Multiplicative perturbations of linear Volterra equations
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- by Abdelaziz Rhandi
- Proc. Amer. Math. Soc. 119 (1993), 493-501
- DOI: https://doi.org/10.1090/S0002-9939-1993-1169047-0
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Abstract:
We prove that the following problems are essentially equivalent: \[ \begin {array}{*{20}{c}} {{{[\operatorname {VO} ]}_{CT}}} & {\qquad \qquad u(t) = x + \int _0^t {a(t - s)CTu(s) ds,} } \\ {{{[\operatorname {VO} ]}_{TC}}} & {\qquad \qquad v(t) = y + \int _0^t {a(t - s)TCv(s) ds,} } \\ \end {array} \] where $T$ is an unbounded closed linear operator in a Banach space $X$ with dense domain $D(T),\;C$ is a bounded linear operator on $X$, and $a \in L_{\operatorname {loc} }^1([0,\infty ),\mathbb {R})$, which is exponentially bounded. We give some applications of our abstract theorem to second-order differential operators on the line.References
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Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 493-501
- MSC: Primary 47N20; Secondary 45D05, 47D03
- DOI: https://doi.org/10.1090/S0002-9939-1993-1169047-0
- MathSciNet review: 1169047