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In this paper, quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides are considered, with the quadrature methods generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, and the smoothness of the involved functions is carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.

The Regularizing Properties of Multistep Methods for First Kind Volterra Integral Equations with Smooth Kernels