Abstract: In this paper, we study a class of Sturm-Liouville problem, where the boundary conditions involve eigenparameters. Firstly, by defining a new inner product which depends on the transmission conditions, we obtain a new Hilbert space, on which the concerned operator $A$ is self-adjoint. Then we construct the fundamental solutions to the problem, obtain the necessary and sufficient conditions for eigenvalues, and prove that the eigenvalues are simple. Finally, we investigated Green's functions of such problem.

Key words: Sturm-Liouville problem, Eigenparameter-dependent boundary condition, Green's function