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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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References:
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