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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