CUBO A Mathematical Journal Vol.10, N o ¯ 01, (43–66). March 2008 Arc-wise Essentially Tangentially Regular Set-valued Mappings and their Applications to Nonconvex Sweeping Process Messaoud Bounkhel King Saud University, Department of Mathematics P.O. Box 2455, Riyadh 11451, Saudi Arabia. e-mail: bounkhel@ksu.edu.sa ABSTRACT Recently, Borwein and Moors (1998) introduced a new class of tangentially reg- ular sets in IR n (called arc-wise essentially smooth sets). They characterized the sets S of this class in terms of arc-wise essential smoothness of the distance function dS . Very recently, the author (2002) gave an appropriate extension of this class to any Banach space X and he extended the above characterization to any Banach space X with a uniformly Gâteaux differentiable norm. In this paper we extend the concept of arc-wise essentially smooth sets to set-valued mappings C : [0,T ]⇉X (T > 0) and we will use this concept to establish an important application to nonconvex sweeping process. RESUMEN Ricientemente Borwein y Moors (1998) introducem una nueva clase de conjuntos tangencialmente regulares en IR n chamados conjuntos essencialmente suaves por arcos). Ellos caracterizan los conjuntos S de esta clase en terminos de la suavidad de la distancia por arco de la función dS . Ricientemente, el autor (2002) dió una 44 Messaoud Bounkhel CUBO 10, 1 (2008) extensión apropriada de esta clase en cualquer espacio de Banach X y extiende la caractarización anterior a cualquer espacio de Banach X y con una norma de Gâteaux uniformemente diferenciable. En este art́ıculo extendemos el concepto de conjunto essencialmente suave por arcos para el conjunto de aplicaciones C : [0,T ]⇉X (T > 0) y usaremos este concepto para establecer una importante aplicación a procesos no convexos generales. Key words and phrases: Nonconvex Sweeping Process, Tangentially Regular Sets. Math. Subj. Class.: 34G25. 1 Introduction In [9] Borwein and Moors introduced, in IR n , the concept of arc-wise essential smoothness for sets and for functions. They characterized the class of all sets S which are arc-wise essentially smooth in terms of arc-wise essential smoothness of the distance function dS . Their definitions and results were strongly based on the finite dimensional structure. In [10] the author gave an appropriate extension of the arc-wise essentially smooth concept for sets and functions in any Banach space and he extended the above characterization of the class of arc-wise essentially smooth sets in any Banach space with a uniformly Gâteaux differentiable norm. In this paper we intend to extend the concept of arc-wise essentially smooth sets to set-valued mappings and to give some applications of this new concept of regularity of set-valued mappings. The paper is organized as follows. In section two we recall some notations and preliminaries that are used in the paper. Section three is devoted to introduce and to study the new concept of arc-wise essentially tangentially regular set-valued mappings. Many examples of this class of set-valued mappings are given in this section. We prove in this section various characterizations of arc-wise essentially tangentially regular set-valued mappings. The main characterization is given in Theorem 3.3 which establishes a relationship between arc-wise essentially tangential regularity of a set-valued mapping C and the arc-wise essentially smoothness of the distance function to the images of the set-valued mapping C. In the last section, we give an important application of this characterization to the nonconvex sweeping process. 2 Preliminaries Throughout, X will be a real Banach space and X ∗ its topological dual. By 〈 ·, · 〉 we will denote the canonical pairing between these spaces. Recall that a function f from X into IR is Lipschitz around x0 ∈ X if there exist two real numbers K > 0 and δ > 0 such that |f(x′) − f(x)| ≤ K‖x′ − x‖ for all x′,x ∈ x0 + δIB, CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 45 where IB denotes the closed united ball of X centered at the origin. We will say that f is locally Lipschitz over X if it is Lipschitz around any point of X. Recall also that the usual directional derivative of f at x0 in the direction v is, f ′ (x0; v) := lim t→0 t −1 [ f(x0 + tv) − f(x0) ] , when this limit exists. For a locally Lipschitz function f : X → IR, we recall that the Clarke generalized directional derivative (resp. the lower Dini directional derivative) of f at x0 ∈ X in the direction v is given by, f 0 (x0; v) := lim sup x→x0 t↓0 t −1 [ f(x + tv) − f(x) ] , ( resp. f − (x0; v) := lim inf t↓0 t −1 [ f(x0 + tv) − f(x0) ] . ) One always has f − (x0; v) ≤ f 0 (x0; v). The reverse inequality is not true in general (take for example f(x) = −‖x‖). The functions f satisfying the equality form in the last inequality are called directionally regular at x0 in the direction v. Recall also that a locally Lipschitz function f : X → IR is strictly differentiable (in short s.d.) at x0 in the direction v if f 0 (x0; v) = −f 0 (x0; −v). It is not difficult to check that, if f is s.d. at x0 in the direction v, then one has f 0 (x0; v) = f − (x0; v) = f ′ (x0; v) = −f 0 (x0; −v) and so it is directionally regular at x0 in the direction v. Recall now, that the Clarke subdifferential (resp. Fréchet subdifferential ) of f at x0 ∈ X is defined by ∂ C f(x0) = {x ∗ ∈ X∗ : 〈 x ∗ ,v 〉 ≤ f0(x0; v), for all v ∈ X}, (resp. ∂ F f(x0) = {x ∗ ∈ X∗ : ∀ǫ > 0,∃δ > 0 : 〈 x ∗ ,x−x0 〉 ≤ f(x)−f(x0)+ǫ‖x−x0‖, ∀x ∈ x0+δIB}). Let S be a nonempty subset of X. We will let d(·,S) (or dS (·)) stand for the usual distance function to S, i.e., d(x,S) := inf u∈S ‖x − u‖. Recall (see [20]) that the Clarke tangent cone and the contingent cone of S at some point x ∈ S are given respectively by TS(x) = {v ∈ X : d 0 C (x; v) = 0}, (2.3) 46 Messaoud Bounkhel CUBO 10, 1 (2008) and KS (x) = {v ∈ X : d − S (x; v) = 0}. (2.4) Note that one always has TS(x) ⊂ KS (x). The sets S for which one has an equality in the last inclusion, will be called tangentially regular at x (see [20] for this definition). Let us recall (see for instance [12]) that the Clarke normal cone (resp. Fréchet normal cone ) of S at x ∈ S is defined by N C S (x) = {x ∗ ∈ X∗ : 〈 x ∗ ,v 〉 ≤ 0, for all v ∈ TS (x)}, (resp. N F S (x) = {x ∗ ∈ X∗ : ∀ǫ > 0,∃δ > 0 : 〈 x ∗ ,x ′ − x 〉 ≤ ǫ‖x′ − x‖, ∀x′ ∈ x + δIB}). The following proposition is needed in the sequel. It was proved for the first time by Kruger [25] (see also Iofee [26].) Proposition 2.1 [12] Let S be a nonempty closed subset in X and let x ∈ S. Then ∂ F dS (x) = N F S (x) ∩ IB. Let I be an interval and let Ω be an open subset of X. By absolutely continuous mapping one means a mapping x : I → Ω such that x(t) = x(a) + ∫ t a x ′ (s)ds, for all t ∈ I, with x ′ ∈ L1X (I) and a ∈ I. We will denote by AC(I, Ω) the familly of all these mappings. Remark 2.1 It is well known (see for instance [15]) that F ◦ x(·), the composition of a locally Lipschitz mapping F : Ω → Y with an absolutely continuous mapping x : I → Ω, is an absolutely continuous mapping, whenever the space Y is reflexive. For more details concerning absolutely continuous mappings we refer the reader to Brézis [15]. 3 Arc-wise essentially tangentially regular set-valued mappings We start with the following definition of arc-wise essentially tangentially regular set-valued mappings: Definition 3.1 Let I :=]0, 1[ and let C : I⇉X be a set-valued mapping with nonempty closed values. We will say that C is arc-wise essentially tangentially regular and we will write C ∈ AWET R(I,X), if for each x ∈ AC(I,X), the set {t ∈ I : x(t) ∈ C(t) and x′(t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} has null measure. CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 47 In this paper we use the name arc-wise essential tangential regularity instead of arc-wise essential smoothness (used in [10] and [7, 8, 9]) because it seems for us that is more significant. Remark 3.1 As one always has KS (x) = TS (x) = X, for each x ∈ intS (the topological interior of S), we can take x only in bd C(t) (the boundary of C(t)), in Definition 3.1, that is, C is arc-wise essentially tangentially regular if and only if for each x ∈ AC(I,X) one has µ ( {t ∈ ] 0, 1 [ : x(t) ∈ bdC(t) and x′(t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} ) = 0. Example 3.1 1. It is easy to see that all set-valued mappings C : I⇉X with closed tangentially regular values are arc-wise essentially tangentially regular. 2. Let S be a fixed set in X which is arc-wise essentially smooth in the sense of [9, 10]. Then using Proposition 4.1 in [10] we can check that the constant set-valued mapping C : I⇉X with C(t) = S is arc-wise essentially tangentially regular. 3. Let S be a fixed set in X which is tangentially regular at each of its points except one point x0 ∈ S. Define the set-valued mapping C as the translation of the set S in the direction v(t), that is, C(t) = S + v(t), with v ∈ AC(I,X). (1) Assume now that v satisfies ±v′(t) 6∈ KS(x0) \ TS (x0), a.e. on I. Then C is an arc-wise essentially tangentially regular set-valued mapping. Indeed, for any x ∈ AC(I,X) we can easily check that µ ( {t ∈ I : x(t) ∈ bdC(t) and x′(t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} ) = µ ({t ∈ I : x(t) = v(t) + x0 and v ′ (t) or − v′(t) ∈ KS (x0) \ TS (x0)}) = 0. Take for example X = IR 2 , S1 is the epigraph of the absolute value function, and take S is the closure of the complement of S1. Take v(t) = (t, 2t), for all t ∈ I \ N and v(t) = (t, 1) for all t ∈ N, where N is a subset of I with null measure. Using what precedes we can easily check that the set-valued mapping C in (1) associated with the set S and v is arc-wise essentially tangentially regular. 4. More general and with the same manner we can prove that the set-valued mapping C in (1) is arc-wise essentially tangentially regular whenever the set S is tangentially regular at each of its points except on a countable set {xn} and with v satisfies ±v′(t) 6∈ KS (xn) \ TS (xn), for all n and a.e. on I. (2) 48 Messaoud Bounkhel CUBO 10, 1 (2008) 5. The condition (2) on v cannot be removed in the last example. Take for example S is the closure of the complement of the epigraph of the absolute value function and take v(t) = (t, 1), for all t ∈ I. The condition (2) is not satisfied and we can check that for some x ∈ AC(I,X) one has µ ( {t ∈ I : x(t) ∈ bdC(t) and x′(t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} ) = 1, and so the set-valued mapping C in this case is not arc-wise essentially tangentially regular. From this example we can conclude that a set-valued mapping with values C(t) tangentially regular except on countable set is not necessarily arc-wise essentially tangentially regular. 6. Let C0 be the Cantor ternary set with 0 ∈ C0. Let C(t) = C0 + t. We claim that C 6∈ AWET R((0, 1),IR). Let x(t) = t. Then {t ∈ (0, 1) : x(t) ∈ C(t) with − x′(t) or x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} = {t ∈ (0, 1) : 0 ∈ C0 with x ′ (t) 6= 0} = (0, 1). and so µ({t ∈ (0, 1) : x(t) ∈ C(t) with − x′(t) or x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))}) 6= 0, which ensures that C is not AWET R((0, 1),IR). A first question, which arises naturally, is to ask whether the epigraph set-valued mapping t⇉C(t) := epift is arc-wise essentially tangentially regular, where epift := {(x,r) ∈ X × IR : f(t,x) ≤ r}. To give an answer to this question we introduce the following concepts. Let f : IR×X → IR be a function from IR × X to IR. We define the following directional derivatives of f at (t0,x0) ∈ IR × X in a direction v ∈ X by f 0 ((t0,x0); v) := lim sup x→x0 (δ,t)↓(0,t0 ) δ −1 [ f(t,x + δv) − f(t,x) ] , and f − ((t0,x0); v) := lim inf (δ,t)↓(0,t0) δ −1 [ f(t,x0 + δv) − f(t,x0) ] . It is clear that if the two above limits exist then the Clarke and the lower Dini direc- tional derivatives of ft0 (·) := f(t0, ·) at x0 in the direction v exist and equal respectively to f 0 ((t0,x0); v) and f − ((t0,x0); v). The converse is not true in general, take for example f(t,x) = f1(t)f2(x) with f1 is not right continuous. CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 49 Definition 3.2 We will say that f is arc-wise essentially directionally regular and we will write f ∈ AWEDR(I × X), if for each x ∈ AC(I,X), the set {t ∈ I : f−((t,x(t)); x′(t)) 6= f0((t,x(t)); x′(t))} has null measure. Example 3.2 1. Any mapping f defined as follows f(t,x) = f1(t) + f2(x), is arc-wise essentially directionally regular whenever f2 is directionally regular and without any assumptions on f1. 2. Any mapping f defined as follows f(t,x) = f1(t)f2(x), is arc-wise essentially directionally regular whenever f2 is directionally regular and f1 is continuous. Recall that (see for instance [20]) for a function f : IR × X → IR one has for all t ∈ IR Kepi ft ((x,ft(x)) = epif − t (x; ·), (3) and Tepi ft ((x,ft(x)) = epif 0 t (x; ·). (4) Now, we are ready to state the following result showing that the epigraph set-valued mapping C(t) := epift is arc-wise essentially tangentially regular whenever f is arc-wise essentially directionally regular. Theorem 3.1 Let I be an open interval and let f : I × X → IR be a locally Lipschitz function from I ×X to IR. Then the set-valued mapping C : I → X ×IR defined by t⇉epift is arc-wise essentially tangentially regular whenever the function f is arc-wise essentially directionally regular. Proof. Put C(t) = epift and suppose that f is arc-wise essentially directionally regular. Let (x,r) ∈ AC(I,X × IR) and put J1 := {t ∈ I : f − ((t,x(t)); x ′ (t)) = f 0 ((t,x(t)); x ′ (t))}, and J2 := {t ∈ I : (x(t),r(t)) ∈ bdC(t) and (x ′ (t),r ′ (t)) ∈ KC(t)(x(t),r(t)) \ TC(t)(x(t),r(t))}. 50 Messaoud Bounkhel CUBO 10, 1 (2008) First, we have µ(J1) = 1 because f is arc-wise essentially directionally regular. Assume that there exists some t0 ∈ J1 ∩ J2. Then f 0 ((t0,x(t0)); x ′ (t0)) and f − ((t0,x(t0)); x ′ (t0)) exist and coincide and they equal to f 0 t0 (x(t0); x ′ (t0)) = f − t0 (x(t0); x ′ (t0)). We also have x ′ (t0) and r ′ (t0) exist and such that (x(t0),r(t0)) ∈ bdC(t0) ( that is, r(t0) = ft0 (x(t0)), because the boundary of C(t0) is the graph of ft0 ) and (x ′ (t0),r ′ (t0)) ∈ KC(t)(x(t0),r(t0)) \ TC(t)(x(t0),r(t0)). (5) Further, (3) and (4) yield r ′ (t0) ≥ f − t0 (x(t0); x ′ (t0)) = f 0 t0 (x(t0); x ′ (t0)) which means (x ′ (t0); r ′ (t0)) ∈ TC(t0)(x(t0),r(t0)) that is a contradiction with (5). Therefore, we obtain J1 ∩J2 = ∅ and hence as µ(J1) = 1 we get µ(J2) = 0. So, C is arc-wise essentially tangentially regular. 2 The following theorem states a necessary condition on f for the arc-wise essential tan- gential regularity of the epigraph set-valued mapping epift. Its proof follows some ideas from [9]. Theorem 3.2 Let I be an open interval and let f : I × X → IR be a locally Lipschitz function from I × X to IR. If the set-valued mapping C : I → X × IR defined by t⇉epift is arc-wise essentially tangentially regular, then the function f is arc-wise essentially strictly differentiable function ( i.e., f ∈ AWESD(X,IR)), in the following sense: for any x ∈ AC(I,X), one has µ ( {t ∈ I : f0t (x(t); −x ′ (t)) 6= −f0t (x(t); x ′ (t))} ) = 0. (6) Proof. Suppose that C is arc-wise essentially tangentially regular and fix any x ∈ AC(I,X). Since f is Lipschitz then by Remark 2.1 the function θ(t) := f(t,x(t)) ∈ AC(I,X) and so θ ′ (t) exists a.e. on I. Fix now any t ∈ I such that x′(t) and θ′(t) exist. Then, by (4) one has (x ′ (t),f 0 (x(t); x ′ (t)) ∈ TC(t)(x(t),f(x(t))). By (3), one also has (x(t),ft(x(t))) ∈ C(t) and (x ′ (t),f ′ t(x(t); x ′ (t)) ∈ KC(t)(x(t),f(x(t))). Put E := E1 ∪ E2 where E1 := {s ∈ E3 : (x ′ (s),θ ′ (s)) ∈ KC(s)(x(s),θ(s)) \ TC(s)(x(s),θ(s))}, E2 := {s ∈ E3 : (−x ′ (s),−θ′(s)) ∈ KC(s)(x(s),θ(s)) \ TC(s)(x(s),θ(s))}, and E3 := {s ∈ I : (x(s),θ(s)) ∈ C}. CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 51 As C is arc-wise essentially tangentially regular one gets by Definition 3.1 that µ(E) = 0. If one assumes further that t /∈ E, we obtain (x ′ (t),θ ′ (t)) ∈ TC(t)(x(t),θ(t)) = epif 0 t (x(t); ·), and (−x′(t),−θ′(t)) ∈ TC(t)(x(t),θ(t)) = epif 0 t (x(t); ·). This means respectively f 0 t (x(t); x ′ (t)) ≤ θ′(t) and f 0 t (x(t); −x ′ (t)) ≤ −θ′(t), which yields f 0 t (x(t); x ′ (t)) ≤ −f0t (x(t); −x ′ (t)) and hence, because the reverse inequality always holds, one gets f 0 t (x(t); x ′ (t)) = −f0t (x(t); −x ′ (t)). Thus, the set Ẽ := {s ∈ I : f0t (x(s); x ′ (s)) 6= −f0t (x(s); −x ′ (s))}, is included in E and so µ(Ẽ) = 0. The proof then is complete. 2 Remark 3.2 Combining Theorems 3.1-3.2 we get the following inclusion: AWEDR(X,IR) ⊂ AWESD(X,IR). (7) This means that any arc-wise essentially directionally regular is arc-wise essentially strictly differentiable in the sense of (6). This inclusion is strict. Take for example the function f in Example 3.2 part (2) with f1 is not continuous. The following lemma will be used in the sequel. Lemma 3.1 Let f be a locally Lipschitz function defined from X into IR and let x0,v ∈ X. Then f is s.d. at x0 in the direction v if and only if 〈 ∂ C f(x0),v 〉 = {f0(x0; v)} iff 〈 ∂ C f(x0),v 〉 is a singleton set. Here 〈 ∂ C f(x),v 〉 := { 〈 x ∗ ,v 〉 : x ∗ ∈ ∂Cf(x)}. Proof. It is clear that it suffices to prove the following relation: 〈 ∂ C f(x0),v 〉 = [−f0(x0; −v),f 0 (x0; v)]. By (b) in Proposition 2.1.2 in [20] one has f 0 (x0; v) = max 〈 ∂ C f(x0),v 〉 and hence −f0(x0; −v) = min 〈 ∂ C f(x0),v 〉 and as the set 〈 ∂ C f(x0),v 〉 is convex, one obtains the desired equality. 2 When the function f does not depend on t, that is, f : X → IR, it is easy to see (by Proposition 3.1 in [10]) that the concept of arc-wise essential strict differentiability in the sense of Theorem 3.2, is equivalent to the concept of arc-wise essential smoothness in the sense of [10]. The next corollary summarizes further characterizations of arc-wise essentially smooth functions. 52 Messaoud Bounkhel CUBO 10, 1 (2008) Corollary 3.1 Let I be an open interval and let f : X → IR be a locally Lipschitz function. Then the following assertions are equivalent: 1. f is arc-wise essentially smooth in the sense of [10]; 2. f is arc-wise essentially strictly differentiable; 3. epif is arc-wise essentially tangentially regular; 4. f is arc-wise essentially directionally regular; 5. for each x ∈ AC( ] 0, 1 [ ,X) µ ( {t ∈ ] 0, 1 [ : 〈 ∂ C f(x(t); x ′ (t)) 〉 = {f0(x(t); x′(t))}}) = 1. 6. for each x ∈ AC( ] 0, 1 [ ,X) µ ( {t ∈ ] 0, 1 [ : f 0 (x(t); x ′ (t)) = (f ◦ x)′(t)} ) = 1; 7. for each x ∈ AC( ] 0, 1 [ ,X) µ ( {t ∈ ] 0, 1 [ : f 0 (x(t); x ′ (t)) = f ′ (x(t); x ′ (t))}) = 1. Proof. The following equivalences follow from Theorems 3.1-3.2, Lemma 3.1 and by what precedes the corollary: (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5). (6) ⇔ (7) : Fix any x ∈ AC( ] 0, 1 [ ,X) and fix also any t ∈]0, 1[ where x′(t) exists. If we put δ = min{t, 1 − t}, the Lipschitz behavior of f ensures for all s ∈] − δ,δ[ s −1 [(f ◦ x)(t + s) − (f ◦ x)(t)] = s−1[f(x(t) + sx′(t)] + ǫ(s) (8) with lim s→0 ǫ(s) = 0. Therefore, for any such t , (f ◦ x)′(t) exists if and only if f′(x(t); x′(t)) exists. The equivalence then holds. (4) ⇔ (6) : Fix any x ∈ AC( ] 0, 1 [ ,X) and t ∈]0, 1[ such that x′(t) and (f ◦x)′(t) exist. Note that the set of such points t has 1 as Lebesgue measure because x and f ◦ x are absolutely continuous, and note also that, by (8), for any such t (f ◦ x)′(t) = f−(x(t); x′(t)). So, the equivalence follows. 2 Using Theorem 3.1, we get the following examples of arc-wise essentially tangentially regular set-valued mappings: CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 53 1. The translation of the epigraph of directionally regular functions in any direction of y-axis, i.e., C(t) = epif2 + (0,f1(t)), with f2 is a directionally regular function and f1 is an arbitrary function. 2. The set-valued mapping C : IR⇉IR 2 C(t) = {(x,f1(t)r) : f2(t) ≤ r}, where f2 : IR → IR is directionally regular and f1 : IR → IR is continuous with f1 6≡ 0. Now we are going to establish a characterization of the class of arc-wise essentially tangen- tially regular set-valued mappings C, in terms of the distance function to the images of the set-valued mapping C. Its proof follows some ideas from [10]. It will be used to give an important application to nonconvex sweeping processes. In the proof of this theorem, we need the following characterization of the contingent cone KC (x). A vector v ∈ KC (x) if and only if there exist two sequences {tn}n∈IN of positive real numbers converging to zero and {vn}n∈IN in X converging to v such that x + tnvn ∈ C, for each n ∈ IN. Theorem 3.3 Let C : I⇉X be a set-valued mapping with nonempty closed values. Assume that dC(·)(·) is arc-wise essentially strictly differentiable. Then C is arc-wise essentially tan- gentially regular. If, in addition, X is a Banach space with uniformly Gâteaux differentiable norm, then C is arc-wise essentially tangentially regular if and only if dC(·)(·) is arc-wise essentially strictly differentiable. Proof. 1) Assume that dC(·)(·) ∈ AWESD(X,IR), i.e., for each x ∈ AC( ] 0, 1 [ ,X), the set A := {t ∈ ] 0, 1 [ : dC(t)(·) is not s.d. at x(t) in the direction x ′ (t) } has null measure. We will show that C is arc-wise essentially tangentially regular, i.e., µ(B) = 0 where B := {t ∈ ] 0, 1 [ : x(t) ∈ C(t) and x′(t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))}). It is enough to prove that B ⊂ A. Let t0 /∈ A. If x(t0) 6∈ C(t0), then t0 6∈ B. So let us suppose that x(t0) ∈ C(t0). If x ′ (t) and −x′(t) 6∈ KC(t)(x(t))\TC(t)(x(t)), then t0 6∈ B. So, assume that x ′ (t) or − x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t)). This ensures d − C(t0) (x(t0); x ′ (t0)) = 0 or d − C(t0) (x(t0); −x ′ (t0)) = 0. Since dC(t0) is s.d. at x(t0) ∈ C(t0) in the direction x ′ (t0), i.e., we have d 0 C(t0) (x(t0); x ′ (t0)) = −d 0 C(t0) (x(t0); −x ′ (t0)). On the other hand, the strict differentiability ensures the directional regularity, that is, d − C(t0) (x(t0); x ′ (t0)) = d 0 C(t0) (x(t0); x ′ (t0)) and d − C(t0) (x(t0); −x ′ (t0)) = d 0 C(t0) (x(t0); −x ′ (t0)) and hence d − C(t0) (x(t0); x ′ (t0)) =d 0 C(t0) (x(t0); x ′ (t0)) =−d 0 C(t0) (x(t0); −x ′ (t0)) =−d − C(t0) (x(t0); −x ′ (t0)). 54 Messaoud Bounkhel CUBO 10, 1 (2008) So in both cases d − C(t0) (x(t0); x ′ (t0)) = 0 or d − C(t0) (x(t0); −x ′ (t0)) = 0, we obtain d 0 C(t0) (x(t0); x ′ (t0)) = d 0 C(t0) (x(t0); −x ′ (t0)) = 0, that is, x ′ (t0) ∈ TC(t0)(x(t0)) and −x ′ (t0) ∈ TC(t0)(x(t0)). Thus, both directions x ′ (t0) and −x ′ (t0) lie in TC(t0)(x(t0)) and hence t0 /∈ B. Consequently, each t0 /∈ A does not lie in B. This completes the proof of the inclusion B ⊂ A. 2) Assume now that X is a Banach space with uniformly Gâteaux differentiable norm and assume that C is arc-wise essentially tangentially regular. Then, for each fixed x in AC( ] 0, 1 [ ,X) by Definition 3.1 we have µ(Bx) = 0, where Bx = B 1 x ∪ B 2 x, B 1 x := {t ∈ ] 0, 1 [ : x(t) ∈ C(t) and x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))} and B 2 x := {t ∈ ] 0, 1 [ : x(t) ∈ C(t) and −x′(t) ∈ KC(t)(x(t)) \ TC(t)(x(t))}. Put A := {t ∈ ] 0, 1 [ : dC(t) is not s.d. at x(t) in the dir. x ′ (t) }. It is not difficult to check that A = {t ∈ ] 0, 1 [ : x(t) ∈ bdC(t), dC(t) is not s.d. at x(t) in the dir. x ′ (t) }. Indeed, if t ∈ ] 0, 1 [ with x(t) ∈ (X \ C(t)) ∪ intC(t) and dC(t) is not s.d. at x(t) in the direction x ′ (t), then (−dC(t)) is not s.d. at x(t) in the direction x ′ (t) and so (−dC(t)) is not directionally regular at x(t) in the direction x ′ (t), which is impossible, because x(t) ∈ (X \C(t))∪intC(t), and Theorem 8 in [2]. Put now Dx′ := {t ∈ ] 0, 1 [ : x ′ (t) exists }, hence µ(A \ Dx′ ) = 0. (9) Put also I := Ir ∪ Il with Ir ( resp. Il ) denotes the set of all isolated points in A ∩ Dx′ relatively to the right topology ( resp. the left topology). It is not difficult to check that I is countable and hence µ(I) = 0. Fix t0 ∈ (A ∩ Dx′ ) \ I. Then there exist two sequences of real positive numbers (λn)n and (ǫn)n converging to zero such that for n sufficiently large t0 + λn and t0 − ǫn lie in (A ∩ Dx′ ) \ I and hence x(t0 + λn) ∈ bd C(t0 + λn) and x(t0 − ǫn) ∈ bd C(t0 − ǫn), for n sufficiently large. Put vn := λ −1 n [ x(t0 + λn) − x(t0) ] and wn := ǫ −1 n [ x(t0 − ǫn) − x(t0) ] . Clearly, vn → x ′ (t0) and wn → −x ′ (t0) and for n sufficiently large x(t0) + λnvn ∈ bd C(t0 + λn) ⊂ C(t0) + RCλnIB ⊂ clC(t0) = C(t0) CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 55 and x(t0) + ǫnwn ∈ bd C(t0 − ǫn) ⊂ C(t0) + RCǫnIB ⊂ cl C(t0) = C(t0). It follows (by the characterization given above of the contingent cone) that x ′ (t0) and −x ′ (t0) lie in KC(t0)(x(t0)). Now, we distinguish two cases. Firstly, if x ′ (t0) ∈ KC(t0)(x(t0)) \ TC(t0)(x(t0)), then t0 ∈ Bx. Secondly, if x ′ (t0) ∈ TC(t0)(x(t0)), then −x ′ (t0) ∈ KC(t0)(x(t0))\ TC(t0)(x(t0)) ( because, if −x′(t0) ∈ TC(t0)(x(t0)), we would have d 0 C(t0) (x(t0); x ′ (t0)) = −d0 C(t0) (x(t0); −x ′ (t0)) = 0, so dC(t0) would be s.d. at x(t0) in the direction x ′ (t0), which would contradict that t0 ∈ A ) . Hence t0 ∈ Bx. Thus (Dx′ ∩ A) \ I ⊂ Bx and hence µ((Dx′ ∩ A) \ I) = 0. (10) Finally, according to (9) and (10), we obtain µ(A) = 0. This ensures that dC(t0) ∈ AWESD(X,IR) and hence the proof is finished. 2 The following corollary follows from Theorem 3.3 and Lemma 3.1. It will be used in the next section. Corollary 3.2 Let H be a Hilbert space. A set-valued mapping C : I⇉cl(H) is arc-wise essentially tangentially regular if and only if for each x ∈ AC(I,H) one has µ ({ t ∈ I : 〈 ∂ C dC(t)(x(t)),x ′ (t) 〉 6= {d0C(t)(x(t); x ′ (t))} }) = 0. (11) 4 Applications to nonconvex sweeping process Throughout this section, we will let H (resp. cl(H)) denote a separable Hilbert space (resp. the collection of all nonempty closed sets in H). Let F : H⇉H be a set-valued mapping from H to H. We will say that F is Hausdorff upper semicontinuous (for more details on Hausdorff upper semicontinuity see [23, 16]) if for any y ∈ H one has lim sup x→x̄ e(F(x),F(y)) ≤ e(F(x̄),F(y)), wheree e(A,B) := sup a∈A [ inf b∈B ‖b − a‖ ] = sup a∈A dB(a). In all the sequel T > 0, I := [0,T ], and C : IR⇉cl(H) will denote a L ′ -Lipschitz set-valued mapping (L ′ > 0) with nonempty closed values, i.e., for any y ∈ H and any t,s ∈ I |d(y,C(t)) − d(y,C(s))| ≤ L′|t − s|. 56 Messaoud Bounkhel CUBO 10, 1 (2008) We prove in the following theorem our main application of the concept of arc-wise essentially tangentially regular set-valued mappings. It proves a stability result for noncon- vex sweeping processes with nonconvex noncontinous perturbation. Let us note that our assumption on F requiring the inclusion in the subdifferential of some function was intro- duced for the first time in the work by [14] and by many other authors (see for instance [1, 3, 4, 11, 28, 30]). Theorem 4.1 Assume that C : [0,T ]⇉H is arc-wise essentially tangentially regular and it has ball compact values. Let F : H⇉H be Hausdorff u.s.c. on H contained in the subdifferential of a directionally regular locally Lipschitz function ψ : H → IR. Let {xn(·)}n be a bounded sequence in AC(I,H) (that is, ‖xn(t)‖ ≤ M, for some M > 0, for any n and any t ∈ I) such that (NSPP)        x ′ n(t) ∈ −N F C(t)(xn(t)) + fn(t) + bn(t) a.e. on [0,T ]; fn(t) ∈ F(xn(θn(t))) and bn(t) ∈ rn(t)IB a.e. on [0,T ] xn(t) ∈ C(t), ∀t ∈ [0,T ]; xn(0) = x0 ∈ C(0), where fn,bn ∈ L 2 (I,H) and rn(t) → 0 + uniformly on I, and θn(t) → t for all t ∈ [0,T ], and ‖x′n(t)‖ ≤ L ′′ a.e. on [0,T ]. Then there exist b ∈]0,T ] and x ∈ AC([0,b],H) such that    x ′ (t) ∈ −NC C(t) (x(t)) + F(x(t)) a.e. on [0,T ]; x(t) ∈ C(t), ∀t ∈ [0,T ]; x(0) = x0 ∈ C(0), Proof. Let α > 0 such that ψ is Lipschitz on x0+αIB with ratio L > 0. Put b := min{ α L′′ ,T} and I := [0,b]. Let (fn)n and (bn)n in L 2 (I,H) such that fn(t) ∈ F(xn(θn(t))) and bn(t) ∈ rn(t)IB a.e. on I. So we have by (NSPP) −x′n(t) + fn(t) + bn(t) ∈ N F C(t)(xn(t)) a.e. on I Since {xn(·)}n is bounded sequence in AC(I,H) and C has ball compact values we get the set {xn(t) : n ≥ 1} is relatively strongly compact in H. Thus, as ‖x ′ n(t)‖ ≤ L ′′ , we get by Ascoli-Arzela’s theorem xn → s x in AC(I,H), x ′ n → w x ′ in L 2 (I,H). Since ‖xn(t) − x0‖ ≤ ∫ t 0 ‖x′n(s)‖ds ≤ L ′′ b ≤ α, for all t ∈ I we obtain fn(t) ∈ F(xn(θn(t))) ⊂ ∂ψ(xn(θn(t))) ⊂ LIB, CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 57 and so we get for n0 large enough ‖ − x′n(t) + fn(t) + bn(t)‖ ≤ L ′′ + L + 1 n0 for all n ≥ n0. Thus, Proposition 2.1 ensures for σ := L ′ + L + 1 n0 and for a.e. t ∈ I −x′n(t) + fn(t) + bn(t) ∈ N F C(t)(xn(t)) ∩ σIB = σ∂ F dC(t)(xn(t)). We can thus apply Castaing techniques (see for instance [17]). The convergence of the sequences {rn}n and {xn}n to 0 and x respectively and the weak convergence of the sequences {x′n}n and {fn}n to x ′ and f, and using Mazur’s lemma yield −x′(t) + f(t) ∈ σ∂CdC(t)(x(t)) and f(t) ∈ ∂ C ψ(x(t)). (12) Now, since the function ψ is directionally regular we obtain, by Corollary 3.1 (ψ ◦ x)′(t) = ψ0(x(t); x′(t)) = 〈 f(t),x ′ (t) 〉 and so ∫ b 0 ψ 0 (x(t); x ′ (t))dt = ∫ b 0 〈 f(t),x ′ (t) 〉 dt. On one hand, as fn(t) ∈ ∂ψ(xn(θn(t))) one has 〈 fn(t),x ′ n(t) 〉 ≤ ψ0(xn(θn(t)); x ′ n(t)), because ψ is regular. On the other hand, since ψ is directionally regular we get ψ 0 (xn(θn(t)); x ′ n(t)) = ψ ′ (xn(θn(t)); x ′ n(t)) and ψ 0 (x(t); x ′ (t)) = ψ ′ (x(t); x ′ (t)) and so by Theorem 2.1 in [2] we obtain lim sup n ∫ b 0 ψ 0 (xn(θn(t)); x ′ n(t))dt = lim sup n ∫ b 0 ψ ′ (xn(θn(t)); x ′ n(t))dt ≤ ∫ b 0 ψ ′ (x(t); x ′ (t))dt = ∫ b 0 ψ 0 (x(t); x ′ (t))dt. Consequently, we get lim sup n ∫ b 0 〈 fn(t),x ′ n(t) 〉 dt ≤ ∫ b 0 〈 f(t),x ′ (t) 〉 dt. Coming back to (12) and using the fact C is arc-wise essentially tangentially regular and the fact that x(t) ∈ C(t) for all t ∈ I, we get (by Corollary 3.2) for a.e. t ∈ I 〈 f(t) − x′(t),x′(t) 〉 = σ 〈 ∂ C dC(t)((x(t)),x ′ (t) 〉 = σd 0 C(t)(x(t); x ′ (t)) = 0 and 58 Messaoud Bounkhel CUBO 10, 1 (2008) 〈 bn(t) + fn(t) − x ′ n(t),x ′ n(t) 〉 = σ 〈 ∂ C dC(t)((xn(t)),x ′ n(t) 〉 = σd 0 C(t)(xn(t); x ′ n(t)) = 0, which gives ‖x′(t)‖2 = 〈 f(t),x ′ (t) 〉 and ‖x′n(t)‖ 2 = 〈 bn(t) + fn(t),x ′ n(t) 〉 . Therefore, ‖x′n‖ 2 L2 = ∫ b 0 〈 bn(t) + fn(t),x ′ n(t) 〉 dt and ‖x′‖2L2 = ∫ b 0 〈 f(t),x ′ (t) 〉 dt. Finally, we have lim sup n ‖x′n‖ 2 L2 ≤ ∫ b 0 〈 f(t),x ′ (t) 〉 dt = ‖x′‖2L2. Since x ′ n → w x ′ in L 2 (I,H) and using the weak l.s.c. of the norm, together with the last inequality we get ‖x′n‖L2 → ‖x ′‖L2. Now, using the fact that L 2 (I,H) is a Hilbert space we conclude the strong convergence of x ′ n to x ′ in L 2 (T,H). Put now ζn(t) := −x ′ n(t) + bn(t) + fn(t), a.e. on I. We have d(ζn(t), F (x(t)) − x ′(t)) = d(ζn(t) + x ′(t), F (x(t))) ≤ ‖bn(t)‖ + ‖x ′ n (t) − x′(t)‖ + d (fn(t), F (x(t))) , ≤ ‖bn(t)‖ + ‖x ′ n (t) − x′(t)‖ + e (F (xn(θn(t))), F (x(t))) → 0 as n → +∞, because of the Hausdorff u.s.c. of F and since xn(θn(t)) → x(t) on I and x ′ n(t) → x ′ (t) a.e. on I. So given ǫ > 0, we can find n0 ≥ 1 such that for all n ≥ n0 we have ζn(t) + x ′ (t) ∈ F(x(t)) + ǫIB. Since ǫ > 0 was arbitrary and F has closed values we get Γ(t) := lim sup{ζn(t)}n≥1 ⊂ F(x(t)) − x ′ (t) a.e. on I. Let ζ be a measurable selection of Γ, i.e., ζ(t) ∈ Γ(t) a.e. on I. Then, we get for a.e. on I ζ(t) ∈ Γ(t) ⊂ cow lim sup{ζn(t)}n≥1 ⊂ co w lim sup σ∂ F dC(t)(xn(t)) ⊂ σ∂ C dC(t)(xn(t)). Therefore, we get for a.e. on I ζ(t) + x ′ (t) ∈ F(x(t)) and ζ(t) ∈ NCC(t)(x(t)), which ensures x ′ (t) ∈ −NCC(t)(x(t)) + F(x(t)) a.e. on I. CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 59 The proof then is complete. 2 Using our stability result for sweeping processes, we prove a new existence result for nonconvex sweeping process with nonconvex and noncontinuous perturbation. First we recall the definition of r-prox-regularity (see [27]) (or equivalently r-proximal smoothness (see [21])) for subsets which is a generalization of convex subsets. Definition 4.1 Let S be a closed nonempty subset in H. We will say that S is r-prox- regular (or r-proximally smooth) if dS is continuously Gâteaux differentiable on the tube U(r) := {u ∈ H : 0 < dS (u) < r}. The following properties of uniformly prox-regular sets are necessary in the sequel. Proposition 4.1 [13] Let S be a r-prox-regular nonempty closed subset in H. Then fol- lowing holds 1. S is tangentially regular at each point x ∈ S. 2. for any x ∈ S and any ξ ∈ ∂F dS (x) one has 〈ξ,x′ − x〉 ≤ 2 r ‖x′ − x‖2 + dS (x ′ ) for all x′ ∈ H with dS (x ′ ) < r. 3. The Clarke and the Fréchet subdifferentials of the distance function dS coincide at each point x ∈ S, that is, ∂CdS (x) = ∂ F dS (x) for all x ∈ S. Therefore, in the sequel of all the paper we will denote ∂dS (x) for both subdifferentials for r-prox-regular sets. 4. The Clarke and the Fréchet normal cones coincide at each point x ∈ S, that is, N C S (x) = N F S (x) for all x ∈ S. Hence, we will use the notation NS (x) for both normal cones for r-prox-regular sets. Note that the converse in the second assertion (even in the finite dimensional setting) is not true in general. For more details and examples, we refer the reader to [13]. Now, we are ready to state the following new existence result for prox-regular sweeping processes with nonconvex and noncontinuous perturbations. Theorem 4.2 Let r : I →]0, +∞] such that ∫ T 0 dt r(t) < ∞. Assume that C : I⇉cl(H) has r(t)-prox-regular and ball compact values for almost every t in I. Let F : H⇉H be Haus- dorff u.s.c. on H contained in the subdifferential of a directionally regular locally Lipschitz function ψ : H → IR. Then there exist b ∈]0,T ] such that the following nonconvex sweeping process with nonconvex noncontinuous perturbation (NSPP)    x ′ (t) ∈ −NC(t)(x(t)) + F(x(t)) a.e. on [0,b]; x(t) ∈ C(t), ∀t ∈ [0,b]; x(0) = x0 ∈ C(0), 60 Messaoud Bounkhel CUBO 10, 1 (2008) has at least one solution. To prove this theorem we need the following propositions: Proposition 4.2 Let r : I →]0, +∞] such that ∫ T 0 dt r(t) < ∞. Assume that C : I⇉cl(H) has r(t)-prox-regular values and let h ∈ L2(I,H) with ‖h(t)‖ ≤ m a.e. on I. Then the following sweeping process (SP)    x ′ (t) ∈ −NC(t)(x(t)) + h(t) a.e. on I; x(t) ∈ C(t), ∀t ∈ I; x(0) = x0 ∈ C(0), has one and only one solution satisfying ‖x′(t)‖ ≤ L′ + 2m a.e. on I. Proof. Put u(t) := x(t) + ∫ t 0 h(s)ds, K(t) := C(t) − ∫ t 0 h(s)ds. Then (SP) is equivalent to (SP ′ )    u ′ (t) ∈ −NK(t)(u(t)) a.e. on I; u(t) ∈ K(t), ∀t ∈ I; u(0) = x0 ∈ K(0). By Theorem 4.1 in [12] (SP ′ ) has one and only one solution u satisfying ‖u(t)‖ ≤ L′ + m a.e. on I. This completes the proof. 2 Note that Theorem 4.1 in [12] is given for set-valued mappings C with r-prox-regular values with r does not depend on t but an inspection of the proof of Theorem 4.1 in [12] shows that it is also true if we take C(t) is r(t)-prox-regular for almost every t in I and with r satisfies ∫ T 0 dt r(t) < ∞. Proposition 4.3 Let r : I →]0, +∞] such that ∫ T 0 dt r(t) < ∞. Assume that C : I⇉cl(H) has r(t)-prox-regular values for almost every t in I. Let x0,y0 ∈ C(0), and f,g ∈ L 2 (I,H), and let x and y be two solutions of the two following problems, respectively (SPf )    x ′ (t) ∈ −NC(t)(x(t)) + f(t) a.e. on I; x(t) ∈ C(t), ∀t ∈ I; x(0) = x0 ∈ C(0), and (SPg)    y ′ (t) ∈ −NC(t)(y(t)) + g(t) a.e. on I; y(t) ∈ C(t), ∀t ∈ I; y(0) = y0 ∈ C(0), and satisfying ‖x′(t)‖ ≤ δf and ‖y ′ (t)‖ ≤ δg, for a.e. on I, CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 61 with δf,δg > 0. Then for p(t) = ∫ t 0 2 r(τ ) max{δf + Lf,δg + Lg}dτ one has ‖x(t) − y(t)‖ ≤ ‖x0 − y0‖e p(t) + ∫ t 0 ‖f(τ) − g(τ)‖ep(t)−p(τ )dτ for all t ∈ I, where Lf and Lg are constants depending on f and g respectively. Proof. By (SPf ) and (SPg) we have for a.e. t ∈ I −x′(t) + f(t) ∈ NC(t)(x(t)), with x(0) = x0 and −y′(t) + g(t) ∈ NC(t)(y(t)), with y(0) = y0 and ‖f(t) − x′(t)‖ ≤ δf + Lf and ‖g(t) − y ′ (t)‖ ≤ δg + Lg. So, by part (1) in Proposition 2.1 we get −x′(t) + f(t) ∈ δ∂dC(t)(x(t)) and − y ′ (t) + g(t) ∈ δ∂dC(t)(y(t)), where δ := max{δf +Lf,δg +Lg}. Now, by using the property of the uniform prox-regularity of the values of C recalled in part (3) in Proposition 2.1, we obtain 〈 − x′(t) + f(t) + y′(t) − g(t),x(t) − y(t) 〉 ≥ −2δ r(t) ‖x(t) − y(t)‖2. Hence 〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ≤ 〈 f(t) − g(t),x(t) − y(t) 〉 + 2δ r(t) ‖x(t) − y(t)‖2, and hence 〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ‖x(t) − y(t)‖ ≤ ‖f(t) − g(t)‖ + 2δ r(t) ‖x(t) − y(t)‖, (3.1) whenever x(t) 6= y(t). Put s(t) := ‖x(t) − y(t)‖, a function which is Lipschitz continuous on I, as the composition of two Lipschitz mappings. Let t be in the set of full measure in which s ′ (t), x ′ (t), and y ′ (t) exist and for which C(t) is r(t)-prox-regular. Then s ′ (t) =        〈 x ′ (t) − y′(t),x(t) − y(t) 〉 ‖x(t) − y(t)‖ , if x(t) 6= y(t) 0, otherwise. Thus, the relation (3.1) ensures for a. e. t ∈ I s ′ (t) ≤ ‖f(t) − g(t)‖ + 2δ r(t) s(t). 62 Messaoud Bounkhel CUBO 10, 1 (2008) We rewrite this inequality in the form ( s ′ (t) − 2δ r(t) s(t) ) e −p(t) ≤ ‖f(t) − g(t)‖e−p(t), where p(t) = ∫ t 0 2δ r(τ) dτ. As the left side is the derivative of the function t 7→ s(t)e−p(t), we can write s(t)e −p(t) − s(0) ≤ ∫ t 0 ‖f(τ) − g(τ)‖e−p(τ )dτ and then ‖x(t) − y(t)‖ = s(t) ≤ ‖x(0) − y(0)‖ep(t) + ∫ t 0 ‖f(τ) − g(τ)‖ep(t)−p(τ )dτ. This completes the proof. 2 Now, we are ready to prove Theoerem 4.2. Proof of Theoerem 4.2. Let α > 0 such that ψ is Lipschitz on x0 + αIB with ratio L > 0. Put γ(t) = ∫ t 0 2(L′+L) r(τ ) dτ and b = min{ α 2(eγ(T )L+L′) ,T}. Now, we consider a sequence of mappings defined on I := [0,b] and prove that a subsequence converges to a solution of (NSPP). For very n ∈ IN put Ink := [0, t n k ], t n k := kb n , k ∈ {1, . . . ,n} and we are going to construct fn,xn : I → H. Pick y n 0 ∈ F(x0) and define fn on I n 1 = [0, b n ] by fn(t) = y n 0 for all t ∈ I n 1 . Then consider the problem (SPPn,0)    x ′ (t) ∈ −NC(t)(x(t)) + fn(t) a.e. on I n 1 ; x(t) ∈ C(t), ∀t ∈ In1 ; x(0) = x0. By Proposition 4.1, problem (SPn) has a unique solution xn ∈ AC(I n 1 ,H) with ‖xn(t)‖ ≤ L ′ + 2L. Let y0 ∈ AC(I n 1 ,H) be the unique solution of (SPn,0)    x ′ (t) ∈ −NC(t)(x(t)) a.e. on I n 1 ; x(t) ∈ C(t), ∀t ∈ In1 ; x(0) = x0. Then Proposition 4.2 ensures ‖xn(t) − y0(t)‖ ≤ e γ(t) ∫ t 0 ‖fn(τ)‖dτ for all t ∈ I n 1 . Also ‖y0(t) − x0‖ ≤ ∫ t 0 ‖y′0(τ)‖dτ ≤ L ′ t for all t ∈ In1 . CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 63 Therefore, we get ‖xn(t) − x0‖ ≤ ‖xn(t) − y0(t)‖ + ‖y0(t) − x0‖ ≤ eγ(T ) ∫ t 0 ‖fn(τ)‖dτ + L ′ t ≤ (eγ(T )L + L′)t ≤ α 2n , which ensures that ‖xn(t) − x0‖ < α n on I n 1 . Assume now that fn and xn have defined on the interval I n k and we will extend these mappings to the interval I n k+1, for all k ∈ {1, . . . ,n}. Taking y n k ∈ F(xn(t n k )), we define fn on (t n k, t n k+1] by fn(t) = y n k . Again let xn ∈ AC(I n k+1,H) be the unique solution of (SPPn,k)    x ′ (t) ∈ −NC(t)(x(t)) + fn(t) a.e. on I n k+1; x(t) ∈ C(t), ∀t ∈ Ink+1; x(0) = xn(t n k ). Then as above we have ‖xn(t) − x0‖ < α(k+1) n on I n k+1. Indeed, let yn,k ∈ AC(I n k+1,H) be the unique solution of (SPn,k).    x ′ (t) ∈ −NC(t)(x(t)) a.e. on I n k+1; x(t) ∈ C(t), ∀t ∈ Ink+1; x(0) = xn(t n k ). By Proposition 4.2 we have ‖xn(t) − yn,k(t)‖ ≤ e γ(t) ∫ t 0 ‖fn(τ)‖dτ ≤ e γ(T ) L ′ t for all t ∈ In1 . Also, we have for all t ∈ Ink+1 ‖yn,k(t) − x0‖ ≤ ‖yn,k(t) − yn,k(0)‖ + ‖xn(t n k ) − x0‖ < ∫ t 0 ‖y′n,k(τ)‖dτ + kα n < L ′ t + kα n . Therefore, we get for all t ∈ Ink+1 ‖xn(t) − x0‖ < (e γ(T ) L ′ + L ′ )t + kα n < (k + 1)α 2n + kα n < (k + 1)α n . So we have obtained two sequences of mappings (fn)n and (xn)n, defined on I. Let θn : I → I be defined by θn(t) = t n k, if t ∈ (t n k, t n k+1] and θn(t n 0 ) = 0. Then by our construction we have x ′ n(t) ∈ −NC(t)(xn(t)) + F(xn(θn(t))) and ‖x ′ n(t)‖ ≤ L ′ + 2L a.e. on I and θn(t) → t for all t ∈ I . Furthermore we have ‖xn(t) − x0‖ < α, for all t ∈ I. Thus, Theorem 4.1 completes the proof. 2 64 Messaoud Bounkhel CUBO 10, 1 (2008) We close the paper with a direct and important corollary of Theorem 4.2. It establishes an existence result for the following differential inclusion: (∗) { x ′ (t) ∈ −NC (x(t)) + F(x(t)) a.e. on [0,b]; x(t) ∈ C, ∀t ∈ [0,b]; x(0) = x0, First, we recall that this type of differential inclusion has been introduced by Henry [24] for studying some economic problems. In the case when F is an u.s.c set-valued mapping, he proved an existence result of (∗) under the convexity assumption on the set C and on the images of the set-valued mapping F . This result has been extended by Cornet [22] by assuming the tangential regularity assumption on the set C and the convexity on the images of F with the u.s.c of F . Thibault in [29], proved an existence result of (∗) for any closed subset C (without any assumption on C), which also required the convexity of the images of F and the u.s.c. of F . Recently, the author proved in [11], without any assumption of convexity on the images of F , the existence of solutions of (∗), but a heavy price was payed for the absence of the convexity. The price is the continuity of F and a standard tangential condition. Noting that all the results mentioned above in [11, 24, 22, 29] are given in the finite dimensional setting. The question arises whether we can drop the assumption of convexity of the images of F , without assuming any tangential condition and without the continuity of F , and if possible in the infinite dimensional setting. Our next corollary establishes a positive answer to this question. Corollary 4.1 Let r ∈]0, +∞] and C be a uniformly prox-regular set in H which is ball compact. Let F : H⇉H be Hausdorff u.s.c. on H contained in the subdifferential of a directionally regular locally Lipschitz function ψ : H → IR. Then for any x0 ∈ C there exists b ∈]0,T ] such that the nonconvex sweeping process with nonconvex noncontinuous perturbation (∗) has at least one solution. Acknowledgement: The author would like to thank the referees for their careful and thorough reading of the paper. Received: November 2006. Revised: March 2007. References [1] F. Ancona and G. Colombo, Existence of solutions for a class of non convex differ- ential inclusions, Rend. Sem. Mat. Univ. Padova, Vol. 83 (1990), pp.71–76. [2] G. Balder, Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals, Nonlinear Anal. 1987 , 1399–1404. CUBO 10, 1 (2008) Arc-wise Essentially Tangentially Regular ... 65 [3] H. Benabdellah, Sur une Classe d’equations differentielles multivoques semi continues superieurement a valeurs non convexes, Sém. d’Anal. convexe, exposé No. 6, 1991. [4] H. Benabdellah, C. Castaing and A. Salvadori, Compactness and discretization methods for differential inclusions and evolution problems, Atti. Semi. Mat. Fis. Modena, Vol. XLV, (1997), pp.9–51 . [5] J.M. Borwein, Minimal CUSCO and subgradients of Lipschitz functions, in Fixed Point Theory and its Applications, (J.-B. Baillon and M. Thera eds.), Pitman Lecture Notes in Math., Longman, Essex, (1991), 57–82. [6] J.M. Borwein, S.P. Fitzpatrick and J.R. Giles, The differentiability of real func- tions on normed linear space using generalised gradients, J. Math. Anal. Appl., 128 (1987), 512–534. [7] J.M. Borwein and W.B. Moors, null sets and essentially smooth Lipschitz functions, SIAM J. Optim., 8 (1998), no. 2, 309–323. [8] J.M. Borwein and W.B. Moors, Essentially smooth Lipschitz functions, J. Funct. Anal., 149 (1997), no. 2, 305–351. [9] J.M. Borwein and W.B. Moors, A chain rule for Lipschitz functions, SIAM J. Op- tim., 8 (1998), no. 2, 300–308. [10] M. Bounkhel, On arc-wise essentially smooth mappings between Banach spaces, J. Optim. Volume, 51 (2002), no. 1, 11–33. [11] M. Bounkhel, Existence Results of Nonconvex Differential Inclusions, J. Portugaliae Mathematica, 59, No. 3, pp.283–310, 2002. [12] M. Bounkhel and L. Thibault, On various notions of regularity of sets, Nonlinear Anal.: Theory, Methods and Applications, Vol. 48, No. 2, 223–246 (2002). [13] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal., 6 (2005), no. 2, 359–374. [14] A. Bressan, A. Cellina, and G. Colombo, Upper semicontinuous differential in- clusions without convexity, Proc. Amer. Math. Soc., Vol. 106, (1989), pp.771–775. [15] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North Holland, Amesterdam, 1973. [16] T. Cardinali, A. Fiacca, and N. S. Papageorgiou, An existence theorem for evolution inclusions involving opposite monotonicities, J. Math. Anal. Appl., 222, no. 1, 1–14 (1998). [17] C. Castaing, T.X. DucHa, and M. Valadier, Evolution equations governed by the sweeping process, Set-valued Analysis, Vol. 1, pp.109–139, 1993. 66 Messaoud Bounkhel CUBO 10, 1 (2008) [18] J.P.R. Christensen, On sets of Haar measure zero in Abelian groups, Israel J. Math., 13 (1972), 255–260. [19] J.P.R. Christensen, Topological and Borel structure, American Elsevier, New Tork, 1974. [20] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. [21] F.H. Clarke, R.J. Stern and P.R. Wolenski, Proximal smoothness and the lower C 2 property, J. Convex Analysis, Vol. 2, No. 1/2, (1995), 117–144. [22] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. Vol. 96, (1983), No. 1, pp.130–147. [23] F.S. De Blasi and J. Myjak, Continuous approximations for multifunctions, Pacific J. Math., 123 (1986), 9–32. [24] C. Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl., Vol. 41 (1973), pp.179–186. [25] A.Y. Kruger, ǫ-semidifferentials and ǫ-normal elements, Depon. VINITI, No. 1331-81, Moscow, 1981 (in Russian). [26] A.D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math . Soc., 2 (1990), 175-192. [27] R.A. Poliquin, R.T. Rockafellar, and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., Vol. 352, No. 11, (2000), 5231–5249. [28] P. Rossi, Viability for upper semicontinuous differential inclusions without convexity, Diff. and Integr. Equations, Vol. 5 , (1992), No. 2, pp.455–459. [29] L. Thibault, Sweeping Process with regular and nonregular sets, Preprint, Montpellier II, (2000). [30] X.D.H. Truong, Existence of Viable solutions of nonconvex differential inclusions, Atti. Semi. Mat. Fis. Modena, Vol. XLVII, (1999), pp.457–471. [31] M. Valadier, Entrainement unilateral, lignes de descente, fonctions lipschitziennes non pathologiques, C. R. Acad. Sci. Paris, 308 Serie I (1989), 241–244. [32] M. Valadier, Lignes de descente de fonctions lipschitziennes non pathologiques, Sem. d’Anal. Convexe Montpellier (1988) exposé n 0 . 9. ArcWise2007.pdf