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Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line


Author: P. P. B. Eggermont
Journal: Math. Comp. 53 (1989), 157-189
MSC: Primary 65R20; Secondary 45D05, 47H17
DOI: https://doi.org/10.1090/S0025-5718-1989-0969485-X
MathSciNet review: 969485
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Abstract: We consider Galerkin methods for monotone Abel-Volterra integral equations of the second kind on the half-line. The $ {L^2}$ theory follows from Kolodner's theory of monotone Hammerstein, equations. We derive the $ {L^\infty }$ theory from the $ {L^2}$ theory by relating the $ {L^2}$- and $ {L^\infty }$-spectra of operators of the form $ x \to b \ast (ax)$ to one another. Here $ \ast $ denotes convolution, and $ b \in {L^1}$ and $ a \in {L^\infty }$. As an extra condition we need $ b(t) = O({t^{ - \alpha - 1}})$, with $ \alpha > 0$. We also prove the discrete analogue. In particular, we verify that the Galerkin matrix satisfies the "discrete" conditions.


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DOI: https://doi.org/10.1090/S0025-5718-1989-0969485-X
Article copyright: © Copyright 1989 American Mathematical Society

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