A relaxation theorem for differential inclusions in Banach spaces. (English) Zbl 0647.34011

Let X be a separable Banach space; \(P_{k(c)}(X)\) denote the collection of all nonempty compact (convex) subsets of X equipped with the Hausdorff metric h. For a multifunction \(F:[0,b]\times X\to P_ k(X)\) the connection between the set P of solutions (in the Carathéodory sense) of the Cauchy problem \(\dot x(t)\in F(t,x(t))\) \(x(0)=x_ 0\) and the set \(P_ c\) of solutions of the problem \(\dot x(t)\in \overline{conv} F(t,x(t))\), \(x(0)=x_ 0\) is considered. The main result. Let \(F(\cdot,x)\) be measurable for all \(x\in X\) and \(F(t,x)\subseteq G(t)\) a.e., where \(G: [0,b]\to P_{kc}(X)\) is integrally bounded and \(h(F,(t,x),F(t,y))\leq w(t,\| x-y\|)\) for all \(x,y\in X\) a.e., where w is a Kamke function. Then \(P_ c=\bar P\) in \(C_ X([0,b])\). {Reviewer’s remark: Similar results are described also in the recent monograph of A. A. Tolstonogov [Differential inclusions in a Banach space (Russian) (Novosibirsk, 1986)].}
Reviewer: V.Obuhovskii


34A60 Ordinary differential inclusions
34G20 Nonlinear differential equations in abstract spaces


Cauchy problem