Nonexpansive multi-valued maps in $\ell^2$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:51:35Z http://mathoverflow.net/feeds/question/47105 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47105/nonexpansive-multi-valued-maps-in-ell2 Nonexpansive multi-valued maps in $\ell^2$ TCL 2010-11-23T15:19:27Z 2011-04-17T22:22:13Z <p>Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-y\|$$ for all $x,y\in C$, where $D$ is the Hausdorff metric defined by $$D(A,B)=\inf\lbrace r>0: N_r(A)\supset B, N_r(B)\supset A\rbrace,$$ $N_r(S) =\lbrace x\in C: d(x,S)\lt r\rbrace$ being the $r$-neighborhood of $S$. </p> <p>The question is whether $T$ has a fixed point, i.e. a point $x\in C$ such that $x\in Tx$. </p> <p>The answer is yes if $T$ is a contraction, i.e. replacing the $\le \|x-y\|$ by $\le \lambda \|x-y\|$ for some $0\le\lambda&lt;1$ in the definition; or if $Tx$ is compact for each $x$.</p> http://mathoverflow.net/questions/47105/nonexpansive-multi-valued-maps-in-ell2/51556#51556 Answer by Dirk for Nonexpansive multi-valued maps in $\ell^2$ Dirk 2011-01-09T16:40:31Z 2011-01-09T16:40:31Z <p>Probably the paper <a href="http://downloads.hindawi.com/journals/fpta/2010/581728.pdf" rel="nofollow">Fixed Point Properties Related to Multivalued Mappings</a> answers your question...</p>