Sturm-Liouville problems with nonlinear dependence on the spectral parameter in the boundary conditions

In this study, asymptotic estimates of the eigenvalues and eigenfunctions are obtained for the Sturm-Liouville problem on a finite interval with boundary conditions depending polynomially on the spectral parameter. Besides, the derivations of the eigenfunctions are derived. Using the eigenfunctions and their derivations, Green’s function of the related problem is calculated. The error bounds in the obtained results are sharper than the existing results in the literature.