Abstract In this paper, we consider a second-order integro-differential equation with unbounded operator coefficients in a Hilbert space, which is an abstract hyperbolic equation perturbed by an integral term with a difference-type kernel of a special form. This equation arises in the description of various viscoelastic systems. Using the system of generalized eigenvectors of the operator sheaf associated with the equation considered, we construct a p-basis in the orthogonal sum of Hilbert spaces. Using this basis, we find a representation of the solution of the equation. We also discuss possible applications to problems of viscoelasticity.
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机译：摘要 本文研究了希尔伯特空间中具有无界算子系数的二阶积分微分方程，该方程是受具有特殊形式的差分型核的积分项扰动的抽象双曲方程。这个方程出现在各种粘弹性系统的描述中。使用与所考虑的方程相关的算子层的广义特征向量系统，我们在希尔伯特空间的正交和中构造了一个 p 基。在此基础上，我们找到了方程解的表示。我们还讨论了粘弹性问题的可能应用。
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