Abstract We consider the Cauchy problem for a linear integro-differential equation whose differential part contains only the first derivative multiplied by a small positive parameter and whose integral part is the sum of two Volterra integral operators, one with slowly and one with rapidly varying kernel. Earlier, this Cauchy problem has only been considered for the case in which the integral part of the equation does not contain an operator with slowly varying kernel. We develop the Lomov regularization method for the new class of problems and use it to construct the asymptotics of the solution and obtain a convergence estimate.

中文翻译：

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微分部分为零算子且核慢速变化的积分-微分方程解的正则化渐近性

摘要 我们考虑线性积分微分方程的柯西问题，其微分部分仅包含乘以一个小的正参数的一阶导数，其积分部分是两个 Volterra 积分算子的和，一个具有缓慢变化的内核，一个具有快速变化的内核。早些时候，这个柯西问题只在方程的积分部分不包含具有缓慢变化核的算子的情况下被考虑。我们为新类别的问题开发了 Lomov 正则化方法，并使用它来构造解的渐近线并获得收敛估计。