On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems
This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associ...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2014/948597 |
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| author | Ahmed Fadili Hamid Bounit |
| author_facet | Ahmed Fadili Hamid Bounit |
| author_sort | Ahmed Fadili |
| collection | DOAJ |
| description | This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a C0-semigroup. Next, a necessary and sufficient condition for Lp-admissibility p∈1,∞ of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2). |
| format | Article |
| id | doaj-art-fed6bcfbb1d045298eb18cceb59cb850 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-fed6bcfbb1d045298eb18cceb59cb8502025-02-03T06:00:41ZengWileyInternational Journal of Differential Equations1687-96431687-96512014-01-01201410.1155/2014/948597948597On the Complex Inversion Formula and Admissibility for a Class of Volterra SystemsAhmed Fadili0Hamid Bounit1Department of Mathematics, Faculty of Sciences, Ibn Zohr University, BP 8106, 8000 Agadir, MoroccoDepartment of Mathematics, Faculty of Sciences, Ibn Zohr University, BP 8106, 8000 Agadir, MoroccoThis paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a C0-semigroup. Next, a necessary and sufficient condition for Lp-admissibility p∈1,∞ of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).http://dx.doi.org/10.1155/2014/948597 |
| spellingShingle | Ahmed Fadili Hamid Bounit On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems International Journal of Differential Equations |
| title | On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems |
| title_full | On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems |
| title_fullStr | On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems |
| title_full_unstemmed | On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems |
| title_short | On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems |
| title_sort | on the complex inversion formula and admissibility for a class of volterra systems |
| url | http://dx.doi.org/10.1155/2014/948597 |
| work_keys_str_mv | AT ahmedfadili onthecomplexinversionformulaandadmissibilityforaclassofvolterrasystems AT hamidbounit onthecomplexinversionformulaandadmissibilityforaclassofvolterrasystems |