Abstract This paper studies, for a differential variational inequality involving alocally prox-regular set, a regularization process with a family of classical differen-tial equations whose solutions converge to the solution of the differential variationalinequality. The concept of local prox-regularity will be termed in a quantified way, as(r, α)-prox-regularity.
Keywords Differential inclusions · Differential variational inequalities ·Regularization · Proximal normal cone · Locally prox-regular set ·Sweeping process · Perturbation
The paper is devoted to showing how a regularization process provides a family ofsolutions of differential equations converging uniformly to the unique solution of thevariational differential inequality
{x(t) ∈ −N (C; x(t)) + f (t, x(t)) a.e. t ∈ [0, T ]x(0) = x0 ∈ C
(P)
M. Mazade (B) · L. ThibaultDépartement de Mathématiques CC 051, Université Montpellier 2,Place Eugène Bataillon, 34095 Montpellier, Francee-mail: [email protected]
under the local prox-regularity (see the next section for details) of the set C of a Hilbertspace H . In the study, f : [0, T ] × H → H is assumed to be a mapping which ismeasurable with respect to t and locally Lipschitz continuous with respect to x , andN (C; ·) denotes the normal cone of C .
The study of differential inclusions as
{x(t) ∈ −N (C; x(t)) + F(t, x(t)) a.e. t ∈ [0, T ]x(0) = x0 ∈ C
(1.1)
often appears in modelization in various fields. When C is a closed convex subset ofthe Hilbert space H and F : [0, T ] × H ⇒ H is a set-valued mapping, we may alsowrite this differential inclusion in the variational form that for some z(t) ∈ F(t, x(t))one has for almost every t ∈ [0, T ]
{ 〈−x(t) + z(t), y − x(t)〉 ≤ 0 for all y ∈ Cx(0) = x0 ∈ C.
(1.2)
Henry [18] introduced, with F independent of the time t , for the study of planningprocedures in mathematical economy the differential inclusion
− x(t) ∈ projTC (x(t))(F(x(t))) a.e. t ∈ [0, T ] and x(0) = x0 ∈ C, (1.3)
where projTC (x(t))(F(x(t))) is the metric projection of the image F(x(t)) onto thetangent cone TC (x(t)). In [18] the space H is finite dimensional, the closed set Cis convex, and the set-valued mapping F is upper semicontinuous with nonemptycompact convex values. In [12], Cornet reduced, as done in [18] under the convexityof C , the preceding problem to the existence of a solution of the differential inclusion
− x(t) ∈ N (C; x(t)) + F(x(t)) a.e. t ∈ [0, T ] and x(0) = x0 ∈ C (1.4)
when the closed set C is merely Clarke tangentially regular. Of course, differentialinclusion (1.1) is subject to the constraint that the state x(t) must stay inside the setC . Such constraints also appear for the dynamic system of optimal control problems.Serea made use of (1.1) to study a Mayer control problem. More precisely, in [33] isconsidered the minimization of the function g (x(T ; t0, x0, u(·))) under the controlledvariational inequality
x(t) ∈ −N (C; x(t)) + f (x(t), u(t)) a.e. t ∈ [t0, T ] and x(t0) = x0. (1.5)
In the following, x(·; T, t0, x0, u(·)) denotes the solution of (1.5), for each control
u(·) and each initial state x0 ∈ C ⊂ Rn at initial time t0 ∈ [0, T ]. Through this
model, Serea showed, under the global uniform r -prox-regularity of the set C andsome additional assumptions, that the value function Vg(·, ·) with
Vg(t, x) := supu(·)∈U(t)
g (x(T ; t, x, u(·))),
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Locally prox-regular sets 245
is the unique viscosity solution (in a suitable sense) of the Hamilton–Jacobi differentialinclusion
{∂V∂t (t, x) + H(x, ∂V
∂x (t, x)) − 〈 ∂V∂x (t, x), N (C; x)〉 � 0 for (t, x) ∈ [0, T [×C
V (T, x) = g(x) for all x ∈ C.
In the latter, the Hamiltonian H is defined by
H(x, p) := minu∈U
〈p, f (x, u)〉 for all (x, p) ∈ C × Rn
and U(t) denotes the set of all measurable mappings (that is, controls) from [t, T ] intothe compact set U .
Differential inclusions in the form (1.1) have been also recently involved for thecrowd motion problem in the papers [21–23,37] by Maury and Venel. The problemconsiders n persons, the ith person being identified to a planar disc centered at xi in R
2,with a radius ρ > 0 independent of i . The vector of positions x := (x1, . . . , xn) ∈ R
2n
of the n persons is required to be in the set of feasible configurations
C := {x ∈ R2n : Di, j (x) := ‖xi − x j‖ − 2ρ ≥ 0, ∀ i �= j },
which puts into practice the fact that overlapping is forbidden. Define V0(xi ) as thespontaneous velocity that each person xi would wish in the absence of other people.Then V (x) := (V0(x1), · · · , V0(xn)) ∈ R
2n corresponds to the spontaneous velocityof the n-tuple of persons. The vector field V (·) is assumed to be locally Lipschitzcontinuous on R
2n . The non-overlapping leads to introduce the set of feasible velocitiesof x as
The actual velocity field at time t ∈ [0, T ] is then the closest to V (x(t)), that is,x(t) = projG(x(t))
(V (x(t))
), where projG(x(t)) is as above the metric projection onto
the closed convex set G(x(t)). This can be shown to amount to the differential inclusion
x(t) ∈ −N (C; x(t)) + V (x(t)) and x(0) = x0 ∈ C. (1.6)
Some differential inclusions similar to (1.1) have been treated in the Hilbert settingwith the theory of maximal monotone operators A, see, e.g., [4]. In this context andwhen A is the subdifferential in the sense of convex analysis of a lsc function ϕ, sayA = ∂ϕ, the studied problem is written in the form
x(t) + ∂ϕ(x(t)) + F(t, x(t)) � 0 and x(0) = x0
where the set-valued mapping F : [0, T ] × H → H is single valued and globallyLipschitz continuous with respect to x , and F(·, x) ∈ L2([0, T ], H) for all x ∈Dom ∂ϕ.
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246 M. Mazade, L. Thibault
Another point of view comes from the work of Moreau (see [26–29]), who intro-duced and thoroughly studied a differential inclusion such as (1.1) with F = 0, calledsweeping process, replacing the fixed set C by a moving set C(t) for all t ∈ [0, T ].The considered problem is then the differential inclusion
x(t) ∈ −N (C(t); x(t)) and x(0) = x0 ∈ C(0) (1.7)
where C(t) are closed and convex sets of a Hilbert space H . Later Castaing et al. [5,6]and Castaing and Montero Marques [7] considered perturbed sweeping processes inthe form
{x(t) ∈ −N (C(t); x(t)) + F(t, x(t)) a.e. t ∈ [0, T ]x(0) = x0,
(1.8)
where all the sets C(t) are either convex or complements of open convex sets (seealso the previous paper [36]), the case of general nonconvex closed sets of R
n hasbeen developped for (1.7) by Benabdellah [1], Colombo and Goncharov [10] andThibault [34], and for (1.8) by Thibault [34] (see also Haddad et al. [17]). In theHilbert setting, under the global prox-regularity of the sets C(t) the existence anduniqueness of a solution has been established by Colombo and Goncharov for (1.7), andby Bounkhel and Thibault [3] and Edmond and Thibault in [15,16] for the perturbedsweeping processes (1.8). These results provide solutions for the study of models suchas (1.5) and (1.6) considering the particular case C(t) := C for all t ∈ [0, T ], witha fixed uniformly r -prox-regular set. Recently, Mazade and Thibault [24] showedin the context of a Hilbert space the well-posedness of the differential variationalinequality (P) under merely the local prox-regularity of the set C . Inclusion (P)
corresponds to (1.8) for C fixed and F(·, ·) single-valued, say F(t, x) = { f (t, x)}, withf : [0, T ] × H → H measurable with respect to t and locally Lipschitz continuouswith respect to x .
In [3,10,15,16], existence and uniqueness results for (1.7) have been established byproving the convergence of the Moreau catching-up algorithm, built with a discretiza-tion (ti ) of time, taking x(t0) = x0 for t0 = 0 and x(ti+1) = projC(ti+1)
(xi ) whichobviously makes sense in the convex case. It also holds under uniform prox-regularitywith appropriate subdivisions. Concerning (P) with a locally prox-regular set C anda suitable final time T , existence and uniqueness of solution has been obtained in [24]through the discretization 0 = t0 < t1 < · · · < tn = T of the interval [0, T ] and thedifferential inclusions
{xn(t) ∈ −∂δC (xn(t)) + f (t, xn(ti )) a.e. t ∈ [ti , ti+1]xn(0) = x0 ∈ C
(1.9)
for which the solution xn(·) is due to [20] according to the primal lower nice propertyof the indicator function δC . Another approach for (1.7) was to consider a solution asthe limit of solutions of classical differential equations. In the convex case, Moreau[26] first regularized the normal cone of the convex set C(t) and then studied theassociated differential equations
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Locally prox-regular sets 247
{xλ(t) = − 1
λ∇d2
C(t)(xλ(t))xλ(0) = x0
(Rλ)
and proved that the family (xλ(·))λ of solutions of (Rλ) converges to a solution of(1.7). The study of (Rλ) has been also adapted to uniformly prox-regular sets in [35].
In the present paper, our aim is to show in contrast to the approach of [24] that aregularization procedure can also be achieved for the general dynamical system (P)
governed by the additional vector field f (t, ·) and when the set C is assumed to belocally prox-regular (i.e., under the same assumption in [24]). More precisely, insteadof the catching-up algorithm (1.9) which is at the heart of the approach in [24], weconsider in the present paper the associated regularized differential equation
{xλ(t) = (−1/2λ)∇d2
C (xλ(t)) + f (t, xλ(t))xλ(0) = x0 ∈ C
(Eλ)
and we prove that it has on an appropriate interval containing 0 (independent of λ) aunique solution xλ(·) and that the family (xλ(·))λ converges uniformly to the solutionof (P). This is done through a quantified local notion of prox-regularity for the set Cthat we considered in [24].
The paper is organized as follows. The next section recalls the above mentionedconcept of quantified local prox-regularity of C at x0. In addition to the propertiesestablished in [24], we provide for such sets some other properties which are at thebase of the convergence of the family (xλ(·))λ above. The third section is devoted tothe study of properties of the solution xλ(·) of the differential equation (Eλ) and tothe proof of the convergence of (xλ(·))λ to the local unique solution of our differentialvariational inequality (P).
2 Preliminaries and locally prox-regular sets
Throughout, H is a real Hilbert space. For a point x ∈ H and for a real η > 0 we denoteby B(x, η) (resp. B[x, η]) the open (resp. closed) ball centered at x with radius η. ByB or BH we will denote the closed unit ball of H centered at zero, i.e., B := B[0, 1].Let C be a nonempty closed subset of a Hilbert space H and y ∈ H . The distancefrom y to C , denoted by dC (y) or d(y, C), is given by
dC (y) := inf{ ||x − y|| : x ∈ C }.
One defines the (possibly empty) set of nearest points of y in C by
ProjC (y) := { x ∈ C : dC (y) = ||y − x || }.
When ProjC (y) is a singleton, we will write projC (y) in place of ProjC (y) to emphasizethis singleton property and we will denote by projC (x) the single point of ProjC (x)
that is ProjC (x) = {projC (x)}. If x ∈ ProjC (y), and s ≥ 0, then the vector s(y − x) iscalled (see, e.g., [8]) a proximal normal to C at x . The set of all vectors obtainable in
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248 M. Mazade, L. Thibault
this way is a cone which is termed the proximal normal cone of C at x . It is denoted byN P (C; x). Observing that a nonzero vector v ∈ N P (C; x) if and only if that for someρ > 0 one has x ∈ ProjC
(x + ρ
‖v‖v), and translating this as ρ2 ≤ ‖x + ρ
‖v‖v− x ′‖2 for
all x ′ ∈ C , we obtain that the inclusion v ∈ N P (C; x) is equivalent to the existenceof some real σ ≥ 0 such that
〈v, x ′ − x〉 ≤ σ‖x ′ − x‖2 for all x ′ ∈ C. (2.1)
Inequality (2.1) can be also localized in the sense that it holds for some σ > 0, i.e.,v ∈ N P (C; x) if and only if there exist some γ ≥ 0 and η > 0 such that
〈v, x ′ − x〉 ≤ γ ‖x ′ − x‖2 for all x ′ ∈ C ∩ B(x, η). (2.2)
One also defines the Mordukhovich limiting normal cone by
N L(C; x) := { v ∈ H : ∃ vn →w v, vn ∈ N P (C; xn), xn →C
x }.
where vn →w v means that the sequence (vn)n converges weakly to v and xn →C xmeans that xn → x and xn ∈ C for all n ∈ N. It clearly appears in the definitionabove that N L(C; x) is the Painlevé-Kuratowski weak sequential outer (or superior)limit of N P (C; x ′) as x ′ → x , where for a set-valued mapping M : U ⇒ H from atopological space U into H the Painlevé-Kuratowski weak sequential outer limit ofM at x ∈ U is the set
seq Lim supx ′→x
M(x ′) := { v ∈ H : ∃ vn →w v, vn ∈ M(xn), xn → x }.
It is worth pointing out that for x outside the closed set C one has N L(C; x) = ∅.Hence (since 0 ∈ N P (C; x) for all x ∈ C)
Dom N P (C; ·) = Dom N L(C; ·) = C,
where for a set valued-mapping M : U ⇒ H we denote by Dom M its (effective)domain: Dom M := {x ∈ U : M(x) �= ∅}.
The elements in the Mordukhovich limiting normal cone can also be obtained asweak limits of sequences of Fréchet normal vectors. A vector v ∈ H is a Fréchetnormal of C at x ∈ C whenever for any real ε > 0 there exists some real η > 0 suchthat
〈v, x ′ − x〉 ≤ ε‖x ′ − x‖ for all x ′ ∈ C ∩ B(x, η).
Denoting by N F (C; x) the cone of all Fréchet normals to C at x ∈ C and puttingN F (C; x) = ∅ for x �∈ C , it is also known (see e.g., [25,32]) that
N L(C; x) = seq Lim supx ′→x
N F (C; x ′) for all x ∈ C.
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Locally prox-regular sets 249
Taking (2.1) into account, we always have
N P (C; x) ⊂ N F (C; x) ⊂ N L(C; x) for all x ∈ C. (2.3)
After those preliminaries concerning normal cones, we can now present a view of thelocal prox-regularity property for sets. For a large development of the concept of localprox regularity of sets, the reader is referred to [31]. In this paper, our analysis needsthe quantification considered in [24] of the local prox-regularity introduced in [31]through the constants that this local concept involves in its definition.
Definition 2.1 For positive real numbers r and α, the closed set C is said to be(r, α)-prox-regular at a point x ∈ C provided that for any x ∈ C ∩ B(x, α) and anyv ∈ N L(C; x) such that ||v|| < r , one has
x = projC (x + v).
The set C is r -prox-regular (resp. prox-regular) at x when it is (r, α)-prox-regular atx for some real α > 0 (resp. for some numbers r > 0 and α > 0).
It is not difficult to see that the latter (r, α)-prox-regularity property of C at x ∈ Cis equivalent to requiring that
x ∈ ProjC (x+rv) for all x ∈ C ∩ B(x, α) and v ∈ N L(C; x) ∩ B. (2.4)
We first observe that inclusion (2.4) means that for each positive real number t < rand each x ∈ C ∩ B(x, α)
C ∩ B[x + tv, t] = {x} for any v ∈ N L(C; x) with ‖v‖ = 1. (2.5)
This means that any unit normal vector in N L(C; x) can be realized (see, e.g., [9,11]and [31]) by a t-ball for any positive real number t < r . Further, translating (like for(2.1)) the same inclusion x ∈ ProjC (x +rv) as ‖rv‖2 ≤ ‖x +rv− x ′‖2 for all x ′ ∈ C ,it is easily seen that C is (r, α)-prox-regular at x if and only if for any x ∈ C ∩ B(x, α)
one has
〈v, x ′ − x〉 ≤ 1
2r‖x ′ − x‖2 for all v ∈ N L(C; x) ∩ B and x ′ ∈ C. (2.6)
This combined with (2.1) and (2.3) gives that
N P (C; x) = N F (C; x) = N L(C; x) for all x ∈ C ∩ B(x, α), (2.7)
whenever the set C is (r, α)-prox-regular at x ∈ C . So, for such an (r, α)-prox-regularset and for x ∈ C ∩ B(x, α) we will sometimes write N (C; x) in place of anyone ofthe four normal cones above.
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250 M. Mazade, L. Thibault
Recall now that a set-valued mapping T : H ⇒ H is monotone on a subset O ⊂ Hprovided that
〈v1 − v2, x1 − x2〉 ≥ 0 for all vi ∈ T (xi ), xi ∈ O, i = 1, 2.
When there exists σ > 0 such that T + σ I is monotone on O , where I denotes theidentity mapping from H into H , the set-valued mapping T is said to be hypomonotone(or σ -hypomonotone) on O . This hypomonotonicity corresponds to having
〈v1 − v2, x1 − x2〉 ≥ −σ ||x1 − x2||2 whenever vi ∈ T (xi ), xi ∈ O, i = 1, 2.
Concerning the quantified concept of (r, α)-prox-regularity of sets of the Hilbert spaceH , we have the following properties related to the hypomonotonicity of the truncatednormal cone.
Proposition 2.1 Let C be a closed subset of H, and x ∈ C. The following hold.
(a) If there exist positive real numbers r and α such that C is (r, α)-prox-regular atx , then the set-valued mapping N L(C, ·) ∩ B is 1
r -hypomonotone on B(x, α).(b) Suppose that there exist 0 < r < α such that the set-valued mapping N L (C, ·) ∩ B
is 1r -hypomonotone on B(x, α). Then for the real number σ := 1
2 (α − r) > 0 theset C is ( r
2 , σ )-prox-regular at x .
Proof See [24] for details. ��Proposition 2.5 below provides other useful properties of (r, α)-prox-regular sets.
These properties will be involved in showing in the next section the existence ofsolutions of differential variational inequalities with locally prox-regular sets. Beforestating the proposition let us prove the following lemma and some other properties.
Lemma 2.1 Assume that the closed set C is (r, α)-prox-regular at x and let 0 < r ′ <
r . For all xi ∈ ProjC (ui ) with xi ∈ B(x, α) ∩ C and dC (ui ) < r ′, i=1,2, we have
‖x1 − x2‖ ≤ r
r − r ′ ‖u1 − u2‖. (2.8)
Proof The (r, α)-prox regularity of the set C at x yields, according to (a) ofProposition 2.1, that
〈v1 − v2, x1 − x2〉 ≥ −1
r||x1 − x2||2 (2.9)
for all vi ∈ N P (C, xi ) ∩ B, and xi ∈ C ∩ B(x, α), i=1,2. Let xi ∈ ProjC (ui ), wheredC (ui ) < r ′. As r
r ′ (ui − xi ) ∈ N P (C, xi ) with ‖ rr ′ (ui − xi )‖ < r , in view of (2.9),
we obtain
⟨ r
r ′ (u1 − x1) − r
r ′ (u2 − x2), x1 − x2
⟩≥ −||x1 − x2||2.
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Locally prox-regular sets 251
Fig. 1 Examples of localquantified enlargements
Consequently, we have
〈u1 − u2, x1 − x2〉 − ||x1 − x2||2 ≥ −r ′
r||x1 − x2||2,
which means that
〈u1 − u2, x1 − x2〉 ≥(
1 − r ′
r
)||x1 − x2||2.
Applying Cauchy–Schwarz we conclude
||u1 − u2|| ≥(
1 − r ′
r
)||x1 − x2||,
and this can be written as
||x1 − x2|| ≤ r
r − r ′ ||u1 − u2||.
��For positive real numbers r, α and the closed set C of H with x ∈ C , the above
lemma leads to introduce the following local quantified enlargement RC (x, r, α) ofthe closed set C around x as
RC (x, r, α) := {x + tv : x ∈ C ∩ B(x, α), t ∈ [0, r [, v ∈ N P (C; x) ∩ B}.
Observe that the inclusion 0 ∈ N P (C; x) ∩ B yields
x ∈ RC (x, r, α). (2.10)
On Fig. 1, each shaded area describes the quantified enlargement RC (xi , ri , αi )
for i = 0, 1, 2, 3. Notice that the set C is (ri , αi )-prox-regular at xi for i = 1, 2 with
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252 M. Mazade, L. Thibault
r1 = ‖y1 − x1‖ and r2 = +∞. It is also (r ′3, α
′3)-prox-regular at x3 but for some
r ′3 < r3 and α′
3 < α3.The next proposition describes RC (x, r, α) as a restriction of the (global) open
r -enlargement set {u ∈ H : dC (u) < r} of C .
Proposition 2.2 If the closed set C is (r, α)-prox-regular at x , then
Proof If u ∈ RC (x, r, α), necessarily ProjC (u) ⊂ B(x, α), and dC (u) < r . Indeedu = x + tv for some x ∈ C ∩ B(x, α), t < r , and v ∈ N P (C, x)∩B. Then, the (r, α)-prox-regularity of C at x leads to x = projC (u). Consequently, ProjC (u) ⊂ B(x, α).Moreover dC (u) = ‖u − x‖ = ‖tv‖ < r . Conversely, if u ∈ H satisfies the propertiesof the right member of (2.11), let x ∈ ProjC (u) ∩ B(x, α). If dC (u) �= 0, then we canwrite u as
u = x + tv where t := dC (u) < r, and v := 1
t(u − x) ∈ N P (C; x) ∩ B,
which yields u ∈ RC (x, r, α). If dC (u) = 0, then choosing t = 0 and v = 0 leadsalso to u ∈ RC (x, r, α). ��
Proposition 2.3 below shows in particular that any element of the (global) openr -enlargement closest enough to x belongs to RC (x, r, α). The proposition providesalso a Lipschitz property on the set of such elements.
Proposition 2.3 Let C be closed and (r, α)-prox-regular at x . Define OC (r) := {u ∈H : dC (u) < r}. Then the following hold.
(a) For all 0 < r ′ < r, ProjC (·) is single valued and rr−r ′ -Lipschitz continuous on
B(x, α/2) ∩ OC (r ′).(b) For all 0 < r ′ ≤ r, B(x, α/2) ∩ OC (r ′) ⊂ RC (x, r ′, α).
Proof To prove (a), let ui ∈ Dom ProjC (·) ∩ B(x, α/2) ∩ OC (r ′), i = 1,2. Recall thatwe have
ProjC (ui ) = ProjC∩B(x,α)(ui ) because x ∈ C and ui ∈ B(x, α/2). (2.12)
Combining this with (2.8), we see that ProjC (·) is single valued and rr−r ′ -Lipschitz
continuous on B(x, α/2)∩OC (r ′)∩Dom ProjC (·). Take any u ∈ B(x, α/2)∩OC (r ′).Choose by Lau theorem [19, Theorem 4] (or Edelstein theorem [14]) a sequence
un ∈ Dom ProjC (·) converging to u with un ∈ B(x, α/2) ∩ OC (r ′). For each integern put pn := projC (un). The inequality ‖pn − pm‖ ≤ r
r−r ′ ‖un − um‖ ensures that(pn)n converges to some element p, which belongs to C according to the closednessof C . Writing for all x ∈ C
‖un − pn‖ ≤ ‖un − x‖ hence ‖u − p‖ ≤ ‖u − x‖
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Locally prox-regular sets 253
we see that p ∈ ProjC (u). Then u ∈ Dom ProjC (·). Therefore B(x, α/2) ∩ OC (r ′) ⊂B(x, α/2) ∩ OC (r ′) ∩ Dom ProjC (·), which entails that ProjC (·) is single valued and
As a direct consequence of what just precedes, according to (2.12) we can write
B(x, α/2) ∩ OC (r ′) = {u ∈ H : dC (u) < r ′ and u ∈ B(x, α/2)} ⊂ RC (x, r ′, α).
which is assertion (b), as desired. ��We establish now that RC (x, r, α) is open whenever the set C is (r, α)-prox-regular
at x .
Proposition 2.4 Let C be closed and (r, α)-prox-regular at x . Then for all 0 < r ′ ≤r, RC (x, r ′, α) is a nonempty open set.
Proof Observe that RC (x, r ′, α) is nonempty by (2.10). Fix u ∈ RC (x, r ′, α). Thenthere exists y ∈ B(x, α) ∩ C and v ∈ N P (C, y) ∩ B such that
u = y + t v where t ∈ [0, r ′[. (2.13)
Since y ∈ B(x, α) ∩ C , there exists η > 0 such that B[y, η] ⊂ B(x, α). Observealso that ‖u − y‖ < r ′ according to (2.13). By continuity of the norm we may chooseη′ > 0 such that
‖u − y‖ < r ′ for all u ∈ B(u, η′), y ∈ B(y, η′). (2.14)
Put η′′ := min{η, η′}. Since (dC (u)+‖u − u‖)2 −d2C (u) →u→u 0, consider a positive
real ε < η′′ such that for all u ∈ B(u, ε)
(1 − t
r
)−1 ((dC (u) + ε)2 − d2
C (u))
< (η′′)2. (2.15)
Fix any point u ∈ B(u, ε). Choose by Lau theorem ([19, Theorem 4]) a sequenceun ∈ Dom ProjC (·) converging to u and such that yn ∈ ProjC (un) exists for everyinteger n. Observing that
we see that limn→∞ ‖u − yn‖ = dC (u). Moreover for each n the equality
‖yn − y‖2 = ‖yn − u‖2 − ‖u − y‖2 + 2 〈u − y, yn − y〉
and the inclusion u − y ∈ N P (C, y) ensure according to (2.6) and to (2.13)
‖yn − y‖2 ≤ ‖yn − u‖2 − ‖u − y‖2 + t
r‖yn − y‖2
123
254 M. Mazade, L. Thibault
since C is (r, α)-prox-regular at x . So we get
(1 − t
r
)‖yn − y‖2 ≤ ‖yn − u‖2 − ‖u − y‖2
≤ (‖yn − u‖ + ‖u − u‖)2 − ‖u − y‖2
≤ (‖yn − u‖ + ‖u − u‖)2 − d2C (u).
The last member in the latter inequality converging to (dC (u) + ‖u − u‖)2 − d2C (u)
when n → ∞, we obtain on one hand by (2.15) that
‖yn − y‖ < η′′ for all n ≥ n1, for some n1 ∈ N
hence yn ∈ B(y, η′′) ∩ C ⊂ B(x, α) ∩ C , for all n ≥ n1. On the other hand, since un
converges to u and u ∈ B(u, ε), we may suppose that
un ∈ B(u, ε) for all n ≥ n1.
According to (2.14) we have also for all n ≥ n1
dC (un) ≤ ‖un − y‖ < r ′.
Applying Lemma 2.1 we get
‖yn − ym‖ ≤ r
r − r ′ ‖un − um‖ for all n ≥ n1
and this implies that (yn)n is a Cauchy sequence because un → u. Denoting by ythe limit of the sequence (yn)n , we get that y ∈ B[y, η′′] ∩ C ⊂ B(x, α) ∩ C and‖u − y‖ = limn→∞ ‖u − yn‖ = dC (u). When dC (u) > 0, we can write u as
u = y + ‖u − y‖ (u − y)
‖u − y‖ ∈ RC (x, r ′, α),
and the case dC (u) = 0 gives
u = y = y + t ′v′ where t ′ = 0 and v′ = 0.
So u ∈ RC (x, r ′, α). This concludes the openness of RC (x, r, α). ��Now let us recall the definitions of subdifferentials of functions. Let f : H → R ∪
{+∞} be an extended real-valued function and let x ∈ dom f , that is, f (x) < +∞.Each one of the above normal cones (see, e.g., [25,32]) leads to a subdifferentialthrough the normal cone to the epigraph epi f of f , where
epi f := { (x, ρ) ∈ H × R : f (x) ≤ ρ }.
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Locally prox-regular sets 255
So, the proximal subdifferential, the Fréchet subdifferential, the Mordukhovich limitingsubdifferential are the (possibly empty) subsets of H given by
∂? f (x) = { v ∈ H : (v,−1) ∈ N ? (epi f ; (x, f (x)))},
where “?” stands for P, F, L respectively. By convention anyone of the above sub-differentials of f at a point x �∈ dom f is empty. The proximal and Fréchet subdif-ferentials have amenable analytical descriptions. Indeed, for x ∈ dom f it is knownthrough (2.2) that v ∈ ∂P f (x) if and only if there exist γ ≥ 0 and η > 0 such that
〈v, x ′ − x〉 ≤ f (x ′) − f (x) + γ ||x ′ − x ||2 for all x ′ ∈ B(x, η).
Analogously, v ∈ ∂F f (x) if and only if for any real ε > 0 there exists some η > 0such that
〈v, x ′ − x〉 ≤ f (x ′) − f (x) + ε||x ′ − x || for all x ′ ∈ B(x, η).
Conversely, considering the indicator function δC of the set C given by δC (x ′) = 0if x ′ ∈ C and δC (x ′) = +∞ otherwise, the four normal cones to C at x ∈ C can berecovered from the concept of subdifferential by the following equalities
N ?(C; x) = ∂?δC (x),
where as above ? stands for P, F, L respectively. Concerning Proj(·, C) and the func-tion d2
C , we have the following properties on RC (x, r, α).
Proposition 2.5 Let C be closed and (r, α)-prox regular at x and let 0 < r ′ < r andlet RC (x, r, α) as defined above. Then the following properties hold.
(a) For all xi ∈ B(x, α) such that xi ∈ ProjC (ui ) with ui ∈ RC (x, r ′, α), one has
||x1 − x2|| ≤ r
r − r ′ ||u1 − u2|| (2.16)
and
〈u1 − u2, x1 − x2〉 ≥(
1 − r ′
r
)||x1 − x2||2. (2.17)
(b) ProjC (·) is single valued and monotone on RC (x, r, α), and Lipschitz continuouson RC (x, r ′, α) with r
r−r ′ as a Lipschitz constant.
(c) d2C is C1,1 on RC (x, r, α), that is ∇d2
C is locally Lipschitz continuous onRC (x, r, α), and ∇d2
C (x) = 2(x − projC (x)), for all x ∈ RC (x, r, α).(d) d2
C + rr−r ′ || · ||2 is convex on any open convex set included in RC (x, r ′, α).
Proof (a) Let 0 < r ′ < r and let xi ∈ B(x, α) such that xi ∈ ProjC (ui ) withui ∈ RC (x, r ′, α). By definition of RC (x, r ′, α), we have ui = x ′
i + tivi for somex ′
i ∈ C ∩ B(x, α), ti < r ′, and vi ∈ N P (C, x ′i ) ∩ B. As C is (r, α)-prox-regular
at x , one has x ′i = projC (ui ), hence xi = x ′
i , since xi ∈ ProjC (ui ) by assumption.
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256 M. Mazade, L. Thibault
Besides, ui − xi = tivi , with dC (ui ) = ||(ui − xi )|| ≤ ti < r ′ hence the vector(ui − xi ) is a proximal normal to C at xi ∈ B(x, α) ∩ C . According to Lemma2.1, we obtain
||x1−x2||≤ r
r −r ′ ||u1−u2|| and 〈u1−u2, x1−x2〉 ≥(
1 − r ′
r
)||x1 − x2||2.
(b) Let O ⊂ RC (x, r ′, α) be an open convex subset of RC (x, r ′, α). Suppose that theFréchet subdifferential of d2
C is nonempty at u1 and u2 where ui ∈ O, i = 1, 2.From [2], we know that ProjC (ui ) is nonempty valued and actually single valued,according to a).
Putting xi = projC (ui ), we deduce from [13] that
∂F (d2C )(ui ) ⊂ ∂FδC (xi ) ∩ {2(ui − xi )},
namely ∂F (d2C )(ui ) = 2(ui − xi ) with 2(ui − xi ) ∈ ∂FδC (xi ) = N F (C; xi ). From
r−r ′ || · ||2 is convex on O, hence ∂F d2C (u) = ∂Ld2
C (u), for allu ∈ O. This implies that ∂F d2
C is nonempty-valued on O and so does ProjC . Part (a)allows us to conclude that ProjC is single valued, monotone and Lipschitz continuouson RC (x, r ′, α), with r
r−r ′ as a Lipschitz constant, for all 0 < r ′ < r . As d2C + r ′
r−r ′ ||·||2is convex on any convex subset of RC (x, r ′, α) and ∂F d2
C (x) = 2(x − projC (x)), forall x ∈ RC (x, r ′, α), 0 < r ′ < r , we conclude that d2
C is C1,1 on RC (x, r, α). ��
3 Uniqueness and existence of a solution of (P) by regularization
We start this section by stating the theorem on the regularization of the differentialvariational inequality (P). Here the mapping f will be assumed to be Bochner measur-able with respect to t . Following [38, p. 130], a mapping ϕ from an interval I into H isBochner measurable (called also strongly measurable in [38]) whenever ϕ is the limitalmost everywhere of a sequence of simple mappings ϕn , i.e., ϕn = ∑kn
i=1 ai,n1Ai,n
where ai,n ∈ H and Ai,n is a Lebesgue measurable subset of I .
Theorem 3.1 Assume that the closed set C is (r, α)-prox-regular at u0 ∈ C. Fix apositive real r ′ < r and let η := min{α/2, r ′}. Let f : [T0, T ′] × B(u0, η) → H bea mapping which is Bochner measurable with respect to t ∈ [T0, T ′] and such that:
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Locally prox-regular sets 257
(i) - there exists K > 0 such that for all t ∈ [T0, T ′],
‖ f (t, u0)‖ ≤ K ;
(ii) - there exists a non-negative real number k such that for all t ∈ [T0, T ′] and forall (x, y) ∈ B(u0, η) × B(u0, η),
|| f (t, x) − f (t, y)|| ≤ k||x − y||.
Let β := K + 2ηk and T ∈]T0, T ′] such that β(T − T0) < η/2. Under theseassumptions, for any λ > 0, the differential equation over [T0, T ] × B(u0, η)
{uλ(t) = (−1/2λ)∇d2
C (uλ(t)) + f (t, uλ(t))uλ(T0) = u0
(Eλ)
is well defined and has a unique solution uλ(·) on [T0, T ], and the family(uλ(·))0<λ<r/(2β) converges uniformly on [T0, T ] as λ ↓ 0 to a solution of the dif-ferential variational inequality (P). Further, this solution stays in B(u0, η) and thesolution inside this ball is unique.
The proof will be achieved through several lemmas. So, suppose in the rest of thepaper:
⎧⎪⎪⎨⎪⎪⎩
− C is (r, α)−prox − regular at u0 ∈C;− Assumptions (i) and (ii) of Theorem 3.1 are fulfilled
for r′ ∈ ]0, r [ fixed and η := min{α/2, r′};−β and T ∈ ]T0, T′] are as in the statement of Theorem 3.1.
Set
c := r
r − r ′ .
We first observe that the assumptions of the theorem above imply that
‖ f (t, x)‖ ≤ β, for all (t, x) ∈ [T0, T ] × B(u0, η). (3.1)
We also observe that B(u0, η) ⊂ RC (u0, r ′, α). Indeed, if u ∈ B(u0, η), we havedC (u) ≤ ‖u − u0‖ < r ′. So, by (b) of Proposition 2.3 we obtain
B(u0, η) ⊂ {u ∈ H : dC (u) < r ′ and u ∈ B(u0, α/2)} ⊂ RC (u0, r ′, α).
Propositions 2.5 and 2.3 then ensure that projC is well defined on the whole ballB(u0, η) and c-Lipschitz continuous on B(u0, η) with (1/2)∇d2
C (x) = x − projC (x).
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258 M. Mazade, L. Thibault
Further, the mapping (1/2)∇d2C (·) is (1 + r
(r−r ′) )-Lipschitz continuous on B(u0, η).Indeed, for u1, u2 ∈ B(u0, η) we have according to (2.16)
Since η < r and η < α, with C(r, α)-prox-regular at u0, according to (2.11), we alsohave for all x ∈ B(u0, η) that projC (x) ∈ B(u0, α).
The following lemma establishes a preparatory equality concerning the derivativeof the distance function from an absolutely continuous mapping to the set C .
Lemma 3.1 Let z : [T0, T ′[→ B(u0, η) be an absolutely continuous mapping andlet g(t) := d(z(t), C), for all t ∈ [T0, T ′]. Then for a.e. t ∈ [T0, T ′]
g(t)g(t) = 〈z(t) − projC (z(t)), z(t)〉.
Proof Put ϕ(t) := (1/2)d2C (z(t)), where z : [T0, T ′] → B(u0, η). The function g is
absolutely continuous since d(·, C) is 1-Lipschitz continuous. Fix t ∈ ]T0, T ′[ suchthat g and z are derivable at t . Then we get for s ∈ ]0, T ′ − t[ small enough,
1
2s
(g2(t + s) − g2(t)
)= 1
s
(ϕ(t + s) − ϕ(t)
).
Since ϕ and z are derivable at t and d2C is Fréchet differentiable at z(t) ∈ B(u0, η) ⊂
RC (u0, r ′, α) according to (c) of Proposition 2.5 we have
lims↓0
1
s
(ϕ(t + s) − ϕ(t)
)= ϕ′(t) = 〈∇((1/2)d2
C (z(t))), z(t)〉.
By the relation (see Proposition 2.5 again)
∇((1/2)d2C (x)) = x − projC (x) for allx ∈ B(u0, η)
we then obtain
g(t)g(t) = ∇(
1
2g2
)(t) = 〈∇((1/2)d2
C (z(t))), z(t)〉 = 〈z(t) − projC (z(t)), z(t)〉.��
By the assumptions on the set C and the mapping f , the mapping(−1/2λ)∇d2
C (·) + f (t, ·) is Lipschitz continuous on B(u0, η). For any real λ > 0,consider on B(u0, η) the differential equation
{uλ(t) = (−1/2λ)∇d2
C (uλ(t)) + f (t, uλ(t))uλ(T0) = u0.
(E ′λ)
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Locally prox-regular sets 259
Let
Tλ := sup
{τ ∈ [T0, T ] : (E ′
λ) has a solution uλ(·) on [T0, τ [withuλ([T0, τ [) ⊂ B(u0, η)
}. (3.2)
By the Lipschitz continuity property on B(u0, η) of the mapping(−1/2λ)∇d2
C (·)+ f (t, ·) and by assumption (i i) of Theorem 3.1 the set whose supre-mum is taken in (3.2) is nonempty, and on [T0, Tλ[ the differential equation (E ′
In the following, we consider the solution uλ : [T0, Tλ[ → B(u0, η) of (E ′λ) and
we set
zλ(t) := f (t, uλ(t)). (3.3)
Then by (b) of Proposition 2.5
uλ(t) = −1
λ
(uλ(t) − projC (uλ(t))
) + zλ(t), a.e. t ∈ [T0, Tλ[. (3.4)
We also recall the following form of Gronwall’s lemma for absolutely continuoussolutions of differential inequalities which directly follows from [4, Lemma A.4], bytaking there as functions φ, m, and constant a:
φ(t) := ζ(t) −t∫
t0
b(s) exp
⎛⎝
t∫s
c(τ ) dτ
⎞⎠ ds, m(t) := c(t) and a := ζ(t0).
Lemma 3.2 (Gronwall’s lemma) Let b, c, ζ : [t0, t1] → [0,+∞[ be three real-valued Lebesgue integrable functions. If the function ζ(·) is absolutely continuous onthe interval [t0, t1] and if for almost all t ∈ [t0, t1]
ζ (t) ≤ b(t) + c(t)ζ(t),
then for all t ∈ [t0, t1]
ζ(t) ≤ ζ(t0) exp
⎛⎝
t∫t0
c(τ ) dτ
⎞⎠ +
t∫t0
b(s) exp
⎛⎝
t∫s
c(τ ) dτ
⎞⎠ ds.
Lemma 3.3 Set gλ(t) := d(uλ(t), C) for any t ∈ [T0, Tλ[. Then gλ is locallyabsolutely continuous on [T0, Tλ[ and
gλ(t) ≤ −1
λgλ(t) + β, a.e. t ∈ [T0, Tλ[.
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260 M. Mazade, L. Thibault
Further for all t ∈ [T0, Tλ[ one has
gλ(t) ≤ βe−t/λ
t∫T0
es/λ ds.
Proof Consider t ∈ ]T0, Tλ[ where gλ(t) and uλ(t) exist and satisfy (E ′λ). Taking
z(t) = uλ(t) in Lemma 3.1, as uλ(t) ∈ B(u0, η), we obtain
gλ(t)gλ(t) = 〈uλ(t), uλ(t) − projC (uλ(t))〉= −1
λ〈uλ(t) − projC (uλ(t)), uλ(t) − projC (uλ(t))〉
+ 〈zλ(t), uλ(t) − projC (uλ(t))〉,
the second equality being due to the equality uλ(t) = − 1λ(uλ(t)−projC (uλ(t)))+zλ(t)
according to (3.4). Since gλ(t) = ||uλ(t) − projC (uλ(t))|| = d(uλ(t), C) we deduce
gλ(t)gλ(t) = −1
λg2λ(t) + 〈zλ(t), uλ(t) − projC (uλ(t))〉
≤ −1
λg2λ(t) + ‖zλ(t)‖‖uλ(t) − projC (uλ(t))‖
= −1
λg2λ(t) + gλ(t)||zλ(t)||.
Then using (3.1) and (3.3) we get
gλ(t) ≤ −1
λgλ(t) + β, if gλ(t) > 0.
The last inequality is still valid whenever gλ(t) = 0 because it leads togλ(t) = 0. Indeed the equality gλ(t) = 0 ensures for |s| small enough that
1
s
[gλ(t + s) − gλ(t)
]= 1
sdC (uλ(t + s)),
hence, since gλ(t) exists, one deduces that
gλ(t)= lims↑0
1
sdC (uλ(t + s))≤0 and gλ(t)= lim
s↓0
1
sdC (uλ(t + s))≥0, so gλ(t)=0.
Thus, for almost every t ∈ [T0, Tλ[
gλ(t) ≤ −1
λgλ(t) + β,
which is the first inequality of our lemma. Further, observing that gλ(T0) =d(uλ(T0), C) = d(u0, C) = 0 and applying Lemma 3.2 with b(·) = β and c(·) = − 1
λ,
we also obtain for all t ∈ [T0, Tλ[ that
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Locally prox-regular sets 261
gλ(t) ≤ βe−t/λ
t∫T0
es/λ ds.
��We prove in the next lemma that the derivative of the solution uλ(·) of (E ′
λ) isbounded, the upper bound being independent of t .
Lemma 3.4 For almost every t ∈ [T0, Tλ[, one has:
||uλ(t) − zλ(t)|| = (1/λ)gλ(t) ≤ (β/λ)e−t/λ
t∫T0
es/λ ds
and
||uλ(t) − zλ(t)|| ≤ β.
Consequently
||uλ(t)|| ≤ 2β.
Proof As uλ(·) is a solution of (E ′λ), one has
uλ(t) = −1
λ(uλ(t) − projC (uλ(t))) + zλ(t).
So
uλ(t) − zλ(t) = −1
λ
(uλ(t) − projC (uλ(t))
).
This yields
||uλ(t) − zλ(t)|| = 1
λ
∣∣∣∣∣∣∣∣(uλ(t) − projC (uλ(t)))
∣∣∣∣∣∣∣∣ = (1/λ)gλ(t).
Applying Lemma 3.3, we obtain
||uλ(t) − zλ(t)|| ≤ (β/λ)e−t/λ
t∫T0
es/λ ds
= e−t/λ β
λ[λet/λ − λeT0/λ]
= β[1 − e(T0−t)/λ] ≤ β.
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262 M. Mazade, L. Thibault
To obtain the last inequality in the lemma, it suffices to note that
||uλ(t)|| ≤ ||uλ(t) − zλ(t)|| + ||zλ(t)|| ≤ 2β.
��The last inequality of Lemma 3.4 tells us that uλ(·) is Lipschitz continuous on
[T0, Tλ[ with 2β as Lipschitz constant. We deduce that on one hand
limt↑Tλ
uλ(t) exists in H,
and we can extend uλ(·) into a (2β)-Lipschitz mapping uλ(·) on [T0, Tλ] by puttinguλ(Tλ) = limt↑Tλ uλ(t) and uλ(t) = uλ(t) for every t ∈ [T0, Tλ[.
On the other hand because β(T − T0) < η/2 we have
||uλ(Tλ) − u0|| ≤ 2β(Tλ − T0) ≤ 2β(T − T0) < η,
and hence uλ(·) is a solution over [T0, Tλ] of (E ′λ) with uλ([T0, Tλ]) ⊂ B(u0, η).
Further Tλ = T , otherwise if Tλ < T with the last estimation ||uλ(Tλ) − u0|| < η,one could extend uλ on the right of Tλ in a solution uλ on some interval open onthe right [T0, τ [ with τ ∈ ]Tλ, T ] and uλ([T0, Tλ]) ⊂ B(u0, η). This would be incontradiction with the definition of Tλ.
Therefore, for any λ > 0, the differential equation on [T0, T ] defined by
{uλ(t) = (−1/2λ)∇d2
C (uλ(t)) + f (t, uλ(t))uλ(T0) = u0
(Eλ)
has a unique solution that we denote uλ(·) ( in place of uλ(·))) on [T0, T ] withuλ([T0, T ]) ⊂ B(u0, η) (here the equation is denoted (Eλ) instead of (E ′
λ) aboveto emphasize that the interval on which the equation is considered is [T0, T ] indepen-dent of λ). The next step is to make sure that (uλ)λ>0 satisfies the Cauchy criterion asλ ↓ 0. Recall that c = r
r−r ′ .
Lemma 3.5 For all positive numbers λ,μ < r/2β, one has for all t ∈ [T0, T ]
||uλ(t) − uμ(t)||2 ≤ 2(λ + μ)β2
t∫T0
exp(
2(k + βc2/r)(t − s))
ds.
Proof Recall that projC (·) is well defined and c-Lipschitz continuous on B(u0, η) andthat the differential equation (Eλ) entails (see (3.4))
projC (uλ(t)) = λ(uλ(t) − zλ(t)) + uλ(t) where we recall that zλ(t) = f (t, uλ(t)).
Thus by the inequality ‖uλ(t) − zλ(t)‖ ≤ β in Lemma 3.4 we get for a.e. t ∈ [T0, T ]r
β
(− uλ(t) + zλ(t)
)∈ NC (projC (uλ(t))) and
∥∥∥∥ r
β
(− uλ(t) + zλ(t)
)∥∥∥∥ ≤ r.
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Locally prox-regular sets 263
Since uλ(t) ∈ B(u0, η) ⊂ RC (u0, r, α), description (2.11) ensures that projC (uλ(t)) ∈B(u0, α). According to the hypomonotonicity property of N (C; ·) (see (a) in Propo-sition 2.1) for a.e. t ∈ [T0, T ]
By Lemma 3.4, we know that ||uλ(t) − zλ(t)|| ≤ β and ||uμ(t) − zμ(t)|| ≤ β.Moreover, since uλ(·), uμ(·) ∈ B(u0, η) and f (t, ·) is k-Lipschitz continuous onB(u0, η), we can write
||zλ(t) − zμ(t)|| = || f (t, uλ(t)) − f (t, uμ(t))||≤ k||uλ(t) − uμ(t)||.
Hence we obtain
1
2
d
dt
[||uλ(t) − uμ(t)||2
]≤
(βc2
r+ k
)||uλ(t) − uμ(t)||2 + (λ + μ)β2.
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264 M. Mazade, L. Thibault
Since uλ(T0)−uμ(T0) = 0, according to Lemma 3.2, we see that for m := 2(
βc2
r + k)
we have
||uλ(t) − uμ(t)||2 ≤ 2(λ + μ)β2
t∫T0
em(t−s) ds,
which is the desired inequality of the lemma. ��Once the uniform Cauchy criterion has been established, we can study the conver-
gence of (uλ)λ>0.
Lemma 3.6 The family (uλ)0<λ<r/2β converges uniformly on [T0, T ] as λ ↓ 0 to asolution of
{−u(t) + f (t, u(t)) ∈ N (C; u(t)), a.e. t ∈ [T0, T ]u(T0) = u0.
Proof By the inequality of Lemma 3.5, for all λ,μ < r/2β, the family (uλ) clearlyverifies the uniform Cauchy criterion, thus (uλ(·)) converges uniformly on [T0, T ] toa continuous mapping u(·) as λ ↓ 0. Further by the second inequality in Lemma 3.3we have
d(uλ(t), C) = gλ(t) ≤ βe−t/λ
t∫T0
es/λds = βe−t/λλ(et/λ − eT0/λ) ≤ βλ
and hence (by the closedness of C)
u(t) ∈ C for all t ∈ [T0, T ]. (3.5)
As we already said before, (2.11) ensures that projC (uλ(t)) ∈ B(u0, α).Besides,
−uλ(t) + zλ(t) ∈ N (C; projC (uλ(t)))
by (c) of Proposition 2.5, then by (2.6) and the second inequality of Lemma 3.4 weget
⟨zλ(t) − uλ(t), x ′ − projC (uλ(t))
⟩ ≤ β
2r
∣∣∣∣∣∣∣∣x ′ − projC (uλ(t))
∣∣∣∣∣∣∣∣2
, for all x ′ ∈ C.
(3.6)
Lemma 3.4 again ensures that ||uλ(t)|| ≤ 2β, a.e. t ∈ [T0, T ], then we can extract asequence (λn), λn ↓ 0, such that uλn (·) converges weakly in L1([T0, T ], H) to somemapping g(·) ∈ L1([T0, T ], H). So for any t ∈ [T0, T ], fixing any z ∈ H and writing
123
Locally prox-regular sets 265
⟨z,
t∫T0
uλn (s) ds〉 =T∫
T0
〈z1[T0,t](s), uλn (s)
⟩ds
we see that
t∫T0
uλn (s)ds →t∫
T0
g(s)ds weakly in H.
As (uλn (t)) converges strongly in H to u(t), it results that
u(t) = u0 +t∫
T0
g(s)ds.
Consequently, u(·) is absolutely continuous with u(t) = g(t), for almost all t and
uλn (·) → u(·), weakly in L1([T0, T ], H).
Set z(t) := f (t, u(t)) and I = [T0, T ]. Then we get
Applying Mazur’s lemma, there exist for each n ∈ N some integer r(n) > n and real
numbers sk,n ≥ 0 with∑r(n)
k=n sk,n = 1, such that(∑r(n)
k=n sk,n(zλk − uλk ))
converges
strongly to z(·) − u(·) in L1([T0, T ], H). Extracting a subsequence, we may supposethat, for some negligible N ⊂ I , the derivatives u(t) and uλn (t) exist for all t ∈ I\Nand that
r(n)∑k=n
sk,n(zλk (t) − uλk (t)) → z(t) − u(t), for all t ∈ I \ N .
We may also suppose that the inequalities in Lemma 3.4 hold for all t ∈ I\N and allλn with n ∈ N. Fix any t ∈ I\N . First we have by Lemma 3.4 the estimation
∣∣∣∣∣∣r(n)∑k=n
sk,n⟨zλk (t) − uλk (t), u(t) − projC (uλ(t))
⟩∣∣∣∣∣∣
≤ β
r(n)∑k=n
sk,n||u(t) − projC (uλk (t))||. (3.7)
123
266 M. Mazade, L. Thibault
Since projC (·) is Lispchitz continuous on B(u0, η), and u(t) ∈ C by (3.5), we see that
and since projC (uλn (t)) → u(t) as n → ∞, using (3.10) it follows that
⟨z(t) − u(t), x ′ − u(t)
⟩ ≤ β
2r||x ′ − u(t)||2 for all x ′ ∈ C,
which entails
− u(t) + z(t) ∈ N P (C; u(t)). (3.11)
The inclusion being true for all t ∈ I \ N , the proof is complete. ��Remaining of the proof of Theorem 3.1 According to Lemma 3.6 it remains to estab-lish the uniqueness to conclude the Proof of Theorem 3.1. To do so, we prove firstthat
‖u(t) − z(t)‖ ≤ ‖z(t)‖ a.e. t ∈ [T0, T ], (3.12)
123
Locally prox-regular sets 267
where we recall that z(t) = f (t, u(t)). Indeed, put φ(τ) := ∫ τ
T0z(s) ds and fix any
t ∈ [T0, T ] where −u(t)+ z(t) �= 0 and where φ(t) exists with φ(t) = z(t). Note that
−u(t) + z(t)
‖ − u(t) + z(t)‖ ∈ N P (C; u(t)). (3.13)
By definition of proximal normal cone, there exists some σt > 0 such that for s < t
⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , u(s) − u(t)
⟩≤ σt ||u(s) − u(t)||2,
(note that u(s) ∈ C by (3.5)). Then writing
⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , u(s) − u(t)
⟩
=⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , u(s) − φ(s)−(u(t)−φ(t))
⟩+⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , φ(s)−φ(t)
⟩,
we get
⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , u(s) − φ(s) − (u(t) − φ(t))
⟩
≤⟨ −u(t) + z(t)
|| − u(t) + z(t)|| , φ(t) − φ(s)
⟩+ σt ||u(s) − u(t)||2.
Dividing by t − s, the latter inequality gives
⟨ −u(t) + z(t)
|| − u(t) + z(t)|| ,u(s) − φ(s) − (u(t) − φ(t))
t − s
⟩
≤⟨ −u(t) + z(t)
|| − u(t) + z(t)|| ,φ(t) − φ(s)
t − s
⟩+ σt
||u(s) − u(t)||2t − s
so
⟨u(t) − z(t)
||u(t) − z(t)|| ,u(s) − φ(s) − (u(t) − φ(t))
s − t
⟩
≤∥∥∥∥φ(s) − φ(t)
s − t
∥∥∥∥ + σt
∥∥∥∥u(s) − u(t)
s − t
∥∥∥∥ · ‖u(s) − u(t)‖.
Taking the limit for s ↑ t we obtain
⟨u(t) − z(t)
||u(t) − z(t)|| , u(t) − z(t)
⟩≤ ‖z(t)‖, i.e., ||u(t) − z(t)|| ≤ ||z(t)||.
123
268 M. Mazade, L. Thibault
This inequality being also true for z(t) − u(t) = 0, it follows that
||u(t) − z(t)|| ≤ ||z(t)|| a.e. t ∈ [T0, T ].
Consider now two solutions u1(·) and u2(·) with ui : [T0, T ] → B(u0, η). Thehypomonotonicity property of the normal cone on C ∩ B(u0, η) (see Proposition 2.1)yields for almost all t ∈ [T0, T ]
〈u1(t) − f (t, u1(t)) − u2(t) + f (t, u2(t)), u1(t) − u2(t)〉≤ 1
r
(|| f (t, u1(t))|| + || f (t, u2(t))||
)||u1(t) − u2(t)||2.
As || f (t, u(t))|| ≤ β (see (3.1)), we obtain
〈u1(t) − u2(t), u1(t) − u2(t)〉 ≤ 2β
r||u1(t) − u2(t)||2
+〈 f (t, u1(t)) − f (t, u2(t)), u1(t) − u2(t)〉.
By assumption, f (t, ·) is k-Lipschitz continuous on B(u0, η), so
1
2
d
dt
(||u1(t) − u2(t)||2
)≤ 1
r
(2β + rk
)‖u1(t) − u2(t)‖2.
According to Gronwall’s lemma (see Lemma 3.2), it follows that u1(·) = u2(·) on[T0, T ]. The proof of the theorem is then complete. ��Acknowledgments We thank both referees for their careful reading which allows us to improve thepresentation of the paper.
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