*(English)*Zbl 1165.34004

The paper deals with the first-order differential inclusion subject to the periodic conditions

and also with the first-order differential inclusion subject to the non-periodic conditions

Here, $I=[0,1]$, $F:I\times {\mathbb{R}}^{n}\to {2}^{{\mathbb{R}}^{n}}$ is a Carathéodory multifunction, $b:I\to \mathbb{R}$ is continuous and does not vanish on the whole interval $I$. Further, $A$, $B$ are $n\times n$ matrices with real elements such that $det(A+B)\ne 0$, and either $det\left(A\right)\ne 0$, $\parallel {A}^{-1}B\parallel <1$, or $det\left(B\right)\ne 0$, $\parallel {B}^{-1}A\parallel <1$. The authors provide new sufficient conditions under which solutions of problem (1) or problem (2) exist. The results apply to differential inclusions that may have a right-hand side with a super-linear growth in its second variable and also apply to systems of first-order differential inclusions. The proofs are based on novel differential inequalities and the Leray-Schauder nonlinear alternative. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained.

##### MSC:

34A60 | Differential inclusions |

34B15 | Nonlinear boundary value problems for ODE |

47N20 | Applications of operator theory to differential and integral equations |