Well-posedness of second order differential equations in Hölder continuous function spaces

By using operator-valued Ċ α -Fourier multiplier results on vector-valued Hölder continuous function spaces C α (ℝ; X) proved by Arendt, Batty and Bu, we obtain a necessary and sufficient condition for the C α -well-posedness for the following second order differential equations: u״ (t) = Au(t) + Bu...

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Bibliographic Details
Published in:Quaestiones mathematicae Vol. 42; no. 10; pp. 1379 - 1391
Main Authors: Bu, Shangquan, Cai, Gang
Format: Journal Article
Language:English
Published: Grahamstown Taylor & Francis 26.11.2019
Taylor & Francis Ltd
Subjects:
Banach spaces
Continuity (mathematics)
Differential equations
Fourier multiplier
Function space
Hölder continuous function spaces
Linear operators
Mathematical analysis
Operators (mathematics)
second order differential equations
Well posed problems
well-posedness
ISSN:1607-3606, 1727-933X
Online Access:Get full text
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Summary:By using operator-valued Ċ α -Fourier multiplier results on vector-valued Hölder continuous function spaces C α (ℝ; X) proved by Arendt, Batty and Bu, we obtain a necessary and sufficient condition for the C α -well-posedness for the following second order differential equations: u״ (t) = Au(t) + Bu׳ (t) + f (t), (t ∈ ℝ), where A and B are closed linear operators on a Banach space X satisfying D(A) ⊂ D(B). We give a concrete example that our abstract result may be applied.
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ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2018.1514330