We construct classes of equations in a Hilbert space whose sets of solutions are invariant under a group isomorphic to a one-parameter group of unitary operators. It is shown that the unbounded solutions of these equations are unstable. The applications of the obtained results to nonlinear mechanics are presented.

中文翻译：

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希尔伯特空间中的方程，其解集在同构于单参数酉算子群的群下是不变的

我们在希尔伯特空间中构造方程组，其解集在同构于单参数酉算子组的群下是不变的。结果表明，这些方程的无界解是不稳定的。介绍了所得结果在非线性力学中的应用。