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

Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary... We consider the eigenvalue problem y (4)(λ,x) − (gy′)′(λ,x) = λ 2 y(λ,x) with separated boundary conditions B j (λ)y = 0 for j = 1,…,4, where g ∈ C 1[0, a] is a real valued function, B j (λ)y = y [p j ](a j ) or B j (λ)y = y [pj](a j ) + iϵ j αλy [qj ] (aj ), aj = 0 for j = 1, 2 and a j = a for j = 3, 4, α > 0, ϵ j ∈ {−1, 1}. We will associate to the above eigenvalue problem a quadratic operator pencil L(λ) = λ 2 M − iαλK − A in the space , where and are bounded self-adjoint operators and k is the number of boundary conditions which depend on λ. We give necessary and sufficient conditions for the operator A to be self-adjoint. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quaestiones Mathematicae Taylor & Francis

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

, Volume 34 (3): 14 – Sep 1, 2011
14 pages

# References (28)

Publisher
Taylor & Francis
Copyright Taylor & Francis Group, LLC
ISSN
1727-933X
eISSN
1607-3606
DOI
10.2989/16073606.2011.622913
Publisher site
See Article on Publisher Site

### Abstract

We consider the eigenvalue problem y (4)(λ,x) − (gy′)′(λ,x) = λ 2 y(λ,x) with separated boundary conditions B j (λ)y = 0 for j = 1,…,4, where g ∈ C 1[0, a] is a real valued function, B j (λ)y = y [p j ](a j ) or B j (λ)y = y [pj](a j ) + iϵ j αλy [qj ] (aj ), aj = 0 for j = 1, 2 and a j = a for j = 3, 4, α > 0, ϵ j ∈ {−1, 1}. We will associate to the above eigenvalue problem a quadratic operator pencil L(λ) = λ 2 M − iαλK − A in the space , where and are bounded self-adjoint operators and k is the number of boundary conditions which depend on λ. We give necessary and sufficient conditions for the operator A to be self-adjoint.

### Journal

Quaestiones MathematicaeTaylor & Francis

Published: Sep 1, 2011

Keywords: Primary: 34B07; Secondary: 34L99; 47E05; Fourth order differential equation; eigenvalue dependent boundary conditions; quadratic operator pencil; self-adjoint operator