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Flow invariance for differential delay equations


Author: Naoki Tanaka
Journal: Proc. Amer. Math. Soc. 143 (2015), 2459-2468
MSC (2010): Primary 47J35; Secondary 47H06, 47H20
DOI: https://doi.org/10.1090/S0002-9939-2015-12437-9
Published electronically: January 9, 2015
MathSciNet review: 3326028
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Abstract: The flow invariance problem for the functional differential equation $ u'(t)\in Au(t)+F(u_t)$ for $ t\geq 0$ with initial condition $ u_0=\phi \in \frak {D}$ is solved in a Banach space $ X$, where $ A$ is a quasi-dissipative operator in $ X$ and $ F$ is a continuous operator from a closed set $ \frak {D}$ in the so-called initial-history space $ \frak {X}$ into $ X$ satisfying a dissipativity condition in the following sense: There exists $ \omega _F\geq 0$ such that $ [\phi (0)-\hat {\phi }(0),~F(\phi )-F(\hat {\phi })]_{+}\leq \omega _F\Vert\phi -\hat {\phi }\Vert _{\frak {X}}$ for $ \phi , \hat {\phi }\in \frak {D}$ satisfying that $ \Vert\phi -\hat {\phi }\Vert _{\frak {X}}\leq \Vert\phi (0)-\hat {\phi }(0)\Vert _X$, where $ [x,\xi ]_{+}=\lim _{h\to 0+}(\Vert x+h\xi \Vert _X-\Vert x\Vert _X)/h$ for $ x,\xi \in X$.


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Additional Information

Naoki Tanaka
Affiliation: Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka 422-8529, Japan
Email: tanaka.naoki@shizuoka.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2015-12437-9
Received by editor(s): July 28, 2013
Received by editor(s) in revised form: December 6, 2013, December 25, 2013, and December 27, 2013
Published electronically: January 9, 2015
Additional Notes: The author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 25400134
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society

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