Publisher Summary The differential system and the boundary conditions are allowed to depend holomorphically on the eigenvalue parameter. The boundary conditions consist of terms at the endpoints and at interior points of the underlying interval and of an integral term. Such boundary eigenvalue problems are considered in suitable Sobolev spaces, so that both the differential operators and the boundary operators define bounded operators on Banach spaces. In a canonical way, holomorphic Fredholm operator valued function is associated to such a boundary eigenvalue problem with the variable being the eigenvalue parameter. This operator function consists of two components—the first one is the differential operator function, and the second one is the boundary operator function. Operator functions defined in this way are called boundary eigenvalue operator functions. The adjoint operator function of a boundary eigenvalue operator function defines the adjoint boundary eigenvalue problem.