On a class of controlled functional differential inclusions (Q2871253)

The authors consider the following initial value problem for a class of functional differential inclusion with causal operator NEWLINE\[NEWLINE\left\{\begin{aligned} & x'(t)\in F(t,x(t), (Qx)(t)), \,\, t\in [0,b),\\ &x|_{[-\sigma,0]}=\phi\in C([-\sigma,0],\mathbb R^N), \end{aligned}\right. NEWLINE\]NEWLINE where \(\sigma\geq 0,\) \(Q\) is a causal operator and \(F:[-\sigma,b]\times \mathbb R^N\times \mathbb R^M\to K_c(\mathbb R^N)\) is a Carathéodory function (\(K_c(\mathbb R^N)\) is the set of all nonempty compact convex subsets of \(\mathbb R^N\)).NEWLINENEWLINEThe existence of solutions and some properties of the solution set are established. An application to an optimal control problem is also presented.