Existence results for boundary value problems for fourth-order differential inclusions with nonconvex valued right hand side. (English) Zbl 1117.34005

Arara, A.; Benchohra, M.; Ntouyas, S. K.; Ouahab, A.

Arch. Math., Brno 40, No. 3, 219-227 (2004).

The authors investigate the solvability of the BVP for the fourth order boundary inclusion \[ L_4 y(t) + a(t) y(t) \in F(t, y(t)), \quad t\in [0,T], \tag{\(\ast \)} \] where \(L_4\) is a disconjugate fourth order differential operator \[ L_4 y(t) = (a_3(t)(a_2(t)(a_1(t)(a_0(t)y)')')')' \] with continuous functions \(a_i(t)> 0\), \(i=0,\dots ,3\), \(a_1(t)=a_3(t)\), \(a(t)\geq 0\), and the set-valued right-hand side \(F\: [0, T] \times \mathbb R\to \mathcal P(\mathbb R)\), \(\mathcal P(\mathbb R)\) being a family of nonempty subsets of \(\mathbb R\), which is not necessarily convex valued. The main results of the paper present additional conditions on \(F\) which guarantee that the periodic BVP \((\ast )\), \(L_i y(0) = L_i y(T)\), \(i=0, 1,2,3\), has a solution. Here \(L_0 y = a_0(t) y\), \(L_i y = a_i(t) (L_{i-1} y(t))'\) are quasiderivatives of \(y\). The research is motivated by the paper of M. Švec [Arch. Math., Brno 33, 167–171 (1997; Zbl 0914.34015)], where a convex-valued right-hand side is considered. The results of the paper are proved using various fixed point theorems for multivalued maps.

Reviewer: Ondřej Došlý (Brno)

MSC:

34A60	Ordinary differential inclusions
34B15	Nonlinear boundary value problems for ordinary differential equations
47H10	Fixed-point theorems

Keywords:

contraction multivalued map; fixed point; decomposable values

Citations:

Zbl 0914.34015