Abstract We study a linear nonlocal problem for an evolution equation in a Banach space. The standard semigroup approach is used but integral averaging over time is used instead of the traditional initial condition. It is assumed that the evolution semigroup associated with the abstract differential equation is superstable (quasinilpotent), i.e., has an infinite negative exponential type. A theorem about unique solvability of the posed nonlocal problem is proved. It is shown that the solution can be represented by a convergent Neumann series. Some corollaries are noted. The case in which the semigroup is nilpotent is treated separately. The class of examples of superstable semigroups that are of interest in mathematical physics is outlined.

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具有超稳定半群的演化方程的非局部问题的可解性

摘要 我们研究了 Banach 空间中演化方程的线性非局部问题。使用标准的半群方法，但使用随时间的积分平均而不是传统的初始条件。假设与抽象微分方程相关的演化半群是超稳定的（拟幂等），即具有无穷负指数型。证明了所提出的非局部问题的唯一可解性定理。结果表明，该解可以用收敛的诺依曼级数表示。注意到了一些推论。半群是幂零的情况单独处理。概述了数学物理学中感兴趣的超稳定半群的例子。