GENERAL EXISTENCE RESULTS FOR NONCONVEX THIRD ORDER DIFFERENTIAL INCLUSIONS

In this paper we prove the existence of solutions to the following third order differential inclusion: � x (3) (t) ∈ F(t,x(t), u x(t), ¨ x(t)) +G(x(t), u x(t), ¨


Introduction
The origins of boundary and initial value problems for differential inclusions are in the theory of differential equations and serve as models for a variety of applications including control theory.
In [6] Hopkins studied an existence result for the third order differential inclusion (ThODI) x (3) (t) ∈ G(x(t), ẋ(t), ẍ(t)), x(0) = x 0 , ẋ(0) = u 0 , ẍ(0 where G is set-valued mapping with an upper semi-continuous compact valued included in the subdifferential of a convex lower semi-continuous function g : R d → R, that is, G(x, y, z) ⊂ ∂ C g(z).In this paper we prove the existence of viable solutions for the general form of the third order differential inclusion (GThODI) x (3) (t) ∈ F (t, x(t), ẋ(t), ẍ(t)) + G(x(t), ẋ(t), ẍ(t)), a.e. on [0, T ] x(0) = x 0 , ẋ(0) = u 0 , ẍ(0) = v 0 , and ẍ(t) ∈ S, ∀t ∈ [0, T ], where is an upper semi-continuous set-valued mapping with G(x, y, z) ⊂ ∂ C g(z) where g : H → R is a uniformly regular function over S and locally Lipschitz, and S is a ball compact subset of a separable Hilbert space H.This general problem covers (T hODI) in three different ways.First, it extends (T hODI) from finite dimensional setting to separable Hilbert spaces.Secondly, it extends g to the case of uniformly regular function (not necessary convex) and it also covers (ThODI) by taking F = 0. Problem (GThODI) includes as a special case the following differential variational inequality: Given T > 0 and three points x 0 , u 0 , v 0 ∈ H. where symmetric, bounded, and elliptic form on H × H.We use our main theorem to prove that (DVI) has at least one solution.
This paper is organized as follows.In Section 2, we recall some definitions and results that will be needed in the paper.In Section 3, we prove our main existence theorem, by constructing a sequence of approximate solutions and showing its convergence to the solution of the given problem.Section 4 contains the application to differential variational inequalities.

Preliminaries
Throughout the paper H will denote a separable Hilbert space.We need to recall, from [1], some notation and definitions that will be used in all the paper.open subset.We will say that f is uniformly regular over O with respect to β ≥ 0 (we will also say β-uniformly regular) if for all x ∈ O and for all ξ ∈ ∂ P f (x) one has Here ∂ P f (x) denotes the proximal subdifferential of f at x (for its definition the reader is referred, for instance, to [3]).We say that f is uniformly regular over a closed set S if there exists an open set O containing S such that f is uniformly regular over O.
The class of functions that are uniformly regular over sets is so large, it contains convex sets, p-convex sets and epigraph of lower-C 2 functions.The following proposition gives some properties for uniformly regular locally Lipschitz functions over sets needed in the sequel.For the proof of these results we refer the reader to [1,4].
Proposition 2.2.Let f : H → R be a locally Lipschitz function and ∅ = S ⊂ domf .If f is uniformly regular over S, then the following hold: (i) The proximal subdifferential of f is closed over S as a set-valued mapping, that is, for every x n → x with x n ∈ S and every (ii) The proximal subdifferential of f coincides with ∂ C f (x) the Clark subdifferential of f (see for instance [3] for the definition of (iii) The proximal subdifferential of f is upper hemicontinuous over S, i.e, the support function ) is u.s.c.over S for every v ∈ H (where σ(v, S) = sup s∈S v, s ); (iv) For any absolutely continuous map x : [0, T ] → S one has

Existence results for third order differential inclusions
We start with the following technical lemmas.Their proofs follow the same lines as in the proof of Theorem 2.3 in [2].
i } satisfying for some rank ν m ≥ 0 the following assertions: (1) 0 = t m 0 , t m νm ≤ a < T with a < α and Lemma 3.2.Let P (t, x, y, z) = F (t, x, y, z) + G(x, y, z).Under the same assumptions in Lemma 3.1, we can construct the sequence of the step functions υ m , u m , x m , f m , c m and θ m with the following properties: and with M as in Lemma 3.1.Furthermore, Now we are in a position to state and prove the main result in this section.
Theorem 3.3.Let S be a nonempty subset of H and let g : H → R be a locally Lipschitz function which is uniformly regular over S with constant β ≥ 0. Assume that (1) S is ball compact.
( Then, for any v 0 ∈ S, u 0 , x 0 ∈ H, there exists a ∈]0, T [ such that has an absolutely continuous solution on [0, a].In other words, there exists a ∈]0, T [ such that (GThODI) has an absolutely continuous solution on [0, a].
Proof.Let L > 0 and ρ > 0 be two positive scalars such that g is Lipschitz over υ 0 + ρB with ratio L. Since S is ball compact, K 0 = S ∩ (υ 0 + ρB) is compact in H. Let M and a be two positive scalars such that . By applying Lemma 3.2, there exist sequences of step functions υ m , u m , x m , f m , y m , c m and θ m with the following properties: Also, and We want to prove that υ m converges to a solution of the given differential inclusion.First, we mention that the sequence f m can be constructed with the relative compactness property in the space of bounded functions (see [7]).Therefore, without loss of generality we can suppose that there is a bounded function f such that We note that and hence υ m is continuous on the nodes t m i .Therefore, the sequence of mappings υ m is equi-Lipschitz with ratio M + 1 on all [0, a].On the other hand, we have 0 , and hence we get x m (t) − x(t) = 0, and um → u, ẋm → ẋ in the weak topology of L 2 ([0, a]; H).
Also, we have that (υ m • θ m ), (u m • θ m ) and (x m • θ m ) converge uniformly on [0, a] to υ, u, and x respectively.Since υ m (θ m (t)) = υ m i ∈ K 0 , by closedness of K 0 we get, υ(t) ∈ K 0 ⊂ S. Indeed, By construction, we have and so by the continuity of F and the closedness of its values we obtain where the above equality follows from the uniform regularity of g over S and the part (ii) in Proposition 2.2.The weak convergence of υm and Mazur's lemma entail for almost all t ∈ [0, a] So, for any ξ ∈ H we have where the last inequality follows from upper hemicontinuity of the proximal subdifferential of uniformly regular functions (part (iii) in Proposition 2.2) and the uniform convergence on [0, a] of f m and c m to f and 0 respectively, and the fact that υ m (θ m (t)) → υ(t) on K 0 .Thus, by the convexity and the closedness of proximal subdifferential of uniformly regular functions, we have As υ is absolutely continuous and g is uniformly regular locally Lipschitz function over S we get by part (iv) in Proposition 2.2 On the other hand, since By adding, we obtain On the other hand, the weak lower semi-continuity of the norm ensures Therefore, we get The proof then is complete.
We end this section with some important corollaries.The first one is an extension of the main result in [6] from finite dimensional spaces to separable Hilbert spaces and from the case of convex functions to uniform regular functions.Our proof is completely different to the one given in [6].
Corollary 3.4.Let S be a nonempty closed subset of H and let g : H → R be a locally Lipschitz function which is uniformly regular over S with constant β ≥ 0. Assume that (1) S is ball compact.
(2) G : H × H × H → H is a u.s.c.set valued mapping with compact values and G(x, y, z) ⊂ g(z), for all x, y, z ∈ S.
Proof.In [1], the author proved in Theorem 4.1 that the function d S is uniformly regular over S. Thus, we can apply Theorem 3.3 with g := d S and the set-valued mapping G := ∂ C d S which satisfies the hypothesis of Theorem 3.3 then we get a solution x of the third order differential inclusion ) and so x is a solution of (T hOSP P ).

Application to differential variational inequalities
In this section we are interested with the following differential variational inequality: Given T > 0 and three points x 0 , u 0 , v 0 ∈ H. where S = {x ∈ H : Λ(x) ≤ 0} (Λ : H → R is a C 1 convex function), a(•, •) is a real bilinear, symmetric, bounded, and elliptic form on H × H. Let A be a linear and bounded operator on H associated with a(•, •), that is, a(u, v) = Au, v , ∀u, v ∈ H.We prove the existence of solutions of (DVI).To do that, we recall first (see for example [5]) that Consequently, all the assumptions of Corollary 3.5 are satisfied and so the proof is complete.