We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation. The existence of a contraction $$C_0 $$ -semigroup is proved under certain assumptions about the kernels of integral operators. Examples of exponential and fractional-exponential (Rabotnov functions) kernels of integral operators satisfying the above assumptions are given.

中文翻译：

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由 Volterra 积分微分方程生成的半群

我们研究抽象的积分微分方程，这些方程是粘弹性问题的算子模型。我们基于与线性演化方程的单参数半群研究相关的方法来呈现结果。所提出的方法也可用于研究包含 Volterra 卷积形式的积分项的其他积分微分方程。给出了一种将Hilbert空间中具有算子系数的模型积分微分方程的原始初值问题化简为一阶微分方程的Cauchy问题的方法。收缩$$C_0 $$ -semigroup 的存在是在关于积分运算符核的某些假设下证明的。