## 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

### MSC:

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

Cauchy problem