Comparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problems

and

with φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

Classically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

The existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

It is shown that in the Hartree approximation the energy functional of the helium atom reaches its minimum and that the corresponding minimizing function is a solution of the Hartree equation.

Relaxation oscillations of a forced van der Pol oscillator are considered. The limiting behaviour of such a solution is investigated using a priori estimates. This simplifies previous approaches of the problem.

A new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

A Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.

In this paper a combination is given of the two methods mentioned in the title with the aim of optimising the definition of the absolute temperature with respect to physical transparency and mathematical simplicity. The mathematical tools used are the elements of vector analysis, i.e. “usual” mathematics (“normale” Mathematik) in the sense of Born [1, p. 162, cf. also 22, p. 579].

The existence of periodic solutions is proved for first order vector ordinary and functional differential equations when the right-hand side satisfies a one-sided growth restriction of Wintner type together with some conditions of asymptotic nature. Special cases in the line of Landesman-Lazer and of Winston are explicited.

It is well known that the Dirichlet problem for hyperbolic equations is a classical “not well posed” problem. Here we consider the Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for the hyperbolic equation uxy = 0 in all positions of the square and a class of rectangles. We also get a partial answer to the problem which deals with a ray that moves from any point on the boundary of a rectangle and is reflected on the boundary such that the angle between every ray and its reflection is π/2.

We show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.

We show that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a (right) Baer semigroup ℛS in which each element of (S) has an idempotent generator f such that fe = efe for all idempotents e with eS ∈ ℛ(S).

Formulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

The semilinear elliptic differential operator L[u] = Δu + c(x, u) is studied and sufficient conditions are derived for all solutions of uL[u] ≦ 0 with suitable boundary conditions to be oscillatory in unbounded domains of Rn. Here, unbounded domains to be considered are cones, strips and cylinders in Rn. The results are based on the conditions for the non-existence of positive solutions of ordinary differential inequalities.

This paper is devoted to the study of the weak respectively strong convergence of solutions of a variational inequality, with nonlinear partial differential operators of the generalized divergence form and of semimonotone type, under a perturbation of the domain of definition. In this study we use abstract convergence theorems given by Stroescu and Vivaldi, convergence concepts defined according to Stummel and compactness theorems of the natural imbedding of the Cartesian product of Sobolev spaces into the direct sum of Lp spaces, also by Stummel.