Brouwer fixed point theorem in ( L 0 )
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Brouwer fixed point theorem in (L)d Samuel Drapeau1 , Martin Karliczek1 , Michael Kupper2* and Martin Streckfuß1 * Correspondence: [email protected] 2 Universität Konstanz, Universitätsstraße 10, Konstanz, 78464, Germany Full list of author information is available at the end of the article
Abstract The classical Brouwer fixed point theorem states that in Rd every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L0 = L0 (, A, P) be the set of random variables. We consider (L0 )d as an L0 -module and show that local, sequentially continuous functions on L0 -convex, closed and bounded subsets have a fixed point which is measurable by construction. MSC: 47H10; 13C13; 46A19; 60H25 Keywords: conditional simplex; fixed points in (L0 )d
Introduction The Brouwer fixed point theorem states that a continuous function from a compact and convex set in Rd to itself has a fixed point. This result and its extensions play a central role in analysis, optimization and economic theory among others. To show the result, one approach is to consider functions on simplexes first and use Sperner’s lemma. Recently, Cheridito et al. [], inspired by the theory developed by Filipović et al. [] and Guo [], studied (L )d as an L -module, discussing concepts like linear independence, σ -stability, locality and L -convexity. Based on this, we define affine independence and conditional simplexes in (L )d . Showing first a result similar to Sperner’s lemma, we obtain a fixed point for local, sequentially continuous functions on conditional simplexes. From the measurable structure of the problem, it turns out that we have to work with local, measurable labeling functions. To cope with this difficulty and to maintain some uniform properties, we subdivide the conditional simplex barycentrically. We then prove the existence of a measurable completely labeled conditional simplex, contained in the original one, which turns out to be a suitable σ -combination of elements of the barycentric subdivision along a partition of . Thus, we can construct a sequence of conditional simplexes converging to a point. By applying always the same rule of labeling using the locality of the function, we show that this point is a fixed point. Due to the measurability of the labeling function, the fixed point is measurable by construction. Hence, even though we follow the constructions and methods used in the proof of the classical result in Rd (cf. []), we do not need any measurable selection argument. In probabilistic analysis theory, the problem of finding random fixed points of random operators is an important issue. Given C , a compact convex set of a Banach space, a continuous random operator is a function R : × C → C satisfying (i) R(·, x) : → C is a random variable for any fixed x ∈ C , (ii) R(ω, ·) : C → C is a continuous function for any fixed ω ∈ . For R there exists a random fixed point which is a random variable ξ : → C such that ξ (ω) = R(ω, ξ (ω)) for any ω (cf. [–])
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