u(t)∈C(t), for all t∈[0, T

Nova S´erie

EXISTENCE RESULTS FOR FIRST AND SECOND ORDER NONCONVEX SWEEPING PROCESS WITH DELAY*

M. Bounkhel and M. Yarou

Abstract: In this paper, we prove several existence results for functional differen- tial inclusions governed by nonconvex sweeping process of first and second order with perturbations depending on all the variables and with delay.

1 – Introduction

In this paper, we present some existence results for functional differential inclusions governed by nonconvex sweeping process of first and second order

(FOSPD)











˙

u(t)∈ −N_C(t)(u(t)) +F₁(t, ut) a.e. on [0, T] ; u(t)∈C(t), for all t∈[0, T] ; u(s) =T(0)u(s) =ϕ(s), for all s∈[−τ,0] ; and

(SOSPD)











¨

u(t)∈ −N_K(u(t))( ˙u(t)) +F₂(t, ut,u˙t) a.e. on [0, T] ;

˙

u(t)∈K(u(t)), for all t∈[0, T] ;

T(0) ˙u=ϕ, on [−τ,0] ;

where τ, T > 0, C: [0, T]⇒H (resp. K: H⇒H) is a set-valued mapping tak- ing values in a Hilbert space H, and F₁: I×C0⇒H (resp. F₂: I×C0×C0⇒H) is a set-valued mapping with convex compact values. HereN(C(t);u(t)) (resp.

Received: December 15, 2002; Revised: June 21, 2003.

AMS Subject Classification: 49J52, 46N10, 58C20.

Keywords: uniformly prox-regular set; nonconvex sweeping processes; nonconvex differential inclusions.

* This Work was supported for the first author by the research center project No. Math/

1422/22.

N(K(u(t)); ˙u(t))) denotes the Clarke normal cone to C(t) (resp. K(u(t)))) at u(t) (resp. ˙u(t)) and C0 := C([−τ,0], H) is the Banach space of all continuous mappings from [−τ,0] to H equipped with the norm of uniform convergence.

For everyt ∈I, the function u_t is given by u_t(s) =T(t)u(s) =u(t+s),for all s∈[−τ,0]. Such problems have been studied in several papers (see for example [13, 15, 19]). In [13], some topological properties of solutions set for (FOSPD) problem in the convex case are established, and in [15], the compactness of the solutions set is obtained in the nonconvex case when H=R^d, using impor- tant properties of uniformly r-prox regular sets developed recently in [8, 9, 21].

The (SOSPD) has been considered in [19] with a perturbation not depending of the third variable,F(t,T(t)u), continuous and with compact values.

For more details on functional differential inclusions for nonconvex sweeping process and related subjects, see [1, 2, 3, 4, 6, 7, 9, 14, 17, 18, 20, 22, 23, 24].

Our main purpose in this paper is to prove existence results for (FOSPD) and (SOSPD) whenC has uniformly r-prox regular values and H is a separable Hilbert space. The paper is organized as follows. In Section 2, we recall some definitions and fundamental results needed in the sequel of the paper. Section 3 is devoted to prove (Theorem 3.1) the existence of approximate solutions for the (FOSPD) under the boundedness ofF. Under two different assumptions onF we prove the existence of absolutely continuous solutions of (FOSPD) by proving the convergence of the approximate solutions established in Theorem 3.1. The last section is concerned with the existence of solutions for (SOSPD). Our approach is based on a classical method which consists to subdivide the interval [0, T] in a sequence of subintervalsInand to reformulate our problem with delay (SOSPD) to a family of problems without delay (SOSP) on each In and next we use an existence result given in [4] on each subinterval to get a solution onI_n. Finally, we prove the convergence of this family of solutions to a solution of (SOSPD).

We point out that a different approach with more restrictive assumptions, is given in [10, 11] to prove the existence of solutions for the same problem (SOSPD).

It consists to make use the existence of solutions for the first order problem (FOSPD) and the fixed point approach.

2 – Preliminaries and fundamental results

Throughout the paper,H will denote areal separable Hilbert space. LetSbe a nonempty closed subset ofH, we denote bydS(·) or d(S,·) the usual distance function to the subsetS. We recall that the proximal normal cone ofS atx∈S

is defined by

N^P(S;x) = ⁿξ ∈H: ∃α >0 : x∈Proj(x+αξ, S)^o, where

Proj(u, S) := ⁿy ∈S: dS(u) =ku−yk^o.

Recall now that for a given r ∈ ]0,+∞], a subset S is uniformly r-proximally regular if and only if for all y∈S and all ξ∈N^P(S;y), ξ6= 0 one has

¿ ξ

kξk, x−y À

≤ 1

2rkx−yk² ,

for allx∈S (see [21]). We make the convention ¹_r = 0 for r= +∞ (in this case, the uniformr-prox-regularity is equivalent to the convexity of S).

In order to make clear the concept ofr-prox-regular sets, we state the following concrete examples: The union of two disjoint intervals [a, b] and [c, d] is r-prox- regular with r = ^c−b₂ . The finite union of disjoint closed convex sets in R² is r-prox-regular with r depends on the distances between the sets (we refer the reader to [5] for a different application of this concept in Variational Inequalities.

A general study of the class ofr-prox-regular sets with more concrete examples is given in a forthcoming paper by the first author).

Let K:H⇒H be a set-valued mapping from H to H. We will say that K is Hausdorff-continuous (resp. Hausdorff–Lipschitz with ratio λ > 0) if for any x∈H one has

xlim⁰→xH(K(x), K(x⁰)) = 0

³resp.

H(K(x), K(x⁰))≤λkx−x⁰k, for all x, x⁰∈H ^´.

Here H stands the Hausdorff distance relative to the norm associated with the Hilbert spaceH defined by

H(A, B) := maxⁿsup

a∈A

d_B(a), sup

b∈B

d_A(b)^o .

Let ϕ:X⇒Y be a set-valued mapping defined between two topological vector spaces X and Y, we say that ϕ is upper semi-continuous (in short u.s.c.) at x∈dom(ϕ) :={x⁰∈X: ϕ(x⁰) 6=∅} if for any open set O containing ϕ(x) there exists a neighborhoodV ofx such thatϕ(V)⊂O.

We will deal with a finite delay τ > 0. If u: [−τ, T] → H, then for every t ∈ [0, T], we define the function u_t(s) = u(t+s), s ∈ [−τ,0] and the Banach

space CT := C([−τ, T], H) (resp. C0:= C([−τ,0], H)) of all continuous mappings from [−τ, T] (resp. [−τ,0]) toH with the norm given by

kϕkC_T := maxⁿkϕ(s)k: s∈[−τ, T]^o

³resp.

kϕkC0 := maxⁿkϕ(s)k: s∈[−τ,0]^{o ´}.

Clearly, if u ∈ CT, then u_t ∈ C₀, and the mapping u → u_t is continuous in the sense of the uniform convergence.

The following propositions summarize some important consequences of the uniform prox-regularity needed in the sequel of the paper. For the proofs we refer to [9].

Proposition 2.1 ([9]). LetS be a nonempty closed subset ofH and x∈S.

Then the following hold

1) ∂^PdS(x) =N_S^P(x)∩B, where∂^PdS(x)is the proximal subdifferential of the distance function (see [9] for the definition of the proximal subdifferential).

2) IfS is uniformlyr-prox-regular, then for any x∈H with d_S(x)< r, then Proj_S(x) 6= ∅ and unique, and the proximal subdifferential ∂^Pd_S(x) is a closed convex set in H.

3) If S is uniformlyr-prox-regular, then for any x∈S and any ξ∈∂^Pd_S(x) one has

hξ, x⁰−xi ≤ 2

r kx⁰−xk²+d_S(x⁰) for all x⁰ ∈H with d_S(x⁰)< r . As a direct consequence of part (2) in the previous proposition we have

∂^PdS(x) = ∂^CdS(x) and hence N^C(S;x) = N^P(S;x), whenever S is uniformly r-prox-regular set. So, we will denote N(S;x) :=N^C(S;x) =N^P(S;x) and

∂dS(x) :=∂^PdS(x) =∂^CdS(x) for such class of sets.

Proposition 2.2 ([4, 7]). Let r∈]0,∞]and Ωbe an open subset inH and let C: Ω⇒H be a Hausdorff-continuous set-valued mapping. Assume that C has uniformly r-prox-regular values. Then, the set-valued mapping given by (z, x)⇒∂d_C(z)(x)fromΩ×H(endowed with the strong topology) toH(endowed with the weak topology) is upper semicontinuous, which is equivalent to the u.s.c.

of the function (z, x)7→ σ(∂d_C(z)(x), p) for any p∈H. Here σ(S, p) denotes the support function associated withS, i.e., σ(S, p) = sup

s∈S

hs, pi.

3 – First order perturbed nonconvex sweeping process with delay

In all this section, let T > 0, I := [0, T], r ∈ ]0,+∞], and C: I⇒H be an absolutely continuous set-valued mapping, that is, for anyy∈H and anyt, t⁰ ∈I (3.1) |d_C(t)(y)−d_C(t0)(y)| ≤ |v(t)−v(t⁰)|,

withv:I→Ris an absolutely continuous function, i.e., there existsg∈L¹(I,R) such thatv(t) =v(0) +^R₀^tg(s)ds, for allt∈I. Note that the functionvdoes not depend ony. The following proposition provides an approximate solution for the (FOSPD) under consideration.

Theorem 3.1. Assume thatC(t)is uniformlyr-prox-regular for everyt∈I. LetF:I×C0⇒Hbe a set-valued mapping with convex compact values inH such that F(t,·) is u.s.c. on C0 for any fixed t ∈ I and F(·, ϕ) admits a measurable selection onI for any fixedϕ∈ C₀. Assume thatF(t, ϕ)⊂lBfor all(t, ϕ)∈I×C₀, for somel >0. Then, for anyϕ∈ C0withϕ(0)∈C(0)and for anynlarge enough there exists a continuous mapping u_n: [−τ, T]→ H which enjoys the following properties:

1) ˙u_n(t)∈ −N(u_n(θ_n(t));C(θ_n(t))) +F(ρ_n(t),T(ρ_n(t))u_n), a.e.t∈I, where θn, ρn:I →I with θn(t)→t and ρn(t)→t for all t∈I.

2) ku˙n(t)k ≤(l+ 1)( ˙v(t) + 1), a.e. t∈I.

Proof: We prove the conclusion of our theorem whenF is globally u.s.c. on I× C0 and then as in [13], we can proceed by approximation to prove it when F(t,·) is u.s.c. onC0for any fixedt∈I andF(·, ϕ) admits a measurable selection onI for any fixed ϕ∈ C0.

First, observing that (3.1) ensures for t≤t⁰ (3.2) |d_C(t⁰₎(y)−d_C(t)(y)| ≤

Z t⁰

t |v(s)|˙ ds ,

we may suppose (replacing ˙v by |v|˙ if necessary) that ˙v(t) ≥ 0 for all t ∈ I. We construct via discretization the sequence desired of continuous mappings {un}n inCT.

For every n∈N, we consider the following partition ofI: (3.3) tⁿ_i := iT

2ⁿ (0≤i≤2ⁿ) and I_iⁿ:= ]tⁿ_i, tⁿ_i+1] if 0≤i≤2ⁿ−1 .

Put

(3.4) µ_n:= T

2ⁿ, ²ⁿ_i :=

Z tⁿ_i+1 tⁿ_i

˙

v(s)ds and ²_n:= max

0≤i<2ⁿ{µn+²ⁿ_i}. As²_n→0, we can fix n₀≥1 satisfying for everyn≥n₀

(3.5) 2µ_n< r

(2l+ 1) and 2²_n<min

½ 1, r

(4l+ 3)

¾ ,

(3.6) First, we put u_n(s) :=ϕ(s), for alls∈[−τ,0] and for alln≥n₀. For every n≥n₀, we define by induction:

(3.7) u_n(tⁿ_i+1) :=uⁿ_i+1= Proj_C(tⁿ

i+1)

³uⁿ_i −µ_nf₀(tⁿ_i,T(tⁿ_i)un)^´, wheref₀(tⁿ_i,T(tⁿ_i)un) is the minimal norm element of F(tⁿ_i,T(tⁿ_i)un), i.e., (3.8) kf₀(tⁿ_i,T(tⁿ_i)un)k = minⁿkyk: y∈F(tⁿ_i,T(tⁿ_i)un)^o ≤ l and

T(tⁿ_i)un:= (un)tⁿ_i .

The above construction is possible despite the nonconvexity of the images of C.

Indeed, we can show that for everyn≥n₀ we have d_C(tⁿ

i+1)

³uⁿ_i −µnf₀(tⁿ_i,T(tⁿ_i)un)^´ ≤ lµn+v(tⁿ_i+1)−v(tⁿ_i) ≤ (l+ 1)²n ≤ r 2 and hence as C has uniformlyr-prox-regular values, by Proposition 2.1 one can choose for all n ≥ n₀ a point uⁿ_i+1= Proj_C(tⁿ

i+1)(uⁿ_i −µ_nf₀(tⁿ_i,T(tⁿ_i)un)). Note that from (3.7) and (3.2) one deduces for every 0≤i <2ⁿ

(3.9) ^°°_°uⁿ_i+1−^³uⁿ_i −µnf₀(tⁿ_i,T(tⁿ_i)un)^´°°_°≤ lµn+²ⁿ_i ≤ (l+ 1) (µn+²ⁿ_i) . (3.10) By construction we have uⁿ_i ∈C(tⁿ_i), for all 0≤i <2ⁿ.

For every n≥n₀, these (uⁿ_i)0≤i≤2ⁿ and (f0(tⁿ_i,T(tⁿ_i)un)0≤i≤2ⁿ are used to construct two mappingsu_n and f_n from I toH by defining their restrictions to each intervalI_iⁿ as follows:

for t= 0, set f_n(t) :=f₀ⁿ and u_n(t) :=uⁿ₀ =ϕ(0), for all t∈I_iⁿ(0≤i≤2ⁿ), set fn(t) :=f_iⁿ and (3.11) un(t) := uⁿ_i +a(t)−a(tⁿ_i)

²ⁿ_i +µn

(uⁿ_i+1−uⁿ_i +µnf_iⁿ) + (t−tⁿ_i)f_iⁿ ,

wheref_iⁿ :=f₀(tⁿ_i,T(tⁿ_i)un) and a(t) :=v(t) +t for allt∈I. Hence for everyt andt⁰ inI_iⁿ (0≤i≤2ⁿ) one has

u_n(t⁰)−u_n(t) = a(t⁰)−a(t)

²ⁿ_i +µ_n (uⁿ_i+1−uⁿ_i +µ_nf_iⁿ) + (t⁰−t)f_iⁿ . Thus, in view of (3.9), if t, t⁰ ∈I_iⁿ (0≤i <2ⁿ) witht≤t⁰, one obtains

(3.12) kun(t⁰)−un(t)k ≤ (l+1)^³a(t⁰)−a(t)^´+l(t⁰−t) ≤ (2l+1)^³a(t⁰)−a(t)^´, and, by addition this also holds for allt, t⁰ ∈I witht≤t⁰. This inequality entails thatu_n is absolutely continuous.

Coming back to the definition ofu_nin (3.11), one observes that for 0≤i <2ⁿ

˙

u_n(t) = a(t)˙

²ⁿ_i +µ_n(uⁿ_i+1−uⁿ_i +µ_nf_iⁿ) +f_iⁿ for a.e. t∈I_iⁿ . Then one obtains, in view of (3.9), for a.e. t∈I

(3.13) ku˙n(t)−fn(t)k ≤ (l+ 1)^³v(t) + 1˙ ^´ , which proves the part 2) of the theorem.

Now, let θ_n,ρ_n be defined from I toI by θ_n(0) = 0, ρ_n(0) = 0, and (3.14) θ_n(t) =tⁿ_i+1, ρ_n(t) =tⁿ_i if t∈I_iⁿ (0≤i <2ⁿ) .

Then, by (3.7), the construction of u_n and f_n, and the properties of proximal normal cones to subsets, we have for a.e. t∈I

fn(t) ∈ F^³ρn(t),T(ρn(t))un

´ and

(3.15) u˙_n(t)−f_n(t) ∈ −N^³C(θn(t));u_n(θn(t))^´ .

These last inclusions ensure part 1) of the theorem and then the proof is complete.

Now, we are able to state our first existence result for (FOSPD).

Theorem 3.2. Assume that the assumptions of Theorem 3.1 are satisfied.

Assume that C(t) is strongly compact for every t ∈ I. Then for every ϕ ∈ C₀ withϕ(0)∈C(0), there exists a continuous mapping u: [−τ, T]→H such that uis absolutely continuous on I and satisfies:

(FOSPD)











˙

u(t)∈ −N_C(t)(u(t)) +F(t, u_t) a.e. on I;

u(t)∈C(t), ∀t∈I;

u(s) =T(0)u(s) =ϕ(s), ∀s∈[−τ,0] ; and

ku(t)k ≤˙ (l+ 1)^³v(t) + 1˙ ^´ a.e. on I .

Proof: Letϕ∈ C0withϕ(0)∈C(0). By Theorem 3.1 there exists a sequence of continuous mappings {un} enjoying the properties 1) and 2) in Theorem 3.1.

Letn₀ ∈Nsatisfying (3.5). Then by (3.1), (3.4), and (3.12) we get for anyn≥n₀ and anyt∈I

(3.16)

d(un(t), C(t)) ≤ kun(t)−u_n(tⁿ_i)k+d(un(tⁿ_i), C(t))

≤ (2l+ 1)^³a(t)−a(tⁿ_i)^´+^³v(t)−v(tⁿ_i)^´

≤ (2l+ 1) (²ⁿ_i +µ_n) +²ⁿ_i ≤ 2(l+ 1)²_n .

Since C(t) is strongly compact and ²_n→ 0, (3.16) implies that the set {un(t) : n≥n₀} is relatively strongly compact inHfor allt∈I. Thus, by Arzela–Ascoli’s theorem we can extract a subsequence of the sequence{un}nstill denoted{un}n, which converges uniformly on [−τ, T] to a continuous function u which clearly satisfiesu₀ =ϕ. Now by lettingn→+∞ we get for allt∈I

(3.17) u(t)∈C(t) .

On one hand, it follows from our construction in the proof of Theorem 3.1 that for allt∈I

(3.18) H^³C(θn(t)), C(t)^´≤ |v(θn(t))−v(t)| ≤ ²_n → 0,

and by (3.12), (3.5), and the uniform convergence of{un}n tou overI we get (3.19) kun(θn(t))−u(t)k ≤ kun(θn(t))−u(θn(t))k+ku(θn(t))−u(t)k → 0. Now, using the same technique in [13] and the relations (3.5) and (3.12) we obtain

n→∞lim kT(ρn(t))un− T(t)unk= 0 in C0 .

Therefore, as the uniform convergence of un tou in [−τ, T] implies that T(t)un

converges toT(t)u uniformly on [−τ,0],we conclude that (3.20) T(ρn(t))un−→ T(t)u=u_t in C₀ .

On the other hand, fromf_n(t)∈F(ρn(t),T(ρn(t))un) and (3.12), the sequences (f_n) and ( ˙u_n) are bounded sequences in L^∞(I, H). Then by extracting subse- quences (because L^∞(I, H) is the dual space of the separable space L¹(I, H)), we may suppose without loss of generality that fn and ˙un weakly-? converge in L^∞(I, H) to some mappings f and ω respectively. Then, for allt∈I one has

u(t) = lim

n→∞un(t) = ϕ(0) + lim

n→∞

Z t 0

˙

un(s)ds = x₀+ Z t

0

ω(s)ds , which proves thatu is absolutely continuous and ˙u(t) =ω(t) for a.e. t∈I.

Using now Mazur’s lemma, we obtain:

˙

u(t)−f(t) ∈ ^\

n

coⁿu˙_k(t)−f_k(t) : k≥n^o a.e. t∈I . Fix suchtinI and any ξ inH, the last relation above yields

hu(t)˙ −f(t), ξi ≤ inf

n sup

k≥n

hu˙_k(t)−f_k(t), ξi. By (3.13), (3.15), and Proposition 2.1 part (1) we have for a.e. t∈I

˙

u_n(t)−f_n(t) ∈ −N^³C(θn(t)); u_n(θn(t))^´∩δ(t)B = −δ(t)∂d_C(θn(t))(un(θn(t))), whereδ(t) := (l+ 1) ( ˙v(t) + 1). Hence, according to this last inclusion and propo- sition 2.2 we get

hu(t)˙ −f(t), ξi ≤ δ(t) lim sup

n σ^³−∂d_C(θn(t)(u_n(θ_n(t)));ξ^´

≤ δ(t)σ^³−∂d_C(t)(u(t)); ξ^´ Since∂d_C(t)(u(t)) is closed convex, we obtain

˙

u(t)−f(t) ∈ −δ(t)∂d_C(t)(u(t))⊂ −N_C(t)(u(t)) and then

˙

u(t) ∈ −N_C(t)(u(t)) +f(t)

becauseu(t)∈C(t). Finally, from (3.20) and the global upper semicontinuity of F and the convexity of its values and with the same techniques used above we can prove that

f(t) ∈ F(t,T(t)u) =F(t, ut) a.e. t∈I . Thus, the existence is proved.

Under different assumptions another existence result for (FOSPD) is also proved in the following theorem.

Theorem 3.3. Assume that the assumptions of Theorem 3.1 are satisfied.

Assume also thatF(t, ϕ)⊂ K ⊂lBfor every(t, ϕ)∈I×C0, whereK is a strongly compact set in H. Then for every ϕ∈ C0 with ϕ(0)∈C(0), there exists a con- tinuous mapping u: [−τ, T]→H such that u is absolutely continuous on I and satisfies: _









˙

u(t)∈ −N_C(t)(u(t)) +F(t, ut) a.e. on I;

u(t)∈C(t), ∀t∈I;

u(s) =T(0)u(s) =ϕ(s), ∀s∈[−τ,0] ; and

ku(t)k ≤˙ (l+ 1)^³v(t) + 1˙ ^´ a.e. on I .

Proof: Letϕ∈ C0withϕ(0)∈C(0). By Theorem 3.1 there exists a sequence of continuous mappings {un} enjoying the properties 1) and 2) in Theorem 3.1.

Let n₀ ∈ N satisfying (3.5). Let us show that the sequence (un)n satisfies the Cauchy property in the space of continuous mappingsC(I, H) endowed with the norm of uniform convergence. Fix m, n ∈N such that m ≥n≥n₀ and fix also t∈I witht6=t_m,ifori= 0, ...,2^m andt6=t_n,j forj= 0, ...,2ⁿ. Observe by (3.2), (3.4), and (3.12) that

(3.21)

d_C(θn(t))(u_m(t)) = d_C(θn(t))(u_m(t))−d_C(θm(t))(u_m(θ_m(t)))

≤ v(θn(t))−v(θm(t)) +kum(θm(t))−um(t)k

≤ Z θn(t)

θm(t)

˙

v(s)ds + (2l+1)

· Z _θm(t)

t

˙

v(s)ds + (θm(t)−t)

¸

≤ ²_n+ (2l+ 1)²_m

and hence, by (3.5) we get d_C(θn(t))(um(t)) < r. Set δ(t) := (l+1) ˙a(t). Then, (3.15), (3.21), and Proposition 2.1 part (3) entail

Du˙_n(t)−f_n(t), un(θn(t))−u_m(t)^E ≤

≤ 2δ(t)

r kun(θn(t))−u_m(t)k²+ δ(t)d_C(θn(t))(um(t))

≤ 2δ(t) r

hkun(t)−um(t)k+kun(θn(t))−un(t)kⁱ²+ δ(t)^³²n+ (2l+1)²m

´, and this yields by (3.4) and (3.12)

Du˙n(t)−fn(t), un(θn(t))−um(t)^E ≤

≤ 2δ(t) r

hkun(t)−um(t)k+ (2l+1)²n

i2

+ δ(t) (2l+1) (²n+²m) . (3.22)

Now, let us define gn(t) :=

Z t 0

fn(s)ds for allt∈I. Observe that for allt∈I the set{gn(t) :n≥n₀}is contained in the strongly compact setTKand so it is rela- tively strongly compact in H. Then, as kfn(t)k ≤ l a.e. on I, Arzela–Ascoli’s theorem yields the relative strong compactness of the set {gn:n≥n₀} with respect to the uniform convergence in C(I, H) and so we may assume without loss of generality that (gn) converges uniformly to some mapping g. Also, we may suppose that (fn) weakly converges in L¹(I, H) to some mapping f. Then, for allt∈I,

g(t) = lim

n gn(t) = lim

n

Z t 0

fn(s)ds = Z t

0

f(s)ds , which gives thatg is absolutely continuous and ˙g=f a.e. on I.

Put now wn(t) := un(t)−gn(t) for all n ≥ n₀ and all t ∈ I and put ηn :=

max{²n,kgn−gk∞}. Then by (3.13) and (3.22) one gets Dw˙_n(t), wn(θn(t))−w_m(t)^E =

= ^Dw˙n(t), un(θn(t))−um(t)^E+^Dw˙n(t), gn(θn(t))−gm(t)^E

≤ 2δ(t) r

hkwn(t)−w_m(t)k+kgn(t)−g_m(t)k+ (2l+1)²_nⁱ² + δ(t) (2l+1) (²n+²_m) +δ(t)kgn(θn(t))−g_m(t)k

≤ 2δ(t) r

hkwn(t)−w_m(t)k+ (ηn+η_m) + (2l+1)η_nⁱ² + 2δ(t) (2l+1) (ηn+ηm).

This last inequality ensures by (3.13)

Dw˙n(t), wn(t)−wm(t)^E ≤ ^Dw˙n(t), wn(t)−wn(θn(t))^E + 2δ(t) (2l+1) (ηn+ηm) + 2δ(t)

r

hkwn(t)−w_m(t)k+ (ηn+η_m) + (2l+1)η_n

¸2

≤ 4δ(t) (2l+1) (ηn+ηm) + 2δ(t)

r

hkwn(t)−w_m(t)k+ (η_n+η_m) + (2l+1)η_nⁱ². In the same way, we also have

Dw˙_m(t), wm(t)−wn(t)^E ≤ 4δ(t) (2l+1) (ηn+η_m) + 2δ(t)

r

hkwn(t)−wm(t)k+ (η_n+η_m) + (2l+1)η_mⁱ². It then follows from both last inequalities that we have for some positive constant αindependent of m, nand t (note thatkwn(t)k ≤lT+kϕ(0)k+^R₀^Tv(s)˙ ds) 2^Dw˙_m(t)−w˙_n(t), w_m(t)−w_n(t)^E ≤ α δ(t) (η_n+η_m) + 8δ(t)

r kwm(t)−w_n(t)k² , and so, for some positive constantsβ and γ independent of m, n andt

d dt

³kwm(t)−w_n(t)k^2´ ≤ βa(t)˙ kwm(t)−w_n(t)k²+γa(t) (η˙ n+η_m) . As kwm(0)−w_n(0)k² = 0, the Gronwall inequality yields for allt∈I

kwm(t)−wn(t)k² ≤ γ(ηn+ηm) Z t

0

·

˙ a(s) exp

µ β

Z t s

˙ a(u)du

¶¸

ds

and hence for some positive constantK independent ofm, n andt we have kwm(t)−wn(t)k² ≤ K(ηn+ηm) .

The Cauchy property in C(I, H) of the sequence (w_n)_n= (u_n−g_n)_n is thus established and hence this sequence converges uniformly to some mapping w.

Therefore the sequence (un)n constructed in Theorem 3.1 converges uniformly to u:=w+g. Following the same arguments in the proof of Theorem 3.2 we prove the conclusion of the theorem, i.e., the limit mappingu is continuous on [−τ, T] and absolutely continuous onI and satisfies











˙

u(t)∈ −N_C(t)(u(t)) +F(t, ut) a.e. on I;

u(t)∈C(t), ∀t∈I;

u(s) =T(0)u(s) =ϕ(s), ∀s∈[−τ,0] ; and

ku(t)k ≤˙ (l+ 1) ( ˙v(t) + 1) a.e. on I .

Remark 3.1. Our results in this section generalizes many results given in [13, 15, 24]. Theorem 3.2 extends the one given in [13] to the case of absolutely continuous set-valued mappings with nonconvex values, and Theorem 3.3 extends Theorem 3.1 in [24] and Theorem 2.1 in [15] given only in the finite dimensional setting. Note that the proof here is completely different of those given in [15, 24]

and it allows us to obtain the result in the infinite dimensional setting. It is interesting to point out that our assumptions onFare different to those supposed in Theorem 2.1 in [15]. They supposed thatFhas compact values and satisfies the linear growth condition and in our Theorem 3.3, F is supposed to be contained in a compact set. In a forthcoming paper, we extend our results to the case when F satisfies some linear growth condition.

We end this section with a uniqueness result. We need first the following lemma. Its proof follows directly from the third part of Proposition 2.1.

Lemma 3.1. Letr∈]0,+∞]and letS be a uniformlyr-prox-regular subset ofH. Then for any x₁, x₂ ∈S and any ξ_i∈∂d_S(xi) (i= 1,2), one has

hξ1−ξ₂, x₁−x₂i ≥ −4

rkx1−x₂k² .

Theorem 3.4. Under the hypothesis of either Theorem 3.2 or Theorem 3.3 and if in addition, there exist a positive functiong∈L¹(I,R) satisfying:

1) g(t)≤ 4

r(l+ 1) ( ˙v(t) + 2), for a.e. t∈I,

2) ∀t∈I, ∀x₁, x₂∈ CT, ∀yi∈F(t,T(t)xi), i= 1,2 one has Dy₁(t)−y₂(t), x₁(t)−x₂(t)^E ≥ g(t)kx₁(t)−x₂(t)k² , then(FOSPD) has a unique solution.

Proof: Letu₀, u₁ be two solutions of (FOSPD) with the initial valuesϕ₁, ϕ₂ inC0 withϕi(0)∈C(0) (i= 1,2), i.e., for i= 1,2, one has



















˙

u_i(t)∈ −N_C(t)(ui(t)) +f_i(t), a.e. t∈I , f_i(t)∈F(t,T(t)ui), a.e. t∈I , u_i(t)∈C(t), for all t∈I ,

u_i =ϕ_i, on [−τ,0],

with

ku˙i(t)k ≤(l+ 1) ( ˙v(t) + 1), a.e. t∈I . As fori= 1,2 and for a.e.t∈I,one haskfi(t)k ≤l, then one gets

−u˙i(t) +fi(t) ∈ N_C(t)(ui(t))∩(l+1) ( ˙v(t) + 2)B a.e. t∈I . This ensures by Proposition 2.1–(1) that for i= 1,2 and for a.e. t∈I

−u˙i(t) +fi(t) ∈ (l+1) ( ˙v(t) + 2)∂d_C(t)(ui(t)). Using the previous lemma one obtains for a.e. t∈I

¿

−^³u˙₁(t)−u˙₂(t)^´+^³f₁(t)−f₂(t)^´, u₁(t)−u₂(t) À

≥ 4

r(l+1) ( ˙v(t)+2)ku₁(t)−u₂(t)k². Now by the second hypothesis of the theorem, one has

Df₁(t)−f₂(t), u₁(t)−u₂(t)^E ≥ g(t)ku1(t)−u₂(t)k² . Additionning the two last inequalities one gets

−^Du˙₁(t)−u˙₂(t), u₁(t)−u₂(t)^E ≥

·

g(t)−4

r(l+1) ( ˙v(t) + 2)

¸

ku1(t)−u₂(t)k² ,

which can be rewritten as d

dt µ1

2ku₁(t)−u₂(t)k²

¶

≤ α(t)ku₁(t)−u₂(t)k² ,

with α(t) =^h4_r(l+ 1) ( ˙v(t) + 2)−g(t)ⁱ≥0 a.e. on I. This ensures by Gronwall inequality that for a.e.t∈I

ku1(t)−u₂(t)k² ≤ ku1(0)−u₂(0)k² exp

· 2

Z t 0

α(s)ds

¸ .

Hence, if u₁ and u₂ are two solutions of (FOSPD) with the same initial value ϕ₁ =ϕ₂ ∈ C(0), we get u₁=u₂ on [−τ,0] and u₁(0) =u₂(0)∈C(0). Therefore, by the last inequality we obtain u₁ = u₂ on [0, T] and so u₁ = u₂ on [−τ, T].

Thus the uniqueness is established.

4 – Second Order Perturbed Sweeping Process with delay

In all this section we letr ∈]0,∞],x₀ ∈H,u₀ ∈K(x₀),V₀ be an open neigh- borhood ofx₀inH, andζ >0 such thatx₀+ζB ⊂ V0,and letK: cl(V0)→Hbe a Lipschitz set-valued mapping with ratioλ >0 taking nonempty closed uniformly r-prox-regular values inH.

First we state the following result used in our main proofs. It is a direct consequence of Theorem 3.1 in [4] by takingG(t, x, u) ={0}.

Theorem 4.1. Assume that:

(i) ∀x ∈cl(V₀), K(x) ⊂ K₁ ⊂lB, K₁ is a convex compact set in H, and l >0 ;

(ii) F: [0,∞[×H×H→His scalarily upper semi-continuous on[0,^ζ_l]×gph(K) with nonempty convex weakly compact values ;

(iii) F(t, x, u)⊂%(1 +kxk+kuk)B, ∀(t, x, u)∈[0,^ζ_l]×gph(K).

Then, for anyT∈]0,^ζ_l],there exists a Lipschitz mapping x:I= [0, T]→cl(V0) such that:











¨

x(t)∈ −N_K(x(t))( ˙x(t)) +F(t, x(t),x(t)),˙ a.e. on I;

˙

x(t)∈K(x(t)), ∀t∈I;

x(0) =x₀, x(0) =˙ u₀,

with kx(t)k ≤˙ l, k¨x(t)k ≤lλ+ 2 (1 +α+l)% a.e. on I.

Remark 4.1. We point out that the solution mappingx obtained in Theo- rem 4.1 is differentiable everywhere onI.

Now let us state the existence result for the second order perturbed sweeping process with delay (SOSPD).

Theorem 4.2. Assume that (i) and the following conditions hold:

(ii)⁰ F: [0,+∞[×C0×C0⇒His scalarily upper semi-continuous on[0,^ζ_l]×C0×C0, taking convex weakly compact values in H, and

(iii)⁰ F(t, ϕ, φ)⊂%(1 +kϕ(0)k+kφ(0)k)B, ∀(t, ϕ, φ)∈[0,^ζ_l]×C₀×C₀. Then for everyT ∈]0,^ζ_l]and for everyφ∈ C0 verifyingφ(0) =u₀,there exists a Lipschitz mapping x: [0, T]→cl(V₀) such that:











¨

x(t)∈ −N_K(x(t))( ˙x(t)) +F(t,T(t)x,T(t) ˙x), a.e. on [0, T] ;

˙

x(t)∈K(x(t)), ∀t∈[0, T] ;

T(0)x=ϕ and T(0) ˙x=φ on [−τ,0], with ϕ(t) =x₀+

Z t

0φ(s)ds, for all t∈[−τ,0], and kx(t)k ≤˙ l and k¨x(t)k ≤ lλ+ 2(1 +α+l)% a.e. on [0, T].

Proof: Without loss of generality, we may take T = 1. Letφ∈ C0 satisfying φ(0) =u₀,and putϕ(t) :=x₀+

Z t

0φ(s)dsfor allt∈[−τ,0]. Let (Pn) be a subdi- vision of [0,1] defined by the points: tⁿ_i := _nⁱ (i= 0,1, ..., n). For every (t, x, u)∈ [−τ,tⁿ₁]×gph(K),we definef₀ⁿ: [−τ,tⁿ₁]×cl(V0)→Handg₀ⁿ: [−τ,tⁿ₁]×K(cl(V0))→H by

f₀ⁿ(t, x) =







ϕ(t), t∈[−τ,0],

ϕ(0) +n t(x−ϕ(0)), t∈[0, tⁿ₁], and

gⁿ₀(t, u) =







φ(t), t∈[−τ,0],

φ(0) +n t(u−φ(0)), t∈[0, tⁿ₁].

We have f₀ⁿ(¹_n, x) =x and g₀ⁿ(_n¹, u) =u for all (x, u)∈gph(K). Observe that the mapping (x, u) 7→ (T(tⁿ₁)f₀ⁿ(·, x),T(tⁿ₁)g₀ⁿ(·, u)) from gph(K) to C0× C0 is

nonexpansive. Indeed, we have for all (x, y)∈H×H

°

°

°

°T^³1_n^´f₀ⁿ(·, x)− T^³1_n^´f₀ⁿ(·, y)

°

°

°

°_C

0

= sup

s∈[−τ,0]

°

°

°

°

f₀^n³s+_n¹, x^´−f₀^n³s+_n¹, y^´

°

°

°

°

= sup

s∈[−τ+_n¹,_n¹]

°

°

°f₀ⁿ(s, x)−f₀ⁿ(s, y)^°°_°

= sup

0≤s≤¹_n

°

°

°n s(x−ϕ(0))−n s(y−ϕ(0))^°°_°

= sup

0≤s≤¹_n

kn s(x−y)k

= kx−yk . In the same way, we get for all (u, v)∈H×H

°

°

°

°T^³_n^1´g₀ⁿ(·, u)− T^³_n^1´gⁿ₀(·, v)

°

°

°

°_C

0

= ku−vk.

Hence the mapping (x, u)7→(T(tⁿ₁)f₀ⁿ(·, x),T(tⁿ₁)g₀ⁿ(·, u)) from gph(K) to C₀×C₀ is nonexpansive and so the set-valued mapping F₀ⁿ: [0,¹_n]×gph(K)⇒H defined by: F₀ⁿ(t, x, u) =F(t,T(_n¹)f₀ⁿ(·, x),T(_n¹)g₀ⁿ(·, u)) is scalarily upper semi-continuous on [0,¹_n]×gph(K) because F is also scalarily upper semi-continuous on [0,¹_n]× C0× C0, with nonempty convex weakly compact values inH and satisfies

F₀ⁿ(t, x, u) = F µ

t,T^³_n^1´f₀ⁿ(·, x),T^³_n^1´g₀ⁿ(·, u)

¶

⊂ %^³1 +kxk+kuk^´, for all (t, x, u)∈[0,_n¹]×gph(K) becauseT(¹_n)f₀ⁿ(0, x) =x andT(¹_n)g₀ⁿ(0, u) =u.

HenceF₀ⁿverifies conditions of Theorem 4.1 [4], provides a Lipschitz differentiable solution y₀ⁿ: [0,_n¹]→cl(V0) to the problem



















¨

y₀ⁿ(t)∈ −N_K(yⁿ

0(t))( ˙y₀ⁿ(t)) +F µ

t, T^³1_n^´f₀ⁿ(·, y₀ⁿ(t)),T^³_n^1´f₁ⁿ(·,y˙₀ⁿ(t))

¶ ,

a.e.on ^h0,¹_nⁱ;

˙

y₀ⁿ(t)∈K(y₀ⁿ(t)), ∀t∈^h0,_n¹ⁱ; y₀ⁿ(0) =x₀ =ϕ(0), y˙ⁿ₀(0) =u₀=φ(0) .

Further we haveky˙₀ⁿ(t)k ≤l and ky¨ⁿ₀(t)k ≤lλ+ 2(1 +α+l)%. Set

yn(t) =







ϕ(t), ∀t∈[−τ,0], y₀ⁿ(t), ∀t∈^h0,_n¹ⁱ.

Then,yn is well defined on [−τ,_n¹],withyn=ϕ on [−τ,0] and

˙ y_n(t) =







φ(t) ∀t∈[−τ,0],

˙

yⁿ₀(t) ∀t∈ⁱ0,_n^1h, and



















¨

yn(t)∈ −N_K(yn(t))( ˙yn(t)) +F µ

t, T^³_n^1´f₀ⁿ(·, yn(t)),T^³_n^1´g₀ⁿ(·,y˙n(t))

¶ ,

a.e. on ^h0,¹_nⁱ;

˙

y_n(t)∈K(yn(t)), ∀t∈^h0,_n¹ⁱ; yn(0) =x₀=ϕ(0), y˙n(0) =u₀ ,

with ky˙_n(t)k ≤l and k¨y_n(t)k ≤lλ+ 2(1 +α+l)% a.e. t∈[0,_n¹].

Suppose that y_n is defined on [−τ,_n^k] (k≥1) with y_n = ϕ on [−τ,0] and satisfies:

y_n(t) =



























yⁿ₀(t) = x₀+ Z t

0y˙_n(s)ds ∀t∈^h0,_n¹ⁱ, yⁿ₁(t) := y_n^³_n^1´+

Z t

1 n

˙

y_n(s)ds ∀t∈^h_n¹,_n²ⁱ,

· · ·

yⁿ_k−1(t) := y_n^³k−_n^1´+ Z t

k−1 n

˙

y_n(s)ds ∀t∈^hk−_n¹,_n^ki, andyn is a Lipschitz solution of























y_n(t) = yⁿ_k−1(t) := y_n^³k−_n^1´+ Z t

k−1 n

˙

y_n(s)ds ∀t∈^hk−_n¹,_n^ki,

¨

y_n(t)∈ −N_K(yn(t))( ˙y_n(t)) +F µ

t, T^³k_n^´f_k−1ⁿ (·, yn(t)),T^³k_n^´g_k−1ⁿ (·,y˙_n(t))

¶ , a.e. ^hk−1_n ,^k_nⁱ,

˙

y_n(t)∈K(yn(t)), ∀t∈^hk−1_n ,^k_nⁱ ,

wheref_k−1ⁿ and gⁿ_k−1 are defined for any (x, u)∈gph(K) as follows

f_k−ⁿ ₁(t, x) =











y_n(t), t∈^h−τ,^k−_n¹ⁱ,

yn

³_k−

1 n

´+n^³t−^k−_n^1´ µ

x−yn

³_k−

1 n

´¶

, t∈^hk−_n¹,^k_nⁱ,