On the Cauchy problem for lower semicontinuous differential inclusions

We provide a new proof of a classical result by A. Bressan on the Cauchy problem for first-order differential inclusions with null initial condition. Our approach allows us to prove the result directly for $k$-th 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 $Mcdotmax{a,a^k}le b$, and let $B$ and $X$ be the closed balls in $ r^n$, centered at the origin with radius $b$ and $M$, respectively. Let $F:[0,a] imes B^k o2^{X}$ be a multifunction with nonempty closed values, such that $F$ is $call([0,a])otimescalb(B^k)$-measurable, and for all $tin[0,a]$ the multifunction $F(t,cdot,)$ is lower semicontinuous. Then there exists $uin W^{k,infty}([0,a], r^n)$ such that $u^{(k)}(t)in F(t,u(t), u^prime(t),ldots,u^{(k-1)}(t))$ a.e. in $[0,a]$, and $u^{(i)}(0)=0_{ r^n}$ for all $i=0,ldots, k-1$.