Search

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg-Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg-Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem. AMS classification : 34B40: Ordinary Differential Equations, Boundary value problems on infinite intervals. 35J60: Nonlinear PDE of elliptic type. 35P15: Estimation of eigenvalues, upper and lower bound.

The present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

SynopsisIn this paper, we prove the existence of at least one solution to the problemwhere ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

SynopsisIn this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

SynopsisThe main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditionsr = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.