Function space flow invariance for functional-differential equations of retarded type
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- by James H. Lightbourne PDF
- Proc. Amer. Math. Soc. 77 (1979), 91-98 Request permission
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 $t \in [0,b]$.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 77 (1979), 91-98
- MSC: Primary 34K05
- DOI: https://doi.org/10.1090/S0002-9939-1979-0539637-7
- MathSciNet review: 539637