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

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 ^{[q_j ]} (a_j ), a_j = 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.