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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On linear Volterra integral equations of convolution type


Author: John S. Lew
Journal: Proc. Amer. Math. Soc. 35 (1972), 450-456
MSC: Primary 45D05
DOI: https://doi.org/10.1090/S0002-9939-1972-0308699-8
Erratum: Proc. Amer. Math. Soc. 43 (1974), 490.
MathSciNet review: 0308699
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Abstract: Let A be the set of all complex-valued locally integrable functions defined on $ [0, + \infty )$, and let T be the topology for A determined by the seminorms $ {t_r}(f) = \smallint_0^r {\vert f(x)\vert dx} $ for $ r = 1,2, \cdots $ , so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution. Then the map $ f \to f'$ from each element to its quasi-inverse is a homeomorphism of (A, T) onto itself. For each f, g in A the equation $ v = f + g ^\ast v$ has a unique solution in A which depends T-continuously on f, g, and is the T-limit of Picard approximations. The set of all f in A with $ f'$ in $ {L^1}[0, + \infty )$ is a set of first category in (A, T) but an open subset of A with the metric $ {\left\Vert {f - g} \right\Vert _1}$. For each series $ \sum\nolimits_{n = 1}^\infty {{p_n}{z^n}} $ converging in some neighborhood of $ z = 0$, and each element f in A, the series $ \sum\nolimits_{n = 1}^\infty {{p_n}{f^{ \ast n}}} $ converges in T to some element $ {p^ \ast }(f)$ in A.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0308699-8
Keywords: Linear Volterra integral equation, convolution kernel, locally integrable function, quasi-inverse, Picard iteration, integrable resolvent
Article copyright: © Copyright 1972 American Mathematical Society