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A generalized KKMF principle. (English) Zbl 1080.54028
Let \(E\) be a vector space and \(A\subset E\) an arbitrary subset. A function \(F: A\to 2^E\) is called a KKM map provided \(\text{conv}\{x_1,x_2,\dots, x_k\}\subset\bigcup^k_{j=1} F(x_j)\) for each finite subset \(\{x_1,x_2,\dots, x_k\}\subset A\). Consider a subset \(X\) of a topological vector space \(Y\). A family \(\{(C_i, K_i)\}_{i\in I}\) of pairs of sets is said to be coercing for a map \(F: X\to Y\) if and only if:
(1) for each \(i\in I\), \(C_i\) is contained in a compact convex subset of \(X\), and \(K_i\) is a compact subset of \(Y\), (2) for each \(i,j\in I\), there exists \(k\in I\) such that \(C_i\cup C_j\subseteq C_k\), (3) for each \(i\in I\), there exists \(k\in I\) with \(\bigcap_{x\in C_k} F(x)\subseteq K_i\).
In the present paper the authors prove the following generalization of the celebrated Knaster-Kuratowski-Mazurkiewicz-Fan principle: Theorem 3.1. Let \(E\) be a Hausdorff topological vector space, \(Y\) a convex subsets of \(E\), \(X\) a non-empty subset of \(Y\), and \(F: X\to Y\) a KKM map with compactly closed (in \(Y\)) values. If \(F\) admits a coercing family, then \(\bigcap_{x\in X} F(x)\neq\emptyset\).

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
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