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Periodic $ L\sp 2$-solutions of an integrodifferential equation in a Hilbert space

Author: Olof J. Staffans
Journal: Proc. Amer. Math. Soc. 117 (1993), 745-751
MSC: Primary 45J05; Secondary 34K15, 47G10, 47N20
MathSciNet review: 1111439
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Abstract: Let $ A$ be a closed, densely defined operator in a Hilbert space $ X$, and let $ \mu ,\;\nu $, and $ \eta $ be finite, scalar-valued measures on $ {\mathbf{R}}$. Consider the abstract integrodifferential equation

$\displaystyle \int_{\mathbf{R}} {\frac{d} {{dt}}u(t - s)\mu (ds) + \int_{\mathb... ...+ \int_{\mathbf{R}} {Au(t - s)\eta (ds) = f(t),\qquad t \in {\mathbf{R}},} } } $

where $ f$ is a $ 2\pi $-periodic $ {L^2}$ function with values in $ X$. We give necessary and sufficient conditions for this equation to have a mild $ 2\pi $-periodic $ {L^2}$-solution with values in $ X$ for all $ f$, as well as necessary and sufficient conditions for it to have a strong solution for all $ f$. Furthermore, we give necessary and sufficient conditions for the operator mapping $ f$ into the periodic solution $ u$ to be compact. These results are applied to prove existence of periodic solutions of a nonlinear equation.

References [Enhancements On Off] (What's this?)

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Keywords: Integrodifferential equation, well-posedness, periodic solutions, $ {L^2}$-multipliers, compact solution operator
Article copyright: © Copyright 1993 American Mathematical Society