On the Cauchy problem for lower semicontinuous differential inclusions

We provide a new proof of a classical result by Bressan on the Cauchy problem for first-order differential inclusions with null initial condition. Our approach allows us to prove the result directly for kth order differential inclusions, under weaker regularity assumptions on the involved multifunction. Our result is the following: let a, b, M be positive real numbers, with \(M\cdot\max\{a,a^{k}\}\le b\), and let B and X be the closed balls in \(\mathbf{R}^{n}\), centered at the origin with radius b and M, respectively. Let \(F:[0,a]\times B^{k}\to2^{X}\) be a multifunction with nonempty closed values, such that F is \(\mathcal{L}([0,a])\otimes\mathcal{B}(B^{k})\)-measurable, and for all \(t\in[0,a]\) the multifunction \(F(t,\cdot)\) is lower semicontinuous. Then there exists \(u\in W^{k,\infty}([0,a],\mathbf{R}^{n})\) such that \(u^{(k)}(t)\in F(t,u(t), u^{\prime}(t),\ldots,u^{(k-1)}(t))\) a.e. in \([0,a]\), and \(u^{(i)}(0)=0_{\mathbf{R}^{n}}\) for all \(i=0,\ldots, k-1\).