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Minimality of the system of root functions of Sturm-Liouville problems with decreasing affine boundary conditions

Y. N. Aliyev (2007)

Colloquium Mathematicae

We consider Sturm-Liouville problems with a boundary condition linearly dependent on the eigenparameter. We study the case of decreasing dependence where non-real and multiple eigenvalues are possible. By determining the explicit form of a biorthogonal system, we prove that the system of root (i.e. eigen and associated) functions, with an arbitrary element removed, is a minimal system in L₂(0,1), except for some cases where this system is neither complete nor minimal.

Monotonicity of the principal eigenvalue related to a non-isotropic vibrating string

Behrouz Emamizadeh, Amin Farjudian (2014)

Nonautonomous Dynamical Systems

In this paper we consider a parametric eigenvalue problem related to a vibrating string which is constructed out of two different materials. Using elementary analysis we show that the corresponding principal eigenvalue is increasing with respect to the parameter. Using a rearrangement technique we recapture a part of our main result, in case the difference between the densities of the two materials is sufficiently small. Finally, a simple numerical algorithm will be presented which will also provide...

New method for computation of discrete spectrum of radical Schrödinger operator

Ivan Úlehla, Miloslav Havlíček (1980)

Aplikace matematiky

A new method for computation of eigenvalues of the radial Schrödinger operator - d 2 / d x 2 + v ( x ) , x ≥ 0 is presented. The potential v ( x ) is assumed to behave as x - 2 + ϵ if x → 0 + and as x - 2 - ϵ if x → + ∞ , ϵ ≥ 0 . The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function z ( x , ℵ ) . It is shown that the eigenvalues are the discontinuity points of the function z ( ∞ , ℵ ) . Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison...

New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito, Nguyen Van Minh, Jong Son Shin (2001)

Studia Mathematica

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where e i s p ( f ) ¯ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*) has a bounded...

Nonhermitian systems and pseudospectra

Lloyd N. Trefethen (2005/2006)

Séminaire Équations aux dérivées partielles

Four applications are outlined of pseudospectra of highly nonnormal linear operators.

Nonlinear eigenvalue problems for fourth order ordinary differential equations

Jolanta Przybycin (1995)

Annales Polonici Mathematici

This paper was inspired by the works of Chiappinelli ([3]) and Schmitt and Smith ([7]). We study the problem ℒu = λau + f(·,u,u',u'',u''') with separated boundary conditions on [0,π], where ℒ is a composition of two operators of Sturm-Liouville type. We assume that the nonlinear perturbation f satisfies the inequality |f(x,u,u',u'',u''')| ≤ M|u|. Because of the presence of f the considered equation does not in general have a linearization about 0. For this reason the global bifurcation theorem of...

Nonlinear systems with mean curvature-like operators

Pierluigi Benevieri, João Marcos do Ó, Everaldo Souto de Medeiros (2007)

Banach Center Publications

We give an existence result for a periodic boundary value problem involving mean curvature-like operators. Following a recent work of R. Manásevich and J. Mawhin, we use an approach based on the Leray-Schauder degree.

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