The KKM principle implies many fixed point theorems. (English) Zbl 1049.47047
The author shows that most of the well-known fixed point theorems for Kakutani maps or Browder maps defined on topological vector spaces are simple consequences of the KKM principle. In particular, in the case of Browder maps, in section 2 he obtains a new general fixed point theorem that extends properly the classical Browder theorem and a corollary that subsumes a number of generalizations of the Browder fixed point theorem. Moreover, from the Ky Fan lemma, he derive a result on the existence of maximal elements and fixed points. In section 3, his main result leads to a fixed point theorem for Kakutani maps with image set of the Zima type. From this result, many well-known results follow.
Reviewer: Rita Pini (Milano)
MSC:
| 47H04 | Set-valued operators |
| 47H10 | Fixed-point theorems |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
| 54C60 | Set-valued maps in general topology |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 55M20 | Fixed points and coincidences in algebraic topology |
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