Let d denote the dimension of the vector space consisting of all solutions of the equation − (p(t)y′)′ + q(t)y = 0, a ≤ t < ∞; that lie in the function space L2[a, ∞). By means of certain bounds on the solutions of this equation, sufficiency criteria are obtained for the cases d = 0 and d = 2.

The asymptotic behaviour of semigroups of nonlinear contractions which have a set of fixed points containing a ball of finite codimension is studied. It is shown that the ω-limit sets of such semigroups are finite dimensional tori, and that an analogue of the classical Kronecker-Weil theorem holds for such semigroups.

We give some results on a boundary-value problem for an ordinary differential equation whose coefficients are in the B*-algebra C(K), where K is a compact metric space. We deduce the existence of a countable number of eigenvalues and corresponding eigenfunctions, the latter being complete in a certain sense. There follows an expansion result and some remarks on a self-adjoint realisation of the associated differential operator.

Using an algebraic approach to the nth order linear differential equations (see [3] for n = 2 and [13 and 14] for n ≧ 2) it is shown the essence of the relation between the limit circle classification and boundedness of solutions of y″ = q(t)y. On this example it is demonstrated that if a problem can be formulated in the rank of the approach, then it is only a technical matter to answer it, e.g. the relationship studied here is based on a standard fact from the theory of functions.

This paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

The inequality considered is

where p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L 2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L 2(a, b).

The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

We consider the eigenfunction expansions associated with a symmetric differential operator M[·] of order 2n with coefficients defined on an open interval (a, b). Each singular endpoint of (a, b) is assumed to be of limit-n type. A direct convergence theory is established for the eigenfunction series expansion of a function y in a set Termwise differentiation of the series is established for the derivatives of order up to n. For O ≤ i ≤ n − 1, the i-fold differentiated series converges absolutely and uniformly to y(i) on compact intervals; the n−fold differentiated series converges to yn in the mean. The expansion theory is valid also when an essential spectrum is present. An explicit formula is given for the calculation of the spectral matrix.

General formulae are obtained for the reflection and transmission of harmonic acoustic waves by a curved interface between two media when the frequency is high. In addition to refracted rays there turn up tunnelling rays, if the surface is concave to the source, which are emitted from an evanescent region when the phenomenon of total internal reflection would be anticipated. Uniformly valid formulae dealing with the transition from refraction to tunnelling in both transmission and reflection are derived.

The theory is applied to the circular cylinder and to the top-hat circular jet. In the latter case it is suggested that radiation may tend to be more significant at inclinations of 50°-65° (downstream) and 25°-40° (upstream) to the axis of the cylinder. The augmentation due to tunnelling rays in propagation upstream is mentioned.

If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

The groups of the title are characterized arithmetically.

In this paper we show that some of Bass' results on the normal structure of the stable general linear group can be extended to infinite dimensional linear groups over non-commutative Noetherian rings.

Given the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

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.

In a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

A generic bifurcation theory is developed which is somewhat different from the approach in [4]. We put special emphasis on equations satisfying additional symmetry properties and on the non-generic bifurcation sets arising in this context. We apply our results on the von Kármán equations for the buckling of a rectangular plate under a compressive thrust and a normal load.

The inequality considered in this paper is

where N is the real-valued symmetric differential expression defined by

General properties of this inequality are considered which result in giving an alternative account of a previously considered inequality

to which (*) reduces in the case p = q = 0, r = 1.

Inequality (*) is also an extension of the inequality

as given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.