The authors consider the following discontinuous Sturm-Liouville eigenvalue problems with eigenvalue parameters both in the equation and in one of the boundary conditions:
with the boundary conditions at
and the transition conditions
where for , for ; is a given real-valued function which is continuous in and in ; , are real numbers with , and . With an operator theoretic approach, some classical properties and asymptotic approximate formulae for eigenvalues and normalized eigenfunctions are obtained.