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On the solution set of differential inclusions in Banach space. (English) Zbl 0735.34017

Some results for differential inclusions in Banach spaces are given. Separability of the space is not assumed. The main results of the paper contain: the Filippov theorem, a relaxation theorem, results concerning continuous dependence of the solution sets on parameters and initial values and differentiability of the solution set.

The relaxation theorem for ${x}^{\text{'}}\left(t\right)\in F\left(t,x\left(t\right)\right)$, $F:\left[a,b\right]×{\Omega }↦{2}^{X}$, is proved under the following assumptions: 1) $F\left(·,x\right)$ is measurable for each $x\in {\Omega }$, $F$ has closed values. 2) $F$ is Lipschitz in the second variable. 3) For every continuous function $x\left(·\right):\left[a,b\right]↦{\Omega }$, ${\int }_{a}^{b}F\left(t,x\left(t\right)\right)d\mu \ne \varnothing$.

Then $c{l}_{C}{S}_{F}\left({x}_{0}\right)=c{l}_{C}{S}_{\overline{co}F}\left({x}_{0}\right)$, where ${S}_{F}\left({x}_{0}\right)$ denotes the solution set for ${x}^{\text{'}}\left(t\right)\in F\left(t,x\left(t\right)\right)$, $x\left(a\right)={x}_{0}$ and ${S}_{\overline{co}F}\left({x}_{0}\right)$ denotes the solution set for ${x}^{\text{'}}\left(t\right)\in \overline{co}F\left(t,x\left(t\right)\right)$, $x\left(a\right)={x}_{0}$.

##### MSC:
 34A60 Differential inclusions 34G20 Nonlinear ODE in abstract spaces