VIABILITY PROBLEM WITH PERTURBATION IN HILBERT SPACE

This paper deals with the existence result of viable solutions of the differential inclusion u x(t) 2 f(t,x(t)) + F(x(t)) x(t) 2 K on (0,T), where K is a locally compact subset in separable Hilbert space H, (f(s,·))s is an equicontinuous family of measurable functions with respect to s and F is an upper semi-continuous set-valued mapping with compact values contained in the Clarke subdifferential @cV (x) of an uniformly regular function V.


Introduction
Existence result of local solution for differential inclusion with upper semi-continuous and cyclically monotone right hand-side whose values in finite-dimensional space, was first established by Bressan, Cellina and Colombo (see [6]). The authors exploited rich properties of subdifferential of convex lower semi-continuous function; in order to overcome the weakly convergence of derivatives of approximate solutions, they used the basic relation (see [7]) problem, where F is not cyclically monotone but contained in the Clarke subdifferential of locally Lipschitz uniformly regular function. However under very strong assumptions namely, the space of states is finite-dimensional and the following tangential condition where T K (x) is the contingent cone at x to K.
Recently, Morchadi and Sajid (see [8]) proved an exact viability version of the work of Ancona and Colombo assuming the same hypotheses and the following tangential condition ∀(t, x) ∈ R × K, ∃v ∈ F (x) such that Remark that in all the above works, the convexity assumption of V and/or the finite-dimensional hypothesis of the space of states were widely used in the proof.
This paper is devoted to establish a local solution of the probleṁ x(t) ∈ K ⊂ H, where K is a locally compact subset of a separable Hilbert space H, F is an upper semi-continuous multifunction, ∂ c V denotes the Clarke subdifferential of a locally lipschitz function V and the set {f (s, .) : s ∈ R} is equicontinuous, where for each x ∈ K, s → f (s, x) is measurable and the same tangential condition (1.1). One case deserves mentioning: when f is globally continuous, the condition (1.1) is weaker than the following To remove the convexity assumption of V and the finite-dimensional hypothesis of H, we rely on some properties of Clarke subdifferential of uniformly regular function and the local compactness of K.

Preliminaries and statement of the main result
Let H be a real separable Hilbert space with the norm · and the scalar product < ·, · > . For x ∈ H and r > 0 let B(x, r) be the open ball centered at x with radius r andB(x, r) be its closure and put B = B(0, 1).
Let us recall the definition of the Clarke subdifferential and the concept of regularity that will be used in the sequel.
where V ↑ (x, h) is the generalized Rockafellar directional derivative given by ∂ p V (x) denotes the proximal subdifferential of V at x which is the set of all y ∈ H for which there exist δ, σ > 0 such that for all x ∈ x + δB We say that V is uniformly regular over closed set S if there exists an open set U containing S such that V is uniformly regular over U . For more details on the concept of regularity, we refer the reader to [4]. Proposition 2.3. [3,4] Let V : H → R be a locally Lipschitz function and S a nonempty closed set. If V is uniformly regular over S, then the following conditions holds: (a) The proximal subdifferential of V is closed over S, that is, for every x n → x ∈ S with x n ∈ S and every ξ n → ξ with ξ n ∈ ∂ p V (x n ) one has ξ ∈ ∂ p V (x). (b) The proximal subdifferential of V coincides with the Clarke subdifferential of V for any point x. (c) The proximal subdifferential of V is upper hemicontinuous over S, that is, the support function x → σ(v, ∂ p V (x)) is u.s.c. over S for every v ∈ H. Now let us state the main result.
Let V : H → R be a locally Lipschitz function and β-uniformly regular over K ⊂ H. Assume that (H1) K is a nonempty locally compact subset in H; (H2) F : K → 2 H is an upper semi-continuous set valued map with compact values satisfying (H3) f : R × H → H is a function with the following properties:  For any x 0 ∈ K, consider the problem:

Proof of the main result
Choose r > 0 such that In the sequel, we will use the following important Lemma. It will play a crucial role in the proof of the main result.
Let (s, y) ∈ [0, T ] × K 0 . By the tangential condition, there exists v ∈ F (y) and h s,y ∈]0, ε] such that Consider the subset then the dominated convergence theorem applied to the sequence (χ [t,t+hs,y] f (·, ·)) t of functions shows that the function Hence On the other hand, since x ∈ K, we have Thus Now, we are able to prove the main result. Our approach consists of constructing, in a first step, a sequence of approximate solutions and deduce, in a second step, from available estimates that a subsequence converges to a solution of (2.1).
On the other hand, by (ii), (iv), (H2), (3.1) and (3.3) we have Note that the function G is differentiable on t and dG dt (t, y) = f (t, y). EJQTDE, 2007 No. 7, p. 8 We have On the other hand Hence and x(t)) .

EJQTDE, 2007 No. 7, p. 9
On the other hand we have Since the family {f (s, ·) : s ∈ R} is equicontinuous, then there exists k 0 such that for all k ≥ k 0 and for all s ∈ R, By (3.5), the last term converges to 0. This completes the proof of the Claim.
So, the convexity and the closedness of the set ∂ c V (x(t)) ensurė Proposition 3.5. The application x(.) is a solution of the problem (2.1).