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(2) $u_2()+hu_1(0)=0, u_2(1)+Hu_1(1)=0. $ Here $P=\\begin{pmatrix} a \u0026 b \\\\ (p_1) \u0026 (p_2) \\end{pmatrix} \\in {C^1[1,1]}^4 : $ : real-valued and h, H are real constants, and λ corresponds to an eigenvalue. We denote the set of eigenvalues of (1)-(2) by ${\\lambda _n)P,h,H)}_(n\\in Z)$ under an appropriate numbering. For $Q=\\begin{pmatrix} a \u0026 b \\\\ (q_1) \u0026 (q_2) \\end{pmatrix} \\in {C^1[0,1]}^4$ and $h, H, H^*, J, J^* \\in R\\setminus {-1,1} (H\\\neq H^*, J\\\neq J^*)$, we obtain the following result on continuous dependence of coefficients and boundary conditions upon eigenvalues : If $\\delta _0\\equiv \\sum _n^\\infty =_(-\\infty )(\\mid \\lambda _n(Q,h,J)-\\lambda _n(P,h,H)\\mid + (\\mid \\lambda _n(Q,h,J^*)-\\lambda _n(P,h,H^*)\\mid $ is sufficientrly small, then $\\parallel Q-P\\parallel _{[c^1[0,1]]^4}\\leqq +\\mid J-H\\mid +\\mid J^*-H^*\\mid \\leqq M\\delta _0$ for some constant M\u003e0. Moreover we get $\\parallel Q-P\\parallel _{[c^1[0,1]]^4}\\leqq M$ . We show also that for given $\\mu _n, \\mu _n ^* \\in C (n\\in Z)$, there exists a unique $(Q,h,J^*)\\in {C^1[0,1]}^4 \\times (R\\setminus {-1,1})^2$ satisfying $\\lambda _n(Q, h, J)=\\mu _n$ and $\\mu _n(Q,h,j^*)=\\mu _n^*$ under appropriate assumptions on $\\mu _n, \\mu _n^*(n\\in Z)$ $(e.g. \\sum _n^\\infty =_(-\\infty )(\\mid \\mu _n - \\lambda _n(P,h,H^*)\\mid)$ is sufficiently small.). We prove these results by the principle of contraction mappings and, in order to apply the principle, we establish a priori estimates of solutions to some hyperbolic systems and results on perturbation of Riesz bases.", "subitem_description_type": "Abstract"}]}, "item_4_identifier_registration": {"attribute_name": "ID登録", "attribute_value_mlt": [{"subitem_identifier_reg_text": "10.15083/00040996", "subitem_identifier_reg_type": "JaLC"}]}, "item_4_publisher_20": {"attribute_name": "出版者", "attribute_value_mlt": [{"subitem_publisher": "The University of Tokyo"}]}, "item_4_source_id_10": {"attribute_name": "書誌レコードID", "attribute_value_mlt": [{"subitem_source_identifier": "AA10538733", "subitem_source_identifier_type": "NCID"}]}, "item_4_source_id_8": {"attribute_name": "ISSN", "attribute_value_mlt": [{"subitem_source_identifier": "02897520", "subitem_source_identifier_type": "ISSN"}]}, "item_4_subject_15": {"attribute_name": "日本十進分類法", "attribute_value_mlt": [{"subitem_subject": "410", "subitem_subject_scheme": "NDC"}]}, "item_4_text_34": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"subitem_text_value": "Departmental Bulletin Paper"}]}, "item_4_text_4": {"attribute_name": "著者所属", "attribute_value_mlt": [{"subitem_text_value": "Department of Mathematics, College of Arts and Sciences, University of Tokyo"}]}, "item_creator": {"attribute_name": "著者", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "Yamamoto, Masahiro"}], "nameIdentifiers": [{"nameIdentifier": "139881", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "ファイル情報", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "2017-06-27"}], "displaytype": "detail", "download_preview_message": "", "file_order": 0, "filename": "scp038005.pdf", "filesize": [{"value": "3.0 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_free", "mimetype": "application/pdf", "size": 3000000.0, "url": {"label": "scp038005.pdf", "url": "https://repository.dl.itc.u-tokyo.ac.jp/record/41005/files/scp038005.pdf"}, "version_id": "ad4e0893-515d-4e64-92d2-64bce3c07453"}]}, "item_language": {"attribute_name": "言語", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"resourcetype": "departmental bulletin paper", "resourceuri": "http://purl.org/coar/resource_type/c_6501"}]}, "item_title": "Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order", "item_titles": {"attribute_name": "タイトル", "attribute_value_mlt": [{"subitem_title": "Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order"}]}, "item_type_id": "4", "owner": "1", "path": ["7300", "7301"], "permalink_uri": "https://doi.org/10.15083/00040996", "pubdate": {"attribute_name": "公開日", "attribute_value": "2008-11-19"}, "publish_date": "2008-11-19", "publish_status": "0", "recid": "41005", "relation": {}, "relation_version_is_last": true, "title": ["Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order"], "weko_shared_id": null}
Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order
https://doi.org/10.15083/00040996
https://doi.org/10.15083/000409963d584204-d3d7-4f99-97b1-b8c572319f90
名前 / ファイル | ライセンス | アクション |
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scp038005.pdf (3.0 MB)
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Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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公開日 | 2008-11-19 | |||||
タイトル | ||||||
タイトル | Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order | |||||
言語 | ||||||
言語 | eng | |||||
資源タイプ | ||||||
資源 | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | departmental bulletin paper | |||||
ID登録 | ||||||
ID登録 | 10.15083/00040996 | |||||
ID登録タイプ | JaLC | |||||
著者 |
Yamamoto, Masahiro
× Yamamoto, Masahiro |
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著者所属 | ||||||
著者所属 | Department of Mathematics, College of Arts and Sciences, University of Tokyo | |||||
抄録 | ||||||
内容記述タイプ | Abstract | |||||
内容記述 | We consider an eigenvalue problem (1)-(2) : (1) $\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} du(x)/dx +P(x)u(x)=\lambda u(x)$ $(0\leqq x\leqq 1 ; u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}$. (2) $u_2()+hu_1(0)=0, u_2(1)+Hu_1(1)=0. $ Here $P=\begin{pmatrix} a & b \\ (p_1) & (p_2) \end{pmatrix} \in {C^1[1,1]}^4 : $ : real-valued and h, H are real constants, and λ corresponds to an eigenvalue. We denote the set of eigenvalues of (1)-(2) by ${\lambda _n)P,h,H)}_(n\in Z)$ under an appropriate numbering. For $Q=\begin{pmatrix} a & b \\ (q_1) & (q_2) \end{pmatrix} \in {C^1[0,1]}^4$ and $h, H, H^*, J, J^* \in R\setminus {-1,1} (H\ eq H^*, J\ eq J^*)$, we obtain the following result on continuous dependence of coefficients and boundary conditions upon eigenvalues : If $\delta _0\equiv \sum _n^\infty =_(-\infty )(\mid \lambda _n(Q,h,J)-\lambda _n(P,h,H)\mid + (\mid \lambda _n(Q,h,J^*)-\lambda _n(P,h,H^*)\mid $ is sufficientrly small, then $\parallel Q-P\parallel _{[c^1[0,1]]^4}\leqq +\mid J-H\mid +\mid J^*-H^*\mid \leqq M\delta _0$ for some constant M>0. Moreover we get $\parallel Q-P\parallel _{[c^1[0,1]]^4}\leqq M$ . We show also that for given $\mu _n, \mu _n ^* \in C (n\in Z)$, there exists a unique $(Q,h,J^*)\in {C^1[0,1]}^4 \times (R\setminus {-1,1})^2$ satisfying $\lambda _n(Q, h, J)=\mu _n$ and $\mu _n(Q,h,j^*)=\mu _n^*$ under appropriate assumptions on $\mu _n, \mu _n^*(n\in Z)$ $(e.g. \sum _n^\infty =_(-\infty )(\mid \mu _n - \lambda _n(P,h,H^*)\mid)$ is sufficiently small.). We prove these results by the principle of contraction mappings and, in order to apply the principle, we establish a priori estimates of solutions to some hyperbolic systems and results on perturbation of Riesz bases. |
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書誌情報 |
Scientific papers of the College of Arts and Sciences, the University of Tokyo 巻 38, p. 69-130, 発行日 1988 |
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ISSN | ||||||
収録物識別子タイプ | ISSN | |||||
収録物識別子 | 02897520 | |||||
書誌レコードID | ||||||
収録物識別子タイプ | NCID | |||||
収録物識別子 | AA10538733 | |||||
フォーマット | ||||||
内容記述タイプ | Other | |||||
内容記述 | application/pdf | |||||
日本十進分類法 | ||||||
主題 | 410 | |||||
主題Scheme | NDC | |||||
出版者 | ||||||
出版者 | The University of Tokyo |