By using a Hilbert space decomposition theorem for two polar cones it is shown that the method of the hypercircle can be extended to determine solutions to best approximation problems involving unilateral constraints. The method is applied to abstract boundary value problems for linear operators involving such constraints.

Associated with a Cauchy problem for an eikonal equation is a structure of caustics and wave fronts. The geometrical properties of these structures are of considerable interest and one would like to classify them. The first step in such a programme is to identify those solutions whose structures have certain (structural) stability properties or robustness. These are structures which survive small perturbations of the initial conditions. (It turns out that they also survive small perturbations of the equation.) This problem for caustics has been tackled with considerable success by a number of authors, in particular by Arnold. Here we develop the analogous, but different, theory for wave fronts and give an algebraic characterization of those local solutions of eikonal equations which have stable wave front structures. From this result it is easy to deduce the qualitative form of those stable wave front structures which occur when the propagation occurs in a space of dimension not greater than five. They correspond to Thorn's elementary catastrophes, as demonstrated by Zakalyukin.

For a certain class of first order systems of differential equations several theorems are derived which give sufficient conditions for an appropriate sesquilinear form to be identically zero on suitable spaces of solutions of the system. As a consequence for second order systems limit-point criteria are obtained which include rather general criteria in the case of second order equations. The method used involves sequences of auxiliary functions and is most expedient for the proof of interval limit-point criteria. The theory is also applicable to second order equations with complex coefficients yielding sufficient conditions for the existence of solutions which are not of integrable square.

In this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation

This paper studies stability of steady state solutions in a non-linear viscoelastic fluid. The main technique is to imbed the equation of motion in singularly perturbed equations and apply an energy method and the parabolic maximum principle.

In this paper, the formally J-symmetric Sturm—Liouville operator with complex-valued coefficients is considered, and as a preliminary two criteria are established for the J-selfadjointness of certain of its extensions. The main purpose of the work is to give results on the location of the point spectra of such operators. Some of the results are extensions of results known for real-valued coefficients, whilst others are new.

We give a formula (4) for a variety of ordinary linear differential equations of order n with distributional coefficients. There appear as coefficients distributions of order k ≦ n/2, i.e. these distributions are kth distributional derivatives of locally L-integrable functions. With a suitable transformation (7) the differential equations can be transformed into first order systems (8) with integrable coefficients. From this follows the existence of a continuous solution, which can be uniquely determined by proper initial conditions.

The coefficients in the differential equations considered are chosen as general as possible but such that a transformation into a system with integrable coefficients can be performed, and that all products are defined by Leibniz' formula.

A uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.

In a previous publication we investigated certain idempotent residuated mappings and showed how these could be used to provide a solution to the problem of finding a Baer semigroup coordinatisation of bounded modular lattices. Here we use essentially the same idempotents to provide a coordinatisation of bounded distributive lattices. Specifically, we prove that a bounded lattice L is distributive if and only if it can be coordinatised by a Baer semigroup S such that if eS, fS, gS ∈ R(S) with eS ∩ fS = eS ∩ gS then there are idempotents ē, , ∈ S such that ēS = eS, S = fS, S = gS and ē commutes with both and .

We provide a sufficient and “almost” necessary condition for the existence of a periodic solution of the equation

where F is nondecreasing in u and has a small linear growth as |u|→∞.

We study bifurcation problems in the presence of continuous groups of symmetries and obtain theorems on the existence and uniqueness of solutions. We also briefly consider some applications.

We prove that a finite semigroup which is decomposable into a direct product of cyclic semigroups which are not groups is uniquely so decomposable, and show how the non-group cyclic direct factors of a finite semigroup, if they exist, can be found.