Topological structure of solution sets to multi-valued asymptotic problems
It is considered the Cauchy problem \[ \dot{x}(t)\in F(t,x(t)),\quad x(0)=x_{0}, \tag{1} \] with \(t\in J= [0,\infty),\) and \(F:J\times \mathbb{R}^{n}\mapsto \mathbb{R}^{n}\) is a set-valued map with nonempty, compact values, \(F(\cdot,x)\) is measurable for all \(x\in \mathbb{R}^{n}\) and there exists a locally integrable function \(\eta:J\to J\) such that, for every \(t\in J\) and all \(x,y \in \mathbb{R}^{n}, d_{H}(F(t,x),F(t,y))\leq \eta(t)|x-y|\). Here, \(d_{H}\) stands for Hausdorff distance. The topological structure of a solution set to problem (1) is studied. In particular, if \(\int_{0}^{\infty}\eta(t) dt<\infty\) then the existence of an entirely bounded solution on \(J\) is proved.
scientific article; zbMATH DE number 1453203
scientific article