## On Nash theorem.(English)Zbl 1057.54029

The authors prove a result they call an infimum principle for a family of continuous functions $$g_s: X\times Y\to \mathbb{R}$$, $$s\in S$$, from a product of a compact topological space $$X$$ with a simplicial structure and corresponding convexity structure and a topological space $$Y$$. From this they derive extensions of a series of classical results including: the Helly infimum principle [W. Kulpa, Topology Appl. 116, No. 2, 227–233 (2001; Zbl 1001.52005)]; the Schauder fixed point theorem [A. Granas and J. Dugundji, Fixed point theory (Springer Monographs in Mathematics, New York, NY: Springer) (2003; Zbl 1025.47002)]; the SchauderTychonoff fixed point theorem; the Ky Fan minimax inequality; the Nash equilibrium theorem; the von Neumann minimax principle; the Kakutani fixed point theorem [all these: Granas and Dugundji, loc. cit.]; and the Gale-Nikaido theorem [H. Nikaido, Convex structures and economic theory (New York-London: Academic Press XII) (1968; Zbl 0172.44502)].

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 55M20 Fixed points and coincidences in algebraic topology 91A99 Game theory 52A01 Axiomatic and generalized convexity

### Citations:

Zbl 1001.52005; Zbl 1025.47002; Zbl 0172.44502
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