We extend a result of J. F. Toland concerning bifurcation from infinity and we made some applications to variational inequalities.

Let

be Hermite's function of order λ and let h = h(λ) be the largest real zero of Hλ(t). Set

In this paper we establish the inequality

where

Equality holds for S = ½. The result is also fairly accurate as S→0 and S→1. The proof is analytical except in the ranges −1·1 ≦ h ≦ −0·1 and where the argument is concluded by means of a computer.

The following deduction is made elsewhere [2, Theorem A]. If u(x) is subharmonic in Rm(m ≧ 2) and the set E where u(x) > 0 has at least k components, where k ≧ 2, then the order ρ of u(x) is at least ϕ(1/k). In particular, if ρ < 1, E is connected. This result fails for ρ = 1.

An inequality of C. R. Putnam involving a Dirichlet functional in the singular theory of the Sturm-Liouville differential equation is generalised. The corresponding result for the Sturm-Liouville theory over a compact interval is established and this is extended to the singular case by means of a Tauberian argument.

This paper investigates the one-dimensional propagation of weak discontinuities, that is acceleration waves, in a homogeneous and isotropic half-space composed of an arbitrary number of non-linearly hyperelastic layers. The transmission and reflection coefficients are evaluated in terms of the initial condition at the boundary, and the steepening of the waves to form a shock is discussed. The results are specialised to the case of periodic layering.

The abstract non-linear non-autonomous functional differential equation

is considered. An evolution operator is associated with the solutions of this equation and existence and stability results are obtained.

In a previous publication [1] we determined the structure of some new types of ordered inverse semigroup in which the ordering need not be the natural ordering. At the moment, very little is known about ordered regular semigroups and the purpose of this paper is to generalise the structure theorems in [1] to corresponding theorems on ordered regular semigroups. The results obtained provide an interesting bridge between the theory of semigroups and that of ordered semigroups, these theories currently having very little in common.