The existence of periodic solutions of non-linear Schrödinger equations with periodic potential is investigated. The main result obtained is an intermediate value type theorem for such non-linear differential operators.

This paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equation

Sufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].

This paper is concerned with some properties of an ordinary symmetric matrix differential expression M, denned on a certain class of vector-functions, each of which is defined on the real line. For such a vector-function F we have M[F] = −F“ + QF on R, where Q is an n × n matrix whose elements are reasonably behaved on R. M is classified in an equivalent of the limit-point condition at the singular points ± ∞, and conditions on the matrix coefficient Q are given which place M, when n> 1, in the equivalent of the strong limit-point for the case n = 1. It is also shown that the same condition on Q establishes the integral inequality for a certain class of vector-functions F.

Using the theory of the weighted Sobolev space H1,2(μ), on a bounded domain Ω, in Rn, the existence and regularity of solutions u in K to the variational inequality

is established for various convex subsets K of H1, 2(μ). The growth conditions imposed on the functions A and B give the differential inequality degenerate elliptic structure, extending the results on regularity for inequalities of elliptic type.

A general formalism for the description of high-frequency gravitational radiation is presented, and a justification is provided for the often made assumption that the energy of an isotropic field of gravitational radiation in a cosmological model may be represented by a perfect fluid.

The perturbed linear ordinary differential equation

is considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

In this paper, we determine the semigroup End D of endomorphisms of the infinite dihedral group D, and give a multiplication table for it. We determine the additive structure of the near-rings E(D) generated by the endomorphisms of D, A(D) generated by the automorphisms of D and I(D) generated by the inner automorphisms of D, and determine their radicals and all their maximal right ideals.

Dual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

Let S and T be formally symmetric ordinary differential operators defined on a real interval I. It is assumed that the order of S is constant and everywhere strictly higher than the possibly varying order of T. The main result of this paper (Theorem 2.3) gives necessary and sufficient conditions for maximality of the deficiency indices of the differential relation Su = Tv considered in a Hilbert space with a scalar product which is a Dirichlet integral (see section 2) belonging to S. The conditions generalise those given in [5] for less general choices of operators S and T. For certain choices of Dirichlet integral they are explicit integrability conditions on the coefficients of the Dirichlet integral and the operator T.

Let f, g be two functions of two Besov spaces (or Sobolev spaces), we look for the Besov spaces to which the product f × g belongs so that the multiplication is a continuous mapping.

The paper provides conditions which enstlre that the Schrödinger operator

defined on an exterior domain has no eigenvalues on a certain half-ray. These conditions are in terms of weak local assumptions on

The proof uses Kato's ideas [16] in conjunction with the physicists' “commutator proof” of the quantum mechanical virial theorem.

We investigate the spectral theory for a class of pseudodifferential operators which includes all constant coefficient differential operators, and also operators such as The operators considered are of the form Su(x) = Au(x)+q(x)u(x), where A is an operator which corresponds in the Fourier transform plane to a multiplication operator, and q(x) is a potential term. We prove an eigenfunction expansion theorem for S and derive some results concerning the spectrum of S.

Let (1) x′ = f(t, x) be any differential equation and S0 the set of solutions of (1) with open domain. It is known that for every g ∊ S0 a non-continuable (= saturated) ∊ S0 exists which is an extension of g. Usually is represented in the form is a sequence in S0 defined by some sort of a variant of what is called ‘recursive definition’ in set theory. It will be shown that a function

exists (P(S0) is the power set of S0) such that the above-mentioned variant can be given the form: There exists a sequence in S0 such that

Many known results about the stability of selfadjointness are extended to results about the stability of the deficiency index of closed symmetric operators on Hilbert space under perturbation.

This note sets out from the observation that it is not, in general, possible to express a homogeneous cubic polynomial in five variables as a sum of cubes of seven linear forms. Some of the geometry, to which particular cubics which do happen to be so expressible give rise, is described. Further particularisations are mentioned, and one such cubic investigated in some detail.

The object of the present note is to study functions f(z) which are bounded and analytic in |z| > 1, with Taylor expansion

where the sequence {λn} of integers satisfies the Hadamard gap condition

Here we give necessary and sufficient conditions foi a prime ι to divide the class number of the Galois closure of a pure field of degree ι over the rationals. The work extends that of Honda in [4] and that of the first author in [8].