Viability for differential inclusions on graphs
Let \(X\) be a Banach space, \(I\) a nonempty bounded interval and let \(K:I \to X\) and \(F:\mathcal{K}\) \(\to X\) be two multifunctions with nonempty values, where \(\mathcal{K}\) is the graph of \(K\). The authors consider the following Cauchy problem \[ u'(t)\in F(t,u(t)), \quad u(\tau)=\xi. \] A solution to the previous problem on \([\tau,T]\subset I\) is a function \(u \in W^{1,1}([\tau,T],X)\) satisfying \((t,u(t)) \in \mathcal{K}\), \(u(\tau)=\xi\) and \(u'(t)\in F(t,u(t))\) for a.e. \(t \in [\tau,T]\). The authors prove several necessary and sufficient conditions in order that \(\mathcal{K}\) be viable with respect to \(F\), which means that for each \((\tau, \xi) \in \mathcal{K}\) there exists \(T> \tau\) such that \([\tau,T]\subset I\) and the problem above has at least a solution \(u\) on \([\tau,T]\).
scientific article; zbMATH DE number 5529276