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Results tagged with set-valued-analysis
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user 165275
Questions about maps into the powerset of a set (called set-valued maps, multivalued maps, or relations), corresponding concepts of continuity (like upper and lower semicontinuity), inclusion problems (like differential inclusions), maximal monotone maps, hyperspaces (families of subsets of a set, endowed with the Hausdorff distance or Vietoris topology), etc.
1
vote
Accepted
Continuity of Kakutani fixed points
I assume that you mean that $F$ is upper semicontinuous on the product space. Then in particular (since $X$ is compact, Hausdorff and $F$ has closed values), $F$ has a closed graph. This implies that …
- 1,869
8
votes
Accepted
Homotopy type of the Hausdorff metric
In J. Andres, M. Väth, Calculation of Lefschetz and Nielsen Numbers in Hyperspaces for Fractals and Dynamical Systems, Proc. Amer. Math. Soc. 135 (2007), 479-487, it was shown (esssentially, the resul …
- 1,869
1
vote
Existence of a global solution to a differential inclusion that does not blow up
Consider the initial value $x(t_0)=x_0$. Let $B$ be the closed ball around $x_0$ with radius $1$. Let $\rho$ denote a retraction onto $B$. Then $G=F\circ\rho$ satisfies $G|_B=F|_B$ and is upper semi-c …
- 1,869
2
votes
Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mi...
I guess that if $v$ is $P$-integrable then the answer is positive, and actually the set is compact.
Indeed, what you are looking for in this case is the compactness of the Aumann integral of the measu …
- 1,869