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

Abstract

Fourth order eigenvalue problems with periodic and separated boundary conditions are considered. One of the separated boundary conditions depends linearly on the eigenvalue parameter λ. These problems can be represented by an operator polynomial L(λ) = λ 2 M -iαλK -A, where α > 0, M and K are self-adjoint operators.Necessary and sufficient conditions are given such that A is self-adjoint.MSC: Primary 34B07; secondary 34B08; 47E05

Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

“…Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order differential operators whose boundary conditions may depend on the eigenvalue parameter have been investigated in 4,5,7,8,9,10,12,13,14 .…”

Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

“…However, they have been attracting a lot of attention. Investigations on eigenvalue parameter for dependent boundary conditions in higher order boundary value problems have seen recent developments, see for example [1][2][3]5,8,[10][11][12][14][15][16][17][18][19][20]22,23,[25][26][27].…”

“…The operators M, K and A are coefficients of powers of λ, k is the number of boundary conditions depending on λ while I is an interval. Applying separation of variables to the stability of elastic rod problems investigated in [12,14,16,17,20,[25][26][27], we get fourth order eigenvalue problems with boundary conditions depending on the eigenvalue parameter λ with the differential equation y (4) − (gy ) = λ 2 y (1.2) depending quadratically on λ.…”

Stability of a flexible missile described by asymptotics of the eigenvalues of fourth order boundary value problems

“…Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order differential operators whose boundary conditions may depend on the eigenvalue parameter have been investigated in [5,6,8,9,10,11,13,14,15].…”

“…Separation of variables leads the stability of elastic rod problems investigated in [5,6,8,9,14,15] to fourth order eigenvalue problems with eigenvalue parameter dependent boundary conditions, where the differential equation…”