The boundary-value problem

Abstract. In this paper small transverse vibrations of a string of inhomo-geneous stiffness in a damping medium with the left end fixed and the right end equipped with a concentrated mass are considered. By means of the Li-ouville transformation the corresponding differential equation is reduced to a Sturm–Liouville problem with parameter-dependent boundary conditions and parameter-dependent potential. This problem is considered as a spec-tral problem for the corresponding quadratic operator pencil. The inverse problem, i.e. the determination of the potential and the boundary conditions by the given spectrum and length of the string, is solved for weakly damped strings (having no purely imaginary eigenvalues). Uniqueness of the solutio