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.

Functional and functional differential equations in a Banach space X are related to systems of operators A(t) in C = C(−r, 0; X), given by

Conditions are sought on F such that A(t) generates an evolution system U(t, s)φThis system gives the segments of solution for φ in a certain domain which is determined.

Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δpu = f(u), p ≧ 2. Sufficient conditions for existence of spherically symmetric entire solutions of Δpu = f(r, u) are also given.

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.

It is shown that there exists an uncountable number of rings R with the properties (1) if r∈R then 2r = 0 and r2 = 0, (2) if r ∈ R then there exist s, t in R with r = st.

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.

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

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 collisionless gas flows through a finite rectangular duct which reflects molecules diffusely. The transmission probability Q of the duct involves the solution of a pair of coupled integral equations. Complementary variational principles have been employed which supply upper and lower bounds to Q. Numerical calculations have been made for a variety of duct shapes and compared, where appropriate, to those of other authors.

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

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

This is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

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.

Some results are given for Weyl's m-coefficient in the algebra B = C(K). Included are a discussion of a theorem of Hille, a generalisation of Weyl's L.P./L.C. classification, and an example.

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].

A quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.

Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:

First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.

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.

In this work we present an algorithm for computing an integrable function almost everywhere on (0,1) when its moments are known. The method is based on the use of certain delta-shaped sequences, and can be adjusted to take advantage of the local smoothness of the function.

As an application, we give an algorithm for the pointwise inversion of the Laplace transform which utilizes the values of the image function at equidistant points.