On bases of eigenfunctions of boundary problem associated with small vibrations of damped nonsmooth inhomogeneous string

The initial–boundary problem is considered for a nonsmooth inhomogeneous string with the left end fixed and the right one equipped with a massive ring moving with damping in the direction orthogonal to the length of the string. The asymptotic behaviour of the eigenvalues of the corresponding boundary problem is investigated. The eigenvectors of the problem are those of a dissipative operator acting in the Hilbert space $H=\hat{W}_2^1[0,l]\oplus L_2[0,l]\oplus\mathbb{C},$ where $\hat{W}_2^1[0,l]$ is the subspace of functions in Sobolev space $W_2^1[0,l],$ which vanish at the origin. It is proved that the set of the normalized rootvectors of this operator is a Riesz basis in $H.$ Under certain conditions the initial–boundary problem admits a unique solution.