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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
##### Keywords:
KKM maps; KKM theorem; fixed point; Hausdorff topological space
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##### References:
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