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A generalization to multifunctions of Fan’s best approximation theorem. (English) Zbl 0672.47043

Let E be a locally convex Hausdorff topological vector space, S a nonempty subset of E and p a continuous seminorm on E. The following Reich’s theorem is well-known:
If S is approximatively compact and f: \(S\to E\) is continuous with f(S) relatively compact then there exists \(x\in S\) satisfying \(p(f(x)-x)=\min \{p(f(x)-y)|\) \(y\in S\}.\)
The authors generalize this result for multifunctions.
Reviewer: Ioan A.Rus

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H05 Monotone operators and generalizations
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