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

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.