Equations in Hilbert Spaces Whose Sets of Solutions are Invariant Under a Group Isomorphic to a One-Parameter Group of Unitary Operators

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.