## 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:

  Allen, G., Variational inequalities, complementarity problems, and duality theorems, J. math. anal. appl., 58, 1-10, (1977) · Zbl 0383.49005  Ben-El-Mechaiekh, H.; Deguire, P.; Granas, A., Points fixes et coincidences pour LES applications multivoques (applications de Ky Fan), C. R. acad. sci. Paris Sér. I math., 295, 257-259, (1982) · Zbl 0525.47042  Ben-El-Mechaiekh, H.; Deguire, P.; Granas, A., Points fixes et coincidences pour LES applications multivoques II (applications de type φ et φ*), C. R. acad. sci. Paris Sér. I math., 295, 337-340, (1982) · Zbl 0525.47042  Border, K.C., Fixed point theorems with applications to economics and game theory, (1985), Cambridge Univ. Press · Zbl 0558.47038  Browder, F.E., The fixed point theory of multivalued mappings in topological vector spaces, Math. ann., 177, 283-301, (1968) · Zbl 0176.45204  Dugundji, J.; Granas, A., KKM maps and variational inequalities, Ann. scuola norm. sup. Pisa cl. sci., 5, 679-682, (1978) · Zbl 0396.47037  Dugundji, J.; Granas, A., Fixed point theory, vol. 1, Monografie matematyczne, vol. 61, (1982), PWN Warszawa · Zbl 0483.47038  Ding, X.P.; Tan, K.K., On equilibria of non-compact generalized games, J. math. anal. appl., 177, 226-238, (1993) · Zbl 0789.90008  Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. ann., 142, 305-310, (1961) · Zbl 0093.36701  Fan, K., A minimax inequality and applications, (), 103-113  Fan, K., Somme properties of convex sets related to fixed point theorems, Math. ann., 266, 519-537, (1984) · Zbl 0515.47029  Florenzano, M., L’équilibre économique général transitif et intransitif: problème d’existence, Monographie du séminaire d’économétrie, (1981), Presses du CNRS Paris  Haddad, G., Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. math., 39, 83-100, (1981) · Zbl 0462.34048  Isac, G.; Bulavski, V.; Kalashnikov, V., Exceptional families, topological degree, and complemetarity problems, J. global optim., 10, 207-225, (1997) · Zbl 0880.90127  Isac, G.; Obuchowska, W.T., Functions without exceptional family of elements and complementarity problems, J. optim. theory appl., 99, 147-163, (1998) · Zbl 0914.90252  Karamardian, S., Generalized complementarity problem, J. optim. theory appl., 8, 161-168, (1971) · Zbl 0218.90052  Knaster, B.; Kuratowski, C.; Mazurkiewicz, S., Ein beweis des fixpunktsatszes für n-dimensionale simplexes, Fund. math., 14, 132-137, (1929) · JFM 55.0972.01  Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. math. anal. appl., 97, 151-201, (1983) · Zbl 0527.47037  Park, S., Eighty years of the Brouwer fixed point theorem, ()
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