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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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