A generalization of the Flow-box Theorem is proven. The assumption of a C 1 vector field f is relaxed to the condition that f be locally Lipschitz continuous. The theorem holds in any Banach space.

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ŽThe notion of an arc field on a locally complete but not necessarily locally . compact metric space is introduced as a generalization of a vector field on a manifold. Generalizing the Cauchy᎐Lipschitz Theorem, sufficient conditions on arc fields are given under which the existence and uniqueness of solution curves and flows are proven. A continuous analog of an iterated function system is given as an example. ᮊ