A generalization of Banach's fixed point theorem applied to non-linear stochastic evolution equations

Deck, Thomas

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URL: http://ub-madoc.bib.uni-mannheim.de/1595
URN: urn:nbn:de:bsz:180-madoc-15951
Document Type: Working paper
Year of publication: 1998
Publication language: German
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Subject headings (SWD): Banach-Algebra , Stochastische nichtlineare Differentialgleichung , Cauchy-Integral
Abstract: Banach's fixed point theorem for contraction operators on Banach spaces is generalized to inductive limits of Banach spaces. Within the framework of white noise analysis such spaces and (generalized) contraction operators arise naturally in the context of non-linear stochastic integral equations. In order to apply the fixed point theorem we establish topological isomorphisms between spaces of continuous mappings with values in generalized random variables, and those with values in U-functionals. As an application we prove that the Cauchy problem for a dass of non-linear stochastic heat equations is well-posed. The same method also applies to stochastic Volterra equations, stochastic reaction-diffusion equations and anticipating stochastic differential equations.
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