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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 $\bigcap_{x\in X} F(x)\ne\emptyset$.

54H25Fixed-point and coincidence theorems in topological spaces
47H04Set-valued operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54C60Set-valued maps (general topology)
Full Text: DOI
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