On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition

Abstract

which contradicts condition (1.3). Therefore, λ * ∈ R. The entire function occurring on the left-hand side in Eq. (2.1) does not vanish for nonreal λ. Consequently, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points.

Some global results for nonlinear fourth order eigenvalue problems

“…The operator L is self-adjoint, discrete and semibounded from below in H [1,16] and therefore L + λI (I : H → H is the identity operator) is invertible for negative λ with sufficiently large absolute value. Without loss of generality we may assume that this is already the case for λ = 0 It follows that λ = 0 cannot be an eigenvalue of the operator L 1 Thus L −1 1 exists and is an integral operator, the kernel of which we denote by ( ) Using the method of paper [32] we can prove that…”

“…In the following, we will denote byˆ + the unique eigenfunction of (17) For each integer > 0 and each ν = + or −, there exists an unbounded continuum of solutionsD ν of (16), such that…”

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

“…in the space 2 (0, ) ⊕ C 4 with the problem (1), (2). We observe that (8) is an operator representation of the eigenvalue problem (1), (2) in the sense that a function satisfies (1), (2) if and only if ( , )̃= 0.…”

“…Other recent results on fourth order differential operators whose boundary conditions depend on the eigenvalue parameter but which are represented by linear operator pencils, include spectral asymptotics and basis properties, see [6][7][8].…”

Oscillation properties for the equation of vibrating beam with irregular boundary conditions

“…Moreover, we assume that the equation which corresponds to the eigenvalue λ n , n 1 has n − 1 zeros in (0, 1). Using an extension of the Prüfer transformation, this case has been studied by Banks and Kurowski [1,2] for q 0 (and in some particular cases when (1.5) is disfocal) and in the recent papers by Kerimov and Aliev [7,8] also for q 0. In the case of regular boundary conditions, it was shown in [6] the existence of a finite number of negative eigenvalues and a sequence of positive and simple eigenvalues, whose corresponding eigenfunctions (to the positive eigenvalues) have the same oscillatory properties as in the Sturm-Liouville problem.…”