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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The null space and the range of a convolution operator in a fading memory space


Author: Olof J. Staffans
Journal: Trans. Amer. Math. Soc. 281 (1984), 361-388
MSC: Primary 34K25; Secondary 45F05
DOI: https://doi.org/10.1090/S0002-9947-1984-0719676-X
MathSciNet review: 719676
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Abstract: We study the convolution equation $ (\ast)$

$\displaystyle \mu \; \ast \;x^{\prime}(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty < t\, < \infty )$

, as well as a perturbed version of $ (\ast)$, namely $ (\ast\ast)$

$\displaystyle \mu \;\ast\;x^{\prime}(t) + v\,\ast\;x(t) = \,F(x)\,(t)\quad ( - \infty < t < \infty ).$

Here $ x$ is a $ {{\mathbf{R}}^n}$-valued function on $ ( - \infty ,\infty ),x^{\prime}(t) = dx(t)/dt$, and $ \mu $ and $ \nu $ are matrix-valued measures. If $ \mu $ and $ \nu $ are supported on $ [0,\infty )$, with $ \mu $ atomic at zero, then $ (\ast)$ can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that $ \mu $ and $ \nu $ are supported on $ [0,\infty )$, prescribe an initial condition of the type $ x(t) = \xi (t)\,(t \leqslant 0)$, and require $ (\ast)$ and $ (\ast\ast)$ to hold only for $ t \geqslant 0$. Instead we permit $ (\ast)$ and $ (\ast\ast)$ to be of "Fredholm" type, i.e. $ \mu $ and $ \nu $ need not vanish on $ ( - \infty ,0)$, we restrict the growth rate of $ x$ and $ f$ at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of $ (\ast)$ and $ (\ast\ast)$ on the whole real line, satisfying conditions like $ \vert x(t)\vert \leqslant C\eta (t)\;( - \infty < t < \infty )$, where $ C$ is a constant, depending on $ x$, and $ \eta $ is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of $ (\ast)$ has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.

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DOI: https://doi.org/10.1090/S0002-9947-1984-0719676-X
Article copyright: © Copyright 1984 American Mathematical Society