A Peano type theorem for a class of nonconvex-valued differential inclusions
This note is concerned with the existence of continuously differentiable solutions for the differential inclusion \[ x'(t) \in \{f_{1}(t,x(t)),\dots, f_{k}(t,x(t))\}, \quad x(0) = x_{0}, \tag{1} \] where \(f_{1},\dots,f_{k}:\Omega\subset \mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\) and \(\Omega\) is an open set containing \((0,x_{0}).\) The existence of a continuously differentiable solution is proved under a continuity condition on the associated multiple-valued function in the sense of Almgren \[ f=\sum_{1}^{k}\mu(f_{i})\to Q_{k}(\mathbb{R}^n). \] Here, \[ Q_{k}(\mathbb{R}^n)=\{\sum_{1}^{k}\mu(x_{i}): x_{i}\in\mathbb{R}^{n}, i= 1,\dots,k\}, \] where \(\mu(x_{i})\) denotes the Dirac measure at \(x_{i}.\) \(Q_{k}(\mathbb{R}^n)\) is equipped with the metric \[ S(\sum_{1}^{k}\mu(x_{i}),\sum_{1}^{k}\mu(y_{i}))=\min \{\max_{i=1,\dots,k}|x_{i}-y_{\sigma(i)}|: \sigma\, \text{ix a permutation of}\, \{1,\dots,k\}\}. \] Theorem. Let the the associated multiple-valued function in the sense of Almgren \[ f=\sum_{1}^{k}\mu(f_{i})\to Q_{k}(\mathbb{R}^n) \] be continuous. Then there exist \(T >0\) and a continuously differentiable solution \(x:[0,T]\to \mathbb{R}^n\) to the differential inclusion (1).
scientific article; zbMATH DE number 5529274