We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – generalized positive definite kernel (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.

中文翻译：

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具有正定核的希尔伯特空间中二阶积分微分方程的均匀多项式稳定性

我们关注具有正定核的希尔伯特空间中二阶积分微分方程的多项式稳定性和能量的可积性，其中记忆可以是振荡的或符号变化的，也可以不是局部绝对连续的（没有任何控制条件）内核的衍生物）。对于这个稳定性问题，不能应用现有正定核理论的工具。为了解决这个问题，我们引入并研究了一个新的数学概念——广义正定核(GPDK)。借助 GPDK 及其属性，我们获得了进化方程多项式稳定性的有效标准，具有如此一般但更复杂和有用的记忆。此外，与现有的正定核相比，GPDK 允许我们直接表达相关核的衰减率。