Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions

Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

“…For rather generic boundary conditions, a quadratic operator pencil has been associated in [3], and we will now recall notations and results from [3] which are relevant in our case.…”

“…and [ / ] means that we omit those terms of the Leibniz expansion which contain a function ( ) with > 4 − . Since the coefficient of (3) in (1) is zero, we have 0 ( ) = 1, see [10, (8.2.3)].…”

“…Again, this problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coefficient operators are self-adjoint. In [3], we have considered this fourth order differential equation with general two-point boundary conditions which depend linearly on the eigenvalue parameter. Necessary and sufficient conditions such that the associated operator pencil consists of self-adjoint operators have been obtained.…”

Fourth-order Birkhoff regular problems with eigenvalue parameter dependent boundary conditions

“…Salaff considered in [14] an arbitrary m th order linear differential expression and m linearly independent, homogeneous, two-point boundary conditions and proved that if the problem is self-adjoint, then it is Birkhoff regular. [6][7][8][9][10] to fourth-order boundary problems with eigenvalue parameter dependent boundary conditions, where the differential equation…”

Sixth order differential operators with eigenvalue dependent boundary conditions

“…For the same differential operator as in [19], we have investigated a more general class of eigenvalue parameter dependent boundary conditions. Necessary and sufficient conditions for the associated operator pencil to consist of selfadjoint operators have been obtained in [21], while in [22,23] we have continued the work in the direction of [19] to find the asymptotic distribution of eigenvalues for boundary conditions which lead to self-adjoint operator representations. In this paper we start to extend this investigation to a corresponding problem for a sixth order differential equation.…”