On the solution set of differential inclusions in Banach space (Q1178011)

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 \(x'(t)\in F(t,x(t))\), \(F:[a,b]\times\Omega\mapsto 2^ X\), is proved under the following assumptions: 1) \(F(.,x)\) is measurable for each \(x\in\Omega\), \(F\) has closed values. 2) \(F\) is Lipschitz in the second variable. 3) For every continuous function \(x(.): [a,b]\mapsto\Omega\), \(\int^ b_ a F(t,x(t))d\mu\neq\emptyset\). Then \(cl_ CS_ F(x_ 0)=cl_ CS_{\overline{co}F}(x_ 0)\), where \(S_ F(x_ 0)\) denotes the solution set for \(x'(t)\in F(t,x(t))\), \(x(a)=x_ 0\) and \(S_{\overline{co}F}(x_ 0)\) denotes the solution set for \(x'(t)\in\overline{co}F(t,x(t))\), \(x(a)=x_ 0\).