The nonlocal boundary value problem with eigenparameter dependent
boundary conditions is studied in this paper. Firstly, we give the
asymptotic expressions of the general solution for the equation
corresponding to the initial conditions with eigenparameters, then we
prove the multiplicity of eigenvalues some properties of the eigenvalues
and eigenfunctions. Finally, the asymptotic formulas of eigenvalues and
eigenfunctions are obtained under certain mild conditions. Our method is
to incorporate the perturbation theory and asymptotic analysis in the
framework of classical Sturm-Liouville problems, which provides a new
sight for the investigating of the Sturm-Liouville problems with
eigenparameter in boundary conditions.