Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions

Abstract

A regular fourth order differential equation with λ-dependent boundary conditions is considered. For four distinct cases with exactly one λ-independent boundary condition, the asymptotic eigenvalue distribution is presented.

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

“…In [5], we have considered the differential equation (1) with the same boundary conditions 3 , 4 at as in this paper but only one -dependent boundary conditions at 0. We observe that the first two terms in the eigenvalue expansion coincide with those in Case 1 of [5], which differs from the present case that the -term is absent in the boundary condition (10). However, the third and fourth terms are similar but different.…”

“…Necessary and sufficient conditions such that the associated operator pencil consists of self-adjoint operators have been obtained. In [4,5], we have derived eigenvalue asymptotics associated with particular boundary conditions. In this paper, we are considering the case of separable boundary conditions where all four of these boundary conditions depend on the eigenvalue parameter.…”

“…In Section 4, we prove that the boundary value problem under investigation is Birkhoff regular, which implies that the eigenvalues for general are small perturbations of the eigenvalues for = 0. Hence, in Section 5, we derive the first four terms of the asymptotics of the eigenvalues and compare them to those obtained for the boundary conditions considered in [5].…”

Asymptotic Behavior of Eigenvalue and Eigenfunction of a Six Order Boundary Value Problem

“…It is well known that many researchers have investigated the spectral properties of the Sturm-Liouville operator generated by the separate boundary condition (Aigunov, 1996) and many researchers have found asymptotic formula for the Sturm-Liouville operator's eigenvalues and functions in the case of periodic andanti-periodic boundary conditions (Menken, 2010;Naimark, 1967;Moller and Zinsou, 2012;Jwamer and Aigounv, 2010;Aigounov and Tamila, 2009;Aigunov, 1996 andTamarkin, 1928). Many researchers have been interested in the ongoing Sturm-Liouville issue in recent years as we see N.B.…”

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.…”