The aim of this paper is to give a functional analytic treatment of the homogeneous and inhomogeneous linear transport equation in the case that the parameter c occurring in that equation equals 1. The larger part of the paper is devoted to the study of a certain operator T−1 A in the space L2(– 1, 1). A peculiarity not arising in the case c < 1 (treated amongst others by Hangelbroek) is that, for c = 1, the operator T−1A has a double eigenvalue 0 and that it is no longer hermitian. The Spectral Theorem is used to diagonalise the operator as far as possible, and full-range and half-range formulae are derived. The results are applied inter alia to give a new treatment of the Milne problem concerning the propagation of light in a stellar atmosphere.

Consider the multiparameter system

where ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equation

where B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

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.

The simultaneous integral equations of Noble and Cooke, for inter alia the Dirichlet problem for an annular disc, are transformed into equations closely analogous to those recently given by Clements and Love for the corresponding Neumann problem. The transformed equations are relatively simple, and do not involve any artificial preliminary dissection of the known functions. They are also uncoupled, and admit iterative solution for virtually all radius ratios.

In the case of the conducting annular disc, iteration of these equations isused to obtain two series for the capacity, an interim one with all terms positive, and a final one with all terms after the first negative. The final series is more rapidly convergent as well as neater. It is used to provide a family of inequalities which enclose the capacity and determine it to high accuracy with very few terms, failing only when the radius ratio is close to 1.

By a new method it is proved that a non-linear elliptic boundary value problem of rather general type admits a weak solution lying between a given weak lower solution ϕ and a given weak upper solution ψ≧ϕ

Some characterisations of commutatitivity for C*-algebras are given in terms of inequalities involving sums and products of self-adjoint elements, and optimal constants are obtained for the corresponding inequalities for non-commutative C*-algebras.

The paper discusses the asymptotic behaviour of solutions of the functional differential equation

where a is a complex constant, 0<λ<1, and b is a constant such that Re b = 0, but b ≠ 0.

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

The present paper is devoted to the study of the weak respectively strong convergence of solutions of variational inequalities, with non-linear partial differential operators of the generalised divergence form and of monotone type, under a perturbation of the domain of the definition. In this study there are used convergence concepts defined according to [ 22] and abstract convergence theorems given in [15 and 16].

We use the notations and results of Part I of this paper [see 7]. Sections and formulae of that paper will be referred to as section I.4, formula (1.5), etc. In particular, T, A, TN0, K, σ, F and Λ(λ) will have the same meaning as in [7].

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.

It is shown that classical solutions backward in time to the alternatives to the Navier-Stokes equations proposed by Ladyzhenskaya cannot exist for all time. Estimates are obtained for the maximum interval of existence.