Electronic Journal of Qualitative Theory of Differential Equations 2005, No.11, 1-22;http://www.math.u-szeged.hu/ejqtde/
On a time-dependent subdifferential
evolution inclusion with a nonconvex
upper-semicontinuous perturbation
S. Guillaume
1, A. Syam
21
D´epartement de math´ematiques, Universit´e d’Avignon,
33, rue Pasteur, 84000 Avignon, France,
sophie.guillaume@univ-avignon.fr
2
D´epartement de math´ematiques et informatique,
Universit´e Cadi Ayyad, 618 Marrakech, Maroc,
syam@fstg-marrakech.ac.ma
Abstract. We investigate the existence of local approximate and strong solutions for a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation.
Keywords. Convex function, Evolution equation, Nonconvex perturbation, Subdifferential, Upper-semicontinuous operator.
AMS (MOS) Subject Classifications. 35G25, 47J35, 47H14
1
Introduction
For a given family of convex lower-semicontinuous functions (ft)
t∈[0,T], defined on a
sepa-rable real Hilbert space X with range in R∪ {∞}, and a family of multivalued operators
(B(t, .))t∈[0,T] onX, we shall prove an existence theorem for evolution equations of type:
u′(t) +∂ft(u(t)) +B(t, u(t))∋0, t∈[0, T]. (1)
For each t, ∂ft denotes the ordinary subdifferential of convex analysis. The operator
B(t, .) :X→
→X is a multivalued perturbation of∂ft, dependent on the time t.
t), Attouch-Damlamian [2] and Yamada [18]. The study of caseB(t, .) nonmonotone and upper-semicontinuous with convex closed values has been developed under some assump-tions of compactness on domft = {x∈ X | ft(x) <∞} the effective domain of ft. For
example, Attouch-Damlamian [1] have studied the casef independent of time. Otani [15] has extended this result with more general assumptions (the convex function ft depends
on time). He has also studied the case where −B(t, .) is the subdifferential of a lower semicontinuous convex function, see [14].
In this article, the operator B(t, .) will be assumed upper-semicontinuous with compact values which are not necessary convex, and it is not assumed be a contraction map. Never-theless, −B(t, .) will be assumed cyclically monotone. Cellina and Staicu [7] have studied this type of inclusion when ft and B(t, .) are not dependent ont.
This paper is organized as follows. In Section 2 we recall some definitions and results on time-dependent subdifferential evolution inclusions and upper-semicontinuity of oper-ators which will be used in the sequel. We also introduce the assumptions of our main result. In Section 3 we obtain existence of approximate solutions for the problem (1) and give properties of these solutions. In Section 4 we establish existence theorem for the problem (1). We particularly study two cases where the family (ft)
t satisfies more
restricted assumptions. Examples illustrate our results in Section 5.
2
Perturbed problem
Assume that X is a real separable Hilbert space. We denote by k.k the norm associated with the inner product h., .i and the topological dual space is identified with the Hilbert space. Let T >0 and (ft)
t∈[0,T] be a family of convex lower-semicontinuous (lsc, in short)
proper functions on X. We will denote by ∂ft the ordinary subdifferential of convex
analysis.
Definition 2.1 A function u: [0, T]→X is said strong solution of
u′ +∂ft(u) +B(t, u)∋0
if 1: (i) there exists β ∈L2(0, T;X) such that β(t)∈B(t, u(t)) for a.e. t∈[0, T],
(ii) u is a solution of
u′(t) +∂ft(u(t)) +β(t)∋0 for a.e. t ∈[0, T]
u(t)∈domft for any t ∈[0, T].
The aim result of this article is, for each u0 ∈ domf0, the existence of a local strong
solution u of u′ +∂ft(u) +B(t, u) ∋ 0 with u(0) = u
0, when the values of the
upper-semicontinuous multiapplication B(t, .) are not convex.
We shall consider the following assumption on (ft)
t∈[0,T], see Kenmochi [10, 11]:
(H0): for each r >0, there are absolutely continuous real-valued functions hr and kr on
[0, T] such that: (i) h′
r ∈L2(0, T) and k′r∈L1(0, T),
(ii) for each s, t∈[0, T] with s6t and each xs ∈domfs with kxsk6r there
exists xt∈domft satisfying
kxt−xsk6|hr(t)−hr(s)|(1 +|fs(xs)|1/2)
ft(x
t)6fs(xs) +|kr(t)−kr(s)|(1 +|fs(xs)|).
or the slightly stronger assumption,see Yamada [18], denoted by(H), when (ii) holds for any s, t in [0, T].
The following existence theorem have been proved in [19]:
Theorem 2.1 Let T >0 and β ∈ L2(0, T;X). Let u
0 ∈ domf0. If (H0) holds, then the
problem
u′(t) +∂ft(u(t)) +β(t)∋0, a.e. t∈[0, T]
u(t)∈domft, t∈[0, T]
u(0) =u0
has a unique solution u: [0, T]→X which is absolutely continuous.
Furthermore, we have the following type of energy inequality, see [11, Chapter 1]: if ku(t)k< r for t∈[0, T], then
ft(u(t))−fs(u(s)) + 1 2
Z t
s k
u′(τ)k2dτ 6 1
2
Z t
s k
β(τ)k2dτ +
Z t
s
cr(τ) [ 1 +|fτ(u(τ))|] dτ
(2) for any s6t in [0, T], where cr :τ 7→4|h′r(τ)|2 +|kr′(τ)| is an element of L1(0, T).
Let us add a compactness assumption on each ft by using the following definition:
Definition 2.2 A function f : X → R∪ {+∞} is said of compact type if the set {x ∈
X | |f(x)|+kxk2 ≤c} is compact at each level c.
Denote byL2
w(0, T;X) the spaceL2(0, T;X) endowed with the weak topology. Under this
compactness assumption on each ft, the map
p:
L2
w(0, T;X) → C([0, T];X)
β 7→ u
is continuous and maps bounded set into relatively compact sets following [9, proposition 3.3], β and u being defined in Theorem 2.1.
Definition 2.3 Let E1 and E2 be two Hausdorff topological sets. A multivalued operator
B : E1→→E2 is said upper-semicontinuous (usc in short) at x ∈ DomB if for all
neigh-borhood V2 of the subset Bx of E2, there exists a neighborhood V1 of x in E1 such that
B(V1)⊂ V2.
Furthermore, if E1 and E2 are two Hausdorff topological spaces with E2 compact and
B : E1→→E2 is a multivalued map with Bx closed for any x ∈ E1, then B is usc if and
only if the graph of B is closed in E1 ×E2. We introduce following conditions on the
multifunction B : [0, T]×X−→ −→X:
(Bo) : (i) Dom(∂ft)⊂DomB(t, .) for any t∈[0, T],
(ii) there exist nonnegative constants ρ, M such that kx−u0k 6 ρ implies
B(t, x)⊂MBX for any t ∈[0, T] and x∈Dom∂ft.
(B) :
(i) Dom(∂ft)⊂DomB(t, .) and the set B(t, x) is compact for any t ∈[0, T] and
x∈Dom(∂ft),
(ii) there exist a nonnegative realρ and a convex lsc function ϕ :X →R such that
kx−u0k6ρ implies B(t, x)⊂ −∂ϕ(x) for any t∈[0, T] and x∈Dom(∂ft),
(iii) for a.e. t∈[0, T], the restriction of B(t, .) to Dom(∂ft) is usc,
(iv) for each r > 0, there is a nonnegative real-valued function gr on [0, T]2 such
that
(a) limt→s−gr(t, s) = 0,
(b) for eachs, t∈[0, T]witht 6sand eachxs∈Dom∂fsandβs∈B(s, xs)
with kxsk ∨ kβsk6r there exists xt∈DomB(t, .) and βt ∈B(t, xt) satisfying
kxt−xsk ∨ kβt−βsk6gr(t, s).
By convexity, the function ϕ of (B)(ii) is M-Lipschitz continuous on some closed ball
u0+ρBX and the inclusion ∂ϕ(x)⊂MBX holds for anyx∈u0+ρBX. In fact, we could
takeϕ with extended real values andu0 in the interior of the effective domain ofϕ. Thus,
(B)(ii) implies (Bo)(ii).
The condition (B)(ii) means that −B(t, .) is cyclically monotone uniformly in t. An example is the multiapplication B(t, .) :Rn→→Rn defined by
β = (β1, . . . , βn)∈B(t, x) ⇐⇒ β1 ∈
{1} if x1 <0
{−1,1} if x1 = 0
{−1} if x1 >0
and β2 =· · ·=βn= 0
for any x = (x1, . . . , xn) ∈ Rn. When B(t, .) = −∂ψt with ψt : X → R∪ {+∞} a lsc
proper function, then ψt is convex if this operator is monotone and (B)(ii) is equivalent
to the existence of a real constant αt with ψt = ϕ+αt. In this case we deals with the
condition (ii) could be extended to a function ϕt which depends on the time t, and also
with a nonconvex function: for example, a convex composite function, see [8].
The condition (B)(iii) is always satisfied if B(t, .) or −B(t, .) is a maximal monotone operator of X, and more generally if they are φ-monotone of order 2.
The condition (B)(iv) is always satisfied if B(t, .) = B : X→
→X is not depending on the
time t. It can also be written for any t6s in [0, T]:
lim
t→s−
e(gphB(t, .)∩rBX2,gphB(s, .)) = 0,
estanding for the excess between two sets. WhenB(t, .) or−B(t, .) is the subdiffferential of a convex lsc function ψt which satisfies (H
0), the condition (iv) is satisfied.
3
Existence of approximate solutions
For any realλ >0 andt∈[0, T], the functionft
λ shall denote theMoreau-Yosida proximal
function of index λ of ft, and we set
Jλt = (I+λ∂ft)−1 , Dfλt =λ−1(I−Jλt).
We first prove the approximate result of existence :
Theorem 3.1 Let (ft)
t∈[0,T] be a family of proper convex lsc functions on X with each
ft of compact type. Assume that (H) and (B
o) are satisfied. For eachu0 ∈domf0, there
exists T0 ∈]0, T] such that u′+∂ft(u) +B(t, u)∋0 has at least an approximate solution
x : [0, T0] → X with x(0) = u0 in the following sense: there exist sequences (xn)n of
absolutely continuous functions from [0, T0] to X, (un)n and (βn)n of piecewise constant
functions from [0, T0] to X which satisfy:
1. for a.e. t∈[0, T0]
x′
n(t) +∂ft(xn(t)) +βn(t)∋0
xn(0) =u0 and βn(t)∈B(θn(t), un(t))
where 06t−θn(t)62−nT,
2. there exists N ∈N such that for any n >N:
∀t∈[0, T] kxn(t)−u0k6ρ and kβn(t)k6M,
3. (xn)n and (un)n converge uniformly to x on [0, T0], (βn)n converges weakly to β in
L2(0, T
0;X), (x′n)n converges weakly to x′ in L2(0, T0;X) and x is the solution of
x′(t) +∂ft(x(t)) +β(t)∋0, x(0) =u
3.1
Proof of Theorem 3.1
Lemma 3.1 We can find a set{zt :t∈[0, T]}andρ0 >0such thatzt ∈ρ0B, ft(zt)≤ρ0
for every t∈[0, T].
Proof. Letz0 ∈domf0 and r >0 such that r >kz0k ∨ |f0(z0)|. For all t∈[0, T], there
exists zt∈domft satisfying
kzt−z0k6|hr(t)−hr(0)|(1 +|f0(z0)|1/2)
ft(z
t)6f0(z0) +|kr(t)−kr(0)|(1 +|f0(z0)|).
The lemma holds with ρ0 = (r+kh′rkL1(1 +r1/2) )∨(r+kk′
rkL1(1 +r) ). Lemma 3.2 [18, Proposition 3.1]. Letx∈X andλ >0. The mapt7→Jt
λxis continuous
on [0, T].
Proof. ¿From Kenmochi [11, Chapter 1, Section 1.5, Theorem 1.5.1], there is a nonneg-ative constant α such that ft(x)>−α(kxk+ 1) for allx∈X and t∈[0, T]. Thus,
ft λ(x)−
1
2λkx−J
t
λxk2 =ft(Jλtx)>−α(1 +kJλtxk),
which implies
kx−Jλtxk2 62λα(1 +kJλtx−xk+kxk) + 2λfλt(x). (3) Since 2λft
λ(x)62λft(zt) +kzt−xk2 62λρ0+ (ρ0+kxk)2by Lemma 3.1, we can conclude:
sup{kJλtxk |t ∈[0, T], λ∈]0,1], x∈rB}<∞
sup{|ft(Jλtx)| |t∈[0, T], x∈rB}<∞
for any r >0.
Lett ∈[0, T] andr>kJt
λxk. By assumption (H0), for eachs ∈[0, T] with s>tthere
exists xs∈domfs satisfying
kJt
λx−xsk6|hr(t)−hr(s)|(1 +|ft(Jλtx)|1/2)
fs(x
s)6ft(Jλtx) +|kr(t)−kr(s)|(1 +|ft(Jλtx)|).
Since λ−1(x−Js
λx)∈∂fs(Jλsx), we have
fs(Jλsx) + 1
λhx−J
s
λx, xs−Jλsxi6fs(xs)6ft(Jλtx) +|kr(t)−kr(s)|(1 +|ft(Jλtx)|).
Hence, for any s>t, we have 1
λhx−J
s
λx, Jλtx−Jλsxi
6 1
λhx−J
s
λx, Jλtx−xsi+ft(Jλtx)−fs(Jλsx) +|kr(t)−kr(s)|(1 +|ft(Jλtx)|)
6 kDfλs(x)k|hr(t)−hr(s)|(1 +|ft(Jλtx)|1/2) +ft(Jλtx)−fs(Jλsx)
By symmetry it is true for any s∈[0, T]. In the same way fort, s in [0, T], we have
Adding these two inequalities we obtain
1
there exists an absolutely continuous function v on [0, T1] satisfying:
* v(0) =u0 and lim supt→0+f
we have kβn
kk6M. We then set
βkn(t) =
βn
k−1(t) ift ∈[tn0, tnk[
βn
k if t∈[tnk, T]
Let us set xn
k = p(βkn). By unicity it follows xnk(t) = xnk−1(t) if t ∈ [0, tnk]. Furthermore,
kβn
k(t)k6M for any t∈[0, T]. By Lemma 3.3,
∀t∈[0, T] kxkn(t)−u0k6
ρ
2.
We then set
xn:=xn2n
−1 = 2n X
k=0
xnk χ[tn k,t
n
k+1[ and βn:=β
n 2n
−1 = 2n X
k=0
βknχ[tn k,t
n k+1[,
where χ[tn k,t
n
k+1[(t) = 1 if t∈[t
n
k, tnk+1[ , and = 0 otherwise. For all t∈[0, T[, there exists
06k 62n with t∈[tn
k, tnk+1[ and we set
θn(t) =tnk and θn(T) =T.
So, xn : [0, T] → X is an absolutely continuous function and βn : [0, T] → X is a
measurable map which satisfy for a.e. t∈[0, T]
x′
n(t) +∂ft(xn(t)) +βn(t)∋0
xn(0) =u0 and βn(t)∈B(θn(t), un(t))
where we set un(t) =Jθ
n(t)
n xn(θn(t)). By construct, there exists N ∈ Nsuch that for any
n >N:
∀t ∈[0, T] kxn(t)−u0k6ρ and kβn(t)k6M.
A subsequence of (βn)n, again denoted by (βn)n, converges weakly toβ inL2(0, T;X). By
continuity of the mapp, the sequence xn=p(βn) converges uniformly to a curvex=p(β)
on [0, T] and a subsequence of (x′
n)n converges weakly to x′ in L2(0, T;X).
In other words, the curve x is the solution of x′(t) +∂ft(x(t)) +β(t) ∋ 0, x(0) = u 0 on
[0, T].
Letn∈N⋆ and t∈[0, T]. We have
kun(t)−x(t)k6kxn(θn(t))−x(θn(t))k+kx(θn(t))−x(t)k+kJnθn(t)x(t)−x(t)k. (4)
Under the assumption (H0), there existsun,t∈domfθn(t) satisfying
kun,t−x(t)k6|hr(θn(t))−hr(t)|(1 +r1/2)
fθn(t)
Lemma 3.5 For almost anyt ∈[0, T]we have ft(x(t)) = lim inf
Proof. By lower semicontinuity of ft, the inequality
ft(x(t))6lim inf
Lemma 3.6 We have the inequality
Since kun
k −u0k6kun(tnk)−xn(tnk)k+ρ/26ρ for n large enough, we have ft
n k(un
k)>0.
Furthermore, inequality (5) assures thatftn k(xn
k(tnk))6f0(u0)+ρ 6r. Using the definition
of the Moreau-Yosida approximate, we obtain
23n
2 ku
n
k −xnk(tnk)k2 =f tn
k
n (xnk(tnk))−ft
n k(un
k)6r.
Thus, kun
k−xnk(tnk)k6
√
r2−3n+1 and
2n
−1
X
k=1
kunk−xnk(tnk)k6√r2−n+1.
Consequently,
lim
n→+∞
2n
−1
X
k=1
kunk−xnk(tnk)k= 0.
By continuity of ϕ and convergence of (xn)n tox, we obtain
lim inf
n→+∞
Z T
0 h
βn(s), x′n(s)ids>ϕ(u0)−ϕ(x(T)).
Since β(s) ∈ −∂ϕ(x(s)) almost everywhere, hβ(s), x′(s)i =−(ϕ◦x)′(s) holds for a.e. s
and we obtain the inequality (6) .
4
Existence of strong solutions
We now prove the existence of strong solutions.
4.1
General case
Let us set
Φ(x, y) =
Z T
0 h
y(t), x′(t)idt−fT(x(T)) +f0(u0)
for any absolutely continuous function x : [0, T] → X with x′ ∈ L2(0, T;X) and any
fonction y∈L2(0, T;X).
Theorem 4.1 Let (ft)
t∈[0,T] be a family of proper convex lsc functions on X with each ft
of compact type which satifies the assumption (H). Assume that for any sequence(xn)n in
H1(0, T;X) which converges uniformly to the absolutely continuous function x with the
weak convergence of (x′
n)n to x′ in L2, and for any (yn)n which converges weakly to y in
L2 with y
n(t)∈∂ft(xn(t)) for almost all t, there exists nk →+∞ such that
lim inf
k→+∞Φ(xnk, ynk)>Φ(x, y).
Then, for each u0 ∈domf0, there exists T0 ∈]0, T] such thatu′+∂ft(u) +B(t, u)∋0has
Proof. Consider x an approximate solution. We prove x′(t) +∂ft(x(t)) +B(t, x(t))∋0
Step 2. - Under the assumption on Φ, it follows by lower semicontinuity of fT :
Let us set Ft(x) = ∂ft(x)∩Y(t) and Gt(x) = B(t, x)∩ Γ(t) for any x ∈ Dom(∂ft)
and t ∈ [0, T]. The multimaps Ft and Gt are upper semicontinuous on Dom(∂ft) with
compact values in X. Let us denote by e the excess between two sets. We have:
d(−x′n(t), Ft(x(t)) +Gt(x(t))) 6 d(yn(t), Ft(x(t))) +d(βn(t), Gt(x(t)))
6 e(Ft(xn(t)), Ft(x(t))) +kβn(t)−αtnk+d(αtn, Gt(x(t)))
6 e(Ft(xn(t)), Ft(x(t))) +gr(θn(t), t) +e(Gt(znt), Gt(x(t))).
The upper-semicontinuity of Ft and Gt assures that
lim
n→+∞d(−x ′
n(t), Ft(x(t)) +Gt(x(t))) = 0.
Since (x′
n)n converges to x′ a.e. on [0, T], the equality d(−x′(t), Ft(x(t)) +Gt(x(t))) = 0
holds for a.e. t∈[0, T] and we obtain by closedness ofFt(x(t)) +Gt(x(t)):
−x′(t)∈Ft(x(t)) +Gt(x(t)) for a.e. t∈[0, T].
Consequently, x is a local solution to x′+∂ft(x) +B(t, x)∋0 with x(0) =u
0.
4.2
Two particular cases
We consider two particular cases for which we can apply Theorem 4.1. These cases contains those of ft not depending on t.
First,
Corollary 4.1 Let (ft)
t∈[0,T] be a family of proper convex lsc functions on X with each
ft of compact type. Let u
0 ∈domf0. Assume that ft=g◦Ft where g is a proper convex
lsc function on a Hilbert space Y and(Ft)
t∈[0,T] is a family of differentiable maps fromX
to Y such that (DFt)
t is equilipschitz continuous on a neighborhood of u0 and such that:
1. for each r >0, there is absolutely continuous real-valued function br on [0, T] such
that: (a) b′
r ∈L2(0, T),
(b) for each s, t∈[0, T], supkxk rkFt(x)−Fs(x)k6|b
r(t)−br(s)|,
2. the qualification condition R+[domg−F0(u0)]−DF0(u0)X =Y holds,
3. for each r > 0, there exists a negligible subset N of [0, T] such that the mapping
t 7→ Ft(x) admits a derivative ∆t(x) on [0, T]\N for any x ∈ rB
X and ∆t is
continuous on rBX for any t∈[0, T]\N,
4. the mapping (t, x) 7→ DFt(x) is bounded on [0, T]×rB
X for each r > 0 and it is
Assume that(H)and(B)are satisfied. Then, there existsT0 ∈]0, T]such thatu′+∂ft(u)+
B(t, u)∋0 has at least a strong solution u: [0, T0]→X with u(0) =u0.
Remark under assumption 1., the mapping t 7→ Ft(x) is absolutely continuous and
ad-mits a derivative at a.e. t ∈ [0, T] for each x. With the uniformly inequality 1.(b), we can hope that the almost derivability of t 7→ Ft(x) at t is uniform in x ∈ rB
X thanks
to the regularity of Ft at x. Illustrate the importance of differentiability of Ft by the
following example : F(t, x) = h(t −x) where X = Y = R and the real function h is
convex, Lipschitz continuous and non differentiable on [0, T].
Proof of Corollary 4.1. Consider x : [0, T] → X an approximate solution. Let us set
yn(t) = −x′n(t)−βn(t) and y(t) = −x′(t)−β(t) for a.e. t in [0, T],zn(t) =Ft(xn(t)) and
z(t) = Ft(x(t)) for a.e. t ∈ [0, T]. Then, z
n and z are absolutely continuous on [0, T],
thus are derivable at a.e. t ∈[0, T] and
zn′(t) = ∆t(xn(t)) +DFt(xn(t))x′n(t) , z′(t) = ∆t(x(t)) +DFt(x(t))x′(t).
Under the qualification condition, we have for any x∈X
∂ft(x) =DFt(x)⋆∂g(Ft(x)).
Let us writeyn(t) =DFt(xn(t))⋆wn(t) andy(t) =DFt(x(t))⋆w(t) wherewn(t)∈∂g(zn(t))
and w(t) ∈ ∂g(z(t)) for almost all t ∈ [0, T]. Hence, g ◦ zn and g ◦ z are absolutely
continuous with hwn(t), zn′(t)i= (g◦zn)′(t) for almost allt ∈[0, T]. So,
hyn(t), x′n(t)i=
d dt(f
t◦x
n)(t)− hwn(t),∆t(xn(t)).
and Φ(xn, yn) = −
RT
0 hwn(t),∆ t(x
n(t))idt.
Let r > kxn(t)k ∨ kx(t)k for any n ∈ N and t ∈ [0, T]. By continuity of ∆t on rBX,
(∆t(x
n(t)))n converges to ∆t(x(t)) for a.e t ∈ [0, T]. Next, for a.e. t and any x ∈ rBX,
we have k∆t(x)k 6 |b′
r(t)|. By Lebesgue’s theorem ∆.(xn(.)) converges to ∆.x(.)) in
L2(0, T, Y). Since a subsequence of (w
n)n converges weakly to w, we can apply
Theo-rem 4.1.
For example, if Ft is the affine mapping x 7→A(t)x+b(t) where A(t) : X →Y is linear
continuous and b(t)∈Y, the assumption of Corollary 4.1 becomes :
1. b is absolutely continuous on [0, T] and there is absolutely continuous real-valued function a on [0, T] such that:
(a) a′ ∈L2(0, T),
(b) for each s, t∈[0, T],kA(t)−A(s)k6|a(t)−a(s)|.
3. for eachr >0, there exists a negligible subset N of [0, T] such that A′(t) is
contin-uous on rBX for any t∈[0, T]\N.
Second, we use the conjugate of ft.
Lemma 4.1 Let (ft)
t∈[0,T]be a family of proper convex lsc functions on X satisfying(H).
Assume that :
for each r > 0, there exists a negligible subset N of [0, T] such that for any
t∈[0, T]\N, the mappings 7→(fs)⋆(y)admits a derivativeγ˙(t, y)att for any
y∈Dom∂(ft)⋆.
Let x : [0, T]→ X be an absolutely continuous function and y : [0, T] → Y be such that
y(t)∈∂ft(x(t)) for a.e. t∈ [0, T]. For almost all t∈[0, T], we have
˙
γ(t, y(t)) + d
dtf
t(x(t)) =hy(t), x′(t)i. (8)
Proof. Let s and t be in [0, T]\N where N is a suitable negligible subset of [0, T]. We have :
(fs)⋆(y(s))−(ft)⋆(y(s))6(fs)⋆(y(s))−(ft)⋆(y(t))− hy(s)−y(t), x(t)i
since x(t)∈∂(ft)⋆(y(t)). From ft(x(t)) + (ft)⋆(y(t)) =hy(t), x(t)i, we deduce
(fs)⋆(y(s))−(ft)⋆(y(s))6ft(x(t))−fs(x(s)) +hy(s), x(s)−x(t)i.
In the same way, for almost every t, s in [0, T] we have
(fs)⋆(y(s))−(ft)⋆(y(s))6ft(x(t))−fs(x(s)) +hy(s), x(s)−x(t)i.
Changing the role of s and t, we also have:
(ft)⋆(y(t))−(fs)⋆(y(t))6fs(x(s))−ft(x(t)) +hy(t), x(t)−x(s)i
6(ft)⋆(y(s))−(fs)⋆(y(s)) +hy(t)−y(s), x(t)−x(s)i.
The function t 7→ ft(x(t)) being absolutely continuous on [0, T], see [11, Chapter 1], we
obtain (8).
The existence of ˙γ implies some regularity on the domain of (ft)⋆. For example, consider
(ft)⋆(y) = h(t − y) where X = Y = R and the real function h is convex, Lipschitz
continuous and non differentiable on [0, T]. Then, we can not apply above lemma. The domain of (ft)⋆ changes witht. But, we can apply Corollary 4.1 sinceft(x) =tx+h⋆(−x).
However, we have the absolutely continuity of s 7→(fs)⋆(y) in the following sense :
Proposition 4.1 Let t ∈[0, T], y ∈Y, η > 0 and r >0 such that if |t−s|6η, the set
∂(fs)⋆(y)∩rB
Proof. 1) Lemma 3.1 with β = ρo assures that (ft)⋆(y)> hy, zti −ft(zt) >−kykβ−β
for any t∈[0, T] and y∈X. For y∈∂ft(x), it follows
−α(kxk+ 1)6ft(x) =hy, xi −(ft)⋆(y)6kyk[kxk+β ] +β.
So, there is a nonnegative constant β such that (ft)⋆(y)>−β(kyk+ 1) for ally ∈X and
t ∈[0, T].
Furthermore, for each r > 0, there is a nonnegative constant c such that |ft(x)| 6
c(kyk+ 1) for all x∈rBX, t ∈[0, T] and y∈∂ft(x).
2) Let t be fixed in [0, T] and y ∈ ∂ft(x). Let r > kxk and s ∈ [t, T]. Under the
assumption (H0), there exists xs ∈domfs satisfying
kx−xsk6|hr(t)−hr(s)|(1 +|ft(x)|1/2)
fs(x
s)6ft(x) +|kr(t)−kr(s)|(1 +|ft(x)|).
By definition of conjugate of a convex function, it follows
(ft)⋆(y)−(fs)⋆(y) 6 hy, x−x
si+fs(xs)−ft(x)
6 kyk|hr(t)−hr(s)|(1 +|ft(x)|1/2) +|kr(t)−kr(s)|(1 +|ft(x)|).
We conclude thanks to 1):
(ft)⋆(y)−(fs)⋆(y) 6 kyk|hr(t)−hr(s)|(1 +|ft(x)|1/2) +|kr(t)−kr(s)|(1 +|ft(x)|)
6 kyk|hr(t)−hr(s)|(1 +√c+
p
ckyk) +|kr(t)−kr(s)|(1 +c+ckyk)
3) Let y ∈ Y, r > 0 and s, t ∈ [0, T]. If the intersections of ∂(ft)⋆(y) and ∂(fs)⋆(y)
with rBX are non empty, let xs ∈ ∂(fs)⋆(y) and xt ∈ ∂(ft)⋆(y) with r > kxsk ∨ kxtk.
Step 2) implies
|(fs)⋆(y)−(ft)⋆(y)|6
kyk|hr(t)−hr(s)|(1 +|fs(xs)|1/2 ∨ |ft(xt)|1/2) +|kr(t)−kr(s)|(1 +|fs(xs)| ∨ |ft(xt)|).
By 1) we conlude:
|(fs)⋆(y)−(ft)⋆(y)|6kyk|hr(t)−hr(s)|(1 +c1/2+ (ckyk)1/2) +|kr(t)−kr(s)|(1 +c+ckyk).
Corollary 4.2 Let (ft)
t∈[0,T] be a family of proper convex lsc functions on X with each
ft of compact type. Assume that (H) and (B) are satisfied and that:
1. for each r > 0, there exists a negligible subset N of [0, T] such that for any t
in [0, T]\N, the mapping s 7→ (fs)⋆(y) admits a derivative γ˙(t, y) at t for any
2. for any (yn)n which converges weakly to y in L2(0, T;X) with yn(t) ∈ ∂ft(xn(t))
where (xn)n converges uniformly, there exists nk →+∞ such that
lim inf
k→+∞
Z T
0
˙
γ(t, ynk(t))dt> Z T
0
˙
γ(t, y(t))dt.
Then, for each u0 ∈domf0, there exists T0 ∈]0, T] such thatu′+∂ft(u) +B(t, u)∋0has
at least a strong solution u: [0, T0]→X with u(0) =u0.
The assumption 2. is true when ˙γ(t, .) is lsc and convex on Dom∂(ft)⋆.
Proof. Lemma 4.1 implies Φ(x, y) =
Z T
0
˙
γ(t, y(t))dt. By assumption 2., we can apply
Theorem 4.1.
5
Examples of families
(
f
t)
t5.1
Rafle
SeeCastaing, Valadier and Moreau [4, 17, 13]. Let (C(t))t∈[0,T]a family of nonempty closed
convex subsets ofXwhose intersection with bounded closed sets is compact. Consider the indicator function ft =δ
C(t) of C(t). Assume that for each r >0, there is an absolutely
continuous real-valued function ar on [0, T] such that:
(i)a′
r ∈L2(0, T);
(ii) for each s, tin [0, T], we have e(C(s)∩rBX, C(t))6|ar(s)−ar(t)|.
Under the assumption (B), we can apply Theorem 4.1: Assume that for any sequence
(xn)n in H1(0, T;X) which converges uniformly to the absolutely continuous function x
with the weak convergence of (x′
n)n to x′ in L2, and for any (yn)n which converges weakly
to y in L2 with y
n(t)∈NC(t)(xn(t)) for almost all t, there exists nk →+∞ such that
lim inf
k→+∞
Z T
0 h
x′nk(t), ynk(t)idt> Z T
0 h
x′(t), y(t)idt.
Then, for each u0 ∈C(0), there exists T0 ∈]0, T] such that u′+NC(t)(u) +B(t, u)∋0has
at least a strong solution u: [0, T0]→X with u(0) =u0.
Corollary 4.1 becomes:
Corollary 5.1 Let u0 ∈C(0) and(Ft)t∈[0,T] be a family of differentiable maps fromX to
an other Hilbert space Y such that (DFt)
t is equilipschitz continuous on a neighborhood
of u0. Assume that C(t) = (Ft)−1(C), C being a nonempty closed convex set. Under the
assumptions (B) and:
1. for each r >0, there is absolutely continuous real-valued function br on [0, T] such
(a) b′
r ∈L2(0, T),
(b) for each s, t∈[0, T], supkxk rkFt(x)−Fs(x)k6|b
r(t)−br(s)|,
2. the qualification condition R+[C−F0(u0)]−DF0(u0)X =Y holds,
3. for each r > 0, there exists a negligible subset N of [0, T] such that the mapping
t 7→ Ft(x) admits a derivative ∆t(x) on [0, T]\N for any x ∈ rB
X and ∆t is
continuous on rBX for any t∈[0, T]\N,
4. the mapping (t, x) 7→ DFt(x) is bounded on [0, T]×rB
X for each r > 0 and it is
continuous at t for each x,
there exists T0 ∈]0, T] such that u′+NC(t)(u) +B(t, u)∋0 has at least a strong solution
u: [0, T0]→X with u(0) =u0.
On the other hand, (ft)⋆is the support function ofC(t), denoted byσ
C(t). Moreau have
proved in [13] that when C is absolutely continuous, the map t 7→ σC(t)(y) is absolutely
continuous on [0, T] for anyy∈D, whereDis the domain ofσC(t) which is not dependent
of t. Corollary 4.2 becomes:
Corollary 5.2 Assume that (B) is satisfied and that:
1. for each r > 0, there exists a negligible subset N of [0, T] such that for any t
in [0, T]\N, the mapping s 7→ σC(s)(y) admits a derivative γ˙(t, y) at t for any
y∈Dom∂σC(t).
2. for any (yn)n which converges weakly to y in L2(0, T;X) with yn(t) ∈ NC(t)(xn(t))
where (xn)n converges uniformly, there exists nk →+∞ such that
lim inf
k→+∞
Z T
0
˙
γ(t, ynk(t))dt> Z T
0
˙
γ(t, y(t))dt.
Then, for each u0 ∈domf0, there exists T0 ∈]0, T] such that u′+NC(t)(u) +B(t, u)∋ 0
has at least a strong solution u: [0, T0]→X with u(0) =u0.
By example, consider the affine map Ft(x) = a(t)x+b(t) where a(t) ∈ R⋆
+ is derivable
nonincreasing at t ∈ [0, T] and b(t) ∈ Y is absolutely continuous at t ∈ [0, T]. We can apply both corollary 4.1 and 4.2 since
˙
γ(t, y) = −1
a(t)2 [a(t)hy, b
′(t)i+a′(t)(σ
C(y)− hy, b(t)i)]
for a.e. t ∈[0, T] and any y∈Y. So, ˙γ(t, .) is convex l.s.c. on X and
lim
n→+∞
Z T
0
˙
γ(t, yn(t))dt=
Z T
0
˙
5.2
Viscosity
Let f : X → R∪ {+∞} be a convex lsc proper function of compact type. Consider
ft(x) =f(x) +ε(t) 2 kxk
2 whereεis an absolutely continuous real-valued function on [0, T]
with nonnegative values and ε′ ∈L1(0, T).
We can write ft = g ◦Ft with g(x, r) = f(x) +r for any (x, r) ∈ X ×R and Ft(x) =
(x,ε(t)2 kxk2) for any x∈X. By absolutely continuity of ε, ∆t exists and is continuous on
X for a.e. t∈[0, T] and
∆t(x) = (0,ε
′(t)
2 kxk
2).
Furthermore,
DFt(x)y= (y, ε(t)hx, yi)
and DFt satifies assumption 5. We can apply Corollary 4.1.
On the other hand, (ft)⋆ = (f⋆)
ε(t) is aC1-function onX and, for anyy ∈X, the map
t 7→(ft)⋆(y) is absolutely continuous on [0, T] with for a.e. t∈[0, T]
˙
γ(t, y) =−ε
′(t)
2 kD(f
t)⋆(y)
k2.
By definition of yn, xn(t) =D(ft)⋆(yn(t)) holds for a.e. t∈[0, T] and any n ∈N, hence
˙
γ(t, yn(t)) =−
ε′(t)
2 kxn(t)k
2.
In the same way,
˙
γ(t, y(t)) =−ε
′(t)
2 kx(t)k
2.
By uniform convergence of (xn)n tox on [0, T] it follows
lim
n→+∞
Z T
0
˙
γ(t, yn(t))dt=
Z T
0
˙
γ(t, y(t))dt.
We can apply Corollary 4.2.
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