On state-dependent sweeping process in Banach spaces

In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous.


1.
Introduction. The existence of solutions for sweeping processes has been studied by many authors since the pioneering work by J.J. Moreau in the 70's (see [29]). He expressed that sweeping process by the following evolution differential inclusion −u(t) ∈ N C(t) (u(t)), a.e. t ∈ [0, T ]; u(0) = u 0 ∈ C(0), where C(t) is a closed convex set in a Hilbert space H and N C(t) (.) is the normal cone to C(t) in the sense of convex analysis, see also [30] and [31]. Then, some contributions in the context of nonconvex sets C(t) were given in a series of papers, see for instance [4,17,18,34,35,36].
In the infinite Hilbert space, the authors in [9,20,21,34,3] showed the existence of solutions of (1) when the sets C(t) are prox-regular, this property got around the convexity of these sets and it is well adapted to the resolution of sweeping processes.
The study of such differential inclusions was motivated by its various applications to mechanical problems. See [28,29,30,31]. friction, see [27]. The first work in the convex case was done in the thesis of Chraibi in the finite dimensional space R 3 (see [15]), and in a Hilbert space by Kunze and Marques in [27]. Then, in the case when C(t, x) belongs to the more general class of prox-regular sets, some results were established with various type of perturbations in finite dimensional and Hilbert spaces, see [14,12,22,2,32,23]. The same problem with the class of subsmooth sets which strictly contains the class of proxregular sets, was established in [24] with an upper semicontinuous perturbation in Hilbert spaces, and in [25] with a mixed semicontinuous perturbation in a finite dimensional space. Recently, some existence results for this kind of problems with positively α-far sets were proved by Jourani and Vilches in [26].
In the framework of separable reflexive uniformly smooth Banach spaces, a detailed study of the inclusion (1) has been made by Bernicot and Venel in [6] where the process is governed by the proximal normal cone.
For first order sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces, we refer the reader to [7] for a time dependent sweeping process, to [8] for a time and state dependent non-perturbed sweeping process and to [1] for the perturbed process.
Our aim in this paper is to prove the result in [6] by considering the state dependent process. In fact, we are interested in the existence of solutions, in a separable reflexive uniformly smooth Banach space E, of the following differential inclusion where C : [0, T ] × E → E is a nonempty closed and r-prox-regular valued setvalued mapping, and F : [0, T ] × E → E a separately scalarly upper semicontinuous set-valued mapping with nonempty convex weakly compact values not necessarily bounded. The proof of our result uses ideas from [6] and [24].
The organization of this paper is as follows. In the next section we give some definitions and notation needed in the sequel and in section 3 we present our main theorem. 2. Preliminaries and notation. From now on (E, . ) is a Banach space, E is its topological dual and ., . their duality product. B E (x, r) and B E (x, r) are respectively the closed and the open ball of E of center x ∈ E and radius r > 0 while B E and B E are the closed and the open unit ball and S E is the unit sphere of E. We denote by B(E) the Borel tribe on E.
Let I := [0, T ] (T > 0). We denote by C E (I) the Banach space of all continuous mappings u : [0, T ] → E, endowed with the sup-norm . C . We also denote by L(I) the σ-algebra of Lebesgue measurable subsets of I. (L 1 E (I), . 1 ) is the quotient Banach space of Lebesgue-Bochner integrable E-valued mappings and (L ∞ E (I), . ∞ ) is the quotient Banach space of essentially bounded E-valued mappings.
We said that a mapping u : I → E is absolutely continuous if there is a mapping v ∈ L 1 E (I) such that u(t) = u(0) + t 0 v(s)ds, ∀t ∈ I, in this case v =u a.e. For A ⊂ E, co(A) denotes the convex hull of A and co(A) its closed convex hull. It is well known that if K is a nonempty subset of E, then where δ * (x , K) denotes the support function associated with K, that is Let A, B be two subsets of E, the Hausdorff distance between A and B is defined Let x be a point in A ⊂ E. We recall (see [16]) that the proximal normal cone of A at x is defined by We will adapt the notation N A (.) instead of N P A (.). We refer to [5] Lemma 1.1 for the following geometric lemma.
Lemma 2.1. Let E be a Banach space, A be a closed subset of E and s > 0. Then for x ∈ A and v ∈ E such that x ∈ P A (x + sv) we have x ∈ P A (x + λsv) for all λ ∈ (0, 1).
Proof. Let u ∈ E and z ∈ P A (u), then we have z ∈ P A (u+t(z −u)), for all t ∈ (0, 1). Indeed, for all t ∈ (0, 1), put u t = u + t(z − u). We have On the other hand, for any y ∈ A Hence z ∈ P A (u t ). Now, Let x ∈ E and v ∈ E such that x ∈ P A (x + sv), by what precedes if we take t = 1 − λ we obtain This finishes the proof of the Lemma.
In the following we give the definition and some important consequences of the prox-regularity needed in the sequel. For the proof and more details we refer the reader to [5] and [33].
Proposition 2.3. Let A be a nonempty closed and r-prox-regular subset of E and x ∈ A. Then we have the following assertions.
(i) For all x ∈ E with d(x, A) < r, the projection of x onto A is well-defined and continous, that is, . Next we recall some properties of the geometric structure of Banach spaces and we refer the reader to [19] for more details.
Definition 2.4. The Banach space (E, . ) is said to be uniformly smooth if its norm is uniformly Fréchet differentiable away of 0, it means that the limit Proposition 2.5. Let E be a uniformly smooth Banach space and p ∈ (1, ∞) be an exponent. The function x −→ x p is C 1 over the whole space E.
Definition 2.6. For E a uniformly smooth Banach space and p ∈ (1, ∞), we denote Definition 2.7. (see [6]) Let I be an interval of R. A separable reflexive uniformly smooth Banach space E is said to be "I-smoothly weakly compact" for an exponent Proposition 2.8. All separable Hilbert space H is I-smoothly weakly compact for p = 2.
The following proposition describes a useful property of weak continuity of the projection operator. For the proof we refer the reader to [6]. Proposition 2.9. Let (E, . ) be a separable, reflexive and uniformly smooth Banach space. Let C n , C : I → E be set-valued mappings taking nonempty closed values and satisfying sup t∈I H(C n (t), C(t)) −→ n→∞ 0.
We assume that for an exponent p ∈ [2, ∞) and a bounded sequence Then the projection P C(.) is weakly continuous in L ∞ E (I) (relatively to the directions given by the sequence (v n ) n ) in the following sense: for all s > 0 and for any bounded 3. Main result. Now, we are able to prove our main existence theorem.
and E be a separable, reflexive, uniformly smooth Banach space which is I-smoothly weakly compact for an exponent p ∈ [2, ∞). Let F : I × E → E be a set-valued mapping with nonempty convex weakly compact values such that ) is scalarly upper semicontinuous, that is for each e ∈ E, the scalar function δ * (e, F (t, .)) is upper semicontinuous on E. Furthermore, we suppose that for some real constant m ≥ 0 Let r > 0 and C : I × E → E be a set-valued mapping taking nonempty closed and r-prox-regular values. We assume that the following assumptions are satisfied.
(ii) For any bounded A ⊂ E, the set C(I × A) is relatively ball compact, i.e., the intersection of C(I × A) with any closed ball is relatively compact. Then for any u 0 ∈ C(0, u 0 ), the differential inclusion has a Lipschitz solution u : I → E. Moreover, we have for almost every t ∈ I Proof.
Step 1. Let n 0 ∈ N * such that For all n ≥ n 0 , consider the partition ({t n,0 }, Since F is scalarly measurable and has convex weakly compact values and H is separable, then for all x ∈ E, the mapping t → f (t, x) is Lebesgue measurable (see [13]), and by (4) it is Lebesgue-integrable.

Furthermore, we have
tn,i f (s, u n,i ) ds tn,i f (s, u n,i ) ds Hence the finite sequence {u n,i : i = 0, ...., n} is well defined satisfying (7) and (8).
It is clear that u n is a continuous mapping, that u n (t n,i+1 ) = u n,i+1 and that for almost every t ∈ I n,i For almost every t ∈ I n,i we put tn,i f (s, u n,i ) ds .
First, let us check that ∆ n (t) is a bounded vector. Using relation (8) we have for almost every t ∈ I n,i That is, ∆ n (t) ≤ M a.e. t ∈ I n,i .
Next, consider the vector v = u n,i − tn,i+1 tn,i f (s, u n,i ) ds, and put for every t ∈ I n,i C n (t) = C(t n,i+1 , u n,i ). Since C has r-prox-regular values, relations (9) and (7) give Observe that by relation (10), we have Then, applying Lemma 2.1 to relation (11) that is, Step 3. Existence of limit mapping. We will prove the convergence of the sequence (u n (.)) ⊂ C E (I).
By relations (4) and (10), we have for almost every t ∈ I, This shows that (u n (.)) is uniformly bounded by α. So (u n (.)) is a bounded sequence of C E (I) since for every t ∈ I u n (t) ≤ u 0 + t 0 u n (s) ds ≤ u 0 + T α =: β, and it is clear that (u n (.)) is equicontinuous. Let us prove that for every fixed t, the sequence (u n (t)) n≥n0 is relatively compact.
Set for every t ∈ I n,i , θ n (t) = t n,i+1 , δ n (t) = t n,i , and observe that that is, lim n→∞ δ n (t) = t. By the same calculus we have lim n→∞ θ n (t) = t. Then relation (7) and the definition of u n show that u n (θ n (t)) ∈ C θ n (t), u n (δ n (t)) ∀t ∈ I.
This last relation with (14) implies that Then, hypothesis (ii) ensures the relative compactness of the sequence (u n (θ n (t))). But, since for every t ∈ I u n (θ n (t)) − u n (t) ≤ α|θ n (t) − t| → 0 as n → +∞, we have that the sequence (u n (t)) n≥n0 is also relatively compact. By Ascoli-Arzelà theorem, we get that (u n ) is relatively compact. By extracting a subsequence (that we do not relabel), we conclude that (u n ) converges uniformly to some mapping u ∈ C E (I).
If F is a single-valued mapping Theorem 3.1 reads as Corollary 3.2. Let I = [0, T ] (T > 0) and E be a separable, reflexive, uniformly smooth Banach space which is I-smoothly weakly compact for an exponent p ∈ [2, ∞).
Let f : I × E → E be a Carathéodory mapping, that is for each x ∈ E f (., x) is Lebesgue-measurable and for each t ∈ I f (t, .) is continuous on E. Furthermore, we suppose that for some real constant m ≥ 0 f (t, x) ≤ m, ∀(t, x) ∈ I × E.

Comments.
(1) Our main theorem is new even when the sets in the process are assumed to be convex.
(2) In [27], [14], [12] and [32], the authors used in their proofs the implicit algorithm u n,i = P C(tn,i,un,i) (u n,i−1 ) thanks to Darbo or Schauder fixed point theorems applied to the continuous mapping v → P C(t,v) (u). The proof of the continuity of this mapping uses the Euclidean or Hilbert structure of the space. In our study, we have not been able to prove this continuity in the context of Banach spaces, that is an open question that attracts our interest.
(3) In the proof of our main theorem it was clear that the inequality k 2 < 1 was necessary to obtain our result, and this inequality cannot be relaxed. In fact, in [27] the authors showed by concrete examples that this kind of problems may have no solution when k 2 ≥ 1.
(4) In the vein of [27], [1] and others, let us consider for example E = L p (Z) which is a separable, reflexive, uniformly smooth Banach space and I-smoothly weakly compact for p ≥ 2 (see [6]), and let J : E → E the duality mapping of E. If we consider C and F as in Theorem 3.1 and we consider the problem of finding an absolutely continuous mapping u : I → E such that there exists a measurable mapping g : I → E satisfying for almost every t ∈ I, u(t) ∈ C(t, u(t)), g(t) ∈ F (t, u(t)), u(0) = u 0 ∈ C(0, u 0 ) and u(t) + g(t), J u(t) + g(t) − J u(t) + g(t) + w − u(t) ≤ w − u(t), J u(t) + g(t) + w − u(t) for all w ∈ C(t, u(t)). Then we can show that this problem can be rewritten as our differential inclusion (P F ). Consequently, Theorem 3.1 ensures the existence of a Lipschitz solution to the latter.