Given a cone \(K\subset\mathbb{R}^ n\) and a set-valued differential equation \(\dot x\in F(t,x)\), \(t\in J=[0,a]\), the author wants to find an absolutely continuous solution which is minimal with respect to the ordering induced by \(K\). The assumptions on \(F\) are the following: (1) \(\| F(t,x)\|\leq c(t)(1+\| x\|)\), \(c\in L^ 1(J)\); (2) \(f(t,x)\in F(t,x)\subset f(t,x)+K\) where \(f\) is quasimonotone (i.e. \(f(t,x+y)\in f(t,x)+T_ K(y))\) with \(T_ K(y)=\{\lambda(z-y):\lambda\geq 0,z\in K\}\) for closed convex \(K\). The first result he shows is the following. Let \(D\subset X=\mathbb{R}^ n\) be closed and convex, \(F:J\times D\) into \(2^ X\) as in (1) above and such that \(F\) is compact-convex valued, measurable in \(t\), upper semicontinuous on \(x\). Assume also that \(F(t,x)\cap T_ D(x)\neq\emptyset\). Then the differential system has an absolutely continuous solution on \(J\) for every \(x_ 0\in D\). The previous result allows the author to prove the following very interesting comparison result. Let \(X=\mathbb{R}^ n\), \(K\neq\{0\}\) a cone of \(\mathbb{R}^ n\). Let \(F:J_ xX\to 2^ X\) as in the previous theorem (plus assumption (2)). If \(v:J\to X\) is absolutely continuous, \(v'\in F(t,v)+K\) a.e., \(v(0)\in x_ 0+K\), then \(x'\in F(t,x)\), \(x(0)=x_ 0\) has a solution \(x\leq v\) (i.e., \(v(t)\in x(t)+K\) on \(J)\). The author answers in the affirmative some open questions he posed in Part I[ibid. 14, No. 1-2, 38-47 (1988; Zbl 0655.34014)].