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Approximation and Léray–Schauder type results for multimaps in the S-KKM class. (English) Zbl 1132.47045
Having as starting point the well-known result of Ky Fan [Math. Z. 112, 234–240 (1969; Zbl 0185.39503)] that, if \(C\) is a nonempty, compact, convex subset of a normed space \(E\), then for any continuous mapping \(f\) from \(C\) to \(E\) there exists an \(x_0\in C\) such that \[ \left\| x_0-f(x_0)\right\|=\inf_{y\in C}\left\| f(x_0)-y\right\|, \] the author establishes Fan type approximation results for \(s\)-KKM multimaps in Hausdorff locally convex spaces and then, as an application, obtains a Leray–Schauder type result. Approximation results in normed spaces are also obtained as corollaries of the main results.
MSC:
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H04 Set-valued operators
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Full Text: Euclid