SECOND-ORDER DIFFERENTIAL INCLUSIONS WITH ALMOST CONVEX RIGHT-HAND SIDES

We study the existence of solutions of a boundary second order differential inclusion under conditions that are strictly weaker than the usual assumption of convexity on the values of the right-hand side.


Introduction
The existence of solutions for second order differential inclusions of the form ü(t) ∈ F (t, u(t), u(t))(t ∈ [0, 1]) with boundary conditions, where F : [0, 1] × E × E ⇉ E is a convex compact multifunction, Lebesgue-measurable on [0, 1], upper semicontinuous on E × E and integrably compact in finite and infinite dimensional spaces has been studied by many authors see for example [1], [7].Our aim in this article is to provide an existence result for the differential inclusion with two-point boundary conditions in a finite dimensional space E of the form where F : E × E ⇉ E is an upper semicontinuous multifunction with almost convex values, i.e., the convexity is replaced by a strictly weaker condition.
For the first order differential inclusions with almost convex values we refer the reader to [5].
After some preliminaries, we present a result which is the existence of W 2,1 E ([a, b])-solutions of (P F ) where F is a convex valued multifunction.Using this convexified problem we show that the differential inclusion (P F ) has solutions if the values of F are almost convex.As an example of the almost convexity of the values of the right-hand side, notice that, if F (t, x, y) is a convex set not containing the origin then the boundary of F (x, y), ∂F (x, y), is almost convex.

Notation and preliminaries
Throughout, (E, .) is a real separable Banach space and E ′ is its topological dual, B E is the closed unit ball of E and σ(E, E ′ ) the weak topology on E. We denote by L Recall that a mapping v : [a, b] → E is said to be scalarly derivable when there exists some mapping v : [a, b] → E (called the weak derivative of v) such that, for every x ′ ∈ E ′ , the scalar function x ′ , v(•) is derivable and its derivative is equal to x ′ , v(•) .The weak derivative v of v when it exists is the weak second derivative.
By W 2,1 E ([a, b]) we denote the space of all continuous mappings in C E ([a, b]) such that their first derivatives are continuous and their second weak derivatives belong to L 1 E ([a, b]).For a subset A ⊂ E, co(A) denotes its convex hull and co(A) its closed convex hull.
Let X be a vector space, a set K ⊂ X is called almost convex if for every ξ ∈ co(K) there exist λ 1 and λ Note that every convex set is almost convex.

The Main result
We begin with a lemma which summarizes some properties of some Green type function.It will after be used in the study of our boundary value problems (see [1], [7] and [3]).
Then the following assertions hold. ( , except on the diagonal, and its derivative is given by ) and for the mapping Furthermore, the mapping u f is derivable, and its derivative uf satisfies The following is an existence result for a second order differential inclusion with boundary conditions and a convex valued right hand side.It will be used in the proof of our main theorem.
Obviously S and X are convex.Let us prove that S is a σ and by the relation (3.3) ) and the function G is uniformly continuous we get the equicontinuity of the sets X and { uf : u f ∈ X}.On the other hand, for any u f ∈ X and for all t ∈ [a, b] we have by the relations (3.1), (3.2) and (3.3) that is, the sets X(t) and { uf (t) : u f ∈ X} are relatively compact in the finite dimensional space E. Hence, we conclude that X is relatively compact EJQTDE, 2011 No. 34, p. 4 Consequently, the sequence (u fn ) converges to u f in C E ([a, b]).By the same arguments, we prove that the sequence ( ufn ) with . EJQTDE, 2011 No. 34, p. 5 Step 2. Observe that a mapping u : For any Lebesgue-measurable mappings v, w : [a, b] → E, there is a Lesbegue-measurable selection s ∈ S such that s(t) ∈ F (v(t), w(t)) a.e.Indeed, there exist sequences (v n ) and (w n ) of simple E-valued functions such that (v n ) converges pointwise to v and (w n ) converges pointwise to w for E endowed by the strong topology.Notice that the multifunctions F (v n (.), w n (.)) are Lebesgue-measurable.Let s n be a Lesbegue-measurable selection of F (v n (.), w n (.)).As )) to some mapping s ∈ S.Here we may invoke the fact that S is a weakly compact metrizable set in the separable Banach space L using the pointwise convergence of (v n (•)) and (w n (•)) to v(•) and (w(•)) respectively, the upper semicontinuity of F and the compactness of its values we get Step 3. Let us consider the multifunction Φ : S ⇉ S defined by where u f ∈ X.In view of Step 2, Φ(f ) is a nonempty set.These considerations lead us to the application of the Kakutani-ky Fan fixed point theorem to the multifunction Φ(.).It is clear that Φ(f ) is a convex weakly compact subset of S. We need to check that Φ is upper semicontinuous on the convex weakly compact metrizable set S. Equivalently, we need to prove that the graph of Φ is sequentially weakly compact in S × S. Let (f n , g n ) be a sequence in the graph of Φ. (f n ) ⊂ S. By extracting a subsequence we may EJQTDE, 2011 No. 34, p. 6 It follows that the sequences (u fn ) and ( ufn ) converge pointwise to u f and uf respectively.On the other hand, g n ∈ Φ(f n ) ⊂ S. We may suppose that (g n ) converges weakly to some element g ∈ S. As g n (t) ∈ F (u fn (t), ufn (t)) a.e., by repeating the arguments given in Step 2, we obtain that g(t) ∈ F (u f (t), uf (t)) a.e.This shows that the graph of Φ is weakly compact in the weakly compact set S× S. Hence Φ admits a fixed point, that is, there exists f ∈ S such that f ∈ Φ(f ) and so )-solution of the problem (P F ). Compactness of the solutions set follows easily from the compactness in Step 1, and the preceding arguments.Now, we present an existence result of solutions to the problem (P F ) if we suppose on F a linear growth condition.Theoreme 3.4.Let E be a finite dimensional space and F : E × E ⇉ E be a convex compact valued multifunction, upper semicontinuous on E × E. Suppose that there is two nonnegative functions p and For the proof of our Theorem we need the following Lemma.
)-solution of (P F ).Then, there exists a measurable mapping f : and hence, In the same way we have These last inequalities show that By the definition of u and consider the multifunction ).Then F 0 inherits the hypotheses on F , and furthermore, for all (x, y) ∈ E×E Consequently, F 0 satisfies all the hypotheses of Proposition 3.3.Hence, we conclude the existence of a W 2,1 E ([a, b])-solution of the problem (P F 0 ).Now, let us prove that u is a solution of (P F 0 ) if and only if u is a solution of (P F ).If u is a solution of (P F 0 ), there exists a measurable mapping f 0 such that u = u f 0 and f 0 (t) ∈ F 0 (u(t), u(t)), a.e., with for almost every t ∈ [a, b] f 0 (t) ≤ β(t) = α(p(t) + q(t)).
EJQTDE, 2011 No. 34, p. 9 Suppose now that u is a solution of (P F ).By Lemma 3.5, we have for all Then, F (u(t), u(t)) = F 0 (u(t), u(t)), that is, u is a solution of (P F 0 ).Now we are able to give our main result.
Theoreme 3.6.Let E be a finite dimensional space and F : E × E ⇉ E be an almost convex compact valued multifunction, upper semicontinuous on E × E and satisfying the following assumptions: (1) there is two nonnegative functions p, q F (x, ξy) ⊆ ξF (x, y) for all (x, y) ∈ E × E and for every ξ > 0.
For the proof we need the following result.Theoreme 3.7.Let F : E × E ⇉ E be a multifunction upper semicontinuous on E × E. Suppose that the assumption (2) in Theorem 3.6 is also satisfied.Let v 0 ∈ E and let x : [a, b] → E, be a solution of the problem and assume that there are two constants λ 1 and λ 2 , satisfying 0 ≤ λ 1 ≤ 1 ≤ λ 2 , such that for almost every t ∈ [a, b], we have Then there exists t = t(τ ), a nondecreasing absolutely continuous map of the interval [a, b] onto itself, such that the map x(τ ) = x(t(τ )) is a solution of the problem (P F ). Moreover x(a) = x(b) = v 0 . Proof.

1 E
([a, b]) the space of all Lebesgue-Bochner integrable E valued mappings defined on [a, b].Let C E ([a, b]) be the Banach space of all continuous mappings u : [a, b] → E endowed with the sup-norm, and C 1 E ([a, b]) be the Banach space of all continuous mappings u : [a, b] → E with continuous derivative, equipped with the norm u C 1 = max{ max t∈[a,b] u(t) , b max t∈[a,b] , s)f (s), z(s) ds.In particular, for z(•) = χ [a,b] (•)e j , where χ [a,b] (•) stands for the characteristic function of [a, b] and (e j ) a basis of E, we obtain lim n→∞ b a G(t, s)f n (s), χ [a,b] (s)e j ds = b a G(t, s)f (s), χ [a,b] (s)e j ds, or equivalently lim n→∞ b a G(t, s)f n (s)ds, e j = b a G(t, s)f (s)ds, e j , which entails lim n→∞

Lemma 3 . 5 .
Let E be a finite dimensional space.Suppose that the hypotheses of Theorem 3.4 are satisfied.If u is a solution in W 2,1 E ([a, b]) of the problem (P F ), then for all t ∈ [a, b] we have

C 1 E
we conclude that for all t ∈ [a, b] u(t) ≤ α and u(t) ≤ α b .EJQTDE, 2011 No. 34, p. 8 Proof of Theorem 3.4.Let us consider the mapping ϕ κ : E → E defined by fact that for all t ∈ [a, b] u(t) = v 0 + b b − a b a G(t, s)f 0 (s)ds, and u(t) = b b − a b a ∂G ∂t (t, s)f 0 (s)ds, we obtain
1 E ([a, b]).Now, application of the Mazur's trick to (s ′ n ) provides a sequence (z n ) with z n ∈ co{s ′ m : m ≥ n} such that (z n ) converges almost every where to s.Then, for almost every t ∈ [a, b] k≥0 {z n (t) : n ≥ k} ⊂ k≥0 co{s ′ n (t) : n ≥ k}.As s ′ n (t) ∈ F (v n (t), w n (t)), we obtain s(t) ∈ k≥0 co( n≥k