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

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.

### MSC:

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

Zbl 0914.34015
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