compact convex values, we can apply Theorem 1.4.1 in [1] to find that
(3.8) f(t)∈coF(x(t))⊂∂CV(x(t)) a.e.(I).
Next, we apply Lemma 2.2 in [3] and deduce from (3.8) that
(3.9) d
dt(V ◦x)(t) =x(t), f(t) a.e.(I) which implies
(3.10) V(x(T))−V(x0) =
T
0 x(t), f(t)dt.
By a standard argument (e.g. [10]) by (3.2), the weak convergence of xn(·) to x(·) in L2(I, H), the weak convergence of fn(·) to f(·) in L2(I, H), and the uniform convergence ofxn(·) to x(·) we have
(3.11) x(t)∈ −NC(x(t)) +f(t) +g(t) a.e.(I).
By Proposition 2.3 we have
(3.12) x(t), x(t)=x(t), g(t) +f(t) a.e.(I).
On the other hand, from (3.5), (3.7) and (3.10) we deduce that (3.13) lim sup
n→∞
T
0 xn(t), xn(t)dt= lim
n→∞
T
0 xn(t), g(t)dt+
+ lim sup
n→∞
T
0 xn(t), fn(t)dt≤
T
0 x(t), g(t)dt+
T
0 x(t), f(t)dt.
It follows from (3.12) and (3.13) that lim sup
n→∞
T
0 xn(t), xn(t)dt≤
T
0 x(t), x(t)dt.
By the weak lower semicontinuity of the norm (e.g., Proposition III. 30 in [4]) xn(·) converges to x(·) in the strong topology of L2(I, H). Therefore there exists a subsequence also denoted xn(·) which converges pointwise a.e. in I to x(·).
It remains to prove that
(3.14) x(t)∈ −NC(x(t)) +F(x(t)) +g(t) a.e.(I).
DefineX(t) := cl{xn(θn(t)); n∈N},t∈I. Obviously, X(t)⊂H is compact.
Since F(·) is upper semicontinuous with compact values, F(X(t)) ⊂ H is compact.
Define Y(t) := cl{xn(t)−fn(t)−g(t); n∈N}∪{0} for almost all t∈I and G(t, x) := −NC(x)∩Y(t), x ∈H. ThenG(t,·) is upper semicontinuous onC∩(x0+rB) with compact values because the set-valued map−NC(·) has closed graph andY(t)⊂H is compact (e.g. Theorem 1.1.1 in [1]).
Since for almost allt∈I,−xn(t) +fn(t) +g(t)∈ −G(t, x(t)) one has d(xn(t), G(t, x(t))+F(x(t))+g(t))≤d∗(G(t, xn(t))+F(xn(θn(t)))+g(t),
G(t, x(t)) +F(x(t)) +g(t))≤
≤d∗(G(t, xn(t)), G(t, x(t)) + d∗(F(xn(θn(t))), F(x(t))), where d∗(A, B) = sup{d(a, B), a∈A}.
It follows from the upper semicontinuity of G(t,·) anf F(·) that
nlim→∞d(xn(t), G(t, x(t)) +F(x(t)) +g(t)) = 0 a.e.(I) and sincexn(t) converges to x(t) for almost all t∈I, we obtain
x(t)∈G(t, x(t)) +F(x(t)) +g(t) a.e.(I), i.e., (3.14) holds and the proof is complete.
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Received 19 September 2006 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania
