The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability

Abstract

We study discretization in classes of integro-differential equations $\begin{gathered} u'(t) + \int_0^t {(\lambda _1 a_1 (t - \tau ) + \lambda _2 a_2 (t - \tau ) + \cdots + \lambda _n a_n (t - \tau ))} u(\tau )d\tau = 0,t > 0, \hfill \\ u(0) = 1,\lambda _j \geqslant 1,j = 1,2,...,n, \hfill \\ \end{gathered} $ , where the functions a j (t), 1 ⩽ j ⩽ n, are completely monotonic on (0,∞) and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted l 1(ρ; 0,∞) norm, where ρ is a given weight function. Applications to the weighted l 1 stability of the numerical solutions of a related equation in Hilbert space are given.