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Generic existence of a solution for a differential equation in a scale of Banach spaces


Author: Tomás Domínguez Benavides
Journal: Proc. Amer. Math. Soc. 86 (1982), 477-484
MSC: Primary 34G20; Secondary 54C50
DOI: https://doi.org/10.1090/S0002-9939-1982-0671219-9
MathSciNet review: 671219
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Abstract: Let $ \left\{ {{X_s}:\alpha \leqslant s \leqslant \beta } \right\}$ be a scale of Banach spaces, $ J$ a real interval, $ U$ an open subset of $ J \times {X_s}$ for some $ s$. In this paper we prove that the existence of solutions for

$\displaystyle x' = A(t)x + f(t,x),\quad x({t_0}) = {x_0},$

is a generic property, when $ A(t)$ is an operator satisfying

$\displaystyle {\left\vert {A(t)} \right\vert _{L({X_{s'}};{X_s})}} \leqslant M{(s' - s)^{ - 1}}\quad (M > 0\;{\text{independent}}\;{\text{of}}\;s,s',t)$

in the scale $ \left\{ {{X_s}} \right\}$ and $ f:J \times U \to {X_\beta }$ is continuous.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0671219-9
Keywords: Differential equation, scale of Banach spaces, Baire category, residual subset, generic property
Article copyright: © Copyright 1982 American Mathematical Society

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