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Proceedings of the American Mathematical Society

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Function space flow invariance for functional-differential equations of retarded type


Author: James H. Lightbourne
Journal: Proc. Amer. Math. Soc. 77 (1979), 91-98
MSC: Primary 34K05
MathSciNet review: 539637
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Abstract: Let $ \mathcal{C}$ denote the Banach space of continuous functions $ \phi :[ - r,0] \to {{\mathbf{R}}^n}$, let $ \Omega \subset \mathcal{C}$ be closed, and let $ f:[0,\infty ) \times \Omega \to {{\mathbf{R}}^n}$ be continuous. In this note we establish necessary and sufficient conditions for function space flow invariance for the functional differential equation: $ x'(t) = f(t,{x_t})$ for $ t \geqslant 0,{x_0} = \phi \in \Omega $. That is, for each $ \phi \in \Omega $ there exist $ b > 0$ and a solution $ x:[ - r,b] \to {{\mathbf{R}}^n}$ such that $ {x_t} \in \Omega $ for each $ t \in [0,b]$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539637-7
Article copyright: © Copyright 1979 American Mathematical Society