Approximate selections, best approximations, fixed points, and invariant sets

Abstract

In the first section of this note we prove an approximate selection theorem for upper semicontinuous mappings defined on paracompact subsets of a Hausdorff topological vector space. In Section 2 we study continuity properties of setvalued metric projections. In the third section we combine the results obtained in the previous sections and establish some new fixed point theorems. In [36] we presented a set-valued version of Fan’s fixed point theorem for inward singlevalued mappings [ 171. We assumed there that the set-valued mapping in question was continuous. Now we are able to show that the same result is true for upper semicontinuous mappings. We also improve recent theorems due to Fitzpatrick and Petryshyn [19] and extend a theorem of Lim’s [27]. In the last section we consider invariance criteria that are similar to the inwardness conditions used in Section 3. In particular, we relate Nagumo’s subtangency condition [31] with Browder’s local support cones [6].