Existence of solutions for a certain differential inclusion of third order

The existence of solutions of a boundary value problem for a third order differential inclusion is investigated. New results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.


Introduction
This paper is concerned with the following boundary value problem Problem (1.1) occurs in hydrodynamic and viscoelastic plates theory.For the motivation of the study of this class of problem we refer to [1] and references therein.
The present paper is motivated by a recent paper of Bartuzel and Fryszkowski ( [1]), where it is considered problem (1.1) and a version of the Filippov EJQTDE, 2009 No. 6, p. 1 lemma for this problem is provided.The aim of our paper is to present two other existence results for problem (1.1).Our results are essentially based on a nonlinear alternative of Leray-Schauder type and on Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values.The methods used are standard, however their exposition in the framework of problem (1.1) is new.
We note that two other existence results for problem (1.1) obtained by the application of the set-valued contraction principle due to Covitz and Nadler jr. may be found in our previous paper [3].
The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.

Preliminaries
In this section we sum up some basic facts that we are going to use later.
Let (X, d) be a metric space with the corresponding norm |.| and let I ⊂ R be a compact interval.Denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all nonempty subsets of X and by B(X) the family of all Borel subsets of X.If A ⊂ I then χ A (.) : I → {0, 1} denotes the characteristic function of A. For any subset A ⊂ X we denote by A the closure of A.
Recall that the Pompeiu-Hausdorff distance of the closed subsets A, B ⊂ X is defined by As usual, we denote by C(I, X) the Banach space of all continuous functions x(.) : I → X endowed with the norm |x(.)|C = sup t∈I |x(t)| and by L 1 (I, X) the Banach space of all (Bochner) integrable functions x(.) : I → X endowed with the norm |x(. is a bounded subset of X for all bounded sets B in X. T (.) is said to be compact if T (B) is relatively compact for any bounded sets B in X. T (.) is said to be totally compact if T (X) is a compact subset of EJQTDE, 2009 No. 6, p. 2 X. T (.) is said to be upper semicontinuous if for any open set D ⊂ X, the set {x ∈ X; T (x) ⊂ D} is open in X. T (.) is called completely continuous if it is upper semicontinuous and totally bounded on X.
It is well known that a compact set-valued map T (.) with nonempty compact values is upper semicontinuous if and only if T (.) has a closed graph.
We recall the following nonlinear alternative of Leray-Schauder type and its consequences.Corollary 2.2.Let B r (0) and B r (0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : B r (0) → P(X) be a completely continuous set-valued map with compact convex values.Then either i) the inclusion x ∈ T (x) has a solution, or ii) there exists x ∈ X with |x| = r and λx ∈ T (x) for some λ > 1.
Corollary 2.3.Let B r (0) and B r (0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : B r (0) → X be a completely continuous single valued map with compact convex values.Then either i) the equation x = T (x) has a solution, or ii) there exists x ∈ X with |x| = r and x = λT (x) for some λ < 1.
We recall that a multifunction T (.) : X → P(X) is said to be lower semicontinuous if for any closed subset C ⊂ X, the subset {s ∈ X; G(s) ⊂ C} is closed.
If F (., .): I × R → P(R) is a set-valued map with compact values and x(.) ∈ C(I, R) we define We say that F (., .) is of lower semicontinuous type if S F (.) is lower semicontinuous with closed and decomposable values.Then G(.) has a continuous selection (i.e., there exists a continuous mapping g(.) : S → L 1 (I, R) such that g(s) ∈ G(s) ∀s ∈ S).
A set-valued map G : I → P(R) with nonempty compact convex values is said to be measurable if for any x ∈ R the function t → d(x, G(t)) is measurable.
A set-valued map F (., .): I ×R → P(R) is said to be Carathéodory if t → F (t, x) is measurable for any x ∈ R and x → F (t, x) is upper semicontinuous for almost all t ∈ I. F (., .) is said to be L 1 -Carathéodory if for any l > 0 there exists h l (.) ∈ L 1 (I, R) such that sup{|v|; v ∈ F (t, x)} ≤ h l (t) a.e.(I), ∀x ∈ B l (0).Theorem 2.5.( [5]) Let X be a Banach space, let F (., .): I ×X → P(X) be a L 1 -Carathéodory set-valued map with S F = ∅ and let Γ : L 1 (I, X) → C(I, X) be a linear continuous mapping.
Then the set-valued map Γ • S F : C(I, X) → P(C(I, X)) defined by has compact convex values and has a closed graph in C(I, X) × C(I, X).
In what follows I = [−1, 1] and let AC 2 (I, R) be the space of two times differentiable functions x(.) : I → R whose second derivative exists and is absolutely continuous on I. On

The main results
We are able now to present the existence results for problem (1.1).We consider first the case when F (., .) is convex valued.

Theorem 2 . 1 .
([6]) Let D and D be the open and closed subsets in a normed linear space X such that 0 ∈ D and let T : D → P(X) be a completely continuous set-valued map with compact convex values.Then either i) the inclusion x ∈ T (x) has a solution, or ii) there exists x ∈ ∂D (the boundary of D) such that λx ∈ T (x) for some λ > 1.