In this paper we examine the influence of capillarity on existence and uniqueness of travelling wave solutions in an isothermal system of van der Waals fluids. Existence and non-uniqueness theorems are proved using phase-space analysis and topological methods.

In this paper we consider differential-boundary operators T over a finite interval depending on a complex parameter. A differential-boundary operator admits boundary conditions in the differential part. The boundary part contains multipoint boundary conditions and integral conditions. For Birkhoff-regular boundary conditions we prove that every Lp -function is expansible into a series with respect to the eigenfunctions and the associated functions of the differential-boundary operator. Here the Birkhoff-regularity only depends on the boundary conditions at the endpoints of the interval, i.e. T is Birkhoff-regular if and only if T0 is Birkhoff-regular where T0 arises from T by omitting the boundary part in the differential equations, the interior point boundary conditions and the integral condition.

Consider the Abel integral operator

where 0 < α < 1. Suppose u is in H1(0, 1) of H1-norm ≦E, and f is an element of L2(0, 1) such that ∥Au – f∥L−2 < ε. We give a regularised approximate solution uβ(f) of the equation

which satisfies

and can be computed simply by performing some integrations. The preceding error estimate can be sharpened by strengthening regularity conditions on u

We consider the strongly-damped Klein–Gordon equation in ℝ3 in the case where the initial data possess radial symmetry. With the latter assumption we are able to extend the result of [2] which assumed a bounded spatial domain. Specifically, we construct a global weak solution v of theundamped equation for high powers p which can be approximated arbitrarily closely (for small α) by the global strong solutions of the damped equation found by Aviles and Sandefur [1].

Contracted inverse semigroup rings are studied subject to the restriction that the semilattices of the given inverse semigroups satisfy a certain finiteness condition, introduced in 1980 by Teply, Turman and Quesada. Results are obtained on semiprimitivity, primitivity, primeness, decomposition into a direct sum of ideals, and chain conditions on one-sided ideals.

[Proc. Roy. Soc. Edinburgh Sect. A91 (1982), 205–212]

[Proc. Roy. Soc. Edinburgh Sect. A106 (1987), 209–219]