  In this paper, the inverse Sturm-Liouville problem with distribution
potential and with polynomials of the spectral parameter in one of the boundary
conditions is considered. We for the first time prove local solvability and
stability of this inverse problem in the general non-self-adjoint case, taking
possible splitting of multiple eigenvalues into account. The proof is based on
the reduction of the nonlinear inverse problem to a linear equation in the
Banach space of continuous functions on some circular contour. Moreover, we
introduce the generalized Cauchy data, which will be useful for investigation
of partial inverse Sturm-Liouville problems with polynomials in the boundary
conditions. Local solvability and stability of recovering the potential and the
polynomials from the generalized Cauchy data are obtained. Thus, the results of
this paper include the first existence theorems for solution of the inverse
Sturm-Liouville problems with polynomial dependence on the spectral parameter
in the boundary conditions in the case of multiple eigenvalues. In addition,
our stability results can be used for justification of numerical methods.
