On the Stability of Linear Multistep Methods for Volterra Convolution Equations

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Abstract
Linear multistep methods for ordinary differential equations generate convolution quadrature rules. We investigate the stability of such methods when applied to Volterra integral equations of the second kind and Volterra integro-differential equations. It is shown that the classical stability concepts for ordinary differential equations (strong stability, A-stability, A(α)-stability) carry over to convolution equations with positive definite or completely monotonic kernels.