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Topological structure of solution sets to multi-valued asymptotic problems. (English) Zbl 0974.34045
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.

MSC:
34C30 Manifolds of solutions of ODE (MSC2000)
34A60 Ordinary differential inclusions
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