Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces
In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$ -Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$ -well-posedness for the third order differential equations $au^{\prime \prime \p...
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| Published in: | Canadian mathematical bulletin Vol. 62; no. 4; pp. 715 - 726 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Canada
Canadian Mathematical Society
01.12.2019
Cambridge University Press |
| Subjects: | |
| ISSN: | 0008-4395, 1496-4287 |
| Online Access: | Get full text |
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| Abstract | In this paper, by using operator-valued
${\dot{C}}^{\unicode[STIX]{x1D6FC}}$
-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the
$C^{\unicode[STIX]{x1D6FC}}$
-well-posedness for the third order differential equations
$au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$
, (
$t\in \mathbb{R}$
), where
$A,B$
are closed linear operators on a Banach space
$X$
such that
$D(A)\subset D(B)$
,
$a\in \mathbb{C}$
and
$0<\unicode[STIX]{x1D6FC}<1$
. |
|---|---|
| AbstractList | In this paper, by using operator-valued
${\dot{C}}^{\unicode[STIX]{x1D6FC}}$
-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the
$C^{\unicode[STIX]{x1D6FC}}$
-well-posedness for the third order differential equations
$au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$
, (
$t\in \mathbb{R}$
), where
$A,B$
are closed linear operators on a Banach space
$X$
such that
$D(A)\subset D(B)$
,
$a\in \mathbb{C}$
and
$0<\unicode[STIX]{x1D6FC}<1$
. In this paper, by using operator-valued \({\dot{C}}^{\unicode[STIX]{x1D6FC}}\)-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the \(C^{\unicode[STIX]{x1D6FC}}\)-well-posedness for the third order differential equations \(au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)\), (\(t\in \mathbb{R}\)), where \(A,B\) are closed linear operators on a Banach space \(X\) such that \(D(A)\subset D(B)\), \(a\in \mathbb{C}\) and \(0<\unicode[STIX]{x1D6FC}<1\). |
| Author | Cai, Gang Bu, Shangquan |
| Author_xml | – sequence: 1 givenname: Shangquan surname: Bu fullname: Bu, Shangquan email: sbu@math.tsinghua.edu.cn organization: Department of Mathematical Science, University of Tsinghua, Beijing 100084, China Email: sbu@math.tsinghua.edu.cn – sequence: 2 givenname: Gang surname: Cai fullname: Cai, Gang email: caigang-aaaa@163.com organization: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China Email: caigang-aaaa@163.com |
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| Cites_doi | 10.1007/978-3-0348-5075-9 10.1002/mma.1576 10.1007/s11464-014-0368-4 10.1007/s002080100231 10.1007/s11856-017-1496-9 10.1016/j.jde.2006.07.018 10.1023/A:1021778428222 10.1002/mana.201200168 10.1016/j.jmaa.2005.06.031 10.1016/j.exmath.2015.07.003 10.1016/S0096-3003(00)00112-0 10.4064/sm160-1-2 10.4064/sm215-3-1 |
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| DOI | 10.4153/S0008439518000048 |
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| Keywords | 34K30 47D06 34G10 Ċ -Fourier multiplier degenerate differential equation 47A10 C -well-posedness Hölder continuous function space |
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| References | Bose (S0008439518000048_r4) 2002; 126 S0008439518000048_r7 Bose (S0008439518000048_r3) 1998; 99 S0008439518000048_r6 S0008439518000048_r5 S0008439518000048_r9 S0008439518000048_r10 S0008439518000048_r12 S0008439518000048_r2 S0008439518000048_r11 S0008439518000048_r14 S0008439518000048_r1 S0008439518000048_r13 Haase (S0008439518000048_r8) 2005 |
| References_xml | – ident: S0008439518000048_r2 doi: 10.1007/978-3-0348-5075-9 – ident: S0008439518000048_r11 doi: 10.1002/mma.1576 – ident: S0008439518000048_r5 doi: 10.1007/s11464-014-0368-4 – volume-title: The Functional Calculus for Sectorial Operators year: 2005 ident: S0008439518000048_r8 – ident: S0008439518000048_r9 doi: 10.1007/s002080100231 – ident: S0008439518000048_r14 doi: 10.1007/s11856-017-1496-9 – ident: S0008439518000048_r10 doi: 10.1016/j.jde.2006.07.018 – volume: 99 start-page: 423 year: 1998 ident: S0008439518000048_r3 article-title: Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1021778428222 – ident: S0008439518000048_r13 doi: 10.1002/mana.201200168 – ident: S0008439518000048_r7 doi: 10.1016/j.jmaa.2005.06.031 – ident: S0008439518000048_r6 doi: 10.1016/j.exmath.2015.07.003 – volume: 126 start-page: 341 year: 2002 ident: S0008439518000048_r4 article-title: Exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure publication-title: Appl. Math. Comput. doi: 10.1016/S0096-3003(00)00112-0 – ident: S0008439518000048_r1 doi: 10.4064/sm160-1-2 – ident: S0008439518000048_r12 doi: 10.4064/sm215-3-1 |
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| Snippet | In this paper, by using operator-valued
${\dot{C}}^{\unicode[STIX]{x1D6FC}}$
-Fourier multiplier results on vector-valued Hölder continuous function spaces, we... In this paper, by using operator-valued \({\dot{C}}^{\unicode[STIX]{x1D6FC}}\)-Fourier multiplier results on vector-valued Hölder continuous function spaces,... |
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| SubjectTerms | Banach spaces Continuity (mathematics) Differential equations Function space Linear operators Mathematical analysis Operators (mathematics) Well posed problems |
| Title | Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces |
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