On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space.
The authors study a Cauchy problem for a first-order differential inclusion of the type \[ \dot{x}(t) \in F(x(t)) \quad\text{for a.e. }t \geq 0, \qquad x(0)=x_0 \in \mathbb{R}^n, \] where \(F:\mathbb{R}^n \to \mathbb{R}^n\) is a multifunction, and, given a closed set \(K \subset \mathbb{R}^n\), they give sufficient conditions for the existence of points \(x_0 \in K\) and solutions to the problem which remain in \(K\), i.e., \(x(t) \in K, \forall t \geq 0\). The authors give different conditions for the solvability of the problem, in particular, they present two different, and often independent methods: in the first part, they prove an existence result by using homology theory and, in the second part, they use retracts. All results obtained are illustrated by examples and a particular case of equations is also considered.
scientific article; zbMATH DE number 1884210
scientific article