Arcwise connectedness of sets of solutions to differential inclusions (Q1776277)

The author studies some topological properties of the set of the solutions of a differential inclusion. In particular, he considers the Cauchy problem \[ x' \in F(t,x), \qquad x(0)=\xi, \] where \(F:[0,+\infty) \to 2^{\mathbb R^n}\) is a multifunction with nonempty, compact values and Lipschitzian with respect to \(x\). Denoting by \(X\) the space of continuous functions \(x:[0,+\infty) \to \mathbb R^n\) with derivatives \(x' \in L^1_{\text{loc}}([0,+\infty) , \mathbb R^n)\) endowed with the distance \(d\) defined by the formula \[ d(x,y)= \|x(0)-y(0) \|+ \sum_{n=0}^{+\infty} \frac{1}{2^n} \frac {\int_0^n \|x'(t)-y'(t) \|\,dt}{1+\int_0^n \|x'(t)-y'(t) \|\, dt}, \] under suitable assumptions on \(F\), the author proves that the set of solutions \[ S_F(\xi)=\{x \in X: x(0)=\xi, x'(t)\in F(t,x(t)) \quad\text{a.e. }t \in [0,+\infty)\} \] is arcwise connected in \((X,d)\).