On stability of an approximate solution of the Cauchy problem for some first-order integrodifferential equations

The Cauchy problem for a first-order evolutionary equation with memory with the time derivative of the Volterra integral term and difference kernel in the finite-dimensional Banach space is considered. The fundamental difficulties of the approximate solution of such problems are caused by nonlocality with respect to time when the solution at the current time depends on the entire history. Transformation of the first-order integrodifferential equation to a system of evolutionary local equations with the approximation of the difference kernel by a sum of exponential functions is used. For the weakly coupled system of local equations with additional ordinary differential equations, estimates of stability of solution with respect to initial data and right-hand side are obtained using the concept of logarithmic norm. Similar estimates are obtained for the approximate solution using two-level time approximations.