On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis
Given the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2018/4168609 |
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| _version_ | 1832556438820487168 |
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| author | Marat V. Markin |
| author_facet | Marat V. Markin |
| author_sort | Marat V. Markin |
| collection | DOAJ |
| description | Given the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R. |
| format | Article |
| id | doaj-art-c393cd1847284e679a7b302552766635 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c393cd1847284e679a7b3025527666352025-02-03T05:45:25ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252018-01-01201810.1155/2018/41686094168609On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real AxisMarat V. Markin0Department of Mathematics, California State University, Fresno, 5245 N. Backer Avenue, M/S PB 108, Fresno, CA 93740-8001, USAGiven the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.http://dx.doi.org/10.1155/2018/4168609 |
| spellingShingle | Marat V. Markin On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis International Journal of Mathematics and Mathematical Sciences |
| title | On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis |
| title_full | On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis |
| title_fullStr | On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis |
| title_full_unstemmed | On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis |
| title_short | On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis |
| title_sort | on the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis |
| url | http://dx.doi.org/10.1155/2018/4168609 |
| work_keys_str_mv | AT maratvmarkin onthedifferentiabilityofweaksolutionsofanabstractevolutionequationwithascalartypespectraloperatorontherealaxis |