Asymptotic representation of the set of \(\delta\)-solutions to a differential inclusion (Q1974367)

The author deals with absolutely continuous solutions to the differential inclusions \[ \dot x(t)= F(t,x(t)),\quad t\in [a,b],\quad \dot x(t)= \text{co }F(t, x(t)),\quad t\in [a,b], \] where \(F: [a,b]\times \mathbb{R}^n\to\mathbb{R}^n\) satisfies the Carathéodory conditions. The basic result is the equation \[ H_{\text{co}}({\mathcal B})= \bigcap_{\delta,\varepsilon> 0} \overline{H_\delta({\mathcal B}^\varepsilon)}; \] here, \({\mathcal B}\subseteq C^n[a,b]\) is a bounded closed set, \({\mathcal B}^\varepsilon\) is the \(\varepsilon\)-neighborhood of \({\mathcal B}\), \(H_\delta({\mathcal B}^\varepsilon)\) is a set of solutions to the inclusion \[ \dot x(t)= F(t,x(t))^{\eta(t,\delta)},\quad t\in [a,b], \] and \(\eta(t,\delta): [a,b]\times (0,\infty)\to [0,\infty)\) is a function satisfying the following conditions: (a) \(\eta(\cdot,\varepsilon)\in L^1[a,b]\) for each \(\varepsilon> 0\); (b) \(\eta(t,\cdot)\) is nondecreasing for almost all \(t\); (c) the function \[ \varphi(t,x,\delta)= \max_{y\in \overline{B(x,\delta)}} h(F(t,x), F(t,y)) \] (\(h(\cdot,\cdot)\) is the Hausdorff metric) is uniformly continuous at point \(\delta= 0\) in \(x\) on the set \(U({\mathcal B})= \{x(t): x\in{\mathcal B}, a\leq t\leq b\}\) with respect to \(\eta\); the latter means that for each \(\varepsilon> 0\) there exists \(\delta= \delta(t,\varepsilon)> 0\) such that \(\varphi(t, x,\delta(\varepsilon, t))\leq \eta(t,\varepsilon)\), \(x\in U({\mathcal B})\) holds.