Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 37, 1-9; http://www.math.u-szeged.hu/ejqtde/

Mixed semicontinuous perturbation of a second order nonconvex sweeping

process

Dalila Azzam-Laouir

Laboratoire de Mathématiques Pures et Appliquées

Université de Jijel, Algérie

Email: azzam_{−}d@yahoo.com

Abstract

We prove a theorem on the existence of solutions of a second order differential inclusion governed by a class of nonconvex sweeping process with a mixed semicontinuous perturbation.

Keywords and phrases: Differential inclusions; Fixed-point theorem; Mixed semicontinuity; Sweeping process.

2000 Mathematics Subject Classification: 34A60, 34B15, 47H10.

1 Introduction

The existence of solutions for the second order differential inclusions governed by the sweeping process

(P_{F}) \begin{cases}−ü(t) ∈ N_{K(u(t))}(u̇(t)) + F (t, u(t), u̇(t)), a.e. t ∈ [0, T], \\ u̇(t) ∈ K(u(t)), \\ u(0) = u_{0}; u̇(0) = v_{0}, \end{cases}

where N_{K(u(t))}(·) denotes the normal cone to K(u(t)), has been thoroughly studied (when the sets K(x) are convex or nonconvex) by Castaing for the first time when F ≡ {0} see [5], and later by many other authors see for example [2], [3], [6], [10] and [12]. Note that in this literature some existence results are established for the problem (P_{F}) with lower and upper semicontinuous perturbations.

Our aim in this paper is to prove existence results for (P_{F}) when F is a mixed semicontinuous set-valued map and K(x) are nonconvex sets. For the first order sweeping process with a mixed semicontinuous perturbation we refer the reader to [9], and to [13] for the sweeping process with non regular sets and to [1] for second order differential inclusions with mixed semicontinuous perturbations.

After some preliminaries, we present our main result in the finite dimensional space H whenever the sets K(x) are uniformly ρ-prox-regular (ρ > 0) and the set-valued mapping F is mixed semicontinuous, that is, F (·, ·, ·) is measurable and for every t ∈ [0, T], at each (x, y) ∈ H × H where F (t, x, y) is convex the set-valued map F (t, ·, ·) is upper semicontinuous, and whenever F (t, x, y) is not convex F (t, ·, ·) is lower semicontinuous on some neighborhood of (x, y).

2 Definition and preliminary results

Let H be a real Hilbert space and let S be a nonempty closed subset of H. We denote by d(·, S) the usual distance function associated with S, i.e., d(u, S) := inf_{y∈S} ∥u − y∥. For

any x ∈ H and r ≥ 0 the closed ball centered at x with radius r will be denoted by \overline{B}_{H}(x, r). For x = 0 and r = 1 we will put \overline{B}_{H} in place of \overline{B}_{H}(0, 1). L([0, T]) is the σ-algebra of Lebesgue-measurable sets of [0, T] and B(H) is the σ-algebra of Borel subsets of H. By L_{H}^{1}([0, T]) we denote the space of all Lebesgue-Bochner integrable H-valued mappings defined on [0, T] and by C_{H}([0, T]) the Banach space of all continuous mappings u : [0, 1] → H, endowed with the sup norm We need first to recall some notation and definitions that will be used in all the paper. Let x be a point in S. We recall ( see [8]) that the proximal normal cone to S at x is defined by N_{S}^{P} (x) := ∂^{P} ψ_{S}(x), where ψ_{S} denotes the indicator function of S, i.e., ψ_{S}(x) = 0 if x ∈ S and +∞ otherwise. Note that the proximal normal cone is also given by

N_{S}^{P} (x) = {ξ ∈ H : ∃α > 0 s.t. x ∈ Proj_{S}(x + αξ)},

where

Proj_{S}(u) := {y ∈ S : d(u, S) := ∥u − y∥}.

Recall now that for a given ρ ∈]0, +∞] the subset S is uniformly ρ-prox-regular (see [11]) or equivalently ρ -proximally smooth ( see [8]) if and only if every nonzero proximal normal to S can be realized by ρ-ball, this means that for all \overline{x} ∈ S and all 0 ≠ ξ ∈ N_{S}^{P} (\overline{x}) one has

⟨\frac{ξ}{∥ξ∥}, x − \overline{x}⟩ ≤ \frac{1}{2ρ} ∥x − \overline{x}∥^{2} ,

for all x ∈ S. We make the convention \frac{1}{ρ} = 0 for ρ = +∞. Recall that for ρ = +∞ the uniform ρ-prox-regularity of S is equivalent to the convexity of S. The following proposition summarizes some important consequences of the uniform prox-regularity needed in the sequel. For the proof of these results we refer the reader to [11].

Proposition 2.1 Let S be a nonempty closed subset in H and x ∈ S. The following assertions hold:

(1) ∂^{P} d(x, S) = N_{S}^{P} (x) ∩ \overline{B}_{H};

(2) let ρ ∈]0, +∞]. If S is uniformly ρ-prox-regular, then (2.1) for all x ∈ H with d(x, S) < ρ; one has Proj_{S}(x) ≠ ∅; (2.2) the proximal subdifferential of d(·, S) coincides with its Clarke subdifferential at all points x ∈ H satisfying d(x, S) < ρ. So, in such a case, the subdiferential ∂d(x, S) := ∂^{P} d(x, S) = ∂^{C}d(x, S) is a closed convex set in H; (2.3) for all x_{i} ∈ S and all v_{i} ∈ N_{S}^{P} (x_{i}) with ∥v_{i}∥ ≤ ρ (i = 1, 2) one has

⟨v_{1} − v_{2}, x_{1} − x_{2}⟩ ≥ − ∥x_{1} − x_{2}∥^{2} .

As a consequence of (2.3) we get that for uniformly ρ-prox-regular sets, the proximal normal cone to S coincides with all the normal cones contained in the Clarke normal cone at all points x ∈ S, i.e., N_{S}^{P} (x) = N_{S}^{C}(x). In such a case, we put N_{S}(x) := N_{S}^{P} (x) = N_{S}^{C}(x). Here and above ∂^{C}d(x, S) and N_{S}^{C}(x) denote respectively the Clarke subdifferential of d(·, S) and the Clarke normal cone to S (see [8]).

Now, we recall some preliminaries concerning set-valued mappings. Let T > 0. Let C : [0, T] ⇉ H and K : H ⇉ H be two set-valued mappings. We say that C is absolutely continuous provided that there exists an absolutely continuous nonnegative function a : [0, T] → ℝ_{+} with a(0) = 0 such that

|d(x, C(t)) − d(y, C(s))| ≤ ∥x − y∥ + |a(t) − a(s)|

for all x, y ∈ H and all s, t ∈ [0, T]. We will say that K is Hausdorff-continuous (resp. Lipschitz with ratio λ > 0) if for any x ∈ H one has

lim_{x′→x} H(K(x), K(x′)) = 0

(resp. if for any x, x′ ∈ H one has

H(K(x), K(x′)) ≤ λ∥x − x′∥.)

We close this section with the following theorem in [4], which is an important closedness property of the subdifferential of the distance function associated with a set-valued mapping.

Theorem 2.1 Let ρ ∈]0, +∞], Ω be an open subset in H, and K : Ω ⇉ H be a Hausdorffcontinuous set-valued mapping. Assume that K(z) is uniformly ρ-prox-regular for all z ∈ Ω. Then for a given 0 < δ < ρ, the following holds: ”for any \overline{z} ∈ Ω, \overline{x} ∈ K(\overline{z}) + (ρ − δ)\overline{B}_{H}, x_{n} → \overline{x}, z_{n} → \overline{z} with z_{n} ∈ Ω (x_{n} not necessarily in K(z_{n})) and ξ_{n} ∈ ∂d(x_{n}, K(z_{n})) with ξ_{n} →^{w} \overline{ξ} one has \overline{ξ} ∈ ∂d(\overline{x}, K(\overline{z})).” Here →^{w} means the weak convergence in H.

Remark 2.1 As a direct consequence of this theorem, we have for every ρ ∈]0, +∞], for a given 0 < δ < ρ, and for every set-valued mapping K : Ω ⇉ H with uniformly ρprox regular values, the set-valued mapping (z, x) ↦ ∂d(x, K(z))) is upper semicontinuous from {(z, x) ∈ Ω × H : x ∈ K(z) + (ρ − δ)} to H endowed with the weak topology, which is equivalent to the upper semicontinuity of the function (z, x) ↦ σ(∂d(x, K(z))), p) on {(z, x) ∈ Ω × H : x ∈ K(z) + (ρ − δ)}, for any p ∈ H. Here σ(S, p) denotes the support function to S defined by σ(S, p) = sup_{s∈S}⟨s, p⟩.

3 Existence results under mixed semicontinuous perturbation.

Our existence result is stated in a finite dimensional space H under the following assumptions. (H_{1}) For each x ∈ H, K(x) is a nonempty closed subset in H and uniformly ρ-prox-regular for some fixed ρ ∈]0, +∞]; (H_{2}) K is Lipschitz with ratio λ > 0; (H_{3}) l = sup_{x∈H} |K(x)| < +∞.

The proof of our main theorem uses existence results for the first order sweeping process, the selection theorem proved in Tolstonogov [14] and the Kakutani fixed point theorem for set-valued mappings. We begin by recalling them.

Proposition 3.1 (Proposition 1.1 in [7]) Let H be a finite dimensional space, T > 0 and let C : I := [0, T] ⇉ H be a nonempty closed valued set-valued mapping satisfying the following assumptions. (A_{1}) For each t ∈ I, C(t) is ρ-prox-regular for some fixed ρ ∈]0, +∞]; (A_{2}) C(t) varies in an absolutely continuous way, that is, there exists a nonnegative absolutely continuous function v : I → ℝ such that

|d(x, C(t)) − d(y, C(s))| ≤ ∥x − y∥ + |v(t) − v(s)|

for all x, y ∈ H and s, t ∈ I. Then for any mapping h ∈ L_{H}^{1}([0, T]), the differential inclusion

\begin{cases}−u̇(t) ∈ N_{C(t)}(u(t)) + h(t), a.e. t ∈ [0, T], \\ u(0) = u_{0} ∈ C(0) \end{cases}

admits one and only one absolutely continuous solution u(·) and

∥u̇(t) + h(t)∥ ≤ |v̇(t)| + ∥h(t)∥.

Further, let m be a nonnegative Lebesgue-integrable function defined on [0, T] and let

K = {h ∈ L_{H}^{1}([0, T]) : ∥h(t)∥ ≤ m(t) a.e}.

Then the solutions set {u_{h} : h ∈ K}, where u_{h} is the unique absolutely continuous solution of the above inclusion, is compact in C_{H}([0, T]), and the mapping h ↦ u_{h} is continuous on K when K is endowed with the weak topology w(L_{H}^{1}([0, T]), L_{H}^{∞}([0, T])).

For the proof of our theorem we will also need the following theorem which is a direct consequence of Theorem 2.1 in [14].

Theorem 3.1 Let H be a finite dimensional space and let M : [0, T] × H × H ⇉ H be a closed valued set-valued mapping satisfying the following hypotheses.

(i) M is L([0, T]) ⊗ B(H) ⊗ B(H)-measurable; (ii) for every t ∈ [0, 1], at each (x, y) ∈ H × H such that M(t, x, y) is convex, M(t, ·, ·) is upper semicontinuous, and whenever M(t, x, y) is not convex, M(t, ·, ·) is lower semicontinuous on some neighborhood of (x, y); (iii) there exists a Caratheodory function ζ : [0, 1] × H × H → ℝ_{+} which is integrably bounded and such that M(t, x, y) ⋂ \overline{B}_{H}(0, ζ(t, x, y)) ≠ ∅ for all (t, x, y) ∈ [0, 1] × H × H. Then for any ε > 0 and any compact set K ⊂ C_{H}([0, T]) there is a nonempty closed convex valued multifunction Φ : K ⇉ L_{H}^{1}([0, T]) which has a strongly-weakly sequentially closed graph such that for any u ∈ K and ϕ ∈ Φ(u) one has

ϕ(t) ∈ M(t, u(t), u̇(t));
∥ϕ(t)∥ ≤ ζ(t, u(t), u̇(t)) + ε,

for almost every t ∈ [0, T].

Now we are able to prove our main result.

Theorem 3.2 Let H be a finite dimensional space, K : H ⇉ H be a set-valued mapping satisfying assumptions (H_{1}), (H_{2}) and (H_{3}). Let T > 0 and let F : [0, T]×H ×H ⇉ H be a set-valued mapping satisfying hypotheses (i) and (ii) of Theorem 3.1 and the following one (iv) there exist nonnegative Lebesgue-integrable functions m, p and q defined on [0, T] such that

F (t, x, y) ⊂ (m(t) + p(t)∥x∥ + q(t)∥y∥)\overline{B}_{H}

for all (t, x, y) ∈ [0, T] × H × H. Then for all u_{0} ∈ H and v_{0} ∈ K(u_{0}), there exist two Lipschitz mappings u, v : [0, T] → H such that

\begin{cases}u(t) = u_{0} + ∫_{0}^{t} v(s)ds, ∀t ∈ [0, T]; \\ −v̇(t) ∈ N_{K(u(t))}(v(t)) + F (t, u(t), v(t)), a.e on [0, T]; \\ v(t) ∈ K(u(t)), ∀t ∈ [0, T]; \\ u(0) = u_{0}; v(0) = v_{0} \end{cases}

with ∥v̇(t)∥ ≤ λl + 2(m(t) + p(t)(∥u_{0}∥ + lT) + q(t)l) a.e. t ∈ [0, T]. In other words, there is a Lipschitz solution u : [0, T] → H to the Cauchy problem (P_{F}).

Proof. Step 1. Put I := [0, T], M(t) = m(t) + p(t)(∥u_{0}∥ + lT) + q(t)l, and let us consider the sets

X = {u ∈ C_{H}(I) : u(t) = u_{0} + ∫_{0}^{t} u̇(s)ds, ∀t ∈ I and ∥u̇(t)∥ ≤ l, a.e.on I},

U = {v ∈ C_{H}(I) : v(t) = v_{0} + ∫_{0}^{t} v̇(s)ds, ∀t ∈ I and ∥v̇(t)∥ ≤ λl + 2M(t), a.e.on I},

K = {h ∈ L_{H}^{1}(I) : ∥h(t)∥ ≤ M(t), a.e.on I}.

It is clear that K is a convex w(L_{H}^{1}(I), L_{H}^{∞}(I))-compact subset of L_{H}^{1}(I), and by Ascoli-Arzelà theorem X and U are convex compact sets in C_{H}(I). Observe now, that for all f ∈ X the set valued mapping K ∘ f is Lipschitz with ratio λl. Indeed, for all t, t′ ∈ I

H((K ∘ f)(t), (K ∘ f)(t′)) = H(K(f(t)), K(f(t′)))
≤ λ∥f(t) − f(t′)∥
= λ∥x_{0} + ∫_{0}^{t} ḟ(s)ds − x_{0} − ∫_{0}^{t′} ḟ(s)ds∥
≤ λ ∫_{t′}^{t} |ḟ(s)|ds ≤ λl|t − t′|.

By Proposition 3.1, for all (f, h) ∈ X × K, there exists a unique solution u_{f,h} to the problem

(P) \begin{cases}−u̇_{f,h}(t) ∈ N_{K(f(t))}(u_{f,h}(t)) + h(t), a.e. on I; \\ u_{f,h}(t) ∈ K(f(t)), ∀t ∈ I; \\ u_{f,h}(0) = v_{0}, \end{cases}

and for almost all t ∈ I, ∥u̇_{f,h}(t)∥ ≤ λl + 2M(t), i.e., u_{f,h} ∈ U.

Let us consider the mapping A : X × K → U defined by A(f, h) = u_{f,h}, where u_{f,h} is the unique solution of (P). We wish to show that A is continuous. Let (f_{n}, h_{n})_{n} be a sequence in X × K such that (f_{n})_{n} converges uniformly to f ∈ X and (h_{n})_{n} converges w(L_{H}^{1}(I), L_{H}^{∞}(I)) to h ∈ K, and since (u_{f_{n},h_{n}})_{n} ⊂ U we may suppose that it converges uniformly to some mapping v ∈ U. For each n ∈ ℕ we have

(P_{n}) \begin{cases}−u̇_{f_{n},h_{n}}(t) ∈ N_{K(f_{n}(t))}(u_{f_{n},h_{n}}(t)) + h_{n}(t), a.e. on I; \\ u_{f_{n},h_{n}}(t) ∈ K(f_{n}(t)), ∀t ∈ I; \\ u_{f_{n},h_{n}}(0) = v_{0}. \end{cases}

Since u_{f_{n},h_{n}}(t) ∈ K(f_{n}(t)) for all t ∈ I, it follows from the Lipschitz property of K

d(u_{f_{n},h_{n}}(t), K(f(t))) ≤ λ∥f_{n}(t) − f(t)∥ → 0

and hence, one obtains v(t) ∈ K(f(t)), because the set K(f(t)) is closed. According to (P_{n}) one has

u̇_{f_{n},h_{n}}(t) + h_{n}(t) ∈ −N_{K(f_{n}(t))}(u_{f_{n},h_{n}}(t)),

and

∥u̇_{f_{n},h_{n}}(t) + h_{n}(t)∥ ≤ λl + M(t) := c(t), i.e., u̇_{f_{n},h_{n}}(t) + h_{n}(t) ∈ c(t)\overline{B}_{H}.

Therefore we get by Proposition 2.1 (1)

u̇_{f_{n},h_{n}}(t) + h_{n}(t) ∈ −c(t)∂d(u_{f_{n},h_{n}}(t), K(f_{n}(t))), a.e. on I.

Now, as (u̇_{f_{n},h_{n}} + h_{n})_{n} converges weakly to v̇ + h ∈ L_{H}^{1}(I), Mazur’s lemma ensures that for a.e. t ∈ I

v̇(t) + h(t) ∈ ⋂_{n} \overline{co}\{u̇_{f_{k},h_{k}}(t) + h_{k}(t) : k ≥ n\}.

Fix such t in I and any μ in H, then the last relation gives

⟨v̇(t) + h(t), μ⟩ ≤ lim sup_{n} σ(−c(t)∂d(u_{f_{n},h_{n}}(t), K(f_{n}(t))), μ)
≤ σ(−c(t)∂d(v(t), K(f(t))), μ),

where the second inequality follows from Remark 2.1 and Theorem 2.1. As the set ∂d(v(t), K(f(t))) is closed and convex (see Proposition 2.1), we obtain

v̇(t) + h(t) ∈ −c(t)∂d(v(t), K(f(t))) ⊂ −N_{K(f(t))}(v(t)),

because v(t) ∈ K(f(t)). This can be rephrased as

\begin{cases}−v̇(t) ∈ N_{K(f(t))}(v(t)) + h(t), a.e. on I; \\ v(t) ∈ K(f(t)), ∀t ∈ I; \\ v(0) = v_{0}. \end{cases}

In other words, v is of the form u_{f,h} with

\begin{cases}−u̇_{f,h}(t) ∈ N_{K(f(t))}(u_{f,h}(t)) + h(t), a.e. on I; \\ u_{f,h}(t) ∈ K(f(t)), ∀t ∈ I; \\ u_{f,h}(0) = v_{0}. \end{cases}

We conclude that A is continuous. Hence, the mapping P : X × K → C_{H}(I) defined by P (f, h), where for all t ∈ I

P (f, h)(t) = u_{0} + ∫_{0}^{t} A(f, h)(s)ds = u_{0} + ∫_{0}^{t} u_{f,h}(s)ds

is also continuous when X is endowed with the topology of uniform convergence and K is endowed with the weak topology. Observe that for all t ∈ I, u_{f,h}(t) ∈ K(f(t)) and then by (H_{3}), we have ∥u_{f,h}(t)∥ ≤ l, we conclude that P (f, h) ∈ X.

Step 2. By Theorem 3.1, there is a nonempty closed convex valued set-valued mapping Φ : X ⇉ L_{H}^{1}(I) such that for any u ∈ X and ϕ ∈ Φ(u)

ϕ(t) ∈ F (t, u(t), u̇(t)) and ∥ϕ(t)∥ ≤ m(t) + p(t)∥u(t)∥ + q(t)∥u̇(t)∥

for almost all t ∈ I. Since u ∈ X, we have ∥u̇(t)∥ ≤ l and

∥u(t)∥ = ∥u_{0} + ∫_{0}^{t} u̇(s)ds∥ ≤ ∥u_{0}∥ + lT,

hence

∥ϕ(t)∥ ≤ m(t) + p(t)(∥u_{0}∥ + lT) + q(t)l = M(t).  (3.1)

The relation (3.1) shows that Φ has w(L_{H}^{1}(I), L_{H}^{∞}(I))-compact values in L_{H}^{1}(I). Now, let us consider the set-valued mapping Ψ : X ⇉ X defined by

Ψ(f) = {P (f, h) : h ∈ Φ(f)}.

It is clear that Ψ has nonempty convex values since Φ has nonempty convex values. Furthermore, for all f ∈ X, Ψ(f) is compact in X. Indeed, Let (v_{n})_{n} be a sequence in Ψ(f), then, for each n, there is h_{n} ∈ Φ(f) such that v_{n} = P (f, h_{n}). Since (h_{n})_{n} ⊂ Φ(f), by extracting a subsequence (that we do not relabel) we may suppose that (h_{n})_{n} w(L_{H}^{1}(I), L_{H}^{∞}(I))-converges to some mapping h ∈ Φ(f), and by the continuity of P we get v_{n} = P (f, h_{n}) → v = P (f, h) ∈ X. This shows the compactness of Ψ(f). We will prove that Ψ is upper semicontinuous, or equivalently the graph of Ψ gph(Ψ) = {(x, y) ∈ X × X : y ∈ Ψ(x)} is closed. Let (x_{n}, y_{n})_{n} be a sequence in gph(Ψ) converging to (x, y) ∈ X × X. For all n ∈ ℕ, y_{n} ∈ Ψ(x_{n}), so, there is h_{n} ∈ Φ(x_{n}) such that y_{n} = P (x_{n}, h_{n}). since (h_{n})_{n} ⊂ K, by extracting a subsequence (that we do not relabel) we may suppose that (h_{n})_{n} w(L_{H}^{1}(I), L_{H}^{∞}(I))-converges to some mapping h ∈ K. As the sequence (x_{n})_{n} converges uniformly to x ∈ X and since gph(Φ) is strongly-weakly sequentially closed we conclude that h ∈ Φ(x). On the other hand, by the continuity of the mapping P we get

y = lim_{n→+∞} y_{n} = lim_{n→+∞} P (x_{n}, h_{n}) = P (x, h).

Hence (x, y) ∈ gph(Ψ). This says that Ψ is upper semicontinuous. An application of Kakutani Theorem gives a fixed point of Ψ, that is, there is f ∈ X such that f ∈ Ψ(f), which means that there is h ∈ Φ(f) such that f = P (f, h). Consequently

f(t) = u_{0} + ∫_{0}^{t} u_{f,h}(s)ds and h(t) ∈ F (t, f(t), u_{f,h}(t))

with

\begin{cases}−u̇_{f,h}(t) ∈ N_{K(f(t))}(u_{f,h}(t)) + h(t), a.e. on I; \\ u_{f,h}(t) ∈ K(f(t)), ∀t ∈ I; \\ u_{f,h}(0) = v_{0}, \end{cases}

or, by putting u = f

\begin{cases}−ü(t) ∈ N_{K(u(t))}(u̇(t)) + F (t, u(t), u̇(t)), a.e. on I; \\ u̇(t) ∈ K(u(t)), ∀t ∈ I; \\ u(0) = u_{0}; u̇(0) = v_{0}, \end{cases}

with, for almost all t ∈ I, ∥ü(t)∥ ≤ λl + 2M(t). This finish the proof of our theorem. ■

References

[1] D. Azzam-Laouir, S. Lounis and L. Thibault, Existence solutions for second-order differential inclusions with nonconvex perturbations. Appl. Anal. Vol. 86. No. 10. October 2007, 1199-1210.

[2] M. Bounkhel, General existence results for second order nonconvex sweeping process with unbounded perturbations. Portugaliae Mathmatica 60 (3) (2003), 269-304.

[3] M. Bounkhel and D. Laouir-Azzam, Existence results on the second-order nonconvex sweeping process with perturbations. Set-Valued Anal. 12: 291-318, 2004.

[4] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. Volume 6, Number 2, 2005.

[5] C. Castaing, Quelques problèmes d’évolution du second ordre, Sem. Anal. Convexe, Montpellier. 1998, exposé No. 5.

[6] C. Castaing, T. X. Duc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal. 1 (1993), 109-139.

[7] C. Castaing, A. Salvadori and L. Thibault, Functional evolution equations governed by nonconvex sweeping process. J. Nonlinear Convex Anal. Vol 2, number 1, 2001.

[8] F.H. Clarke, R.L. Stern and P.R. Wolenski, Proximal smoothness and the lower C^{2} property. J. Convex Anal. Vol 2 (1995), No 1/2, 117-144.

[9] T. Haddad and L. Thibault, Mixed semicontinuous perturbation of nonconvex sweeping process. to appear.

[10] M. D. P. Monteiro-Marques, Differential inclusions in nonsmooth mechanical problem, shocks and dry friction. Birkauser, 1995.

[11] R.A. Poliquin, R.T. Rockafellar and L. Thibault, Local differentiability of ditance functions. Trans. Math. Soc, Vol. 352 (2000), No 11, 5231-5249.

[12] A. Syam, Contributions aux inclusions différentielles. Thèse, Université Montpellier II. Montpellier 1993.

[13] L. Thibault, Sweeping process with regular and nonregular sets. J. Diff. Equa. 193, 1-26 (2003).

[14] A. A. Tolstonogov, Solutions of a differential inclusion with unbouded right-hand side. (Russian) Sibirsk. Mat. Zh. 29 no. 5, 212-225, 241 translation in Siberian. Math. J. 29, no. 5, (1988). 857-868.

(Received June 29, 2008)