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Zbl 0957.47040
Lim, Teck-Cheong
A fixed point theorem for weakly inward multivalued contractions. (English)
[J] J. Math. Anal. Appl. 247, No.1, 323-327 (2000). ISSN 0022-247X

Using transfinite induction the author proves the following fixed point theorem: \par Let $D$ be a nonempty closed subset of a Banach space $X$ and let $T$ be a mapping assigning to $x\in D$ a nonempty closed set $T(x)\subset X$. Assume that $T$ is contractive with respect to the Hausdorff metric and that $Tx$ is contained in the closure of $x+\{\lambda(z-x)\mid z\in D$, $\lambda\ge 1\}$ for each $x\in D$. Then there is an $x\in D$ such that $x\in Tx$.
[Christian Fenske (Giessen)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H04 Set-valued operators
47H09 Mappings defined by "shrinking" properties

Keywords: fixed point theorems; multivalued contraction; weakly inward mapping; transfinite induction

Cited in: Zbl 1113.47044

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